27edo: Difference between revisions
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=== Ground's tertian chord table === | === Ground's tertian chord table === | ||
This table uses [[Ground's_composition_theory#Interval_name_shorthand|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. | This table uses [[Ground's_composition_theory#Interval_name_shorthand|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. | ||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="line-height:1.2; text-align:center;" | ||
! style="width:1.5em;" | Fifth | ! style="width:1.5em;" | Fifth | ||
! style="width:1.5em;" | | ! style="width:1.5em;" | | ||
| Line 23: | Line 23: | ||
! style="width:1.5em;" | | ! style="width:1.5em;" | | ||
|- | |- | ||
!19\27 | ! style="height:3em;" |19\27 | ||
| colspan="2" style="background:#333;" | | | colspan="2" style="background:#333;" | | ||
| colspan="2" style="background:#333;" | | | colspan="2" style="background:#333;" | | ||
| Line 31: | Line 31: | ||
| colspan="2" style="background:#333;" | | | colspan="2" style="background:#333;" | | ||
|- | |- | ||
!18\27 | ! style="height:3em;" |18\27 | ||
| style="background:#333;" |<small>(tj< | | style="background:#333; line-height:0.8;" |<small>(tj<br>1<br>inv)</small> | ||
| colspan="2" style="background:#333;" |(j 1inv) | | colspan="2" style="background:#333;" |(j 1inv) | ||
| colspan="2" | | | colspan="2" | | ||
| colspan="2" class="thl" |classic<br>a | | colspan="2" class="thl" |classic<br>a | ||
| colspan="2" |<small>10:13:16</small><br>a <small>(2inv) | | colspan="2" style="line-height:0.8;" |<small>10:13:16<br>(17:22:27)</small><br>a <small>(2inv)</small> | ||
| colspan="2" style="background:#333;" |(m 2inv) | | colspan="2" style="background:#333;" |(m 2inv) | ||
| style="background:#333;" |<small>(tm< | | style="background:#333; line-height:0.8;" |<small>(tm<br>2<br>inv)</small> | ||
|- | |- | ||
! style="height:3em;" | 17\27 | ! style="height:3em;" |17\27 | ||
| colspan="2" |(sj 1inv) | | colspan="2" |(sj 1inv) | ||
| colspan="2" |sensic<br>simic | | colspan="2" |sensic<br>simic | ||
Revision as of 04:09, 12 November 2026
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 chord 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 | ||||||
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) | |
