289edo

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289edo

290edo →

Prime factorization	172
Step size	4.15225 ¢
Fifth	169\289 (701.73 ¢)
Semitones (A1:m2)	27:22 (112.1 ¢ : 91.35 ¢)
Consistency limit	9
Distinct consistency limit	9

Theory

289edo is the optimal patent val for 13-limit history temperament, which tempers out 364/363, 441/440 and 676/675, and provides a good tuning for the 11-limit version also, and is also the optimal patent val for sextilififths in both the 11- and 13-limit. It is uniquely consistent in the 9-odd-limit, and tempers out the schisma, 32805/32768 in the 5-limit; 4375/4374 and 65625/65536 in the 7-limit; 441/440 and 4000/3993 in the 11-limit; and 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079 in the 13-limit.

289 is 17 squared. In light of containing 17edo as a subset, 289edo supports the chlorine temperament, which tempers out the septendecima [-52 -17 34⟩ and the ragisma 4375/4374.

Prime harmonics

Approximation of prime harmonics in 289edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.22	-0.15	-1.35	+0.93	-1.77	-1.15	+1.45	-1.28	+0.18	+0.99
	Relative (%)	+0.0	-5.4	-3.7	-32.6	+22.4	-42.7	-27.7	+34.9	-30.9	+4.3	+23.7
Steps (reduced)		289 (0)	458 (169)	671 (93)	811 (233)	1000 (133)	1069 (202)	1181 (25)	1228 (72)	1307 (151)	1404 (248)	1432 (276)

Regular temperament properties

289edo has decent 11 and 13-limit interpretations despite not being consistent.

Rank two temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	20\289	83.045	21/20	Sextilififths
17	93\289 (8\289)	386.159 (33.218)	[-23 5 9 -2⟩ (100352/98415)	Chlorine