525edo: Difference between revisions
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+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]] | ||
Revision as of 14:19, 16 August 2022
| ← 524edo | 525edo | 526edo → |
Theory
525edo is distinctly consistent through the 25-odd-limit. It tempers out the schisma, 32805/32768, and [8 77 -56⟩ in the 5-limit; 250047/250000, 703125/702464 and [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 [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.
525's divisors are 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.24 | -0.03 | +0.32 | -0.46 | +0.62 | +0.19 | -0.37 | +0.30 | -1.01 | +0.11 |
| Relative (%) | +0.0 | -10.5 | -1.2 | +13.9 | -20.2 | +26.9 | +8.2 | -16.2 | +13.0 | -44.0 | +4.7 | |
| Steps (reduced) |
525 (0) |
832 (307) |
1219 (169) |
1474 (424) |
1816 (241) |
1943 (368) |
2146 (46) |
2230 (130) |
2375 (275) |
2550 (450) |
2601 (501) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [512 -323⟩ | [⟨525 832]] | +0.0759 | 0.0759 | 3.32 |
| 2.3.5 | 32805/32768, [8 77 -56⟩ | [⟨525 832 1219]] | +0.0546 | 0.0689 | 3.02 |
| 2.3.5.7 | 32805/32768, 250047/250000, [21 3 1 -10⟩ | [⟨525 832 1219 1474]] | +0.0128 | 0.0940 | 4.11 |
| 2.3.5.7.11 | 3025/3024, 24057/24010, 32805/32768, 102487/102400 | [⟨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 | [⟨525 832 1219 1474 1816 1943]] | +0.0030 | 0.1164 | 5.09 |
Rank-2 temperaments
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 3 | 218\525 (43\525) |
498.29 (98.29) |
4/3 (18/17) |
Term |
| 3 | 109\525 (66\525) |
249.14 (150.86) |
15/13 (12/11) |
Hemiterm (525f) |
| 7 | 218\525 (7\525) |
498.29 (16.00) |
4/3 (99/98) |
Septant |
| 21 | 256\525 (6\525) |
585.14 (13.71) |
91875/65536 (126/125) |
Akjayland |