User:Overthink/49edo and octave compression
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49edo is a strongly sharp-tending system in the up to 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) | |