Syntonic–kleismic equivalence continuum: Difference between revisions
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All temperaments in the continuum satisfy (81/80)<sup>''k''</sup> ~ {{monzo|-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. | All temperaments in the continuum satisfy (81/80)<sup>''k''</sup> ~ {{monzo|-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 | This continuum can be expressed as the relationship between 81/80 and the [[enneadeca]] ({{Monzo|-14 -19 19}}). That is, (81/80)<sup>''n''</sup> ~ {{monzo|-14 -19 19}}. In this case, ''n'' = 3''k'' - 19. | ||
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Revision as of 15:14, 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 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 | 19edo | 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
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|]