Porcupine family: Difference between revisions

Yourmusic Productions (talk | contribs)
m Nautilus: extend beyond optimal patent val as 44 is considerably closer to ideal tuning.
Update keys
Line 26: Line 26:
[[Comma list]]: 250/243
[[Comma list]]: 250/243


[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}]
{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}


: mapping generators: ~2, ~10/9
: mapping generators: ~2, ~10/9
Line 46: Line 46:
Comma list: 55/54, 100/99
Comma list: 55/54, 100/99


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


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


Gencom: [2 10/9; 55/54, 100/99]  
: gencom: [2 10/9; 55/54, 100/99]  


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.8867
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.8867
Line 63: Line 63:
Comma list: 55/54, 100/99, 512/507
Comma list: 55/54, 100/99, 512/507


Sval mapping: [{{val| 1 5 8 8 2 }}, {{val| 0 -6 -10 -8 3 }}]
Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}


: sval mapping generators: ~2, ~65/44
: sval mapping generators: ~2, ~65/44
Line 82: Line 82:
[[Comma list]]: 64/63, 250/243
[[Comma list]]: 64/63, 250/243


[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -3 -5 6 }}]
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}


{{Multival|legend=1| 3 5 -6 1 -18 -28 }}
{{Multival|legend=1| 3 5 -6 1 -18 -28 }}
Line 90: Line 90:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 109: Line 109:
Comma list: 55/54, 64/63, 100/99
Comma list: 55/54, 64/63, 100/99


Mapping: [{{val| 1 2 3 2 4 }}, {{val| 0 -3 -5 6 -4 }}]
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.1055
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.1055
Line 131: Line 131:
Comma list: 40/39, 55/54, 64/63, 66/65
Comma list: 40/39, 55/54, 64/63, 66/65


Mapping: [{{val| 1 2 3 2 4 4 }}, {{val| 0 -3 -5 6 -4 -2 }}]
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.4425
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.4425
Line 157: Line 157:
Comma list: 55/54, 64/63, 91/90, 100/99
Comma list: 55/54, 64/63, 91/90, 100/99


Mapping: [{{val| 1 2 3 2 4 6 }}, {{val| 0 -3 -5 6 -4 -17 }}]
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 162.6361
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 162.6361
Line 181: Line 181:
Comma list: 55/54, 64/63, 100/99, 196/195
Comma list: 55/54, 64/63, 100/99, 196/195


Mapping: [{{val| 1 2 3 2 4 1 }}, {{val| 0 -3 -5 6 -4 20 }}]
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.3781
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.3781
Line 198: Line 198:
Comma list: 55/54, 64/63, 65/63, 100/99
Comma list: 55/54, 64/63, 65/63, 100/99


Mapping: [{{val| 1 2 3 2 4 3 }}, {{val| 0 -3 -5 6 -4 5 }}]
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.6778
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.6778
Line 217: Line 217:
[[Comma list]]: 28/27, 126/125
[[Comma list]]: 28/27, 126/125


[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -9 }}]
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}


{{Multival|legend=1| 3 5 9 1 6 7 }}
{{Multival|legend=1| 3 5 9 1 6 7 }}
Line 224: Line 224:


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis]] ([[unchanged-interval basis]]): 2.7
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7


{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
Line 235: Line 235:
Comma list: 28/27, 55/54, 77/75
Comma list: 28/27, 55/54, 77/75


Mapping: [{{val| 1 2 3 4 4 }}, {{val| 0 -3 -5 -9 -4 }}]
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit eigenmonzo basis (unchanged-interval basis): 2.7
* 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.7


{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
Line 251: Line 251:
Comma list: 28/27, 40/39, 55/54, 66/65
Comma list: 28/27, 40/39, 55/54, 66/65


Mapping: [{{val| 1 2 3 4 4 4 }}, {{val| 0 -3 -5 -9 -4 -2 }}]
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.6312
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.6312


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit eigenmonzo basis (unchanged-interval basis): 2.7
* 13- and 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 38bceff }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 38bceff }}
Line 269: Line 269:
[[Comma list]]: 225/224, 250/243
[[Comma list]]: 225/224, 250/243


[[Mapping]]: [{{val| 1 2 3 5 }}, {{val| 0 -3 -5 -16 }}]
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


{{Multival|legend=1| 3 5 16 1 17 23 }}
{{Multival|legend=1| 3 5 16 1 17 23 }}
Line 277: Line 277:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5


{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
Line 288: Line 288:
Comma list: 55/54, 100/99, 225/224
Comma list: 55/54, 100/99, 225/224


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


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.3207
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.3207
Line 294: Line 294:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
: Eigenmonzo basis (unchanged-interval basis): 2.7/5
: eigenmonzo basis (unchanged-interval basis): 2.7/5


{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51 }}
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51 }}
Line 305: Line 305:
Comma list: 55/54, 65/64, 91/90, 100/99
Comma list: 55/54, 65/64, 91/90, 100/99


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


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.4782
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.4782
Line 320: Line 320:
[[Comma list]]: 250/243, 525/512
[[Comma list]]: 250/243, 525/512


[[Mapping]]: [{{val| 1 2 3 1 }}, {{val| 0 -3 -5 13 }}]
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


{{Multival|legend=1| 3 5 -13 1 -29 -44 }}
{{Multival|legend=1| 3 5 -13 1 -29 -44 }}
Line 328: Line 328:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.3
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3


{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
Line 339: Line 339:
Comma list: 55/54, 100/99, 525/512
Comma list: 55/54, 100/99, 525/512


Mapping: [{{val| 1 2 3 1 4 }}, {{val| 0 -3 -5 13 -4 }}]
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 165.9246
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 165.9246
Line 345: Line 345:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: Eigenmonzo basis (unchanged-interval basis): 2.3
: eigenmonzo basis (unchanged-interval basis): 2.3


{{Optimal ET sequence|legend=1| 7, 22d, 29, 65ce }}
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65ce }}
Line 356: Line 356:
Comma list: 55/54, 65/64, 100/99, 105/104
Comma list: 55/54, 65/64, 100/99, 105/104


Mapping: [{{val| 1 2 3 1 4 3 }}, {{val| 0 -3 -5 13 -4 5 }}]
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 166.0459
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 166.0459
Line 362: Line 362:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: Eigenmonzo basis (unchanged-interval basis): 2.3
: eigenmonzo basis (unchanged-interval basis): 2.3


{{Optimal ET sequence|legend=1| 7, 22d, 29, 65cef }}
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65cef }}
Line 375: Line 375:
[[Comma list]]: 36/35, 160/147
[[Comma list]]: 36/35, 160/147


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -1 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}


{{Multival|legend=1| 3 5 1 1 -7 -12 }}
{{Multival|legend=1| 3 5 1 1 -7 -12 }}
Line 383: Line 383:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5


{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
Line 394: Line 394:
Comma list: 22/21, 36/35, 80/77
Comma list: 22/21, 36/35, 80/77


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


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.7684
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.7684
Line 409: Line 409:
[[Comma list]]: 21/20, 175/162
[[Comma list]]: 21/20, 175/162


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -2 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}


{{Multival|legend=1| 3 5 2 1 -5 -9 }}
{{Multival|legend=1| 3 5 2 1 -5 -9 }}
Line 429: Line 429:
[[Comma list]]: 50/49, 245/243
[[Comma list]]: 50/49, 245/243


[[Mapping]]: [{{val| 2 1 1 2 }}, {{val| 0 3 5 5 }}]
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}


: mapping generators: ~7/5, ~9/7
: mapping generators: ~7/5, ~9/7
Line 446: Line 446:
Comma list: 50/49, 55/54, 99/98
Comma list: 50/49, 55/54, 99/98


Mapping: [{{val| 2 1 1 2 4 }}, {{val| 0 3 5 5 4 }}]
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.5281
Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.5281
Line 459: Line 459:
Comma list: 50/49, 55/54, 65/63, 99/98
Comma list: 50/49, 55/54, 65/63, 99/98


Mapping: [{{val| 2 1 1 2 4 3 }}, {{val| 0 3 5 5 4 6 }}]
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 436.3087
Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 436.3087
Line 472: Line 472:
Comma list: 40/39, 50/49, 55/54, 66/65
Comma list: 40/39, 50/49, 55/54, 66/65


Mapping: [{{val| 2 1 1 2 4 6 }}, {{val| 0 3 5 5 4 2 }}]
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.1856
Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.1856
Line 485: Line 485:
Comma list: 50/49, 245/243, 385/384
Comma list: 50/49, 245/243, 385/384


Mapping: [{{val| 2 1 1 2 12 }}, {{val| 0 3 5 5 -7 }}]
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.3289
Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.3289
Line 501: Line 501:
[[Comma list]]: 49/48, 250/243
[[Comma list]]: 49/48, 250/243


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -6 -10 -3 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}


: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20
Line 518: Line 518:
Comma list: 49/48, 55/54, 245/242
Comma list: 49/48, 55/54, 245/242


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -6 -10 -3 -8 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.8017
Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.8017
Line 531: Line 531:
Comma list: 49/48, 55/54, 91/90, 100/99
Comma list: 49/48, 55/54, 91/90, 100/99


Mapping: [{{val| 1 2 3 3 4 5 }}, {{val| 0 -6 -10 -3 -8 -19 }}]
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.9123
Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.9123
Line 544: Line 544:
Comma list: 40/39, 49/48, 55/54, 66/65
Comma list: 40/39, 49/48, 55/54, 66/65


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -6 -10 -3 -8 -4 }}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 82.0342
Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 82.0342
Line 560: Line 560:
[[Comma list]]: 250/243, 686/675
[[Comma list]]: 250/243, 686/675


[[Mapping]]: [{{val| 1 5 8 10 }}, {{val| 0 -9 -15 -19 }}]
{{Mapping|legend=1| 1 5 8 10 | 0 -9 -15 -19 }}


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
Line 577: Line 577:
Comma list: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675


Mapping: [{{val| 1 5 8 10 8 }}, {{val| 0 -9 -15 -19 -12 }}]
Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 454.5050
Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 454.5050
Line 590: Line 590:
Comma list: 55/54, 91/90, 100/99, 169/168
Comma list: 55/54, 91/90, 100/99, 169/168


Mapping: [{{val| 1 5 8 10 8 9 }}, {{val| 0 -9 -15 -19 -12 -14 }}]
Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}


Optimal tuning (CTE): ~2 = 1\1, ~13/10 = 454.4798
Optimal tuning (CTE): ~2 = 1\1, ~13/10 = 454.4798
Line 603: Line 603:
[[Comma list]]: 250/243, 1728/1715
[[Comma list]]: 250/243, 1728/1715


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -9 -15 -4 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}


: mapping generators: ~2, ~36/35
: mapping generators: ~2, ~36/35
Line 620: Line 620:
Comma list: 55/54, 100/99, 352/343
Comma list: 55/54, 100/99, 352/343


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -9 -15 -4 -12 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}


Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.7019
Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.7019
Line 633: Line 633:
Comma list: 55/54, 65/63, 100/99, 352/343
Comma list: 55/54, 65/63, 100/99, 352/343


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -9 -15 -4 -12 -7 }}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}


Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.5751
Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.5751
Line 643: Line 643:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine]]
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]