308edo: Difference between revisions

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== Theory ==
== Theory ==
308et only is consistent in the 5-limit. Ignoring the harmonics 7, 11 and 13, 308et is strong in the 2.3.5.17.19.23.29.31 subgroup.
308edo only is [[consistent]] in the [[5-odd-limit]]. Ignoring the [[harmonic]]s [[7/1|7]], [[11/1|11]] and [[13/1|13]], it is strong in the 2.3.5.17.19.23.29.31 [[subgroup]].


308et tempers out following commas:
Using the [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] [[19683/19600]], [[65625/65536]], and [[390625/388962]] in the 7-limit, and [[243/242]], 1375/1372, [[6250/6237]], [[9801/9800]], and 14700/14641 in the 11-limit.
 
7-limit commas: 4096000/4084101, [[390625/388962]], 26873856/26796875, [[19683/19600]], [[78125000/78121827]], [[65625/65536]], [[1640558367/1638400000]]
 
11-limit commas: 806736/805255, 1835008/1830125, 14700/14641, 26214400/26198073, 166698/166375, [[243/242]], [[131072/130977]], [[6250/6237]], 107495424/107421875, 9765625/9732096, 137781/137500, 180224/180075, 1375/1372, 17537553/17500000, 47265625/47258883, [[9801/9800]], 539055/537824, 202397184/201768035


Using the 308d val, it supports [[unidec]] and [[gammic]].
Using the 308d val, it supports [[unidec]] and [[gammic]].
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=== Subsets and supersets ===
=== Subsets and supersets ===
308 factors into 2<sup>2</sup> x 7 x 11, with subset edos {{EDOs| 2, 4, 7, 11, 14, 22, 28, 44, 77, and 154}}.
Since 308 factors into 2<sup>2</sup> × 7 × 11, 308edo has subset edos {{EDOs| 2, 4, 7, 11, 14, 22, 28, 44, 77, and 154 }}.


==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]]
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! 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| -122 77 }}
| {{monzo| -122 77 }}
| {{val| 308 488 }}
| {{mapping| 308 488 }}
| 0.2070
| 0.2070
| 0.2071
| 0.2071
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| 2.3.5
| 2.3.5
| {{monzo| -36 11 8 }}, {{monzo| -7 22 -12 }}
| {{monzo| -36 11 8 }}, {{monzo| -7 22 -12 }}
| {{val| 308 488 715 }}
| {{mapping| 308 488 715 }}
| 0.2241
| 0.2241
| 0.1708
| 0.1708
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| 128\308<br>(4\308)
| 128\308<br>(4\308)
| 498.70<br>(15.58)
| 498.70<br>(15.58)
| 4/3<br>(2048/2025)
| 4/3<br>(126/125)
| [[Oquatonic]]
| [[Oquatonic]] (308)
|}
|}
 
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{Todo| review }}

Revision as of 10:54, 21 February 2024

← 307edo 308edo 309edo →
Prime factorization 22 × 7 × 11
Step size 3.8961 ¢ 
Fifth 180\308 (701.299 ¢) (→ 45\77)
Semitones (A1:m2) 28:24 (109.1 ¢ : 93.51 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

308edo only is consistent in the 5-odd-limit. Ignoring the harmonics 7, 11 and 13, it is strong in the 2.3.5.17.19.23.29.31 subgroup.

Using the patent val nonetheless, the equal temperament tempers out 19683/19600, 65625/65536, and 390625/388962 in the 7-limit, and 243/242, 1375/1372, 6250/6237, 9801/9800, and 14700/14641 in the 11-limit.

Using the 308d val, it supports unidec and gammic.

Prime harmonics

Approximation of prime harmonics in 308edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.66 -0.60 +1.30 +1.93 +1.03 +0.24 -1.41 -1.00 -1.01 +0.42
Relative (%) +0.0 -16.8 -15.4 +33.5 +49.5 +26.5 +6.1 -36.2 -25.7 -25.8 +10.8
Steps
(reduced) 		308
(0) 		488
(180) 		715
(99) 		865
(249) 		1066
(142) 		1140
(216) 		1259
(27) 		1308
(76) 		1393
(161) 		1496
(264) 		1526
(294)

Subsets and supersets

Since 308 factors into 22 × 7 × 11, 308edo has subset edos 2, 4, 7, 11, 14, 22, 28, 44, 77, and 154.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-122 77⟩ [⟨308 488]] 0.2070 0.2071 5.32
2.3.5 [-36 11 8⟩, [-7 22 -12⟩ [⟨308 488 715]] 0.2241 0.1708 4.38

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 9\308 35.06 128/125 Gammic (308d)
28 128\308
(4\308) 		498.70
(15.58) 		4/3
(126/125) 		Oquatonic (308)

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

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