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Diaschismic family: Difference between revisions

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The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34 EDO]] is a good tuning choice, with [[46 EDO]], [[56 EDO]], [[58 EDO]] or [[80 EDO]] being other possibilities. Both [[12 EDO]] and [[22 EDO]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.
The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.


== Srutal (12&34, aka diaschismic) ==
== Srutal (12&34, aka diaschismic) ==
Line 66: Line 66:
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


Vals: {{Val list| 34d, 46, 80, 126, 206cd }}
Optimal GPV sequence: {{Val list| 34d, 46, 80, 126, 206cd }}


Badness: 0.035315
Badness: 0.035315
Line 85: Line 85:
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


Vals: {{Val list| 34d, 46, 80, 206cd, 286bcde }}
Optimal GPV sequence: {{Val list| 34d, 46, 80, 206cd, 286bcde }}


Badness: 0.025286
Badness: 0.025286
Line 103: Line 103:
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


Vals: {{Val list| 34d, 46, 80, 126, 206cd }}
Optimal GPV sequence: {{Val list| 34d, 46, 80, 126, 206cd }}


Badness: 0.018594
Badness: 0.018594
Line 147: Line 147:
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 709.091]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 709.091]


Vals: {{Val list| 10e, 12, 22, 34d, 56d }}
Optimal GPV sequence: {{Val list| 10e, 12, 22, 34d, 56d }}


Badness: 0.020343
Badness: 0.020343
Line 160: Line 160:
POTE generator: ~3/2 = 708.919
POTE generator: ~3/2 = 708.919


Vals: {{Val list| 12, 22 }}
Optimal GPV sequence: {{Val list| 10e, 12, 22 }}


Badness: 0.027642
Badness: 0.027642
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98
Mapping: [{{val| 2 0 11 12 26 1 5 }}, {{val| 0 1 -2 -2 -6 2 1 }}]
POTE generator: ~3/2 = 708.806
Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
Badness: 0.020899
==== Pajarina ====
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 64/63, 78/77, 99/98
Mapping: [{{val| 2 0 11 12 26 36 }}, {{val| 0 1 -2 -2 -6 -9 }}]
POTE generator: ~3/2 = 706.133
Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
Badness: 0.022327
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98
Mapping: [{{val| 2 0 11 12 26 36 5 }}, {{val| 0 1 -2 -2 -6 -9 1 }}]
POTE generator: ~3/2 = 706.410
Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
Badness: 0.018375


=== Pajarous ===
=== Pajarous ===
Line 178: Line 217:
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = 709.091
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = 709.091


Vals: {{Val list| 10, 12e, 22, 120bce, 142bce }}
Optimal GPV sequence: {{Val list| 10, 12e, 22, 120bce, 142bce }}


Badness: 0.028349
Badness: 0.028349
Line 191: Line 230:
POTE generator: ~3/2 = 710.240
POTE generator: ~3/2 = 710.240


Vals: {{Val list| 10, 22, 54f, 76bdf }}
Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdf }}


Badness: 0.025176
Badness: 0.025176
Line 204: Line 243:
POTE generator ~3/2 = 710.818
POTE generator ~3/2 = 710.818


Vals: {{Val list| 10, 22f, 32f, 54f }}
Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54f }}


Badness: 0.0274
Badness: 0.027355


=== Pajaric ===
=== Pajaric ===
Line 217: Line 256:
POTE generator: ~3/2 = 705.524
POTE generator: ~3/2 = 705.524


Vals: {{Val list| 10, 12, 22e, 34de }}
Optimal GPV sequence: {{Val list| 10, 12, 22e, 34de }}


Badness: 0.023798
Badness: 0.023798
Line 230: Line 269:
POTE generator: ~3/2 = 707.442
POTE generator: ~3/2 = 707.442


Vals: {{Val list| 10, 12f, 22ef, 34def }}
Optimal GPV sequence: {{Val list| 10, 12f, 22ef, 34def }}


Badness: 0.0205
Badness: 0.020461


=== Hemipaj ===
=== Hemipaj ===
Line 243: Line 282:
POTE generator: ~11/8 = 546.383
POTE generator: ~11/8 = 546.383


Vals: {{Val list| 20, 22, 68d, 90d }}
Optimal GPV sequence: {{Val list| 20, 22, 68d, 90d }}


Badness: 0.038890
Badness: 0.038890
Line 256: Line 295:
POTE generator: ~64/55 = 246.907
POTE generator: ~64/55 = 246.907


Vals: {{Val list| 10, 24d, 34d }}
Optimal GPV sequence: {{Val list| 10, 24d, 34d }}


Badness: 0.048885
Badness: 0.048885
Line 269: Line 308:
POTE generator: ~15/13 = 246.907
POTE generator: ~15/13 = 246.907


Vals: {{Val list| 10, 24d, 34d }}
Optimal GPV sequence: {{Val list| 10, 24d, 34d }}


Badness: 0.028755
Badness: 0.028755
Line 311: Line 350:
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]


Vals: {{Val list| 12, 46, 58, 104c, 162ce }}
Optimal GPV sequence: {{Val list| 12, 46, 58, 104c, 162ce }}


Badness: 0.025034
Badness: 0.025034
Line 330: Line 369:
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]


Vals: {{Val list| 46, 58, 104c, 162cef }}
Optimal GPV sequence: {{Val list| 46, 58, 104c, 162cef }}


Badness: 0.018926
Badness: 0.018926
Line 348: Line 387:
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]


Vals: {{Val list| 46, 58, 104c }}
Optimal GPV sequence: {{Val list| 46, 58, 104c }}


Badness: 0.016425
Badness: 0.016425
Line 378: Line 417:
POTE generator: ~3/2 = 707.609
POTE generator: ~3/2 = 707.609


Vals: {{Val list| 22, 56, 78, 212be, 290bbe }}
Optimal GPV sequence: {{Val list| 22, 56, 78, 212be, 290bbe }}


Badness: 0.045270
Badness: 0.045270
Line 408: Line 447:
POTE generator: ~3/2 = 705.087
POTE generator: ~3/2 = 705.087


Vals: {{Val list| 12, 68, 80 }}
Optimal GPV sequence: {{Val list| 12, 68, 80 }}


Badness: 0.040191
Badness: 0.040191
Line 421: Line 460:
POTE generator: ~3/2 = 705.301
POTE generator: ~3/2 = 705.301


Vals: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}


Badness: 0.041137
Badness: 0.041137
Line 434: Line 473:
POTE generator: ~3/2 = 705.334
POTE generator: ~3/2 = 705.334


Vals: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}


Badness: 0.028631
Badness: 0.028631
Line 471: Line 510:
: Eigenmonzos: 2, 11/7
: Eigenmonzos: 2, 11/7


Vals: {{Val list| 22, 58, 80, 138cde, 218cde }}
Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde, 218cde }}


Badness: 0.025987
Badness: 0.025987
Line 484: Line 523:
POTE generator: ~9/7 = 434.756
POTE generator: ~9/7 = 434.756


Vals: {{Val list| 22, 58, 80, 138cde }}
Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}


Badness: 0.023679
Badness: 0.023679
Line 497: Line 536:
POTE generator: ~9/7 = 434.816
POTE generator: ~9/7 = 434.816


Vals: {{Val list| 22, 58, 80, 138cde }}
Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}


Badness: 0.020273
Badness: 0.020273
Line 523: Line 562:
POTE generator: ~8/7 = 235.096
POTE generator: ~8/7 = 235.096


Vals: {{Val list| 10, 36e, 46, 102, 148, 342bcdd }}
Optimal GPV sequence: {{Val list| 10, 36e, 46, 102, 148, 342bcdd }}


Badness: 0.045127
Badness: 0.045127
Line 536: Line 575:
POTE generator: ~8/7 = 235.088
POTE generator: ~8/7 = 235.088


Vals: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}


Badness: 0.028874
Badness: 0.028874
Line 549: Line 588:
POTE generator: ~8/7 = 235.088
POTE generator: ~8/7 = 235.088


Vals: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}


Badness: 0.019304
Badness: 0.019304
Line 584: Line 623:
POTE generator: ~33/32 = 52.680
POTE generator: ~33/32 = 52.680


Vals: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}
Optimal GPV sequence: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}


Badness: 0.026489
Badness: 0.026489
Line 597: Line 636:
POTE generator: ~33/32 = 52.654
POTE generator: ~33/32 = 52.654


Vals: {{Val list| 22f, 24f, 46, 68, 114 }}
Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}


Badness: 0.028057
Badness: 0.028057
Line 610: Line 649:
POTE generator: ~33/32 = 52.647
POTE generator: ~33/32 = 52.647


Vals: {{Val list| 22f, 24f, 46, 68, 114 }}
Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}


Badness: 0.018716
Badness: 0.018716
Line 623: Line 662:
POTE generator: ~33/32 = 52.730
POTE generator: ~33/32 = 52.730


Vals: {{Val list| 22fh, 24fh, 46, 68, 114, 182bef }}
Optimal GPV sequence: {{Val list| 22fh, 24fh, 46, 68, 114, 182bef }}


Badness: 0.017540
Badness: 0.017540
Line 651: Line 690:
POTE generator: ~11/9 = 351.863
POTE generator: ~11/9 = 351.863


Vals: {{Val list| 24, 34d, 58, 150cdee, 208ccddee }}
Optimal GPV sequence: {{Val list| 24, 34d, 58 }}


Badness: 0.041459
Badness: 0.041459
Line 664: Line 703:
POTE generator: ~11/9 = 351.886
POTE generator: ~11/9 = 351.886


Vals: {{Val list| 24, 34d, 58, 150cdeef, 208ccddeeff }}
Optimal GPV sequence: {{Val list| 24, 34d, 58, 150cdeef, 208ccddeeff }}


Badness: 0.023791
Badness: 0.023791
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220
Mapping: [{{val| 2 0 11 -15 -1 9 5 }}, {{val| 0 2 -4 13 5 -1 2 }}]
POTE generator: ~11/9 = 351.857
Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
Badness: 0.020536


== Anguirus ==
== Anguirus ==
Line 692: Line 744:
POTE generator: ~8/7 = 247.816
POTE generator: ~8/7 = 247.816


Vals: {{Val list| 10, 24, 34, 58d, 92de }}
Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92de }}


Badness: 0.049253
Badness: 0.049253
Line 705: Line 757:
POTE generator: ~8/7 = 247.691
POTE generator: ~8/7 = 247.691


Vals: {{Val list| 10, 24, 34, 58d, 92def }}
Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92def }}


Badness: 0.030829
Badness: 0.030829
Line 733: Line 785:
POTE generator: ~33/32 = 50.130
POTE generator: ~33/32 = 50.130


Vals: {{Val list| 2, 22d, 24 }}
Optimal GPV sequence: {{Val list| 2, 22d, 24 }}


Badness: 0.063483
Badness: 0.063483
Line 746: Line 798:
POTE generator: ~33/32 = 50.535
POTE generator: ~33/32 = 50.535


Vals: {{Val list| 22df, 24 }}
Optimal GPV sequence: {{Val list| 22df, 24 }}


Badness: 0.045731
Badness: 0.045731
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