Syntonic–kleismic equivalence continuum: Difference between revisions
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The '''syntonic | {{Technical data page}} | ||
The '''syntonic–kleismic equivalence continuum''' (or '''syntonic–enneadecal equivalence continuum''') is a [[equivalence continuum|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 {{nowrap|(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, {{nowrap|(81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap|''k'' {{=}} 3''n'' − 19}}. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | '' | ! rowspan="2" | ''n'' | ||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
! colspan="2" | Comma | ! colspan="2" | Comma | ||
| Line 16: | Line 17: | ||
|- | |- | ||
| 0 | | 0 | ||
| 19 & 19c | | 19 & 19c | ||
| [[1162261467/1073741824]] | | [[19-comma|1162261467/1073741824]] | ||
| {{monzo|-30 19}} | | {{monzo|-30 19}} | ||
|- | |- | ||
| 1 | | 1 | ||
| | | 7c & 12c | ||
| [[71744535/67108864]] | | [[71744535/67108864]] | ||
| {{monzo|-26 15 1}} | | {{monzo|-26 15 1}} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[High badness temperaments#Hogzilla|Hogzilla]] | | [[High badness temperaments #Hogzilla|Hogzilla]] | ||
| [[4428675/4194304]] | | [[4428675/4194304]] | ||
| {{monzo|-22 11 2}} | | {{monzo|-22 11 2}} | ||
|- | |- | ||
| 3 | | 3 | ||
| [[High badness temperaments#Stump|Stump]] | | [[High badness temperaments #Stump|Stump]] | ||
| [[273375/262144]] | | [[273375/262144]] | ||
| {{monzo|-18 7 3}} | | {{monzo|-18 7 3}} | ||
| Line 51: | Line 52: | ||
|- | |- | ||
| 7 | | 7 | ||
| [[ | | [[Sensipent family#Sensipent|Sensipent]] | ||
| [[78732/78125]] | | [[78732/78125]] | ||
| {{monzo|2 9 -7}} | | {{monzo|2 9 -7}} | ||
| Line 61: | Line 62: | ||
|- | |- | ||
| 9 | | 9 | ||
| 19 & 51c | | 19 & 51c | ||
| [[129140163/125000000]] | | [[129140163/125000000]] | ||
| {{monzo|-6 17 -9}} | | {{monzo|-6 17 -9}} | ||
| Line 78: | Line 79: | ||
Examples of temperaments with fractional values of ''k'': | Examples of temperaments with fractional values of ''k'': | ||
{| class="wikitable" | |||
|+ style="font-size: 105%;" | Notable temperaments of fractional ''n'' | |||
|- | |||
! Temperament !! ''n'' !! Comma | |||
|- | |||
| [[Unsmate]] || 9/2 = 4.5 || {{monzo| -24 2 9 }} | |||
|- | |||
| [[Sycamore]] || 11/2 = 5.5 || {{monzo| -16 -6 11 }} | |||
|- | |||
| [[Counterhanson]] || 25/4 = 6.25 || {{monzo| -20 -24 25 }} | |||
|- | |||
| [[Enneadecal]] || 19/3 = 6.{{overline|3}} || {{monzo| -14 -19 19 }} | |||
|- | |||
| [[Egads]] || 51/8 = 6.375 || {{monzo| -36 -52 51 }} | |||
|- | |||
| [[Acrokleismic]] || 32/5 = 6.4 || {{monzo| 22 33 -32 }} | |||
|- | |||
| [[Parakleismic]] || 13/2 = 6.5 || {{monzo| 8 14 -13 }} | |||
|- | |||
| [[Countermeantone]] || 20/3 = 6.{{overline|6}} || {{monzo| 10 23 -20 }} | |||
|- | |||
| [[Mowgli]] || 15/2 = 7.5 || {{monzo| 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 {{nowrap| ''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|legend=1| 1 2 2 | 0 -4 3 }} | |||
: mapping generators: ~2, ~16/15 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1202.3403{{c}}, ~16/15 = 126.0002{{c}} | |||
: [[error map]]: {{val| +2.340 -1.275 -3.633 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 125.6610{{c}} | |||
: error map: {{val| 0.000 -4.599 -9.331 }} | |||
{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }} | |||
[[Badness]] (Sintel): 2.04 | |||
[ | == 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: 442.2674 cents | [[POTE generator]]: ~675/512 = 442.2674 cents | ||
{{Optimal ET sequence|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 | == Counterhanson == | ||
{{See also| Ragismic microtemperaments #Counterkleismic }} | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: {{monzo| -20 -24 25 }} = 298023223876953125/296148833645101056 | |||
[[Mapping]]: [{{val| 1 -5 -4 }}, {{val| 0 25 2 4}}] | |||
[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081 | |||
{{Optimal ET sequence|legend=1| 19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }} | |||
[[Badness]]: 0.317551 | |||
== Countermeantone == | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: {{monzo| 10 23 -20 }} = 96402615118848/95367431640625 | |||
[[Mapping]]: [{{val| 1 10 12 }}, {{val| 0 -20 -23 }}] | |||
[[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913 | |||
{{Optimal ET sequence|legend=1| 19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c }} | |||
[[Badness]]: 0.373477 | |||
== Mowgli == | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: {{monzo| 0 22 -15 }} | |||
[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 15 22 }}] | |||
[[Optimal tuning]] ([[POTE]]): ~27/25 = 126.7237 | |||
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161 }} | |||
[[Badness]]: 0.653871 | |||
== Oviminor == | |||
{{See also| Ragismic microtemperaments #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]]: {{monzo| -134 -185 184 }} | |||
[[Mapping]]: [{{val| 1 50 51 }}, {{val| 0 -184 -185 }}] | |||
[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501 | |||
{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }} | |||
[ | [[Badness]]: 32.0 | ||
[[Category:19edo]] | [[Category:19edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Latest revision as of 09:18, 14 July 2025
- 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
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: 1.061630
The temperament finder - 5-limit 19 & 8c
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: 0.317551
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: 0.373477
Mowgli
Subgroup: 2.3.5
Comma list: [0 22 -15⟩
Mapping: [⟨1 0 0], ⟨0 15 22]]
Optimal tuning (POTE): ~27/25 = 126.7237
Optimal ET sequence: 19, 85c, 104c, 123, 142, 161
Badness: 0.653871
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: 32.0