1289edo: Difference between revisions

Revision as of 12:53, 16 November 2024

1289edo is consistent to the 9-odd-limit. As an equal temperament, it tempers out [-16 35 -17⟩ (minortone comma) in the 5-limit. Using the patent val, it tempers out 3025/3024, 180224/180075, 2460375/2458624 and 50014503/50000000 in the 11-limit; 1716/1715, 4096/4095, 91125/91091 and 5282739/5281250 in the 13-limit. In the 2.3.13.23.29.31 subgroup it tempers out 19344/19343, in the 2.3.5.7.11.23.31 subgroup 19251/19250.

Approximation of prime harmonics in 1289edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.016	+0.032	+0.297	-0.193	+0.124	+0.242	+0.392	+0.120	+0.058	+0.038
Error	Relative (%)	+0.0	-1.7	+3.5	+32.0	-20.7	+13.3	+26.0	+42.1	+12.9	+6.2	+4.1
Steps (reduced)		1289 (0)	2043 (754)	2993 (415)	3619 (1041)	4459 (592)	4770 (903)	5269 (113)	5476 (320)	5831 (675)	6262 (1106)	6386 (1230)

1289edo is the 209th prime edo.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-2043 1289⟩	[⟨1289 2043]]	+0.0049	0.0049	0.53
2.3.5	[-16 35 -17⟩, [91 -12 -31⟩	[⟨1289 2043 2993]]	−0.0014	0.0097	1.04
2.3.5.7	2460375/2458624, 78125000/78121827, 12884901888/12867859375	[⟨1289 2043 2993 3619]]	−0.0275	0.0461	4.95
2.3.5.7.11	3025/3024, 180224/180075, 2460375/2458624, 50014503/50000000	[⟨1289 2043 2993 3619 4459]]	−0.0109	0.0530	5.69
2.3.5.7.11.13	3025/3024, 1716/1715, 4096/4095, 91125/91091, 5282739/5281250	[⟨1289 2043 2993 3619 4459 4770]]	−0.0146	0.0491	5.27
2.3.5.7.11.13.17	3025/3024, 1716/1715, 4096/4095, 2500/2499, 37180/37179, 3536379/3536000	[⟨1289 2043 2993 3619 4459 4770 5269]]	−0.0210	0.0481	5.17
2.3.5.7.11.13.17.19	3025/3024, 1716/1715, 2376/2375, 4096/4095, 2500/2499, 270864/270725, 75735/75712	[⟨1289 2043 2993 3619 4459 4770 5269 5476]]	−0.0299	0.0508	5.46

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	142\1289	132.196	[-38 5 13⟩	Astro
1	196\1289	182.467	10/9	Minortone
1	238\1289	221.567	8388608/7381125	Fortune

* 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 65: / Line 74: @@
 === 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 84: / Line 100: @@
 | 8388608/7381125
 | [[Fortune]]
-{{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