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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}}).
{{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>''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 {{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]] ({{Monzo|-14 -19 19}}). That is, (81/80)<sup>''n''</sup> ~ {{monzo|-14 -19 19}}. In this case, ''n'' = 3''k'' - 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'' &minus; 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" | ''k''
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
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| 0
| 0
| 19 &amp; 19c
| 19 &amp; 19c
| [[1162261467/1073741824]]
| [[19-comma|1162261467/1073741824]]
| {{monzo|-30 19}}
| {{monzo|-30 19}}
|-
|-
| 1
| 1
| Lalayo
| 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}}
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|-
|-
| 7
| 7
| [[Sensi]]
| [[Sensipent family#Sensipent|Sensipent]]
| [[78732/78125]]
| [[78732/78125]]
| {{monzo|2 9 -7}}
| {{monzo|2 9 -7}}
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Examples of temperaments with fractional values of ''k'':
Examples of temperaments with fractional values of ''k'':


* 19 &amp; 8c (''k'' = 3.5)
{| class="wikitable"
* [[High badness temperaments#Unsmate|Unsmate]] (''k'' = 4.5)
|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''
* [[Sycamore family#Sycamore|Sycamore]] (''k'' = 5.5)
|-
* [[Counterhanson]] (''k'' = 25/4 = 6.25)
! Temperament !! ''n'' !! Comma
* [[Enneadecal]] (''k'' = 19/3 = 6.{{overline|3}})
|-
* [[Very high accuracy temperaments#Egads|Egads]] (''k'' = 51/8 = 6.375)
| [[Unsmate]] || 9/2 = 4.5 || {{monzo| -24 2 9 }}
* [[Acrokleismic]] (''k'' = 32/5 = 6.4)
|-
* 19 &amp; 506 (''k'' = 58/9 = 6.{{overline|4}})
| [[Sycamore]] || 11/2 = 5.5 || {{monzo| -16 -6 11 }}
* [[Parakleismic]] (''k'' = 6.5)
|-
* [[Countermeantone]] (''k'' = 20/3 = 6.{{overline|6}})
| [[Counterhanson]] || 25/4 = 6.25 || {{monzo| -20 -24 25 }}
* [[Mowgli]] (''k'' = 7.5)
|-
| [[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 }}


== Lalayo ==
: mapping generators: ~2, ~16/15
[[Comma list]]: {{Monzo|-26 15 1}} = 71744535/67108864


[[Mapping]]: [{{val| 1 2 -4 }}, {{val| 0 -1 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 }}


[[POTE generator]]: ~4/3 = 505.348 cents
{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }}


{{Val list|legend=1| 7c, 12c, 19 }}
[[Badness]] (Sintel): 2.04


[[Badness]]: 0.803397
== Lalasepyo (8c &amp; 11) ==
[[Subgroup]]: 2.3.5


== 8c &amp; 19 ==
[[Comma list]]: {{monzo| -32 10 7 }} = 4613203125/4294967296
[[Comma list]]: {{Monzo|-32 10 7}} = 4613203125/4294967296


[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}]
[[Mapping]]: [{{val| 1 -1 6 }}, {{val| 0 7 -10 }}]
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[[POTE generator]]: ~675/512 = 442.2674 cents
[[POTE generator]]: ~675/512 = 442.2674 cents


{{Val list|legend=1| 8c, 11, 19 }}
{{Optimal ET sequence|legend=1| 8c, 11, 19 }}


[[Badness]]: 1.061630
[[Badness]]: 1.061630
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[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 &amp; 506 ==
== Counterhanson ==
[[Comma list]]: {{Monzo| 38 61 -58 }}
{{See also| Ragismic microtemperaments #Counterkleismic }}
 
[[Mapping]]: [{{val| 1 26 28 }}, {{val| 0 -58 -61 }}]
 
[[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]
[[Subgroup]]: 2.3.5


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


[[Mapping]]: [{{val|1 -5 -4}}, {{val|0 25 24}}]
[[Mapping]]: [{{val| 1 -5 -4 }}, {{val| 0 25 2 4}}]


[[POTE generator]]: ~6/5 = 316.081 cents
[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.081


{{Val list|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]]: 0.317551
[[Badness]]: 0.317551


== Countermeantone ==
== Countermeantone ==
[[Comma list]]: {{Monzo|10 23 -20}} = 96402615118848/95367431640625
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 10 23 -20 }} = 96402615118848/95367431640625


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


[[POTE generator]]: ~104976/78125 = 504.913 cents
[[Optimal tuning]] ([[POTE]]): ~104976/78125 = 504.913


{{Val list|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]]: 0.373477
[[Badness]]: 0.373477


== Mowgli ==
== Mowgli ==
[[Comma list]]: {{Monzo|0 22 -15}}
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 0 22 -15 }}


[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 15 22 }}]
[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 15 22 }}]


[[POTE generator]]: ~27/25 = 126.7237 cents
[[Optimal tuning]] ([[POTE]]): ~27/25 = 126.7237


{{Val list|legend=1| 19, 85c, 104c, 123, 142, 161 }}
{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161 }}


[[Badness]]: 0.653871
[[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:Theory]]
[[Category:Temperament]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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