684edo: Difference between revisions

Revision as of 13:02, 16 November 2024

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
Error	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)

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.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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.

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

@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 27: / Line 36: @@
 | 0.0800
 | 4.56
-{{comma basis end}}
+|}
 * 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]].
@@ Line 33: / Line 42: @@
 Note: 11-limit temperaments supported by [[342edo|342et]] are not shown.
-{{rank-2 begin}}
+{| 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
@@ Line 46: / Line 62: @@
 | 500/429<br />(144/143)
 | [[Semihemienneadecal]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct