239edo: Difference between revisions

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== Theory ==
== Theory ==
239et tempers out [[2401/2400]], [[5120/5103]], and 29360128/29296875 in the 7-limit, [[support|supporting]] the [[hemififths]] temperament, providing an excellent tuning. It also supports and provides a good tuning for [[quasiorwell]] and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], 5632/5625, and 12005/11979.
239edo has a sharp tendency, with [[prime harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[5120/5103]], and 29360128/29296875 in the 7-limit, [[support]]ing the [[hemififths]] temperament and providing an excellent tuning. It also supports and provides a good tuning for [[quasiorwell]] and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], [[5632/5625]], and 12005/11979.
 
239edo is the 52nd [[prime edo]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|239}}
{{Harmonics in equal|239}}
=== Subsets and supersets ===
239edo is the 52nd [[prime edo]].


== 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 8ve <br>stretch (¢)
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| 379 -239 }}
| {{monzo| 379 -239 }}
| [{{val| 239 379 }}]
| {{mapping| 239 379 }}
| -0.307
| -0.307
| 0.307
| 0.307
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| 2.3.5
| 2.3.5
| {{monzo| 3 -18 11}}, {{monzo| 32 -7 -9 }}
| {{monzo| 3 -18 11}}, {{monzo| 32 -7 -9 }}
| [{{val| 239 379 555 }}]
| {{mapping| 239 379 555 }}
| -0.247
| -0.247
| 0.265
| 0.265
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 5120/5103, 29360128/29296875
| 2401/2400, 5120/5103, 29360128/29296875
| [{{val| 239 379 555 671 }}]
| {{mapping| 239 379 555 671 }}
| -0.204
| -0.204
| 0.241
| 0.241
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4000/3993, 5120/5103
| 2401/2400, 3025/3024, 4000/3993, 5120/5103
| [{{val| 239 379 555 671 827 }}]
| {{mapping| 239 379 555 671 827 }}
| -0.220
| -0.220
| 0.218
| 0.218
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Neptune]] (7-limit)
| [[Neptune]] (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


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Hemififths]]
[[Category:Hemififths]]
[[Category:Quasiorwell]]
[[Category:Quasiorwell]]
[[Category:Alphaquarter]]
[[Category:Alphaquarter]]
[[Category:Prime EDO]]

Revision as of 12:56, 25 March 2024

← 238edo 239edo 240edo →
Prime factorization 239 (prime)
Step size 5.02092 ¢ 
Fifth 140\239 (702.929 ¢)
Semitones (A1:m2) 24:17 (120.5 ¢ : 85.36 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

239edo has a sharp tendency, with prime harmonics 3 through 11 all tuned sharp. The equal temperament tempers out 2401/2400, 5120/5103, and 29360128/29296875 in the 7-limit, supporting the hemififths temperament and providing an excellent tuning. It also supports and provides a good tuning for quasiorwell and alphaquarter. In the 11-limit, it tempers out 3025/3024, 4000/3993, 5632/5625, and 12005/11979.

Prime harmonics

Approximation of prime harmonics in 239edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.97 +0.30 +0.21 +0.98 -2.03 +0.48 -1.28 -0.66 -0.29 -0.27
Relative (%) +0.0 +19.4 +5.9 +4.2 +19.6 -40.5 +9.6 -25.5 -13.1 -5.7 -5.3
Steps
(reduced) 		239
(0) 		379
(140) 		555
(77) 		671
(193) 		827
(110) 		884
(167) 		977
(21) 		1015
(59) 		1081
(125) 		1161
(205) 		1184
(228)

Subsets and supersets

239edo is the 52nd prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [379 -239⟩ [⟨239 379]] -0.307 0.307 6.12
2.3.5 [3 -18 11⟩, [32 -7 -9⟩ [⟨239 379 555]] -0.247 0.265 5.27
2.3.5.7 2401/2400, 5120/5103, 29360128/29296875 [⟨239 379 555 671]] -0.204 0.241 4.80
2.3.5.7.11 2401/2400, 3025/3024, 4000/3993, 5120/5103 [⟨239 379 555 671 827]] -0.220 0.218 4.34

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 3\239 15.06 121/120 Yarman I (239)
1 11\239 35.15 1990656/1953125 Gammic (5-limit)
1 7\239 55.23 33/32 Escapade / alphaquarter
1 35\239 175.73 72/65 Quadrafifths (239f)
1 54\239 271.13 90/77 Quasiorwell (239)
1 70\239 351.46 49/40 Hemififths (7-limit)
1 79\239 396.65 44/35 Squarschmidt
1 83\239 416.74 14/11 Unthirds (239f)
1 116\239 582.43 7/5 Neptune (7-limit)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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