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
where the division by 4 in the denominator stems from the octave cropping.
Since the Pythagorean fifth in cents reads \( \log(3/2) \cdot 100/(\log(\sqrt[12]{2})) = 701.95500 \), 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:
And an alterated meantone fifth in cents calculates to:
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 | Gъ |
15 | 9754.84 | 154.84 | Dъ |
16 | 10451.61 | 851.61 | Aъ |
17 | 11148.39 | 348.39 | Eъ |
18 | 11845.16 | 1045.16 | Bъ |
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 .