584edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== | |||
== Theory == | |||
584edo is only [[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[3/1|3]] is quite large. With reasonable approximations to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], and [[17/1|17]], it commends itself as a 2.9.5.7.11.13.17 [[subgroup]] tuning. | |||
If we use the harmonic 3 instead, we notice the better-tuned 584d [[val]] is [[enfactoring|enfactored]], with the same tuning as [[292edo]]. Therefore, we are left with the [[patent val]], which tempers out 48828125/48771072 and 67108864/66976875, [[support]]ing [[hemiluna]]. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|584}} | {{Harmonics in equal|584}} | ||
==Scales== | |||
== 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| 463 -292 }} | |||
| {{mapping| 584 926 }} | |||
| −0.2476 | |||
| 0.2475 | |||
| 12.05 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 3 -18 11 }}, {{monzo| 21 -20 4 }} | |||
| {{mapping| 584 926 1356 }} | |||
| −0.1633 | |||
| 0.2346 | |||
| 11.42 | |||
|- | |||
| 2.3.5.7 | |||
| 1500625/1492992, 1605632/1594323, 235298/234375 | |||
| {{mapping| 584 926 1356 1639 }} | |||
| −0.0319 | |||
| 0.3052 | |||
| 14.85 | |||
|- | |||
| 2.3.5.7.11 | |||
| 5632/5625, 160083/160000, 26411/26244, 968000/964467 | |||
| {{mapping| 584 926 1356 1639 2020 }} | |||
| +0.0111 | |||
| 0.2862 | |||
| 13.93 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 2080/2079, 1001/1000, 4096/4095, 85750/85293, 983125/979776 | |||
| {{mapping| 584 926 1356 1639 2020 2161 }} | |||
| +0.0145 | |||
| 0.2613 | |||
| 12.72 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 47\584 | |||
| 96.58 | |||
| 200/189 | |||
| [[Hemiluna]] (584, 7-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 | |||
== Scales == | |||
* [[Hemiluna14]] | * [[Hemiluna14]] | ||
== Music == | |||
; [[User:Francium|Francium]] | |||
* [https://www.youtube.com/watch?v=7qF6IwKB8Iw ''Are You From The Moon?''] (2023) – hemiluna in 584edo tuning | |||
[[Category:Listen]] | |||