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Syntonic–kleismic equivalence continuum: Difference between revisions

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|-
|-
| 0
| 0
| 19 & 19c
| [[Graywood]]
| [[19-comma|1162261467/1073741824]]
| [[19-comma|1162261467/1073741824]]
| {{Monzo| -30 19 }}
| {{Monzo| -30 19 }}
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|-
|-
| 9
| 9
| 19 & 51c
| [[Xenial]]
| [[129140163/125000000]]
| [[129140163/125000000]]
| {{Monzo| -6 17 -9 }}
| {{Monzo| -6 17 -9 }}
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|}
|}


== Parakleismic (5-limit) ==
== Graywood ==
{{Main| Parakleismic }}
Named by [[CompactStar]] in 2024, graywood tempers out the [[19-comma]], corresponding to {{nowrap| ''n'' {{=}} 0 }}. It takes [[19edo]]'s closed [[circle of fifths]], but adds an independent generator for [[prime interval|prime]] [[5/1|5]]. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. [[38edo|38c-edo]] or [[57edo|57c-edo]], are not nearly as accurate as 19 itself.
: ''For extensions, see [[Ragismic microtemperaments #Parakleismic]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 1162261467/1073741824
 
{{Mapping|legend=1| 19 30 0 | 0 0 1 }}
: mapping generators: ~2187/2048, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2187/2048 = 63.2773{{c}}, ~5/4 = 381.7568{{c}}
: [[error map]]: {{val| +2.268 -3.637 -0.020 }}
* [[CWE]]: ~2187/2048 = 63.1579{{c}}, ~5/4 = 382.7889{{c}}
: error map: {{val| 0.000 -7.218 -3.525 }}
 
{{Optimal ET sequence|legend=1| 19 }}
 
[[Badness]] (Sintel): 32.4
 
== Hogzilla ==
: ''For extensions, see [[Semaphoresmic clan #Helayo]].''
 
Hogzilla is similar to [[godzilla]] in that it is generated by a [[semitwelfth]]. It corresponds to {{nowrap| ''n'' {{=}} 2 }}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 4428675/4194304
 
{{Mapping|legend=1| 1 0 11 | 0 2 -11 }}
: mapping generators: ~2, ~2048/1215
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5490{{c}}, ~2048/1215 = 949.3637{{c}}
: [[error map]]: {{val| +2.549 -3.228 -1.275 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2048/1215 = 947.2462{{c}}
: error map: {{val| 0.000 -7.463 -6.022 }}
 
{{Optimal ET sequence|legend=1| 14, 19 }}
 
[[Badness]] (Sintel): 9.96
 
== Stump ==
: ''For extensions, see [[Marvel temperaments #Triton]] and [[Sensamagic clan #Pycnic]].''


The 5-limit version of parakleismic tempers out the [[parakleisma]]. It corresponds to {{nowrap| ''n'' {{=}} 13/2 }}, and 13 generator steps give the interval class of [[3/1|3]].  
Stump splits the [[3/1|3rd]] [[harmonic]] into three equal parts, each for [[~]][[64/45]]. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.  


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


[[Comma list]]: 1224440064/1220703125
[[Comma list]]: 273375/262144


{{Mapping|legend=1| 1 -8 -8 | 0 13 14 }}
{{Mapping|legend=1| 1 0 6 | 0 3 -7 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.971{{c}}, ~5/3 = 884.7383{{c}}
* [[WE]]: ~2 = 1202.6288{{c}}, ~64/45 = 633.1214{{c}}
: [[error map]]: {{val| -0.029 -0.127 +0.253 }}
: [[error map]]: {{val| +2.629 -2.591 -2.391 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/3 = 884.8576{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 631.6779{{c}}
: error map: {{val| 0.000 -0.106 +0.293 }}
: error map: {{val| 0.000 -6.921 -8.059 }}


{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
{{Optimal ET sequence|legend=1| 17, 19, 207bbccc }}


[[Badness]] (Sintel): 1.02
[[Badness]] (Sintel): 4.71


== Negri ==
== Negri (5-limit) ==
{{Main| Negri }}
: ''For extensions, see [[Semaphoresmic clan #Negri]].''
: ''For extensions, see [[Semaphoresmic clan #Negri]].''


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[[Badness]] (Sintel): 2.04
[[Badness]] (Sintel): 2.04


== Counterhanson ==
== Xenial ==
: ''For extensions, see [[Ragismic microtemperaments #Counterkleismic]].''
: ''For extensions, see [[Starling temperaments #Xenial]] and [[Sensamagic clan #Xenia]].''
 
Named by [[User:Xenllium|Xenllium]] in 2026, xenial splits the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. It corresponds to {{nowrap| ''n'' {{=}} 9 }}. Its [[ploidacot]] is zeta-enneacot, and from this it derives its name.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 129140163/125000000
 
{{Mapping|legend=1| 1 -6 -12 | 0 9 17 }}
: mapping generators: ~2, ~9/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2802{{c}}, ~9/5 = 1011.2914{{c}}
: [[error map]]: {{val| +0.280 -2.013 +2.278 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.0762{{c}}
: error map: {{val| 0.000 -2.269 +1.982 }}
 
{{Optimal ET sequence|legend=1| 19, 70, 89, 108, 127 }}
 
[[Badness]] (Sintel): 8.84
 
== Lalasepyo (8c & 11) ==
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 4613203125/4294967296
 
{{Mapping|legend=1| 1 -1 6 | 0 7 -10 }}
: mapping generators: ~2, ~675/512
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5641{{c}}, ~675/512 = 443.2124{{c}}
: [[error map]]: {{val| +2.564 -2.033 -3.053 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~675/512 = 442.2692{{c}}
: error map: {{val| 0.000 -6.071 -9.006 }}
 
{{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]


== Unsmate ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -20 -24 25 }}
[[Comma list]]: 17578125/16777216


{{Mapping|legend=1| 1 -5 -4 | 0 25 24 }}
{{Mapping|legend=1| 1 -6 4 | 0 9 -2 }}
: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~1875/1024


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0419{{c}}, ~6/5 = 316.0916{{c}}
* [[WE]]: ~2 = 1201.8892{{c}}, ~1875/1024 = 1012.5428{{c}}
: [[error map]]: {{val| +0.042 +0.126 -0.282 }}
: [[error map]]: {{val| +1.889 -0.405 -3.843 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0021{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1875/1024 = 1011.0348{{c}}
: error map: {{val| 0.000 +0.097 -0.344 }}
: error map: {{val| 0.000 -2.642 -8.383 }}
 
{{Optimal ET sequence|legend=1| 6b, 13, 19, 89c, 108c, 127c, 146cc }}
 
[[Badness]] (Sintel): 10.8


{{Optimal ET sequence|legend=1| 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}
== Parakleismic ==
{{Main| Parakleismic }}
: ''For extensions, see [[Ragismic microtemperaments #Parakleismic]] and [[Starling temperaments #Paraguay]].''


[[Badness]] (Sintel): 7.45
The 5-limit version of parakleismic tempers out the [[parakleisma]]. It corresponds to {{nowrap| ''n'' {{=}} 13/2 }}, and 13 generator steps give the interval class of [[3/1|3]].  


== Countermeantone ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 10 23 -20 }}
[[Comma list]]: 1224440064/1220703125


{{Mapping|legend=1| 1 10 12 | 0 -20 -23 }}
{{Mapping|legend=1| 1 -8 -8 | 0 13 14 }}
: mapping generators: ~2, ~78125/52488
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9478{{c}}, ~78125/52488 = 695.0566{{c}}
* [[WE]]: ~2 = 1199.971{{c}}, ~5/3 = 884.7383{{c}}
: [[error map]]: {{val| -0.052 -0.301 +0.562 }}
: [[error map]]: {{val| -0.029 -0.127 +0.253 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~78125/52488 = 695.0846{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/3 = 884.8576{{c}}
: error map: {{val| 0.000 -0.264 +0.631 }}
: error map: {{val| 0.000 -0.106 +0.293 }}


{{Optimal ET sequence|legend=1| 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c }}
{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}


[[Badness]] (Sintel): 8.76
[[Badness]] (Sintel): 1.02


== Mowgli ==
== Mowgli ==
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[[Badness]] (Sintel): 15.3
[[Badness]] (Sintel): 15.3


== Lalasepyo (8c & 11) ==
== Enneadecal (5-limit) ==
: ''For extensions, see [[Ragismic microtemperaments #Enneadecal]].''
 
The 5-limit version of enneadecal tempers out the [[enneadeca]], which simply equates a stack of nineteen [[6/5]] minor thirds with five [[2/1|octaves]]. It corresponds to {{nowrap| ''n'' {{=}} 19/3 }}, with a 19th-octave period and a generator of a [[3/2|perfect fifth]].
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 19073486328125/19042491875328
 
{{Mapping|legend=1| 19 0 14 | 0 1 1 }}
: mapping generators: ~648/625, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9861{{c}}
: [[error map]]: {{val| +0.013 +0.044 -0.095 }}
* [[CWE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9900{{c}}
: error map: {{val| 0.000 -0.035 -0.113 }}
 
{{Optimal ET sequence|legend=1| 19, 95, 114, 133, 152, 171, 323, 494, 665, 1159, 1824, 2983, 7125c }}
 
[[Badness]] (Sintel): 1.12
 
== Countermeantone ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 4613203125/4294967296
[[Comma list]]: {{monzo| 10 23 -20 }}


{{Mapping|legend=1| 1 -1 6 | 0 7 -10 }}
{{Mapping|legend=1| 1 -10 -11 | 0 20 23 }}
: mapping generators: ~2, ~675/512
: mapping generators: ~2, ~78125/52488


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.5641{{c}}, ~675/512 = 443.2124{{c}}
* [[WE]]: ~2 = 1199.9478{{c}}, ~78125/52488 = 695.0566{{c}}
: [[error map]]: {{val| +2.564 -2.033 -3.053 }}
: [[error map]]: {{val| -0.052 -0.301 +0.562 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~675/512 = 442.2692{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~78125/52488 = 695.0846{{c}}
: error map: {{val| 0.000 -6.071 -9.006 }}
: error map: {{val| 0.000 -0.264 +0.631 }}
 
{{Optimal ET sequence|legend=1| 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c }}
 
[[Badness]] (Sintel): 8.76
 
== Counterhanson ==
: ''For extensions, see [[Ragismic microtemperaments #Counterkleismic]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -20 -24 25 }}
 
{{Mapping|legend=1| 1 -5 -4 | 0 25 24 }}
: mapping generators: ~2, ~6/5


{{Optimal ET sequence|legend=1| 8c, 11, 19 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0419{{c}}, ~6/5 = 316.0916{{c}}
: [[error map]]: {{val| +0.042 +0.126 -0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0021{{c}}
: error map: {{val| 0.000 +0.097 -0.344 }}


[[Badness]] (Sintel): 24.9
{{Optimal ET sequence|legend=1| 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c]
[[Badness]] (Sintel): 7.45


== Oviminor ==
== Oviminor ==
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