18edo
18 Equal Divisions of the Octave also known as The Third-Tone System.
Contents
Basic Properties
18-EDO divides the octave into 18 equal parts of ~66.667 cents each. It does not approximate the 3rd harmonic at all, unless a >30¢-error is considered acceptable, and it approximates the 5th and 7th harmonics equally with 12-TET. It does, however, render a most accurate tuning of 9/8, 7/6, 21/16, 15/11, 12/7, 16/9, and 13/7. It is also the smallest EDO to approximate the harmonic series chord 5:6:7 without tempering out 36/35 (and thus without using the same interval to approximate both 6/5 and 7/6).
In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach, meaning to avoid the usual closed-voice "root-3rd-5th" type of chord and instead use chords which are either more compressed or more stretched out. 18-EDO may be treated as a temperament of the 17-limit 4*18 subgroup just intonation subgroup 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as 72 does on the full 17-limit, and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.
However, less accurate approximations can be used, and 18edo can be treated as a 7-limit exotemperament with the mapping <18 29 42 51|. This maps 3/2 to 733.33¢ and 7/4 to 1000¢; as a result, 28/27 is tempered out, and weird things happen: 9/8 and 7/6 are both mapped to 266.67¢, while 8/7 gets mapped below both of them to 200¢, making for a rather disordered 7-limit tonality diamond, but hey, whatever floats your boat!
18-EDO contains sub-EDOs 2, 3, 6, and 9, and itself is half of 36-EDO and one-fourth of 72-EDO. It bears some similarities to 13-EDO (with its very flat 4ths and nice subminor 3rds), 11-EDO (with its very sharp minor 3rds, two of which span a very flat 5th), 16-EDO (with its sharp 4ths and flat 5ths), and 17-EDO and 19-EDO (with its narrow semitone, three of which comprise a whole-tone). It is an excellent tuning for those seeking a forceful deviation from the common practice.
Representations of Just Intervals
Degree | Cents | pions | 7mus | 5L3s Notation | Nearest Ratio | Error | 17-Limit Ratios* |
0 | C | 1/1 | 0 | 1/1 | |||
1 | 66.67 | 70.67 | 85.33 (55.55_{16}) | Db | 27/26 | +1.329 | 78/75, 75/72 |
2 | 133.33 | 141.33 | 170.67 (AA.AB_{16}) | C# | 27/25 | +0.096 | 51/55, 42/39 |
3 | 200 | 212 | 256 (100_{16}) | D | 9/8 | -3.910 | 9/8 |
4 | 266.67 | 282.67 | 341.33 (155.55_{16}) | Eb | 7/6 | -0.204 | 75/64 |
5 | 333.33 | 353.33 | 426.67 (1AA.AB_{16}) | D# | 17/14 or 40/33 | -2.796 +0.293 | 39/32 |
6 | 400 | 424 | 512 (200_{16}) | E | 5/4 or 44/35 | +13.686 +3.822 | 64/55 |
7 | 466.67 | 494.67 | 597.33 (255.55_{16}) | F | 21/16 | -4.114 | 21/16 |
8 | 533.33 | 565.33 | 682.67 (2AA.AB_{16}) | Gb | 15/11 | -3.617 | 102/75 |
9 | 600 | 636 | 768 (300_{16}) | F# | 17/12 or 24/17 | -3.000 +3.000 | 17/12 |
10 | 666.67 | 706.67 | 853.33 (355.55_{16}) | G | 22/15 | +3.617 | 75/51 |
11 | 733.33 | 777.33 | 938.67 (3AA.AB_{16}) | Hb | 32/21 | +4.114 | 32/21 |
12 | 800 | 848 | 1024 (400_{16}) | G# | 8/5 or 35/22 | -13.686 -3.822 | 51/32 |
13 | 866.67 | 919.67 | 1109.33 (455.55_{16}) | H | 28/17 or 33/20 | +2.796 -0.293 | 64/39 |
14 | 933.33 | 989.33 | 1194.67 (4AA.AB_{16}) | A | 12/7 | +0.204 | 55/32 |
15 | 1000 | 1060 | 1280 (500_{16}) | Bb | 16/9 | +3.910 | 16/9 |
16 | 1066.67 | 1131.67 | 1365.33 (555.55_{16}) | A# | 50/27 | -0.096 | 39/21 |
17 | 1133.33 | 1201.33 | 1440.67 (5AA.AB_{16}) | B | 52/27 | -1.329 | 75/39 |
18 | 1200 | 1272 | 1536 (600_{16}) | C | 2/1 | 0 | 2/1** |
- based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament
Notation
18edo can be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 10\18. This is only 4¢ worse that the best approximation, which becomes the up-fifth. Using this 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this.
The first way preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
The second way preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 18edo "on the fly".
Degree | Cents | pions | 7mus | Up/down notation using the narrow 5th of 10\18,
with major wider than minor |
Up/down notation using the narrow 5th of 10\18,
with major narrower than minor | ||||
---|---|---|---|---|---|---|---|---|---|
0 | 0 | perfect unison | P1 | D | perfect unison | P1 | D | ||
1 | 67 | 71 | 85 (55_{16}) | up unison, downminor 2nd | ^1, vm2 | D^, Ev | up unison, downmajor 2nd | ^1, vM2 | D^, Ev |
2 | 133 | 141 | 171 (AB_{16}) | minor 2nd | m2 | E | major 2nd | M2 | E |
3 | 200 | 212 | 256 (100_{16}) | mid 2nd | ~2 | E^ | mid 2nd | ~2 | E^ |
4 | 267 | 283 | 341 (155_{16}) | major 2nd, minor 3rd | M2, m3 | E#, Fb | minor 2nd, major 3rd | m2, M3 | Eb, F# |
5 | 333 | 353 | 427 (1AB_{16}) | mid 3rd | ~3 | Fv | mid 3rd | ~3 | Fv |
6 | 400 | 424 | 512 (200_{16}) | major 3rd | M3 | F | minor 3rd | m3 | F |
7 | 467 | 495 | 597 (255_{16}) | upmajor 3rd, down 4th | ^M3, v4 | F^, Gv | upminor 3rd, down 4th | ^m3, v4 | F^, Gv |
8 | 533 | 565 | 683 (2AB_{16}) | perfect 4th | P4 | G | perfect 4th | P4 | G |
9 | 600 | 636 | 768 (300_{16}) | up 4th, down 5th | ^4, v5 | G^, Av | up 4th, down 5th | ^4, v5 | G^, Av |
10 | 667 | 707 | 853 (355_{16}) | perfect 5th | P5 | A | perfect 5th | P5 | A |
11 | 733 | 777 | 939 (3AB_{16}) | up 5th, downminor 6th | ^5, vm6 | A^, Bv | up fifth, downmajor 6th | ^5, vM6 | A^, Bv |
12 | 800 | 848 | 1024 (400_{16}) | minor 6th | m6 | B | major 6th | M6 | B |
13 | 867 | 919 | 1109 (455_{16}) | mid 6th | ~6 | B^ | mid 6th | ~6 | B^ |
14 | 933 | 989 | 1195 (4AB_{16}) | major 6th, minor 7th | M6, m7 | B#, Cb | minor 6th, major 7th | m6, M7 | Bb, C# |
15 | 1000 | 1060 | 1280 (500_{16}) | mid 7th | ~7 | Cv | mid 7th | ~7 | Cv |
16 | 1067 | 1131 | 1365 (555_{16}) | major 7th | M7 | C | minor 7th | m7 | C |
17 | 1133 | 1201 | 1441 (5AB_{16}) | upmajor 7th, down 8ve | ^M7, v8 | C^, Dv | upminor 7th, down 8ve | ^m7, v8 | C^, Dv |
18 | 1200 | 1272 | 1536 (600_{16}) | perfect 8ve | P8 | D | perfect 8ve | P8 | D |
For alternative notations, see Ups and Downs Notation -"Supersharp" EDOs (pentatonic and nonatonic fifth-generated) and Ups and Downs Notation - Natural Generators (heptatonic third-generated).
Useful Moment-of-Symmetry Scales
Note: This list excludes scales found in 9-EDO.
Pentatonic:
3L2s Father Pentatonic: 4 4 3 4 3
Hexatonic:
4L2s Bicycle: 4 4 1 4 4 1
2L4s Rice Hexatonic: 2 5 2 2 5 2
Heptatonic:
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2
Octatonic:
5L3s Father Octatonic: 3 1 3 3 1 3 3 1
2L6s Rice Octatonic: 2 2 3 2 2 2 3 2
Decatonic:
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
Dodecatonic:
6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1
Application to Guitar
18-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!
The "Father Octatonic" scale maps very simply to a 6-string guitar tuned in "reverse-standard" tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21 (all of which are most evenly open-tuned using a series of sharpened 4ths and a minor or neutral 3rd, and whose scales thus often require position-shifting and/or larger stretches of the hand).
Commas
18 EDO tempers out the following commas. (Note: This assumes the val < 18 29 42 51 62 67 |.)
Comma | Monzo | Value (Cents) | Name 1 | Name 2 |
---|---|---|---|---|
128/125 | | 7 0 -3 > | 41.06 | Diesis | Augmented Comma |
| 23 6 -14 > | 3.34 | Vishnuzma | Semisuper | |
50/49 | | 1 0 2 -2 > | 34.98 | Tritonic Diesis | Jubilisma |
686/675 | | 1 -3 -2 3 > | 27.99 | Senga | |
875/864 | | -5 -3 3 1 > | 21.90 | Keema | |
1728/1715 | | 6 3 -1 -3 > | 13.07 | Orwellisma | Orwell Comma |
16875/16807 | | 0 3 4 -5 > | 6.99 | Mirkwai | |
3136/3125 | | 6 0 -5 2 > | 6.08 | Hemimean | |
99/98 | | -1 2 0 -2 1 > | 17.58 | Mothwellsma | |
100/99 | | 2 -2 2 0 -1 > | 17.40 | Ptolemisma | |
65536/65219 | | 16 0 0 -2 -3 > | 8.39 | Orgonisma | |
385/384 | | -7 -1 1 1 1 > | 4.50 | Keenanisma | |
9801/9800 | | -3 4 -2 -2 2 > | 0.18 | Kalisma | Gauss' Comma |
91/90 | | -1 -2 -1 1 1 > | 19.13 | Superleap |