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 | 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. The just value of m is 1.0908441588…
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 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⟩ |
Examples of temperaments with fractional values of n:
- Python (n = 1/2 = 0.5)
- Ripple (n = 5/4 = 1.25)
- Diminished (n = 4/3 = 1.3)
- Augmented (n = 3/2 = 1.5)
- Passion (n = 5/3 = 1.6)
- Quintaleap (n = 5/2 = 2.5)
Compton (12&72)
- and Compton family
Subgroup: 2.3.5
Comma list: [-19 12⟩ = 531441/524288
Mapping: [⟨12 19 28], ⟨0 0 -1]]
⟨⟨0 12 19]]
POTE generator: ~5/4 = 384.882
Optimal ET sequence: 12, 48, 60, 72, 84
Badness: 0.094494
Python (12&79)
Subgroup: 2.3.5
Comma list: [-23 16 -1⟩ = 43046721/41943040
Mapping: [⟨1 0 -23], ⟨0 -1 -16]]
⟨⟨1 16 23]]
POTE generator: ~4/3 = 500.970
Optimal ET sequence: 12, …, 79, 91, 103
Badness: 0.295079
Quintaleap (12&121)
Subgroup: 2.3.5
Comma list: [37 -16 -5⟩ = 137438953472/134521003125
Mapping: [⟨1 2 1], ⟨0 -5 16]]
⟨⟨5 -16 -37]]
POTE generator: ~135/128 = 99.267
Optimal ET sequence: 12, …, 85, 97, 109, 121, 133, 278c, 411bc, 544bc
Badness: 0.444506
Undim (12&152)
Subgroup: 2.3.5
Comma list: [41 -20 -4⟩
Mapping: [⟨4 0 41], ⟨0 1 -5]]
⟨⟨4 -20 -41]]
POTE generator: ~3/2 = 702.6054
Optimal ET sequence: 12, …, 104, 116, 128, 140, 152, 610, 772, 924c, 1076bc, 1228bc
Badness: 0.241703
Quindromeda (12&205)
Subgroup: 2.3.5
Comma list: [56 -28 -5⟩
Mapping: [⟨1 2 0], ⟨0 -5 28]]
POTE generator: ~4428675/4194304 = 99.526
⟨⟨5 -28 -56]]
Optimal ET sequence: 12, …, 181, 193, 205, 217, 422
Badness: 0.399849
Sextile (12&270)
Subgroup: 2.3.5
Comma list: [71 -36 -6⟩
Mapping: [⟨6 0 71], ⟨0 1 -6]]
POTE generator: ~3/2 = 702.2356
⟨⟨6 -36 -77]]
Optimal ET sequence: 12, …, 222, 234, 246, 258, 270, 1068, 1338, 1608, 1878, 4026bc
Badness: 0.555423