308edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|308}}
{{EDO intro|308}}
== 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.
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.
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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
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]].
===Prime harmonics===
 
=== Prime harmonics ===
{{Harmonics in equal|308}}
{{Harmonics in equal|308}}
===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}}.
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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>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
| 2.3
|{{monzo|-122 77}}
| {{monzo| -122 77 }}
|{{val|308 488}}
| {{val| 308 488 }}
| 0.2070
| 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}}
| {{val| 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<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! 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>(2048/2025)
|[[Oquatonic]]
| [[Oquatonic]]
|}
|}
{{Todo| review }}

Revision as of 14:18, 16 May 2023

← 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

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.

308et tempers out following commas:

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.

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

308 factors into 22 x 7 x 11, with 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
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 9\308 35.06 128/125 Gammic (308d)
28 128\308
(4\308) 		498.70
(15.58) 		4/3
(2048/2025) 		Oquatonic
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