684edo: Difference between revisions

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== Theory ==
== 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.
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 ===
=== Prime harmonics ===
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768
| 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768
| [{{val| 684 1084 1588 1920 2366 2531 }}]
| {{mapping| 684 1084 1588 1920 2366 2531 }}
| +0.0994
| +0.0994
| 0.0558
| 0.0558
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005
| 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005
| [{{val| 684 1084 1588 1920 2366 2531 2796 }}]
| {{mapping| 684 1084 1588 1920 2366 2531 2796 }}
| +0.0744
| +0.0744
| 0.0800
| 0.0800
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 11-limit temperaments supported by [[342edo|342et]] are not shown.
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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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| [[Semihemienneadecal]]
| [[Semihemienneadecal]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 08:22, 22 October 2023

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

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 it is distinct

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