389edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | == Theory == | ||
389edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps, which makes it a [[dual-fifth system]]. Otherwise, it has a reasonable approximation to harmonics [[7/1|7]], [[9/1|9]], and [[17/1|17]], with optional additions of either [[5/1|5]] or [[11/1|11]] and [[15/1|15]], making it suitable for a 2.9.5.7.17 or 2.9.15.7.11.17 [[subgroup]] interpretation. | |||
=== Odd harmonics === | |||
=== | {{Harmonics in equal|389}} | ||
=== Subsets and supersets === | |||
389edo is the 77th [[prime edo]]. | |||
The solstice leap day scale with 94 notes uses 269\389 as a generator. | === Miscelleneous properties === | ||
389edo represents the ''north solstice'' (summer in the northern hemisphere) ''leap year cycle 69/389'' as devised by Sym454 inventor Irvin Bromberg. The outcome scale uses 327\389, or 62\389 as its generator. The solstice leap day scale with 94 notes uses 269\389 as a generator. Since this is a [[maximal evenness]] scale, temperament can be generated by simply merging the numerator and the denominator. | |||
==== Solstice leap day (94 & 295) ==== | |||
295 seems to precede 389. | |||
Subgroup: 2.5.7.11.17 | |||
POTE generator: 370.1796c | |||
Comma list: 250000/248897, 2100875/2097152, 4096000/4092529 | |||
== 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|Comma List]] | ||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |- | ||
| | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
| | | {{monzo| 20 -17 3 }}, {{monzo| -39 -12 25 }} | ||
| | | {{mapping| 389 617 903 }} | ||
| | | −0.19 | ||
|0. | | 0.500 | ||
| | | 16.2 | ||
|- | |- | ||
|2.5.7 | | 2.3.5 | ||
|2100875/2097152, {{monzo|0 | | 2109375/2097152, {{monzo|-7 44 -27 }} | ||
| | | {{mapping| 389 616 903 }} (389b) | ||
|0.12 | | +0.46 | ||
|0.131 | | 0.451 | ||
|4.2 | | 14.6 | ||
|- | |||
| 2.5.7 | |||
| 2100875/2097152, {{monzo| 0 52 -43 }} | |||
| {{mapping| 389 903 1092 }} | |||
| +0.12 | |||
| 0.131 | |||
| 4.2 | |||
|- | |||
| 2.5.7.11.17 | |||
| 6664/6655, 156250/155771, 180625/180224, 184960/184877 | |||
| {{mapping| 389 903 1092 1346 1590 }} | |||
| +0.03 | |||
| 0.177 | |||
| 5.7 | |||
|} | |} | ||
== Scales == | == Scales == | ||
* Solstice[69] | * Solstice[69] | ||
* SolsticeDay[94] | * SolsticeDay[94] | ||
Latest revision as of 22:54, 20 February 2025
| ← 388edo | 389edo | 390edo → |
389 equal divisions of the octave (abbreviated 389edo or 389ed2), also called 389-tone equal temperament (389tet) or 389 equal temperament (389et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 389 equal parts of about 3.08 ¢ each. Each step represents a frequency ratio of 21/389, or the 389th root of 2.
Theory
389edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps, which makes it a dual-fifth system. Otherwise, it has a reasonable approximation to harmonics 7, 9, and 17, with optional additions of either 5 or 11 and 15, making it suitable for a 2.9.5.7.17 or 2.9.15.7.11.17 subgroup interpretation.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.39 | -0.71 | -0.19 | -0.31 | +0.87 | -1.45 | +0.68 | -0.07 | -1.37 | +1.20 | +1.03 |
| Relative (%) | +45.0 | -23.0 | -6.1 | -10.1 | +28.1 | -47.1 | +22.0 | -2.3 | -44.4 | +38.9 | +33.4 | |
| Steps (reduced) |
617 (228) |
903 (125) |
1092 (314) |
1233 (66) |
1346 (179) |
1439 (272) |
1520 (353) |
1590 (34) |
1652 (96) |
1709 (153) |
1760 (204) | |
Subsets and supersets
389edo is the 77th prime edo.
Miscelleneous properties
389edo represents the north solstice (summer in the northern hemisphere) leap year cycle 69/389 as devised by Sym454 inventor Irvin Bromberg. The outcome scale uses 327\389, or 62\389 as its generator. The solstice leap day scale with 94 notes uses 269\389 as a generator. Since this is a maximal evenness scale, temperament can be generated by simply merging the numerator and the denominator.
Solstice leap day (94 & 295)
295 seems to precede 389.
Subgroup: 2.5.7.11.17
POTE generator: 370.1796c
Comma list: 250000/248897, 2100875/2097152, 4096000/4092529
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [20 -17 3⟩, [-39 -12 25⟩ | [⟨389 617 903]] | −0.19 | 0.500 | 16.2 |
| 2.3.5 | 2109375/2097152, [-7 44 -27⟩ | [⟨389 616 903]] (389b) | +0.46 | 0.451 | 14.6 |
| 2.5.7 | 2100875/2097152, [0 52 -43⟩ | [⟨389 903 1092]] | +0.12 | 0.131 | 4.2 |
| 2.5.7.11.17 | 6664/6655, 156250/155771, 180625/180224, 184960/184877 | [⟨389 903 1092 1346 1590]] | +0.03 | 0.177 | 5.7 |
Scales
- Solstice[69]
- SolsticeDay[94]