Schismic–countercommatic equivalence continuum
The schismic–countercommatic equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with the Pythagorean countercomma ([65 -41⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments supported by 41edo.
All temperaments in the continuum satisfy (32805/32768)n ~ [65 -41⟩. 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 41edo 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 10.1575233481…, and temperaments having n near this value tend to be the most accurate ones.
The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, 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 example:
For a similar but perhaps more intuitive and practical concept, see Schismic–Pythagorean equivalence continuum.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -7 | Merman | 1121008359375 / 1099511627776 | [-40 15 7⟩ |
| -6 | Ampersand | 34171875 / 33554432 | [-25 7 6⟩ |
| -5 | Magic | 3125 / 3072 | [-10 -1 5⟩ |
| -4 | Tetracot | 20000 / 19683 | [5 -9 4⟩ |
| -3 | Rodan | 131072000 / 129140163 | [20 -17 3⟩ |
| -2 | Hemififths | 858993459200 / 847288609443 | [35 -25 2⟩ |
| -1 | Kwai | (32 digits) | [50 -33 1⟩ |
| 0 | Countercomp | (40 digits) | [65 -41⟩ |
| 1 | Cotoneum | (50 digits) | [80 -49 -1⟩ |
| 2 | Newt | (58 digits) | [95 -57 -2⟩ |
| 3 | 41 & 282 | (68 digits) | [110 -65 -3⟩ |
| 4 | 41 & 335 | (76 digits) | [125 -73 -4⟩ |
| 5 | 41 & 388 | (86 digits) | [140 -81 -5⟩ |
| 6 | 41 & 441 | (94 digits) | [155 -89 -6⟩ |
| 7 | 41 & 453 | (104 digits) | [170 -97 -7⟩ |
| 8 | 41 & 506 | (112 digits) | [185 -105 -8⟩ |
| 9 | 41 & 559 | (122 digits) | [200 -113 -9⟩ |
| 10 | 41 & 571 | (130 digits) | [215 -121 -10⟩ |
| 11 | 41 & 624 | (140 digits) | [-230 129 11⟩ |
| 12 | 41 & 677 | (148 digits) | [-245 137 12⟩ |
| 13 | 41 & 730 | (158 digits) | [-260 145 13⟩ |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
Examples of temperaments with fractional values of n:
- Septimin (n = −112)
- Shibboleth (n = −92)
- Pluto (n = −72)
- 3737 & 5585 (n = 313 = 10.3)
- 1277 & 2513 (n = −212)
Rodan (5-limit)
Subgroup: 2.3.5
Comma list: 131072000/129140163
Mapping: [⟨1 1 -1], ⟨0 3 17]]
Optimal tuning (POTE): ~729/640 = 234.528
Optimal ET sequence: 5, 31c, 36c, 41, 46, 87, 220, 307
Badness: 0.168264
Hemififths (5-limit)
Subgroup: 2.3.5
Comma list: 858993459200/847288609443
Mapping: [⟨1 1 -5], ⟨0 2 25]]
Optimal tuning (POTE): ~655360/531441 = 351.476
Optimal ET sequence: 41, 58, 99, 239, 338, 915b, 1253bc
Badness: 0.372848
Kwai (5-limit)
Subgroup: 2.3.5
Comma list: [50 -33 1⟩ = 5629499534213120/5559060566555523
Mapping: [⟨1 0 -50], ⟨0 1 33]]
Optimal tuning (POTE): ~3/2 = 702.630
Optimal ET sequence: 41, 111, 152
Badness: 0.636715
Countercomp
- See also: Countercomp family and 41-comma
Subgroup: 2.3.5
Comma list: [65 -41⟩
Mapping: [⟨41 65 0], ⟨0 0 1]]
Optimal tuning (POTE): ~5/4 = 386.668
Optimal ET sequence: 41, 123, 164, 205, 369, 574, 779, 2132bc
Badness: 0.934310
Cotoneum (5-limit)
Subgroup: 2.3.5
Comma list: [80 -49 -1⟩
Mapping: [⟨1 0 80], ⟨0 1 -49]]
Optimal tuning (POTE): ~3/2 = 702.315
Optimal ET sequence: 41, 135c, 176, 217, 475, 1167, 1642, 2117b
Badness: 1.240078
Newt (5-limit)
Subgroup: 2.3.5
Comma list: [95 -57 -2⟩
Mapping: [⟨1 1 19], ⟨0 2 -57]]
Optimal tuning (POTE): ~[47 -28 -1⟩ = 351.114
Optimal ET sequence: 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc
Badness: 1.528465