18edo

Theory

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.

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.

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 Just Intervals

Degree	Cents	Nearest Ratio	Error	17-Limit Ratios
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**

Commas

18 EDO tempers out the following commas. (Note: This assumes the val < 18 29 42 51 62 67 |.)

Ratio	Monzo	Cents	Color Name	Name 1	Name 2
128/125	[7 0 -3⟩	41.06	Trigu	Diesis	Augmented Comma
???	[23 6 -14⟩	3.34	Sasa-sepbigu	Vishnuzma	Semisuper
50/49	[1 0 2 -2⟩	34.98	Biruyo	Tritonic Diesis	Jubilisma
686/675	[1 -3 -2 3⟩	27.99	Trizo-agugu	Senga
875/864	[-5 -3 3 1⟩	21.90	Zotriyo	Keema
1728/1715	[6 3 -1 -3⟩	13.07	Triru-agu	Orwellisma	Orwell Comma
16875/16807	[0 3 4 -5⟩	6.99	Quinru-aquadyo	Mirkwai
3136/3125	[6 0 -5 2⟩	6.08	Zozoquingu	Hemimean
99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemisma
65536/65219	[16 0 0 -2 -3⟩	8.39	Satrilu-aruru	Orgonisma
385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
9801/9800	[-3 4 -2 -2 2⟩	0.18	Bilorugu	Kalisma	Gauss' Comma
91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap

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).

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