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27 tone equal tempertament


If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444... cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the third, fifth and 7/4 sharply.

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13 2/3 cents. The result is that 6/5, 7/5 and especially 7/6 are all tuned more accurately than this.

27edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with 22edo tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.

Though the 7-limit tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit diamond is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest harmonic entropy possible and thus is, in theory, most dissonant, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 EDOsteps, just one EDOstep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with pythagorean scale, but also to many classically trained violinists. et voila

Intervals


Intervals

Degree
Cents value
Approximate
Ratios*
Solfege
ups and downs notation
0
0
1/1
do
P1
perfect unison
D
1
44.44
36/35, 49/48, 50/49
di
^1, m2
minor 2nd
Eb
2
88.89
16/15, 21/20, 25/24
ra
^^1, ^m2
upminor 2nd
Eb^
3
133.33
14/13, 13/12
ru
~2
mid 2nd
Evv
4
177.78
10/9
reh
vM2
downmajor 2nd
Ev
5
222.22
8/7, 9/8
re
M2
major 2nd
E
6
266.67
7/6
ma
m3
minor 3rd
F
7
311.11
6/5
me
^m3
upminor 3rd
F^
8
355.56
16/13
mu
~3
mid 3rd
F^^
9
400
5/4
mi
vM3
downmajor 3rd
F#v
10
444.44
9/7, 13/10
mo
M3
major 3rd
F#
11
488.89
4/3
fa
P4
perfect 4th
G
12
533.33
49/36, 48/35
fih
^4
up 4th
G^
13
577.78
7/5, 18/13
fi
^^4
double-up 4th
G^^
14
622.22
10/7, 13/9
se
vv5
double-down 5th
Avv
15
666.67
72/49, 35/24
sih
v5
down fifth
Av
16
711.11
3/2
so/sol
P5
perfect 5th
A
17
755.56
14/9, 20/13
lo
m6
minor 6th
Bb
18
800
8/5
le
^m6
upminor 6th
Bb^
19
844.44
13/8
lu
~6
mid 6th
Bvv
20
888.89
5/3
la
vM6
downmajor 6th
Bv
21
933.33
12/7
li
M6
major 6th
B
22
977.78
7/4, 16/9
ta
m7
minor 7th
C
23
1022.22
9/5
te
^m7
upminor 7th
C^
24
1066,67
13/7, 24/13
tu
~7
mid 7th
C^^
25
1111.11
40/21
ti
vM7
downmajor 7th
C#v
26
1155.56
35/18, 96/49, 49/25
da
M7
major 7th
C#
27
1200
2/1
do
P8
8ve
D
*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
quality
color
monzo format
examples
minor
blue
{a, b, 0, 1}
7/6, 7/4
"
fourthward white
{a, b}, b < -1
32/27, 16/9
upminor
green
{a, b, -1}
6/5, 9/5
mid
emerald
{a, b, 0, 0, 0, 1}
13/12, 13/8
"
ochre
{a, b, 0, 0, -1}
16/13, 24/13
downmajor
yellow
{a, b, 1}
5/4, 5/3
major
fifthward white
{a, b}, b > 1
9/8, 27/16
"
red
{a, b, 0, -1}
9/7, 12/7
All 27edo chords can be named using ups and downs. Here are the blue, green, jade, yellow and red triads:
color of the 3rd
JI chord
notes as edosteps
notes of C chord
written name
spoken name
blue
6:7:9
0-6-16
C Eb G
Cm
C minor
green
10:12:15
0-7-16
C Eb^ G
C.^m
C upminor
jade
18:22:27
0-8-16
C Evv G
C~
C mid
yellow
4:5:6
0-9-16
C Ev G
C.v
C downmajor or C dot down
red
14:18:27
0-10-16
C E G
C
C major or C
For a more complete list, see Ups and Downs Notation - Chord names in other EDOs. See also the 22edo page.

Rank two temperaments

List of 27edo rank two temperaments by badness
List of edo-distinct 27e rank two temperaments
Periods
per octave
Generator
Temperaments
1
1\27
Quartonic/Quarto
1
2\27
Octacot/Octocat
1
4\27
Tetracot/Modus/Wollemia
1
5\27
Machine/Kumonga
1
7\27
Myna/Coleto/Minah
1
8\27
Beatles/Ringo
1
10\27
Sensi/Sensis
1
11\27
Superpyth
1
13\27
Fervor
3
1\27
Semiaug/Hemiaug
3
2\27
Augmented/Augene/Ogene
3
4\27
Oodako
9
1\27
Terrible version of Ennealimmal
/ Niner

Commas

27 EDO tempers out the following commas. (Note: This assumes the val < 27 43 63 76 93 100 |.)
Comma
Monzo
Value (Cents)
Name 1
Name 2
Name 3
128/125
| 7 0 -3 >
41.06
Diesis
Augmented Comma

20000/19683
| 5 -9 4 >
27.66
Minimal Diesis
Tetracot Comma

78732/78125
| 2 9 -7 >
13.40
Medium Semicomma
Sensipent Comma

4711802/4709457
| 1 -27 18 >
0.86
Ennealimma


686/675
| 1 -3 -2 3 >
27.99
Senga


64/63
| 6 -2 0 -1 >
27.26
Septimal Comma
Archytas' Comma
Leipziger Komma
50421/50000
| -4 1 -5 5 >
14.52
Trimyna


245/243
| 0 -5 1 2 >
14.19
Sensamagic


126/125
| 1 2 -3 1 >
13.79
Septimal Semicomma
Starling Comma

4000/3969
| 5 -4 3 -2 >
13.47
Octagar


1728/1715
| 6 3 -1 -3 >
13.07
Orwellisma
Orwell Comma

420175/419904
| -6 -8 2 5 >
1.12
Wizma


2401/2400
| -5 -1 -2 4 >
0.72
Breedsma


4375/4374
| -1 -7 4 1 >
0.40
Ragisma


250047/250000
| -4 6 -6 3 >
0.33
Landscape Comma


99/98
| -1 2 0 -2 1 >
17.58
Mothwellsma


896/891
| 7 -4 0 1 -1 >
9.69
Pentacircle


385/384
| -7 -1 1 1 1 >
4.50
Keenanisma


91/90
| -1 -2 -1 1 0 1 >
19.13
Superleap



Music


Music For Your Ears play by Gene Ward Smith The central portion is in 27edo, the rest in 46edo.
Sad Like Winter Leaves by Igliashon Jones
Superpythagorean Waltz by Igliashon Jones
Galticeran Sonatina by Joel Taylor
miniature prelude and fugue by Kosmorsky
Chicago Pile-1 by Chris Vaisvil
Tetracot Perc-Sitar by Dustin Schallert
Tetracot Jam by Dustin Schallert
Tetracot Pump by Dustin Schallert all in 27edo
27-EDO Guitar 1 by Dustin Schallert