248edo: Difference between revisions
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+RTT table and rank-2 temperaments |
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{{EDO intro|248}} | {{EDO intro|248}} | ||
== Theory == | |||
248et tempers out [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. | 248et tempers out [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. | ||
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|248|}} | {{Harmonics in equal|248|}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 287 -181 }} | |||
| [{{val| 248 393 }}] | |||
| +0.108 | |||
| 0.108 | |||
| 2.23 | |||
|- | |||
| 2.3.5 | |||
| 32805/32768, {{monzo| 12 32 -27 }} | |||
| [{{val| 248 393 576 }}] | |||
| -0.041 | |||
| 0.228 | |||
| 4.70 | |||
|- | |||
| 2.3.5.7 | |||
| 3136/3125, 32805/32768, 420175/419904 | |||
| [{{val| 248 393 576 696 }}] | |||
| +0.066 | |||
| 0.270 | |||
| 5.58 | |||
|- | |||
| 2.3.5.7.11 | |||
| 441/440, 3136/3125, 8019/8000, 41503/41472 | |||
| [{{val| 248 393 576 696 858 }}] | |||
| +0.036 | |||
| 0.249 | |||
| 5.15 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 441/440, 729/728, 847/845, 1001/1000, 3136/3125 | |||
| [{{val| 248 393 576 696 858 918 }}] | |||
| +0.079 | |||
| 0.275 | |||
| 5.69 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 5\248 | |||
| 24.19 | |||
| 686/675 | |||
| [[Sengagen]] | |||
|- | |||
| 1 | |||
| 103\248 | |||
| 498.39 | |||
| 4/3 | |||
| [[Helmholtz]] | |||
|- | |||
| 2 | |||
| 77\248<br>(47\248) | |||
| 372.58<br>(227.42) | |||
| 26/21<br>(154/135) | |||
| [[Essence]] | |||
|- | |||
| 2 | |||
| 103\248 | |||
| 498.39 | |||
| 4/3 | |||
| [[Bischismic]] | |||
|- | |||
| 8 | |||
| 117\248<br>(7\248) | |||
| 566.13<br>(33.87) | |||
| 104/75<br>(49/48) | |||
| [[Octowerck]] | |||
|- | |||
| 31 | |||
| 103\248<br>(1\248) | |||
| 498.39<br>(4.84) | |||
| 4/3<br>(385/384) | |||
| [[Birds]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||