584edo: Difference between revisions
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Rework; cleanup; clarify the title row of the rank-2 temp table |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|584}} | {{EDO intro|584}} | ||
==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" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo| 463 -292}} | | {{monzo| 463 -292 }} | ||
|{{ | | {{mapping| 584 926 }} | ||
| -0.2476 | | -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 | | -0.1633 | ||
| 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 | | -0.0319 | ||
| 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 | ||
| Line 55: | Line 61: | ||
|+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> | ! 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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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 | |||
Revision as of 10:07, 25 October 2023
| ← 583edo | 584edo | 585edo → |
Theory
584edo is only consistent to the 5-odd-limit and the error of harmonic 3 is quite large. With reasonable approximations to harmonics 5, 7, 9, 11, 13, and 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 enfactored, with the same tuning as 292edo. Therefore, we are left with the patent val, which tempers out 48828125/48771072 and 67108864/66976875, supporting hemiluna.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.785 | -0.012 | -1.018 | -0.485 | -0.633 | -0.117 | +0.772 | -0.161 | +0.432 | -0.233 | +0.493 |
| Relative (%) | +38.2 | -0.6 | -49.5 | -23.6 | -30.8 | -5.7 | +37.6 | -7.8 | +21.0 | -11.3 | +24.0 | |
| Steps (reduced) |
926 (342) |
1356 (188) |
1639 (471) |
1851 (99) |
2020 (268) |
2161 (409) |
2282 (530) |
2387 (51) |
2481 (145) |
2565 (229) |
2642 (306) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [463 -292⟩ | [⟨584 926]] | -0.2476 | 0.2475 | 12.05 |
| 2.3.5 | [3 -18 11⟩, [21 -20 4⟩ | [⟨584 926 1356]] | -0.1633 | 0.2346 | 11.42 |
| 2.3.5.7 | 1500625/1492992, 1605632/1594323, 235298/234375 | [⟨584 926 1356 1639]] | -0.0319 | 0.3052 | 14.85 |
| 2.3.5.7.11 | 5632/5625, 160083/160000, 26411/26244, 968000/964467 | [⟨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 | [⟨584 926 1356 1639 2020 2161]] | +0.0145 | 0.2613 | 12.72 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 47\584 | 96.58 | 200/189 | Hemiluna (584, 7-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
Music
- Are You From The Moon? (2023) – hemiluna in 584edo tuning