27edo
From XenReference
27edo, or 27 equal divisions of the octave, is the equal tuning with a step size of (1200/27) ≈ 44.4¢, 27 of which stack to the perfect octave 2/1. It is best known as a Superpyth tuning alongside 22edo with a distinct sharp tendency in the 7-limit, and approximating prime 13 well instead of 11.
Theory
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +9.2 | +13.7 | +9.0 | -18.0 | +3.9 | -16.1 | +13.6 | -6.1 | -7.4 | +10.5 | +15.3 | +15.4 | +21.8 | +1.2 | +15.4 |
| Relative (%) | 0.0 | +20.6 | +30.8 | +20.1 | -40.5 | +8.8 | -36.1 | +30.6 | -13.6 | -16.5 | +23.7 | +34.5 | +34.6 | +49.1 | +2.6 | +34.6 | |
| Steps (reduced) |
27 (0) |
43 (16) |
63 (9) |
76 (22) |
93 (12) |
100 (19) |
110 (2) |
115 (7) |
122 (14) |
131 (23) |
134 (26) |
141 (6) |
145 (10) |
147 (12) |
150 (15) |
155 (20) | |
Useful aberrismic scales in 27edo include blackdye, pinedye, and their mutual superset trackdye.
Ground's tertian triad table
This table uses ground's interval name shorthand. It shows the bounding fifth on the vertical axis and the step difference between the two stacked thirds on the horizontal axis, with the center being no difference.
| Fifth | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19\27 | 8:10:13 a (1inv) |
|||||||||||
| 18\27 | (tj 1 inv) |
(j 1inv) | classic a |
10:13:16 (17:22:27) a (2inv) |
(m 2inv) | (tm 2 inv) | ||||||
| 17\27 | (sj 1inv) | sensic simic |
13:16:20 a |
sensic taphric |
(um 2inv) | |||||||
| 16\27 | 6:7:9 um |
10:12:15 m |
18:22:27 n |
4:5:6 j |
14:18:21 sj |
|||||||
| 15\27 | 15:17:22 tim |
13:>15:19 tm |
15:18:22 dm |
13:16:19 dj |
15:19:22 tj |
13:<17:19 tlj | ||||||
| 14\27 | 7:8:10 | 19:22:27 d |
classic d |
7:9 :10 | ||||||||
| 13\27 | 18:20:25 | 15:17:21 | 18:21:25 d |
5:6:7 d |
18:22:25 | 15:19:21 | ||||||
g_
The chords in the 15\27 row all share the quality of their unstable fifth. The chords in the 16\27 row share the quality of their stable fifth (~3/2), except for neutral, which I think is much closer in quality to the 15\27 row due to its unstable third. It's also intermediate im function between the diminished and augmented chords.
Multiples
54edo
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +9.2 | -8.5 | +9.0 | +4.2 | +3.9 | +6.2 | -8.6 | -6.1 | -7.4 | +10.5 | -6.9 | -6.8 | -0.4 | +1.2 | -6.8 |
| Relative (%) | 0.0 | +41.2 | -38.4 | +40.3 | +19.1 | +17.6 | +27.7 | -38.8 | -27.2 | -33.1 | +47.3 | -31.0 | -30.8 | -1.8 | +5.2 | -30.8 | |
| Steps (reduced) |
54 (0) |
86 (32) |
125 (17) |
152 (44) |
187 (25) |
200 (38) |
221 (5) |
229 (13) |
244 (28) |
262 (46) |
268 (52) |
281 (11) |
289 (19) |
293 (23) |
300 (30) |
309 (39) | |
81edo
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -5.7 | -1.1 | -5.9 | -3.2 | +3.9 | -1.3 | -1.2 | -6.1 | -7.4 | -4.3 | +0.5 | +0.6 | +7.0 | +1.2 | +0.6 |
| Relative (%) | 0.0 | -38.2 | -7.6 | -39.6 | -21.4 | +26.4 | -8.4 | -8.2 | -40.9 | -49.6 | -29.0 | +3.4 | +3.8 | +47.3 | +7.8 | +3.8 | |
| Steps (reduced) |
81 (0) |
128 (47) |
188 (26) |
227 (65) |
280 (37) |
300 (57) |
331 (7) |
344 (20) |
366 (42) |
393 (69) |
401 (77) |
422 (17) |
434 (29) |
440 (35) |
450 (45) |
464 (59) | |
