275edo: Difference between revisions
+subsets and supersets; style; -redundant category |
Cleanup |
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== Theory == | == Theory == | ||
If harmonic 5 is used, 275et tends very sharp. It tempers out {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 19 10 -15 }} ( | If [[harmonic]] [[5/1|5]] is used, 275et tends very sharp. It [[tempering out|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]], [[1575/1573]] 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]], | 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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|275|intervals=prime | {{Harmonics in equal|275|intervals=prime}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
| Line 28: | Line 28: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 436 -275 }} | | {{monzo| 436 -275 }} | ||
| | | {{mapping| 275 436 }} | ||
| -0.1863 | | -0.1863 | ||
| 0.1862 | | 0.1862 | ||
| Line 35: | Line 35: | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }} | | {{monzo| 24 -21 4 }}, {{monzo| 19 10 -15 }} | ||
| | | {{mapping| 275 436 639 }} | ||
| -0.4184 | | -0.4184 | ||
| 0.3618 | | 0.3618 | ||
| Line 42: | Line 42: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 6144/6125, 10976/10935, 9882516/9765625 | | 6144/6125, 10976/10935, 9882516/9765625 | ||
| | | {{mapping| 275 436 639 772 }} | ||
| -0.3051 | | -0.3051 | ||
| 0.3698 | | 0.3698 | ||
| Line 49: | Line 49: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 441/440, 4000/3993, 6144/6125, 10976/10935 | | 441/440, 4000/3993, 6144/6125, 10976/10935 | ||
| | | {{mapping| 275 436 639 772 952 }} (275e) | ||
| -0.4096 | | -0.4096 | ||
| 0.3912 | | 0.3912 | ||
| Line 56: | Line 56: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 364/363, 441/440, 676/675, 6144/6125, 10976/10935 | | 364/363, 441/440, 676/675, 6144/6125, 10976/10935 | ||
| | | {{mapping| 275 436 639 772 952 1018 }} (275e) | ||
| -0.4158 | | -0.4158 | ||
| 0.3574 | | 0.3574 | ||
| Line 66: | Line 66: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 101: | Line 101: | ||
| [[Hendecatonic]] | | [[Hendecatonic]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||