Music of Sacred Temperament

In the beginning was the Note.

All the Music was made by the Note,
and without the Note was not any music made that was made.
There was a man sent from God, whose name was John.
The same came for a witness, to bear witness of the Light of the Music.
In the daytime, the light was from the sun.
In the night, the shadow was from the stars.
The day was made of twelve hours of daytime and of twelve hours of night.
So was the Music.
The Note was made of twelve tones namely all the tones and semitones.
And sequenced in the major scale or in the minor scale.
The Music was in twenty-four keys, and without them was not any music that was made.
Email : mocfujita@aol.com

---since March 7th, 2019
ver: Nov 26th, 2012 (now July 3rd, 2022) -- Anna Magdalena -- Carl Philipp Emanuel -- The Uncertainty -- Johann Adolph Scheibe
-- Keyserlingk -- the Calov Bible; eine Amusa --Richter and Gould
index - next page - Japanese Version
MUSIC OF INTELLECT the Goldberg Variations --- Music of Sweet Temperament the Well-Tempered Clavier Book 2

The Well-Tempered Clavier
or
P reludes and
Fugues through all the tones and semitones
including those with a major third or Ut Re Mi
as well as those with a minor third or Re
Mi Fa. For the profit and use of
musical youth desirous of learning
and especially for the pastime
of those already skilled in
this study composed and prepared by
Johann Sebastian Bach
at present
Capellmeister to
His Serene
Highness
the Prince
of Anhalt-Cothen,
and director
of His
Chamber Music.
Anno
1722
here...... Music of New Testament ......here
For those of you who want to listen to all of wtc1
Here it is! mp3 by TruePianos
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MUSIC OF INTELLECT the Goldberg Variations
Music of Sweet Temperament the Well-Tempered Clavier Book 2
The Well-Tempered Clavier
composed and prepared by Johnann Sebastian Bach in 1722
text and data arranged by Iori Fujita
September 2001
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Let me talk about the Well-Tempered Clavier.

One or ? There are two books called as the Well-Tempered Clavier, Book 1 and Book 2. Which one do you like better? I love both. But some preludes and fuges of each book impress me a lot more than others. According to the history, the Book 2 was not named as such by J. S. Bach. It was named by Johann Christov Altnikol, his son-in-law. We don't know why Bach had completed the second book about 20 years after of the Book 1.

Nature Many people argue about the meaning of "Well-Tempered". Some say it dose not mean "Equal-Tempered". The "Equal-temperament" is thought to be a compromise. But the Nature has sometimes a fuzzy aspect. On the Uncertainty Principle by German physicist Werner Heisenberg ( published in 1927 ), the position and the velocity of a subatomic particle cannot both be measured exactly, at the same time, even in theory. In future, someone would find a new music theory for this temperament issue.

Ourselves J. S. Bach gave the Well-Tempered Clavier to "musical youth desirous of learning and especially for the pastime of those already skilled in this study". Then why do we have to listen to the performances of great pianists? Great pianists should have played it for thier own pastime. I think that the value of the Well-Tempered Clavier will emerge when we play it by ourselves like singing hymns in our church. Oh, no, we cannot play the piano well. Then there is a way. We can transcript the score on our notebook and enjoy the transcription work itself. Oh, no, we cannot hear the music. Right. But if you transcript the score into midi data, you can listen to the music freely.

Genius Sviatoslav Richter and Glenn Gould are outstanding geniuses. They created Bach's music through the piano. No other pianists can do the same performance like those. Anyway,we can never listen to J. S. Bach's perfromance itself. So, Sviatoslav Richter and Glenn Gould are the missionaries for the New Temperament. How can I explain the differences among other pianists except Sviatoslav Richter and Glenn Gould?
Sviatoslav Richter and Glenn Gould

Outstanding Geniuses
Sviatoslav Richter and Glenn Gould

A History of the Well Tempered Clavier Performance by Sviatoslav Richter

Sviatoslav Richter did not play the WTC in public in 1980's nor in 1990's.
Data from "Sviatoslav Richiter Notebooks and Conversations" by Bruno Monsaingeon 1998
There may be some differences between the graphs and the chart below.

Programs of concerts (only about the WTC)
yearmondaylocationBook 123456789 101112131415161718192021222324
1937Conservatory? ???????????? ????????????
194610Moscow1 & 2 oooooooooooo oooooooooooo
1954510 Budapest 1oooo oo
196211 Perugia 1oooooooo
196211Perugia1 oooooooo
196212Paris1 oooooooo
196322London1 oooooooo
196329Brussels1 oooooooo
1963223Geneva1 oooooooo
196342Bern1 oooooooo
196344Strasbourg1 oooooooo
1969117 Leningrad 1 oooooooooooo
1969118 Leningrad 1 oooooooooooo
196942Paris1 oooooooooooo
196943Leningrad1 oooooooooooo
1969420Moscow1 oooooooooooo
1969421Moscow1 oooooooooooo
1969529Turin1 oooooooooooo
1969530Turin1 oooooooooooo
196965 Bologna 1 oooo oooo
1969616 Florence 1 oooo oooo
1969620Lausanne1 oooo oooo
196975Tours1 oooooooooooo
196976Tours1 oooooooooooo
1969820 Locarno 1 oooooooo oooooooo
1969104King's Lynn1 oooo oooo
1969107Newcastle1 oooo oooo
1969109Northhampton1 oooo oooo
19691015Ely oooo oooo
yearmondaylocationBook 123456789 101112131415161718192021222324
197335Vienna2 oooooooooooo o
1973311Vienna2 oooooooooooo
1973316Budapest2 oooooooooooo o
1973318Budapest2 oooooooooooo
1973511Turku2 oooooooo
1973513Helsinki2 oooooooo
1973515Leningrad2 oooooooo
1973628Paris2 oooooooooooo
1973630Meslay2 oooooooooooo
197377Meslay2 oooooooooooo
1973710Paris2 oooooooooooo
1973712Gourdon2 oooo oooo
1973715Carcassonne2 oooo oooo
1973718Provence2 oooooooooooo
1973726Innsbruck2 oooooooooooo
1973728Innsbruck2 oooooooooooo
197387Innsbruck1 oooooooooooo
1973810Innsbruck1 oooooooooooo
1973730Ansbach2 oooooooooooo
1973731Ansbach2 oooooooooooo
197575Tours1&2 121 121
yearmondaylocationBook 123456789 101112131415161718192021222324

Sviatoslav Richter recorded the Well Tempered Clavier two times in his life. One is of the year 1970, another is of the year 1973 (from a live recording). The first recording was recorded in one day or another. The second was at live for two day concert for the WTC 1. There seemed to be no complicated editting procedures. Richter was a genuine live performer. The second version "Great Legacies of Sviatoslav Richter J.S.Bach THE WELL-TEMPERED CLAVIER" was selected in later days around 1995 by Richiter himself for coming generations (as his words).
Sviatoslav Richter was born in Zhitomir, in the Ukraine, on March 20, 1915. Neuhaus, his teatcher, said, "Here is the pupil for whom I have waited all my life. In my opinion, he is a genius." It was not until May 1960 that he was allowed to travel to the West. The leading piano teacher at the Juilliard School, Rosina Lhevinne, praised it: "Richter is an inspired poet of music ... an exceptional phenomenon of the twentieth century." Richter was an excellent sight-reader and could immediately play pieces he had never seen or heard. Richter died in Moscow on August 1, 1997.

In some cases he felt that others had played these pieces so well that he had nothing else to say. So he dared not play the Goldberg Variations nor Kreisleriana. He surely loved those works. Almost all works for piano of J.S.Bach and Roberto Schumann were played by him in concert or for recording except those two great works. Richiter never played them.

Who are "others" for those specific works?
"I've heard G.G. play in the concert hall, and have also listened to his recording. Some day I'd like to play it (the Goldberg Variations) myself - if ever I manage to master it.", he noted in 1973.*

He noted also in 1972, "My recording of Part One is undoubtedly more successful than that of Part Two. In the second volume, it's the most significant preludes and fugues that have turned out the worst - those in E flat minor (No.8) , F sharp minor (No.14), and this weighs heavily on my conscience."*

Again Richiter was distressed in 1975 listening his own record, "Unlike Part One, which is fairly successful, this recording of Part Two is riddled with mistakes, above all - unfortunately - in the most important preludes and fugues, such as in F sharp minor and B flat minor."*
* from "Sviatoslav Richiter Notebooks and Conversations"
written by Bruno Monsaingeon 1998
translated by Stewart Spencer

In comparison of two versions for the WTC Book2 No.14 Prelude and Fugue, I became to think that the 1973 version could confirm him that the WTC was his own exclusive possession. Sviatoslav Richiter has the Well Tempered Clavier as Glenn Gould has the Goldberg Variations.


Listen! - S. Richiter 1st --- Glenn Gould --- S. Richiter 2nd - Listen!

A History of the Well Tempered Clavier Performance by Glenn Gould

Programs of concerts, broadcastings or recordings
(only about the WTC)
yearmondaylocationBook 123456789 101112131415161718192021222324
1946410Toronto2 o
1947410Toronto2 o o
19521021Radio2 o
1953216Ottawa2 o
1954226Radio2 o oo
1954114Montreal2 o o
195778Recording2 o o
19626,9-Recording1 oooooooo
196334TV2 f
19636,8-Recording1 oooo oooo
19652,8-Recording1 oooooooo
19668,9-Recording2 oooooooo
19661113Radio1 p
19661129Radio2 ffff
19671,2-Rec Again2 oooooooo
1967312Radio1 oooo
19699,1210Recording2 oooo oooo
1970218TV2 o oo
19711-Recording2 oooooooo
yearmondaylocationBook 123456789 101112131415161718192021222324

p : only prelude, f : only fugue
Data from "Glenn Gould Life and Variations"
by Otto Friedrich 1989
Glenn Gould was born in Toronto, Canada on September 25, 1932. He won first prize in the 1946 "Piano Trophy Competition". He played the Prelude and Fugue in B flat minor from the Well Tempered Clavier Book 1. This was the only piano competition in which Gould took part.
After that Gould never played any Prelude nor Fugue from the Well Tempered Clavier Book 1 on concert. It is very different from the case of Richiter. Richiter played much more times Book 1 than Book 2, even though he started his study of WTC from Book 2.

In 1962 Gould began recording the Well Tempered Clavier in the Columbia's New York studio. It took nine years to complete in total. The Book 1 was released in 1965. The Book 2 in 1972.
He might have said, "I would like to be a recording producer for Richiter, if he wants me to be. His music is great, but recording was so bad." Richiter would have replied, "I am a performing pianist. Recording should be a result of performance." Regretfully there is the best performance of S. Richiter's Well Tempered Clavier Book 2 beautiful No.14 Fsharp minor on the CD live-recorded in 1973 in Innsbruck with many annoying coughs from the audience instead of Gould's joyful hummings.

WTC 1 - Prelude No.24
Before his recording, Glenn Gould listened to the performance of the WTC Book 1 the 24th Prelude by Leopold Stokowski's transcription with his orchestra. Of course almost everyone knew it.
Stokowski Style
Then Gould realized that it was a losing game to play that Prelude with the similar style, and that a complete new way of playing was necessary for him, which was completely different from the orchestration of Leopold Stokowski. So he decided to play it without refrains and with staccatos of walking base. It was interesting but not great. Gould's performance was very much to the point. Walking base plays a major role in this Prelude, which Sviatoslav Richter would have never understood.

After that, Sviatoslav Richter recorded it in 1970. It was like Stokowski's piano version, but it was much more great and spiritual.

Gould regreted it but it was too late. After all the WTC Book 1 belonged to Richter. Anyway Gould was glad to find out that in the Book 2 especially on the 14th Prelude and Fugue he had won the game against Sviatoslav Richter.

Gould Style
Richter Style

Glenn Gould passed away in 1982.
Actually there are many questions to be answered.
The Well-Tempered Clavier
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The period of Köthen: 1717-1723

Bach worked at the small Court of Anhalt-Köthen with the title of Capellmeister. On 1 March 1719 Bach travelled to Berlin to acquire a splendid new harpsichord with two keyboards for Köthen court.
0n 22 January 1720, when Bach dedicated the first Clavierbuchlein to his nine-old son Wilhelm Friedmann, a vital new element entered the conception of his keyboard works, namely instruction. Playing and composition, practice and theory, instruction and delectation: all should be integrated.
In the summer of the same year 1720, his wife, Maria Barbara passe away suddenly when Bach was out of the town with his Prince. His wife had been buried in his absence.Afterwards four their children left motherless.

But he continued his work, composing and playing cantatas for his Prince. In order to perform these works there were singers from nearby Courts. He found one of these, Anna Magdalena. She attracted Bach's attention with her fine soprano voice. In 1721, Anna Magdalena at the age of 20 and Bach at 35 married. His second wife Anna Magdalena kept their home happy with his children and raised thier children. One of thier three daughters Elizabeto Uriana married with Johann Christov Altnikol who was one of pupils of J. S. Bach and afterwards put the name of the Well-Tempered Clavier Book 2.

And also she tried hard to copy his manuscripts. She was an important assistant for Bach.

In 1723, Bach left with his family for Leipzig. There he was to remain for the rest of his life.

The days for the Book 2

Around 1744, Bach was devoted into special music works with the most profound statements, especially "the Art of Fuge 'Die Kunst der Fuge' BWV 1080" in which the concept of distinction between keys was out of matters for Bach. And as a great maestro Bach recieved many visitors as his son Carl-Phillip Emanuel wrote " no musician of any consequence passing through Leipzig would fail to call upon my father".
click here - sample sound start - click here

The Well-Tempered Clavier
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The Sonnets No. 8
William Shakespeare 1609
 Music to hear, why hear'st thou music sadly?
Sweets with sweets war not, joy delights in joy:
Why lov'st thou that which thou receiv'st not gladly,
Or else receiv'st with pleasure thine annoy?
If the true concord of well-tuned sounds,
By unions married do offend thine ear,
They do but sweetly chide thee, who confounds
In singleness the parts that thou shouldst bear:
Mark how one string sweet husband to another,
Strikes each in each by mutual ordering;
Resembling sire, and child, and happy mother,
Who all in one, one pleasing note do sing:
Whose speechless song being many, seeming one,
Sings this to thee, 'Thou single wilt prove none'.

1. Secret of Temperament
2. Great Performers' Temperament System
3. Inharmonicity of the Piano's Tone Structure
4. The Sacred Temperament
1. Secret of Temperament


Prelude No.1 C major
The first set of arpeggio (green) of each bar is tuned on the Equal Temperament, and the second (blue) is on the Pure Temperament. Can you hear the difference?


( the first arpeggio )
Greens sound interesting, and Blues sound simple even in good sense. I am familiar with Greens. There are natural vibrations and tensions. On the other hand, Blues are harmonious. Groups of notes are combined with each other and became one harmony. It is good. But I would like to identify each note more clearly in the polyphonic music in order to feel the structure. In Greens each note is independent. At the same time I can feel the harmony of notes as well.

You can see the beats on the Green Wave - Equal temperament.
C - E is especially different on each temperament.

Anyway the difference is very delicate. But if you are interested in this matter, it should mean a lot. Listen to the last part of this demo, Brown part is a mixture of Green and Blue. It sounds queerly. The difference certainly exists.

There are many kinds of Temperament Systems. Pure, Mean-Tone, Kirnberger-1, Kirnberger-3, Werckmeister-3, Silbermann, and others. Now ordinary keyboards are tuned basicaly on the Equal-Temperament System.

( comment: This Pure Temperament (blue) is formed on C as the basic note. In other words, it is the C major scale Pure Temperamnet. So, there are 12 kinds of the Pure Temperament. In this case, Prelude No.1 is on the C major, so the harmony is very good. But it is too dificult to manage for all keys. Then others Temperament methods were created. On th Equal Temperament, I can play this Prelude as C# major without any harmonic problems.)

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
Pure261.62279.06294.32313.94327.03348.83367.90392.43418.59436.03470.92490.54523.24
W3261.62275.62292.50310.07328.04348.83367.91391.22413.43437.99465.11492.07523.25

Let's hear the C-E-G chord.


These notes are synthesized using sine curves with certain frequencies and are conbined into a chord. The first part is on the pure temperament, the second is on the werckmeister-3 temperament, and the last is on the equal temperament. The first one sound like a one tone, which can be said to be on a pure harmony. But it can not be identified as a chord. So the first part ( the pure temperament ) is not interesting for me. The difference between the second one and the third one is the numbers of thier beats. Which do you prefer?
2. Great Performers' Temperament System
(1) Helmut Walcha

Helmut Walcha used a historical chembalo "Jan Ruckers der Jungere, Antwerpen, 1640" for recording the Well-Tempered Clavier Book 1 in 1974.

Tuning for this chembalo for Helmut Walcha by frequency (Hz) compared with the Equal Temperament:

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
Walcha246.81261.13276.26293.03311.05329.06348.94369.44391.18415.15439.01465.72493.68

Frequency numbers were obtained from the 24th fuge of the Book 1.
There is a sequence of single notes. It is very easy to get wave data of each note. Using the FFT method makes it possible.
Note "A" Look closely!

What I could get from the data above is the fact that every frequency is different. But Walcha's "A" is very low from the standard "A" ( A = 440 Hz ). Then I found that that chembalo was tuned lower by about a half tone. How lower? So, Walcha's "A#" is standard "A".

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
Walcha
+halftone		261.13276.26293.03311.05329.06348.94369.44391.18415.15439.01465,72493.68
+/--0.49-0.92-0.53-0.07-0.56-0.28-0.55-0.81-0.15-0.99-0.44-0.22

The result is clear. Walcha used an equal-tempered chembalo. Of course there should be a subtle and delicate tuning by professionals.

Again, I listened to the first prelude, then I confirmed that his chembalo was tuned lower by just a half tone. In other words, I should think that he played the prelude No.1 C-dur on H-dur ( transposed ) without seeing his keyboard performance. Is that right?

(2) Sviatoslav Richter

Sviatoslav Richter played the piano for the Well-Tempered Clavier Book 1. The sound is impressive with spiritual echoes. The piano sound is found to be more complicated than that of chembalo for frequency analysis.

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
S.R.267.87283.00302.40317.49336.17357.54378.50401.46425.48449.05476.85505.29537.17
S.R.-2.39%261.62276.40295.34310.08328.33349.20369.18392.09415.55438.57465.72493.50525.15
+/-0.00-0.78+1.82-1.04-1.29-0.02-0.81+0.10+0.20-1.43-0.44-0.33+1.90
W3 +/-0.00-1.56-1.66-1.05-1.58-0.39-2.08-0.77-1.87-2.01-1.05-1.810.00
W3 means the Werckmeister-3 system on the fundamental "C". So, W3+- is the difference between S.R.-2.39% and W3.

I assume that S. Richiter used a basically equal-tempered piano tuned a little higher than standard.

(3) Glenn Gould

Glenn Gould recorded the Well-Tempered Clavier Book 1 from 1962 to 1965,in New York City, using a concert grand piano. I put out the data as blow.

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
Gould264.23280.08297.38315.04333.41355.77375.65396.77421.00444.42474.42499.89528.46
+/-+2.61+2.90+3.72+3.92+3.79+6.55+5.66+4.78+5.70+4.42+8.26+6.01+5.21

What do you think? Look at "A". Gould's "A" is a little higher than standard. Actually about % higher. So, let's make a -1.045 % lowerGould's Temperament.

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66 311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
Gould
-1.045%		261.50277.18294.30311.78329.96352.09371.76392.66416.64440.00469.51494.72522.99
+/--0.120.00-0.64+0.66-0.34+1.87+1.77+0.67+1.340.00+3.35+1.84-1.26

Gould notes here are staccatos, so their durations are very short. And there were wave distortions. Anyway there might be some frequency counting errors.
But I would say, "Gould used an Equal-Tempered Piano".

(4) Data for the Werckmeister-3
For your information, I will show you the Werckmeister-3 system on the fundamental "C", which is widely thought to be the most adequate temperament system nowadays.

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
W3261.62275.62292.50310.07328.04348.83367.91391.22413.43437.99465.11492.07523.25
W3+/-0.00-1.56-1.66-1.05-1.58-0.39-2.08-0.77-1.87-2.01-1.05-1.810.00
W3+/-(Cent)0.00-9.77-6.84-5.86-8.31-1.93-9.77-3.42-7.82-10.26-3.90-6.350.00


3. Inharmonicity of the Piano's Tone Structure

(1) Actual thickness or diameter of strings;

Piano's strings do not necessarily vibrate at perfect natural harmonics due to the thickness and stiffness of strings themselves which are extreamly thicker than theoretical thin-strings for the calculation of harmonics. The frequency ratio of overtone is slightly higher than the theoretically predicted harmonic. This effect is known as the stretching of the upper and lower octaves among skilled tuners. It depends on the specific conditions of each piano, but let me say for example, " The acutual upper double harmonic of center C is 1.2 cnet higher than a mathematically doubled frequency of C. The notes will gradually become higher (or sharper), until at the high C this sharpening may be as high as 20 to 30 cents" This stretching of the octaves had been considered only as an acoustic effect in order to make the music more brilliant and lively. But in fact it follows the physical theory of string vibrations with consideration of acutual thickness and stiffness of each string.

In 1964, Mr. Fletcher estimated it by the formula.

fn = n x f0 x ( 1 + B x n2 )1/2
fn : n-times harmonic overtone
B : a certain parameter for each string depend on a specific piano.

As a result the tone structure of piano has the inevitable non-integer charactaristics. So the ancient theory that the temperament scheam depends on the existance of the ratio of whole numbers will not work for pianos. Now we can break the spell imposed more than 2500 years ago by the Pythagoras' theorem which never allows any irrational numbers for any musical use.

(2) Two or Three strings;

A key will strike two or three strings simultaneously using its connected hammer. Those strings are not tuned on one frequency. Each of them has a slightly different frequency in order to make some beats which will make the sound sustainable and will give it an acoustic richness. You can hear some beats in one single tone sound of piano.

(3) The Temperament for the Piano;

Now we have found that piano has an extended octave temperament for both directions ( upward and downward ) which produces a rich music sound with numbers of wonderful undulation. Inharmonicity is indispensable. Upon this inharmonicity there exists music. The equal temperament can't be the final answer.

4. The Sacred Temperamnet

By the way, even nowadays there are many people who claim that the 12 tone-equal temperament is a product of mere compromise. They say that the pure temperament is the only authentic one, and that the 12 tone-equal temperament makes dirty resonances without being expressed as the ratios of whole numbers, which is, they believe, a fundamental music fact. On the contrary, I think that the basic temperament should be the pure temperament and the 12 tone-equal temperament, and that other various temperament systems are nothing but compromise. Moreover the pure temperament, I think, can be only a little modified version of the 12 tone-equal temperament. Musicians specialized in classic music, especially performers of works of Bach and Mozart and professional piano tuners, might have guilty consciences in basically using the 12 tone-equal temperament system for their daily mission because it has been "a priori" imprinted on their mind as the origin of dirty resonances. For all that, those who hate the 12 tone-equal temperament usually use the unit called "cent", which is very convenient, in order for them to discuss about their favorite temperament sysytem. In fact this "cent" is a logarithmic unit itself and is remote from the "harmonious" ratios of whole numbers. And other people want to talk about the beats bewteen two tones to make them consonant. When, in an octave in which A=440Hz is located, two tones with certain beats are within a harmoniously permissible range, one octave higher tones will have double numbers of beats because they are determined by their frequency difference.

Then, let's start afresh in the basic musical scale. First of all, we can accept the existance of "octave relationship" as a common understanding. It can be said as a true fact that one tone and its octave are in perfect harmony except the case of inharmonicity of the piano.
Then, does a two-octave higher note have three times higher frequency? No. Pure G1 is the note with three times higher frequency from the root note C0. In fact a two-octave higher note has four times higher frequency. In other words, it means "22 times higher". A three-octave higher note has 23 ( = eight ) times higher frequency. The frequency ratio of octaves is not on the sequence of 1, 2, 3...( natural numbers ), but on the exponential sequence ( 2n ). A one-octave lower note has 2-1 ( = 1/2 ) times lower frequency.

The basic factor of the scale concept is on the octave system:

fn = f0 x 2n ( n = a whole number ; f0 = frequency of the root note )
Musical sound is one of the physical phenomena. So naturally it can be reasoned by analogy that "n" will be expanded, leaving the question of numbers of notes in one octave. Anyway those notes in one octave will be distributed on the exponential curve. And the distribution pitch will be an exponential-equal one. We, with historically advance knowledge, will chose twelve notes in one octave. Then straightforwardly the 12 tone equal temperament cames out.
fm = fn x 2m/12 ( m = natural number ; fn is the first note of a certain octave )
f0 is for C0, then f2 is for D0, f7 is for G0 and f11 is for B0. The ratio between two adjacent tones is 12th power root of 2 ( about 1.0594631 ). When "m" equals 12, you get one octave higher C1. When "m" equals 24 or 32, you get two octave higher note or three octave higher one. This formula is perfectly consistent with the octave system.

Additionally it is known as a physical fact that the human hearing ability has a tendency of exponential ( or inversely "logarithmic" ) sensibility over the height or loudness of a sound. About the loudness, there is the Wever-Fechner's law saying, "We feel the tone sensibility going up steadly when the physical stimulation level is going up with exponential pitch. Human hearing ability for the sound height depends on the logarithmic scale of frequency as the one for the loudness." So, we use often the unit called "dB". As well as the case for the loudness, for the height of sounds we have a tendency of exponential sensibility. Especially when the tone frequency is doubled, four times or eight times higher, we accept the differences as one octave up, two octave up or three octave up. About the height of note, we clearly identify the scale. I think that we sense the difference of two notes in one octave according to the logarithmic scale of frequency.


n-tone equal temperament
There has been no explanation here yet about the reason why one octave consists of 12 different tones. There have been many proposals like 6, 7, 9, 10, 17, 19, 22,...., 360 tone equal ( or not-equal ) temperament system. For 53-tone temperament system, one actual keyboard was said to be built in practice. But I think that an octave might have consisted of 12 tones from the very first just like that a week is seven days a priori.

Look at the diagram on the left. I showed the allocation of tone-heights for 2 - 13-tone equal temperament systems. Red lines indicate the places for pure 2nd, pure 3rd and pure 5th on C. For pure 5th, G of 12 tone equal temperament comes very close to it for the first time among increasing n-tone equal temperaments. D is also nearly the same as pure 2nd. In case of D, 6 tone equal temperament gives us the near pure 2nd. It is a matter of course because 6-tone equal temperament is only the half of 12 tone one. For your information, there is no near-G in 6 tone equal temperament.

The credit for the great accomplishment of establishing the music theory with 12 tones belongs to Pythagoras, one of the greatest Greek philosophers. Pythagoras succeded in making the West European world assured that one octave concists of 12 prominent tones. But nobody really knows why twelve. By simply counting "do, re, mi, fa, sol, la, ti ( and do )", we get seven tones in an octave. In this scale there are whole tone intervals and half tone intervals. Divide each whole tone interval to two, then we get 12 tones. In this world there are many kind of music scales in which some part of these 12 tones are selected deliberately depending on their historical background. Some of them have microtones. Many have their own characteristics with subtle differences from the others. Although appreciating the richness related to these differences, let me categorize the tonal systems roughly. Then almost of all those systems will end in the system with 12 tones. In thr West European music, on this 12 tone system, the law of harmony has developed and the music theory has flourished.

But it was a mere coincidence that the musical interval for a pure fifth happened to be very near from that for a fifth of 12 tone equal temperament. And this coincidence made Pythagoras think that this pure fifth relationship was the foundation of the tonal system. This caused a historical error or riddle which even now suffers many people in music. If you used a pure third, which is 5/4 of the root note pitch, in place of a pure fifth, as the standard interval for the accumulation of pitches, you would have understood easily that there would be found no meaningful tonal system. By doubling pure thirds twelve times, you could not get an octave. Even more there was no Wolf. And the insignificant fact that a pure fifth is the first natural harmonics except an octave caught the mind of people who thought a pure fifth as the base of the temperament. In other words, there was no ground for assuming that the accumulations of a pure fifth organize the temperament system and reach to an above octave. All the things people could perceive by their musical experience were that a fifth, a third and a fourth of 12 tone equal temperament are sufficiently close to those made of natural harmonics and that a fifth is used convenienly for tuning violins or keyboard instruments. Guitarists use basically a fourth for tuning thier instruments.
From C, we can get G as a pure fifth. D, which is obtained as a pure fifth of aforesaid G, is a 9 times natural overtone of under root C. This D is strongly related to the root note and is somehow meaningful. A, which is obtained as a pure fifth of that D, is a 27 times natural overtone of under root C. We can make it one octave down, and get a tone with 13.5 times above. This A is utterly different from the 13 times natural overtone. That is to say that only the less than 3 times accumulation of pure fifths on the root note should be considered verifiable. A pure fifth of that A is nothing but a natural overtone on the root D. It is not included in the series of natural overtones of C. But people assumed some concrete meaning of twelve times accumulation of a pure fifth and made it a philosophic issue that the assumption could not reach to a precise octave, as ill luck would have it.

The Pythagorean theorem says that in a right angled triangle the sum of the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Algebraically a2 + b2 = c2
Those a, b and c can be natural numbers. There are sets of natural numbers for these a, b and c. The most famous one is "3, 4 and 5". And "5, 12 and 13", "8, 15 and 17" and others are known. In case of rational numbers, sets of numbers come to the same speaking of ratios, like "1, 4/3 and 5/3" or "4/150, 1/20 and 17/300". Anyway these sets of special natural numbers are called "Pythagorean numbers". It was proved that "Pythagorean numbers" exist in infinite number and can be explore by the following equation.
d(m2 - n2)2 + d(mn)2 = d(m2 + n2)2
d, m, n ; natural number. m, n ; mutually prime, one is even, another is odd.

In Set Theory, Pythagorean numbers are in one-one correspondence with the natural numbers. And the rational numbers, which are written as + ( or - ) m/n, are also in one-one correspondence with the natural numbers. The smallest infinite set is composed of the integers or any set equivalent to it. But ordinary sets of "a, b and c for a2 + b2 = c2" are correspondent with the real numbers ( rational and irrational ). The set of all real numbers is the second smallest infinite set. Pythagoras was an inhabitant of the world of the smallest infinite set. Anyway "3, 4 and 5" on the Pythagorean theorem was a great gift of knowledge by which we can make a precise rectangular angle easily.

But when we look at a simple right angled triangle with two equal sides, we have to face with the root of 2 ( = 21/2) in its hypotenuse, an "irrational" number which is Pythagoras hated extremely. In case of another right angled triangle with an angle of 30 degree, there comes out the root of 3 ( = 31/2).

Why 21/2 is an irrational number or not a rational one?
Assume 21/2 is a rational number, 21/2 = q / p ( p, q ; positive integer and mutually prime )
Square both sides of the above equation; 2 = q2 / p2 and modify them; 2 p2 = q2
Then q2 is an even number, namely a multiple of 2
q2 = q x q means at least one of the first q and the second q. q = q so both of them are multiples of 2 at last.
Now, q = 2n ( n ; positive integer ), 2 p2 = ( 2n )2
That is, 2 p2 = 4 n2 , Then, p2 = 2 n2, so p also shoud be a multiple of 2.
The result that both q and p are multiples of 2 is against the assumption that q and p are mutually prime.
Finally 21/2 can not be a rational number. It is an irrational number.
( Proved by the Method of Reduction to Absurdity )
Although there exists the root of 2 ( = 21/2 or 21/12 x 6 ) as the hypotenuse length of a right angled triangle with two equal sides ( length = 1 ) which is one of the simplest geometric figure or a diagonal half of a square, Pythagoras did never accept the existance of irrational numbers as a vicious concept which Gods would not allow him to have. And he tried to cover up this vicious concept by founding a secret society. Especially he did not allow music infected with irrational numbers. Music should be the most rational concept in the Pythagorean world.
Another important number, the ratio of the circumference of a circle to its diameter ( π ) was not proved to be an irrational number for good or for evil in Greek era. More than two thousand years after from Pythagoras' days, at last in 1761 Mathematician J. H. Lambert proved it out using trigonometric functions and continued fractions that π is an irrational number. Sound is a physical phenomenon. We can not describe it without trigonometric functions like sin or cos. These trigonometric functions can not live without π. If Pythagoras could have known that π, which is closely related to the foundamental figure - circle, is also an irrational number like the root of 2, the unhappy history of temperament might have been avoided.
For your information, the note corresponded to the hypotenuse length of a right angled triangle with two equal sides is F# or Gb against C. F# is called to be the furthest note from C. If you hear the sound of combination of C-F# or A-D#, you will be surprised at its weird and frightening resonance. So, Pythagoras must have been surely glad that the ratios of whole numbers worked very well like "3, 4 and 5" of the Pythagorean theorem and "4, 5 and 6" of the musical temperament of C, E and G. But if irrational numbers would have been considered as a heavenly gift from Gods by Pythagoras, the everlasting history of the equal temperament issue might not have such a guilty conscience like ours.
You may have the impression like that the name ( irrational munber ) indicates a number without reason, a number without sense or a ridiculous number because of the word "irrational". But this "irrational" means frankly the number which is not described as a ratio like integer/integer. This should be called as the number "non ratio-nal".
2(1/12)x 0 2(1/12)x 1 2(1/12)x 2 2(1/12)x 3 2(1/12)x 4 2(1/12)x 5 2(1/12)x 6 2(1/12)x 7 2(1/12)x 8 2(1/12)x 9 2(1/12)x10 2(1/12)x11 2(1/12)x12

When a melody has the root note, for example C, then I may allow some deviations from the 12 tone equal temperament in case of pure G, E or D and in case of pure F because these are prominent pitches. As to the celestial body, planetary motion has certain deviations depending on other bigger planets in its neighborhood. This is called as "Perturbation". In the universe there is no perfect elliptic orbit in fact. This resembled the relation between the equal temperament and the pure one.

Hz/noteCC#DD#EFF#GG#AA#BC
Equal261.62277.18293.66311.12329.62349.22369.99391.99415.30440.00466.16493.88523.25
Pure261.62279.06294.32313.94327.03348.83367.90392.43418.59436.03470.92490.54523.24

J. S. Bach used a fast moving series of arpeggio passeges in the 3rd Prelude C sharp Major of the Well Tempered Clavier Book 1. Human listening ability is even now too much complex and not fully investigated. So, the physical analytic method may be applied. In the digital frequency analysis, the difference between two ferquencies ( ΔF ) and the necessary measuring time to identify that difference ( ΔT ) are said to be inversely proportional. "ΔT = 1 / ΔF" Now the biggest difference among those between the frequency on the pure temperament and that on the equal temperament of the same note in the ordinary octave scale is 3.97Hz ( = 400.00 - 436.03 ) of the note A. The necessary measuring time to identify this difference is;
ΔT = 1 / ΔF = 1 / 3.97 = 0.2518 ( sec ). The less the difference of frequencies, the longer the necessary measuring time. In this case of A, about 0.25sec, in other expression, 1/4 sec is necessary. The length of a note in arpeggio passeges in the 3rd Prelude C sharp Major of Book 1 is about 1/12 sec because the tempo is 2 bars per second. The durations of short notes are 1/3 of the necessary measuring time ( about 1/4 sec ). If this physical measurement can be applied to the human sense of listening, this kind of subtle differences can not be detected by listeners. Here shows the splendid insight of J. S. Bach. In the Book 1, first appears C Major, next C minor and then C sharp Major as the 3rd Prelude and Fugue. This C sharp Major is unfamiliar even now. In Bach's days an so-so appropriately tuned keyboard would have probably made weird sounds for this C sharp Major key work. Bach tried not to make players disappointed at the possible strange harmony on C sharp Major. By using shorter notes of arpeggio, Bach thought that people could enjoy the cheerful and bright sound even if the keyboard was a bit out of tune. Again this by itself is sufficient proof of Bach's excellent penetration.
The right chart is for the equal temperament and for the other temperament systems with the base note A (440Hz). In the scale "C D E F G A B C", either the distance between E and F or the distance between B and C is a little wider for the pure temperament or several other temperaments than for the equal temperament. In other words E and B have flat touches. Especially B has a motive to get closer to next C. In this sense the equal temperament is favorably utilizing "Perturbation".


When one melody shifts its base note or key from C to G, the Perturbation should also change the axis. From a different perspective we do not have to worry about that twelve times accumulation of a pure fifth can not reach to an exact octave. In the field of geometry, the sum of three interior angles of a triangle is 180 degree only in a special case. In non-Euclidean geometry, the sum of three interior angles is not 180 degree.
It might be an extreme exanple, but assuming that the Earth is a perfect sphere, and a triangle is made of following three points on earth, the North Pole, the longitude 0 degree point on the equator and the longitude 90 degree point also on the equator is an equilateral triangle with three right angles on the earth surface. So, the sum of three interior angles of this triangle is 270 degree. Three sides are equal in length and each angle is of 90 degree for people living on the earth.
I think that each key note or its scale has a non-Euclidean contortion, which makes music richer contrary to our anxiety. It can be described in forms of two equations as following.
fm = f0 x 2m/12 x player ( key, m, time )
The temperament of physically sounding character of music.

Fm = fm x listener ( key, m, time )
The temperament of music which human brain is listening to with appreciation.

player (key, m, time )

is a function whose value fluctuates according to the changes of key in the middle of the tune. That is a compensatory function for contortion by each key. Members of an ensemble or an orchestra perform thier instruments while listening at the same time to others' performances or an overall harmony, which is an important competence as musicians. This function works like that. In case many tones collaborate to make a long chord sound or a harmony, there is no need to adhere to the equal temperament. The sounds will be unified by the power of the center or basic tone with competence of musicians and will be naturally harmonized. On the contrary, there are other types of musical expressions like vibrato with vibrating frequency, glissando with sliding up or down frequency, choking of guitar, pitchbend of electric keyboard and so on, which are characterized by thier shakes or divergences from the standard note pitch. Human singing voices can do much more complex and fertile expression in this sense.

listener ( key, m, time )

is also a function with changes in the middle of the tune but a function concerning to the human listening sensibility. It is not merely about the physical function of the human listening sensibility but a musical function through which our brains accept music musically. For example, hear the C Major chord strongly attacked with C, E and G, from the first to the end. At first, there is an intense and rich but complex resonance, right after that, beats start, which is not emerged for the purely tempered piano, but followed by attenuation of the sound of the chord the beats are going to calm themself down and instead well harmonized Major chord comes out. I thought once that it is because of the mutual interference among piano strings, but that, because the same phenomenon happens even for digital pianos, it is adequate to understand this phenomenon due to the characteristics of the human listening sensibility. For me the purely tempered Major chord without that kind of transition, is too plain.
On the other hand, for notes of short duration, the difference between two ferquencies ( ΔF ) and the necessary measuring time to identify that difference ( ΔT ) are said to be inversely proportional; ΔF = 1 / ΔT as mentioned earlier. One of the five notes sequencially continue in one second has its duration of 0.20 sec.
ΔF = 1 / ΔT = 1 / 0.20 = 5 Hz. There is an ambiguity of 5 Hz. But we do not feel nor sense it ambiguous. By analizing the note C of 261.62Hz in the first 0.20 sec by FFT, the range of 4.32Hz from 259.94Hz to 264.28Hz is within -1dB of the peek. The peek is rather flat in wider area. The longer analyzing duration, the steeper the peek. The duration from attack to ceasing gives a range 2Hz within -5.9dB. I think that we are not analyzing the sound by ears and nerves using FFT realtime, but we are reconstructing or creating music referring our own inner note information. That is why we can image sound of music without physical sound waves outside of our ears.
It is not the effect of the pure temperament but the power of music itself that makes the sound beautifully harmonized and tensely resonant. I do not think that music is harmonized thanks to the pure temperament. I think that music has its own potential power to be harmonized if necessary. For the piano, based on its sound characteristics, this player ( key, m, time ) can be set as 1 constantly.

The Uncertainty Theory for the Music

Genesis ( a fiction I made )
11:1 And the whole earth was of twelve notes, and of one temperament.
11:2 And it came to pass, as they journeyed from the east, that they found a plain; and they dwelt there.
11:3 And they said one to another, Go to, let us make our voices musical and also make instruments for music.
11:4 And they said, Go to, let us build us a city and a music tower, whose temperament [may reach] unto heaven; and let us make us a name, lest we be scattered abroad upon the face of the whole earth.
11:5 And the LORD came down to see the tower and the temperament, which the children of men builded.
11:6 And the LORD said, Behold, the people [is] one, and they have all one temperament; and this they begin to do: and now nothing will be restrained from them, which they have imagined to do.
11:7 Go to, let us go down, and there confound their temperament, that they may not understand one another's music.
11:8 So the LORD scattered them abroad from thence upon the face of all the earth: and they left off to make different music.
11:9 Therefore is the name of it called Wolf; because the LORD did there confound the temperament of all the earth: and from thence did the LORD scatter them abroad upon the face of all the earth.
Once upon a time, the equal temperament and the pure temperament should have been the same. There was only one temperament. But since a certain old time, the confusion of temperament system became one of the sins of human being. LORD made a fifth, a fourth and a third never be under one temperament. I think that we have been sufferring for our arrogance, like saying that we, people, tried to unify these all fifth, fourth and third into a heavenly harmony only by our own knowledge. Our predecessors were scared of roars of Wolf which we could not get rid of. We should humbly admit that actual pure fifth, fourth and third are inevitably out of tune from the equal temperament which was given to us by LORD. Those of us who can not help adhering to the thought that only the pure temperament is the trueth and the equal temperament is nothing but a compromise might be staying in a situation without an awareness of this sinfulness of ours.

I feel that J. S. Bach leads us to the music world of Equal Temperament accepting this sin, using humble title, "the Well Tempered Clavier" for this work. Bach surely placed more importance on the fact that music is built on the twelve notes than a subtle distinction relating to many kinds of temperament systems. LORD imposed us a serious confusion about temperament but favorably gave us twelve notes for music. Bach traveled these twelve notes one by one for "Soli Deo Gloria" and hoped us, human beeings, to be saved by music with conscious of sin, so I think.

All the Music was made by the Note,
and without the Note was not any music made that was made.
There was a man sent from God, whose name was John.
The same came for a witness, to bear witness of the Light of the Music.
In the daytime, the light was from the sun.
In the night, the shadow was from the stars.
The day was made of twelve hours of daytime and of twelve hours of night.
So was the Music.
The Note was made of twelve tones namely all the tones and semitones.
And sequenced in the major scale or in the minor scale.
The Music was in twenty-four keys, and without them was not any music that was made.

In the Beginning was the Note.

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A Method for The Well-Tempered Clavier
by J. S. Bach 2001
--- This is a fiction. ---
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Welcome! I am J. S. Bach 2001.
This is a Method for "The Well-Tempered Clavier". I am writing this in the year 1744.

The Well-Tempered Clavier
or
Preludes and
Fugues through all the tones and semitones
including those with a major third or Ut Re Mi
as well as those with a minor third or Re
Mi Fa. For the profit and use of
musical youth desirous of learning
and especially for the pastime
of those already skilled in
this study composed and prepared by
Johann Sebastian Bach
at present
Capellmeister to
His Serene
Highness
the Prince
of Anhalt-Cothen,
and director
of His
Chamber Music.
Anno
1722
Preface
There are three important points in this title that I have to mention about.
First of all,
you should use a well-tempered clavier.

Around the year 1722 when I completed this work, it was very difficult, even for me, of course for you, to keep the tuning of claviers stabilized. You may have an experience of hearing a sound from a badly-tuned clavier on which a young lady of a royal family is fighting to play one of my works in her grand mansion. It is miserable. In case of organs, you do not have to worry about the tuning everytime before your playing, because the tuning of organs should have been all set at the completion of manufacturing of pipes. But claviers have many highly tensioned steel strings whose ends tuning pins are holding. Temperature, humidity, mechanical shock, rough usage and the most important factor "time" affect the tuning conditions. Even after some professional tuned your clavier properly, on the next day some strings would slip. A week later you will have a different tuning. So, you have to be very careful about the sound. It is the best that you know how to tune your clavier. Violinists know how to tune thier own instruments. Singletone instrument players do not have this kind of nuisance.

If you have a good clavier, you will enjoy your practice and music. But if you use a badly-tempered clavier, your practice and study of this Well-Tempered Clavier will not work. Worse, you may lose your ability to identify the tonality and the chord system. And you may say that this work is not interesting because you think that there are many hardly understandable notes and sequences. Or you may say that some of the preludes and fugues are only messes. Before you say those things, please check your clavier. The Well-Tempered Calvier is a new music work which requires a physically well-tempered clavier at least. In near future, we will have some good claviers well tempered by a rigid frame reinforced with steel structure which will give thier tuning a great stability for a longer period. In the next century or the 21th there will be a good clavier with an automatic tuner, then you will not be suffered from bad tunings.

Then what kind of tuning system do you apply? The equal-temperament system? The system of equal temperament, in which the ratio of frequencies of successive semitones is always exactly the twelfth root of two. No kidding. It is impossible for you to tune any instruments on the equal-temperament system. Why? Do you have a precise frequency counter? Without it, you cannot even attempt. In 1722, even in 1744, there was ( or is ) neither any knowledge about the tool of counting frequency nor a logarithm calculation method for figuring out the numbers of beats per second between two tones. The equal-temperament system is an imaginary one at that time or even now ( in 1744 ). So you may use one of the systems which Mr. Werckmeister invented. In the tuning of claviers we should achieve correct and pure temperament that all the keys sound pure and agreeable.

For a clavier I need an independent "3rd and 5th" like "C ( root ) - E", "C - G". In a pure-temperament, "E and G" is one of the neighboring harmonics of C ( root ). On one register, if any, every tone of a clavier has the same charactor. In a course of a polyphonic movement of notes, the perfectly resonant 3rd or 5th on the root will lose a certain power and the line of melody can not keep a steady stream. So they have to have some independence. I taught to one of my pupils, Friedrich Wilhelm Marpurg, that every 3rd should be higher. In chorales or string ensambles each voice or each string instrument has its own character and position by which I can identify the sound as a chorale or a string ensamble.

In order to check your clavier, I recommend you to play the theme of the last fugue, namely Fugue No.24 h-moll ( = B minor ) and to listen to the notes of all the tones and semitones.


listen

There are G-Fis, H-Ais, E-Dis, C-H, Fis-F, D-Cis-C, C-Cis and A-Gis ( in trill ) on the h-moll basic chord ( H-D-Fis; B minor ). All these semitones should sound natural. And if you get the concrete image of this theme as a melody and you can reproduce it in your brain, you will master your own Well-Temperament.

Notes of the musical scale are thought to be based upon the natural harmonics of sounds. Of course, we should respect the natual harmonics. I also find the sound of equal temperament, if exists, unacceptable for vocal or orchestral music. But through the Clavier I found out that in the begining there was the note. I can not accept the existance of the independent tone of Ais in place of B ( in your expression; Bflat ). As you know my family name, BACH is the combination of "B", "A", "C" and "H". In other words, Bflat, A, C and B. This sequence of notes should be the same in every scale. It is an enigma that in Germany "H" is used in place of "B". Perhaps the original expression was "CDEFGABC". In small cases, "cdefgabc". German people were not good at writing "b". Then "b" was distorted into "h" which is " b minus _" or its bottom end was opened. And "b" has almost the same shape of "flat". So, in order to distinguish them with each other, German people began to use "h" instead "b". Fortunately for the Bach family, "H" of "Bach" is one of the note names. So BACH can be a melodic theme. "BACH" is the Word, and at the same time is the Note. Some says that the system of equal temperament, if exists, is a universal compromise which leaves each key equally damaged and none perfectly in tune, and that it also totally removes the differences between keys which composers assumed when writing keyboard music. I do not agree with this statement. The system of equal temperament is not a compromise. It is a principle a priori. As I said, "in the begining there was the note". One year is not exactly 365 days. A year is not 52 weeks. But one year consists of 365 days or 52 weeks. No one says these things are compromises. I would say other temperament systems are compromises. So there are many kinds of compromises. Like the Word, in the begining there was the Note. In the universe, I mean the outer space, there is said to be no air, then no sound. But music should exist even in the space as an information by the Note. The differences between keys of course exist. But they do not depend on the system of temperament. The authenticity of each key is from the melody. Every melody has its original tone hight. You can transpose any melody into any key for your convenience. But the original is the original. Composers may have some specific mood when they are going to write a music on the key like C sharp major ( Cis-dur ) which has seven sharps on the score or A flat minor ( as-mool ) which has seven flats.The score looks weird. Musicians will be tensed up in front of the score. Don't say, " Let's make one halftone higher from C major or one flat from A minor".

The 8-note diatonic scale has two variations the major scale and the minor scale. There are 15 major scales. Each of the 15 major scales has its own relative minor scale the natural minor scale which begins on the sixth note of the major scale. The other two forms of the minor scale are the harmonic minor scale and the melodic minor scale. Well so, there are 60 scales. 15 scales for major, 45 scales for minor. The three forms of the minor scale are the inner variations or melodiac treatments of the basic minor scale or key.

H-dur = Ces-dur, Fis-dur = Ges-dur and Cis-dur = Des-dur for major scales, gis-moll = as-moll, dis-moll = es-moll, and ais-moll = b-moll for minor scales are in enharmonic relationship. I applied Cis-dur ( Csharp Major ) for No.3 Prelude and Fugue in place of Des-dur ( Dflat Major ). H-dur ( Asharp Major ) in place of Ces-dur ( Cflat Major ). es-moll instead of dis-moll.

If you do not agree with the equal-temperament, it is as though you still believe that the sun and the planets revolve around Earth, and there are 15 major scales and 15 minor scales remained. When you understand the meaning of the equal-temperament theory, you will get 12 major scales and 12 minor scales. The equal-temperament theory is not only for the way of tuning. It establishes the Note as the music infromation or language in the world and the universe.

About a hundred years after, someone called Hans von Bulow would praise me by saying, "The Well-Tempered Clavier by J. S. Bach is the Old Testament, and Be*******'s Sonatas are the New Testament for pianists." Excuse me for saying, "No, thank you". If you graciously compare the Well-Tempered Clavier to the Sacred Scripture, I would owe you that you might say "The Well-Tempered Clavier by J. S. Bach is the New Testament". The Old Testament of music might be "Gregorian Chants".

Secondarily,
you should not only listen to but also perform these preludes and fugues.

I composed it "For the profit and use of musical youth desirous of learning and especially for the pastime of those already skilled in this study". You can not play the piano? Then you have to practise. Or instead of playing the piano, you can read and transcribe the score for your use. Ah, you can buy the score and there is no need for you to transcribe it. Well, then transcribe it into your mind.

About a hundred years after, someone called Robert Schumann would descrive the Well-Tempered Clavier as "a daily food for the mind". I will appreciate this statement. And he would add, "I am now analyzing fugues of my beloved 'the Well-Tempered Clavier' by J. S. Bach in detail. This study gives me a lot and makes my mind and heart stronger from the theoretical aspect. Bach is a man of the man. He dose have no insufficiency, no morbidity at all. All has been writen for the eternity." If Robert were here, we could have talked a lot about the music with him.

My intenstion is for musical young people to develop their own musical intelligence from the very beginning along with technique. Wilhelm Friedemann, my eldest son at that time (in the year 1722) eleven years old, learned how to compose through the Well-Tempered Clavier. Listening only is a passive way of study and it is not sufficient. You have to participate into the music. If you invent a new instrument without keyboard which can performe automatically under your direction with inputting every note, it will be great. One of the most important concepts for me is that there are twelve keys, even imaginary, in one octave. The Note is the Word in music. I have covered all the notes possible.

Anna Magdalena, my second wife, a vocalist with fine soprano, became a good housekeeper, and she took a lively interest in my work, often helping me by neatly transcribing my manuscripts. Of course she copied several times the Well-Tempered Clavier for our pupils. Through these transcriptions, Anna Magdalena has been trained to understand my music.

Finally,
this Well-Tempered Clavier is not Book 1 nor Volume 1.

Surely, now ( around 1744 ) there is another plan for a new set of the Well-Tempered Clavier. In my opinion, it is not necessary to have a Book 2 or a Volume 2. In these 20 years, the Well-Tempered Clavier has worked well and is still working. If some revision is possible, I would like to change, for example, No.14 F sharp Minor to another new beautiful one which I have now.
But I have to care about my second wife. Anna Magdalena is always very kind to my ex-wife's children. As you may know that I presented her a special and beautiful music notebook with her name on its cover. At that time the children from my second marriage were small. But they are all born musicians, and I could form both a vocal and an instrumental concert within my family, especially since Anna Magdalena sang with a pure soprano voice.
And she wanted a new complete practice book for our children between Anna and me. It was just like that Wilhelm Friedemann, Maria Barbara's son, had this Well-Tempered Clavier for his learnig clavier. I understand her feeling. Johann Christov Altnikol, my daughter's future husband has been helping his future mother-in-law to compiling the new set of twenty four Preludes and Fugues and I heard that he put a name to this set ""The Well-Tempered Clavier Book 2". I will accept this. But the first one is not "Book 1". Do not tell this to my dear wife, Anna Magdalena. It is a secret between you and me.

The Well-Tempered Clavier is aimed to regulate the musical universe within the Note of 12 tones and 24 keys. In this sense, no Book 2 is necessary unless it has something completely new. It is not the matter of comparison.

Well, anyway enjoy yourself playing the Well-Tempered Clavier.
The Well-Tempered Clavier(Vol.1)INDEX

Clich the name of tune, then you will go to the page you want.
Click "mid" or "mp3", then you can hear the music on this page.
Or for a continuous play on WMP, click " PLAY "
PreludeFugueKey( Key )BWV
Prelude No. 1: mid:mp3 Fugue No. 1: mid:mp3 C MajorC-dur846
Prelude No. 2: mid:mp3 Fugue No. 2: mid:mp3 C Minorc-moll847
Prelude No. 3: mid:mp3 Fugue No. 3: mid:mp3 C# MajorCis-dur848
Prelude No. 4: mid:mp3 Fugue No. 4: mid:mp3 C# Minorcis-moll849
Prelude No. 5: mid:mp3 Fugue No. 5: mid:mp3 D MajorD-dur850
Prelude No. 6: mid:mp3 Fugue No. 6: mid:mp3 D Minord-moll851
Prelude No. 7: mid:mp3 Fugue No. 7: mid:mp3 Eb MajorEs-dur852
Prelude No. 8: mid:mp3 Fugue No. 8: mid:mp3 Eb Minor
D# Minor		es-moll
dis-moll		853
Prelude No. 9: mid:mp3 Fugue No. 9: mid:mp3 E MajorE-dur854
Prelude No.10: mid:mp3 Fugue No.10: mid:mp3 E Minore-moll855
Prelude No.11: mid:mp3 Fugue No.11: mid:mp3 F MajorF-dur856
Prelude No.12: mid:mp3 Fugue No.12: mid:mp3 F Minorf-moll857
Prelude No.13: mid:mp3 Fugue No.13: mid:mp3 F# MajorFis-dur858
Prelude No.14: mid:mp3 Fugue No.14: mid:mp3 F# Minorfis-moll859
Prelude No.15: mid:mp3 Fugue No.15: mid:mp3 G MajorG-dur860
Prelude No.16: mid:mp3 Fugue No.16: mid:mp3 G Minorg-moll861
Prelude No.17: mid:mp3 Fugue No.17: mid:mp3 Ab MajorAs-dur862
Prelude No.18: mid:mp3 Fugue No.18 : mid:mp3 G# Minorgis-moll863
Prelude No.19: mid:mp3 Fugue No.19: mid:mp3 A MajorA-dur864
Prelude No.20: mid:mp3 Fugue No.20: mid:mp3 A Minora-moll865
Prelude No.21: mid:mp3 Fugue No.21: mid:mp3 Bb MajorB-dur866
Prelude No.22: mid:mp3 Fugue No.22: mid:mp3 Bb Minorb-moll867
Prelude No.23: mid:mp3 Fugue No.23: mid:mp3 B MajorH-dur868
Prelude No.24: mid:mp3 Fugue No.24: mid:mp3 B Minorh-moll869
PreludeFugueKey( Key )BWV

mp3 by TruePianos

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Soli Deo Gloria
A Method for The Well-Tempered Clavier
by J. S. Bach 2001
--- This is a fiction. ---
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Prelude No.1 C Major
The object of this Prelude No.1 C major is that the musical youth should utilize their thumbs properly. So it is not possible for anyone to play this without fingering of thumbs. For small children who cannot open their fingers enough to cover an octorve, it is designed in a form of arpeggio and fingers can be moved smoothly on the keyboard.

In early 18th century, it was not common to use thumbs for the keyboard. Without using thumbs, there will no future development for the keyboard music. So, I think it very important to make this kind of etude. Moreover it is possible to study and feel the structure of chords, their sequences and versatility of sonolity of this chord progression.

In order to do your practice, you may try various kinds of arpeggios.

Good or Not? Listen! Good or Not? Listen!
Why these arpeggios above are not good?

Because the original arpeggio is the best.

I have explored every possible arpeggio and encountered this splendid and outstanding pattern. It is unusual. First of all it is beautiful in shape. And it has a peculiar rhythmic ability. Of course the basic pattern in the half bar is "four 16ths and four 16ths". But there are "5 16ths and 3 16ths", "2 16ths, 3 16ths and 3 16ths" or "2 16ths and 6 16ths". No, it is none of them. Sometimes it is free from the beats and gets the calm and spiritual atmosphere.

This arpeggio makes all five voices independent. These five voices are in chord but move by themselves. One of them moves chromatically. Then its chord structure seems to be very complicated. If you look at the chord progression chart, you will think this music should have benn made in the late 20th century. ( Oh, I will not know that. )

The chord progression is as follows;

bar01020304050607080910
1-10 CDm7/CG7/BCAm/CD7/CG/BC/BAm7D7
11-20GGdimDm/FFdimC/EF/EDm7G7 CC7
21-30Fmaj7F#dimDm/AbG7C/GG7sus4G7Ebdim/GC/GG7sus4
31-G7C7F/CG7/CC

How it seems complicated is not the matter. It is the matter of voicing. The inner melodic structure is more important.

AMB is my dear wife Anna Magdalena. She is extremely pleased to play this prelude and listen to it.

Why don't you talk with Mrs Anna Magdalena Bach ?

By the way, about a hundred years after someone called Charles Gounod would use this prelude as the harmonic basis for "Ave Maria". He loves this prelude. I appreciate his adoration. But I am afraid that he does not know the real meaning of this work. Sorry, this prelude is not for backing.

Anna Magdalena Bach - Ave Maria - Uncertainty - next page
Here is another version of Prelude No.1 for you. It is enigmatic. Can you find the secret of it?
midi listen!

Probe and Email : mocfujita@aol.com

A Method for The Well-Tempered Clavier
by A M B 1744
--- This is a fiction. ---
Back to the main page!
An Appreciation of Anna Magdalena Bach to the First Prelude
of the Well Tempered Clavier
Thank you for visiting my page and giving me a chance to tell you my appreciation to the Well Tempered Clavier, especially the First Prelude of it.
Nice to meet you.
I am Anna Magdalena Bach of the year 1744.

my imaginary portrait
at the age of 20 years old
??secret??
By the way additional green handwriting letters
are later printed by Carl Philipp Emanuel.

This Music Notebook is my treasure. My dear great husband, exCapellmeister and now Cantor Johann Sebastian Bach, presented it to me in the year 1725, about three years after our marriage which was celebrated on 3 December 1721. Its cover is made of light yellowish green parchment and with golden letterings, "AMB" and "1725". My husband wrote two Partitas in the first pages of my Music Notebook in his own writing beforehand as a precious gift to me. And the first Partita of them is A moll in other words A minor or AM. The chord of the first measure is naturally also AM namely my initials. B means Bflat in our country and appears in the middle of this Partita three times. I love AM and B.

My Music Notebook is beautiful. It has the size of 25.0cm by 19.5cm, long from side to side. It is not an ordinary scorebook as to its size. There are 75 sheets of score paper. Both sides of each sheet can have only eight staves. In other words, we can use only four great staves on each page using our rastrale or doppelrastrale which are five line writers.
So I can't write many notes in one page. In case of the First Prelude of the Well Tempered Clavier, I had to omit several measures, from 16th measure to 20th, in order to place this prelude just on two facing pages. When I copied this prelude into my Music Notebook in 1733, actually I could play it from memory. Johann's students and our older sons could use one of the entirely copied books of the Well Tempered Clavier to practise it.

Then, why did I include this prelude in my Music Notebook?

This prelude was made of beautiful arpeggios of chords as you know. And I found out that some of chords are very peculiar. I have been interested in those chords of this prelude of the Well Tempered Clavier. They are mysterious, impressive and somehow uneasy. I had not heard them as a beautiful harmony before knowing this prelude. My dear great husband Johann Sebastian taught me that this type of chord is a stack of three minor thirds or (Re mi Fa)s : 1st, min 3rd, flat 5th, double flat 7th. Music theorists of after-ages might name them as "Diminished 7th".
In this Prelude there are four measures which include this type of chord. They are 12th, 14th, 22th and 28th. In case of the 28th measure the chord is the same as of the 22th with the base note of G. 12th, 14th and 22th are exclusively different each other.
The Fifth Circle
Measure:
12th ; GBC#Eor(C#-E-G-B)
14th ; FG#HDor(D-F-G#-H)
22th ; F#CD#Aor(C-D#-F#-A)
"B" is "Bb" and "H" is "B".

This chord is constructed out of a very special harmonic interval, the Tritone, so it may be seen to consist of two Tritones superimposed. The Tritone is either of an augmented 4th or, enharmonically, of a diminished 5th, and is considered one of the dissonant intervals. If I hear,for example, C and F# at the same time, I fear its sound. When the Tritone is used in a drama music, the situation is just for the chance of appearance of haunted monsters. But in case of the chord with four note combination I would not feel any fear but devotedness. Even though there is uneasiness, I would have cross myself when I was playing these parts of the prelude. I can't do crossing myself and playing the keyboard at the same time.

The Chromatic Circle




Carl Philipp Emanuel told me, "I think these chords are the most fruitful and powerful ones among other chords to get pleasant and surprising music transitions when you are trying to change the present music to the remotest key. It can produce numerous harmonious diversities like a springboard to other positions. But I don't use them so frequently." He always think about "the True Art of Clavier Playing" and takes special notice of this type of chord. But he confessed, "No one but our great father can combine closely all three patterns of this chord into one clavier work naturally."

Each note of this chord consists of four notes and is an equal distance apart or in all minor third intervals. By dividing one octave equally into quarters, we can get four notes for this chord. To add additional minor 3rd results in doublings of the previous notes in the chord. So building successive this kind of chord on all of its four notes produces identical copies of the original chord. In the Sacred Temperament there are only three types for this chord. Each of the three fully diminished 7th chords have four possible roots. The notes of these three chords are exactly twelve notes of an octave. Of course I did not omit measures including those chords.

C C# D D# E F F# G G# A B H
There will be much to be studied.
Each of them looks like the Holy Cross.
Wilhelm Friedemann, the oldest son of my family, was given his own original music notebook by his father in which there was also the almost same prelude as the finished version of the Well Tempered Clavier. But this prelude on his music notebook was made of only 24 measures and used only one of these special chords. My great husband had added later three more and used all three types of them.

My son, Johann Christian is now 9 years old. He started playing the clavier much earlier. Carl Philipp Emanuel, his half brother, is giving him musical training. But copied score books of the Well Tempered Clavier and other works are in great demand among our students. So Johann Christian sometimes uses my music notebook to play this prelude. Of course he does not omit to play those measures which were not handprinted. His performance is quite well. He shows great promise for the future.

By the way, you think it strange that I did not talk anything about "My Notebook presented by my honorable husband in 1722", don't you? The case related to that Notebook of 1722 in which Johann Sebastian wrote Clavier Suites from No.1 to No.5, lately named the French Suites, made it possible for me to have here my treasure,this Notebook presented in 1725. To tell the truth, I inscribed the new title on the scorebook originally for composition memorandum of my husband in a flurry but as carefully as possible early in 1723, although I was not accustomed to do such a thing. Additionally I put some pretty illustrations of musical notes in order to make the book look just like a present. The year 1722 had been already writen by my husband. It is me, Anna Magdarena Bach, who wrote the Minuet in the last page of that Notebook. Johann Sebastian told me to add a tune which everyone would think that I had wrote it by myself after writing the title. If the appearance of the score I wrote looks very poper, some would think that my husband would had written it. However well I imitated my husband's pen work, it is obvious that I wrote it. But I made some intentional errors in case. Now when you look at these errors, you might think like, "Probably Anna Magdarena does not like scorewriting so much that she made such silly mistakes you can not look over." Well I have to accept it. There were some looks-difficult comments previously written by my husband on the lower right corner on the title page. I could not erase them, so I left them just as they were.
The Clavier Notebook for Anna Magdarena Bach
the year 1722
B above is a combination of J, S and B.
In the lower right corner,
there are names of three books written
by J. S. Bach with his own hand.
Ante Calvinismus
Curricula of Theology
Anti Melancholism
by Dr. Pfeiffer, theologian
The time was just when our family were just leaving for Leipzig from Cöthen. We had started our newly wed life in Cöthen. And it had been going very well and calmly with my wonderful husband and our good children. But various situations made my husband decide to leave the Court of Cöthen for Leipzig to discharge his new duties there. On leaving for Leipzig, he was anxious whether the Court of Cöthen was going to allow him to bring out scores of his works which he composed during staying in Cöthen, or not. You know, the greatest asset of our family at that time consisted only of the music proficiency of Johann Sebastian himself, our family and, well, music scores. The Court will not usually allow the composer to bring out with him his works which he has made for the Court or the Church as his duty because these works and thier music scores belong to the Court. They would say, "Exceptionally copies only for Johann Sebastian himself, scores of the works dedicated to certain players like works for solo violin, those which have nothing to do with the Court nor the Church, and scores of the works only for your domestic affairs can be your belongings." Even this kind of treatment could be said to be fairly generous. Other lords would have been severer.

Then where are this "the Suits for Clavier" categorized? The Suites for Clavier, lately named as the French Suites by someone are in this Music Scorebook. So Johann Sebastian tought that the book should be for his wife. I thought that it was the best way to get the permission from the Court. When we got the inspection of our belongings by the authority of the Court before our move, they checked them very carefully in case of music materials, especially music scores. Actually we could bring out only a limited number of music scores compared with the total volume of composition done in Cöthen by my husband. I am afraid that the Well Tempered Clavier could have beed confiscated by the Court. But the Well Tempered Clavier already had the title writen by my husband which indicated clearly the purpose of it like

"For the profit and use of musical youth desirous of learningand especially for the pastime of those already skilled in this study composed and prepared by Johann Sebastian Bach at present Capellmeister to His Serene Highness the Prince of Anhalt-Cöthen, and director of His Chamber Music. Anno 1722".

Incidentally, I may add the fact that Johann Sebastian intended to maintain his personal friendship with Prince Leopold of Cöthen, and that he tried to visit there one time or another. In fact, I clearly remember that he officially visited Cöthen in July 1724, November and December 1725 and January 1728 when there were certain events in the Court. While staying in Cöthen, he seemed to try to take back the music scores which he was obliged to leave behind in 1723.
As a result of these certain complications, my husband presented me a new and genuine Music Notebook for me when we became to feel at home in Leipzig. It was very beautifully bound. And I found lovely partitas newly composed and written by his own hand in the first pages of it. The first and old Music Notebook which was said to be mine had returned back to the Composition Memorandum of my husband soon after our arrival at Leipzig. The Suites for Clavier of the first Book made me feel related to me by a strange chance. So I copied No.1 and the first half of No.2 to my genuine Music Notebook while Johann Sebastian was polishing up the Suites. For all these things, I thank God for divine grace.

By the way, after Prince Leopold passed away young in 1728, Johann Sebastian visited Cöthen to attend the funeral ceremony and to perform Cantatas as tribute to the memory of the late Prince Leopold in March 1729. I accompanied my husband. He was busy there for conducting Cantatas and attending events, so with the permission of the Court, I dared to copy only some of scores which Sebastian had left behind in Cöthen. Among them there was the score of "The Six Suites for Unaccompanied Cello" which was composed by Johann Sebastian Bach and dedicated to Christian Ferdinand Abel, the renowned cellist and the principal viola da gamba player, of the 12 member Cöthen Court Orchestra. Mr. Abel confirmed me that autograph scores by Johann Sebastian were registered as the Court Property. But he carefully arranged me to do my job as a copyist there. My husband said, "Anna Magdalena, you can easily copy them without serious errors because the score for the Cello Suites is a single staff one." Still, time was limited and specified there in Cöthen, so I am afraid that some slurs and similar others might have lacked preciseness even though I did nothing but simple copying just as it was.

Anyway, I am sorry to say, Calvinists there seemed to have different minds toward music from ours, which made many of other music scores had been already scattered and lost. The name of Mr. Abel remainds me that he has a son called Carl Friedrich Abel who is now about 20 years old. And he is not only a promosing composer but also a good player of viola da gamba having inherited his father's talent. He has dropped by our house in Leipzig from time to time. My son, Johann Christian, is 10 years younger than he. And they were friends. I imagine that this friendship will contribute them greatly in future.

Johann Christian
9 years old

This is about all that I can tell you here. In these 20 or so years of our mariage, I have continued learning music under the warm direction of my great husband, in spite the fact that I am a woman. Some tease me saying that my handwriting has become somehow like Johann Sebastian's as for my score copying assistant works.
I love all the works of my husband, some of which are of course too difficult for me to comprehend. Among them I love the most this prelude and the aria of the so-called "variations" which was gifted to Count Keyserlingk, Goldberg's master. These works are my lifelong friends. When I play these works on the clavier in my spare time between houseworks or while there are visitors in our house, I am brimful of joy and feel like saying a little prayer of my thankfulness.

From now on, through this prelude,
I would like to share the delight of music
with all of you as ever.
Thank you very much.

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the secret of my imaginary portrait
---------
Johann Christian Bach to Anna Magdalena Bach
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Uncertainty Principle for Temperament
--- by Iori Fujita for the Well Tempered Clavier---
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***** listen *****
Sacred Temperament and the Uncertainty Principle

In quantum physics, the Heisenberg Uncertainty Principle expresses
a limitation on accuracy of simultaneous measurement of observables
such as the position and the momentum of a particle. (1927)
Kurt Gödel's Incompleteness Theorem says that ....
"any precise mathematical system must contain some statements
that are neither provable nor disprovable
by the means allowed within the system". (1931)
In 1916 Einstein published his general theory of relativity. In it he proposed that gravity is not a force but a curved field in the space-time continuum that is created by the presence of mass. People were astonished by the fact that space and time are under relativity. Even though, the world could stay ease within the determinism where there was a general algorithmic procedure for resolving all mathematical questions.
But since Heisenberg's discovery, Isaac Newton's laws of motion has not been used to predict accurately the behavior of single subatomic particles. The world was then suffered from this uncertainty. Gödel gave an additional blow to the people's mental world with the incompleteness.
Relativity, Uncertainty and Incompleteness!
Wow! We are free from the determinism!

Roger Penrose, Professor of Mathematics at Oxford University and a physicist, now happily says, "We cannot create any kind of new artistic sensitivity however we may accumulate many times of calculations. Art is a non-computable physics."

I would like to say, "Music is a non-computable physics, too".

But over the centuries musicians, mathematicians, theorists, thinkers, experts and amateurs have been suffered from the comma which is the difference between a perfectly tuned octave and the octave resulting from a tuned circle of fifths. Many great people have been trying to create the perfect scale in vain. Mathematics easily proves that perfection is not possible. Any solution does not exist. Musicians, especially pianists, have been accused of using the Equal Temperament for thier pianos because the Equal Temperament is said to be an anti-musical compromise which leaves each key equally damaged and none perfectly in tune.
This comma has put a curse on music.
ΔxΔp ≥ h ⁄ (4π)

x ; position
p ; momentum
Δx, Δp ; the standard deviation of x, p
h ; the Planck's constant

If Δx → 0, then Δp → ∞.

An accurate measurement of one observable characteristic involves a large uncertainty in the measurement of the other.

This is the Heisenberg Uncertainty Principle .

Can this be applicable to the sound wave? This is for the behavior of single subatomic particles, isn't it? The Planck's constant "h" is too small to be took into consideration in our daily life.
h = 6.6261 × 10-34 Js

A tennis ball (about 50g) with a 180km/h ( 50m/s ) speed which was served by Ms. Martina Hingis would have any uncertainty about its location? Actually No! In case of Δv=0.01m/s, the quantum deviation Δx is about 2*10-31m or 0.0000000000000000000000000002mm.

Sound is the result of periodic fluctuations in air pressure propagating through the ordinary atmosphere. The sound waves are in the ordinary world. You may think that the Uncertainty Principle has nothing to do with music. But the formula can be transformed into the relation between time and frequency of waves.
ΔxΔp ≥ h ⁄ (4π)
p = mv → Δp = m Δvv = velocity (m/s)
E = m v2/2 → ΔE = m v Δv = m Δv v
ΔE = Δp v = Δp (Δx/Δt) = (Δp Δx)/Δt
ΔE ≥ (h /(4π))/Δt
ΔE Δt ≥ h /(4π)
E = h ν → ΔE = h Δνν = frequency(1/s) = f, E for a photon
h Δν Δt ≥ h /(4π)
Δν Δt ≥ 1 /(4π)
Here, there is no "h". Δν ≥ 1 /(4π Δt)
Δt ≥ 1 /(4π Δν)
Δt and Δν are the standard deviation of t and ν.
The position and momentum of a wave in space have been changed in form to the time and frequency of a wave. Here is a sound wave as a time-varying signal. Now we have found out the fact that it is impossible for us to know the exact frequencies in our sound wave at an exact moment in time. When Δν is 0.1Hz, Δt should be more than 1 /(4π *0.1) sec which is about 0.80 sec.
Middle C of the equal temperament is calculated approximately as 261.6256Hz. If you want to determine C with this precision 0.0001Hz, you need Δt of 800 sec, or 13 min 20 sec. If you want to get the exact Middle C, you need an infinite time and a continuous wave of C.
In other words, theoretically we cannot get the perfect fifth tone with a frequency f ( that is ν ) from a root tone with a frequency f0 by calculating f = f0 × 3/2.
In another treatment, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, also in the case of normally distributed variables, leads to a lower bound of
Δν Δt ≥ 1/( 2π ) for the product of the uncertainties. For different types of wave packets and for other treatments, the uncertainty can be set to a much lower bound.
a normal distribution wave
the minimum wave packet
Δν Δt ≥ 1 /( 4π ) is for the case of the normally distributed variables or the minimum wave packet like a photon and a wave with the standard deviation form.
Δν Δt ≥ 1 is the ordinary interpretation for the frequency and time.
popular sound waves which are not minimum wave packets
In this physical world there can exist no temperament based on the ratios of whole numbers. Even harmony must live with this "uncertainty".
the Fourier Transform Theory:
You may say again that it is only for the subatomic world. But the Fourier Transform Theory reveals that the sound wave cannot get rid of the Uncertainty of Time and Frequency.
Jean-Baptiste-Joseph Fourier was best known for his analysis of heat conduction, and he was also an public administrator of French Covernment and Egyptologist. Fourier, as Egyptologist, accompanied the expedition of Napoleon who invaded Egypt in 1798. His book "The Analytical Theory of Heat" in 1822 showed how the conduction of heat in solid bodies could be analyzed in terms of an infinite mathematical series, the Fourier series. Fourier Analysis for music is based on the concept that sound waves can be approximated by a sum of sinusoids. The frequency of each sinusoid in the series is an integer multiple of the frequency of the basic signal. It is said to be the same as the harmonics of the original waveform.

It is a very useful theory for us to study timbres. There are sawtooth, square, trangle, sine and other types of wave form. Each form corresponds with a certain timbre. By arranging an and bn, we can create any type of wave form.
The equation below shows how a square wave can be made up by adding together pure sine waves at the harmonics of the fundamental frequency.
f ( t) = sin(ωt) − (1/3) sin(3ωt) + (1/5) sin(5ωt) − (1/7) sin(7ωt) +.....

In general; ao = 0 and bn = 0
f ( t) = a1 sin(ωt) + a2 sin(2ωt) + a3 sin(3ωt) + a4 sin(4ωt) + .....

This f ( t) is defined within a certain area. And It should be periodic. So, f ( t) ≠ 0 outside of the area. The sound can't stop while accumulating sin(nωt)s.

This theory is for a periodic and continuous wave. Sound waves are approximately periodic within the time-length while they exist and to some extent continuous before they fade away. The Fourier Analysis is setting forth "the infinite continuity of the wave" as a premise. At least it is a theory for a stationary wave. Sounds are to be classified as non-continuous and pseudo-periodic waves. In other words, it is difficult to describe an actual sound of a limited duration by integer multiples of the frequency of its basic signal.
Then comes the Fourier Transform Theory's turn;
A physical signal, such as sound pressure can be represented as a continuous function of time. This is the time domain representation of the sound. There is an equally valid frequency domain representation.
f ( t) = a sound wave ; for example; the simplest one:
f ( t) = A(t) sin( 2π ν t) ; ν = frequency(1/s)


ω = 2 π ν

F ( ω ) equals the integral from minus infinity to infinity of f of t times the exponential to the minus j ω t dt.
The Fourier transform separates a waveform into sinusoids or sine curve functions of different frequency which sum to the original waveform. It is typically thought of as decomposing a signal into its component frequencies and their amplitudes. The magnitude of this function F ( ω ) is normally called the "frequency response". And the magnitude of the resulting complex-valued function F ( ω ) represents the amplitudes of the respective frequencies ω and shows the frequency distribution.

T1 -T0 = Δt = 0.025 sec
listen!
You may say, "Until t = T0, there was a silence. At t = T0, the sound started with a certain frequency. And at t = T1, the sound stopped. After t = T1, there remained silence. There is no room for the uncertainty."
For the note of 440Hz, the wave with 11 cycles should be descrived as;
t < T0 f ( t) = 0
T0 ≤ t ≤ T1 f ( t) = A sin( 2π ×440 t)
T1 > t f ( t) = 0

the amplitude of F ( ω )
ω0 = 2π ×440 and Δt = 0.025 sec

The red curve indicates the normal distribution with σ = 80Hz. This curve resembles the amplitude of F ( ω ) almost nearly. So, the amplitude of F ( ω ) also has the standard deviation of 80Hz.
Δt × Δf = 0.025 sec × 80 Hz = 2


Δt × Δf = 2
Temperament Examples (Equal Pure Werck3)
The equations above look very simple and make us think that the spectrum would have its peak at 440Hz and that the magnitude of other frequencies should be zero.
The spectrum spreads widely while Δt is short. The longer Δt, the narrower the bandwidth of the spectrum. But even Δt is 1.000 sec, the bandwidth remains about 2Hz.
axis i

440Hz
Δt = 0.50 → 0.05 pitch=0.05
amplitude of the spectrum
normal distribution with
the standard deviation of 2 / Δt

Δt × Δf = 2

An Example of Performing Time

Δt =0.20sec → Δf = 10.0Hz
Δt =1.60sec → Δf = 1.25Hz

The uncertainty clouds cover the differentials of pitches which depend on various temperaments. And this uncertainty might be able to break the historic curse over the music.
Interpretation:
The just noticeable differential in pitch is conveniently expressed in cents, and the standard figure for the human ear is 5 cents. For the pitch A = 440 Hz, "5 cents" is approximately ± 1.3 Hz.
n (cent) = 1200 log10( p/p0) / log10 2 → p = p0 2n/1200
441.3 = 440 × 25/1200
Synchronization:
It is said that two pendulum clocks with a slightly different time lag, if they are attached side by side on one thin wooden board, will be synchronized before long and show the same swing of pendulums. Autoresonance is a remarkable phenomenon of nonlinear physics when a driven nonlinear system stays in resonance with the driving oscillation or wave continuously despite variation of system's parameters.
The system of a piano has the same charactaristics of autoresonance or synchronization as above said. In case of the C Major chord strongly attacked with C, E and G, at first, there is an intense and rich but complex resonance, right after that, beats start, which is not emerged for the purely tempered piano, but followed by attenuation of the sound of the chord the beats are going to calm themself down and instead well harmonized Major chord comes out. It is adequate to understand this phenomenon due to the characteristics of the human listening sensibility. But it is also because of the mutual interference among piano strings, boards and structure of the acoustic piano.
The Sacred Temperament:
The background music of this page is the First Prelude of the Well Tempered Clavier Book 1 with a certain trick. The Prelude starts on C Major and ends on D Major. Its pitch is gradually getting higher. Can you sense out the change?

***** listen ***** midi *****
Verleih mir, Höchster, solche Güte, so wird gewiß mein Singen recht getan;
so klingt es schön in meinem Liede, und ich bet dich in Geist und Wahrheit an;
so hebt dein Geist mein Herz zu dir empor, daß ich dir Psalmen sing im höhren Chor.
 from the Music Notebook for Anna Magdalena Bach (39b)
* temporary translation *
Borrow to me, Holy God, your kindness, thus my singing is certainly done quite well;
thus it sounds beautiful in my song, and I pray you in spirit and truth on;
thus your spirit raises my heart up to you, that I sing Psalms for you at the height of Chor.

In the beginning was the Note.

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A Method for The Well-Tempered Clavier
by J. S. Bach 2001
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Fugue No.1 C Major

Which finger are you going to use for the first note "C quater" ? You use your righthand's thumb. Oh my God!

There are a few clavier works for practice which encourage the use of your left hand. Your left hand have to be awaken up. In this fugue the left hand first. It is like escorting a lady by your left hand. If you practise the violin, your left hand has more important role than your right hand. Why not for the clavier? In the first bar all five fingers of your left hand have to work naturally.

Anyway this is a fugue with four voices. It is not easy for playing compared with the first prelude. And the theme starts on C Major and is afterward developed on several Major and minor keys. Accidental shraps and flats are following as the changes of keys. Or the first note of the recurring theme decides the key for the following bars. All these keys are mixed up and we can identify them only vaguely. But you have to find the meaning of each accidental sharp, flat or natural clearly. It is convenient for you to study it because this is on C Major key and basically you use white keys on your clavier. All black keys which you use are out of C Major scale.
By the way you may find out that this Fugue contains twenty four statements of its subject as the first of a collection of twenty four fugues. Although three of them are a little bit varied, I placed the theme 24 times in this Fugue. My effort has been rewarded. Young Goldberg has already known about it.
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*+*+*+* Goldberg's comment *+*+*+*
A Method for The Well-Tempered Clavier
by J. S. Bach 2001
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Prelude No.2 C Minor
Prelude No. 2 requires a precise synchronization of both hands of yours. And a lot of important notes should be played by your thumbs and little fingers. Those fingers should be connected to your brain at first. After a sufficient practice each of your thumbs and little fingers will have its own independent mind under your control. Then you can play the upper part by your left hand and the lower part by your right hand.

It is rare that this Prelude No. 2 has its own indication for the tempo, Presto at the 28th bar and Adagio at the 34th bar appointed by me, J. S. Bach. Well, "Presto", and you need a speed from the first bar.

The reason why I did not start with A minor after C Major, instead I started with C minor, is not so simple. For the practice, it may be easier to start with A minor, because there is no sharps nor flats as basic key. C minor needs three flats.

One of the reasons is that you have to get used to black keys as early as possible. Using black keys is not difficult at all. Sometimes it is very convenient to your fingers especially in fast passages. The second reason is that starting with A minor for the twelve minor key journey seems old-fashioned to me. For the Aeolian mood, note names, I think, were allocated as a-b-c-d-e-f-g-a. Actually this mood is now "minor". However, in older days it was major in use. As music developed, the Ionian mood became prevailing. This Ionian mood is now C Major. Major Scales are suitable to do versatile composition works for ordinary musicians. Yes, C major is major, and at the same time it was thought to be on the extension of A minor. Now A minor is on the extension of C Major. Anyway C Major and A minor are strongly related with each other. I do not want to follow this way of thinking. So I decided to start like C Major, then C minor.... To tell the truth, C Major and C minor are so distant from each other that I thought I should put a connection in Prelude No. 2. Can you find it? Ok. The final chord or harmony is C Major itself. C-C-G-C-E(with natural).

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A Method for The Well-Tempered Clavier
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Fugue No.2 C Minor

In this fugue it is very important to keep firmly the rhythm of the theme by playing staccatos precisely. The rhythm makes the character of this theme. You need not put any strong accent for these notes with staccato. Many people have a wrong concept like "notes with staccato should be played stronger than others." Notes with staccato should NOT be played stronger than others. But instead, you have to keep the length of the notes with staccato constant. The length with staccato is half of one without staccato. Too short is wrong. When one half time has passed, you lift your specific finger from the key and rest a while. It is rather difficult than you imagine.
On the contrary some teachers may tell you that you should play with slurs, in other words, without staccatos. It is also a good idea. In any case the theme should be identified with a concrete character.

And there is a series of left-hand octave with fast motion in the last seven bars. Here you have to use your thumb and little finger.

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A Method for The Well-Tempered Clavier
by J. S. Bach 2001
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Prelude No.3 C# Major

This prelude is C sharp Major with seven sharps on the score. I would say this is the first clavier work of C sharp Major in history. You may think it very difficult to play. You have use black keys many times. But it is not so difficult. It is a matter of experience.

~
I wrote this prelude in C sharp Major instead of D flat Major which needs only five flats. I thought that next after C should be C sharp naturally. The truth is that C sharp Major is the furthest key from C Major. Of course you can write in D flat Major.

What I am afraid of is that your clavier is not well-tempered. To avoid harmonious troubles to your ears, almost of all notes are short and in arpeggio style. Then the outcome is brilliant note combinations like black keys are transformed into golden keys or silver ones even for a not-so-well-tempered clavier. You may not be bothered by the dispute about the temperament system,and you can enjoy playing it.

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A Method for The Well-Tempered Clavier
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Fugue No.3 C# Major
Through performing this fugue you will get accustomed to black keys.
And you will love them.
Where, do you think, is the center of the keyboard?
Is it C?
No. The center should be D or G sharp regarding the symmetry. Then D is a white key. D is a low profile and sandwiched by two black keys, so its image as the ceter key is weak. G sharp is black one and stands out.
In short I wanted to tocar el centro del clavier or the center of keyboard. But you would say that it is G sharp, not C sharp. Isn't this fugue on C sharp Major? It should have started with a note of C sharp? Ah yes, it is Fugue C sharp Major. G sharp is the perfect 5th of C sharp. Six fugues out of 24 start with the perfect fifth note. There is no problem at all. It's fun.

Fugue010203040506070809101112131415161718192021222324
keyCCC#C#DDEbD#EEFFF#F#GGAbG#AABbBbBB
first noteCCG#C#DDBbD#EECCC#F#GDAbG#AAFBbBF#
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A Method for The Well-Tempered Clavier
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Prelude No.4 C# Minor

There is no word necessary to describe the meaning of this Prelude. C sharp Minor will produce a new world of clavier music. It should be beautiful. It should bring you a comfort.

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A Method for The Well-Tempered Clavier
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Fugue No.4 C# Minor
The first note is a C sharp whole and single note. And afterwords you will experience a five voice fugue for the first time for this well-tempered clavier.

Again the length of the first note determines the whole tempo of this fugue. Keep it in mind.

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A Method for The Well-Tempered Clavier
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Prelude No.5 D Major
Your right hand creates a series of 16th notes, and your left hand keeps the rhythm by 8th notes.
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A Method for The Well-Tempered Clavier
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Fugue No.5 D Major
The subject is rhythmic. Rhythm is important here. But the certain tonal height of the subject is also indispensable.

Which height do you prefer?
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A Method for The Well-Tempered Clavier
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Prelude No.6 D Minor
This prelude was originally written for the Clavier Book for Wilhelm Friedemann Bach. I added the last part to the original and completed it like this.
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A Method for The Well-Tempered Clavier
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Fugue No.6 D Minor
What do you see through the D key if it is transparent ?

If you feel some symmetries between the first half part and the latter, you are in the new world of music.

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A Method for The Well-Tempered Clavier
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Prelude No.7 Eb Major
Regrettably this key was described as "Eflat Major". I wanted to use the keyname "Dsharp Major", which requires nine sharps. As the result there should be two double sharps, "Cdoublesharp and Fdoublesharp". I do not care. But it was not allowed by the scoring rule. It is an over enharmonic. Then you are facing to "Eflat Major". We came here climbing from C, Csharp, to D, but now we should go down one semitone from E to get Dsharp.

So I used a work I previously composed for this No.7. It is a prelude with figue, which might be an unexpected child of the enharmonic system.

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A Method for The Well-Tempered Clavier
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Fugue No.7 Eb Major
I pulled myself together to try to go with Eflat Major.
Then I composed a new fugue for this key.
It is quite simple but anyway I did it.
This fugue is for circling notes.
So have fun.
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A Method for The Well-Tempered Clavier
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Prelude No.8 Eb Minor

In the first bar, you use only black keys. Listen to the sound. It is great, isn't it? Feel good? Ok. Let's talk about the music.

The reason why this prelude is on the Eflat Minor instead of Dsharp Minor is simple. Rewrite it to Dsharp Minor, then you will find many doublesharps on the score in order to make the score more readable. Of course these doublesharps can be replaced by naturals, for example. Anyway in this case I applied Eflat Minor.

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A Method for The Well-Tempered Clavier
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Fugue No.8 D# Minor

As I said, I should have writen Prelude No.8 E flat Minor on the key D sharp Minor. Anyway I understood the meaning of "enharmonic". So, this combination of E flat Minor and D sharp Minor would have some educational effect. To tell the truth, this fugue was originally written on D Minor and transposed. So naturally it is on D sharp Minor.
The theme of this Fugue sounds like a beautiful melody of a song. So to increase the beauty of this Fugue, it won't work to add inverted statements a lot on a cold logic. Then augmentated statements must be used in a proper way. They have inherited the melody itself. You can find them only three times in the second half of theis Fugue. The first appearance is in bar 62 as the lower voice. The second is in bar 67 as the middle voice. And finally the third appearance is in bar 77 as the upper voice, singing with spiritual elevation. The consideration about how this augmentated statement as the upper voice appears beautifully made this Fugue longer than expected.
A fictional dialogue between J.S.B. and W.F.B.
J.S.B. Here comes my first son Wilhelm Friedemann. It's a long time since I saw you last. How are you doing?
W.F.B. Fine, thank you, my great father. You look really great. Well, I am from Dresden. But recently I am going to look for a job in another place. By the way, how is Gottlieb, one of your eligible disciples? His skill of performing the clavier must be advanced.
J.S.B. You are talking about Johann Gottlieb Goldberg, aren't you? Yes. He is now working for Count Keyserlingk as his clavier-player. Then I sent reverently one of the first printed copies of "Keyboard practice or clavier etude consisting of an Aria with different variations for the harpsichord with two manuals" to Count Keyserlingk as a gift. Gottlieb surely plays this work very well.
W.F.B. I hope so. His name is Johann Gottlieb Goldberg, but, you know, sometimes he is called Johann Theophilus Goldberg. "Gottlieb" is equal to Theophilus and it means "love of God" in German, and in Latin it is "Amadeus". So young Goldberg might be called Johann Amadeus Goldberg.
J.S.B. I don't see what you mean by that, my son.
W.F.B. I don't know what I mean, either. But I saw a young maestro of future who had "Theophilus" as his muddle name in my dream of one night of last month. In my dream, he was transcripting this fugue "Fugue No.8 D sharp Minor" into D minor for the three string trio.--listen!-- He also transcripted other two fugues of yours and one of mine. My fugue is the work which I will compose in far future, so I have never heard of it.
His name was Johannes Chrysosstomus Wolfgangus Theophilus M***** and he was called "Amadeus" in short, so he said. In this dream, he was quite interested in fugues and he studied them eagerly. He said, in my dream, that he had met with Johann Christian, now our youngest boy in our family in a foreign city. It is mysterious, isn't it?
J.S.B. Did he played music? Did you hear any music by that Amadeus in your dream?
W.F.B. Yes, I memorized some of his own clavier works. They are beautiful and sound new to me. I can play them, but I can't take down in musical notation. Try to do it as I might, I couln't do it. His works in our brains absolutely refuse to be writen down to the score by me nor by my colleagues. I have been inspired with his music. I can't help giving myself up to improvisation based on his musical imagination.
J.S.B. Ah, my son! You should not have dreamed such a dream. You have to forget about it as soon as possible. Amadeus might be a genius for music, but for you he would be a devil. I awfully afraid that he would ruin your musical talent.
listen! Transcribed Version by V.A.M.
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A Method for The Well-Tempered Clavier
by J. S. Bach 2001
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Prelude No.9 E Major
E Major seems to be very simple but needs four sharps. The first bass note is long E which lasts one bar and three quarters. Can your clavier make this long sustaining sound ?
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A Method for The Well-Tempered Clavier
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Fugue No.9 E Major
This fugue will be said to have a long subject which consists of twenty nine notes. But it doesn't matter at all. Anyway a set of the first two notes, " E and F sharp " is the most important subject in this fugue.
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A Method for The Well-Tempered Clavier
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Prelude No.10 E Minor
This Prelude No. 10 has its own indication for the tempo, Presto by me, J. S. Bach.

It took me some longer days to complete this Prelude No.10 E Minor. At first this work was simple like moving lower sixtieths by left hand with chords by right hand. Afterawrds I added an upper melody part and the latter Prest part. So, I put the indication of tempo, Prest.

You are impressed with this melody, you can hear a flute version.
Here you are !
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A Method for The Well-Tempered Clavier
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Fugue No.10 E Minor

This is the only one fugue with two voices in this Well-Tempered Clavier. Usually fugues have three or more voices.
There are only two voices, and so there are two bars (No.19 and No.38), where the upper part and the lower part form an unison which might be very unusual for a fugue to contain.
FugueNo.010203040506070809101112131415161718192021222324
voices433543333234343444343544
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A Method for The Well-Tempered Clavier
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Prelude No.11 F Major
This is a trill practice for both of your hands. You should think about what kind of trills are suitable.

The Method of Ornaments

Clickable Map

Trillo Mordant Trillo u.
Mordant 		Cadence Doppelt
Cadence 		Idem
Doppelt
Cadence u. Mordant 		Idem Akzent
steigend 		Akzent
fallend 		Akzent u.
Mordant 		Akzent u.
Trillo 		Idem

In order to explain various signs and show the better way of performing ornaments I put the whole list of ornaments in "A little lesson book for clavier (1720)". My children learnt the technique by this book.
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A Method for The Well-Tempered Clavier
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Fugue No.11 F Major
F Major is not unusual. You do not have to use black keys so many times. You can forget about the well temperament mental stress.
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A Method for The Well-Tempered Clavier
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Prelude No.12 F Minor
This prelude has nothing special. But we have come to the turning point of the Well-Tempered Clavier. So it should have some meaning like a healing effect. Oh, that's it.
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A Method for The Well-Tempered Clavier
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Fugue No.12 F Minor
The subject in its first appearance does not have the same note except C. The subject in its second appearance does not have the same note except F. So you will encounter an almost complete chromatic sequence.
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A Method for The Well-Tempered Clavier
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Prelude No.13 F# Major
F sharp Major requires six sharps. As far as I know, nobody has composed a tune on F sharp Major before. Which might be more unusual C sharp Major and F sharp Major ? C sharp Major requires seven sharps, but it can be treated as D flat Major, then it needs only five flats. In fact F sharp Major is the farthest from the most popular C Major.

I wanted to make this prelude look like an easy and lovely one to play.

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A Method for The Well-Tempered Clavier
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Fugue No.13 F# Major
I composed this fugue compact and cute
in order not to make you sick
with six sharps by practising it.
In my family concert this fugue would be played in ensemble.
lute, trumpet and cello
It will be fun.

In case of Clavier, you have to create this atmosphere for yourself.
So, it will be more challenging.

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A Method for The Well-Tempered Clavier
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Prelude No.14 F# Minor
If possible, I would like to replace this prelude with another F sharp minor Prelude, which would be included in the Well-Tempered Clavier Book 2 or Volume 2.
another page for F sharp minor Prelude
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A Method for The Well-Tempered Clavier
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***** listen *****
Prelude No.14 F# Minor from so-called Book 2

a tea cup named as Fujisan
by Hon'ami Koetsu
Edo Period (17th Century) National Treasure (Japan)

"Then Peiwoh changed the key and sang of love. The forest swayed like an ardent swain deep lost in thought. On high, like a haughty maiden, swept a cloud bright and fair; but passing, trailed long shadows on the ground, black like despair."

I love a cup of coffee. Oriental people are said to love a cup of tea.
Whose hand is it? It is a hand of G. Goldberg.
No. It should be a hand of another G.G..

This is the F sharp minor Prelude from Book 2.

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A Method for The Well-Tempered Clavier
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***** listen *****
Fugue No.14 F# Minor from so-called Book 2

Thunder God and Wind God Screens
Fujin Raijin Zu Byoubu
by Tawaraya Sotatsu
Edo Period (17th Century) National Treasure (Kennin-ji Temple, Kyoto, Japan)

"Again the mode was changed; Peiwoh sang of war, of clashing steel and trampling steeds. And in the harp arose the tempest of Lungmen, the dragon rode the lightning, the thundering avalanche crashed through the hills."

The biggest sounds in the world might be from thunders. I was impressed by the fact that this Thunder God has twelve small drums around himself. You have to count them. Why twelve?
In the beginning was the Note.

This is the F sharp minor Fugue from so-called Book 2.

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A Method for The Well-Tempered Clavier
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Fugue No.14 F# Minor
If possible, I would like to replace this fugue with another F sharp minor Fugue, which would be included in the Well-Tempered Clavier Book 2 or Volume 2.
another page for F sharp minor Fugue
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A Method for The Well-Tempered Clavier
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Prelude No.15 G Major
As always G Major is major and popular. So I tried a very special method for the rhythm, 24/16 for the upper staff and 4/4 for the lower staff. You may think that 24/16 is for the right hand and 4/4 for the left hand. My intended point is different. Actually three 16th notes form a triplet on score. But you have to treat them exactly as 16th notes no matter which hand of yours in use for 16th notes, then each 16th note will have an independence and the wave of the 16th notes will appear in front of storng rhythm keeping 8th notes.
So, it is different from Prelude No.6.
A Comment by Count Hermann Carl von Keyserlingk in 1744
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A Comment for The Well-Tempered Clavier
by Count Hermann Carl von Keyserlingk in 1744
--- This is a fiction. ---
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***** listen *****
Prelude No.15 G Major from so-called Book 2
My name is Reichsgraf von Hermann Carl Keyserlingk.

When I was asked to make a comment for the Well Tempered Clavier by my great friend Johann Sebastian Bach, I thought that I was supposed to be in charge of 'the Fifteenth Prelude and Fugue'. It is on G major. I am addicted to G major because 'the Aria and 30 Variations', which J. S. Bach presented to me, is on G major. I love them so much. By the way 'the Fifteenth Prelude and Fugue' of the Well Tempered Clavier is not said to be the better one. Mr. Altonichol has been compiling the Second Volume of the Well Tempered Clavier. And the Fifteenth of the new volume is simply beautiful. Can you hear it? Why doesn't he propose the replacement of the Fifteenth.

I wonder where I can start my comment. Let me see........
I got it!
A story about Mr. Goeorg Erdmann and my musician Goldberg
may have something to do with G major.
It was 13 years ago in the year 1731 when I visited Dantzih and for the first time I met Mr. Georg Erdmann there. Dantzih or Danzig is called by the Polish as Gdansk, pronounced like "Gdanisk". The City of Danzig is located at the mouth of the Vistula River on the Baltic Sea and has ports and shipyards. It was then granted local autonomy and prosperous beautifully as the largest of any eastern European city. There Mr. Georg Erdmann was working as a diplomatic agent or a commissioner of the Embassy of the Emperor of Russia in Danzig. Mr. Erdmann was 48 years old and I was 34. I was 14 years yonger than him, but I was Russian Envoy to Berlin, so he remained reserved at first. In course of courteous conversation at a night party for the diplomatic corps, he happened to talk to me about his old friend, Johann Sebastian Bach. I already heard of Bach's reputation as the Director of Music and Cantor at the Thomas-Schule in Leipzig. Mr. Erdmann and I were similarly disposed toword music. Even after the party two of us talked a lot about various matters and especially on music.
Mr. Georg Erdmann said,
"On 15 March 1700, I remember this as well as if it had happened yesterday, Bach, only just fifteen, and I, two years older than him, left our hometown, Ohrdruf, and went 290 km north to Lüneberg to study at the Latin School. There we paid our tuition and living expenses by ourselves by singing in the matin choir of St. Michael's choir and church. Actually Johann Sebastian was an orphan. And in my case, I wanted to be a lawyer in spite of my father's strong opposition. We had to earn our living. Our good boysoprano voices helped us to live. Anyway it was a great learning period in my friend's musical career. The school had an impressive musical library where there were over 1000 important music manuscripts and prints of the works of about 200 of Germany's greatest composers. Johann Sebastian seemed happy with them. He was fond of reading and copying scores. And I knew that he was a splendid keyboard performer even when he was only 15. So he had an oppotunity to take special lessons from the famous organist, Georg Böhm. He also tought Johann Sebastian the French musical tradition. We soon lost our soprano voices, but he was able to work and earn as a violinist in the orchestra and as a harpsichord accompanist during choir rehearsals. And my father at last consented to my wish to be a lawyer and thankfully began to support my expense in Lüneberg."

Mr. Georg Erdmann continued,
"Of course we studied Latin, Rhetoric and other subjects seriously. He always ranked almost the highest in our school. I did my best, too. We completed the Latin school when he was 18 and I was 20. After graduation we parted at Lüneberg and headed for our respective destinations. I went to college to study Jurisprudence and Politics. And Johann Sebastian went Weimar to work but shortly after he found a post as organist at Arnstadt. More than 10 years after I visited Weimar in 1714 on business and saw his family. At that time he was 28 years old and he had changed his post to an active member of the chamber orchestra and Organist to the Court of Weimar. But I'm sorry I have never had a chance to see him since then. Rumor has it that Johann Sebastian plays an active part in the world of music. Also I corespond with him one time or another. The recent letter from him was one I recieved last year, in 1730. In that letter Johann Sebastian wrote about his domestic situation,

'I am married for the second time, my late wife having died in Cöthen. From the first marriage I have three sons and one daughter living, whom you will graciously remember having seen in Weimar. From the second marriage I have one son and two daughters living. My eldest son is a Studiosus Juris, and of the other two from the first marriage, one is in the prima class, the last class of school and the other in the secunda, and the eldest daughter is also still unmarried. The children of my second marriage are still small, the eldest, a boy, being six years old. But they are all born musicians, and I can assure you that I can already form an ensemble both vocaliter and intrumentaliter within my family, particularly since my present wife sings a good clear soprano, and my eldest daughter, too, joins in not badly.'
I can imagine how they are enjoying thier family life."

Mr. Georg Erdmann added,
"Well, I just purchased a new score book published by J.S.Bach titled 'Clavierübung I' which contains 6 Partitas. Three months ago I ordered it for friendship's sake. I thought it very interesting and beautiful. My clavier performing ability was not well enough for this work. There was a young organist in Marienkiiche or St Mary's Church of Danzig whose father was from Ohrdruf. So I met this young with his father man in our hometown party at the nearest restaurant from the Danzig City Hall. He said that he would de delighted to play it at a music gathering I planned to held in my residence. The music was marvelous and had a good reputation which led me to give free concerts at my favorite coffee shop two times until now. Among the audience of the latest concert there were Mr. and Mrs. Goldberg with a small boy from Königsberg. They said that this music was amaging and that thier son was only 4 years old but glad to listen to it. He just started to play with clavichord at home. Mr. Goldberg said that his son might be a genius and added, 'I am like a doting father'.

Königsberg and Seven Bridges
Königsberg is a next city from Danzig. It is also beautiful and prosperous like here. But it is worldly famous for the Königsberg Bridges Problem, which means 'The river Pregel divides the city into four separate landareas. Seven bridges connect them. People in Königsberg love to take walks along the river and the island for some time, and it has became a Sunday tradition to take the walk of seven bridges. Many of them wondered if it were possible to take a journy across all seven bridges without having to cross any bridge more than once. The elderly say, 'We could do that easily, but then there were only six bridges.' Now many people want to find the solution. So they walk in town over and over and they don't have time to listen to music.' Mr. Goldberg said, 'People there are addicted to this kind of problem or Philosophical matters. I wish my son could live with music like in Danzig.' Later I made a handcopy of the easier peaces from Clavierübung 1, and presented it to this kid named Johann Gottlieb Theophilus Goldberg."

Mr. Georg Erdmann was worried while saying,
"Your Excellency Mr. Count, would you please do me a little favor? My friend Johann Sebastian Bach wrote to me in the same letter,

'....The authorities of Leipzig are odd and little interested in music, so that I must live amid almost continual vexation, envy, and persecution; accordingly I shall be forced, with God's help, to seek my fortune elsewhere. Should Your Honor know or find a suitable post in your city for an old and faithful servant, I beg you most humbly to put in a most gracious word of recommendation for me.'
He seemed to have been in trouble. I'm afraid I am much overrated. I don't know what to do for him. Johann Sebastian complained that the Princess, the bride of the gracious Prince of Cöthen, had seemed to be pleasurable or an amusante and the Prince got absorbed in being with her day and night. This marriage made the Prince overjoyed. Their wedding celebration continued a full five weeks. Johann Sebastian composed two of new cantatas for the celebration party. These things forced, not directly however, Johann Sebastian and his family leave for Leipzig. Without her presence, he would have spent the rest of his life in Cöthen."

In the evening of next week Mr. Georg Erdmann prepared a small concert at the Ambassador's residence in Danzig. The young organist in St Mary's Church played 'Clavierübung I' or 6 Partitas composed by J. S. Bach. The Ambassador and his wife invited honored guests and they were all very glad to listen to the music. Of course the Ambassador invited me to this concert at Mr. Georg Erdmann's discretion. I fell in love with Bach's music. Then I thought I should become acquainted with him.

Six years later, early in 1737, I had another chance to visit Danzig in business from Dresden, because I was discharging my duties as the Russian Ambassador to Dresden from 1733. I was informed beforehand that Mr. Georg Erdmann had passed away the year before. I felt sorry that I could not talk with him about music especially about J. S. Bach. That time, to take the boy Goldberg to Dresden was my assignment. Because Dresden was a center of contemporary music activities. The boy grew up to 10 years old and was amazingly talented in music at the school of St Mary's Church of Danzig by the support of the late Mr. Georg Erdmann. But the poor boy had lost his father and he should live by himself.
According to Mr. Erdmann's last wishes I desided to take care of the boy by making him one of my attendants in my residence in Dresden. The boy was lively and intelligent. On the way to Dresden he said, "I have been lonely but I am all right now. By the way, my Lord, do you know the Königsberg Bridges Problem which was very famous because many people tried and failed to solve it? But Leonhard Euler, the Swiss mathematician and physicist, visited Königsberg and solved it magically in 1735. Euler solved the problem by showing why a one-pass-route could not be found. That's really something. It is not only a new and profound mathematical concept but also easy to understand even for me. The name of city Königsberg will be immortal in the history of mathematics. I appreciate Euler's achievement and feel sorry that he lost the sight of one eye two years ago because of his hard work like his excessive obsevation of the Sun. So for my part I will do my best to be helpful to my Lord and be a good musician."
The Königsberg Solution
Shortly after we arrived at Dresden, I arranged an opportunity for the boy to learn from Wilhelm Friedemann Bach 27 years old, the eldest child of J.S.Bach, who was the organist in Dresden from 1733, the same year when I settled in Dresden.
I assisted Bach to get the title of the Court Composer. Even more I arranged successfully a royal meeting of Bach with King Friedrich.
The boy, Goldberg, sometimes gave me a new piece of information about mathematics, saying, "My Lord, do you remember Prof. Euler? Yes, he solved my town's problem. This time he adopted the symbol π pronouced pai, the ratio of the circumference of a circle to its radius in 1737, yes, two years ago. Beautiful, isn't it? And it is quickly becoming a standard notation. Then I had a dream in which Prof. Euler would reveal an exquisit formula in about coming ten years."

"It looks beautiful but I can't imagin its meaning."

In 1741 J. S. Bach met Goldberg of his age of 14 years old. Even for a short time the boy became one of pupils of Bach and stayed Bach's resident. When Goldberg visited Bach's resident for the first time, he met Anna Magdalena. The boy, Goldberg played some of 'Clavierübung I' for her. Of course he played Fantasia and Allemande from Partita No.3 and Toccata and Allemande from No.6, whose originals were written in her cherished Music Book by her great husband and dedicated to her. Anna Magdalena knew that Fantasia from Partita No.3 was titled "Praeludium" on the key "A M" or "a moll" in her book and that its starting notes were two As. She was very glad to listen to his performance and the boy was welcome to Leipzig.

One of my sons was studying at the Leipzig University, and I heve been there several times. One day when I visited Bach's house in Leipzig, I asked him, half in jest, to compose for me a kind of music which can soothe my insomnia. He said that for healing the insomnia, variations might have a force against it. Johann Sebastian decided to finish these variations under development at that time Johann Gottlieb Goldberg has been an eligible young harpsichordist these days. Goldberg will perform that work well.
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Fugue No.15 G Major
Simple and Easy.
A Comment by Count Hermann Carl von Keyserlingk in 1744
about the Calov Bible and "eine Amusa"
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A Comment for The Well-Tempered Clavier
by Count Hermann Carl von Keyserlingk in 1744
--- This is a fiction. ---
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***** listen *****
Fugue No.15 G Major from so-called Book 2
This is Reichsgraf von Hermann Carl Keyserlingk, again.

Here I would like to make another comment for the Well Tempered Clavier composed by my great friend Johann Sebastian Bach. This time is for 'the Fifteenth Fugue'. It is of course on G major.
The Calov Bible and the WTC cover

The Calov Bible is a commentary Bible written by Dr. Abraham Calov, a distinguished 17th century Lutheran theologian. Abraham Calov was the editor and commentator of the three-volume Bible. Each verse of Scripture was printed in large bold type followed by Calov's commentary in smaller, less bold type.
I happened to know that my friend studied the Calov Bible in earnest and I found out that in the lower right coner of the title page of each of the three volumes was the monogram J.S.Bach 1733 written in his own hand. He had been wanting to get for long and he could finally purchase these three volumes in 1733 through a local-leading bookdealer. In one place of the books, Bach wrote in, "In devotional music, God is always present with His grace." In another place, near the 25th chapter of I Chronicles, he wrote, "This chapter is the true foundation of all God-pleasing church music." He told that until then he had to read them at the church library. Of course he could not write anything nor add his comments on any page. He found the Calov Bible at first at the Lutheran Church of Saint Agnus when he was in Cöthen. He loved it. Especially he liked the cover of it. So he used its design image for the front cover of the Well Tempered Clavier.

eine Amusante

Even though Prince Leopold was basically "a prince who knew music as well as he loved it", in 1721 my friend's life in Cöthen had become less congenial after the prince's marriage on December 11th with a pleasure-loving girl at her age of 19 years old, whom my friend called "eine Amusante" according to the letter of the year 1730 to Mr. Georg Erdmann from Johann Sebastian. I saw the letter when I visited Dantzih in 1731. Her name was Frederica Henrietta von Anhalt-Bernburg. According to my inquiry through diplomatic channels, the situations were as following;
The wedding was followed by 5 weeks of festivities in the town and Johann Senastian provided two cantatas for the occasion. Prince Leopold was smitten by his new bride and he wanted to be always with her. She was attractive and pretty. She liked music. She brought a few of music scores as a part of her bridal belongings. Prince Leopold was over head and ear in love with her. He had gone to another pleasant place where my friend could not visit. But happy days would not last so long. Princess became to be in bad health about one year after. Then Prince looked after her day after day. He did not have time to enjoy music. Tragically, she died two years later, in the spring, on April 4th of 1723. Prince was stricken with her death and his mind was too occupied with her death to think about music and Johann Sebastian Bach. My friend felt deeply sorry for his Lord. On April 19th, he signed a curious document that reads as if he were not yet free from Cöthen, but he could be free within a month. He had already got appointment in Leipzig, so after several weeks he left with his family and their belongings for Leipzig on May 22nd of 1723.


Princess Friederica Henrietta
von Anhalt-Bernburg
with a bright make-up around eyes

Two of Bach's daughters
were named after this Princess.
Christiane Sophie Henriette
was born in 1723
and died in 1726 at three.
Elisabeth Juliane Friederike
was born in 1726
and married with
Mr. Johann Christov Altnikol,
one of the most eligible pupils
of Johann Sebastian.

a part of Bach's letter to his friend Georg Erdmann of the year 1730

eine Amusante or eine Amusa

As I said, in 1731 I saw the letter which had the phrase, "Princess might be eine Amusante". But now, in 1744, people were talking as if to say that Johann Sebastian Bach had named Princess as eine Amusa which is strained to "a - Muse = someone not interested in the Muses or a lady against the muses", "an anti-musical persona" or "unmusikalisch", in other words, "a person who hates music". It is believed that he wrote in the letter, "Princess might be eine Amusa". It is ridiculous. My Friend never wanted to insult a high personage like Princess this way. Actually Johann Sebastian told me that he did not know the word "Amusa" at all. Unfortunately Mr. Georg Erdmann passed away in 1736, so I can not ask him what happened. And I could not find the letter. I do not know where it is.

There are some doubts here.

Who knows the meaning of Amusa ?
Who falsified the word Amusante to Amusa ?
And why ?

Amusante is a French word, an adjective for the feminine. It means "pleasant". So eine Amusante means "a pleasant persona" or "a woman who makes people pleasant". In this case A of the adjective Amusante should be a capital letter according to the German grammar. So in the letter A was written big in script like in "amusante".

Amüsant is a German word which has the almost the same meaning of Amusante. ü is necesary here for German. But my friend did not use it. So this Amusante is a French word, intentionaly selected by him.

For Amusa, A is not necessary to be a capital letter. eine amusa should be the proper expression. "eine amusa" is strange. Moreover only some dilettantes can analyse the meaning of Amusa. On the analogy of "Moralismus - Amoralismus, Symmetrie - Asymmetrie or normal - anormal", "amusa" was translated into "a - musa" which would mean "against the muses". But there has been no word like "musa" in German. Muse is nothing but "muse". It is not "musa". Someone had forcibly interpreted "musa is muse". This must be the only one origin of several kinds of interpretations.

Johann Sebastian wrote in that letter to his old friend Georg Erdmann;
"Should Your Honor know or find a suitable post in your city for an old and faithful servant, I beg you most humbly to put in a most gracious word of recommendation for me--I shall not fail to do my best to give satisfaction and justify your most gracious intercession in my behalf."
Johann Sebastian seemed to be looking for a new suitable job. I do not know if he really meant it. Anyway Mr. Georg Erdmann took his friend's word seriously. Danzig of now is prosperous and people there need the concord music. And there is a beautiful church, St Mary's Church of Danzig.

I guessed;

He tried to talk about the job opening of Cantor or Music Director of the city with some of the city authority showing Bach's letter to them. One of the authority members said, "The statement in the letter 'The musical interests of Prince Leopold had become somewhat lukewarm, especially as the new Princess seemed to be pleasant, ..... so J. S. Bach left Cöthen for Leipzig.' does not easily make sence. I can imagine that Prince fell deeply in love with Princess. But this will make J. S. Bach lose face. It is as if he were jealous of Prince's marriage."
He added another point, "The expression 'eine Amusante' is a mishmash of German and French. In this letter there is only one French word, Amusante, here. We do not jumble them up together. If we write something in German, we write it in German. If we write something in French, we write it all in French. J. S. Bach is said to be the embodiment of the German spirit. So he should not have done this kind of a mishmash in his letter. Can you make a small revision in this letter?"
Mr. Georg Erdmann went back to the Embassy and asked one of his colleagues who was a technician of espionage to alter or delete the word 'Amusante'. The technician told that it was easy to delete the word but the part would be unnatural. He recommended to alter the word to 'Amusa'. Mr. Erdmann didn't know the meaning of this new word 'Amusa'. The technician told him not to worry, pointing that Amusa' looked like 'a-Muse' which means anti-music. "The statement of 'The musical interests of Prince Leopold had become somewhat lukewarm, especially as the new Princess seemed to dislike music, ..... so J. S. Bach left Cöthen for Leipzig.' now makes sense." So he deleted 'nte' skillfully. There remained no trace of falsification. From then on, Princess Frederica Henrietta was an infamous "eine Amusa" named by J. S. Bach.

Of course I think Mr. Erdmann didn't mean any harm to Princess Frederica Henrietta von Anhalt-Bernburg. But her happy years were too short. I can not help pitying the poor Princess. I think I should restore not only her honor but also my friend's innocence to this matter.

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A Method for The Well-Tempered Clavier
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Prelude No.16 G Minor
This is for young girls to play with chatting. You don't know until when it lasts.
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Fugue No.16 G Minor
Do you know a school fugue? If there is that kind of fugue, this is the one. Prelude No.16 is for girls. Then they should study more on this fugue. They will not have a spare time for chatting around.
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Prelude No.17 Ab Major
What is the difference between Melody and Accompaniment? "Melody is in the upper part and played by your right hand. Accompaniment is in the lower part and played by your left hand." is not true. It depends on the player or the listener. A sequence of notes can be Melody or Accompaniment. In this prelude you can learn the truth.
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Fugue No.17 Ab Major
This fugue is for your memory skill-up. I suppose this is the most difficult tune for you to memorize all. Because there is only an impression except the subject. Prelude No.17 is on the contrary easy one to remember. Then this is a balance.
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Prelude No.18 G# Minor
This is the first and only one prelude with 6/8 beat. And this is the only one Minor tune with 6/8 beat in this Wel-Tempered Clavier.
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Fugue No.18 G# Minor
This fugue is with four voices. But it is important to raise some monophonic feeling naturally.
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Prelude No.19 A Major

How oft when thou, my music, music play'st,
Upon that blessed wood whose motion sounds
With thy sweet fingers when thou gently sway'st
The wiry concord that mine ear confounds,
Do I envy those jacks that nimble leap,
To kiss the tender inward of thy hand,
Whilst my poor lips which should that harvest reap,
At the wood's boldness by thee blushing stand.
To be so tickled they would change their state
And situation with those dancing chips,
O'er whom thy fingers walk with gentle gait,
Making dead wood more blest than living lips,
Since saucy jacks so happy are in this,
Give them thy fingers, me thy lips to kiss.

The Sonnets No. 128 William Shakespeare 1609

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Fugue No.19 A Major
The subject of this fugue has unusually rests in itself. You have to recognize these rests as the important parts of the subject.

This is a fugue with three vocies but the subjets apeares early downward four times on A, E, a and e.

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Prelude No.20 A Minor
There is no sharp nor flat on the staff. It is clear, isn't it? By the way I used sharps and flats about fifty times in the score. I really love semitones.
As to the rhythm, you can find a strong characteristic movement even in the first bar whereas the score itself is somehow still.
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Fugue No.20 A Minor
This fugue has one of the longest subjects, with 31 notes and about 3 bar length. It is longer than that of Fugue No.9 E Major. The first part of the subject will be terminated by nine degree fall (F, g-sharp, e). The second and last part is for the remedy of that dramatic fall.
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Prelude No.21 Bb Major
B flat is B. My initial of family name is, as you know, "B". BACH is the combination of "B", "A", "C" and "H", in other words, Bflat, A, C and B. So, Bflat is special for me.
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Fugue No.21 Bb Major
This fugue has the longest subject with 38 notes and about 4 bar length among 24 fugues in this Well-Tempered Clavier. "B" of Bach is actually Bflat and on one of the blackkeys, which should not have been black but golden.

I am too modest to start this fugure with the note "B" (Bflat). So the starting note of this theme is F, dominant of Bflat. But for Fugue No.22, Bflat note will appear from the first.

Ah, let me see. I applied Bflat as the first note for Fugue No.7 Eflat Major. Sorry, I am not so modest as I am supposed to be.

FugueNo.010203040506070809101112131415161718192021222324
keyCMCmC#MC#mDMDmEbMD#mEMEmFMFmF#MF#mGMGmAbMG#mAMAmBbMBbmBMBm
starting
note		CCG#C#DDBbD#EECCC#F#GDAbG#AAFBbBF#
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Prelude No.22 Bb Minor
Then is there mirth in heaven, When earthly things made even Atone together.

AS YOU LIKE IT
William Shakespeare 1600

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Fugue No.22 Bb Minor

"Chris-te e-le-i-son, e-le-i-son"
There are only two five-voice fugues in the Well-Tempered Clavier. This is the second and last one. The subject of this fugue starts from the above sky, then goes down and down. And they will fill the space. It should be difficult for you to play this on an ordinary clavier. You may want to play it on an organ with foot keyboard. But try the clavier, you will find a new possibilty of this instrument.
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Prelude No.23 B Major
There are only 19 bars. This is the second shortest Prelude in the Well-Tempered Clavier. The shortest one is the F Major Prelude ( No. 11 ) which has only 18 bars. But No. 11 has many trills and you will feel longer than this Prelude.

This prelude is the 23rd one in the Well Tempered Clavier. By the way, the pair "Prelude and Fugue" is not unusual. Prelude is supposed to be an introduction to prepare for a succeeding part, in this case Fugue. But as you know, Prelude is usually self-contained, moreover sometimes independent with a rather informal character. It would have been possible to assemble only 24 preludes for the Well Tempered Clavier without fugues. Some students only play preludes for practice because they think that fugues are too complicated and tiresome for them to enjoy.

If the difference between Prelude and Fugue lay in realm of nuance or mere style, I would not have tried to compose 48 pieces for this Well Tempered Clavier. Say, 24 fugues would be sufficient. Here, for Prelude, "harmony" is the key and harmony concerns the building of chord tones played together derived from the scale on which the music is based. It also involves the order in which successions of chords accompany a melody. The shifting harmony will add color, tension, and release to the piece as you experienced in the first prelude C major.
"From harmony to melody" or "melody made on harmony".

Even "harmony as melody".

These days (around the year 1722 or 1744), main stream music is like an easy listening melody accompanied by backing chords. In other words, melody is dominated by its related harmony. Some says that a melody can not exist without apparent harmony or even hidden. It is OK. But it is not enough to fill the universe with music.
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Fugue No.23 B Major
This fugue is on B Major with five sharps. You have to use black keys many times, and you will find that you feel no uneasiness in the well-temperament world.

This fugue is the 23rd one in the Well Tempered Clavier. By the way, the pair "Prelude and Fugue" is not unusual. Fugue is supposed to be a composition in which two or more voices imitate and answer each other contrapuntally. One main theme is introduced and other voices "chase" the first according to a strict and detailed plan. It would have been possible to assemble only 24 fugues for the Well Tempered Clavier without preludes. Some teachers assemble only fugues into one notebook for thier students in order to train their skill with more difficult pieces.

If the difference between Prelude and Fugue lay in realm of difficulty or mere complexity, I would not have tried to compose 48 pieces fot this Well Tempered Clavier. Several preludes are rather difficult to play, so 24 preludes would be sufficient. Here for Fugue, melody or theme is the key. Melody, a set of sequential notes, is monophony and probably derives from the inflections of the human voice. It involves pitch, or the relative highness or lowness of tone.
"From melody to harmony" or "harmony produced by accumulation of melodies". Basically Prelude can be said harmony-oriented,
and Fugue can be said melody-oriented.
Both approaches are necessary.

Even "melody un-subdued by harmony".

Now we have twelve tones, major/minor keys and harmony-oriented / melody-oriented structures. So, there should be 48 pieces. That's all we need for the musical universe.

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A Method for The Well-Tempered Clavier
by J. S. Bach 2001
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Prelude No.24 B minor


In the last day, that great day of the feast, Jesus stood and cried,
saying, If any man thirst, let him come unto me, and drink.
He that believeth on me, as the scripture hath said,
out of his belly shall flow rivers of living water.
John 7:37-38 		I have a lot of things I want to tell you about this prelude. It is the last prelude for this Well-Tempered Clavier. But it opens the door to the infinity of music. The sign of infinity looks like the character eight, 8. And also "eight" indicates an octave. So this prelude opens the door to the next octave reincarnated.

You can hear and feel the tense and comfort of every harmony which every set of two notes makes here. This is the Well-Tempered Music World which only the clavier can create, and which strings can not.

You should play this prelude No.24 B minor in such a way that the left hand plays the prescribed notes while the right hand adds consonances and dissonance so that a well-sounding harmony results for the glory of God and the permissible delight of the soul.
This Prelude No. 24 has its own indications for the tempo
"Andante"
by me, J. S. Bach 1722.

I would like to listen to the Well-Tempered Strings.
But violinists and other stirngs players can not get rid of the Pure-Temperament Spell-bindings.

While I wrote, "I have a lot of things I want to tell you about this prelude." at the beginning of this page, I should not add any farther comment by myself because of some reasons.
Then my eligible son Carl Philipp Emanuel has willingly complied with my request for an in-depth analysis of this prelude No.24. He is one of the prominent music theoreticians of present days, well in 1744.

the Well-Tempered Strings - Carl Philipp Emanuel
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A Method for The Well-Tempered Clavier
by Carl Philipp Emanuel Bach
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The Secret of Prelude No.24 B minor

An Addictive Analysis by Carl Philipp Emanuel Bach
in 1744

My great father Johann Sebastian Bach told me to make some comments about this prelude, the Well Tempered Clavier Prelude No. 24 B minor. I am not so addictive as people may suppose me to be. But I found out its secret that Johann Sebastian had buried under an ordinary face of this beautiful arioso or cantabile. To tell the truth, the walking bass line of the first half of this prelude is made of the Dorian Mode using mainly the notes in the A Major Scale in some minor mood.

Dorian Mode
The bass sequence of the first measure is "B C# D E F# G# A B", which is "Tonic, Whole-Step, Half-Step, Whole-Step, Whole-Step, Whole-Step, Half-Step, Whole-Step". This is called the Dorian Mode. In other words, this pattern of intervals: T-S-T-T-T-S-T is the characteristic of the Dorian Mode. It is symmetric.
T
T-S-T-T-T-S-T
T
T
T
For the reference, T is Tone and S is Semitone. "Dorian" is one of the original "Church Modes" and it has a beautiful minor sound. The Dorian mode is a kind of minor scale because of the flattened third note. But it is not the B Minor scale where G sharp should be in the place of G.
-------
In the C Major Scale, build a scale using "D" as its tonic, then there emerges the "D" Dorian Mode which is a very popular formula "D E F G A B C D". The "D" Dorian Mode is not the same as the D Major scale nor as the D Minor one.
This mode can express "solemnity", "gracefulness" and "modesty" under calm and tranquil circumstances. In other words it is called the mode for "theoria" or "contemplation".
Aristotle wrote in his "POLITICS";
A certain mode produces a moderate and settled temper, which appears to be the peculiar effect of the Dorian. All men agree that the Dorian music is the gravest and manliest. And whereas we say that the extremes should be avoided and the mean followed, and whereas the Dorian is a mean between the other modes, it is evident that our youth should be taught the Dorian music.

My father Johann Sebastian Bach intended to defer his proper respects to the church modes on which Gregorian Chants had been compiled, by adopting the Dorian in his final prelude of the Well Tempered Clavier. Somehow he was afraid of the reaction of churchs because "the Well Tempered Clavier", especially this No.24 Prelude and Fugue, could have been supposed to be a rebellion towards Gregorian Chants or churchs. Even though there was a gradual eclipse of the old church modes upon which Western music had been based, he would have received an informal warning that the Equal Temperament theory was contrary to the teachings of the church saying, "You should not be another Galileo Galilei who lived near Florence under house arrest during his last eight years, from 1634 to 1642." The Catholics only wanted to get the scientists like Galileo out from their sight, so his less illustrious composer father, Vincenzo Galilei, was never in any danger. But the Calvinists were antagonistic to the splendors of the Lutheran liturgy even in music.

When my father was in Kothen holding the position of Capellmeister, his master was the young prince Leopold of Anhalt-Kothen and the son of Calvinism of which founder, Calvin, had said that God's electing purpose be not conditioned by anything in man. The young prince respected Johann Sebastian as a great musician and made a good effort to let him work comfortably and fruitfully in Kothen. But so there was no church music there. My father's theological thinking was based on the Lutheranism. Luther thought music as a gift from God, rather than a human invention, while he put a heavy censorship on all artistic expression in the church. Only music was excepted. Calvin thought that a constant vigilance must be maintained for music like as a prisoner. He had no feeling at all for the developing art of sacred music. Anyway and fortunately the young Prince loved and enjoyed cheerful and instrumental music featuring the latest styles and fashions. My father happily wrote much of his chamber music, violin concertos, sonatas, keyboard music there in Kothen. But Johann Sebastian had an experience that he was arrested and imprisoned for a month in 1716 while he lived in Weimar because of his disobedience against his master at that time. He could not forget it. Always he had to be careful not only about the difference of the theological doctrine between Calvinistm and Lutheranism but also about the changing temperament of his master. On 7th of July, 1720, my mother, Maria Barbara, 36 years old, passed away suddenly while my father was out of town with his master. I remember that the funeral was held in a Calvinist church and my mother was buried in my father's absence. But at that time I was only 6 years old, too young to know the details. My father was in a great shock for a while but he had to take care of children and to continue his job.

Thereafter one of his assigned works was to compose and perform cantatas for the Prince's birthday and the New Year. Anna Magdalena, daughter of Mr. Wilcke, Court-Trumpeter at Weisenfels, came to sing these cantatas as one of singers and attracted my father's attention with her fine soprano voice and her sunny disposition. In December 1721, Anna Magdalena at the age of 20 and my fahter 36 married to make a happy home again. His problem was that thier wedding ceremony would have had to be held in the Calvinist church which the Prince reserved. Johann Sebastian was annoyed but he begged a pardon to the Prince that he could use his house for thier wedding ceremony. Of course he could not insist that the Lutheran church was the right place. Anyway he should be happy with his new bride, so there was a lot of consumption of wine and the wedding was a feast.

John 2:( fictionalized ) ...And when they wanted wine, Maria saith unto him, They have no wine. ... Jesus saith unto them, Fill the waterpots with water. And they filled them up to the brim. And he saith unto them, Draw out now, and bear unto the governor of the feast. And they bare it. When the bridegroom had tasted the water that was made wine, and knew whence it was.

When my father accomplished the Well Temepred Clavier in 1722, he dared not attempt to put an inordinate expression for this work, say "Even-Tempered" or " Equal-Tempered". Let me review the history.

In 1530, Nicolaus Copernicus finished his great book, 'Concerning the Revolutions of the Celestial Spheres' but hesitated to publish it, because he was afraid of being contempted from the church which might have accused him of his novely and absurd opinion. In fact his theory was in opposition to the teachings of the Roman Catholic church, and the book had not been published for 13 years till the year 1543. Andreas Osiander, one of his supporters and a Luteran theologist, added some comments in its preface saying, "... If there is a hypothesis, it needs not to be a truth. It is sufficient that the result of calculation corresponds the data of observation. This theory is only for calculation." He intended to protect Copernicus, but both Luther and Calvin, who were leaders of Protestantism, actually scorned Copernicus as an idiot who was trying to confuse the Astronomy. After all in 1616 the Roman Catholic church forced this book under the Index of Prohibited Books.
In 1609 Johannes Kepler published a book on the results of his work, boldly titling it 'The New Astronomy' about his discovery that the orbits in which the Earth and the other planets of the solar system travel around the sun are elliptical, or oval, in shape. But he died in 1630 before the arrest of Galileo. By the way he wrote another book which contains 'Kepler's Third Law' in 1619. Its title was "the Harmony of the World (Harmonices mundi)". There he discussed about the musical scales and harmonics related to the planetary motion. Especially the correspondence between musical ratios of whole numbers and planetary velocities was, he thought, marvellous.
In 1687 Issac Newton published his major work, 'Principia' (Philosophiae Naturalis Principia Mathematica, or Mathematical Principles of Natural Philosophy), setting forth the theory of gravitation. Newton had not been bothered by religious pressures. Fortunately he lived in the country whose king tried to insist that a Benedictine monk be given a degree without taking any examinations or swearing the required oaths. As to music, in 1704 he wrote another book called "Opticks" which dealt with the theory of light and colour and the relation with musical scale or harmony.
Newton classified the spectrum of the sun-ray into seven colors and placed dividing lines. Then he thought that he found out its relation with music. It admirably represented the Dorian mode. It is astonishing that even Issac Newton believed in ratios of whole numbers as the base of musical harmony.

Johann Sebastian Bach only wrote down decently "Well Tempered" and "For the profit and use of musical youth desirous of learning and especially for the pastime of those already skilled in this study" on the title page behind which, in the last pages, he made an inconspicuous challenge to create a no-one-never-heard-of chromatic mode in the last fugue next to this prelude.
If not humble nor careful, the Calvinists might have accused my father of possibly leading young people to the service of vanity and sensuality with his "heretical" music conception. Even Anna Magdarena had become aware of the existence of all three types of diminished 7th in the Prelude No.1 C Major, which might have been thought that he had intended to dominate the music universe by excluding the original "Church Modes". Someone might have found out that this theme of Fugue No.24 was so peculiar that it could be accused falsely as anti-Calvinism. So he had to be very discreet.

In following year, in 1723, Johann Sebastian had took an examination to be appointed as Thomaskantor. He had to give formal and written assent to the Book of Concord. In the examination he had to answer the questions related to Biblical theology in general and to Lutheran theology as delineated in the Book of Concord denying non-Luteran beliefs. While denying non-Luteran beliefs, he should avoid another arrest and imprisonment caused by his master's displeasure or slanders of Calvinists. Because he already decided to leave Kothen as soon as possible with us uneventfully.

An Analysis of the Prelude No.24
The score below is the one transposed three semitones and one octave up,
namely from B to D(and an octave up).
And there is no sharp nor flat placed on the staff
at the beginning of a piece as the key signature.

So he used mainly the Dorian mode as the principal character
and placed supporting harmonies for it.

------The first measure consists of Dorian itself.
------In the original score it can be called B-Dorian.

------It looks complicated but it is the descending mode of Dorian. C sometimes accompanies sharp and B does flat.

------A Neapolitan quasi-cadence with C sharp (or D flat) is in the third measure to create an ambiguous stop before restarting Dorian in the fourth measure.

------The sixth, seventh and eighth measures consist of Mixolydian Mode. In the key of C, a mixolydian scale is built on G, and consists of "G, A, B, C, D, E, F, G" which means T-T-S-T-T-S-T. This scale is like the major scale except that the seventh step is lowered a half step.

------An Neapolitan quasi-cadence with G sharp (or A flat)

------The 13th and 14th measures are of Dorian mode. Here can be seen T-S-T-T-T-S-T as the characteristic of Dorian Mode. In the original score they can be called E-Dorian.

------An Neapolitan quasi-cadence with C sharp (or D flat) is identical with the 11th measure.

------This is an abbreviated figure of the last two measures. It shows the descending mode of Dorian.

This prelude was built to present my father's homage to the church. So its base is "Church Mode". At first my father Johann Sebastian, I suppose, treated this Dorian sequence as upper melody and added lower suporting chord backing to it in order to unite modes and chords. Then he made the work upside down and got a surprisingly beautiful result.

For the second part of this prelude he tried to create a new fusion of modes and chords with a chromatic flavor where modes and chords can not be separated. It became the introduction to the final fugue.

Thank you for giving me your company to this last.
Carl Philipp Emanuel Bach
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by J. S. Bach 2001
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Fugue No.24 B Minor
This Fugue No.24 has its own indicate for the tempo, Largo
by me, J. S. Bach 1722.
In this first theme there are G-Fis, H-Ais, E-Dis, C-H, Fis-F, D-Cis-C, C-Cis and A-Gis ( in trill ) on the h-moll basic chord ( H-D-Fis; B minor ). Some analysts may say that, for example, G before Fis is an appoggiatura ( a transitional note ) for the next note. But all of these semitones are equally important. I do not intend that the first note of semitone combinations should be subordinate to the second. I put slurs on these combinations in order that you could feel the distances between two notes. You will detect a short beat which indicates the distance. These distances, I think, should be similar in character among G-Fis, H-Ais, E-Dis, C-H, Fis-F, D-Cis-C, C-Cis and A-Gis.
Finally there are no differences among white and black keys, which would be fused into the infinity.

And, , , there are all the twelve notes of the scale in this theme. This is the music world. Where there is devotional music, God with his grace is always present.

S.D.G.
Soli deo sit gloria
"To God alone be glory!"
THE END

OR
"Music in the Air"

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A Critical Essay for The Well-Tempered Clavier
by Johann Adolph Scheibe in 1744
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About Fugue No.24 B Minor
I labeled the great Cantor J. S. Bach A Pedant who spoiled the natural beauty of his creation with "an excess of art". My name, which is notorious among enthusiasts of Bach's music, is Johann Adolph Scheibe.
For me, good music must possess a cantabile melody unobscured by excessive ornaments, angular intervals and above all, the other voices in the composition.
I was too puzzled to undestand the fugue, "Fugue No.24 B Minor" of the Well-Tempered Clavier, which has too many off-scale notes. Are they all appoggiaturas or transitional notes? I don't understand. Even somehow complicated Fugue No.1 C Major has resonable and small number of off-scale notes taking into consideration of the condition that it is on the major scale.
For example, one minor fugue for chembalo, which was compsed by the great Cantor Georg Philipp Telemann, has 539 notes in which there are only 40 off-scale notes. In other words only 7.4% of notes are off-scale. In case of Bach's this fugue has 21.9% of off-scale notes.
"Fugue No.24 B Minor" of the Well-Tempered Clavier so-called Book 2 has 14.4% of off-scale notes.
Johann Christoph Altnikol, the Great Bach's ex-pupil and now 24 yeras old assistant, has been helping Mrs Anna Magdalena Bach to compiling the new set of twenty four Preludes and Fugues and he put a title-name to this secondary set "The Well-Tempered Clavier Book 2" by the year 1744, I mean, this year. I don't know whether Johann Sebastian would authorize this new Well-Tempered Clavier. So, I call it "so-called Book 2". Although Book 1 was not printed nor published, we know that the title page of Book 1 was beautifully drawn by our Cantor by himself. But obviously he didn't make the title page of Book 2.
Mr. Altnikol showed it to me and I found that all of Preludes and Fugues of the Book 2 are of course utterly different from those of the Book 1. Among them, Prelude No.14 F sharp Minor is particularly impressive to me because it has a beautiful cantabile melody. Also you will see that the first Prelude and the last Fugue are moderate and, I should say, not too intricate. Additionally for Prelude No.1 C Major of Book 1, I questioned about the fact that it consists only of chords without any melody and that its chord progression is beyond our comprehension. So I felt relieved after listening to new versions of them. Young Altnikol might have the same feeling. He wanted to protect his great teacher from slanders saying, "Cantor are teaching young people a heretic musicology like music without melody or music without harmony." But anyway the serious problem remains. The problem is that the first Prelude and the last Fugue of Book 1 are still incredibly beautiful and profound.
The Title Page of Book 2

The Well-Tempered Clavier,
Second Part,
consisting of
Preludes and Fugues
through
all the Tones and Semitones,
written by
Johann Sebastian Bach,
Royal Polish and Electoral Saxon Court Composer,
Capellmeister and Directore Chori Musici
 in Leipzig.
dated in 1744

I wrote in the year 1737 in my paper "Der Critischer Musikus", "This gerat man would be the object of admiration for entire nations if only he had more charm, and if he did not deprive his compositions of naturalness through bombastic and confusing character, and obscure their beauty through an excess of art." At that time I was only 29 years old. I was young and audacious. I flattered myself that I was the first professional critic of music in the nations in history. Of course I acknowledged Bach's extraordinary skill as a performer on the organ and the harpsichord and as a productive composer. Moreover I know that, when my father Johann Scheibe made an organ in 1716 at the university church which is called the Paulinerkirche, Johann Sebastian Bach was at Leipzig examining that organ in December 1717. Even at that time Johann Sebastian Bach was famous and admired.

I was born in 1708 and raised in Leipzig, and studied law and philosophy, as well as the piano, the organ and composing. From 1736 until 1739, I lived in Hamburg, where I became to be known as an author and composer. Stylistically I was in the ideals of the Baroque, but I strived for a purer and simpler expression. The new music for me and for our time needed a more alluring and sensitive melody.
Now, music is designed on a grand scale. Like the architecture and painting of our time, music has its structure. We experienced a gradual eclipse of the old church modes which were unstable and had obscure forms over the century. Among them, the modes known as the Greek Ionian and Aeolian became the major and minor scales. Our age brought an increased interest in instrumental music. Keyboard instruments including the clavichord, harpsichord, and organ are now in general use. Then the chord system has been established.

The French composer Jean-Philippe Rameau was known for his contribution to music theory. His system of harmony published in 1722 is based on the natural overtone series. Three or more different notes sounded at the same time produce chords which construct harmony. The root note originates a chord. There are the three most important chords which are the tonic, the dominant seventh and the subdominant.

A composer can create variety and tension in his music by using off-diatonic-scale-notes. Having a central point of reference the tonic makes it possible for the composer to create variety and tension in his music by moving to notes not closely related to the tonic. The notes of the diatonic scale most closely related to the tonic and therefore "rest" tones are the 3rd and 5th; the "active" tones, which create movement, are the 2nd, 4th, 6th, and 7th.
For further variety, composers occasionally use notes and chords from keys or scales other than the predominant one; this is called chromaticism. For a time they might also completely change the tonality, or key, of the music; this is called modulation. Then they return to the original key.

For all the things above, the geat Cantor J. S. Bach is like an all-seeing god.
Johann Sebastian
parodying my family name
might have shouted
"Scheiße!"
which means
"Damn it!".
I am now 37 years old enough to deliberate what to say or what to do. Next year (in the year 1745) I will make a review of the Italian Concerto in which I will apologize courteously saying, "I did this great man an injustice".
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