8ed11/5

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8ed11/5

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Prime factorization	23
Step size	170.626¢
Octave	7\8ed11/5 (1194.38¢) (convergent)
Twelfth	11\8ed11/5 (1876.88¢) (semiconvergent)
Consistency limit	6
Distinct consistency limit	2

8ed11/5 is the equal division of the neutral ninth into eight parts of 170.6 cents each. It is 7edo with ~7 cent compressed octaves. This tuning tempers out the syntonic comma.

Intervals

#	Cents	Approximate ratios
0	0.0	1/1
1	170.6	11/10, 54/49, 10/9
2	341.3	17/14, 39/32, 11/9
3	511.9	4/3, 27/20
4	682.5	3/2, 40/27
5	853.1	18/11, 64/39
6	1023.8	9/5, 29/16, 20/11
7	1194.4	2/1
8	1365.0	11/5

Harmonics

Compared to 7edo, 8ed11/5 offers worse approximations of 2, 3, 5, 11 and 29. But it offers better approximations of 7, 13, 17, 19, 23, 31 and 37. Its performance on those higher harmonics lends it a less dissonant sound than one might expect from its poor approximations of simple ratios.

Most of its errors are also in the same direction: it tunes the 2nd, 3rd, 5th and 11th harmonics all flat. So this may further contribute to making the error sound less nasty in practice than it looks on paper.

Approximation of prime harmonics in 8ed11/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	-5.6	-25.1	-56.3	+43.7	-56.3	-4.3	+43.2	+21.3	+31.7	-28.3	+26.9	+61.8
	Relative (%)	-3.3	-14.7	-33.0	+25.6	-33.0	-2.5	+25.3	+12.5	+18.6	-16.6	+15.7	+36.2
Steps (reduced)		7 (7)	11 (3)	16 (0)	20 (4)	24 (0)	26 (2)	29 (5)	30 (6)	32 (0)	34 (2)	35 (3)	37 (5)

Approximation of prime harmonics in 7edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.0	-16.2	-43.5	+59.7	-37.0	+16.6	+66.5	+45.3	+57.4	-1.0	+55.0	-79.9
	Relative (%)	+0.0	-9.5	-25.3	+34.9	-21.6	+9.7	+38.8	+26.5	+33.5	-0.6	+32.1	-46.6
Steps (reduced)		7 (0)	11 (4)	16 (2)	20 (6)	24 (3)	26 (5)	29 (1)	30 (2)	32 (4)	34 (6)	35 (0)	36 (1)

Scales

Pseudo-octave-repeating

Groovy Pentatonic

• 341.251
• 511.877
• 682.502
• 1023.753
• 1194.379