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The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.
The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.


Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{multival| 3 5 1 }} for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.
Its [[monzo]] is {{monzo| 1 -5 3 }}. The [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.


Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.  
Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.  
Line 45: Line 45:
Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}
Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}


Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
{{Mapping|legend=4| 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]  
Line 84: Line 84:


{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
{{Multival|legend=1| 3 5 -6 1 -18 -28 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 93: Line 91:
[[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|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) 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|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 111: Line 109:
Comma list: 55/54, 64/63, 100/99
Comma list: 55/54, 64/63, 100/99


Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
{{Mapping|legend=0| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 119: Line 117:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: eigenmonzo (unchanged-interval) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
Line 134: Line 132:
Comma list: 40/39, 55/54, 64/63, 66/65
Comma list: 40/39, 55/54, 64/63, 66/65


Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
{{Mapping|legend=0| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 142: Line 140:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: eigenmonzo (unchanged-interval) basis: 2.11
: unchanged-interval (eigenmonzo) basis: 2.11


Tuning ranges:  
Tuning ranges:  
Line 160: Line 158:
Comma list: 55/54, 64/63, 91/90, 100/99
Comma list: 55/54, 64/63, 91/90, 100/99


Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
{{Mapping|legend=0| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 168: Line 166:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
: eigenmonzo (unchanged-interval) basis: 2.13/11
: unchanged-interval (eigenmonzo) basis: 2.13/11


Tuning ranges:  
Tuning ranges:  
Line 184: Line 182:
Comma list: 55/54, 64/63, 100/99, 196/195
Comma list: 55/54, 64/63, 100/99, 196/195


Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
{{Mapping|legend=0| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


Optimal tunings:  
Optimal tunings:  
Line 192: Line 190:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
: eigenmonzo (unchanged-interval) basis: 2.13/7
: unchanged-interval (eigenmonzo) basis: 2.13/7


Optimal ET sequence: {{Optimal ET sequence| 15f, 22f, 37, 59f }}
Optimal ET sequence: {{Optimal ET sequence| 15f, 22f, 37, 59f }}
Line 203: Line 201:
Comma list: 55/54, 64/63, 65/63, 100/99
Comma list: 55/54, 64/63, 65/63, 100/99


Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
{{Mapping|legend=0| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


Optimal tunings:  
Optimal tunings:  
Line 211: Line 209:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: eigenmonzo (unchanged-interval) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


Optimal ET sequence: {{Optimal ET sequence| 7, 15f, 22 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 15f, 22 }}
Line 225: Line 223:


{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
{{Multival|legend=1| 3 5 9 1 6 7 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063


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


{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
Line 242: Line 238:
Comma list: 28/27, 55/54, 77/75
Comma list: 28/27, 55/54, 77/75


Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
{{Mapping|legend=0| 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 (unchanged-interval) basis: 2.7
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


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


Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
{{Mapping|legend=0| 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 (unchanged-interval) basis: 2.7
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Optimal ET sequence: {{Optimal ET sequence| 7d, 8d, 15, 38bceff }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 8d, 15, 38bceff }}
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{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
{{Multival|legend=1| 3 5 16 1 17 23 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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[[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|eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) 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 297: Line 291:
Comma list: 55/54, 100/99, 225/224
Comma list: 55/54, 100/99, 225/224


Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
{{Mapping|legend=0| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 305: Line 299:
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 (unchanged-interval) basis: 2.7/5
: unchanged-interval (eigenmonzo) basis: 2.7/5


Optimal ET sequence: {{Optimal ET sequence| 7d, 15d, 22, 51 }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 15d, 22, 51 }}
Line 316: Line 310:
Comma list: 55/54, 65/64, 91/90, 100/99
Comma list: 55/54, 65/64, 91/90, 100/99


Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
{{Mapping|legend=0| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


Optimal tunings:  
Optimal tunings:  
Line 334: Line 328:


{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
{{Multival|legend=1| 3 5 -13 1 -29 -44 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 343: Line 335:
[[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|eigenmonzo (unchanged-interval) basis]]: 2.3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
Line 354: Line 346:
Comma list: 55/54, 100/99, 525/512
Comma list: 55/54, 100/99, 525/512


Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
{{Mapping|legend=0| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 362: Line 354:
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 (unchanged-interval) basis: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3


Optimal ET sequence: {{Optimal ET sequence| 7, 22d, 29, 65ce }}
Optimal ET sequence: {{Optimal ET sequence| 7, 22d, 29, 65ce }}
Line 373: Line 365:
Comma list: 55/54, 65/64, 100/99, 105/104
Comma list: 55/54, 65/64, 100/99, 105/104


Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
{{Mapping|legend=0| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Optimal tunings:  
Optimal tunings:  
Line 381: Line 373:
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 (unchanged-interval) basis: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3


Optimal ET sequence: {{Optimal ET sequence| 7, 22d, 29, 65cef }}
Optimal ET sequence: {{Optimal ET sequence| 7, 22d, 29, 65cef }}
Line 395: Line 387:


{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
{{Multival|legend=1| 3 5 1 1 -7 -12 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 404: Line 394:
[[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|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
Line 415: Line 405:
Comma list: 22/21, 36/35, 80/77
Comma list: 22/21, 36/35, 80/77


Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 433: Line 423:


{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}
{{Multival|legend=1| 3 5 2 1 -5 -9 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 457: Line 445:


: mapping generators: ~7/5, ~9/7
: mapping generators: ~7/5, ~9/7
{{Multival|legend=1| 6 10 10 2 -1 -5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 473: Line 459:
Comma list: 50/49, 55/54, 99/98
Comma list: 50/49, 55/54, 99/98


Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
{{Mapping|legend=0| 2 1 1 2 4 | 0 3 5 5 4 }}


Optimal tunings:  
Optimal tunings:  
Line 488: Line 474:
Comma list: 50/49, 55/54, 65/63, 99/98
Comma list: 50/49, 55/54, 65/63, 99/98


Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
{{Mapping|legend=0| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}


Optimal tunings:  
Optimal tunings:  
Line 503: Line 489:
Comma list: 40/39, 50/49, 55/54, 66/65
Comma list: 40/39, 50/49, 55/54, 66/65


Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
{{Mapping|legend=0| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}


Optimal tunings:  
Optimal tunings:  
Line 518: Line 504:
Comma list: 50/49, 245/243, 385/384
Comma list: 50/49, 245/243, 385/384


Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
{{Mapping|legend=0| 2 1 1 2 12 | 0 3 5 5 -7 }}


Optimal tunings:  
Optimal tunings:  
Line 539: Line 525:


: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20
{{Multival|legend=1| 6 10 3 2 -12 -21 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 555: Line 539:
Comma list: 49/48, 55/54, 245/242
Comma list: 49/48, 55/54, 245/242


Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 570: Line 554:
Comma list: 49/48, 55/54, 91/90, 100/99
Comma list: 49/48, 55/54, 91/90, 100/99


Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


Optimal tunings:  
Optimal tunings:  
Line 585: Line 569:
Comma list: 40/39, 49/48, 55/54, 66/65
Comma list: 40/39, 49/48, 55/54, 66/65


Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 606: Line 590:


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 9 15 19 3 5 2 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 622: Line 604:
Comma list: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675


Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
{{Mapping|legend=0| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}


Optimal tunings:  
Optimal tunings:  
Line 637: Line 619:
Comma list: 55/54, 91/90, 100/99, 169/168
Comma list: 55/54, 91/90, 100/99, 169/168


Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}
{{Mapping|legend=0| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}


Optimal tunings:  
Optimal tunings:  
Line 655: Line 637:


: mapping generators: ~2, ~36/35
: mapping generators: ~2, ~36/35
{{Multival|legend=1| 9 15 4 3 -19 -33 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 671: Line 651:
Comma list: 55/54, 100/99, 352/343
Comma list: 55/54, 100/99, 352/343


Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}


Optimal tunings:  
Optimal tunings:  
Line 686: Line 666:
Comma list: 55/54, 65/63, 100/99, 352/343
Comma list: 55/54, 65/63, 100/99, 352/343


Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}


Optimal tunings:  
Optimal tunings:  
Line 695: Line 675:


Badness: 0.044739
Badness: 0.044739
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