27edo

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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 zo {a, b, 0, 1} 7/6, 7/4
" fourthward wa {a, b}, b < -1 32/27, 16/9
upminor gu {a, b, -1} 6/5, 9/5
mid tho {a, b, 0, 0, 0, 1} 13/12, 13/8
" thu {a, b, 0, 0, 0, -1} 16/13, 24/13
downmajor yo {a, b, 1} 5/4, 5/3
major fifthward wa {a, b}, b > 1 9/8, 27/16
" ru {a, b, 0, -1} 9/7, 12/7

All 27edo chords can be named using ups and downs. Here are the zo, gu, ilo, yo and ru triads:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-6-16 C Eb G Cm C minor
gu 10:12:15 0-7-16 C Eb^ G C.^m C upminor
ilo 18:22:27 0-8-16 C Evv G C~ C mid
yo 4:5:6 0-9-16 C Ev G C.v C downmajor or C dot down
ru 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