Syntonic–kleismic equivalence continuum: Difference between revisions
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== Mowgli == | == Mowgli == | ||
Comma list: {{Monzo|0 22 -15}} | |||
POTE generator: 126.7237 cents | POTE generator: 126.7237 cents | ||
Mapping: [{{val| 1 0 0 }}, {{val| 0 15 22 }}] | |||
{{Val list|legend=1| 19, 85c, 104c, 123, 142, 161 }} | |||
== 8c & 19 == | |||
Comma list: {{Monzo|-32 10 7}} = 4613203125/4294967296 | |||
POTE generator: 442.2674 cents | POTE generator: 442.2674 cents | ||
Mapping: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}] | |||
{{Val list|legend=1| 8c, 11, 19 }} | |||
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c] | [http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c] | ||
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== 19 & 506 == | == 19 & 506 == | ||
Comma list: {{Monzo| 38 61 -58 }} | |||
POTE generator: 505.1394 cents | POTE generator: 505.1394 cents | ||
Mapping: [{{val| 1 26 28 }}, {{val| 0 -58 -61 }}] | |||
{{Val list|legend=1| 19, 468, 487, 506, 1031 }} | |||
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_506&limit=5 The temperament finder - 5-limit 19 & 506] | [http://x31eq.com/cgi-bin/rt.cgi?ets=19_506&limit=5 The temperament finder - 5-limit 19 & 506] | ||
Revision as of 08:08, 16 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 can 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 | 19 & 19c | 1162261467/1073741824 | [-30 19⟩ |
| 1 | Lalayo | 71744535/67108864 | [-26 15 1⟩ |
| 2 | Hogzilla | 4428675/4194304 | [-22 11 2⟩ |
| 3 | Stump | 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)
- Acrokleismic (k = 32/5 = 6.4)
- 19 & 506 (k = 58/9 = 6.4)
- Parakleismic (k = 6.5)
- Countermeantone (k = 20/3 = 6.6)
- Mowgli (k = 7.5)
Mowgli
Comma list: [0 22 -15⟩
POTE generator: 126.7237 cents
Mapping: [⟨1 0 0], ⟨0 15 22]]
8c & 19
Comma list: [-32 10 7⟩ = 4613203125/4294967296
POTE generator: 442.2674 cents
Mapping: [⟨1 -1 6], ⟨0 7 -10]]
The temperament finder - 5-limit 19 & 8c
19 & 506
Comma list: [38 61 -58⟩
POTE generator: 505.1394 cents
Mapping: [⟨1 26 28], ⟨0 -58 -61]]