User:Overthink/49edo and octave compression
49edo is a strongly sharp-tending system up to the 11-limit. Using the 49fgh val, this sharp tendency extends to the 19-limit. In the pure-octaves tuning, however, the higher primes are tuned too sharp, with 13 being 16.6 cents sharp, 17 being 17.5 cents sharp, and 19 being 20.9 cents sharp. It is therefore beneficial to compress the octave, using tunings such as 114ed5, 127ed6, or 138ed7. Errors on harmonics are shown below.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.4 | +4.5 | -4.7 | +0.0 | +2.1 | +4.1 | -7.1 | +8.9 | -2.4 | +3.7 | -0.3 |
| Relative (%) | -9.7 | +18.3 | -19.4 | +0.0 | +8.6 | +16.7 | -29.1 | +36.6 | -9.7 | +15.2 | -1.1 | |
| Steps (reduced) |
49 (49) |
78 (78) |
98 (98) |
114 (0) |
127 (13) |
138 (24) |
147 (33) |
156 (42) |
163 (49) |
170 (56) |
176 (62) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.8 | +1.7 | +4.5 | -9.5 | +7.8 | +6.6 | +10.7 | -4.7 | +8.6 | +1.3 | -2.3 |
| Relative (%) | +31.9 | +7.0 | +18.3 | -38.9 | +31.7 | +26.9 | +43.9 | -19.4 | +35.0 | +5.5 | -9.4 | |
| Steps (reduced) |
182 (68) |
187 (73) |
192 (78) |
196 (82) |
201 (87) |
205 (91) |
209 (95) |
212 (98) |
216 (102) |
219 (105) |
222 (108) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.2 | +3.2 | -6.4 | -1.9 | +0.0 | +1.8 | -9.5 | +6.4 | -5.1 | +0.9 | -3.2 |
| Relative (%) | -13.0 | +13.0 | -26.1 | -7.7 | +0.0 | +7.4 | -39.1 | +26.1 | -20.7 | +3.7 | -13.0 | |
| Steps (reduced) |
49 (49) |
78 (78) |
98 (98) |
114 (114) |
127 (0) |
138 (11) |
147 (20) |
156 (29) |
163 (36) |
170 (43) |
176 (49) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.8 | -1.4 | +1.3 | +11.7 | +4.4 | +3.2 | +7.3 | -8.2 | +5.0 | -2.3 | -6.0 |
| Relative (%) | +19.6 | -5.7 | +5.3 | +47.9 | +18.2 | +13.0 | +29.8 | -33.8 | +20.4 | -9.3 | -24.4 | |
| Steps (reduced) |
182 (55) |
187 (60) |
192 (65) |
197 (70) |
201 (74) |
205 (78) |
209 (82) |
212 (85) |
216 (89) |
219 (92) |
222 (95) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.8 | +2.2 | -7.6 | -3.4 | -1.7 | +0.0 | -11.5 | +4.3 | -7.2 | -1.3 | -5.5 |
| Relative (%) | -15.7 | +8.9 | -31.3 | -13.8 | -6.8 | +0.0 | -47.0 | +17.7 | -29.5 | -5.4 | -22.5 | |
| Steps (reduced) |
49 (49) |
78 (78) |
98 (98) |
114 (114) |
127 (127) |
138 (0) |
147 (9) |
156 (18) |
163 (25) |
170 (32) |
176 (38) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.4 | -3.8 | -1.2 | +9.1 | +1.8 | +0.5 | +4.5 | -11.0 | +2.2 | -5.1 | -8.9 |
| Relative (%) | +9.9 | -15.7 | -4.9 | +37.4 | +7.4 | +2.1 | +18.6 | -45.1 | +8.9 | -21.0 | -36.3 | |
| Steps (reduced) |
182 (44) |
187 (49) |
192 (54) |
197 (59) |
201 (63) |
205 (67) |
209 (71) |
212 (74) |
216 (78) |
219 (81) |
222 (84) | |
In each of these tunings, it is best to avoid using intervals containing large numbers of factors of 2, such as 9/8 and 16/15, as they tend to be mapped inconsistently due to the error of harmonic 2 accumulating. However, if large powers of 2 are excluded, then most if not all intervals are consistent. For example, 138ed7 is completely consistent in the no-8s no-16s no-20s 22-integer limit! Harmonic 23 is also usable; we just have to be careful with our voicings to avoid inconsistencies. This obviously gives us an abundance of harmonic resources. 49edo's optimal octave compression, especially in higher limits, is quite hard (-3.577 cents in 19-limit WE), but prime 2 is still usable; we just have to be careful with it to avoid inconsistencies.