255edo: Difference between revisions

The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the septendecima, [-52 -17 34⟩, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports the mirkat temperament, and in fact provides the optimal patent val. It also gives the optimal patent val for mirkat in the 11-limit, tempering out 540/539, 1375/1372, 3025/3024 and 8019/8000. In the 13-limit it tempers out 847/845, 625/624, 1575/1573 and 1716/1715.

Approximation of prime harmonics in 255edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.78	-0.43	+0.59	-0.73	+1.83	-1.43	-1.04	+2.31	+1.01	-1.51
Error	Relative (%)	+0.0	-16.5	-9.2	+12.4	-15.5	+38.8	-30.3	-22.2	+49.2	+21.5	-32.0
Steps (reduced)		255 (0)	404 (149)	592 (82)	716 (206)	882 (117)	944 (179)	1042 (22)	1083 (63)	1154 (134)	1239 (219)	1263 (243)

Since 255 factors into 3 × 5 × 17, 255edo has subset edos 3, 5, 15, 17, 51, and 85.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-404 255⟩	[⟨255 404]]	+0.246	0.246	5.22
2.3.5	[8 14 -13⟩, [-36 11 8⟩	[⟨255 404 592]]	+0.226	0.203	4.30
2.3.5.7	16875/16807, 19683/19600, 65625/65536	[⟨255 404 592 716]]	+0.117	0.257	5.46
2.3.5.7.11	540/539, 1375/1372, 8019/8000, 65625/65536	[⟨255 404 592 716 882]]	+0.136	0.233	4.95

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	39\255	183.53	10/9	Mirkat (255f)
1	52\255	244.71	15/13	Subsemifourth (255)
1	67\255	315.29	6/5	Parakleismic (5-limit)
1	74\255	348.24	11/9	Eris (255)
3	82\255 (3\255)	385.88 (14.12)	5/4 (126/125)	Mutt (7-limit)
5	53\255 (2\255)	249.41 (9.41)	81/70 (176/175)	Hemipental / hemipent (255) / hemipentalis (255f)
5	106\255 (4\255)	498.82 (18.82)	4/3 (81/80)	Pental (5-limit)
17	53\255 (7\255)	249.41 (32.94)	[-25 -9 17⟩ (1990656/1953125)	Chlorine (5-limit)

* 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
@@ Line 41: / Line 50: @@
 | 0.233
 | 4.95
-{{comma basis end}}
+|}
 === 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 93: / Line 109: @@
 | {{monzo| -25 -9 17 }}<br />(1990656/1953125)
 | [[Chlorine]] (5-limit)
-{{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
 [[Category:Mirkat]]