The new music theory, as music and mathematics are common denominator, of the novel Glass Bead Game (2)

Below fig.1~3 are referred from an article
The new music theory, as music and mathematics are common denominator, of the novel Glass Bead Game. – Kyo’s Blog (kyonissho.com)

The expression of (fig.x) is each previous number in the former article.

fig.1. (fig.2) sounds have forms in time, [0, 0]time_accord_interval.

fig.2:
A sound experience distributes, it is to past and to future, from a [0, 0]time_accord_interval.
As for imagination, for example, I draw the to-past and the to-future meets in the dark blue in fig.3, as the time-distributed experience meets.

fig.2. (fig.3.3) dk is the kinetic time difference, so-called time in where we experienced. dp is the potential time. difference in where we imagine/create any arising to live.

fig.3. (fig.3.4) common multiple and fundamental difference of sound spins.

fig.3. (fig.3.4): [0, 0]time_accord_interval is applied to {8, 13}wavelength_set, | | of fundamental difference is found between {8, 13}wavelength_set and {13, 17}wavelength_set.

Fig.4 is the score of Bach's Fuga canonica, E♭ major (C minor) for the left hand, and, F major and C major for the right hand make expanded potential sound made; space. Seemingly a simple score, but there is vast potential space behind the score. See from fig.3 to fig.2.

As for fig.1, as an image, Piano sound is close to sign wave, Violin sound is close to saw wave if it is seen with Fourier series, Flute sound is close to both type of the waves.
Although with a synthesizer, naturally any synthesized sound can be made as kinetic space, this time, we think for potential space.

From the beginning of human kind, man/woman saw the thousands of years in the night sky at constellations of twinkling stars, saw several tens of years lifetime at people in front of them, saw the season of the harvest since about B.C. 4000 (a calculation is B.C. 4004 October), and saw 29.5 days of synodic month, that was/is at same time.

The time set is {thousands years, several tens years, a season, 29.5 days}, at same time.

fig.5.0. the horizons of La Gioconda (Mona Lisa).

fig.5.1. only the horizons of fig.5.0. there are two different ruined space behind Mona Lisa.

fig.5.0 and fig.5.1:

At same time, European painted a path of time about the time set {thousands years, several tens years, a season, 29.5 days}. In a paint, it can be so, and music has the common denominator with mathematics.
My teacher Ayao Ide told me the model of Father Jacobus in the novel Glass Bead Game is Burckhardt who named the time of a time force as Renaissance. Burckhardt has been printed in the largest note of Switzerland.
European people trained each time force in the sense of art, leading brain reactions, at least from 14th century, I think it was more before, because Jesus Christ was sensed from the 6th day of Genesis.

...

When a basic sound is f, the equal temperament is f×2^(n/12), it is a binary measure. I try to make a quinary number instrument with oscillator.

fig.6.0. generated by AFSG for iPhone, an oscillator software.

C4 is 261.6 Hz, C#4 is 261.6×(2^(1/12))≒277.15. If C#4 in the quinary number, it is 261.6×(5^(1/12))≒299.15.
In Galois field, the binary number → F2, F2 includes already the Galois' imaginary number to realize any potential sound. When C3 in the binary number of music is added to 299.15, 315.32, it is F(5+√F2) as including Galois' imaginary number to realize any potential sound.

Ref.
fig.8.2~8.8b; F2
I’ve planned 300 hours for the details of Wiles’ Proof of Fermat’s Last Theorem. – Kyo’s Blog (kyonissho.com)

Any stringed instrument can make any sound out of the equal temperament, but no standard, especially for its musical keyboard, of a ternary (base-3), quinary (base-5), septenary (base-7), undecimal (base-11) ... base-x numeral system; f × x^d ← the complement dimension sound of Fx to generate any space.

If only base-2 octave system is used, it is less for all common denominator of music and mathematics.

fig.6.2. a linkage of base-x numeral system; f × x^d ← the complement dimension sound of Fx to generate any space.

fig.6.3 (fig.8.4):
I’ve planned 300 hours for the details of Wiles’ Proof of Fermat’s Last Theorem. – Kyo’s Blog (kyonissho.com)

fig.6.4:
(1) n∋0: f × 7^(0/12) = f×1 = f, this isn't an exponent acceleration, e.g. C1 = C2 = ... = Cn in this case.
But p = 0 of Fp, C1 = C2 = ... = Cn ← the complement, i.e. {0, 10, 100, .... , ∞} | 1 ∋ complement.
(2) Instead of n = 0, e.g. C4 ≒ 261.1Hz, B3 ≒ 246.9, B3 × 2^(1/12) ≒ 261.1Hz ≒ C4, here is an exponent acceleration of equal temperament.
From 2^x to p^x is in Fp, F2 includes Galois imaginary number, Fp+s (s is a number) is requirement for [0, 0]interval of music (see fig.1, 2, and 3).

I consider base-x as below, 〇 is an accord of base-x, 〇 of factor causes Galois' imaginary number.
The base-1 has no complement, i.e. no upper digit, stable.

fig.6.5. sin curve for base-1, base-3, base-5, base-7, base-11. the Galois' imaginary number is in +π.

fig.6.6. the π has been revised. the +π of phase had been lost in sight.

fig.6.7. base-1 (water-blue), and base-2 (purple). base-2 numeral number has already Galois' imaginary number.

fig.6.8. base-2, base-3, base-5, base-7, base-11. base-2's imaginary number appears really on 0 of y. then base-x\base-2 should have +s for [0, 0]interval of music, and seek a sight.

There is no base-x instrument yet, therefore I calculated some base-5 frequencies and have marked on a 88-keys keyboard.

The base-5 C5 shift on the circle of fifth appears near a base-2 E5. At the playing, I regarded an octave between the base-5 C4 and the base-5 C5 (then the 27 keys make the octave), between the base-2 C4 and the base-2 C5 at the same time.

Yet now, my instrument doesn't have the frequencies of base-5, an analog synthesizer has a frequency width as including the Galois' imaginary number.
But some synthesizers generate a frequency around center of two or more keys, e.g. [C1, E1]'s center is near D1 (fig.6.11, 6.12), but it is not base-x itself.

fig.6.11. based a synthesized sound, you may make a new sound. this case, generated by animoog for iPhone.

in case of amsynth of Linux, PORTAMENT is applied.

Then I try to play Fuga canonica in Epidiapreate of Bach with my right hand for the near base-5 and my left hand for the base-2. See fig.4.
If it is a keyborad of base-2 for the left hand, and another keyboard of base-[ , ] of fig.11 type for the right hand, fig.6.9's approach is present one, but very difficult to realize.

fig.7.0. a paint in swaying lights of candle as singing.

fig.7.0:
Even just before the 20th century, the literacy rate was less than 50% even in developed countries at the time.
Furthermore, the literacy rate of women before the second industrial revolution with the machines was lower than men.
I think it is worth to understand what this picture influenced the times.