Quince clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit

FloraC (talk | contribs)

4 user groups
27,269 edits
VisualWikitext
Re-organize
m Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "
Tags: Mobile edit Mobile web edit
 
(11 intermediate revisions by 2 users not shown)
Line 3: Line 3:


== Mercy ==
== 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
[[Subgroup]]: 2.5.7
Line 9: Line 9:
[[Comma list]]: 823543/819200
[[Comma list]]: 823543/819200


{{Mapping|legend=2| 1 3 3 | 0 -7 -2 }}
{{Mapping|legend=3| 1 3 3 | 0 -7 -2 }}


{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}
: mapping generators: ~2, ~343/320
: mapping generators: ~2, ~343/320


Line 28: Line 28:
* ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]
* ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Slendi]]'' (+4000/3969) → [[Octagar temperaments #Slendi|Octagar temperaments]]
* ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]
* ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]
* ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]
* ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]
Line 36: Line 37:


== Countermiracle ==
== 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
[[Subgroup]]: 2.3.5.7
Line 62: Line 65:
Comma list: 441/440, 3388/3375, 6144/6125
Comma list: 441/440, 3388/3375, 6144/6125


Mapping: {{mapping| 1 -21 -4 1 -39 | 0 25 7 2 47 }}
{{Mapping|legend=0| 1 -21 -4 1 -39 | 0 25 7 2 47 }}


Optimal tunings:  
Optimal tunings:  
Line 77: Line 80:
Comma list: 196/195, 352/351, 1001/1000, 6144/6125
Comma list: 196/195, 352/351, 1001/1000, 6144/6125


Mapping: {{mapping| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}
{{Mapping|legend=0| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}


Optimal tunings:  
Optimal tunings:  
Line 92: Line 95:
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224


Mapping: {{mapping| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}
{{Mapping|legend=0| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}


Optimal tunings:  
Optimal tunings:  
Line 107: Line 110:
Comma list: 351/350, 441/440, 3146/3125, 3584/3575
Comma list: 351/350, 441/440, 3146/3125, 3584/3575


Mapping: {{mapping| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}
{{Mapping|legend=0| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}


Optimal tunings:  
Optimal tunings:  
Line 122: Line 125:
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125


Mapping: {{mapping| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}
{{Mapping|legend=0| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}


Optimal tunings:  
Optimal tunings:  
Line 137: Line 140:
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
Comma list: 364/363, 441/440, 3388/3375, 6144/6125


Mapping: {{mapping| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}
{{Mapping|legend=0| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}


Optimal tunings:  
Optimal tunings:  
Line 152: Line 155:
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375


Mapping: {{mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
{{Mapping|legend=0| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}


Optimal tunings:  
Optimal tunings:  
Line 161: Line 164:


Badness (Sintel): 2.08
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 ==
== Subgroup extensions ==
=== Mercy (2.5.7.13) ===
=== 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
Subgroup: 2.5.7.13


Comma list: 343/338, 640/637
Comma list: 343/338, 640/637


Subgroup-val mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
{{Mapping|legend=2| 1 3 3 4 | 0 -7 -2 -3 }}


Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
{{Mapping|legend=4| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 230: Line 190:
Comma list: 170/169, 224/221, 343/338
Comma list: 170/169, 224/221, 343/338


Subgroup-val mapping: {{mapping| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
{{Mapping|legend=2| 1 3 3 4 4 | 0 -7 -2 -3 1 }}


Gencom mapping: {{mapping| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
{{Mapping|legend=4| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}


Optimal tunings:  
Optimal tunings:  
Line 247: Line 207:
Comma list: 170/169, 224/221, 343/338, 476/475
Comma list: 170/169, 224/221, 343/338, 476/475


Subgroup-val mapping: {{mapping| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
{{Mapping|legend=2| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}


Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
{{Mapping|legend=4| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}


Optimal tunings:  
Optimal tunings:  
Line 259: Line 219:
Badness (Sintel): 0.759
Badness (Sintel): 0.759


[[Category:Quince clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Quince clan| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 17:44, 30 September 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The quince clan tempers out the quince comma (monzo: [-15 0 -2 7⟩, ratio: 823543/819200).

Mercy

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

Comma list: 823543/819200

Subgroup-val mapping: [⟨1 3 3], ⟨0 -7 -2]]

Gencom mapping: [⟨1 0 3 3], ⟨0 0 -7 -2]]

mapping generators: ~2, ~343/320

Optimal tunings:

  • WE: ~2 = 1200.2169 ¢, ~343/320 = 116.3116 ¢
error map: ⟨+0.217 +0.156 -0.798]
  • CWE: ~2 = 1200.0000 ¢, ~343/320 = 116.2576 ¢
error map: ⟨0.000 -0.117 -1.341]

Optimal ET sequence: 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd

Badness (Sintel): 0.428

Overview to extensions

Temperaments discussed elsewhere are:

Discussed below is countermiracle.

Countermiracle

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 31 & 176 temperament, and finds the 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).

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- 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 equal-temperament joins.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping: [⟨1 -21 -4 1], ⟨0 25 7 2]]

mapping generators: ~2, ~625/336

Optimal tunings:

  • WE: ~2 = 1199.7944 ¢, ~625/336 = 1083.8973 ¢ (~343/320 = 115.8970 ¢)
error map: ⟨-0.206 -0.203 +1.790 -1.237]
  • CWE: ~2 = 1200.0000 ¢, ~625/336 = 1084.0787 ¢ (~343/320 = 115.9213 ¢)
error map: ⟨0.000 +0.013 +2.237 -0.669]

Optimal ET sequence: 31, 114, 145, 176

Badness (Sintel): 2.59

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 -21 -4 1 -39], ⟨0 25 7 2 47]]

Optimal tunings:

  • WE: ~2 = 1199.7942 ¢, ~144/77 = 1083.8983 ¢ (~77/72 = 115.8959 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0798 ¢ (~77/72 = 115.9202 ¢)

Optimal ET sequence: 31, 114e, 145, 176

Badness (Sintel): 1.29

Countermiraculous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 1001/1000, 6144/6125

Mapping: [⟨1 -21 -4 1 -39 29], ⟨0 25 7 2 47 -28]]

Optimal tunings:

  • WE: ~2 = 1199.4465 ¢, ~144/77 = 1083.6196 ¢ (~77/72 = 115.8268 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1148 ¢ (~77/72 = 115.8852 ¢)

Optimal ET sequence: 31, 83e, 114e, 145, 321ceff

Badness (Sintel): 1.62

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224

Mapping: [⟨1 -21 -4 1 -39 29 33], ⟨0 25 7 2 47 -28 -32]]

Optimal tunings:

  • WE: ~2 = 1199.4013 ¢, ~144/77 = 1083.5835 ¢ (~77/72 = 115.8178 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1219 ¢ (~77/72 = 115.8781 ¢)

Optimal ET sequence: 31, 83e, 114e, 145

Badness (Sintel): 1.50

Counterbenediction

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 3146/3125, 3584/3575

Mapping: [⟨1 -21 -4 1 -39 29 57], ⟨0 25 7 2 47 -59]]

Optimal tunings:

  • WE: ~2 = 1199.9460 ¢, ~144/77 = 1084.0178 ¢ (~77/72 = 115.9283 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0663 ¢ (~77/72 = 115.9337 ¢)

Optimal ET sequence: 31, 145f, 176, 207

Badness (Sintel): 1.88

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125

Mapping: [⟨1 -21 -4 1 -39 29 57 61], ⟨0 25 7 2 47 -59 -63]]

Optimal tunings:

  • WE: ~2 = 1199.9950 ¢, ~144/77 = 1084.0564 ¢ (~77/72 = 115.9386 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0609 ¢ (~77/72 = 115.9391 ¢)

Optimal ET sequence: 31, 176, 207

Badness (Sintel): 1.85

Countermanna

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 -21 -4 1 -39 -102], ⟨0 25 7 2 47 117]]

Optimal tunings:

  • WE: ~2 = 1199.6687 ¢, ~144/77 = 1083.8108 ¢ (~77/72 = 115.8578 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1065 ¢ (~77/72 = 115.8935 ¢)

Optimal ET sequence: 31f, 145, 176, 321ce

Badness (Sintel): 2.21

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375

Mapping: [⟨1 -21 -4 1 -39 -102 -98], ⟨0 25 7 2 47 117 113]]

Optimal tunings:

  • WE: ~2 = 1199.6458 ¢, ~144/77 = 1083.7968 ¢ (~77/72 = 115.8490 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1132 ¢ (~77/72 = 115.8868 ¢)

Optimal ET sequence: 31fg, 145, 321ce

Badness (Sintel): 2.08

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

Comma list: 343/338, 640/637

Subgroup-val mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]

Gencom mapping: [⟨1 0 3 3 0 4], ⟨0 0 -7 -2 0 -3]]

Optimal tunings:

  • WE: ~2 = 1199.2166 ¢, ~14/13 = 116.0181 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 116.2099 ¢

Optimal ET sequence: 10, 21, 31

Badness (Sintel): 0.546

2.5.7.13.17 subgroup

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 343/338

Subgroup-val mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]

Gencom mapping: [⟨1 0 3 3 0 4 4], ⟨0 0 -7 -2 0 -3 1]]

Optimal tunings:

  • WE: ~2 = 1198.5476 ¢, ~14/13 = 115.6294 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.9132 ¢

Optimal ET sequence: 10, 21, 31, 83fg

Badness (Sintel): 0.451

2.5.7.13.17.19 subgroup

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 224/221, 343/338, 476/475

Subgroup-val mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]

Gencom mapping: [⟨1 0 3 3 0 4 4 3], ⟨0 0 -7 -2 0 -3 1 13]]

Optimal tunings:

  • WE: ~2 = 1198.4839 ¢, ~14/13 = 115.5700 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.7299 ¢

Optimal ET sequence: 10h, 21, 31, 52f, 83fg

Badness (Sintel): 0.759

Retrieved from "https://en.xen.wiki/index.php?title=Quince_clan&oldid=238304"