Syntonic–31 equivalence continuum: Difference between revisions
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Mapping: [{{val| 1 2 -9 }}, {{val| 0 -1 27 }}] | Mapping: [{{val| 1 2 -9 }}, {{val| 0 -1 27 }}] | ||
{{Val list|legend=1| 12c, 19c, 31, 43c, 50c | {{Val list|legend=1| 12c, 19c, 31, 43c, 50c }} | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c] | [http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c] | ||
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Mapping: [{{val| 1 7 12 }}, {{val| 0 -14 -25 }}] | Mapping: [{{val| 1 7 12 }}, {{val| 0 -14 -25 }}] | ||
{{Val list|legend=1| 13c, 18bc, 31, 44c, 49bc | {{Val list|legend=1| 13c, 18bc, 31, 44c, 49bc, 75c, 80bc }} | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c] | [http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c] | ||
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Mapping: [{{val| 1 3 0 }}, {{val| 0 -11 18 }}] | Mapping: [{{val| 1 3 0 }}, {{val| 0 -11 18 }}] | ||
{{Val list|legend=1| 8c, 23c, 31, 39c | {{Val list|legend=1| 8c, 23c, 31, 39c, 132, 163 }} | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c] | [http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c] | ||
Revision as of 08:48, 16 March 2021
The syntonic-31 equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with a 31-comma ([-49 31⟩). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 31edo.
All temperaments in the continuum satisfy (81/80)n ~ [-49 31⟩. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 31edo (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 7.46781..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | 31 & 31c | [-49 31⟩ | |
| 1 | 31 & 12c | [-45 27 1⟩ | |
| 2 | Quasimoha | 2353579470675/2199023255552 | [-41 23 2⟩ |
| 3 | Oncle | 145282683375/137438953472 | [-37 19 3⟩ |
| 4 | Sentinel | 8968066875/8589934592 | [-33 15 4⟩ |
| 5 | Tritonic | 553584375/536870912 | [-29 11 5⟩ |
| 6 | Ampersand | 34171875/33554432 | [-25 7 6⟩ |
| 7 | Orson | 2109375/2097152 | [-21 3 7⟩ |
| 8 | Würschmidt | 393216/390625 | [-17 -1 8⟩ |
| 9 | Valentine | 1990656/1953125 | [-13 -5 9⟩ |
| 10 | Myna | 10077696/9765625 | [9 9 -10⟩ |
| 11 | Nusecond | 51018336/48828125 | [5 13 -11⟩ |
| 12 | Cypress | 258280326/244140625 | [1 17 -12⟩ |
| 13 | Diesic | 10460353203/9765625000 | [-3 21 -13⟩ |
| 14 | 31 & 13c | 847288609443/781250000000 | [-7 25 -14⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of n:
- 31 & 70c (n = 11/2 = 5.5)
- Slender (n = 13/2 = 6.5)
- Luna (n = 15/2 = 7.5)
- Semisept (n = 17/2 = 8.5)
- Casablanca (n = 19/2 = 9.5)
31 & 12c
In fifths notation, 5/4 is mapped to the quadruple-diminished fifth.
Comma list: [-45 27 1⟩ = 38127987424935/35184372088832
POTE generator: ~4/3 = 503.0504
Mapping: [⟨1 2 -9], ⟨0 -1 27]]
The temperament finder - 5-limit 31 & 12c
31 & 13c
Comma list: [-7 25 -14⟩ = 847288609443/781250000000
POTE generator: 464.4231
Mapping: [⟨1 7 12], ⟨0 -14 -25]]
The temperament finder - 5-limit 31 & 13c
31 & 70c
Comma list: [-54 18 11⟩ = 18917016064453125/18014398509481984
POTE generator: 154.5972
Mapping: [⟨1 3 0], ⟨0 -11 18]]