Breedsmic temperaments: Difference between revisions

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By adding [[243/242]] (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.
By adding [[243/242]] (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 58: Line 57:
POTE generator: ~11/9 = 351.521
POTE generator: ~11/9 = 351.521


Vals: {{Val list| 17c, 41, 58, 99e }}
Optimal GPV sequence: {{Val list| 17c, 41, 58, 99e }}


Badness: 0.023498
Badness: 0.023498
Line 71: Line 70:
POTE generator: ~11/9 = 351.573
POTE generator: ~11/9 = 351.573


Vals: {{Val list| 17c, 41, 58, 99ef, 157eff }}
Optimal GPV sequence: {{Val list| 17c, 41, 58, 99ef, 157eff }}


Badness: 0.019090
Badness: 0.019090
Line 84: Line 83:
POTE generator: ~49/40 = 351.505
POTE generator: ~49/40 = 351.505


Vals: {{Val list| 58, 140, 198 }}
Optimal GPV sequence: {{Val list| 58, 140, 198 }}


Badness: 0.042487
Badness: 0.042487
Line 97: Line 96:
POTE generator: ~49/40 = 351.502
POTE generator: ~49/40 = 351.502


Vals: {{Val list| 58, 140, 198, 536f }}
Optimal GPV sequence: {{Val list| 58, 140, 198, 536f }}


Badness: 0.021188
Badness: 0.021188
Line 112: Line 111:
POTE generator: ~243/220 = 175.7378
POTE generator: ~243/220 = 175.7378


Vals: {{Val list| 41, 157, 198, 239, 676b, 915be }}
Optimal GPV sequence: {{Val list| 41, 157, 198, 239, 676b, 915be }}


Badness: 0.040170
Badness: 0.040170
Line 125: Line 124:
POTE generator: ~72/65 = 175.7470
POTE generator: ~72/65 = 175.7470


Vals: {{Val list| 41, 157, 198, 437f, 635bcff }}
Optimal GPV sequence: {{Val list| 41, 157, 198, 437f, 635bcff }}


Badness: 0.031144
Badness: 0.031144
Line 133: Line 132:


Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo|171EDO]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo|171EDO]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 158: Line 156:
POTE generator: ~256/245 = 77.227
POTE generator: ~256/245 = 77.227


Vals: {{Val list| 31, 109e, 140e, 171, 202 }}
Optimal GPV sequence: {{Val list| 31, 109e, 140e, 171, 202 }}


Badness: 0.035576
Badness: 0.035576
Line 171: Line 169:
POTE generator: ~117/112 = 77.203
POTE generator: ~117/112 = 77.203


Vals: {{Val list| 31, 109e, 140e, 171 }}
Optimal GPV sequence: {{Val list| 31, 109e, 140e, 171 }}


Badness: 0.036876
Badness: 0.036876
Line 184: Line 182:
POTE generator: ~68/65 = 77.201
POTE generator: ~68/65 = 77.201


Vals: {{Val list| 31, 109eg, 140e, 171 }}
Optimal GPV sequence: {{Val list| 31, 109eg, 140e, 171 }}


Badness: 0.027398
Badness: 0.027398
Line 197: Line 195:
POTE generator: ~22/21 = 77.173
POTE generator: ~22/21 = 77.173


Vals: {{Val list| 31, 109, 140, 171e, 311e }}
Optimal GPV sequence: {{Val list| 31, 109, 140, 171e, 311e }}


Badness: 0.030171
Badness: 0.030171
Line 210: Line 208:
POTE generator: ~22/21 = 77.158
POTE generator: ~22/21 = 77.158


Vals: {{Val list| 31, 109, 140, 311e, 451ee }}
Optimal GPV sequence: {{Val list| 31, 109, 140, 311e, 451ee }}


Badness: 0.028384
Badness: 0.028384
Line 223: Line 221:
POTE generator: ~22/21 = 77.162
POTE generator: ~22/21 = 77.162


Vals: {{Val list| 31, 109g, 140, 311e, 451ee }}
Optimal GPV sequence: {{Val list| 31, 109g, 140, 311e, 451ee }}


Badness: 0.022416
Badness: 0.022416
Line 236: Line 234:
POTE generator: ~45/44 = 38.596
POTE generator: ~45/44 = 38.596


Vals: {{Val list| 31, 280, 311, 342 }}
Optimal GPV sequence: {{Val list| 31, 280, 311, 342 }}


Badness: 0.015633
Badness: 0.015633
Line 249: Line 247:
POTE generator: ~45/44 = 38.588
POTE generator: ~45/44 = 38.588


Vals: {{Val list| 31, 280, 311, 964f, 1275f, 1586cff }}
Optimal GPV sequence: {{Val list| 31, 280, 311, 964f, 1275f, 1586cff }}


Badness: 0.033573
Badness: 0.033573
Line 262: Line 260:
POTE generator: ~45/44 = 38.589
POTE generator: ~45/44 = 38.589


Vals: {{Val list| 31, 280, 311, 653f, 964f }}
Optimal GPV sequence: {{Val list| 31, 280, 311, 653f, 964f }}


Badness: 0.025298
Badness: 0.025298
Line 275: Line 273:


Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with {{multival| 12 34 20 30 52 …}} as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with {{multival| 12 34 20 30 52 …}} as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 300: Line 297:
POTE generator: ~21/20 = 83.167
POTE generator: ~21/20 = 83.167


Vals: {{Val list| 14c, 58, 72, 130, 202 }}
Optimal GPV sequence: {{Val list| 14c, 58, 72, 130, 202 }}


Badness: 0.015867
Badness: 0.015867
Line 313: Line 310:
POTE generator: ~21/20 = 83.116
POTE generator: ~21/20 = 83.116


Vals: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}
Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}


Badness: 0.013046
Badness: 0.013046
Line 326: Line 323:
POTE generator: ~21/20 = 83.168
POTE generator: ~21/20 = 83.168


Vals: {{Val list| 14cf, 58, 72, 130, 202g }}
Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 202g }}


Badness: 0.012657
Badness: 0.012657
Line 334: Line 331:


Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.
Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 359: Line 355:
POTE generator: ~90/77 = 271.111
POTE generator: ~90/77 = 271.111


Vals: {{Val list| 31, 208, 239, 270 }}
Optimal GPV sequence: {{Val list| 31, 208, 239, 270 }}


Badness: 0.017540
Badness: 0.017540
Line 372: Line 368:
POTE generator: ~90/77 = 271.107
POTE generator: ~90/77 = 271.107


Vals: {{Val list| 31, 239, 270, 571, 841, 1111 }}
Optimal GPV sequence: {{Val list| 31, 239, 270, 571, 841, 1111 }}


Badness: 0.017921
Badness: 0.017921
Line 379: Line 375:
{{see also| Qintosec family #Decoid }}
{{see also| Qintosec family #Decoid }}


Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14ths equal temperament|linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]].
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14 equal-step tuning|linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]].
 


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 407: Line 402:
POTE generator: ~16/15 = 111.070
POTE generator: ~16/15 = 111.070


Vals: {{Val list| 10e, 130, 270, 670, 940, 1210, 2150c }}
Optimal GPV sequence: {{Val list| 10e, 130, 270, 670, 940, 1210, 2150c }}


Badness: 0.018735
Badness: 0.018735
Line 420: Line 415:
POTE generator: ~16/15 = 111.083
POTE generator: ~16/15 = 111.083


Vals: {{Val list| 10e, 130, 270, 940, 1210f, 1480cf }}
Optimal GPV sequence: {{Val list| 10e, 130, 270, 940, 1210f, 1480cf }}


Badness: 0.013475
Badness: 0.013475
Line 450: Line 445:
POTE generator: ~33/28 = 283.276
POTE generator: ~33/28 = 283.276


Vals: {{Val list| 72, 161, 233, 305 }}
Optimal GPV sequence: {{Val list| 72, 161, 233, 305 }}


Badness: 0.027959
Badness: 0.027959
Line 463: Line 458:
POTE generator: ~13/11 = 283.294
POTE generator: ~13/11 = 283.294


Vals: {{Val list| 72, 161f, 233f }}
Optimal GPV sequence: {{Val list| 72, 161f, 233f }}


Badness: 0.026942
Badness: 0.026942
Line 493: Line 488:
POTE generator: ~1372/1089 = 392.991
POTE generator: ~1372/1089 = 392.991


Vals: {{Val list| 58, 113, 171 }}
Optimal GPV sequence: {{Val list| 58, 113, 171 }}


Badness: 0.052358
Badness: 0.052358
Line 506: Line 501:
POTE generator: ~180/143 = 392.989
POTE generator: ~180/143 = 392.989


Vals: {{Val list| 58, 113, 171 }}
Optimal GPV sequence: {{Val list| 58, 113, 171 }}


Badness: 0.026974
Badness: 0.026974
Line 519: Line 514:
POTE generator: ~64/51 = 392.985
POTE generator: ~64/51 = 392.985


Vals: {{Val list| 58, 113, 171 }}
Optimal GPV sequence: {{Val list| 58, 113, 171 }}


Badness: 0.023205
Badness: 0.023205
Line 566: Line 561:
POTE generator: ~14/11 = 416.718
POTE generator: ~14/11 = 416.718


Vals: {{Val list| 72, 167, 239, 311, 1316c }}
Optimal GPV sequence: {{Val list| 72, 167, 239, 311, 1316c }}


Badness: 0.022926
Badness: 0.022926
Line 579: Line 574:
POTE generator: ~14/11 = 416.716
POTE generator: ~14/11 = 416.716


Vals: {{Val list| 72, 311, 694, 1005c, 1699cd }}
Optimal GPV sequence: {{Val list| 72, 311, 694, 1005c, 1699cd }}


Badness: 0.020888
Badness: 0.020888
Line 609: Line 604:
POTE generator: ~49/40 = 351.115
POTE generator: ~49/40 = 351.115


Vals: {{Val list| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
Optimal GPV sequence: {{Val list| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}


Badness: 0.019461
Badness: 0.019461
Line 622: Line 617:
POTE generator: ~49/40 = 351.117
POTE generator: ~49/40 = 351.117


Vals: {{Val list| 41, 229, 270, 581, 851, 2283b, 3134b }}
Optimal GPV sequence: {{Val list| 41, 229, 270, 581, 851, 2283b, 3134b }}


Badness: 0.013830
Badness: 0.013830
Line 632: Line 627:


While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.  
While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.  


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 657: Line 651:
POTE generator: ~21/20 = 84.8843
POTE generator: ~21/20 = 84.8843


Vals: {{val list| 99, 212e, 311, 410, 721, 1032, 1343 }}
Optimal GPV sequence: {{val list| 99, 212e, 311, 410, 721, 1032, 1343 }}


Badness: 0.101
Badness: 0.100668


==== 13-limit ====
==== 13-limit ====
Line 670: Line 664:
POTE generator: ~21/20 = 84.8838
POTE generator: ~21/20 = 84.8838


Vals: {{val list| 99, 212ef, 311, 410, 721, 1032 }}
Optimal GPV sequence: {{val list| 99, 212ef, 311, 410, 721, 1032 }}


Badness: 0.0499
Badness: 0.049893


=== Amorous ===
=== Amorous ===
Line 683: Line 677:
POTE generator: ~21/20 = 84.8896
POTE generator: ~21/20 = 84.8896


Vals: {{val list| 99e, 212, 311, 1145c, 1456cd }}
Optimal GPV sequence: {{val list| 99e, 212, 311, 1145c, 1456cd }}


Badness: 0.0489
Badness: 0.048924


==== 13-limit ====
==== 13-limit ====
Line 696: Line 690:
POTE generator: ~21/20 = 84.8910
POTE generator: ~21/20 = 84.8910


Vals: {{val list| 99ef, 212, 311, 834, 1145c }}
Optimal GPV sequence: {{val list| 99ef, 212, 311, 834, 1145c }}


Badness: 0.0347
Badness: 0.034681


=== Pseudoamical ===
=== Pseudoamical ===
Line 709: Line 703:
POTE generator: ~21/20 = 84.9091
POTE generator: ~21/20 = 84.9091


Vals: {{val list| 99, 113, 212, 961ccdeee }}
Optimal GPV sequence: {{val list| 99, 113, 212, 961ccdeee }}


Badness: 0.0858
Badness: 0.085837


==== 13-limit ====
==== 13-limit ====
Line 722: Line 716:
POTE generator: ~21/20 = 84.9127
POTE generator: ~21/20 = 84.9127


Vals: {{val list| 99, 113, 212, 537cdeff, 749ccdeefff }}
Optimal GPV sequence: {{val list| 99, 113, 212, 537cdeff, 749ccdeefff }}


Badness: 0.0470
Badness: 0.047025


=== Pseudoamorous ===
=== Pseudoamorous ===
Line 735: Line 729:
POTE generator: ~21/20 = 84.8917
POTE generator: ~21/20 = 84.8917


Vals: {{val list| 99e, 212e }}
Optimal GPV sequence: {{val list| 99e, 212e }}


Badness: 0.0566
Badness: 0.056583


==== 13-limit ====
==== 13-limit ====
Line 748: Line 742:
POTE generator: ~21/20 = 84.9164
POTE generator: ~21/20 = 84.9164


Vals: {{val list| 99ef, 113, 212ef }}
Optimal GPV sequence: {{val list| 99ef, 113, 212ef }}


Badness: 0.0428
Badness: 0.042826


=== Floral ===
=== Floral ===
Line 761: Line 755:
POTE generator: ~21/20 = 84.8788
POTE generator: ~21/20 = 84.8788


Vals: {{val list| 198, 212, 410 }}
Optimal GPV sequence: {{val list| 198, 212, 410 }}


Badness: 0.0651
Badness: 0.065110


==== 13-limit ====
==== 13-limit ====
Line 774: Line 768:
POTE generator: ~21/20 = 84.8750
POTE generator: ~21/20 = 84.8750


Vals: {{val list| 198, 410 }}
Optimal GPV sequence: {{val list| 198, 410 }}


Badness: 0.0370
Badness: 0.037013


=== Humorous ===
=== Humorous ===
Line 787: Line 781:
POTE generator: ~4096/3993 = 42.4391
POTE generator: ~4096/3993 = 42.4391


Vals: {{val list| 85c, 113, 198, 311, 509, 820 }}
Optimal GPV sequence: {{val list| 85c, 113, 198, 311, 509, 820 }}


Badness: 0.0582
Badness: 0.058249


==== 13-limit ====
==== 13-limit ====
Line 800: Line 794:
POTE generator: ~40/39 = 42.4391
POTE generator: ~40/39 = 42.4391


Vals: {{val list| 85c, 113, 198, 311, 509, 820f }}
Optimal GPV sequence: {{val list| 85c, 113, 198, 311, 509, 820f }}


Badness: 0.0283
Badness: 0.028267


== Septidiasemi ==
== Septidiasemi ==
{{Main| Septidiasemi }}
{{Main| Septidiasemi }}
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 834: Line 828:
POTE generator: ~15/14 = 119.279
POTE generator: ~15/14 = 119.279


Vals: {{Val list| 10, 151, 161, 171, 332 }}
Optimal GPV sequence: {{Val list| 10, 151, 161, 171, 332 }}


Badness: 0.090687
Badness: 0.090687
Line 847: Line 841:
POTE generator: ~15/14 = 119.281
POTE generator: ~15/14 = 119.281


Vals: {{Val list| 10, 151, 161, 171, 332, 835eeff }}
Optimal GPV sequence: {{Val list| 10, 151, 161, 171, 332, 835eeff }}


Badness: 0.045773
Badness: 0.045773
Line 860: Line 854:
POTE generator: ~15/14 = 119.281
POTE generator: ~15/14 = 119.281


Vals: {{Val list| 10, 151, 161, 171, 332, 503ef, 835eeff }}
Optimal GPV sequence: {{Val list| 10, 151, 161, 171, 332, 503ef, 835eeff }}


Badness: 0.027322
Badness: 0.027322
Line 939: Line 933:
POTE generator: ~8/7 = 227.500
POTE generator: ~8/7 = 227.500


Vals: {{Val list| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
Optimal GPV sequence: {{Val list| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


Badness: 0.059260
Badness: 0.059260
Line 952: Line 946:
POTE generator: ~8/7 = 227.493
POTE generator: ~8/7 = 227.493


Vals: {{Val list| 21, 37, 58, 153bcef, 211bccdeeff }}
Optimal GPV sequence: {{Val list| 21, 37, 58, 153bcef, 211bccdeeff }}


Badness: 0.032205
Badness: 0.032205
Line 980: Line 974:
POTE generator: ~100/77 = 454.318
POTE generator: ~100/77 = 454.318


Vals: {{Val list| 37, 103, 140, 243e }}
Optimal GPV sequence: {{Val list| 37, 103, 140, 243e }}


Badness: 0.056514
Badness: 0.056514
Line 993: Line 987:
POTE generator: ~13/10 = 454.316
POTE generator: ~13/10 = 454.316


Vals: {{Val list| 37, 103, 140, 243e }}
Optimal GPV sequence: {{Val list| 37, 103, 140, 243e }}


Badness: 0.027429
Badness: 0.027429
Line 999: Line 993:
== Mintone ==
== Mintone ==
In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo|-3 11 -5 -1}} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58&103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo|-3 11 -5 -1}} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58&103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 1,024: Line 1,017:
POTE generator: ~10/9 = 186.345
POTE generator: ~10/9 = 186.345


Vals: {{Val list| 58, 103, 161, 425b, 586b, 747bc }}
Optimal GPV sequence: {{Val list| 58, 103, 161, 425b, 586b, 747bc }}


Badness: 0.039962
Badness: 0.039962
Line 1,037: Line 1,030:
POTE generator: ~10/9 = 186.347
POTE generator: ~10/9 = 186.347


Vals: {{Val list| 58, 103, 161, 425b, 586bf }}
Optimal GPV sequence: {{Val list| 58, 103, 161, 425b, 586bf }}


Badness: 0.021849
Badness: 0.021849
Line 1,050: Line 1,043:
POTE generator: ~10/9 = 186.348
POTE generator: ~10/9 = 186.348


Vals: {{Val list| 58, 103, 161, 425b, 586bf }}
Optimal GPV sequence: {{Val list| 58, 103, 161, 425b, 586bf }}


Badness: 0.020295
Badness: 0.020295
Line 1,080: Line 1,073:
POTE generator: ~250/189 = 489.252
POTE generator: ~250/189 = 489.252


Vals: {{Val list| 103, 130, 233, 363, 493e, 856be }}
Optimal GPV sequence: {{Val list| 103, 130, 233, 363, 493e, 856be }}


Badness: 0.036785
Badness: 0.036785
Line 1,093: Line 1,086:
POTE generator: ~65/49 = 489.256
POTE generator: ~65/49 = 489.256


Vals: {{Val list| 103, 130, 233, 363 }}
Optimal GPV sequence: {{Val list| 103, 130, 233, 363 }}


Badness: 0.021694
Badness: 0.021694
Line 1,121: Line 1,114:
POTE generator: ~7/5 = 583.387
POTE generator: ~7/5 = 583.387


Vals: {{Val list| 35, 37, 72, 109, 181, 253 }}
Optimal GPV sequence: {{Val list| 35, 37, 72, 109, 181, 253 }}


Badness: 0.032225
Badness: 0.032225
Line 1,134: Line 1,127:
POTE generator: ~7/5 = 583.387
POTE generator: ~7/5 = 583.387


Vals: {{Val list| 37, 72, 109, 181f }}
Optimal GPV sequence: {{Val list| 37, 72, 109, 181f }}


Badness: 0.028683
Badness: 0.028683
Line 1,162: Line 1,155:
POTE generator: ~11/9 = 348.639
POTE generator: ~11/9 = 348.639


Vals: {{Val list| 31, 86ce, 117ce, 148bce }}
Optimal GPV sequence: {{Val list| 31, 86ce, 117ce, 148bce }}


Badness: 0.046181
Badness: 0.046181

Revision as of 12:41, 8 January 2022

This page discusses miscellaneous rank-2 temperaments tempering out the breedsma, [-5 -1 -2 4⟩ = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12EDO, for example) which does not possess a neutral third cannot be tempering out the breedsma.

The breedsma is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, 25/24. We might note also that 49/40 × 10/7 = 7/4 and 49/40 × (10/7)2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.

Temperaments discussed elsewhere include:

Hemififths

Main article: Hemififths

Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with 99EDO and 140EDO providing good tunings, and 239EDO an even better one; and other possible tunings are 160(1/25), giving just 5s, the 7- and 9-odd-limit minimax tuning, or 14(1/13), giving just 7s. It may be called the 41&58 temperament. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS[clarification needed].

By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 5120/5103

Mapping: [⟨1 1 -5 -1], ⟨0 2 25 13]]

Wedgie: ⟨⟨ 2 25 13 35 15 -40 ]]

POTE generator: ~49/40 = 351.477

Minimax tuning:

[[1 0 0 0⟩, [7/5 0 2/25 0⟩, [0 0 1 0⟩, [8/5 0 13/25 0⟩]
Eigenmonzos: 2, 5

Algebraic generator: (2 + sqrt(2))/2

Template:Val list

Badness: 0.022243

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 896/891

Mapping: [⟨1 1 -5 -1 2], ⟨0 2 25 13 5]]

POTE generator: ~11/9 = 351.521

Optimal GPV sequence: Template:Val list

Badness: 0.023498

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 364/363

Mapping: [⟨1 1 -5 -1 2 4], ⟨0 2 25 13 5 -1]]

POTE generator: ~11/9 = 351.573

Optimal GPV sequence: Template:Val list

Badness: 0.019090

Semihemi

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3388/3375, 5120/5103

Mapping: [⟨2 0 -35 -15 -47], ⟨0 2 25 13 34]]

POTE generator: ~49/40 = 351.505

Optimal GPV sequence: Template:Val list

Badness: 0.042487

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 1716/1715

Mapping: [⟨2 0 -35 -15 -47 -37], ⟨0 2 25 13 34 28]]

POTE generator: ~49/40 = 351.502

Optimal GPV sequence: Template:Val list

Badness: 0.021188

Quadrafifths

This has been logged as semihemififths in Graham Breed's temperament finder, but quadrafifths arguably makes more sense.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5120/5103

Mapping: [⟨1 1 -5 -1 8], ⟨0 4 50 26 -31]]

POTE generator: ~243/220 = 175.7378

Optimal GPV sequence: Template:Val list

Badness: 0.040170

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 2401/2400, 3025/3024

Mapping: [⟨1 1 -5 -1 8 10], ⟨0 4 50 26 -31 -43]]

POTE generator: ~72/65 = 175.7470

Optimal GPV sequence: Template:Val list

Badness: 0.031144

Tertiaseptal

Main article: Tertiaseptal

Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. 171EDO makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 65625/65536

Mapping: [⟨1 3 2 3], ⟨0 -22 5 -3]]

Wedgie: ⟨⟨ 22 -5 3 -59 -57 21 ]]

POTE generator: ~256/245 = 77.191

Template:Val list

Badness: 0.012995

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 65625/65536

Mapping: [⟨1 3 2 3 7], ⟨0 -22 5 -3 -55]]

POTE generator: ~256/245 = 77.227

Optimal GPV sequence: Template:Val list

Badness: 0.035576

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 625/624, 3584/3575

Mapping: [⟨1 3 2 3 7 1], ⟨0 -22 5 -3 -55 42]]

POTE generator: ~117/112 = 77.203

Optimal GPV sequence: Template:Val list

Badness: 0.036876

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575

Mapping: [⟨1 3 2 3 7 1 1], ⟨0 -22 5 -3 -55 42 48]]

POTE generator: ~68/65 = 77.201

Optimal GPV sequence: Template:Val list

Badness: 0.027398

Tertia

Subgroup:2.3.5.7.11

Comma list: 385/384, 1331/1323, 1375/1372

Mapping: [⟨1 3 2 3 5], ⟨0 -22 5 -3 -24]]

POTE generator: ~22/21 = 77.173

Optimal GPV sequence: Template:Val list

Badness: 0.030171

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 1331/1323

Mapping: [⟨1 3 2 3 5 1], ⟨0 -22 5 -3 -24 42]]

POTE generator: ~22/21 = 77.158

Optimal GPV sequence: Template:Val list

Badness: 0.028384

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 715/714

Mapping: [⟨1 3 2 3 5 1 1], ⟨0 -22 5 -3 -24 42 48]]

POTE generator: ~22/21 = 77.162

Optimal GPV sequence: Template:Val list

Badness: 0.022416

Hemitert

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 65625/65536

Mapping: [⟨1 3 2 3 6], ⟨0 -44 10 -6 -79]]

POTE generator: ~45/44 = 38.596

Optimal GPV sequence: Template:Val list

Badness: 0.015633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095

Mapping: [⟨1 3 2 3 6 1], ⟨0 -44 10 -6 -79 84]]

POTE generator: ~45/44 = 38.588

Optimal GPV sequence: Template:Val list

Badness: 0.033573

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095

Mapping: [⟨1 3 2 3 6 1 1], ⟨0 -44 10 -6 -79 84 96]]

POTE generator: ~45/44 = 38.589

Optimal GPV sequence: Template:Val list

Badness: 0.025298

Harry

Main article: Harry
See also: Gravity family § Harry

Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament. The period is half an octave, and the generator 21/20, with generator tunings of 9\130 or 14\202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.

Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is ⟨⟨ 12 34 20 30 … ]].

Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with ⟨⟨ 12 34 20 30 52 … ]] as the octave wedgie. 130EDO is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 19683/19600

Mapping: [⟨2 4 7 7], ⟨0 -6 -17 -10]]

Wedgie: ⟨⟨ 12 34 20 26 -2 -49 ]]

POTE generator: ~21/20 = 83.156

Template:Val list

Badness: 0.034077

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 4000/3993

Mapping: [⟨2 4 7 7 9], ⟨0 -6 -17 -10 -15]]

POTE generator: ~21/20 = 83.167

Optimal GPV sequence: Template:Val list

Badness: 0.015867

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363, 441/440

Mapping: [⟨2 4 7 7 9 11], ⟨0 -6 -17 -10 -15 -26]]

POTE generator: ~21/20 = 83.116

Optimal GPV sequence: Template:Val list

Badness: 0.013046

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 289/288, 351/350, 441/440

Mapping: [⟨2 4 7 7 9 11 9], ⟨0 -6 -17 -10 -15 -26 -6]]

POTE generator: ~21/20 = 83.168

Optimal GPV sequence: Template:Val list

Badness: 0.012657

Quasiorwell

In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = [22 -1 -10 1⟩. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)1/8, giving just 7s, or 3841/38, giving pure fifths.

Adding 3025/3024 extends to the 11-limit and gives ⟨⟨ 38 -3 8 64 … ]] for the initial wedgie, and as expected, 270 remains an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 29360128/29296875

Mapping: [⟨1 31 0 9], ⟨0 -38 3 -8]]

Wedgie: ⟨⟨ 38 -3 8 -93 -94 27 ]]

POTE generator: ~1024/875 = 271.107

Template:Val list

Badness: 0.035832

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5632/5625

Mapping: [⟨1 31 0 9 53], ⟨0 -38 3 -8 -64]]

POTE generator: ~90/77 = 271.111

Optimal GPV sequence: Template:Val list

Badness: 0.017540

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095

Mapping: [⟨1 31 0 9 53 -59], ⟨0 -38 3 -8 -64 81]]

POTE generator: ~90/77 = 271.107

Optimal GPV sequence: Template:Val list

Badness: 0.017921

Decoid

See also: Qintosec family § Decoid

Decoid tempers out 2401/2400 and 67108864/66976875, as well as the linus comma, [11 -10 -10 10⟩. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the qintosec temperament.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 67108864/66976875

Mapping: [⟨10 0 47 36], ⟨0 2 -3 -1]]

Mapping generators: ~15/14, ~8192/4725

Wedgie: ⟨⟨ 20 -30 -10 -94 -72 61 ]]

POTE generator: ~16/15 = 111.099

Template:Val list

Badness: 0.033902

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625, 9801/9800

Mapping: [⟨10 0 47 36 98], ⟨0 2 -3 -1 -8]]

POTE generator: ~16/15 = 111.070

Optimal GPV sequence: Template:Val list

Badness: 0.018735

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095

Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]

POTE generator: ~16/15 = 111.083

Optimal GPV sequence: Template:Val list

Badness: 0.013475

Neominor

The generator for neominor temperament is tridecimal minor third 13/11, also known as Neo-gothic minor third.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 177147/175616

Mapping: [⟨1 3 12 8], ⟨0 -6 -41 -22]]

Wedgie: ⟨⟨ 6 41 22 51 18 -64 ]]

POTE generator: ~189/160 = 283.280

Template:Val list

Badness: 0.088221

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 35937/35840

Mapping: [⟨1 3 12 8 7], ⟨0 -6 -41 -22 -15]]

POTE generator: ~33/28 = 283.276

Optimal GPV sequence: Template:Val list

Badness: 0.027959

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 364/363, 441/440

Mapping: [⟨1 3 12 8 7 7], ⟨0 -6 -41 -22 -15 -14]]

POTE generator: ~13/11 = 283.294

Optimal GPV sequence: Template:Val list

Badness: 0.026942

Emmthird

The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 14348907/14336000

Mapping: [⟨1 -3 -17 -8], ⟨0 14 59 33]]

Wedgie: ⟨⟨ 14 59 33 61 13 -89 ]]

POTE generator: ~2744/2187 = 392.988

Template:Val list

Badness: 0.016736

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 1792000/1771561

Mapping: [⟨1 -3 -17 -8 -8], ⟨0 14 59 33 35]]

POTE generator: ~1372/1089 = 392.991

Optimal GPV sequence: Template:Val list

Badness: 0.052358

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 2200/2197

Mapping: [⟨1 -3 -17 -8 -8 -13], ⟨0 14 59 33 35 51]]

POTE generator: ~180/143 = 392.989

Optimal GPV sequence: Template:Val list

Badness: 0.026974

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197

Mapping: [⟨1 -3 -17 -8 -8 -13 9], ⟨0 14 59 33 35 51 -15]]

POTE generator: ~64/51 = 392.985

Optimal GPV sequence: Template:Val list

Badness: 0.023205

Quinmite

The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth".

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1959552/1953125

Mapping: [⟨1 -7 -5 -3], ⟨0 34 29 23]]

Wedgie: ⟨⟨ 34 29 23 -33 -59 -28 ]]

POTE generator: ~25/21 = 302.997

Template:Val list

Badness: 0.037322

Unthirds

The generator for unthirds temperament is undecimal major third, 14/11.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 68359375/68024448

Mapping: [⟨1 -13 -14 -9], ⟨0 42 47 34]]

Wedgie: ⟨⟨ 42 47 34 -23 -64 -53 ]]

POTE generator: ~3969/3125 = 416.717

Template:Val list

Badness: 0.075253

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 4000/3993

Mapping: [⟨1 -13 -14 -9 -8], ⟨0 42 47 34 33]]

POTE generator: ~14/11 = 416.718

Optimal GPV sequence: Template:Val list

Badness: 0.022926

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400

Mapping: [⟨1 -13 -14 -9 -9 -47], ⟨0 42 47 34 33 146]]

POTE generator: ~14/11 = 416.716

Optimal GPV sequence: Template:Val list

Badness: 0.020888

Newt

This temperament has a generator of neutral third (0.2 cents flat of 49/40) and tempers out the garischisma.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 33554432/33480783

Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]

Wedgie: ⟨⟨ 2 -57 -28 -95 -50 95 ]]

POTE generator: ~49/40 = 351.113

Template:Val list

Badness: 0.041878

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 19712/19683

Mapping: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]

POTE generator: ~49/40 = 351.115

Optimal GPV sequence: Template:Val list

Badness: 0.019461

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

Mapping: [⟨1 1 19 11 -10 -20], ⟨0 2 -57 -28 46 81]]

POTE generator: ~49/40 = 351.117

Optimal GPV sequence: Template:Val list

Badness: 0.013830

Amicable

See also: Amity family

The amicable temperament tempers out the amity comma and the canousma in addition to the breedsma, and is closely associated with the canou temperament.

While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1600000/1594323

Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]

Wedgie: ⟨⟨ 20 52 31 36 -7 -74 ]]

POTE generator: ~21/20 = 84.880

Template:Val list

Badness: 0.045473

Amical

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 131072/130977, 1600000/1594323

Mapping: [⟨1 3 6 5 -8], ⟨0 -20 -52 -31 162]]

POTE generator: ~21/20 = 84.8843

Optimal GPV sequence: Template:Val list

Badness: 0.100668

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206

Mapping: [⟨1 3 6 5 -8 -5], ⟨0 -20 -52 -31 162 123]]

POTE generator: ~21/20 = 84.8838

Optimal GPV sequence: Template:Val list

Badness: 0.049893

Amorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 19712/19683

Mapping: [⟨1 3 6 5 14], ⟨0 -20 -52 -31 -149]]

POTE generator: ~21/20 = 84.8896

Optimal GPV sequence: Template:Val list

Badness: 0.048924

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647

Mapping: [⟨1 3 6 5 14 17], ⟨0 -20 -52 -31 -149 -188]]

POTE generator: ~21/20 = 84.8910

Optimal GPV sequence: Template:Val list

Badness: 0.034681

Pseudoamical

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 1600000/1594323

Mapping: [⟨1 3 6 5 -1], ⟨0 -20 -52 -31 63]]

POTE generator: ~21/20 = 84.9091

Optimal GPV sequence: Template:Val list

Badness: 0.085837

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1375/1372, 19773/19712

Mapping: [⟨1 3 6 5 -1 2], ⟨0 -20 -52 -31 63 24]]

POTE generator: ~21/20 = 84.9127

Optimal GPV sequence: Template:Val list

Badness: 0.047025

Pseudoamorous

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 980000/970299

Mapping: [⟨1 3 6 5 7], ⟨0 -20 -52 -31 -50]]

POTE generator: ~21/20 = 84.8917

Optimal GPV sequence: Template:Val list

Badness: 0.056583

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 1875/1859

Mapping: [⟨1 3 6 5 7 10], ⟨0 -20 -52 -31 -50 -89]]

POTE generator: ~21/20 = 84.9164

Optimal GPV sequence: Template:Val list

Badness: 0.042826

Floral

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 14641/14580

Mapping: [⟨2 6 12 10 13], ⟨0 -20 -52 -31 -43]]

POTE generator: ~21/20 = 84.8788

Optimal GPV sequence: Template:Val list

Badness: 0.065110

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580

Mapping: [⟨2 6 12 10 13 19], ⟨0 -20 -52 -31 -43 -82]]

POTE generator: ~21/20 = 84.8750

Optimal GPV sequence: Template:Val list

Badness: 0.037013

Humorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 1600000/1594323

Mapping: [⟨1 3 6 5 3], ⟨0 -40 -104 -62 13]]

POTE generator: ~4096/3993 = 42.4391

Optimal GPV sequence: Template:Val list

Badness: 0.058249

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024

Mapping: [⟨1 3 6 5 3 6], ⟨0 -40 -104 -62 13 -65]]

POTE generator: ~40/39 = 42.4391

Optimal GPV sequence: Template:Val list

Badness: 0.028267

Septidiasemi

Main article: Septidiasemi

Aside from 2401/2400, septidiasemi tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of 15/14). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 2152828125/2147483648

Mapping: [⟨1 -1 6 4], ⟨0 26 -37 -12]]

Wedgie: ⟨⟨ 26 -37 -12 -119 -92 76 ]]

POTE generator: ~15/14 = 119.297

Template:Val list

Badness: 0.044115

Sedia

The sedia temperament (10&161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 939524096/935859375

Mapping: [⟨1 -1 6 4 -3], ⟨0 26 -37 -12 65]]

POTE generator: ~15/14 = 119.279

Optimal GPV sequence: Template:Val list

Badness: 0.090687

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 2200/2197, 3584/3575

Mapping: [⟨1 -1 6 4 -3 4], ⟨0 26 -37 -12 65 -3]]

POTE generator: ~15/14 = 119.281

Optimal GPV sequence: Template:Val list

Badness: 0.045773

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575

Mapping: [⟨1 -1 6 4 -3 4 2], ⟨0 26 -37 -12 65 -3 21]]

POTE generator: ~15/14 = 119.281

Optimal GPV sequence: Template:Val list

Badness: 0.027322

Maviloid

See also: Ragismic microtemperaments § Parakleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1224440064/1220703125

Mapping: [⟨1 31 34 26], ⟨0 -52 -56 -41]]

Wedgie: ⟨⟨ 52 56 41 -32 -81 -62 ]]

POTE generator: ~1296/875 = 678.810

Template:Val list

Badness: 0.057632

Subneutral

See also: Luna family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 274877906944/274658203125

Mapping: [⟨1 19 0 6], ⟨0 -60 8 -11]]

Wedgie: ⟨⟨ 60 -8 11 -152 -151 48 ]]

POTE generator: ~57344/46875 = 348.301

Template:Val list

Badness: 0.045792

Osiris

See also: Metric microtemperaments § Geb

Subgroup: 2.3.5.7

Comma list: 2401/2400, 31381059609/31360000000

Mapping: [⟨1 13 33 21], ⟨0 -32 -86 -51]]

Wedgie: ⟨⟨ 32 86 51 62 -9 -123 ]]

POTE generator: ~2800/2187 = 428.066

Template:Val list

Badness: 0.028307

Gorgik

Subgroup: 2.3.5.7

Comma list: 2401/2400, 28672/28125

Mapping: [⟨1 5 1 3], ⟨0 -18 7 -1]]

Wedgie: ⟨⟨ 18 -7 1 -53 -49 22 ]]

POTE generator: ~8/7 = 227.512

Template:Val list

Badness: 0.158384

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 2401/2400, 2560/2541

Mapping: [⟨1 5 1 3 1], ⟨0 -18 7 -1 13]]

POTE generator: ~8/7 = 227.500

Optimal GPV sequence: Template:Val list

Badness: 0.059260

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 196/195, 364/363, 512/507

Mapping: [⟨1 5 1 3 1 2], ⟨0 -18 7 -1 13 9]]

POTE generator: ~8/7 = 227.493

Optimal GPV sequence: Template:Val list

Badness: 0.032205

Fibo

Subgroup: 2.3.5.7

Comma list: 2401/2400, 341796875/339738624

Mapping: [⟨1 19 8 10], ⟨0 -46 -15 -19]]

Wedgie: ⟨⟨ 46 15 19 -83 -99 2 ]]

POTE generator: ~125/96 = 454.310

Template:Val list

Badness: 0.100511

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 43923/43750

Mapping: [⟨1 19 8 10 8], ⟨0 -46 -15 -19 -12]]

POTE generator: ~100/77 = 454.318

Optimal GPV sequence: Template:Val list

Badness: 0.056514

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 847/845, 1375/1372

Mapping: [⟨1 19 8 10 8 9], ⟨0 -46 -15 -19 -12 -14]]

POTE generator: ~13/10 = 454.316

Optimal GPV sequence: Template:Val list

Badness: 0.027429

Mintone

In addition to 2401/2400, mintone tempers out 177147/175000 = [-3 11 -5 -1⟩ in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58&103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 177147/175000

Mapping: [⟨1 5 9 7], ⟨0 -22 -43 -27]]

Wedgie: ⟨⟨ 22 43 27 17 -19 -58 ]]

POTE generator: ~10/9 = 186.343

Template:Val list

Badness: 0.125672

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 43923/43750

Mapping: [⟨1 5 9 7 12], ⟨0 -22 -43 -27 -55]]

POTE generator: ~10/9 = 186.345

Optimal GPV sequence: Template:Val list

Badness: 0.039962

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 847/845

Mapping: [⟨1 5 9 7 12 11], ⟨0 -22 -43 -27 -55 -47]]

POTE generator: ~10/9 = 186.347

Optimal GPV sequence: Template:Val list

Badness: 0.021849

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 351/350, 441/440, 561/560, 847/845

Mapping: [⟨1 5 9 7 12 11 3], ⟨0 -22 -43 -27 -55 -47 7]]

POTE generator: ~10/9 = 186.348

Optimal GPV sequence: Template:Val list

Badness: 0.020295

Catafourth

See also: Sensipent family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 78732/78125

Mapping: [⟨1 13 17 13], ⟨0 -28 -36 -25]]

Wedgie: ⟨⟨ 28 36 25 -8 -39 -43 ]]

POTE generator: ~250/189 = 489.235

Template:Val list

Badness: 0.079579

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 78408/78125

Mapping: [⟨1 13 17 13 32], ⟨0 -28 -36 -25 -70]]

POTE generator: ~250/189 = 489.252

Optimal GPV sequence: Template:Val list

Badness: 0.036785

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 10985/10976

Mapping: [⟨1 13 17 13 32 9], ⟨0 -28 -36 -25 -70 -13]]

POTE generator: ~65/49 = 489.256

Optimal GPV sequence: Template:Val list

Badness: 0.021694

Cotritone

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625/387072

Mapping: [⟨1 -13 -4 -4], ⟨0 30 13 14]]

Wedgie: ⟨⟨ 30 13 14 -49 -62 -4 ]]

POTE generator: ~7/5 = 583.385

Template:Val list

Badness: 0.098322

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 4000/3993

Mapping: [⟨1 -13 -4 -4 2], ⟨0 30 13 14 3]]

POTE generator: ~7/5 = 583.387

Optimal GPV sequence: Template:Val list

Badness: 0.032225

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 364/363, 385/384, 625/624

Mapping: [⟨1 -13 -4 -4 2 -7], ⟨0 30 13 14 3 22]]

POTE generator: ~7/5 = 583.387

Optimal GPV sequence: Template:Val list

Badness: 0.028683

Quasimoha

For the 5-limit version of this temperament, see High badness temperaments #Quasimoha.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3645/3584

Mapping: [⟨1 1 9 6], ⟨0 2 -23 -11]]

POTE generator: ~49/40 = 348.603

Template:Val list

Badness: 0.110820

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 1815/1792

Mapping: [⟨1 1 9 6 2], ⟨0 2 -23 -11 5]]

POTE generator: ~11/9 = 348.639

Optimal GPV sequence: Template:Val list

Badness: 0.046181

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