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 ({{Monzo| -30 19}}). | 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 ({{Monzo|-30 19}}). | ||
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. | ||
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|- | |- | ||
| 0 | | 0 | ||
| 19 & 19c | | 19 & 19c | ||
| [[1162261467/1073741824]] | | [[1162261467/1073741824]] | ||
| {{monzo|-30 19}} | | {{monzo|-30 19}} | ||
| Line 61: | Line 61: | ||
|- | |- | ||
| 9 | | 9 | ||
| 19 & 51c | | 19 & 51c | ||
| [[129140163/125000000]] | | [[129140163/125000000]] | ||
| {{monzo|-6 17 -9}} | | {{monzo|-6 17 -9}} | ||
| Line 78: | Line 78: | ||
Examples of temperaments with fractional values of ''k'': | Examples of temperaments with fractional values of ''k'': | ||
* 19 & 8c (''k'' = 3.5) | * 19 & 8c (''k'' = 3.5) | ||
* [[High badness temperaments#Unsmate|Unsmate]] (''k'' = 4.5) | * [[High badness temperaments#Unsmate|Unsmate]] (''k'' = 4.5) | ||
* [[Sycamore family#Sycamore|Sycamore]] (''k'' = 5.5) | * [[Sycamore family#Sycamore|Sycamore]] (''k'' = 5.5) | ||
* [[Counterhanson]] (''k'' = 25/4 = 6.25) | |||
* [[Enneadecal]] (''k'' = 19/3 = 6.{{overline|3}}) | * [[Enneadecal]] (''k'' = 19/3 = 6.{{overline|3}}) | ||
* [[Very high accuracy temperaments#Egads|Egads]] (''k'' = 51/8 = 6.375) | |||
* [[Acrokleismic]] (''k'' = 32/5 = 6.4) | * [[Acrokleismic]] (''k'' = 32/5 = 6.4) | ||
* 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]] (''k'' = 7.5) | ||
== | == Lalayo == | ||
[[Comma list]]: {{Monzo|-26 15 1}} = 71744535/67108864 | |||
[[Mapping]]: [{{val| 1 2 -4 }}, {{val| 0 -1 15 }}] | |||
POTE generator: | [[POTE generator]]: ~4/3 = 505.348 cents | ||
{{Val list|legend=1| 7c, 12c, 19 }} | |||
[[Badness]]: 0.803397 | |||
== 8c & 19 == | == 8c & 19 == | ||
[[Comma list]]: {{Monzo|-32 10 7}} = 4613203125/4294967296 | |||
[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}] | |||
POTE generator: 442.2674 cents | [[POTE generator]]: ~675/512 = 442.2674 cents | ||
{{Val list|legend=1| 8c, 11, 19 }} | |||
[[Badness]]: 1.061630 | |||
[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] | ||
== 19 & 506 == | == 19 & 506 == | ||
[[Comma list]]: {{Monzo| 38 61 -58 }} | |||
[[Mapping]]: [{{val| 1 26 28 }}, {{val| 0 -58 -61 }}] | |||
POTE generator: 505.1394 cents | [[POTE generator]]: ~{{monzo|-12 -20 19}} = 505.1394 cents | ||
{{Val list|legend=1| 19, 468, 487, 506, 1031 }} | |||
[[Badness]]: 2.105450 | |||
[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] | ||
== Counterhanson == | |||
[[Comma list]]: {{Monzo|-20 -24 25}} = 298023223876953125/296148833645101056 | |||
[[Mapping]]: [{{val|1 -5 -4}}, {{val|0 25 24}}] | |||
[[POTE generator]]: ~6/5 = 316.081 cents | |||
{{Val list|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }} | |||
[[Badness]]: 0.317551 | |||
== Countermeantone == | |||
[[Comma list]]: {{Monzo|10 23 -20}} = 96402615118848/95367431640625 | |||
[[Mapping]]: [{{val|1 10 12}}, {{val|0 -20 -23}}] | |||
[[POTE generator]]: ~104976/78125 = 504.913 cents | |||
{{Val list|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }} | |||
[[Badness]]: 0.373477 | |||
== Mowgli == | |||
[[Comma list]]: {{Monzo|0 22 -15}} | |||
[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 15 22 }}] | |||
[[POTE generator]]: ~27/25 = 126.7237 cents | |||
{{Val list|legend=1| 19, 85c, 104c, 123, 142, 161 }} | |||
[[Badness]]: 0.653871 | |||
[[Category:19edo]] | [[Category:19edo]] | ||
Revision as of 08:10, 22 May 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)
- 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
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
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
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
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