Syntonic–kleismic equivalence continuum: Difference between revisions
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Examples of temperaments with fractional values of ''n'': | |||
* [[Parakleismic]] (''n'' = 0.5) | |||
* 19 & 506 (''n'' = 1/3 = 0.{{overline|3}}) | |||
== 19 & 506 == | |||
Commas: {{Monzo|38 61 -58}} | |||
POTE generator: 505.1394 cents | |||
Map: [<1 26 28|, <0 -58 -61|] | |||
EDOs: {{EDOs| 19, 38, 57, 468, 487, 506, 525, 544, 1012, 1031 }} | |||
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_506&limit=5 The temperament finder - 5-limit 19 & 506] | |||
[[Category:19edo]] | [[Category:19edo]] | ||
Revision as of 09:33, 14 March 2021
The syntonic-enneadecal equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the enneadeca ([-14 -19 19⟩).
All temperaments in the continuum satisfy (81/80)n ~ [-14 -19 19⟩. 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 19edo (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 0.1309..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Enneadecal | [-14 -19 19⟩ | |
| 1 | Countermeantone | [-10 -23 20⟩ | |
| 2 | Sensi | 78732/78125 | [2 9 -7⟩ |
| 3 | 19 & 169c | [2 31 -22⟩ | |
| 4 | 19 & 162c | [-2 35 -23⟩ | |
| 5 | Unicorn | 1594323/1562500 | [-2 13 -8⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of n:
- Parakleismic (n = 0.5)
- 19 & 506 (n = 1/3 = 0.3)
19 & 506
Commas: [38 61 -58⟩
POTE generator: 505.1394 cents
Map: [<1 26 28|, <0 -58 -61|]