**30.1 Diagram. Equally Tempered Tones** Helmholtz

**30.2 The 31 Note Octave** The Mathematics of Music By John W. Link, 781.1 L6485m

"Our theory will show the need for a type of instrument that is not now available. This is an electronic organ with multiple tones, viz., with 31 notes to the octave. This organ would have a special panel for selecting one out of three tones for each white key and for selecting either of two tones for each black key. In this way, the organ could be played in both Pythagorean and Just Intonations, without returning, or the 31 tones could be especially tuned to any one of the harmonic temperaments. 31 notes per octave is called "meantonal", because the arrangement is borrowed from the meantone temperament".

30.3

30.4 The 53 Part Octave. The Mercator Scale

"Units of divergence from equal temperament is 1/12 diatonic comma, 1.001129891 or 1.955000865 cents. C-D has a ratio of 1.125 or 203.910001731 cents. C-B sharp has a ratio of 1.013643265 or 23.460010384 cents. C-B sharp is, itself, the diesis. The break comma has a ratio of 1.011528852 or 19.844964522 cents. Formulae: 1.01364326541 x 1.01152885212 = 2 (23. 460010384 x 41) + (19.844964522 x 12) = 1200 cents.

30.5

Meantonal: 1.01364326519 x 1.03931824810 x 1.02532294082 = 2 (23. 460010384 x 19) + (66. 764985288 x 10) + (43.304974906 x 2) = 1200 cents.

Meantonal signifies 31 notes to the octave, as in the meantone temperament. In the Pythagorean intonation, there are two kinds of meantonal breaks, the lesser of which occurs between two complex notes, e.g., Ax (A augmented) and D double flat. This irregularity will not occur in any of the harmonic temperaments".

30.6

**30.6.1 Table. Letter Parts of the Pythagorean Intonation**

Found on pg.10 of the book.

(Note: A Hint of Coil Construction: It is necessary to understand that bucking, bifilar, counterwound coils is what is important here. It cancels the mag/electric component, leaving the pure energy component in the axis).

"Arranging letter parts may seem like child's play; yet it is a necessary step in formulating a system. Notice that D is the center note because it is central in the order of fifths, i.e., F, C, G, D, A, E, B. Since the Pythagorean intonation has 53 parts, there must be 26 ascending and 26 descending fifths. Join these end to end; and do not be concerned with the connection in the middle of the coil, which will take care of itself. After the coil has been formed, count forwards or backwards by 12's. The direction is immaterial. So long as one keeps counting, the sets will literally form themselves. For each Pythagorean set, the order of letters is reversed... Let us start, now, with C and count backwards to form the C set. The notes will run: C, D double flat, E flat''', Ax. Obviously, Ax does not belong in the C set, and, just as obviously, the notes have appeared in reverse sequence, for D double flat and E flat''' are lower than C, according to the Pythagorean scheme of things. Let us see what we have: Ax) (E flat''', D double flat, C. We should start again and count forwards to complete this set: (E flat''', D double flat, C, B sharp, A sharp'').

It is thought by some that the pentatonic",

"are a rudimentary species of the Pythagorean; hence, it is interesting that the sets, C, D, E, G, A are sets with five parts, while all the others are sets of four.

Though intonations are formed from fractions of integers, while temperaments are basically irrational, the Pythagorean intonation has the features of a temperament. Everything learned about the Pythagorean can later be applied to formulating the harmonic temperaments.

Mathematically, the Pythagorean intonation is constructed from a single fraction and its inversion, viz., 3/2 and 2/3, the various powers applied to each part of the fraction".

(Note: the following Table is the powers of 2 and 3),

..."because...the calculator is not accurate for the highest powers of either number. To cite one such case, the calculator displays 847,288,609,476 as the 25th power of 3; and, for the antilog of 25 times the log of 3, the reading is 847,288,609,342. But neither is correct".

**30.7 Table. Powers of 2 and 3**

Found on pg. 12 of the book.

**30.8 Table. Pythagorean Intonation, Fractions**
Found on pg. 13 of
the book.

**30.9 Table. Pythagorean Intonation, Ratios and Frequencies**
Found on
pg. 15 of the book.

**30.10 Table. Pythagorean Intonation, Cents**
Found on pg. 17 of the
book.

**30.11 Table. Pythagorean Intonation, Meantonal Fractions**
Found on
pg. 19 of the book.

30.12.1

218 / 311 = 262144 / 177147).

**30.13 Table. Pythagorean Intonation. Meantonal Ratio and Cents**
Found on pg. 21 of the book.

**30.14 Table. Pythagorean Intonation, Meantonal Ratios and Frequencies**
Found on pg. 22 of the book.

**30.15 Table. Pythagorean Intonation, Meantonal Cents**
Found
on pg. 23 of the book.

**30.16 Table. Units of Divergence**
Found on pg. 24 of the book.

**30.17 Table. Pythagorean Intonation, Diatonic and Chromatic Data**
Found
on pg. 26 of the book.

**30.18 Table. Pythagorean Beat Table**
Found on pg. 28 of the book.

**30.19 Table. Pythagorean Beat Table, Complex Intervals**
Found
on pg. 28 of the book.

**30.20 Table. Typical Pythagorean Intervals**
Found on pg. 30 of the
book.

**30.21 Table. Typical Pythagorean Intervals, Ratios and Cents**
Found
on pg. 31 of the book.

The Mathematics of Music

By John W. Link,

781.1 L6485m

30.22 "100" Tone Scale of Poole Helmholtz, pg.475

"The significance of the scales of music influencing atomic matter led me to a Mr. Poole whose keyboard had 100 tones, the same number of tones as Keely's keyboard. Mr. Poole was published in Sillimans American Journal of Arts and Sciences, 1850 vol. ix. pp. 68-83, 199-216 ; 1867, vol. xliv. pp. 1-22. The organ was called the enharmonic organ, of just intonation.This is a Duodenary Arrangement of Mr. Pooles 100 tones":

"The following is a diagram of Mr. Pooles 100 tone arrangement containing the double and triple diatonic scales":

"Mr. Poole was also aware of the alteration by a skhisma, and the consequent reduction of the number of pipes (of the organ). He also refers to the 53 division, but he does not seem to adopt either"...

"It will be seen that Mr. Poole had 100 note to the octave, of which 39 arose from the harmonic sevenths. If the skhisma were neglected there would remain only 36 tertian and 20 septimal, or in all 56 tones to the octave..... With regard to the double diatonic or dichordal scale, which Mr. Poole always solfas as fah, sol, la, se, do, re, mi, fah (where se is the harmonic seventh to do ), so that do is the dominant, he says that the most beautiful, varied, and ornate compositions are made of the elements it contains. It has the capacity in certain styles of music of using with much grace accidentals, or chromatics as they are called; for example, the si the regular leading note to do, and the sol sharp, a diatonic semitone to below la, or the leading note to the relative minor; these chromatics always ascending a diatonic Semitone (15 : 16) to the notes above ( the experimental intonation) In an example given he also admits se to be raised by 27 cents, that is to be the regular fourth of the triple or trichordal scale, and also allows the introduction of the sixth of this scale. Hence if we use the duodenary form and represent the dichordal scale of F by capitals and these permissive additions by small letters we shall have the scheme in the margin. (Note pg. 275) This gives the trichordal scale of C major complete with its grave second, and also one form of its relative A1 minor complete, but both without the harmonic seventh of the dominants, which of course he would be ready to add when the harmony in his view required it. There is also the complete trichordal scale of F major without the grave second. Hence his dichordal scale resolves itself into a means of bringing these three scales into close connection, chiefly by help of the chord of the ninth (C,E1,G7 B flat, D) in the above scheme".

Sensations of Tone By Helmholtz.

30.25 Penrose Tilings and Musical Scales

(Note: At the end of the following page will be found the ratios of

30.26 From Note A 440 to A 432

(Note: 432 Hz agrees perfectly, in scale, with gematria and world grid points). James Furia's works:

Number 432 (pufori no
longer exists)

See; http://www.greatdreams.com/432.htm

Circle of Perfect 5th's View the Music Wheel and adjust your volume, its a musical page and so is James Furia's

30.27 From Note A 440 To A 450

(Note: 450 Hz is calculated by Ray Tomes):

450, Ray Tomes Home Page.

© Copyright. Robert Grace. 1999