Schismic-Pythagorean 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 out both commas and thus tempers out 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.
The Pythagorean comma is the characteristic 3-limit comma tempered out in 12edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For an n that is not coprime with 12, however, the corresponding temperament splits the octave into gcd (n, 12) parts, and splits the interval class of 3 into n/gcd (n, 12). For example:
- Meantone (n = 1) is generated by a fifth with an unsplit octave;
- Diaschismic (n = 2) splits the octave in two, as 2 divides 12;
- Misty (n = 3) splits the octave in three, as 3 divides 12;
- Undim (n = 4) splits the octave in four, as 4 divides 12;
- Quindromeda (n = 5) does not split the octave but splits the fourth in five, as 5 is coprime with 12.
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… The syntonic comma is way larger but way simpler than the schisma. As such, this continuum does not contain as many microtemperaments, but has more useful lower-complexity temperaments.
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
Wedgie: ⟨⟨ 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)
Wedgie: ⟨⟨ 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