Schismic–countercommatic equivalence continuum: Difference between revisions
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The ''' | The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with the [[41-comma|Pythagorean countercomma ({{monzo| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]]. | ||
All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| 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 [[tempering out|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. | All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| 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 [[tempering out|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. | ||
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For a similar but perhaps more intuitive and practical concept, see [[ | For a similar but perhaps more intuitive and practical concept, see [[Schismic–Pythagorean equivalence continuum]]. | ||
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Revision as of 16:17, 10 December 2024
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 = -11/2 = -5.5)
- Shibboleth (n = -9/2 = -4.5)
- Pluto (n = -7/2 = -3.5)
- 3737 & 5585 (n = 31/3 = 10.3)
- 1277 & 2513 (n = 21/2 = 10.5)
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