27edo: Difference between revisions

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Created page with "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 == {{Harmonics in ED|27|53|0}} Useful aberrismic scales in 27edo include blackdye, pinedye, and their mutual superset trackdye. === Ground's chord..."
 
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Useful aberrismic scales in 27edo include [[blackdye]], [[pinedye]], and their mutual superset [[trackdye]].
Useful aberrismic scales in 27edo include [[blackdye]], [[pinedye]], and their mutual superset [[trackdye]].


=== Ground's chord table ===
=== Ground's tertian triad table ===
This table uses [[Ground's_composition_theory#Interval_name_shorthand|ground's interval name shorthand]].
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:1em;" | fif
! style="width:1.5em;" | Fifth
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
! style="width:1em;" |
! style="width:1.5em;" |
|-
|-
! style="height:3em;" | 17\27
! style="height:3em;" |19\27
| colspan="2" |(sj<br>1inv)
| colspan="2" style="background:#333;" |
| colspan="2" style="background:#333;" |
| colspan="2" |<small>8:10:13</small><br>a <small>(1inv)</small>
| colspan="2" |
| colspan="2" style="background:#333;" |
| colspan="2" style="background:#333;" |
|-
! style="height:3em;" |18\27
| style="background:#333; line-height:0.8;" |<small>(tj<br>1<br>inv)</small>
| colspan="2" style="background:#333;" |(j 1inv)
| colspan="2" |
| colspan="2" class="thl" |classic<br>a
| 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)
| style="background:#333; line-height:0.8;" |<small>(tm<br>2<br>inv)</small>
|-
! style="height:3em;" |17\27
| colspan="2" |(sj 1inv)
| colspan="2" |sensic<br>simic
| colspan="2" |sensic<br>simic
| colspan="2" |
| colspan="2" |<small>13:16:20</small><br>a
| colspan="2" |
| colspan="2" |
| colspan="2" |sensic<br>taphric
| colspan="2" |sensic<br>taphric
| colspan="2" |(um<br>2inv)
| colspan="2" |(um 2inv)
|-
|-
! style="height:3em;" | 16\27
! style="height:3em;" | 16\27
|
| style="background:#333;" |
| colspan="2" style="font-size:125%;" |um
| colspan="2" |<small>6:7:9</small><br>um
| colspan="2" style="font-size:125%;" |m
| colspan="2" |<small>10:12:15</small><br>m
| colspan="2" style="font-size:125%;" |n
| colspan="2" class="thl" |<small>18:22:27</small><br>n
| colspan="2" style="font-size:125%;" |j
| colspan="2" |<small>4:5:6</small><br>j
| colspan="2" style="font-size:125%;" |sj
| colspan="2" |<small>14:18:21</small><br>sj
|
| style="background:#333;" |
|-
|-
! style="height:3em;" | 15\27
! style="height:3em;" | 15\27
| colspan="2" style="font-size:125%;" |tim
| colspan="2" |<small>15:17:22</small><br>tim
| colspan="2" style="font-size:125%;" |tm
| colspan="2" |<small>13:&gt;15:19</small><br>tm
| colspan="2" style="font-size:125%;" |dm
| colspan="2" |<small>15:18:22</small><br>dm
| colspan="2" style="font-size:125%;" |dj
| colspan="2" |<small>13:16:19</small><br>dj
| colspan="2" style="font-size:125%;" |tj
| colspan="2" |<small>15:19:22</small><br>tj
| colspan="2" style="font-size:125%;" |tlj
| colspan="2" |<small>13:&lt;17:19</small><br>tlj
|-
|-
! style="height:3em;" | 14\27
! style="height:3em;" | 14\27
|
| style="background:#333;" |
| colspan="2" style="background:#333;" |<small>7:8:10</small>
| colspan="2" |<small>19:22:27</small><br>d
| colspan="2" class="thl" |classic<br>d
| colspan="2" |
| colspan="2" |
| colspan="2" |
| colspan="2" style="background:#333;" |
| colspan="2" |classic<br>dim.
| style="background:#333;" |<small>7:9<br>:10</small>
| colspan="2" |
| colspan="2" |
|
|-
|-
! style="height:3em;" | 13\27
! style="height:3em;" | 13\27
| colspan="2" |
| colspan="2" style="background:#333;" |<small>18:20:25</small>
| colspan="2" |
| colspan="2" style="background:#333;" |<small>15:17:21</small>
| colspan="2" |
| colspan="2" |<small>18:21:25</small><br>d
| colspan="2" |otonal<br>dim.
| colspan="2" |<small>5:6:7</small><br>d
| colspan="2" |
| colspan="2" style="background:#333;" |<small>18:22:25</small>
| colspan="2" |
| colspan="2" style="background:#333;" |<small>15:19:21</small>
|}
|}
{{UserTag|g_|Ground|7766ff|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 ==
== Multiples ==

Latest revision as of 04:27, 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

Approximation of prime harmonics in 27edo
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

Approximation of prime harmonics in 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

Approximation of prime harmonics in 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)
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