Syntonic–kleismic equivalence continuum: Difference between revisions
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The '''syntonic-enneadecal equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the 19-comma ({{ | The '''syntonic-kleismic equivalence continuum''' (or '''syntonic-enneadecal equivalence continuum''') is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the 19-comma ({{monzo| -30 19 }}). | ||
All temperaments in the continuum satisfy (81/80)<sup>'' | All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ {{monzo|-30 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 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
This continuum can be expressed as the relationship between 81/80 and the [[enneadeca]] ({{ | This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, (81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}. In this case, ''k'' = 3''n'' - 19. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments in the continuum | |+ Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | '' | ! rowspan="2" | ''n'' | ||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
! colspan="2" | Comma | ! colspan="2" | Comma | ||
Revision as of 08:41, 31 October 2022
The syntonic-kleismic equivalence continuum (or 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)n ~ [-30 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 6.376…, and temperaments having n near this value tend to be the most accurate ones.
This continuum can also be expressed as the relationship between 81/80 and the enneadeca ([-14 -19 19⟩). That is, (81/80)k ~ [-14 -19 19⟩. In this case, k = 3n - 19.
| n | 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)
- Counterhanson (k = 25/4 = 6.25)
- Enneadecal (k = 19/3 = 6.3)
- Egads (k = 51/8 = 6.375)
- 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)
Lalayo
Comma list: [-26 15 1⟩ = 71744535/67108864
Mapping: [⟨1 2 -4], ⟨0 -1 15]]
POTE generator: ~4/3 = 505.348 cents
Badness: 0.803397
Lalasepyo (8c & 19)
Comma list: [-32 10 7⟩ = 4613203125/4294967296
Mapping: [⟨1 -1 6], ⟨0 7 -10]]
POTE generator: ~675/512 = 442.2674 cents
Badness: 1.061630
The temperament finder - 5-limit 19 & 8c
Counterhanson
Comma list: [-20 -24 25⟩ = 298023223876953125/296148833645101056
Mapping: [⟨1 -5 -4], ⟨0 25 24]]
POTE generator: ~6/5 = 316.081 cents
Badness: 0.317551
19 & 506
Comma list: [38 61 -58⟩
Mapping: [⟨1 26 28], ⟨0 -58 -61]]
POTE generator: ~[-12 -20 19⟩ = 505.1394 cents
Badness: 2.105450
The temperament finder - 5-limit 19 & 506
Countermeantone
Comma list: [10 23 -20⟩ = 96402615118848/95367431640625
Mapping: [⟨1 10 12], ⟨0 -20 -23]]
POTE generator: ~104976/78125 = 504.913 cents
Badness: 0.373477
Mowgli
Comma list: [0 22 -15⟩
Mapping: [⟨1 0 0], ⟨0 15 22]]
POTE generator: ~27/25 = 126.7237 cents
Badness: 0.653871
Oviminor
Subgroup: 2.3.5
Comma list: [-134 -185 184⟩
Mapping: [⟨1 50 51], ⟨0 -184 -185]]
Optimal tuning (CTE): ~6/5 = 315.7501
Badness: 32.0