275edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|275}} == Theory == If harmonic 5 is used, 275et tends very sharp. It tempers out {{monzo| 24 -21 4 }} (vulture comma) and {{monzo| 19 10 -15 }}..." |
+RTT table and rank-2 temperaments |
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If harmonic 5 is used, 275et tends very sharp. It tempers out {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 19 10 -15 }} ([[trisedodge comma]]) in the 5-limit; [[6144/6125]] and [[10976/10935]] in the 7-limit. | If harmonic 5 is used, 275et tends very sharp. It tempers out {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 19 10 -15 }} ([[trisedodge comma]]) in the 5-limit; [[6144/6125]] and [[10976/10935]] in the 7-limit. | ||
The 275e val {{val| 275 436 639 772 '''952''' }} being the best, tempers out [[441/440]], [[4000/3993]], [[14700/14641]], and [[19712/19683]]. This can be extended to the 13-limit through [[364/363]], [[676/675]], [[1001/1000]], and [[2080/2079]]. | The 275e val {{val| 275 436 639 772 '''952''' }} being the best, tempers out [[441/440]], [[4000/3993]], [[14700/14641]], and [[19712/19683]]. This can be extended to the 13-limit through [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]]. | ||
The 275 val {{val| 275 436 639 772 '''951''' }} tempers out [[3025/3024]], [[3773/3750]], [[8019/8000]]. This can be extended to the 13-limit through [[352/351]], 676/675, [[1716/1715]], [[2200/2197]], and [[3584/3575]]. | The 275 val {{val| 275 436 639 772 '''951''' }} tempers out [[3025/3024]], [[3773/3750]], [[8019/8000]]. This can be extended to the 13-limit through [[352/351]], 676/675, [[1716/1715]], [[2200/2197]], and [[3584/3575]]. | ||
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|275|intervals=prime|columns=11}} | {{Harmonics in equal|275|intervals=prime|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<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 436 -275 }} | |||
| [{{val| 275 436 }}] | |||
| -0.1863 | |||
| 0.1862 | |||
| 4.27 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }} | |||
| [{{val| 275 436 639 }}] | |||
| -0.4184 | |||
| 0.3618 | |||
| 8.29 | |||
|- | |||
| 2.3.5.7 | |||
| 6144/6125, 10976/10935, 9882516/9765625 | |||
| [{{val| 275 436 639 772 }}] | |||
| -0.3051 | |||
| 0.3698 | |||
| 8.48 | |||
|- | |||
| 2.3.5.7.11 | |||
| 441/440, 4000/3993, 6144/6125, 10976/10935 | |||
| [{{val| 275 436 639 772 952 }}] (275e) | |||
| -0.4096 | |||
| 0.3912 | |||
| 8.97 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 364/363, 441/440, 676/675, 6144/6125, 10976/10935 | |||
| [{{val| 275 436 639 772 952 1018 }}] (275e) | |||
| -0.4158 | |||
| 0.3574 | |||
| 8.19 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(Reduced) | |||
! Cents<br>(Reduced) | |||
! Associated<br>Ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 6\275 | |||
| 26.18 | |||
| 1594323/1562500 | |||
| [[Sfourth]] (5-limit) | |||
|- | |||
| 1 | |||
| 109\275 | |||
| 485.64 | |||
| 320/243 | |||
| [[Vulture]] (5-limit) | |||
|- | |||
| 1 | |||
| 128\275 | |||
| 558.55 | |||
| 112/81 | |||
| [[Condor]] (275e) | |||
|- | |||
| 5 | |||
| 17\275 | |||
| 74.18 | |||
| 25/24 | |||
| [[Countdown]] (275e) | |||
|- | |||
| 11 | |||
| 114\275<br>(11\275) | |||
| 497.45<br>(48.00) | |||
| 4/3<br>(36/35) | |||
| [[Hendecatonic]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||