342edo: Difference between revisions

Revision as of 12:21, 16 November 2024

342edo is a very strong 11-limit system. It is, as one would expect, distinctly consistent through the 11-odd-limit, but goes no higher; nonetheless, it is a zeta peak edo. A basis for the 11-limit commas consists of 2401/2400, 3025/3024, 4375/4374 and 32805/32768. It is the optimal patent val for 11-limit hemitert temperament, and supports hemiennealimmal.

Approximation of prime harmonics in 342edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.20	-0.35	-0.40	-0.44	+1.58	+0.31	+0.73	-0.20	-1.51	-1.18
Error	Relative (%)	+0.0	-5.7	-9.9	-11.5	-12.6	+45.0	+8.8	+20.9	-5.8	-43.0	-33.5
Steps (reduced)		342 (0)	542 (200)	794 (110)	960 (276)	1183 (157)	1266 (240)	1398 (30)	1453 (85)	1547 (179)	1661 (293)	1694 (326)

342 factors as 2 × 3² × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.

684edo, which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3.5.7.11	2401/2400, 3025/3024, 4375/4374, 32805/32768	[⟨342 542 794 960 1183]]	+0.110	0.0556	1.59
2.3.5.7.11.13	676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712	[⟨342 542 794 960 1183 1265]] (342f)	+0.178	0.1618	4.61
2.3.5.7.11.13	625/624, 729/728, 847/845, 1575/1573, 4096/4095	[⟨342 542 794 960 1183 1266]] (342)	+0.020	0.2061	5.87

342et is lower in relative error than any previous equal temperaments in the 11-limit, being the first to beat 270. Not until 612 do we find a better equal temperament in terms of absolute error, and not until 1848 do we find one in terms of relative error.

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	11\342	38.60	45/44	Hemitert
2	5\342	17.54	99/98	Poseidon
2	50\342	175.44	448/405	Bisesqui
2	124\342 (47\342)	435.09 (164.91)	9/7 (11/10)	Semisupermajor
2	142\342 (29\342)	498.25 (101.75)	4/3 (35/33)	Bipont
3	71\342 (43\342)	249.12 (150.88)	15/13 (12/11)	Hemiterm
6	97\342 (17\342)	340.35 (59.65)	162/133 (88/85)	Semiseptichrome
6	142\342 (28\342)	498.25 (98.25)	4/3 (18/17)	Semiterm
9	63\342 (13\342)	221.05 (45.61)	25/22 (77/75)	Quadraennealimmal
18	71\342 (5\342)	249.12 (17.54)	15/13 (99/98)	Hemiennealimmal
38	142\342 (2\342)	498.25 (7.02)	4/3 (225/224)	Hemienneadecal

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

@@ Line 14: / Line 14: @@
 == 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
@@ Line 36: / Line 45: @@
 | 0.2061
 | 5.87
-{{comma basis end}}
+|}
 * 342et is lower in relative error than any previous equal temperaments in the 11-limit, being the first to beat [[270edo|270]]. Not until [[612edo|612]] do we find a better equal temperament in terms of absolute error, and not until [[1848edo|1848]] do we find one in terms of relative error.
 === Rank-2 temperaments ===
-{{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
 |-
 | 1
@@ Line 107: / Line 123: @@
 | 4/3<br />(225/224)
 | [[Hemienneadecal]]
-{{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