684edo: Difference between revisions

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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 684 factors into {{factorization|684}}, 684edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342 }}.
Since 684 factors into {{factorization|684}}, 684edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342 }}.
== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[5818zpi]]
| 683.938934890938
| 1.75454260429165
| 14.267321
| 1.773752
| 20.109967
| 684edo
| 1200.10714133549
| 18
| 18
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 17: Line 49:
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
Line 45: Line 77:
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
| 18
| 18
| 271\684<br />(5\684)
| 271\684<br>(5\684)
| 475.44<br />(8.77)
| 475.44<br>(8.77)
| 1053/800<br />(1287/1280)
| 1053/800<br>(1287/1280)
| [[Semihemiennealimmal]]
| [[Semihemiennealimmal]]
|-
|-
| 38
| 38
| 151\684<br />(7\684)
| 151\684<br>(7\684)
| 264.91<br />(12.28)
| 264.91<br>(12.28)
| 500/429<br />(144/143)
| 500/429<br>(144/143)
| [[Semihemienneadecal]]
| [[Semihemienneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[5818zpi]]
| 683.938934890938
| 1.75454260429165
| 14.267321
| 1.773752
| 20.109967
| 684edo
| 1200.10714133549
| 18
| 18
|}

Revision as of 10:34, 19 January 2025

← 683edo 684edo 685edo →
Prime factorization 22 × 32 × 19
Step size 1.75439 ¢ 
Fifth 400\684 (701.754 ¢) (→ 100\171)
Semitones (A1:m2) 64:52 (112.3 ¢ : 91.23 ¢)
Consistency limit 17
Distinct consistency limit 17

Template:EDO intro

Theory

684edo divides the steps of 171edo into four. It is consistent to the 17-odd-limit, tempering out 2401/2400, 3025/3024, 4225/4224, 4375/4374, and 32805/32768 in the 13-limit; 1089/1088, 1225/1224, 1701/1700, 2025/2023, 2058/2057, 2500/2499, 8624/8619, and 14875/14872 in the 17-limit.

Prime harmonics

Approximation of prime harmonics in 684edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.201 -0.349 -0.405 -0.441 -0.177 +0.308 +0.733 -0.204 +0.247 +0.578
Relative (%) +0.0 -11.4 -19.9 -23.1 -25.1 -10.1 +17.5 +41.8 -11.6 +14.1 +33.0
Steps
(reduced) 		684
(0) 		1084
(400) 		1588
(220) 		1920
(552) 		2366
(314) 		2531
(479) 		2796
(60) 		2906
(170) 		3094
(358) 		3323
(587) 		3389
(653)

Subsets and supersets

Since 684 factors into 22 × 32 × 19, 684edo has subset edos 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342.

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
5818zpi 683.938934890938 1.75454260429165 14.267321 1.773752 20.109967 684edo 1200.10714133549 18 18

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5.7.11.13 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768 [⟨684 1084 1588 1920 2366 2531]] +0.0994 0.0558 3.18
2.3.5.7.11.13.17 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005 [⟨684 1084 1588 1920 2366 2531 2796]] +0.0744 0.0800 4.56
  • 684et is the first equal temperament past 494 with a lower 13-limit absolute error. The next equal temperament that is better tuned is 764.

Rank-2 temperaments

Note: 11-limit temperaments supported by 342et are not shown.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
18 271\684
(5\684) 		475.44
(8.77) 		1053/800
(1287/1280) 		Semihemiennealimmal
38 151\684
(7\684) 		264.91
(12.28) 		500/429
(144/143) 		Semihemienneadecal

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

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