18edo
| ← 17edo | 18edo | 19edo → |
Theory
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | ||
|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.0 | +31.4 | +13.7 | +31.2 | -18.0 | +26.1 |
| relative (%) | 0 | +47 | +21 | +47 | -27 | +39 | |
| nearest edomapping | 18 | 11 | 6 | 15 | 8 | 13 | |
| fifthspan | 0 | +1 | -6 | +3 | +4 | -7 | |
18 Equal Divisions of the Octave, also known as The Third-Tone System, 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, 7th and 9th harmonics equally well (or equally poorly) as 12-TET 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. 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. 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.
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¢, 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-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.
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.
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 | 5L3s Notation | |||||
|---|---|---|---|---|---|---|---|---|
| with major wider than minor | with major narrower than minor | |||||||
| 0 | 0 | perfect unison | P1
P0 |
D | perfect unison | P1
P0 |
D | C |
| 1 | 67 | up unison, downminor 2nd
up unison, downminor 1st |
^1, vm2
^0, vm1 |
^D, vE | up unison, downmajor 2nd
up unison, downmajor 1st |
^1, vM2
^0, vM1 |
^D, vE | Db |
| 2 | 133 | minor 2nd
minor 1st |
m2
m1 |
E | major 2nd
major 1st |
M2
M1 |
E | C# |
| 3 | 200 | mid 2nd
mid 1st |
~2
~1 |
^E | mid 2nd
mid 1st |
~2
~1 |
^E | D |
| 4 | 267 | major 2nd, minor 3rd
major 1st, minor 2nd |
M2, m3
M1, m2 |
E#, Fb | minor 2nd, major 3rd
minor 1st, major 2nd |
m2, M3
m1, M2 |
Eb, F# | Eb |
| 5 | 333 | mid 3rd
mid 2nd |
~3
~2 |
vF | mid 3rd
mid 2nd |
~3
~2 |
vF | D# |
| 6 | 400 | major 3rd
major 2nd |
M3
M2 |
F | minor 3rd
minor 2nd |
m3
m2 |
F | E |
| 7 | 467 | upmajor 3rd, down 4th
upmajor 2nd, down 3rd |
^M3, v4
^M2, v3 |
^F, vG | upminor 3rd, down 4th
upminor 2nd, down 3rd |
^m3, v4
^m2, v3 |
^F, vG | F |
| 8 | 533 | perfect 4th
perfect 3rd |
P4
P3 |
G | perfect 4th
perfect 3rd |
P4
P3 |
G | Gb |
| 9 | 600 | up 4th, down 5th
up 3rd, down 4th |
^4, v5
^3, v4 |
^G, vA | up 4th, down 5th
up 3rd, down 4th |
^4, v5
^3, v4 |
^G, vA | F# |
| 10 | 667 | perfect 5th
perfect 4th |
P5
P4 |
A | perfect 5th
perfect 4th |
P5
P4 |
A | G |
| 11 | 733 | up 5th, downminor 6th
up 4th, downminor 5th |
^5, vm6
^4, vm5 |
^A, vB | up 5th, downmajor 6th
up 4th, downmajor 5th |
^5, vM6
^4, vM5 |
^A, vB | Hb |
| 12 | 800 | minor 6th
minor 5th |
m6
m5 |
B | major 6th
major 5th |
M6
M5 |
B | G# |
| 13 | 867 | mid 6th
mid 5th |
~6
~5 |
^B | mid 6th
mid 5th |
~6
~5 |
^B | H |
| 14 | 933 | major 6th, minor 7th
major 5th, minor 6th |
M6, m7
M5, m6 |
B#, Cb | minor 6th, major 7th
minor 5th, major 6th |
m6, M7
m5, M6 |
Bb, C# | A |
| 15 | 1000 | mid 7th
mid 6th |
~7
~6 |
vC | mid 7th
mid 6th |
~7
~6 |
vC | Bb |
| 16 | 1067 | major 7th
major 6th |
M7
M6 |
C | minor 7th
minor 6th |
m7
m6 |
C | A# |
| 17 | 1133 | upmajor 7th, down 8ve
upmajor 6th, down 7th |
^M7, v8
^M6, v7 |
^C, vD | upminor 7th, down 8ve
upminor 6th, down 7th |
^m7, v8
^m6, v7 |
^C, vD | B |
| 18 | 1200 | perfect 8ve
perfect 7th |
P8
P7 |
D | perfect 8ve
perfect 7th |
P8
P7 |
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 Just 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** |
- ↑ based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament
Commas
18 EDO 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) |
|---|---|---|---|---|---|
| 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 |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Useful Moment-of-Symmetry Scales
Note: This list excludes scales found in 9-EDO.
Pentatonic
3L 2s Oneiro-Pentatonic (aka Father Pentatonic): 4 4 3 4 3
Hexatonic
4L 2s Doublewide[6] (aka Bicycle): 4 4 1 4 4 1
2L 4s Octokaidecal[8] (aka Rice Hexatonic): 2 5 2 2 5 2
Heptatonic
4L 3s Sixix[7] (aka Mish Heptatonic): 3 2 3 2 3 3 2
Octatonic
5L 3s Oneirotonic (aka Father Octatonic): 3 1 3 3 1 3 3 1
2L 6s Octokaidecal[8] (aka Rice Octatonic): 2 2 3 2 2 2 3 2
Enneatonic
3L 6s Augmented[9]: 4 1 1 4 1 1 4 1 1
Decatonic
8L 2s Octokaidecal[10] (aka Biggie Decatonic): 2 2 1 2 2 2 2 1 2 2
Dodecatonic
3L 9s Augmented[12]: 3 1 1 1 3 1 1 1 3 1 1 1
6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1
Pentadecatonic
Pathological 3L 12s Augmented[15]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 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 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).
Approaches to 18edo
Music
- Fuga a3 in 18ET by Aaron Andrew Hunt
- Prelude in 18et by Chris Vaisvil → composer notes
- Flippertronics by Chris Vaisvil
- Gerbils at the Wheel of Government by Chris Vaisvil (in 9 and 18 edo simultaneously)
- Do Androids Dream Of 18ED2? by Carlo Serafini (blog entry)
- Composition of June 2015 by TomPrice719