# 18edo

(Redirected from 18-edo)
 ← 17edo 18edo 19edo →
Prime factorization 2 × 32
Step size 66.6667¢
Fifth 11\18 (733.333¢)
Semitones (A1:m2) 5:-1 (333.3¢ : -66.67¢)
Dual sharp fifth 11\18 (733.333¢)
Dual flat fifth 10\18 (666.667¢) (→5\9)
Dual major 2nd 3\18 (200¢) (→1\6)
Consistency limit 7
Distinct consistency limit 5

18 equal divisions of the octave (abbreviated 18edo or 18ed2), also called 18-tone equal temperament (18tet) or 18 equal temperament (18et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 18 equal parts of about 66.7 ¢ each. Each step represents a frequency ratio of 21/18, or the 18th root of 2.

18edo is also known as the third-tone system.

## Theory

18edo does not approximate the 3rd harmonic at all, unless a >30¢-error is considered acceptable, and it approximates the 5th, 7th and 9th harmonics equally well (or equally poorly) as 12edo does. It does, however, render more accurate tunings of 7/6, 21/16, 15/11, 12/7, 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. 18edo 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 72edo 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. 18edo also approximates 12:13:14:17:23:27:29 quite well, with the least maximum relative error out of any edos ≤ 100 (the worst-approximated dyad is 23/13, with relative error 18.36%). Hence it can be viewed as an "/3 temperament" (/3 used in the primodality sense), specifically in the 2.9.13/12.7/6.17/12.23/12.29/24 subgroup. As for more simple subgroups, 18edo can be treated as a 2.9.5.7 subgroup temperament.

However, less accurate approximations can be used, and 18edo can be treated as a 7-limit (with 3s) exotemperament with the mapping 18 29 42 51]. This maps 3/2 to 733.33¢, 5/4 to 400¢ 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-odd-limit tonality diamond, but hey, whatever floats your boat! This 7-limit mapping supports 7-limit sixix thus is strongly associated with 18edo's 4L 3s mos.

18edo contains sub-edos 2, 3, 6, and 9, and itself is half of 36edo and one-fourth of 72edo. It bears some similarities to 13edo (with its very flat 4ths and nice subminor 3rds), 11edo (with its very sharp minor 3rds, two of which span a very flat 5th), 16edo (with its sharp 4ths and flat 5ths), and 17edo and 19edo (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.

### Odd harmonics

Approximation of odd harmonics in 18edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +31.4 +13.7 +31.2 -3.9 -18.0 +26.1 -21.6 +28.4 -30.8 -4.1 -28.3
Relative (%) +47.1 +20.5 +46.8 -5.9 -27.0 +39.2 -32.4 +42.6 -46.3 -6.2 -42.4
Steps
(reduced)
29
(11)
42
(6)
51
(15)
57
(3)
62
(8)
67
(13)
70
(16)
74
(2)
76
(4)
79
(7)
81
(9)

## Intervals and 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 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
5L3s Notation
0 0 perfect unison P1 D perfect unison P1 D C
1 67 up unison, downminor 2nd ^1, vm2 ^D, vE up unison, downmajor 2nd ^1, vM2 ^D, vE Db
2 133 minor 2nd m2 E major 2nd M2 E C#
3 200 mid 2nd ~2 ^E mid 2nd ~2 ^E D
4 267 major 2nd, minor 3rd M2, m3 E#, Fb minor 2nd, major 3rd m2, M3 Eb, F# Eb
5 333 mid 3rd ~3 vF mid 3rd ~3 vF D#
6 400 major 3rd M3 F minor 3rd m3 F E
7 467 upmajor 3rd, down 4th ^M3, v4 ^F, vG upminor 3rd, down 4th ^m3, v4 ^F, vG F
8 533 perfect 4th P4 G perfect 4th P4 G Gb
9 600 up 4th, down 5th ^4, v5 ^G, vA up 4th, down 5th ^4, v5 ^G, vA F#
10 667 perfect 5th P5 A perfect 5th P5 A G
11 733 up 5th, downminor 6th ^5, vm6 ^A, vB up fifth, downmajor 6th ^5, vM6 ^A, vB Hb
12 800 minor 6th m6 B major 6th M6 B G#
13 867 mid 6th ~6 ^B mid 6th ~6 ^B H
14 933 major 6th, minor 7th M6, m7 B#, Cb minor 6th, major 7th m6, M7 Bb, C# A
15 1000 mid 7th ~7 vC mid 7th ~7 vC Bb
16 1067 major 7th M7 C minor 7th m7 C A#
17 1133 upmajor 7th, down 8ve ^M7, v8 ^C, vD upminor 7th, down 8ve ^m7, v8 ^C, vD B
18 1200 perfect 8ve P8 D perfect 8ve P8 D C

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, nonotonic 5th-generated, and heptatonic 3rd-generated.

Pentatonic 5th-generated: D * * * E * * G * * * A * * C * * * D (generator = wide 3/2 = 11\18 = perfect 5thoid)

D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D

P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

Nonatonic 5th-generated: A * B * C * D * E * F * G * H * J * A (every other note is a generator, all notes are perfect)

1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4- 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8 - ^8/v9 - 9 - ^9/v10 - 10

heptatonic 3rd-generated: D * * E * F * * G * A * * B * C * * D (generator = 5\18 = perfect 3rd)

D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D

P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb... ("Every good boy deserves fudge and candy")

genchain of thirds: ...A4 - A6 - A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 - d3 - d5...

## Representations of JI intervals

Degree Cents Nearest Ratio Error 17-Limit Ratios [1]
0 0.000 1/1 0 1/1
1 66.667 27/26 +1.329 78/75, 75/72
2 133.333 27/25 +0.096 51/55, 42/39
3 200.000 9/8 -3.910 9/8
4 266.667 7/6 -0.204 75/64
5 333.333 17/14 or 40/33 -2.796 +0.293 39/32
6 400.000 5/4 or 44/35 +13.686 +3.822 64/55
7 466.667 21/16 -4.114 21/16
8 533.333 15/11 -3.617 102/75
9 600.000 17/12 or 24/17 -3.000 +3.000 17/12
10 666.667 22/15 +3.617 75/51
11 733.333 32/21 +4.114 32/21
12 800.000 8/5 or 35/22 -13.686 -3.822 51/32
13 866.667 28/17 or 33/20 +2.796 -0.293 64/39
14 933.333 12/7 +0.204 55/32
15 1000.000 16/9 +3.910 16/9
16 1066.667 50/27 -0.096 39/21
17 1133.333 52/27 -1.329 75/39
18 1200.000 2/1 0 2/1**
1. based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament

## Regular temperament properties

### Uniform maps

13-limit uniform maps between 17.5 and 18.5
Min. size Max. size Wart notation Map
17.5000 17.6323 18bcdddeefff 18 28 41 49 61 65]
17.6323 17.7006 18bcdeefff 18 28 41 50 61 65]
17.7006 17.7775 18bcdeef 18 28 41 50 61 66]
17.7775 17.8731 18bcdf 18 28 41 50 62 66]
17.8731 17.9708 18bdf 18 28 42 50 62 66]
17.9708 17.9815 18bd 18 28 42 50 62 67]
17.9815 17.9885 18d 18 29 42 50 62 67]
17.9885 18.0666 18 18 29 42 51 62 67]
18.0666 18.2411 18e 18 29 42 51 63 67]
18.2411 18.3038 18eff 18 29 42 51 63 68]
18.3038 18.3447 18cceff 18 29 43 51 63 68]
18.3447 18.3556 18ccddeff 18 29 43 52 63 68]
18.3556 18.5000 18ccddeeeff 18 29 43 52 64 68]

### Commas

18edo tempers out the following commas. (Note: This assumes the val 18 29 42 51 62 67].)

Prime
Limit
Ratio[1] Monzo Cents Color name Name(s)
3 (18 digits) [29 -18 564.81 Wa-18 18-comma
5 128/125 [7 0 -3 41.06 Trigu Diesis, Augmented Comma
5 (20 digits) [23 6 -14 3.34 Sasa-sepbigu Vishnuzma, Semisuper
7 50/49 [1 0 2 -2 34.98 Biruyo Tritonic Diesis, Jubilisma
7 686/675 [1 -3 -2 3 27.99 Trizo-agugu Senga
7 875/864 [-5 -3 3 1 21.90 Zotriyo Keema
7 1728/1715 [6 3 -1 -3 13.07 Triru-agu Orwellisma, Orwell Comma
7 16875/16807 [0 3 4 -5 6.99 Quinru-aquadyo Mirkwai
7 3136/3125 [6 0 -5 2 6.08 Zozoquingu Hemimean
11 99/98 [-1 2 0 -2 1 17.58 Loruru Mothwellsma
11 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
11 65536/65219 [16 0 0 -2 -3 8.39 Satrilu-aruru Orgonisma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 9801/9800 [-3 4 -2 -2 2 0.18 Bilorugu Kalisma, Gauss' Comma
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap
1. Ratios longer than 10 digits are presented by placeholders with informative hints

## Scales

Note: This list excludes scales found in 9edo.

3L 2s: 4 4 3 4 3

### Hexatonic

4L 2s: 4 4 1 4 4 1

2L 4s: 2 5 2 2 5 2

### Heptatonic

4L 3s: 3 2 3 2 3 3 2

### Octatonic

5L 3s: 3 1 3 3 1 3 3 1

2L 6s: 2 2 3 2 2 2 3 2

### Enneatonic

3L 6s: 4 1 1 4 1 1 4 1 1

### Decatonic

8L 2s: 2 2 1 2 2 2 2 1 2 2

### Hendecatonic

7L 4s: 2 1 2 2 1 2 2 1 2 1 2

### Dodecatonic

3L 9s: 3 1 1 1 3 1 1 1 3 1 1 1

6L 6s: 2 1 2 1 2 1 2 1 2 1 2 1

Pathological 3L 12s: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1

## Application to guitar

18edo 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 8-note oneirotonic 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).

## Music

Ambient Esoterica
Beheld
Francium
Aaron Andrew Hunt
Noah Jordan
• The Moon (18edo album recorded on the 1/3 tone piano of Sonido 13 / Julian Carrillo)
Mandrake
Claudi Meneghin
Mundoworld
No Clue Music
norokusi
NullPointerException Music
Carlo Serafini
TomPrice719
Chris Vaisvil
Xeno*n*
David Zaydullin