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## Develop the 31-Tone System

At the beginning, we start from a stacking of four perfect fifth above any given note, say C. The frequency ratios are then (notes are Pythagorean notes !):

Note Ratio Cropped by
Octave
C 1 = 1.0000 1.0000
G (3/2)1 = 1.5000 1.5000
D (3/2)2 = 2.2500 1.1250
A (3/2)3 = 3.3750 1.6875
E (3/2)4 = 5.0625 1.2656

Now, from the meantone spirit, we want to make C-E a perfect major third. We therefore multiply the fifth interval by

$\text{MF} = \sqrt[{\Large 4}\;]{\frac{5/4}{(3/2)^4/4}} = 0.9968991874808$

where the division by 4 in the denominator stems from the octave cropping.

$\text{In cents: MFC} = \log(\text{MF}) \cdot \frac{100}{\log(\sqrt{2})} = -5.3765724$

Since the Pythagorean fifth in cents reads $$\log(3/2) \cdot 100/(\log(\sqrt{2})) = 701.95500$$, the diminished fifth will thus read $$701.955 - 5.3765724 = 696.5784276$$

Recall a cent to be the 100th part of a halftone in the Equal Temperament on a logarithmic scale.

This gives us a new table (notes are Quarter-Comma-Meantone notes !):

Note MT Ratio Cropping by
Octave
In Cents Compare: Cents in
Equ. Temp.
C 1 = 1.0000 1.0000 0.00 0.00
G (3/2)1 * (MF)1 = 1.4953 1.4953 696.58 700.00
D (3/2)2 * (MF)2 = 2.2360 1.1180 1393.16 1400.00
A (3/2)3 * (MF)3 = 3.3437 1.6719 2089.74 2100.00
E (3/2)4 * (MF)4 = 5.0000 1.2500 2786.31 2800.00

Let's now continue this table to make a full 12-tone cycle, and start at Eb (this is per convention):

Note MT Ratio Cropping by
Octave
In Cents Compare: Cents in
Equ. Temp.
Eb 1 = 1.0000 1.0000 0.00 0.00
Bb (3/2)1 * (MF)1 = 1.4953 1.4953 696.58 700.00
F (3/2)2 * (MF)2 = 2.2360 1.1180 1393.16 1400.00
C (3/2)3 * (MF)3 = 3.3437 1.6719 2089.74 2100.00
G (3/2)4 * (MF)4 = 5.0000 1.2500 2786.31 2800.00
D (3/2)5 * (MF)5 = 7.4767 1.8692 3482.89 3500.00
A (3/2)6 * (MF)6 = 11.1803 1.3975 4179.47 4200.00
E (3/2)7 * (MF)7 = 16.7185 1.0449 4876.05 4900.00
H (3/2)8 * (MF)8 = 25.0000 1.5625 5572.63 5600.00
F# (3/2)9 * (MF)9 = 37.3837 1.1682 6269.21 6300.00
C# (3/2)10 * (MF)10 = 55.9017 1.7469 6965.78 7000.00
G# (3/2)11 * (MF)11 = 83.5925 1.3061 7662.36 7700.00
(D#) (3/2)12 * (MF)12 = 125.0000 1.9531 8358.94 8400.00

the D# does not exist in the Meantone temperament, it is here to finish the circle. Because we have stacked slightly diminuished fifths, the meantone circle cannot close. With 12 meantone fifth we are 41.06 cents short! This interval has a name, it is called Diesis (some people say "lesser" diesis, because there are other intervals defined in a slightly different way and giving slightly different values, but these are not so important here, so I omit the "lesser").

The basic idea now is to just continue this scheme and see where we end up. Because we leave the 12-tone naming scheme, I for now just number the tones and do not give them names.

Note MT Ratio Cropping by
Octave
In Cents
1 1 = 1.0000 1.0000 0.00
2 (3/2)1 * (MF)1 = 1.4953 1.4953 696.58
3 (3/2)2 * (MF)2 = 2.2360 1.1180 1393.16
4 (3/2)3 * (MF)3 = 3.3437 1.6719 2089.74
5 (3/2)4 * (MF)4 = 5.0000 1.2500 2786.31
6 (3/2)5 * (MF)5 = 7.4767 1.8692 3482.89
7 (3/2)6 * (MF)6 = 11.1803 1.3975 4179.47
8 (3/2)7 * (MF)7 = 16.7185 1.0449 4876.05
9 (3/2)8 * (MF)8 = 25.0000 1.5625 5572.63
10 (3/2)9 * (MF)9 = 37.3837 1.1682 6269.21
11 (3/2)10 * (MF)10 = 55.9017 1.7469 6965.78
12 (3/2)11 * (MF)11 = 83.5925 1.3061 7662.36
13 (3/2)12 * (MF)12 = 125.0000 1.9531 8358.94
14 (3/2)13 * (MF)13 = 186.9186 1.4603 9055.5196
15 (3/2)14 * (MF)14 = 279.5085 1.0918 9752.0980
16 (3/2)15 * (MF)15 = 417.9627 1.6327 10448.6764
17 (3/2)16 * (MF)16 = 625.0000 1.2207 11145.2549
18 (3/2)17 * (MF)17 = 934.5930 1.8254 11841.8333
19 (3/2)18 * (MF)18 = 1397.5425 1.3648 12538.4117
20 (3/2)19 * (MF)19 = 2089.8135 1.0204 13234.9901
21 (3/2)20 * (MF)20 = 3125.0000 1.5259 13931.5686
22 (3/2)21 * (MF)21 = 4672.9649 1.1409 14628.1470
23 (3/2)22 * (MF)22 = 6987.7124 1.7060 15324.7254
24 (3/2)23 * (MF)23 = 10449.0673 1.2755 16021.3039
25 (3/2)24 * (MF)24 = 15625.0000 1.9073 16717.8823
26 (3/2)25 * (MF)25 = 23364.8247 1.4261 17414.4607
27 (3/2)26 * (MF)26 = 34938.5621 1.0662 18111.0391
28 (3/2)27 * (MF)27 = 52245.3363 1.5944 18807.6176
29 (3/2)28 * (MF)28 = 78125.0000 1.1921 19504.1960
30 (3/2)29 * (MF)29 = 116824.1235 1.7826 20200.7744
31 (3/2)30 * (MF)30 = 174692.8107 1.3328 20897.3529
32 (3/2)31 * (MF)31 = 261226.6816 1.9930 21593.9313
33 (3/2)32 * (MF)32 = 390625.0000 1.4901 22290.5097

There is a near hit concerning a closed circle on note number 32. It is only 6.068044 cents too low (215.93 half tones lies in the 17th octave, thus we subtract 17*12 and come to 1193.9313, which is an octave 1200 short by 6.068044). This is really close! And, even better, given such a series of 31 notes you will recognize that it is complete in the sense that you can freely transpose each melody to any base you like.

We make one last step to make things perfect: we add 6.068044 / 31 = 0.195743 cents to each fifth, and by doing so, get a closed circle of 31 tones, containing every possible meantone scale to nearly perfect accuracy, and at the same time freely transposable melodies!. The factor then reads:

$MF' = MF \cdot e^{ 0.195743 \cdot \frac{ \log_2( 2 ) }{1200} } = 0.997011908848$

And an alterated meantone fifth in cents calculates to:

$\text{Fifth} = \text{Meantone fifth} + 0.195743 = 696.578428 + 0.195743 = 696.7741718$

Our final table is now (we show only cents here):

Note Cents Cents Octave
cropped
31-Notation
1 0.00 0.00 C
2 696.77 696.77 G
3 1393.55 193.55 D
4 2090.32 890.32 A
5 2787.10 387.10 E
6 3483.87 1083.87 B
7 4180.65 580.65 F#
8 4877.42 77.42 C#
9 5574.19 774.19 G#
10 6270.97 270.97 D#
11 6967.74 967.74 A#
12 7664.52 464.52 E# = Fъ
13 8361.29 1161.29 B# = Cъ
14 9058.06 658.06
15 9754.84 154.84
16 10451.61 851.61
17 11148.39 348.39
18 11845.16 1045.16
19 12541.94 541.94 F‡
20 13238.71 38.71 C‡
21 13935.48 735.48 G‡
22 14632.26 232.26 D‡
23 15329.03 929.03 A‡
24 16025.81 425.81 E‡ = Fb
25 16722.58 1122.58 B‡ = Cb
26 17419.35 619.35 Gb
27 18116.13 116.13 Db
28 18812.90 812.90 Ab
29 19509.68 309.68 Eb
30 20206.45 1006.45 Bb
31 20903.23 503.23 F
32 21600.00 0.00 C

The smallest step in our new temperament you can e.g. see from note number 1 to note number 20, it is 38.7097 cents. Because it is close to the diesis defined above, this term is used here as well. To make the difference clear, I call it 31-Diesis

 Definition: smallest step in 31-tone temperament = 31-Diesis = 38.7097 cents

Because we have tones that are close to the meantone temperament, I reintroduced the note names in the above table. For the missing notes I use accidentials ‡ to go one 31-diesis up and ъ to go one diesis down. Note that the traditional meantone temperament corresponds to notes 29, 30, 31, 1, 2, 3, 4, 5, 6, 7, 8, 9. Also note that C,D,E,F,... do only closely correspond to notes in the Equal Temperament. So you cannot play "some notes" of 31-music on your mechanical piano!

A ъ is always a diesis above the b as an accidential, the ‡ always a diesis below the #. So the "chromatic scale" goes

    C C‡ C# Db Dъ D D‡ D# Eb Eъ E E‡ E# F F‡ F# Gb Gъ G G‡ G# Ab Aъ A A‡ A# Bb Bъ B B‡ B# C


Despite the tiny chance you are one of the few individuals on this planet who have a 31-tone instrument, you still can compose 31-tone music and let the computer play 31-tone music .