80ed6
| ← 79ed6 | 80ed6 | 81ed6 → |
80 equal divisions of the 6th harmonic (abbreviated 80ed6) is a nonoctave tuning system that divides the interval of 6/1 into 80 equal parts of about 38.8 ¢ each. Each step represents a frequency ratio of 61/80, or the 80th root of 6.
Theory
80ed6 is related to 31edo, but with the 6/1 rather than the 2/1 being just. This stretches the octave by about 2 cents. Like 31edo, 80ed6 is consistent to the 12-integer-limit. It is pretty well optimized for the 11-limit, trading the accuracy of the 5th and 7th harmonics for an improved 3rd harmonic and a massively improved 11th harmonic, which is only 2.5 cents flat of just (in comparison, 31edo's 11th harmonic is 9.4 cents flat). Also improved is the 23rd harmonic, which is now only 0.1 cents sharp of just.
The 13th, 17th, and 19th harmonics are now about halfway between the steps, suggesting the use of 160ed6.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.0 | -2.0 | +4.0 | +5.4 | +0.0 | +4.6 | +6.0 | -4.0 | +7.5 | -2.5 | +2.0 |
| Relative (%) | +5.2 | -5.2 | +10.4 | +14.0 | +0.0 | +11.7 | +15.5 | -10.4 | +19.2 | -6.3 | +5.2 | |
| Steps (reduced) |
31 (31) |
49 (49) |
62 (62) |
72 (72) |
80 (0) |
87 (7) |
93 (13) |
98 (18) |
103 (23) |
107 (27) |
111 (31) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +18.5 | +6.6 | +3.4 | +8.0 | -19.4 | -2.0 | -18.1 | +9.5 | +2.5 | -0.4 | +0.1 | +4.0 |
| Relative (%) | +47.8 | +16.9 | +8.9 | +20.7 | -50.0 | -5.2 | -46.6 | +24.4 | +6.6 | -1.1 | +0.4 | +10.4 | |
| Steps (reduced) |
115 (35) |
118 (38) |
121 (41) |
124 (44) |
126 (46) |
129 (49) |
131 (51) |
134 (54) |
136 (56) |
138 (58) |
140 (60) |
142 (62) | |
Subsets and supersets
Since 80 factors into primes as 24 × 5, 80ed6 has subset ed6's 2, 4, 5, 8, 10, 16, 20, and 40. 160ed6, which doubles it, much corrects its 13th, 17th, and 19th harmonics.