Schismic–commatic equivalence continuum
The schismic-Pythagorean equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with Pythagorean comma ([-19 12⟩). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 12edo.
All temperaments in the continuum satisfy (32805/32768)n ~ [-19 12⟩. Varying n results in different temperaments listed in the table below. It converges to schismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 12edo due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of n is approximately 12.0078623975…, and temperaments having n near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small Kirnberger's atom (the difference between 12 schismas and the Pythagorean comma) is.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Gracecordial | (22 digits) | [-34 20 1⟩ |
| 0 | Compton | 531441/524288 | [-19 12⟩ |
| 1 | Meantone | 81/80 | [-4 4 -1⟩ |
| 2 | Diaschismic | 2048/2025 | [11 -4 -2⟩ |
| 3 | Misty | 67108864/66430125 | [26 -12 -3⟩ |
| 4 | Undim | (26 digits) | [41 -20 -4⟩ |
| 5 | Quindromeda | (34 digits) | [56 -28 -5⟩ |
| 6 | Sextile | (44 digits) | [71 -36 -6⟩ |
| 7 | Heptacot | (52 digits) | [86 -44 -7⟩ |
| 8 | World calendar | (62 digits) | [101 -52 -8⟩ |
| 9 | Quinbisa-tritrigu (12&441) | (70 digits) | [116 -60 -9⟩ |
| 10 | Lesa-quinbigu (12&494) | (80 digits) | [131 -68 -10⟩ |
| 11 | Quadtrisa-legu (12&559) | (88 digits) | [146 -76 -11⟩ |
| 12 | Atomic | (98 digits) | [161 -84 -12⟩ |
| 13 | Quintrila-theyo (12&677) | (106 digits) | [-176 92 13⟩ |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the syntonic-Pythagorean equivalence continuum, which is essentially the same thing. The just value of m is 1.0908441588…
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Python | 43046721/41943040 | [-23 16 -1⟩ |
| 0 | Compton | 531441/524288 | [-19 12⟩ |
| 1 | Schismic | 32805/32768 | [-15 8 1⟩ |
| 2 | Diaschismic | 2048/2025 | [11 -4 -2⟩ |
| 3 | Augmented | 128/125 | [7 0 -3⟩ |
| 4 | Diminished | 648/625 | [3 4 -4⟩ |
| 5 | Ripple | 6561/6250 | [-1 8 -5⟩ |
| 6 | Wronecki | 531441/500000 | [-5 12 -6⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
| n | m | Temperament | Comma |
|---|---|---|---|
| 5/3 = 1.6 | 5/2 = 2.5 | Passion | [18 -4 -5⟩ |
| 5/2 = 2.5 | 5/3 = 1.6 | Quintaleap | [37 -16 -5⟩ |
Python
Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by sixteen fifths octave-reduced, which is a double augmented second (C-Dx). It can be described as 12 & 91, and 103edo is a good tuning.
Subgroup: 2.3.5
Comma list: [-23 16 -1⟩ = 43046721/41943040
Mapping: [⟨1 0 -23], ⟨0 1 16]]
- mapping generators: ~2, ~3
⟨⟨1 16 23]]
Optimal ET sequence: 12, …, 79, 91, 103
Badness: 0.295079
Sextile
The 5-limit version of sextile reaches the interval class of 5 by -6 perfect fifths minus a period of 1/6-octave.
Subgroup: 2.3.5
Comma list: [71 -36 -6⟩
Mapping: [⟨6 0 71], ⟨0 1 -6]]]
- mapping generators: ~4096/3645, ~3
- CTE: ~4096/3645 = 1\6, ~3/2 = 702.2627 (~4428675/4194304 = 97.7373)
- CWE: ~4096/3645 = 1\6, ~3/2 = 702.2434 (~4428675/4194304 = 97.7566)
⟨⟨6 -36 -77]]
Optimal ET sequence: 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc
Badness: 0.555423
Heptacot
Heptacot tempers out the heptacot comma and divides the fifth into seven equal parts, the most notable example being 12edo (7\12).
Subgroup: 2.3.5
Comma list: [86 -44 -7⟩
Mapping: [⟨1 1 6], ⟨0 7 -44]]]
Optimal ET sequence: 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc
Badness: 0.683