308edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|308}}
{{ED intro}}
 
== 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]]
Using the 308d val, it supports [[unidec]] and [[gammic]].


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
=== Prime harmonics ===
{{Harmonics in equal|308}}


Using the 308d val, it supports [[Unidec]] and [[Gammic]].
=== Subsets and supersets ===
===Prime harmonics===
Since 308 factors into {{nowrap|2<sup>2</sup> × 7 × 11}}, 308edo has subset edos {{EDOs| 2, 4, 7, 11, 14, 22, 28, 44, 77, and 154 }}.
{{Harmonics in equal|308}}
===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}}.


==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
! rowspan="2" | [[Subgroup]]
|{{monzo|-122 77}}
! rowspan="2" | [[Comma list]]
|{{val|308 488}}
! rowspan="2" | [[Mapping]]
| 0.2070
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -122 77 }}
| {{mapping| 308 488 }}
| +0.2070
| 0.2071
| 0.2071
| 5.32
| 5.32
|-
|-
|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
| 4.38
| 4.38
Line 44: Line 44:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|9\308
| 9\308
|35.06
| 35.06
|128/125
| 128/125
|[[Gammic]]
| [[Gammic]] (308d)
|-
|-
|28
| 28
|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 />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Latest revision as of 13:31, 13 March 2026

← 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

308 equal divisions of the octave (abbreviated 308edo or 308ed2), also called 308-tone equal temperament (308tet) or 308 equal temperament (308et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 308 equal parts of about 3.9 ¢ each. Each step represents a frequency ratio of 21/308, or the 308th root of 2.

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 distinct

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