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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
***** listen again *****
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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