684edo: Difference between revisions

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'''684EDO''' is the [[EDO|equal division of the octave]] into 684 parts of 1.75439 [[cent]]s each (dividing the steps of [[171edo|171EDO]] into four). It is consistent to the 17-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.
{{Infobox ET}}
{{ED intro}}


[[Category:Edo]]
== 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 ===
{{Harmonics in equal|684|columns=11}}
 
=== 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 }}.
 
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 5818
| steps = 683.938934890938
| step size = 1.75454260429165
| tempered height = 14.267321
| pure height = 7.268914
| integral = 1.773752
| gap = 20.109967
| octave = 1200.10714133549
| consistent = 18
| distinct = 18
}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7.11.13
| 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768
| {{mapping| 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
| {{mapping| 684 1084 1588 1920 2366 2531 2796 }}
| +0.0744
| 0.0800
| 4.56
|}
* 684et is the first equal temperament past [[494edo|494]] with a lower 13-limit absolute error. The next equal temperament that is better tuned is [[764edo|764]].
 
=== Rank-2 temperaments ===
Note: 11-limit temperaments supported by [[342edo|342et]] are not shown.
 
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 18
| 271\684<br>(5\684)
| 475.44<br>(8.77)
| 1053/800<br>(1287/1280)
| [[Semihemiennealimmal]]
|-
| 38
| 151\684<br>(7\684)
| 264.91<br>(12.28)
| 500/429<br>(144/143)
| [[Semihemienneadecal]]
|}
<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
 
== Scales ==
 
* [[15-odd-limit|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64

Latest revision as of 13:31, 13 March 2026

← 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

684 equal divisions of the octave (abbreviated 684edo or 684ed2), also called 684-tone equal temperament (684tet) or 684 equal temperament (684et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 684 equal parts of about 1.75 ¢ each. Each step represents a frequency ratio of 21/684, or the 684th root of 2.

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 Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
5818zpi 683.938935 1.754543 14.267321 7.268914 1.773752 20.109967 1200.107141 0.107141 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

Scales

  • Diamond15: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64
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