Syntonic–kleismic equivalence continuum: Difference between revisions
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| Line 125: | Line 125: | ||
[[Badness]] (Sintel): 2.04 | [[Badness]] (Sintel): 2.04 | ||
== Counterhanson == | == Counterhanson == | ||
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{{Optimal ET sequence|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }} | {{Optimal ET sequence|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }} | ||
[[Badness]]: | [[Badness]] (Sintel): 7.45 | ||
== Countermeantone == | == Countermeantone == | ||
| Line 167: | Line 152: | ||
{{Optimal ET sequence|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }} | {{Optimal ET sequence|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }} | ||
[[Badness]]: | [[Badness]] (Sintel): 8.76 | ||
== Mowgli == | == Mowgli == | ||
: ''For extensions, see [[Hemimean clan #Mowglic]].'' | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: | [[Comma list]]: 31381059609/30517578125 | ||
{{Mapping|legend=1| 1 0 0 | 0 15 22 }} | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[CTE]] and [[POTE]]: ~2 = 1200.000, ~27/25 = 126.724 | |||
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161 }} | {{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 15.3 | ||
== Lalasepyo (8c & 11) == | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: {{monzo| -32 10 7 }} = 4613203125/4294967296 | |||
[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}] | |||
[[POTE generator]]: ~675/512 = 442.2674 cents | |||
{{Optimal ET sequence|legend=1| 8c, 11, 19 }} | |||
[[Badness]] (Sintel): 24.9 | |||
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c] | |||
== Oviminor == | == Oviminor == | ||
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{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }} | {{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 750.8 | ||
[[Category:19edo]] | [[Category:19edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 12:30, 6 February 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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 | 7c & 12c | 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 | Sensipent | 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:
| Temperament | n | Comma |
|---|---|---|
| Unsmate | 9/2 = 4.5 | [-24 2 9⟩ |
| Sycamore | 11/2 = 5.5 | [-16 -6 11⟩ |
| Counterhanson | 25/4 = 6.25 | [-20 -24 25⟩ |
| Enneadecal | 19/3 = 6.3 | [-14 -19 19⟩ |
| Egads | 51/8 = 6.375 | [-36 -52 51⟩ |
| Acrokleismic | 32/5 = 6.4 | [22 33 -32⟩ |
| Parakleismic | 13/2 = 6.5 | [8 14 -13⟩ |
| Countermeantone | 20/3 = 6.6 | [10 23 -20⟩ |
| Mowgli | 15/2 = 7.5 | [0 22 -15⟩ |
Negri (5-limit)
- For extensions, see Semaphoresmic clan #Negri.
The 5-limit version of negri tempers out the negri comma, spliting a perfect fourth into four ~16/15 generators. It corresponds to n = 4. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7.
Subgroup: 2.3.5
Comma list: 16875/16384
Mapping: [⟨1 2 2], ⟨0 -4 3]]
- mapping generators: ~2, ~16/15
- WE: ~2 = 1202.3403 ¢, ~16/15 = 126.0002 ¢
- error map: ⟨+2.340 -1.275 -3.633]
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 125.6610 ¢
- error map: ⟨0.000 -4.599 -9.331]
Optimal ET sequence: 9, 10, 19, 67c, 86c, 105c
Badness (Sintel): 2.04
Counterhanson
Subgroup: 2.3.5
Comma list: [-20 -24 25⟩ = 298023223876953125/296148833645101056
Mapping: [⟨1 -5 -4], ⟨0 25 2 4]]
Optimal tuning (POTE): ~6/5 = 316.081
Optimal ET sequence: 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c
Badness (Sintel): 7.45
Countermeantone
Subgroup: 2.3.5
Comma list: [10 23 -20⟩ = 96402615118848/95367431640625
Mapping: [⟨1 10 12], ⟨0 -20 -23]]
Optimal tuning (POTE): ~104976/78125 = 504.913
Optimal ET sequence: 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c
Badness (Sintel): 8.76
Mowgli
- For extensions, see Hemimean clan #Mowglic.
Subgroup: 2.3.5
Comma list: 31381059609/30517578125
Mapping: [⟨1 0 0], ⟨0 15 22]]
Optimal ET sequence: 19, 85c, 104c, 123, 142, 161, 303
Badness (Sintel): 15.3
Lalasepyo (8c & 11)
Subgroup: 2.3.5
Comma list: [-32 10 7⟩ = 4613203125/4294967296
Mapping: [⟨1 -1 6], ⟨0 7 -10]]
POTE generator: ~675/512 = 442.2674 cents
Optimal ET sequence: 8c, 11, 19
Badness (Sintel): 24.9
The temperament finder - 5-limit 19 & 8c
Oviminor
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.
Subgroup: 2.3.5
Comma list: [-134 -185 184⟩
Mapping: [⟨1 50 51], ⟨0 -184 -185]]
Optimal tuning (CTE): ~6/5 = 315.7501
Optimal ET sequence: 19, …, 1600, 3219, 4819
Badness (Sintel): 750.8