Quince clan: Difference between revisions

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added countermiracle, which is the alternative mapping of 3 for quince on the other side of 31edo
Sorting, units & misc. cleanup
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{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}


: gencom: [2 343/320; 823543/819200]
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~343/320 = 116.291{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 116.291


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}


==== 2.5.7.13 subgroup ====
== Countermiracle ==
Subgroup: 2.5.7.13
The ''countermiracle'' temperament ({{nowrap| 31 & 145 }}) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.
 
Comma list: 343/338, 640/637
 
Sval mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
 
: gencom: [2 14/13; 343/338 640/637]
 
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 116.094
 
{{Optimal ET sequence|legend=1| 10, 21, 31}}
 
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
 
Comma list: 170/169, 224/221, 640/637
 
Sval mapping: {{mapping| 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 }}
 
: gencom: [2 14/13; 170/169 224/221 640/637]
 
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 115.769
 
{{Optimal ET sequence|legend=1| 10, 21, 31 }}
 
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
 
Comma list: 170/169, 343/338, 640/637, 16384/16055
 
Sval mapping: {{mapping| 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 }}
 
: gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]
 
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 115.716
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}


== Countermiracle ==
{{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.}}
The ''countermiracle'' temperament (31 & 145) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.
{{todo|inline=1|explain relationship to miracle|comment=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.}}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 81: Line 35:


{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}
{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}
: mapping generators: ~2, ~343/320


: Mapping generators: ~2, ~343/320
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~343/320 = 115.9169{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 115.9169


{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}


[[Badness]]: 0.102326
[[Badness]] (Smith): 0.102326


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 4 3 3 8 | 0 -25 -7 -2 -47 }}
Mapping: {{mapping| 1 4 3 3 8 | 0 -25 -7 -2 -47 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.9158{{c}}


{{Optimal ET sequence|legend=1| 31, 114e, 145, 176 }}
{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}


Badness: 0.039162
Badness (Smith): 0.039162


==== Countermiraculous ====
==== Countermiraculous ====
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Mapping: {{mapping| 1 4 3 3 8 1 | 0 -25 -7 -2 -47 28 }}
Mapping: {{mapping| 1 4 3 3 8 1 | 0 -25 -7 -2 -47 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8803{{c}}


{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145, 321ceff }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}


Badness: 0.039271
Badness (Smith): 0.039271


===== 17-limit =====
===== 17-limit =====
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Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }}
Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8756
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8756{{c}}


{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145 }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}


Badness: 0.029496
Badness (Smith): 0.029496


==== Counterbenediction ====
==== Counterbenediction ====
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Mapping: {{mapping| 1 4 3 3 8 -2 | 0 -25 -7 -2 -47 59 }}
Mapping: {{mapping| 1 4 3 3 8 -2 | 0 -25 -7 -2 -47 59 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.9335{{c}}


{{Optimal ET sequence|legend=1| 31, 114ef, 145f, 176, 207, 383c, 590cc }}
{{Optimal ET sequence|legend=0| 31, 114ef, 145f, 176, 207, 383c, 590cc }}


Badness: 0.045569
Badness (Smith): 0.045569


===== 17-limit =====
===== 17-limit =====
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Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }}
Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9391
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.9391{{c}}


{{Optimal ET sequence|legend=1| 31, 114efg, 145fg, 176, 207 }}
{{Optimal ET sequence|legend=0| 31, 114efg, 145fg, 176, 207 }}


Badness: 0.036289
Badness (Smith): 0.036289


==== Countermanna ====
==== Countermanna ====
Line 162: Line 115:
Mapping: {{mapping| 1 4 3 3 8 15 | 0 -25 -7 -2 -47 -117 }}
Mapping: {{mapping| 1 4 3 3 8 15 | 0 -25 -7 -2 -47 -117 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8898
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8898{{c}}


{{Optimal ET sequence|legend=1| 145, 176, 321ce }}
{{Optimal ET sequence|legend=0| 145, 176, 321ce }}


Badness: 0.053409
Badness (Smith): 0.053409


===== 17-limit =====
===== 17-limit =====
Line 175: Line 128:
Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }}
Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8832{{c}}


{{Optimal ET sequence|legend=1| 145, 321ce }}
{{Optimal ET sequence|legend=0| 145, 321ce }}


Badness: 0.040898
Badness (Smith): 0.040898


=== Counterrevelation ===
=== Counterrevelation ===
Line 188: Line 141:
Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }}
Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }}


Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~343/320 = 115.9192{{c}}


{{Optimal ET sequence|legend=1| 31, 114, 145e, 176e }}
{{Optimal ET sequence|legend=0| 31, 114, 145e, 176e }}


Badness: 0.064070
Badness (Smith): 0.064070


==== 13-limit ====
==== 13-limit ====
Line 201: Line 154:
Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }}
Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~273/256 = 115.8624{{c}}


{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
{{Optimal ET sequence|legend=0| 31, 83, 114, 145e }}


Badness: 0.057497
Badness (Smith): 0.057497


==== 17-limit ====
==== 17-limit ====
Line 214: Line 167:
Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}
Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~91/85 = 115.8527{{c}}
 
{{Optimal ET sequence|legend=0| 31, 83, 114, 145e }}
 
Badness (Smith): 0.044043
 
== Subgroup extensions ==
=== Mercy (2.5.7.13) ===
Subgroup: 2.5.7.13
 
Comma list: 343/338, 640/637
 
Subgroup-val mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
 
Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 116.094{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31}}
 
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
 
Comma list: 170/169, 224/221, 640/637
 
Subgroup-val mapping: {{mapping| 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 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 115.769{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
 
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
 
Comma list: 170/169, 343/338, 640/637, 16384/16055
 
Subgroup-val mapping: {{mapping| 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 }}


{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 115.716{{c}}


Badness: 0.044043
{{Optimal ET sequence|legend=0| 10, 21, 31, 52f }}


[[Category:Temperament clans]]
[[Category:Temperament clans]]

Revision as of 18:40, 31 January 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, [-15 0 -2 7⟩ = 823543/819200. Quince temperaments include:

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.

Subgroup: 2.5.7

Comma list: 823543/819200

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

mapping generators: ~2, ~343/320

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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~343/320 = 116.291 ¢

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

Countermiracle

The countermiracle temperament (31 & 145) tempers out the trimyna, 50421/50000 and the quince comma, 823543/819200.

Todo: expand

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.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping: [⟨1 4 3 3], ⟨0 -25 -7 -2]]

mapping generators: ~2, ~343/320

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~343/320 = 115.9169 ¢

Optimal ET sequence: 31, 114, 145, 176, 559cc, 735cc

Badness (Smith): 0.102326

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 4 3 3 8], ⟨0 -25 -7 -2 -47]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.9158 ¢

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

Badness (Smith): 0.039162

Countermiraculous

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 4 3 3 8 1], ⟨0 -25 -7 -2 -47 28]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.8803 ¢

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

Badness (Smith): 0.039271

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 4 3 3 8 1 1], ⟨0 -25 -7 -2 -47 28 32]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.8756 ¢

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

Badness (Smith): 0.029496

Counterbenediction

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 4 3 3 8 -2], ⟨0 -25 -7 -2 -47 59]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.9335 ¢

Optimal ET sequence: 31, 114ef, 145f, 176, 207, 383c, 590cc

Badness (Smith): 0.045569

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 4 3 3 8 -2 -2], ⟨0 -25 -7 -2 -47 59 63]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.9391 ¢

Optimal ET sequence: 31, 114efg, 145fg, 176, 207

Badness (Smith): 0.036289

Countermanna

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 4 3 3 8 15], ⟨0 -25 -7 -2 -47 -117]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.8898 ¢

Optimal ET sequence: 145, 176, 321ce

Badness (Smith): 0.053409

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 4 3 3 8 15 15], ⟨0 -25 -7 -2 -47 -117 -113]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.8832 ¢

Optimal ET sequence: 145, 321ce

Badness (Smith): 0.040898

Counterrevelation

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 4 3 3 5], ⟨0 -25 -7 -2 -16]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~343/320 = 115.9192 ¢

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

Badness (Smith): 0.064070

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 4 3 3 5 1], ⟨0 -25 -7 -2 -16 28]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~273/256 = 115.8624 ¢

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

Badness (Smith): 0.057497

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 4 3 3 5 1 1], ⟨0 -25 -7 -2 -16 28 32]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~91/85 = 115.8527 ¢

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

Badness (Smith): 0.044043

Subgroup extensions

Mercy (2.5.7.13)

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 tuning (POTE): ~2 = 1200.000 ¢, ~14/13 = 116.094 ¢

Optimal ET sequence: 10, 21, 31

2.5.7.13.17 subgroup

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 640/637

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 tuning (POTE): ~2 = 1200.000 ¢, ~14/13 = 115.769 ¢

Optimal ET sequence: 10, 21, 31

2.5.7.13.17.19 subgroup

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 343/338, 640/637, 16384/16055

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 tuning (POTE): ~2 = 1200.000 ¢, ~14/13 = 115.716 ¢

Optimal ET sequence: 10, 21, 31, 52f

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