525edo: Difference between revisions
+infobox; cleanup |
+RTT table and rank-2 temperaments |
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== Theory == | == Theory == | ||
525edo is distinctly [[consistent]] through the [[25-odd-limit]]. It tempers out the [[schisma]], 32805/32768, and {{monzo| 8 77 - | 525edo is distinctly [[consistent]] through the [[25-odd-limit]]. It tempers out the [[schisma]], 32805/32768, and {{monzo| 8 77 -56 }} in the 5-limit; [[250047/250000]], [[703125/702464]] and {{monzo| 21 3 1 -10 }} in the 7-limit; [[3025/3024]], 24057/24010, 102487/102400 and 180224/180075 in the 11-limit; [[729/728]], [[1716/1715]], [[2200/2197]], [[4096/4095]] and 14641/14625 in the 13-limit. | ||
It supports the 140 & 525 temperament, with period 35 which sets 7/5 and 10/7 to two "legs" of 35edo (17\35 and 18\35) opposing the tonic and tempers out {{monzo| 34 0 70 -70 }}, setting a circle of thirty-five [[50/49]]'s equal with the octave. In addition, it supports 21st-octave period called [[akjayland]]. | It supports the 140 & 525 temperament, with period 35 which sets 7/5 and 10/7 to two "legs" of 35edo (17\35 and 18\35) opposing the tonic and tempers out {{monzo| 34 0 70 -70 }}, setting a circle of thirty-five [[50/49]]'s equal with the octave. In addition, it supports 21st-octave period called [[akjayland]]. | ||
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|525|columns=11}} | {{Harmonics in equal|525|columns=11}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal 8ve <br>stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 512 -323 }} | |||
| [{{val| 525 832 }}] | |||
| +0.0759 | |||
| 0.0759 | |||
| 3.32 | |||
|- | |||
| 2.3.5 | |||
| 32805/32768, {{monzo| 8 77 -56 }} | |||
| [{{val| 525 832 1219 }}] | |||
| +0.0546 | |||
| 0.0689 | |||
| 3.02 | |||
|- | |||
| 2.3.5.7 | |||
| 32805/32768, 250047/250000, {{monzo| 21 3 1 -10 }} | |||
| [{{val| 525 832 1219 1474 }}] | |||
| +0.0128 | |||
| 0.0940 | |||
| 4.11 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 24057/24010, 32805/32768, 102487/102400 | |||
| [{{val| 525 832 1219 1474 1816 }}] | |||
| +0.0368 | |||
| 0.0969 | |||
| 4.24 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 729/728, 1716/1715, 2200/2197, 3025/3024, 14641/14625 | |||
| [{{val| 525 832 1219 1474 1816 1943 }}] | |||
| +0.0030 | |||
| 0.1164 | |||
| 5.09 | |||
|} | |||
=== 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 | |||
|- | |||
| 3 | |||
| 218\525<br>(43\525) | |||
| 498.29<br>(98.29) | |||
| 4/3<br>(18/17) | |||
| [[Term]] | |||
|- | |||
| 3 | |||
| 109\525<br>(66\525) | |||
| 249.14<br>(150.86) | |||
| 15/13<br>(12/11) | |||
| [[Hemiterm]] (525f) | |||
|- | |||
| 7 | |||
| 218\525<br>(7\525) | |||
| 498.29<br>(16.00) | |||
| 4/3<br>(99/98) | |||
| [[Septant]] | |||
|- | |||
| 21 | |||
| 256\525<br>(6\525) | |||
| 585.14<br>(13.71) | |||
| 91875/65536<br>(126/125) | |||
| [[Akjayland]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] | [[Category:Equal divisions of the octave|###]] | ||
[[Category:Akjayland]] | [[Category:Akjayland]] | ||