584edo: Difference between revisions
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{{Infobox ET}} | |||
{{ | {{ED intro}} | ||
==Theory== | |||
== 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}} | ||
==Regular temperament properties== | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| 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 | |||
|- | |- | ||
|2.3 | ! [[TE error|Absolute]] (¢) | ||
|{{monzo| 463 -292}} | ! [[TE simple badness|Relative]] (%) | ||
|{{ | |- | ||
| | | 2.3 | ||
| {{monzo| 463 -292 }} | |||
| {{mapping| 584 926 }} | |||
| −0.2476 | |||
| 0.2475 | | 0.2475 | ||
| 12.05 | | 12.05 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo| 3 -18 11}}, {{monzo| 21 -20 4}} | | {{monzo| 3 -18 11 }}, {{monzo| 21 -20 4 }} | ||
|{{ | | {{mapping| 584 926 1356 }} | ||
| | | −0.1633 | ||
| 0.2346 | | 0.2346 | ||
| 11.42 | | 11.42 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|1500625/1492992, 1605632/1594323, 235298/234375 | | 1500625/1492992, 1605632/1594323, 235298/234375 | ||
|{{ | | {{mapping| 584 926 1356 1639 }} | ||
| | | −0.0319 | ||
| 0.3052 | | 0.3052 | ||
| 14.85 | | 14.85 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|5632/5625, 160083/160000, 26411/26244, 968000/964467 | | 5632/5625, 160083/160000, 26411/26244, 968000/964467 | ||
|{{ | | {{mapping| 584 926 1356 1639 2020 }} | ||
| +0.0111 | | +0.0111 | ||
| 0.2862 | | 0.2862 | ||
| 13.93 | | 13.93 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|2080/2079, 1001/1000, 4096/4095, 85750/85293, 983125/979776 | | 2080/2079, 1001/1000, 4096/4095, 85750/85293, 983125/979776 | ||
|{{ | | {{mapping| 584 926 1356 1639 2020 2161 }} | ||
| +0.0145 | | +0.0145 | ||
| 0.2613 | | 0.2613 | ||
| 12.72 | | 12.72 | ||
|} | |} | ||
=== 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>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|47\584 | | 47\584 | ||
|96.58 | | 96.58 | ||
|200/189 | | 200/189 | ||
|Hemiluna | | [[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== | == Scales == | ||
* [[Hemiluna14]] | * [[Hemiluna14]] | ||
==Music== | |||
* [https://www.youtube.com/watch?v=7qF6IwKB8Iw Are You From The Moon?] | == Music == | ||
; [[User:Francium|Francium]] | |||
* [https://www.youtube.com/watch?v=7qF6IwKB8Iw ''Are You From The Moon?''] (2023) – hemiluna in 584edo tuning | |||
[[Category:Listen]] | |||