Schismic–countercommatic equivalence continuum: Difference between revisions
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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 tempers both commas and thus tempers 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 tempers both commas and thus tempers 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. | ||
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 02:01, 23 May 2021
The schismic-Mercator equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with counterpyth comma ([65 -41⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
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 both commas and thus tempers 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.
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 | [50 -33 1⟩ | |
| 0 | Counterpyth | [65 -41⟩ | |
| 1 | Cotoneum | [80 -49 -1⟩ | |
| 2 | Newt | [95 -57 -2⟩ | |
| 3 | 41&282 | [110 -65 -3⟩ | |
| 4 | 41&335 | [125 -73 -4⟩ | |
| 5 | 41&388 | [140 -81 -5⟩ | |
| 6 | 41&441 | [155 -89 -6⟩ | |
| 7 | 41&453 | [170 -97 -7⟩ | |
| 8 | 41&506 | [185 -105 -8⟩ | |
| 9 | 41&559 | [200 -113 -9⟩ | |
| 10 | 41&571 | [215 -121 -10⟩ | |
| 11 | 41&624 | [-230 129 11⟩ | |
| 12 | 41&677 | [-245 137 12⟩ | |
| 13 | 41&730 | [-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)