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Syntonic–kleismic equivalence continuum: Difference between revisions

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{{Technical data page}}
{{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 }}).
The '''syntonic–kleismic equivalence continuum''' (or '''syntonic–enneadecal equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|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 {{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.
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 [[support]]ed by [[19edo]] (due to it being the unique equal temperament that tempers out both commas and thus tempers out 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]] ({{monzo| -14 -19 19 }}). That is, {{nowrap|(81/80)<sup>''k''</sup> ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap|''k'' {{=}} 3''n'' &minus; 19}}.
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"
|+ style="font-size: 105%;" | Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
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|-
|-
| 0
| 0
| 19 &amp; 19c
| 19 & 19c
| [[19-comma|1162261467/1073741824]]
| [[19-comma|1162261467/1073741824]]
| {{monzo|-30 19}}
| {{Monzo| -30 19 }}
|-
|-
| 1
| 1
| 7c & 12c
| 7c & 12c
| [[71744535/67108864]]
| [[71744535/67108864]]
| {{monzo|-26 15 1}}
| {{Monzo| -26 15 1 }}
|-
|-
| 2
| 2
| [[High badness temperaments #Hogzilla|Hogzilla]]
| [[Hogzilla]]
| [[4428675/4194304]]
| [[4428675/4194304]]
| {{monzo|-22 11 2}}
| {{monzo|-22 11 2}}
|-
|-
| 3
| 3
| [[High badness temperaments #Stump|Stump]]
| [[Stump]]
| [[273375/262144]]
| [[273375/262144]]
| {{monzo|-18 7 3}}
| {{Monzo| -18 7 3 }}
|-
|-
| 4
| 4
| [[Negri]]
| [[Negri]]
| [[16875/16384]]
| [[16875/16384]]
| {{monzo|-14 3 4}}
| {{Monzo| -14 3 4 }}
|-
|-
| 5
| 5
| [[Magic]]
| [[Magic]]
| [[3125/3072]]
| [[3125/3072]]
| {{monzo|-10 -1 5}}
| {{Monzo| -10 -1 5 }}
|-
|-
| 6
| 6
| [[Hanson]]
| [[Hanson]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo|-6 -5 6}}
| {{Monzo| -6 -5 6 }}
|-
|-
| 7
| 7
| [[Sensipent family#Sensipent|Sensipent]]
| [[Sensipent]]
| [[78732/78125]]
| [[78732/78125]]
| {{monzo|2 9 -7}}
| {{Monzo| 2 9 -7 }}
|-
|-
| 8
| 8
| [[Unicorn]]
| [[Unicorn]]
| [[1594323/1562500]]
| [[1594323/1562500]]
| {{monzo|-2 13 -8}}
| {{Monzo| -2 13 -8 }}
|-
|-
| 9
| 9
| 19 &amp; 51c
| 19 & 51c
| [[129140163/125000000]]
| [[129140163/125000000]]
| {{monzo|-6 17 -9}}
| {{Monzo| -6 17 -9 }}
|-
|-
| …
| …
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| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1}}
| {{Monzo| -4 4 -1 }}
|}
|}


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{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~6/5


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{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
: mapping generators: ~2, ~16/15
: mapping generators: ~2, ~16/15


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[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -20 -24 25 }} = 298023223876953125/296148833645101056
[[Comma list]]: {{monzo| -20 -24 25 }}


[[Mapping]]: [{{val| 1 -5 -4 }}, {{val| 0 25 2 4}}]
{{Mapping|legend=1| 1 -5 -4 | 0 25 2 4}}


[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081
[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081{{c}}


{{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 }}
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[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 10 23 -20 }} = 96402615118848/95367431640625
[[Comma list]]: {{monzo| 10 23 -20 }}


[[Mapping]]: [{{val| 1 10 12 }}, {{val| 0 -20 -23 }}]
{{Mapping|legend=1| 1 10 12 | 0 -20 -23 }}


[[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913
[[Optimal tuning]] ([[POTE]]): 1200.000{{c}}, ~104976/78125 = 504.913{{c}}


{{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 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]] and [[POTE]]: ~2 = 1200.000, ~27/25 = 126.724
* [[CTE]] and [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 126.724{{c}}


{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}
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[[Badness]] (Sintel): 15.3
[[Badness]] (Sintel): 15.3


== Lalasepyo (8c &amp; 11) ==
== Lalasepyo (8c & 11) ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -32 10 7 }} = 4613203125/4294967296
[[Comma list]]: 4613203125/4294967296


[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}]
{{Mapping|legend=1| 1 -1 6 | 0 7 -10 }}


[[POTE generator]]: ~675/512 = 442.2674 cents
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~675/512 = 442.2674{{c}}


{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
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[[Comma list]]: {{monzo| -134 -185 184 }}
[[Comma list]]: {{monzo| -134 -185 184 }}


[[Mapping]]: [{{val| 1 50 51 }}, {{val| 0 -184 -185 }}]
{{Mapping|legend=1| 1 50 51 | 0 -184 -185 }}


[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 315.7501{{c}}


{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}
{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}
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