Xenharmonic Wiki

27edo

Revision as of 02:29, 4 July 2020 by FloraC (talk | contribs) (Some ratios)

Theory

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. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it's 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 the Pythagorean scale, but also to many classically trained violinists.

Intervals

# Cents Approximate Ratios* Ups and Downs Notation Solfege
0 0.00 1/1 P1 perfect unison D do
1 44.44 28/27, 36/35, 49/48, 50/49, 81/80 ^1, m2 up-unison, minor 2nd ^D, Eb di
2 88.89 16/15, 21/20, 25/24 ^m2 upminor 2nd ^Eb ra
3 133.33 15/14, 14/13, 13/12 ~2 mid 2nd vvE ru
4 177.78 10/9 vM2 downmajor 2nd vE reh
5 222.22 8/7, 9/8 M2 major 2nd E re
6 266.67 7/6 m3 minor 3rd F ma
7 311.11 6/5 ^m3 upminor 3rd ^F me
8 355.56 16/13 ~3 mid 3rd ^^F mu
9 400.00 5/4 vM3 downmajor 3rd vF# mi
10 444.44 9/7, 13/10 M3 major 3rd F# mo
11 488.89 4/3 P4 perfect 4th G fa
12 533.33 27/20, 48/35 ^4 up 4th ^G fih
13 577.78 7/5, 18/13 ~4, vd5 mid 4th, updim 5th ^^G, ^Ab fi
14 622.22 10/7, 13/9 vA4, ~5 downaug 4th, mid 5th vG#, vvA se
15 666.67 40/27, 35/24 v5 down fifth vA sih
16 711.11 3/2 P5 perfect 5th A so/sol
17 755.56 14/9, 20/13 m6 minor 6th Bb lo
18 800.00 8/5 ^m6 upminor 6th ^Bb le
19 844.44 13/8 ~6 mid 6th vvB lu
20 888.89 5/3 vM6 downmajor 6th vB la
21 933.33 12/7 M6 major 6th B li
22 977.78 7/4, 16/9 m7 minor 7th C ta
23 1022.22 9/5 ^m7 upminor 7th ^C te
24 1066.67 28/15, 13/7, 24/13 ~7 mid 7th ^^C tu
25 1111.11 15/8, 40/21, 48/25 vM7 downmajor 7th vC# ti
26 1155.56 27/14, 35/18, 96/49, 49/25, 160/81 M7 major 7th C# da
27 1200.00 2/1 P8 8ve D do

* 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. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). 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 vvE G C~ C mid
yo 4:5:6 0-9-16 C vE G Cv C downmajor or C down
ru 14:18:21 0-10-16 C E G C C major or C

For a more complete list, see Ups and Downs Notation #Chords and Chord Progressions. See also the 22edo page.

Selected just intervals by error

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13
Error absolute (¢) 0.00 +9.16 +13.69 +8.95 -17.98 +3.92
relative (%) 0.0 +20.6 +30.8 +20.1 -40.5 +8.8

The following table shows how 15-odd-limit intervals are represented in 27edo. Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢)
7/6, 12/7 0.204
15/11, 22/15 3.617
16/13, 13/8 3.917
6/5, 5/3 4.530
10/9, 9/5 4.626
7/5, 10/7 4.734
14/13, 13/7 5.035
13/12, 24/13 5.239
11/9, 18/11 8.148
8/7, 7/4 8.952
4/3, 3/2 9.156
9/7, 14/9 9.360
13/10, 20/13 9.770
11/10, 20/11 12.774
5/4, 8/5 13.686
15/14, 28/15 13.891
18/13, 13/9 14.395
12/11, 11/6 17.304
14/11, 11/7 17.508
11/8, 16/11 17.985
9/8, 16/9 18.312
15/13, 26/15 18.926
16/15, 15/8 21.602
13/11, 22/13 21.901
Patent val mapping
Interval, complement Error (abs, ¢)
7/6, 12/7 0.204
16/13, 13/8 3.917
6/5, 5/3 4.530
10/9, 9/5 4.626
7/5, 10/7 4.734
14/13, 13/7 5.035
13/12, 24/13 5.239
8/7, 7/4 8.952
4/3, 3/2 9.156
9/7, 14/9 9.360
13/10, 20/13 9.770
5/4, 8/5 13.686
15/14, 28/15 13.891
18/13, 13/9 14.395
11/8, 16/11 17.985
9/8, 16/9 18.312
15/13, 26/15 18.926
13/11, 22/13 21.901
16/15, 15/8 22.842
14/11, 11/7 26.936
12/11, 11/6 27.141
11/10, 20/11 31.671
11/9, 18/11 36.297
15/11, 22/15 40.827

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].)

Prime
Limit 		Ratio Monzo Cents Color Name Name 1 Name 2 Name 3
5 128/125 | 7 0 -3 > 41.06 Trigu Diesis Augmented Comma
" 20000/19683 | 5 -9 4 > 27.66 Saquadyo Minimal Diesis Tetracot Comma
" 78732/78125 | 2 9 -7 > 13.40 Sepgu Medium Semicomma Sensipent Comma
" 4711802/4709457 | 1 -27 18 > 0.86 Satritribiyo Ennealimma
7 686/675 | 1 -3 -2 3 > 27.99 Trizo-agugu Senga
" 64/63 | 6 -2 0 -1 > 27.26 Ru Septimal Comma Archytas' Comma Leipziger Komma
" 50421/50000 | -4 1 -5 5 > 14.52 Quinzogu Trimyna
" 245/243 | 0 -5 1 2 > 14.19 Zozoyo Sensamagic
" 126/125 | 1 2 -3 1 > 13.79 Zotrigu Septimal Semicomma Starling Comma
" 4000/3969 | 5 -4 3 -2 > 13.47 Rurutriyo Octagar
" 1728/1715 | 6 3 -1 -3 > 13.07 Triru-agu Orwellisma Orwell Comma
" 420175/419904 | -6 -8 2 5 > 1.12 Quinzo-ayoyo Wizma
" 2401/2400 | -5 -1 -2 4 > 0.72 Bizozogu Breedsma
" 4375/4374 | -1 -7 4 1 > 0.40 Zoquadyo Ragisma
" 250047/250000 | -4 6 -6 3 > 0.33 Trizogugu Landscape Comma
11 99/98 | -1 2 0 -2 1 > 17.58 Loruru Mothwellsma
" 896/891 | 7 -4 0 1 -1 > 9.69 Saluzo Pentacircle
" 385/384 | -7 -1 1 1 1 > 4.50 Lozoyo Keenanisma
13 91/90 | -1 -2 -1 1 0 1 > 19.13 Thozogu Superleap

Music

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