981edo: Difference between revisions

981edo is a good 13- and 17-limit system, consistent to the 17-odd-limit. As an equal temperament, it tempers out 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4225/4224, 4375/4374, 4459/4455 and 4914/4913 in the 17-limit. It provides the optimal patent val for 13-limit ennealimmic, the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit ennealimmia, which also tempers out 4096/4095.

Approximation of prime harmonics in 981edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.186	+0.231	-0.019	+0.364	-0.161	+0.243	-0.265	+0.472	+0.392	-0.081
Error	Relative (%)	+0.0	+15.2	+18.9	-1.5	+29.8	-13.1	+19.9	-21.7	+38.6	+32.1	-6.7
Steps (reduced)		981 (0)	1555 (574)	2278 (316)	2754 (792)	3394 (451)	3630 (687)	4010 (86)	4167 (243)	4438 (514)	4766 (842)	4860 (936)

Since 981 factors into primes as 3² × 109, 981edo has subset edos 3, 9, 109, and 327.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[1555 -981⟩	[⟨981 1555]]	−0.0586	0.0586	4.79
2.3.5	[1 -27 18⟩, [85 -17 -25⟩	[⟨981 1555 2278]]	−0.0722	0.0515	4.21
2.3.5.7	2401/2400, 4375/4374, [79 -25 -23 5⟩	[⟨981 1555 2278 2754]]	−0.0385	0.0562	4.59
2.3.5.7.11	2401/2400, 4375/4374, 131072/130977, 1771561/1771470	[⟨981 1555 2278 2754 3394]]	−0.0630	0.0545	4.46
2.3.5.7.11.13	2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470	[⟨981 1555 2278 2754 3394 3630]]	−0.0453	0.0636	5.20
2.3.5.7.11.13.17	2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913	[⟨981 1555 2278 2754 3394 3630 4010]]	−0.0473	0.0591	4.83

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	409\981	500.306	8192/6137	Protolangwidge
9	258\981 (40\981)	315.60 (48.93)	6/5 (36/35)	Ennealimmal / ennealimmia

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

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a good 13- and 17-limit system, [[consistent]] to the [[17-odd-limit]]. It [[tempers out]] [[2080/2079]], [[2401/2400]], [[2431/2430]], [[4096/4095]], [[4225/4224]], [[4375/4374]], 4459/4455 and [[4914/4913]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit [[ennealimmic]], the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit [[ennealimmia]], which also tempers out 4096/4095.
+edo is a good 13- and 17-limit system, [[consistent]] to the [[17-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] [[2080/2079]], [[2401/2400]], [[2431/2430]], [[4096/4095]], [[4225/4224]], [[4375/4374]], 4459/4455 and [[4914/4913]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit [[ennealimmic]], the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit [[ennealimmia]], which also tempers out 4096/4095.
 === Prime harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-Since 981 = {{factorization|981}}, 981edo has subset edos 3, 9, 109, and 327.
+Since 981 factors into primes as 3<sup>2</sup> × 109, 981edo has subset edos 3, 9, 109, and 327.
 == Regular temperament properties ==