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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 used to 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
Line 16: Line 17:
|-
|-
| 0
| 0
| 19edo
| 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
| Lala-Yoyo
| [[High badness temperaments #Hogzilla|Hogzilla]]
| [[4428675/4194304]]
| [[4428675/4194304]]
| {{monzo|-22 11 2}}
| {{monzo|-22 11 2}}
|-
|-
| 3
| 3
| Latriyo
| [[High badness temperaments #Stump|Stump]]
| [[273375/262144]]
| [[273375/262144]]
| {{monzo|-18 7 3}}
| {{monzo|-18 7 3}}
Line 51: Line 52:
|-
|-
| 7
| 7
| [[Sensi]]
| [[Sensipent family#Sensipent|Sensipent]]
| [[78732/78125]]
| [[78732/78125]]
| {{monzo|2 9 -7}}
| {{monzo|2 9 -7}}
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|-
|-
| 9
| 9
| 19 & 51c
| 19 &amp; 51c
| [[129140163/125000000]]
| [[129140163/125000000]]
| {{monzo|-6 17 -9}}
| {{monzo|-6 17 -9}}
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|}
|}


Examples of temperaments with fractional values of ''n'':
Examples of temperaments with fractional values of ''k'':


* [[Enneadecal]] (''k'' = 19/3 = 6.{{overline|3}})
{| class="wikitable"
* 19 & 506 (''k'' = 58/9 = 6.{{overline|4}})
|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''
* [[Parakleismic]] (''k'' = 6.5)
|-
* [[Countermeantone]] (''k'' = 20/3 = 6.{{overline|6}})
! 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 &amp; 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]]: 1.061630
 
[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c]
 
== 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.


== 19 & 506 ==
[[Subgroup]]: 2.3.5


Commas: {{Monzo|38 61 -58}}
[[Comma list]]: {{monzo| -134 -185 184 }}


POTE generator: 505.1394 cents
[[Mapping]]: [{{val| 1 50 51 }}, {{val| 0 -184 -185 }}]


Map: [&lt;1 26 28|, &lt;0 -58 -61|]
[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.7501


EDOs: {{EDOs| 19, 38, 57, 468, 487, 506, 525, 544, 1012, 1031 }}
{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=19_506&limit=5 The temperament finder - 5-limit 19 & 506]
[[Badness]]: 32.0


[[Category:19edo]]
[[Category:19edo]]
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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