227edo: Difference between revisions

Revision as of 12:26, 16 November 2024

The equal temperament tempers out 15625/15552 (kleisma) and [61 -37 -1⟩ in the 5-limit; 5120/5103, 65625/65536, and 117649/116640 in the 7-limit, so that it supports countercata. In the 11-limit, it tempers out 385/384, 2200/2187, 3388/3375, and 12005/11979, so that it provides the optimal patent val for 11-limit countercata. In the 13-limit, it tempers out 325/324, 352/351, 625/624, 676/675, and 847/845, and again supplies a good tuning for 13-limit countercata, although 140edo tunes it better in this case.

227edo is accurate for the 13th harmonic, as the denominator of a convergent to log₂13, after 10 and before 5231.

Approximation of prime harmonics in 227edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+1.13	-0.41	-1.43	-1.54	+0.00	+0.77	-1.48	+0.80	+1.26	+2.10
Error	Relative (%)	+0.0	+21.4	-7.8	-27.0	-29.1	+0.0	+14.6	-28.0	+15.1	+23.8	+39.7
Steps (reduced)		227 (0)	360 (133)	527 (73)	637 (183)	785 (104)	840 (159)	928 (20)	964 (56)	1027 (119)	1103 (195)	1125 (217)

227edo is the 49th prime edo.

Main article: Table of 227edo intervals

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[360 -227⟩	[⟨227 360]]	−0.3561	0.3560	6.73
2.3.5	15625/15552, [61 -37 -1⟩	[⟨227 360 527]]	−0.1785	0.3842	7.27
2.3.5.7	5120/5103, 15625/15552, 117649/116640	[⟨227 360 527 637]]	−0.0071	0.4461	8.44
2.3.5.7.11	385/384, 2200/2187, 3388/3375, 12005/11979	[⟨227 360 527 637 785]]	+0.0832	0.4380	8.29
2.3.5.7.11.13	325/324, 352/351, 385/384, 625/624, 12005/11979	[⟨227 360 527 637 785 840]]	+0.0693	0.4010	7.59
2.3.5.7.11.13.17	325/324, 352/351, 385/384, 595/594, 625/624, 3185/3179	[⟨227 360 527 637 785 840 928]]	+0.0324	0.3821	7.23

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	25\227	132.16	121/112	Kastro
1	60\227	317.18	6/5	Countercata
1	94\227	496.92	4/3	Undecental

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

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@@ Line 17: / Line 17: @@
 == 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 60: / Line 69: @@
 | 0.3821
 | 7.23
-{{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 82: / Line 98: @@
 | 4/3
 | [[Undecental]]
-{{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
 == Music ==