389edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''389edo''' divides the octave into parts of 3.0848c each.
{{EDO intro|389}}


== Theory ==
== Theory ==
{{Primes in edo|389|columns=8}}
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.


389edo has two mappings for 3, which makes it a [[dual-fifth system]]. The best approach to this tuning is through a 2.5.7.11.17 subgroup.
=== Odd harmonics ===
=== Relation to a calendar reform ===
{{Harmonics in equal|389}}
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 maximum evenness scale, temperament can be generated by simply merging the numerator and the denominator.


==== Solstice Leap Day (94 & 295) ====
=== 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.
295 seems to precede 389.


Line 20: Line 25:
== 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]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal
! rowspan="2" | Optimal<br>8ve Stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3.5
| 2.3.5
|{{monzo|20 -17 3}}, {{monzo|-39 -12 25}}
| {{monzo| 20 -17 3 }}, {{monzo| -39 -12 25 }}
|[{{val|389 617 903}}]
| {{mapping| 389 617 903 }}
| -0.19
| -0.19
|0.500
| 0.500
|16.2
| 16.2
|-
|-
|2.3.5
| 2.3.5
|2109375/2097152, {{monzo|-7, 44, -27}}
| 2109375/2097152, {{monzo|-7 44 -27 }}
|[{{val|389 616 903}}] (389b)
| {{mapping| 389 616 903 }} (389b)
|0.46
| 0.46
|0.451
| 0.451
|14.6
| 14.6
|-
|-
|2.5.7
| 2.5.7
|2100875/2097152, {{monzo|0, 52, -43}}
| 2100875/2097152, {{monzo| 0 52 -43 }}
|[{{val|389 903 1092}}]
| {{mapping| 389 903 1092 }}
|0.12
| 0.12
|0.131
| 0.131
|4.2
| 4.2
|-
|-
|2.5.7.11.17
| 2.5.7.11.17
|6664/6655, 156250/155771, 180625/180224, 184960/184877
| 6664/6655, 156250/155771, 180625/180224, 184960/184877
|[{{val|389 903 1092 1346 1590}}]
| {{mapping| 389 903 1092 1346 1590 }}
|0.03
| 0.03
|0.177
| 0.177
|5.7
| 5.7
|}
|}
== Scales ==
== Scales ==
* Solstice[69]
* Solstice[69]
* SolsticeDay[94]
* SolsticeDay[94]
== Links ==
https://individual.utoronto.ca/kalendis/leap/index.htm#mod
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 16:40, 6 November 2023

← 388edo 389edo 390edo →
Prime factorization 389 (prime)
Step size 3.08483 ¢ 
Fifth 228\389 (703.342 ¢)
Semitones (A1:m2) 40:27 (123.4 ¢ : 83.29 ¢)
Dual sharp fifth 228\389 (703.342 ¢)
Dual flat fifth 227\389 (700.257 ¢)
Dual major 2nd 66\389 (203.599 ¢)
Consistency limit 3
Distinct consistency limit 3

Template:EDO intro

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

Approximation of odd harmonics in 389edo
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]
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