248edo

Theory

248et tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440, 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It also notably tempers out the quartisma.

It supports bischismic temperament, providing the optimal patent val for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for essence temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords. 248 has divisors 2, 4, 8, 31, 62, and 124.

Prime harmonics

Approximation of prime harmonics in 248edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.34	+0.78	-1.08	+0.29	+1.41	+1.50	-2.35	+0.76	+1.07	+1.74
	Relative (%)	+0.0	-7.1	+16.2	-22.4	+6.1	+29.1	+30.9	-48.6	+15.7	+22.1	+35.9
Steps (reduced)		248 (0)	393 (145)	576 (80)	696 (200)	858 (114)	918 (174)	1014 (22)	1053 (61)	1122 (130)	1205 (213)	1229 (237)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[287 -181⟩	[⟨248 393]]	+0.108	0.108	2.23
2.3.5	32805/32768, [12 32 -27⟩	[⟨248 393 576]]	-0.041	0.228	4.70
2.3.5.7	3136/3125, 32805/32768, 420175/419904	[⟨248 393 576 696]]	+0.066	0.270	5.58
2.3.5.7.11	441/440, 3136/3125, 8019/8000, 41503/41472	[⟨248 393 576 696 858]]	+0.036	0.249	5.15
2.3.5.7.11.13	441/440, 729/728, 847/845, 1001/1000, 3136/3125	[⟨248 393 576 696 858 918]]	+0.079	0.275	5.69

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	5\248	24.19	686/675	Sengagen
1	103\248	498.39	4/3	Helmholtz
2	77\248 (47\248)	372.58 (227.42)	26/21 (154/135)	Essence
2	103\248	498.39	4/3	Bischismic
8	117\248 (7\248)	566.13 (33.87)	104/75 (49/48)	Octowerck
31	103\248 (1\248)	498.39 (4.84)	4/3 (385/384)	Birds