Quince clan: Difference between revisions

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== Mercy ==

Mercy is the no-3 version ~~of miracle. It can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup~~ of [[~~31edo~~]]. Two generators make an [[8/7]]; five generators make a [[7/5]].

Mercy is the no-3 version of [[miracle]]. Two generators make [[~]][[8/7]], five generators make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph.

[[Subgroup]]: 2.5.7

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* ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]

* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]

* ''[[Slendi]]'' (+4000/3969) → [[Octagar temperaments #Slendi|Octagar temperaments]]

* ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]

* ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]

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== Countermiracle ==

The ''countermiracle'' temperament ({{nowrap| 31 & ~~145~~ }}~~) tempers out~~ the ~~trimyna, 50421~~/~~50000 and~~ the [[~~quince comma~~]]~~, 823543/819200~~.

The countermiracle temperament tempers out the porwell comma ([[6144/6125]]), the trimyna comma ([[50421/50000]]) and the quince comma ([[823543/819200]]). It can be described as the {{nowrap| 31 & 176 }} temperament, and finds the [[3/2|perfect fifth]] -25 generators away (normalized for 2.5.7) as opposed to miracle's +6 generators. It has the [[ploidacot]] signature of 22-sheared 25-cot (or omega-heptaseph).

~~{{Todo|inline=1|expand|comment=Explain relationship to~~ miracle. ~~There's clearly ''some'' relationship, since~~ the ~~extensions are named~~ the ~~same way~~, ~~but~~ the ~~nature of that relationship isn't obvious from looking at this page~~.}}

Like miracle, it is naturally an [[11-limit]] temperament with the generator representing [[77/72]], but here the generator does not represent [[15/14]] or [[16/15]]. [[176edo]] makes for a recommendable tuning for both the 7- and 11-limit.

Countermiracle extends less naturally to the [[13-limit|13-]] and [[17-limit]], with countermiraculous (31 & 145), countermanna (145 & 176), and counterbenediction (176 & 207) being possible candidates, each favoring a tuning range shown by the [[temperament merging|equal-temperament joins]].

[[Subgroup]]: 2.3.5.7

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Badness (Sintel): 2.08

~~=== Counterrevelation ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 121/120, 176/175, 50421/50000~~

~~Mapping: {{mapping| 1 -21 -4 1 -11 | 0 25 7 2 16 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.1897{{c}}, ~275/147 = 1084.2521{{c}} (~343/320 = 115.9375{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~275/147 = 1084.0850{{c}} (~343/320 = 115.9150{{c}})

~~{{Optimal ET sequence|legend=0| 31, 114, 145e }}~~

~~Badness (Sintel): 2.12~~

~~==== 13-limit ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 121/120, 176/175, 196/195, 13750/13689~~

~~Mapping: {{mapping| 1 -21 -4 1 -11 29 | 0 25 7 2 16 -28 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.7651{{c}}, ~220/117 = 1083.9253{{c}} (~117/110 = 115.8398{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~220/117 = 1084.1357{{c}} (~117/110 = 115.8643{{c}})

~~{{Optimal ET sequence|legend=0| 31, 83, 114 }}~~

~~Badness (Sintel): 2.38~~

~~==== 17-limit ====~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625~~

~~Mapping: {{mapping| 1 -21 -4 1 -11 29 33 | 0 25 7 2 16 -28 -32 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.6935{{c}}, ~170/91 = 1083.8704{{c}} (~91/85 = 115.8231{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~170/91 = 1084.1464{{c}} (~91/85 = 115.8536{{c}})

~~{{Optimal ET sequence|legend=0| 31, 83, 114 }}~~

~~Badness (Sintel): 2.24~~

== Subgroup extensions ==

=== Mercy (2.5.7.13) ===

This extension may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].

Subgroup: 2.5.7.13

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 == Mercy ==
-Mercy is the no-3 version of miracle. It can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]]. Two generators make an [[8/7]]; five generators make a [[7/5]].
+Mercy is the no-3 version of [[miracle]]. Two generators make [[~]][[8/7]], five generators make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph.
 [[Subgroup]]: 2.5.7
@@ Line 28: / Line 28: @@
 * ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]
 * [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
+* ''[[Slendi]]'' (+4000/3969) → [[Octagar temperaments #Slendi|Octagar temperaments]]
 * ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]
 * ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]
@@ Line 36: / Line 37: @@
 == Countermiracle ==
-The ''countermiracle'' temperament ({{nowrap| 31 & 145 }}) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.
+The countermiracle temperament tempers out the porwell comma ([[6144/6125]]), the trimyna comma ([[50421/50000]]) and the quince comma ([[823543/819200]]). It can be described as the {{nowrap| 31 & 176 }} temperament, and finds the [[3/2|perfect fifth]] -25 generators away (normalized for 2.5.7) as opposed to miracle's +6 generators. It has the [[ploidacot]] signature of 22-sheared 25-cot (or omega-heptaseph).
-{{Todo|inline=1|expand|comment=Explain relationship to miracle. There's clearly ''some'' relationship, since the extensions are named the same way, but the nature of that relationship isn't obvious from looking at this page.}}
+Like miracle, it is naturally an [[11-limit]] temperament with the generator representing [[77/72]], but here the generator does not represent [[15/14]] or [[16/15]]. [[176edo]] makes for a recommendable tuning for both the 7- and 11-limit.
+Countermiracle extends less naturally to the [[13-limit|13-]] and [[17-limit]], with countermiraculous (31 & 145), countermanna (145 & 176), and counterbenediction (176 & 207) being possible candidates, each favoring a tuning range shown by the [[temperament merging|equal-temperament joins]].
 [[Subgroup]]: 2.3.5.7
@@ Line 161: / Line 164: @@
 Badness (Sintel): 2.08
-=== Counterrevelation ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 176/175, 50421/50000
-Mapping: {{mapping| 1 -21 -4 1 -11 | 0 25 7 2 16 }}
-Optimal tunings:
-* WE: ~2 = 1200.1897{{c}}, ~275/147 = 1084.2521{{c}} (~343/320 = 115.9375{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~275/147 = 1084.0850{{c}} (~343/320 = 115.9150{{c}})
-{{Optimal ET sequence|legend=0| 31, 114, 145e }}
-Badness (Sintel): 2.12
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 176/175, 196/195, 13750/13689
-Mapping: {{mapping| 1 -21 -4 1 -11 29 | 0 25 7 2 16 -28 }}
-Optimal tunings:
-* WE: ~2 = 1199.7651{{c}}, ~220/117 = 1083.9253{{c}} (~117/110 = 115.8398{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~220/117 = 1084.1357{{c}} (~117/110 = 115.8643{{c}})
-{{Optimal ET sequence|legend=0| 31, 83, 114 }}
-Badness (Sintel): 2.38
-==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
-Mapping: {{mapping| 1 -21 -4 1 -11 29 33 | 0 25 7 2 16 -28 -32 }}
-Optimal tunings:
-* WE: ~2 = 1199.6935{{c}}, ~170/91 = 1083.8704{{c}} (~91/85 = 115.8231{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~170/91 = 1084.1464{{c}} (~91/85 = 115.8536{{c}})
-{{Optimal ET sequence|legend=0| 31, 83, 114 }}
-Badness (Sintel): 2.24
 == Subgroup extensions ==
 === Mercy (2.5.7.13) ===
+This extension may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].
 Subgroup: 2.5.7.13