User:Contribution/Exploring 31-EDO
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11-limit optimizations of 31-EDO
Euler-product
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.0 | -3.5 | +3.2 | +1.8 | -5.8 | +14.9 | +15.4 | +16.6 | -4.3 | -18.1 | -17.4 |
| Relative (%) | +2.7 | -9.2 | +8.2 | +4.7 | -15.0 | +38.5 | +39.8 | +42.8 | -11.0 | -46.8 | -44.8 | |
| Step | 31 | 49 | 72 | 87 | 107 | 115 | 127 | 132 | 140 | 150 | 153 | |
Tenney-Euclidian regular temperament
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.2 | -3.3 | +3.6 | +2.3 | -5.2 | +15.6 | +16.1 | +17.3 | -3.5 | -17.3 | -16.5 |
| Relative (%) | +3.1 | -8.5 | +9.2 | +5.9 | -13.5 | +40.1 | +41.6 | +44.6 | -9.0 | -44.6 | -42.6 | |
| Step | 31 | 49 | 72 | 87 | 107 | 115 | 127 | 132 | 140 | 150 | 153 | |
121-ED15
121-ED15 offers a good trade-off between these two optimization methods for 11-limit in 31-EDO.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.1 | -3.4 | +3.4 | +2.1 | -5.5 | +15.3 | +15.8 | +17.0 | -3.8 | -17.7 | -16.9 |
| Relative (%) | +2.9 | -8.8 | +8.8 | +5.4 | -14.2 | +39.4 | +40.8 | +43.8 | -9.9 | -45.6 | -43.6 | |
| Steps (reduced) |
31 (31) |
49 (49) |
72 (72) |
87 (87) |
107 (107) |
115 (115) |
127 (6) |
132 (11) |
140 (19) |
150 (29) |
153 (32) | |
90-ED15/2
90-ED15/2 is another good option for helping the 11-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.5 | -2.8 | +4.3 | +3.2 | -4.2 | +16.7 | +17.4 | +18.6 | -2.1 | -15.8 | -15.0 |
| Relative (%) | +3.9 | -7.2 | +11.1 | +8.2 | -10.7 | +43.1 | +44.8 | +48.0 | -5.4 | -40.8 | -38.6 | |
| Steps (reduced) |
31 (31) |
49 (49) |
72 (72) |
87 (87) |
107 (17) |
115 (25) |
127 (37) |
132 (42) |
140 (50) |
150 (60) |
153 (63) | |
170-ED45
170-ED45 is another good option for helping the 11-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.7 | -2.4 | +4.8 | +3.8 | -3.4 | +17.6 | +18.3 | -19.2 | -1.0 | -14.7 | -13.8 |
| Relative (%) | +4.5 | -6.2 | +12.5 | +9.8 | -8.7 | +45.3 | +47.3 | -49.4 | -2.7 | -37.9 | -35.7 | |
| Steps (reduced) |
31 (31) |
49 (49) |
72 (72) |
87 (87) |
107 (107) |
115 (115) |
127 (127) |
131 (131) |
140 (140) |
150 (150) |
153 (153) | |
80-ED6
80-ED6 is another good option for helping the 11-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.0 | -2.0 | +5.4 | +4.6 | -2.5 | +18.5 | -19.4 | -18.1 | +0.1 | -13.4 | -12.5 |
| Relative (%) | +5.2 | -5.2 | +14.0 | +11.7 | -6.3 | +47.8 | -50.0 | -46.6 | +0.4 | -34.6 | -32.4 | |
| Steps (reduced) |
31 (31) |
49 (49) |
72 (72) |
87 (7) |
107 (27) |
115 (35) |
126 (46) |
131 (51) |
140 (60) |
150 (70) |
153 (73) | |
59-ED15/4
59-ED15/4 is another good option for helping the 11-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.3 | -1.5 | +6.1 | +5.4 | -1.4 | -19.1 | -18.1 | -16.8 | +1.5 | -11.9 | -11.1 |
| Relative (%) | +6.0 | -3.9 | +15.9 | +13.9 | -3.6 | -49.3 | -46.8 | -43.3 | +3.9 | -30.8 | -28.5 | |
| Steps (reduced) |
31 (31) |
49 (49) |
72 (13) |
87 (28) |
107 (48) |
114 (55) |
126 (8) |
131 (13) |
140 (22) |
150 (32) |
153 (35) | |