86ed12

← 85ed12

86ed12

87ed12 →

Prime factorization	2 × 43
Step size	50.0227 ¢
Octave	24\86ed12 (1200.55 ¢) (→ 12\43ed12)
Twelfth	38\86ed12 (1900.86 ¢) (→ 19\43ed12)
Consistency limit	6
Distinct consistency limit	6

86 equal divisions of the 12th harmonic (abbreviated 86ed12) is a nonoctave tuning system that divides the interval of 12/1 into 86 equal parts of about 50 ¢ each. Each step represents a frequency ratio of 12^1/86, or the 86th root of 12.

Theory

86ed12 is very nearly identical to 24edo (quarter-tone tuning), but with the 12th harmonic rather than the octave being just. The octave is about 0.546 cents stretched.

Harmonics

Approximation of harmonics in 86ed12

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.5	-1.1	+1.1	+15.0	-0.5	-17.3	+1.6	-2.2	+15.5	+0.6	+0.0
	Relative (%)	+1.1	-2.2	+2.2	+29.9	-1.1	-34.6	+3.3	-4.4	+31.0	+1.1	+0.0
Steps (reduced)		24 (24)	38 (38)	48 (48)	56 (56)	62 (62)	67 (67)	72 (72)	76 (76)	80 (80)	83 (83)	86 (0)

Approximation of harmonics in 86ed12 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+11.5	-16.8	+13.9	+2.2	-2.7	-1.6	+4.8	+16.1	-18.4	+1.1	+24.2	+0.5
	Relative (%)	+23.0	-33.5	+27.7	+4.4	-5.5	-3.3	+9.6	+32.1	-36.8	+2.2	+48.4	+1.1
Steps (reduced)		89 (3)	91 (5)	94 (8)	96 (10)	98 (12)	100 (14)	102 (16)	104 (18)	105 (19)	107 (21)	109 (23)	110 (24)

Subsets and supersets

Since 86 factors into primes as 2 × 43, 86ed12 contains subset ed12's 2ed12 and 43ed12.

See also

• 14edf – relative edf
• 24edo – relative edo
• 38edt – relative edt
• 56ed5 – relative ed5
• 62ed6 – relative ed6
• 198ed304 – close to the zeta-optimized tuning for 24edo