User:2^67-1/TempClean sandbox/Pythagorean tuning
The 3-limit consists of intervals that are either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as [math]2^a \cdot 3^b[/math], where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are 3/2, 4/3, 9/8. Confining intervals to the 3-limit is known as Pythagorean tuning, and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it. A 3-limit interval is also known as a Pythagorean interval.
EDO approximation
EDOs which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the continued fraction for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, …
Another approach is to find edos which have more accurate 3 than all smaller edos. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, …
Approaches
There are two possible approaches to Pythagorean tuning, namely octave-based and tritave-based.
Octave-based Pythagorean tuning is essentially how it is used in the common-practice. This gives MOS sizes of 2, 3, 5 (2L 3s pentic), 7 (5L 2s diatonic), 12 (5L 7s chromatic), 17 (12L 5s enharmonic), 29, 41, and 53.
Tritave-based Pythagorean tuning is an approach described in this paper by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8, 11, 19, 25, 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.
Table of intervals
This is a table of 3-limit (or Pythagorean) intervals which can be obtained by stacking nine (or less) fifths or fourths from the tonic and octave-reducing them.
| Ratio | Monzo | Size (¢) | Color Name | Diatonic Category | ||
|---|---|---|---|---|---|---|
| 1/1 | [0⟩ | 0.000 | w1 | wa unison | P1 | C |
| 2187/2048 | [-11 7⟩ | 113.685 | Lw1 | lawa 1sn | A1 | C# |
| 256/243 | [8 -5⟩ | 90.225 | sw2 | sawa 2nd | m2 | Db |
| 9/8 | [-3 2⟩ | 203.910 | w2 | wa 2nd | M2 | D |
| 19683/16384 | [-14 9⟩ | 317.595 | Lw2 | lawa 2nd | A2 | D# |
| 32/27 | [5 -3⟩ | 294.135 | w3 | wa 3rd | m3 | Eb |
| 81/64 | [-6 4⟩ | 407.820 | Lw3 | lawa 3rd | M3 | E |
| 8192/6561 | [13 -8⟩ | 384.360 | sw4 | sawa 4th | d4 | Fb |
| 4/3 | [2 -1⟩ | 498.045 | w4 | wa 4th | P4 | F |
| 729/512 | [-9 6⟩ | 611.730 | Lw4 | lawa 4th | A4 | F# |
| 1024/729 | [10 -6⟩ | 588.270 | sw5 | sawa 5th | d5 | Gb |
| 3/2 | [-1 1⟩ | 701.955 | w5 | wa 5th | P5 | G |
| 6561/4096 | [-12 8⟩ | 815.640 | Lw5 | lawa 5th | A5 | G# |
| 128/81 | [7 -4⟩ | 792.180 | sw6 | sawa 6th | m6 | Ab |
| 27/16 | [-4 3⟩ | 905.865 | w6 | wa 6th | M6 | A |
| 32768/19683 | [15 -9⟩ | 882.405 | sw7 | sawa 7th | d7 | Bbb |
| 16/9 | [4 -2⟩ | 996.090 | w7 | wa 7th | m7 | Bb |
| 243/128 | [-7 5⟩ | 1109.775 | Lw7 | lawa 7th | M7 | B |
| 4096/2187 | [12 -7⟩ | 1086.315 | sw8 | sawa 8ve | d8 | Cb |
| 2/1 | [1⟩ | 1200.000 | w8 | wa 8ve | P8 | C |