217edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|Comma List]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal 8ve <br>Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 344 -217 }} | | {{monzo| 344 -217 }} | ||
| {{mapping| 217 344 }} | | {{mapping| 217 344 }} | ||
| | | -0.110 | ||
| 0.1101 | | 0.1101 | ||
| 1.99 | | 1.99 | ||
| Line 27: | Line 35: | ||
| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }} | | {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }} | ||
| {{mapping| 217 344 504 }} | | {{mapping| 217 344 504 }} | ||
| | | -0.186 | ||
| 0.1398 | | 0.1398 | ||
| 2.53 | | 2.53 | ||
| Line 34: | Line 42: | ||
| 3136/3125, 4375/4374, 823543/819200 | | 3136/3125, 4375/4374, 823543/819200 | ||
| {{mapping| 217 344 504 609 }} | | {{mapping| 217 344 504 609 }} | ||
| | | -0.043 | ||
| 0.2757 | | 0.2757 | ||
| 4.99 | | 4.99 | ||
| Line 41: | Line 49: | ||
| 441/440, 3136/3125, 4000/3993, 4375/4374 | | 441/440, 3136/3125, 4000/3993, 4375/4374 | ||
| {{mapping| 217 344 504 609 751 }} | | {{mapping| 217 344 504 609 751 }} | ||
| | | -0.131 | ||
| 0.3034 | | 0.3034 | ||
| 5.49 | | 5.49 | ||
| Line 48: | Line 56: | ||
| 364/363, 441/440, 676/675, 3136/3125, 4375/4374 | | 364/363, 441/440, 676/675, 3136/3125, 4375/4374 | ||
| {{mapping| 217 344 504 609 751 803 }} | | {{mapping| 217 344 504 609 751 803 }} | ||
| | | -0.111 | ||
| 0.2808 | | 0.2808 | ||
| 5.08 | | 5.08 | ||
| Line 55: | Line 63: | ||
| 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125 | | 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125 | ||
| {{mapping| 217 344 504 609 751 803 887 }} | | {{mapping| 217 344 504 609 751 803 887 }} | ||
| | | -0.099 | ||
| 0.2616 | | 0.2616 | ||
| 4.73 | | 4.73 | ||
| Line 62: | Line 70: | ||
| 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215 | | 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215 | ||
| {{mapping| 217 344 504 609 751 803 887 922 }} | | {{mapping| 217 344 504 609 751 803 887 922 }} | ||
| | | -0.119 | ||
| 0.2504 | | 0.2504 | ||
| 4.53 | | 4.53 | ||
| Line 69: | Line 77: | ||
| 343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675 | | 343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675 | ||
| {{mapping| 217 344 504 609 751 803 887 922 982 }} | | {{mapping| 217 344 504 609 751 803 887 922 982 }} | ||
| | | -0.158 | ||
| 0.2610 | | 0.2610 | ||
| 4.72 | | 4.72 | ||
|} | |||
* 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat [[72edo|72]] in the 19-limit and [[193edo|193]] in the 23-limit. The next equal temperament that does better in either subgroup is [[243edo|243e]] for absolute error and [[270edo|270]] for relative error. | * 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat [[72edo|72]] in the 19-limit and [[193edo|193]] in the 23-limit. The next equal temperament that does better in either subgroup is [[243edo|243e]] for absolute error and [[270edo|270]] for relative error. | ||
* 23-limit is not the subgroup it does the best, with the no-23 29- and 31-limit approximated even better. | * 23-limit is not the subgroup it does the best, with the no-23 29- and 31-limit approximated even better. | ||
| Line 78: | Line 86: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br>Ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 147: | Line 161: | ||
|- | |- | ||
| 7 | | 7 | ||
| 94\217<br | | 94\217<br>(1\217) | ||
| 519.82<br | | 519.82<br>(5.53) | ||
| 27/20<br | | 27/20<br>(325/324) | ||
| [[Brahmagupta]] | | [[Brahmagupta]] | ||
|- | |- | ||
| 31 | | 31 | ||
| 90\217<br | | 90\217<br>(1\217) | ||
| 497.70<br | | 497.70<br>(5.53) | ||
| 4/3<br | | 4/3<br>(243/242) | ||
| [[Birds]] | | [[Birds]] | ||
|} | |||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Scales == | == Scales == | ||
| Line 179: | Line 193: | ||
>>> r217text = '[paste the above Ringer 217 data here]' | >>> r217text = '[paste the above Ringer 217 data here]' | ||
>>> r217=toneji(r217text) # r217 | >>> r217=toneji(r217text) # r217 | ||
>>> r31s = [[r217[7 * i + j] for i in range(31)] + [r217[j] * 2] for j in range(7)] | >>> r31s = [ [r217[7*i+j] for i in range(31)]+[r217[j]*2] for j in range(7) ] | ||
>>> r31s2 = [ toneji( | >>> r31s2 = [ toneji(':'.join([ str(h) for h in r31 ]),True) for r31 in r31s ] | ||
>>> for i in range(7): | >>> for i in range(7): | ||
print(str(i) + | print(str(i)+'th: ',':'.join([ str(h) for h in r31s2[i] ])) | ||
0th: 274:280:286:293:299:306:313:320:327:334:342:350:358:366:374:383:392:400:409:418:428:438:448:458:468:479:490:500:512:524:535:548 | 0th: 274:280:286:293:299:306:313:320:327:334:342:350:358:366:374:383:392:400:409:418:428:438:448:458:468:479:490:500:512:524:535:548 | ||
1th: 351:359:367:375:384:393:401:410:420:429:439:449:459:469:480:491:502:514:526:536:549:562:574:588:600:614:628:642:656:672:687:702 | 1th: 351:359:367:375:384:393:401:410:420:429:439:449:459:469:480:491:502:514:526:536:549:562:574:588:600:614:628:642:656:672:687:702 | ||
| Line 192: | Line 206: | ||
>>> # using the below code can be used to show that only the 0th and 1th 31nejis are mapped correctly by 31edo's patent val | >>> # using the below code can be used to show that only the 0th and 1th 31nejis are mapped correctly by 31edo's patent val | ||
>>> for i in range(7): # (output omitted to avoid spam) | >>> for i in range(7): # (output omitted to avoid spam) | ||
print(str(i) + | print(str(i)+'th:\n') | ||
worstneji(r31s2[i], 9) | worstneji(r31s2[i],9) | ||
print( | print('\n'*2) | ||
</syntaxhighlight> | </syntaxhighlight> | ||
[[Category:Arch]] | [[Category:Arch]] | ||
[[Category:Birds]] | [[Category:Birds]] | ||
[[Category:Cotoneum]] | [[Category:Cotoneum]] | ||