Syntonic–kleismic equivalence continuum: Difference between revisions
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Examples of temperaments with fractional values of '' | Examples of temperaments with fractional values of ''k'': | ||
* 19 & 8c (''k'' = 3.5) | |||
* [[High badness temperaments#Unsmate|Unsmate]] (''k'' = 4.5) | |||
* [[Sycamore family#Sycamore|Sycamore]] (''k'' = 5.5) | |||
* [[Enneadecal]] (''k'' = 19/3 = 6.{{overline|3}}) | * [[Enneadecal]] (''k'' = 19/3 = 6.{{overline|3}}) | ||
* 19 & 506 (''k'' = 58/9 = 6.{{overline|4}}) | * 19 & 506 (''k'' = 58/9 = 6.{{overline|4}}) | ||
* [[Parakleismic]] (''k'' = 6.5) | * [[Parakleismic]] (''k'' = 6.5) | ||
* [[Countermeantone]] (''k'' = 20/3 = 6.{{overline|6}}) | * [[Countermeantone]] (''k'' = 20/3 = 6.{{overline|6}}) | ||
* [[Mowgli]] (''k'' = 7.5) | |||
== Mowgli == | |||
Commas: {{Monzo|0 22 -15}} | |||
POTE generator: 126.7237 cents | |||
Map: [<1 0 0|, <0 15 22|] | |||
EDOs: {{EDOs| 19, 38, 57, 66c, 76, 85c, 104c, 123, 142, 161 }} | |||
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_104c&limit=5 The temperament finder - 5-limit mowgli] | |||
== 19 & 8c == | |||
Commas: {{Monzo|-32 10 7}} (4613203125/4294967296) | |||
POTE generator: 442.2674 cents | |||
Map: [<1 -1 6|, <0 7 -10|] | |||
EDOs: {{EDOs| 8c, 11, 19, 27c, 30b, 38, 46c, 49b, 57, 76 }} | |||
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c] | |||
== 19 & 506 == | == 19 & 506 == | ||
Revision as of 12:30, 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 19-comma ([-30 19⟩).
All temperaments in the continuum satisfy (81/80)k ~ [-30 19⟩. Varying k results in different temperaments listed in the table below. It converges to meantone as k approaches infinity. If we allow non-integer and infinite k, 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 k is approximately 6.376..., and temperaments having k near this value tend to be the most accurate ones.
This continuum used to be expressed as the relationship between 81/80 and the enneadeca ([-14 -19 19⟩). That is, (81/80)n ~ [-14 -19 19⟩. In this case, n = 3k - 19.
| k | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | 19edo | 1162261467/1073741824 | [-30 19⟩ |
| 1 | Lalayo | 71744535/67108864 | [-26 15 1⟩ |
| 2 | Lala-Yoyo | 4428675/4194304 | [-22 11 2⟩ |
| 3 | Latriyo | 273375/262144 | [-18 7 3⟩ |
| 4 | Negri | 16875/16384 | [-14 3 4⟩ |
| 5 | Magic | 3125/3072 | [-10 -1 5⟩ |
| 6 | Hanson | 15625/15552 | [-6 -5 6⟩ |
| 7 | Sensi | 78732/78125 | [2 9 -7⟩ |
| 8 | Unicorn | 1594323/1562500 | [-2 13 -8⟩ |
| 9 | 19 & 51c | 129140163/125000000 | [-6 17 -9⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of k:
- 19 & 8c (k = 3.5)
- Unsmate (k = 4.5)
- Sycamore (k = 5.5)
- Enneadecal (k = 19/3 = 6.3)
- 19 & 506 (k = 58/9 = 6.4)
- Parakleismic (k = 6.5)
- Countermeantone (k = 20/3 = 6.6)
- Mowgli (k = 7.5)
Mowgli
Commas: [0 22 -15⟩
POTE generator: 126.7237 cents
Map: [<1 0 0|, <0 15 22|]
EDOs: 19, 38, 57, 66c, 76, 85c, 104c, 123, 142, 161
The temperament finder - 5-limit mowgli
19 & 8c
Commas: [-32 10 7⟩ (4613203125/4294967296)
POTE generator: 442.2674 cents
Map: [<1 -1 6|, <0 7 -10|]
EDOs: 8c, 11, 19, 27c, 30b, 38, 46c, 49b, 57, 76
The temperament finder - 5-limit 19 & 8c
19 & 506
Commas: [38 61 -58⟩
POTE generator: 505.1394 cents
Map: [<1 26 28|, <0 -58 -61|]