476edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|476}} == Theory == 476edo is consistent to the 7-odd-limit and the harmonic 3 is about halfway its steps. Using the patent val, it tempers o..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
476edo is consistent to the [[7-odd-limit]] | 476edo is [[consistent]] to the [[7-odd-limit]], but the [[harmonic]] [[3/1|3]] is about halfway its steps, while its [[5/1|5]] and [[7/1|7]] are both tuned flat. To start with, consider the 2.3.5.7 [[patent val]], as well as 2.9.15.21 and 2.9.5.7 [[subgroup]]s. | ||
Using the patent val, the equal temperament [[tempering out|tempers out]] [[2401/2400]] and [[19683/19600]] in the 7-limit, [[support]]ing [[harry]]. The 11-limit 476e val tempers out [[3025/3024]] and [[41503/41472]], whereas the patent val tempers out [[243/242]], [[441/440]], [[540/539]], [[4000/3993]], [[8019/8000]], and [[9801/9800]], supporting 11-limit harry. | |||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
476 factors into | 476 factors into {{factorisation|476}}, with subset edos {{EDOs| 2, 4, 7, 14, 17, 28, 34, 68, 119, and 238 }}. [[952edo]], which doubles it, gives a good correction to the harmonic 3, but unfortunately it is inconsistent in the [[5-odd-limit]]. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.9 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|1509 -476}} | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|476 1509}} | ! rowspan="2" | [[Mapping]] | ||
| | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.9 | |||
| {{monzo| 1509 -476 }} | |||
| {{mapping| 476 1509 }} | |||
| −0.0460 | |||
| 0.0460 | | 0.0460 | ||
| 1.82 | | 1.82 | ||
|- | |- | ||
|2.9.5 | | 2.9.5 | ||
|{{monzo|33 -17 9}}, {{monzo|-65 0 28}} | | {{monzo| 33 -17 9 }}, {{monzo| -65 0 28 }} | ||
|{{mapping|476 1509 1105}} | | {{mapping| 476 1509 1105 }} | ||
| +0.0554 | | +0.0554 | ||
| 0.1482 | | 0.1482 | ||
| 5.88 | | 5.88 | ||
|- | |- | ||
|2.9.5.7 | | 2.9.5.7 | ||
|703125/702464, 4802000/4782969, | | 703125/702464, 4802000/4782969, {{monzo| 25 3 -3 8 }} | ||
|{{mapping|476 1509 1105 1336}} | | {{mapping| 476 1509 1105 1336 }} | ||
| +0.1091 | | +0.1091 | ||
| 0.1586 | | 0.1586 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|205\476 | | 205\476 | ||
|516.81 | | 516.81 | ||
|27/20 | | 27/20 | ||
|[[ | | [[Larry]] (476) | ||
|- | |- | ||
|2 | | 2 | ||
|205\476<br>(33\476) | | 205\476<br>(33\476) | ||
|516.81<br>(83.19) | | 516.81<br>(83.19) | ||
|27/20<br>(21/20) | | 27/20<br>(21/20) | ||
|[[Harry]] | | [[Harry]] (11-limit, 476) | ||
|- | |- | ||
|28 | | 28 | ||
|197\476<br>(6\476) | | 197\476<br>(6\476) | ||
|496.64<br>(15.13) | | 496.64<br>(15.13) | ||
|4/3<br>(105/104) | | 4/3<br>(105/104) | ||
|[[Oquatonic]] | | [[Oquatonic]] (5-limit) | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
<nowiki>* | |||