16ed5/2: Difference between revisions

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== 16ed5/2 as a generator ==

16ED5/2 can also be thought as a [[generator]] of a [[~~cluster~~ temperament]] which tempers out ~~{{monzo|37 -16 -5}} (trisa-quingu comma). It has twelve~~ clusters of notes in an octave~~. There are some extensions for 7 and higher limit including~~ ''~~quintupole~~'' (12~~&~~121), ~~''quinticosiennic'' (12&~~145), and ~~''quintapole'' (12&85)~~.

16ED5/2 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 256/255, 361/360, and 4624/4617, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintaleap'' temperament). This temperament is supported by {{Val list|12, 109, 121, 133, 145}}, and [[157edo|157]] EDOs.

'''~~Trisa-quingu~~ (12&121)</font~~>'''<br~~>

Tempering out 400/399 (equating 20/19 and 21/20) leads ''[[Octagar temperaments #Quintupole|Quintupole]]'' (12&121), and tempering out 476/475 (equating 19/17 with 28/25) leads ''[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]]'' (12&145).

'''~~~~5-limit~~~~'''

Comma: {{monzo|37 -16 -5}} = 137438953472/~~1345210003125~~

; Quintaleap (12&121)

'''5-limit'''

Comma: {{monzo|37 -16 -5}} = 137438953472/134521003125

Mapping: [{{val|1 2 1}}, {{val|0 -5 16}}]

POTE generator: ~135/128 = 99.267

Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc

Badness: 0.444506

'''[[Octagar temperaments #Quintupole|Quintupole]] (12&121)</font~~>'''<br~~>

'''2.3.5.17.19 subgroup'''

Comma list: 256/255, 361/360, 4624/4617

Gencom: [2 18/17; 256/255 361/360 4624/4617]

Gencom mapping: [{{val|1 2 1 5 4}}, {{val|0 -5 16 -11 3}}]

POTE generator: ~18/17 = 99.276

Vals: 12, 109, 121, 133

RMS error: 0.3427 cents

; [[Octagar temperaments #Quintupole|Quintupole]] (12&121)

'''~~~~7-limit~~~~'''

'''7-limit'''

Comma list: 4000/3969, 458752/455625

Mapping: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]

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Vals: 12, 97, 109, 121

Badness: 0.111620

'''~~~~11-limit~~~~'''

'''11-limit'''

Comma list: 896/891, 1375/1372, 4375/4356

Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]

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Vals: 12, 109, 121, 351bde, 472bdee

Badness: 0.056501

'''[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]] (12&145)</font~~>'''<br~~>

'''13-limit'''

'''~~~~7-limit~~~~'''

Comma list: 352/351, 364/363, 625/624, 2704/2695

Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]

POTE generator: ~55/52 = 99.165

Vals: 12f, 109, 121

Badness: 0.038431

'''17-limit'''

Comma list: 256/255, 352/351, 364/363, 375/374, 442/441

Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]

POTE generator: ~18/17 = 99.172

Vals: 12f, 109, 121

Badness: 0.028721

'''19-limit'''

Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374

Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]

POTE generator: ~18/17 = 99.164

Vals: 12f, 109, 121

Badness: 0.023818

; [[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]] (12&145)

'''7-limit'''

Comma list: 5120/5103, 395136/390625

Mapping: [{{val|1 2 1 -1}}, {{val|0 -5 16 46}}]

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Vals: 12, 133, 145, 157, 302c, 459bcc

Badness: 0.158041

'''~~~~11-limit~~~~'''

'''11-limit'''

Comma list: 441/440, 896/891, 78408/78125

Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]

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Vals: 12, 133, 145

Badness: 0.080674

'''[[Marvel temperaments #Quintapole|Quintapole]] (12&85)</font~~>'''<br~~>

'''13-limit'''

Comma list: 196/195, 352/351, 364/363, 78408/78125

Mapping: [{{val|1 2 1 -1 -2 -3}}, {{val|0 -5 16 46 66 81}}]

POTE generator: ~35/33 = 99.307

Vals: 12f, 133, 145

Badness: 0.052464

'''17-limit'''

Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750

Mapping: [{{val|1 2 1 -1 -2 -3 5}}, {{val|0 -5 16 46 66 81 -11}}]

POTE generator: ~18/17 = 99.308

Vals: 12f, 133, 145

Badness: 0.037108

'''19-limit'''

Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475

Mapping: [{{val|1 2 1 -1 -2 -3 5 4}}, {{val|0 -5 16 46 66 81 -11 3}}]

POTE generator: ~18/17 = 99.303

Vals: 12f, 133, 145

Badness: 0.028440

; [[Marvel temperaments #Quintapole|Quintapole]] (12&85)

'''~~~~7-limit~~~~'''

'''7-limit'''

Comma list: 225/224, 7812500/7411887

Mapping: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]

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Vals: 12, 73c, 85, 97d

Badness: 0.192498

'''~~~~11-limit~~~~'''

'''11-limit'''

Comma list: 100/99, 225/224, 85184/84035

Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]

@@ Line 171: / Line 171: @@
 == 16ed5/2 as a generator ==
-ED5/2 can also be thought as a [[generator]] of a [[cluster temperament]] which tempers out {{monzo|37 -16 -5}} (trisa-quingu comma). It has twelve clusters of notes in an octave. There are some extensions for 7 and higher limit including ''quintupole'' (12&amp;121), ''quinticosiennic'' (12&amp;145), and ''quintapole'' (12&amp;85).
+ED5/2 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 256/255, 361/360, and 4624/4617, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintaleap'' temperament). This temperament is supported by {{Val list|12, 109, 121, 133, 145}}, and [[157edo|157]] EDOs.
-'''<font style="font-size: 1.35em">Trisa-quingu (12&amp;121)</font>'''<br>
+Tempering out 400/399 (equating 20/19 and 21/20) leads ''[[Octagar temperaments #Quintupole|Quintupole]]'' (12&amp;121), and tempering out 476/475 (equating 19/17 with 28/25) leads ''[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]]'' (12&amp;145).
-'''<font style="font-size: 1.2em">5-limit</font>'''<br>
-Comma: {{monzo|37 -16 -5}} = 137438953472/1345210003125<br>
+; <font style="font-size: 1.15em">Quintaleap (12&amp;121)</font>
+'''5-limit'''<br>
+Comma: {{monzo|37 -16 -5}} = 137438953472/134521003125<br>
 Mapping: [{{val|1 2 1}}, {{val|0 -5 16}}]<br>
 POTE generator: ~135/128 = 99.267<br>
 Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc<br>
 Badness: 0.444506<br><br>
-'''<font style="font-size: 1.35em">[[Octagar temperaments #Quintupole|Quintupole]] (12&amp;121)</font>'''<br>
+'''2.3.5.17.19 subgroup'''<br>
+Comma list: 256/255, 361/360, 4624/4617<br>
+Gencom: [2 18/17; 256/255 361/360 4624/4617]<br>
+Gencom mapping: [{{val|1 2 1 5 4}}, {{val|0 -5 16 -11 3}}]<br>
+POTE generator: ~18/17 = 99.276<br>
+Vals: 12, 109, 121, 133<br>
+RMS error: 0.3427 cents<br><br>
+; <font style="font-size: 1.15em">[[Octagar temperaments #Quintupole|Quintupole]] (12&amp;121)</font>
 {{See also|34ed7 #34ed7 as a generator}}
-'''<font style="font-size: 1.2em">7-limit</font>'''<br>
+'''7-limit'''<br>
 Comma list: 4000/3969, 458752/455625<br>
 Mapping: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]<br>
@@ Line 188: / Line 199: @@
 Vals: 12, 97, 109, 121<br>
 Badness: 0.111620<br><br>
-'''<font style="font-size: 1.2em">11-limit</font>'''<br>
+'''11-limit'''<br>
 Comma list: 896/891, 1375/1372, 4375/4356<br>
 Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]<br>
@@ Line 194: / Line 205: @@
 Vals: 12, 109, 121, 351bde, 472bdee<br>
 Badness: 0.056501<br><br>
-'''<font style="font-size: 1.35em">[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]] (12&amp;145)</font>'''<br>
+'''13-limit'''<br>
-'''<font style="font-size: 1.2em">7-limit</font>'''<br>
+Comma list: 352/351, 364/363, 625/624, 2704/2695<br>
+Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]<br>
+POTE generator: ~55/52 = 99.165<br>
+Vals: 12f, 109, 121<br>
+Badness: 0.038431<br><br>
+'''17-limit'''<br>
+Comma list: 256/255, 352/351, 364/363, 375/374, 442/441<br>
+Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]<br>
+POTE generator: ~18/17 = 99.172<br>
+Vals: 12f, 109, 121<br>
+Badness: 0.028721<br><br>
+'''19-limit'''<br>
+Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374<br>
+Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]<br>
+POTE generator: ~18/17 = 99.164<br>
+Vals: 12f, 109, 121<br>
+Badness: 0.023818<br><br>
+; <font style="font-size: 1.15em">[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]] (12&amp;145)</font>
+'''7-limit'''<br>
 Comma list: 5120/5103, 395136/390625<br>
 Mapping: [{{val|1 2 1 -1}}, {{val|0 -5 16 46}}]<br>
@@ Line 201: / Line 230: @@
 Vals: 12, 133, 145, 157, 302c, 459bcc<br>
 Badness: 0.158041<br><br>
-'''<font style="font-size: 1.2em">11-limit</font>'''<br>
+'''11-limit'''<br>
 Comma list: 441/440, 896/891, 78408/78125<br>
 Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]<br>
@@ Line 207: / Line 236: @@
 Vals: 12, 133, 145<br>
 Badness: 0.080674<br><br>
-'''<font style="font-size: 1.35em">[[Marvel temperaments #Quintapole|Quintapole]] (12&amp;85)</font>'''<br>
+'''13-limit'''<br>
+Comma list: 196/195, 352/351, 364/363, 78408/78125<br>
+Mapping: [{{val|1 2 1 -1 -2 -3}}, {{val|0 -5 16 46 66 81}}]<br>
+POTE generator: ~35/33 = 99.307<br>
+Vals: 12f, 133, 145<br>
+Badness: 0.052464<br><br>
+'''17-limit'''<br>
+Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750<br>
+Mapping: [{{val|1 2 1 -1 -2 -3 5}}, {{val|0 -5 16 46 66 81 -11}}]<br>
+POTE generator: ~18/17 = 99.308<br>
+Vals: 12f, 133, 145<br>
+Badness: 0.037108<br><br>
+'''19-limit'''<br>
+Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475<br>
+Mapping: [{{val|1 2 1 -1 -2 -3 5 4}}, {{val|0 -5 16 46 66 81 -11 3}}]<br>
+POTE generator: ~18/17 = 99.303<br>
+Vals: 12f, 133, 145<br>
+Badness: 0.028440<br><br>
+; <font style="font-size: 1.15em">[[Marvel temperaments #Quintapole|Quintapole]] (12&amp;85)</font>
 {{See also|42ed11 #42ed11 as a generator}}
-'''<font style="font-size: 1.2em">7-limit</font>'''<br>
+'''7-limit'''<br>
 Comma list: 225/224, 7812500/7411887<br>
 Mapping: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]<br>
@@ Line 215: / Line 262: @@
 Vals: 12, 73c, 85, 97d<br>
 Badness: 0.192498<br><br>
-'''<font style="font-size: 1.2em">11-limit</font>'''<br>
+'''11-limit'''<br>
 Comma list: 100/99, 225/224, 85184/84035<br>
 Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]<br>