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 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 | 17433922005/17179869184 | [-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 | [41 -20 -4⟩ | |
5 | Quindromeda | [56 -28 -5⟩ | |
6 | Sextile | [71 -36 -6⟩ | |
7 | Sepsa-sepgu (12&323) | [86 -44 -7⟩ | |
8 | World calendar | [101 -52 -8⟩ | |
9 | Quinbisa-tritrigu (12&441) | [116 -60 -9⟩ | |
10 | Lesa-quinbigu (12&494) | [131 -68 -10⟩ | |
11 | Quadtrisa-legu (12&559) | [146 -76 -11⟩ | |
12 | Atomic | [161 -84 -12⟩ | |
13 | Quintrila-theyo (12&677) | [-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⟩ |
Temperament | n | m |
---|---|---|
Passion | 5/3 = 1.6 | 5/2 = 2.5 |
Quintaleap | 5/2 = 2.5 | 5/3 = 1.6 |
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]]
- POTE: ~2 = 1\1, ~3/2 = 699.030
Optimal ET sequence: 12, …, 79, 91, 103
Badness: 0.295079
Sextile
- See also: Landscape microtemperaments #Sextile
Subgroup: 2.3.5
Comma list: [71 -36 -6⟩
Mapping: [⟨6 0 71], ⟨0 1 -6]]]
- mapping generators: ~4096/3645, ~3
- POTE: ~4096/3645 = 1\6, ~3/2 = 702.2356
Wedgie: ⟨⟨6 -36 -77]]
Optimal ET sequence: 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc
Badness: 0.555423