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{{interwiki
{{interwiki
| de =  
| de = 18-EDO
| en = 18edo
| en = 18edo
| es =  
| es =  
| ja = 18平均律
| ja = 18平均律
}}
}}
{{Infobox ET}}
{{ED intro}}


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


== Theory ==
== 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 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).
18edo does not include the 3rd or 7th harmonics, and contains the same controversial tuning of [[5/4]] 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. 18-EDO may be treated as a temperament of the 17-limit [[k*N_subgroups|4*18 subgroup]] [[Just_intonation_subgroups|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|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.
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 [[k*N_subgroups|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 interval 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 2.9.5.11 subgroup temperament.


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!
However, less accurate approximations can be used, and 18edo can be treated as a 7-limit (with 3s) exotemperament with the mapping {{val| 18 29 42 51 }}. This maps 3/2 to 733.33{{c}}, 5/4 to 400{{c}} and 7/4 to 1000{{c}}; as a result, 28/27 is tempered out, and unintuitive things happen: 9/8 and 7/6 are both mapped to 266.67{{c}}, while 8/7 gets mapped below both of them to 200{{c}}, making for a rather disordered [[9-odd-limit]] [[tonality diamond]], although this may be serviceable for the more exotemperamental music. This 7-limit mapping [[support]]s 7-limit [[sixix]], and thus is strongly associated with 18edo's [[4L 3s]] [[mos]].


18-EDO contains sub-EDOs [[2edo|2]], [[3edo|3]], [[6edo|6]], and [[9edo|9]], and itself is half of [[36edo|36-EDO]] and one-fourth of [[72edo|72-EDO]]. It bears some similarities to [[13edo|13-EDO]] (with its very flat 4ths and nice subminor 3rds), [[11edo|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.
18edo contains sub-edos [[2edo|2]], [[3edo|3]], [[6edo|6]], and [[9edo|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.


== Intervals and Notation ==
18edo is the basic example of a dual-fifth system (beyond perhaps 11 or 13edo), as the sharp and flat fifths multiply to a good approximation of 9/4. By alternating these fifths, a diatonic scale (5L 1m 1s) is generated which is similar to 19edo's diatonic, but cut short by one step.


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.
=== Odd harmonics ===
{{Harmonics in equal|18}}


The first way preserves the <u>melodic</u> 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.
== Intervals ==


The second way preserves the <u>harmonic</u> 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".
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png|thumb]]


{| class="wikitable"
{| class="wikitable center-all right-2"
|-
! | Degree
! | Cents
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]] using the narrow 5th of 10\18,
 
with major wider than minor
! colspan="3" | Up/down notation using the narrow 5th of 10\18,
 
with major narrower than minor
!5L3s Notation
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0
| style="text-align:center;" | perfect unison
| style="text-align:center;" | P1
| style="text-align:center;" | D
| style="text-align:center;" | perfect unison
| style="text-align:center;" | P1
| style="text-align:center;" | D
|C
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 67
| style="text-align:center;" | up unison, downminor 2nd
| style="text-align:center;" | ^1, vm2
| style="text-align:center;" | ^D, vE
| style="text-align:center;" | up unison, downmajor 2nd
| style="text-align:center;" | ^1, vM2
| style="text-align:center;" | ^D, vE
|Db
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 133
| style="text-align:center;" | minor 2nd
| style="text-align:center;" | m2
| style="text-align:center;" | E
| style="text-align:center;" | major 2nd
| style="text-align:center;" | M2
| style="text-align:center;" | E
|C#
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 200
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^E
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^E
|D
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 267
| style="text-align:center;" | major 2nd, minor 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E#, Fb
| style="text-align:center;" | minor 2nd, major 3rd
| style="text-align:center;" | m2, M3
| style="text-align:center;" | Eb, F#
|Eb
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 333
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | vF
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | vF
|D#
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 400
| style="text-align:center;" | major 3rd
| style="text-align:center;" | M3
| style="text-align:center;" | F
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | m3
| style="text-align:center;" | F
|E
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 467
| style="text-align:center;" | upmajor 3rd, down 4th
| style="text-align:center;" | ^M3, v4
| style="text-align:center;" | ^F, vG
| style="text-align:center;" | upminor 3rd, down 4th
| style="text-align:center;" | ^m3, v4
| style="text-align:center;" | ^F, vG
|F
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 533
| style="text-align:center;" | perfect 4th
| style="text-align:center;" | P4
| style="text-align:center;" | G
| style="text-align:center;" | perfect 4th
| style="text-align:center;" | P4
| style="text-align:center;" | G
|Gb
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 600
| style="text-align:center;" | up 4th, down 5th
| style="text-align:center;" | ^4, v5
| style="text-align:center;" | ^G, vA
| style="text-align:center;" | up 4th, down 5th
| style="text-align:center;" | ^4, v5
| style="text-align:center;" | ^G, vA
|F#
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 667
| style="text-align:center;" | perfect 5th
| style="text-align:center;" | P5
| style="text-align:center;" | A
| style="text-align:center;" | perfect 5th
| style="text-align:center;" | P5
| style="text-align:center;" | A
|G
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 733
| style="text-align:center;" | up 5th, downminor 6th
| style="text-align:center;" | ^5, vm6
| style="text-align:center;" | ^A, vB
| style="text-align:center;" | up fifth, downmajor 6th
| style="text-align:center;" | ^5, vM6
| style="text-align:center;" | ^A, vB
|Hb
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 800
| style="text-align:center;" | minor 6th
| style="text-align:center;" | m6
| style="text-align:center;" | B
| style="text-align:center;" | major 6th
| style="text-align:center;" | M6
| style="text-align:center;" | B
|G#
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 867
| style="text-align:center;" | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^B
| style="text-align:center;" | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^B
|H
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 933
| style="text-align:center;" | major 6th, minor 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B#, Cb
| style="text-align:center;" | minor 6th, major 7th
| style="text-align:center;" | m6, M7
| style="text-align:center;" | Bb, C#
|A
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 1000
| style="text-align:center;" | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | vC
| style="text-align:center;" | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | vC
|Bb
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 1067
| style="text-align:center;" | major 7th
| style="text-align:center;" | M7
| style="text-align:center;" | C
| style="text-align:center;" | minor 7th
| style="text-align:center;" | m7
| style="text-align:center;" | C
|A#
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 1133
| style="text-align:center;" | upmajor 7th, down 8ve
| style="text-align:center;" | ^M7, v8
| style="text-align:center;" | ^C, vD
| style="text-align:center;" | upminor 7th, down 8ve
| style="text-align:center;" | ^m7, v8
| style="text-align:center;" | ^C, vD
|B
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 1200
| style="text-align:center;" | perfect 8ve
| style="text-align:center;" | P8
| style="text-align:center;" | D
| style="text-align:center;" | perfect 8ve
| style="text-align:center;" | P8
| style="text-align:center;" | 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.
 
'''<u>Pentatonic 5th-generated:</u> 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)
 
'''<u>Nonatonic 5th-generated:</u> 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
 
'''<u>heptatonic 3rd-generated:</u>  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 ==
 
{| class="wikitable"
! Degree
! Degree
! Cents
! Cents
Line 264: Line 36:
|-
|-
| 0
| 0
| 0.00
| 0.000
| 1/1
| 1/1
| 0
| 0
Line 270: Line 42:
|-
|-
| 1
| 1
| 66.67
| 66.667
| 27/26
| 27/26
| +1.329
| +1.329
| 78/75, 75/72
| 26/25, 25/24
|-
|-
| 2
| 2
| 133.33
| 133.333
| 27/25
| 27/25
| +0.096
| +0.096
| 51/55, 42/39
| 55/51, 14/13
|-
|-
| 3
| 3
| 200
| 200.000
| 9/8
| 9/8
| -3.910
| -3.910
Line 288: Line 60:
|-
|-
| 4
| 4
| 266.67
| 266.667
| 7/6
| 7/6
| -0.204
| -0.204
Line 294: Line 66:
|-
|-
| 5
| 5
| 333.33
| 333.333
| 17/14 or 40/33
| 17/14 or 40/33
| -2.796 +0.293
| -2.796 +0.293
Line 300: Line 72:
|-
|-
| 6
| 6
| 400
| 400.000
| 5/4 or 44/35
| 5/4 or 44/35
| +13.686 +3.822
| +13.686 +3.822
Line 306: Line 78:
|-
|-
| 7
| 7
| 466.67
| 466.667
| 21/16
| 21/16
| -4.114
| -4.114
Line 312: Line 84:
|-
|-
| 8
| 8
| 533.33
| 533.333
| 15/11
| 15/11
| -3.617
| -3.617
| 102/75
| 34/25
|-
|-
| 9
| 9
| 600
| 600.000
| 17/12 or 24/17
| 17/12 or 24/17
| -3.000 +3.000
| -3.000 +3.000
Line 324: Line 96:
|-
|-
| 10
| 10
| 666.67
| 666.667
| 22/15
| 22/15
| +3.617
| +3.617
| 75/51
| 25/17
|-
|-
| 11
| 11
| 733.33
| 733.333
| 32/21
| 32/21
| +4.114
| +4.114
Line 336: Line 108:
|-
|-
| 12
| 12
| 800
| 800.000
| 8/5 or 35/22
| 8/5 or 35/22
| -13.686 -3.822
| -13.686 -3.822
Line 342: Line 114:
|-
|-
| 13
| 13
| 866.67
| 866.667
| 28/17 or 33/20
| 28/17 or 33/20
| +2.796 -0.293
| +2.796 -0.293
Line 348: Line 120:
|-
|-
| 14
| 14
| 933.33
| 933.333
| 12/7
| 12/7
| +0.204
| +0.204
Line 354: Line 126:
|-
|-
| 15
| 15
| 1000
| 1000.000
| 16/9
| 16/9
| +3.910
| +3.910
Line 360: Line 132:
|-
|-
| 16
| 16
| 1066.67
| 1066.667
| 50/27
| 50/27
| -0.096
| -0.096
| 39/21
| 13/7
|-
|-
| 17
| 17
| 1133.33
| 1133.333
| 52/27
| 52/27
| -1.329
| -1.329
| 75/39
| 25/13
|-
|-
| 18
| 18
| 1200
| 1200.000
| 2/1
| 2/1
| 0
| 0
| 2/1**
| 2/1**
|}
|}
<references/>
<references />
{{Clear}}
== Notation ==
=== Ups and downs 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{{c}} worse that the best approximation, which becomes the up-fifth.
{{Mavila}}
{| class="wikitable center-all right-2"
! Degree
! Cents
! colspan="3" | [[Ups and downs notation|Up/down notation]] using the narrow 5th of 10\18, <br> with major wider than minor
! colspan="3" | Up/down notation using the narrow 5th of 10\18, <br> 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.
 
'''<u>Pentatonic 5th-generated:</u> 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)
 
'''<u>Nonatonic 5th-generated:</u> 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
 
'''<u>heptatonic 3rd-generated:</u>  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...
 
===Sagittal notation===
This notation is a subset of the notations for EDOs [[36edo#Sagittal notation|36]] and [[72edo#Sagittal notation|72]] and a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====
 
{{Sagittal chart|Evo}}
 
====Revo flavor====


[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]]
{{Sagittal chart}}


== Commas ==
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=18}}


18 EDO [[tempering out|tempers out]] the following [[Comma|commas]]. (Note: This assumes the [[val]] &lt; 18 29 42 51 62 67 |.)
=== Commas ===
18et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 18 29 42 51 62 67 }}.)


{| class="wikitable" style="text-align:center;"
{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Ratio]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
! [[Color notation/Temperament Names|Color Name]]
! [[Color name]]
! Name 1
! Name(s)
! Name 2
|-
|-
| 128/125
| 3
| style="text-align:left;" | <nowiki> | 7 0 -3 </nowiki>&gt;
| [[536870912/387420489|(18 digits)]]
| style="text-align:right;" | 41.06
| {{monzo| 29 -18 }}
| 564.81
| Wa-18
| 18-comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Trigu
| Diesis
| Augmented comma, diesis
| Augmented Comma
|-
|-
| style="text-align:center;" | ???
| 5
| style="text-align:left;" | <nowiki> | 23 6 -14 </nowiki>&gt;
| [[6115295232/6103515625|(20 digits)]]
| style="text-align:right;" | 3.34
| {{monzo| 23 6 -14 }}
| 3.34
| Sasa-sepbigu
| Sasa-sepbigu
| Vishnuzma
| [[Vishnuzma]], Semisuper comma
| Semisuper
|-
|-
| 50/49
| 7
| style="text-align:left;" | <nowiki> | 1 0 2 -2 </nowiki>&gt;
| [[50/49]]
| style="text-align:right;" | 34.98
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Biruyo
| Tritonic Diesis
| Jubilisma, tritonic diesis
| Jubilisma
|-
|-
| 686/675
| 7
| style="text-align:left;" | <nowiki> | 1 -3 -2 3 </nowiki>&gt;
| [[686/675]]
| style="text-align:right;" | 27.99
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Trizo-agugu
| Senga
| Senga
|
|-
|-
| 875/864
| 7
| style="text-align:left;" | {{Monzo| -5 -3 3 1 }}
| [[875/864]]
| style="text-align:right;" | 21.90
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Zotriyo
| Keema
| Keema
|
|-
|-
| 1728/1715
| 7
| style="text-align:left;" | {{Monzo| 6 3 -1 -3 }}
| [[1728/1715]]
| style="text-align:right;" | 13.07
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma
| Orwellisma
| Orwell Comma
|-
|-
| 16875/16807
| 7
| style="text-align:left;" | {{Monzo| 0 3 4 -5 }}
| [[16875/16807]]
| style="text-align:right;" | 6.99
| {{monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|
|-
|-
| 3136/3125
| 7
| style="text-align:left;" | {{Monzo| 6 0 -5 2 }}
| [[3136/3125]]
| style="text-align:right;" | 6.08
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Zozoquingu
| Hemimean
| Hemimean comma
|
|-
|-
| 99/98
| 11
| style="text-align:left;" | {{Monzo| -1 2 0 -2 1 }}
| [[99/98]]
| style="text-align:right;" | 17.58
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Loruru
| Mothwellsma
| Mothwellsma
|
|-
|-
| 100/99
| 11
| style="text-align:left;" | {{Monzo| 2 -2 2 0 -1 }}
| [[100/99]]
| style="text-align:right;" | 17.40
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Luyoyo
| Ptolemisma
| Ptolemisma
|
|-
|-
| 65536/65219
| 11
| style="text-align:left;" | {{Monzo| 16 0 0 -2 -3 }}
| [[65536/65219]]
| style="text-align:right;" | 8.39
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Satrilu-aruru
| Orgonisma
| Orgonisma
|
|-
|-
| 385/384
| 11
| style="text-align:left;" | {{Monzo| -7 -1 1 1 1 }}
| [[385/384]]
| style="text-align:right;" | 4.50
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|
|-
|-
| 9801/9800
| 11
| style="text-align:left;" | {{Monzo| -3 4 -2 -2 2 }}
| [[9801/9800]]
| style="text-align:right;" | 0.18
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Bilorugu
| Kalisma
| Kalisma
| Gauss' Comma
|-
|-
| 91/90
| 13
| style="text-align:left;" | {{Monzo| -1 -2 -1 1 0 1 }}
| [[91/90]]
| style="text-align:right;" | 19.13
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
| 
|}
|}
<references/>


== Useful Moment-of-Symmetry Scales ==
== Octave stretch or compression ==
18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]]. Suitable shrunk versions of 18edo include [[zpi|61zpi]], [[ed12|65ed12]] and [[ed6|47ed6]].


Note: This list excludes scales found in [[9-EDO]].
== Scales ==
Note: This list excludes scales found in [[9edo]].


=== Pentatonic ===
=== Pentatonic ===


3L2s Father Pentatonic: 4 4 3 4 3
[[3L 2s]]: 4 4 3 4 3


=== Hexatonic ===
=== Hexatonic ===


4L2s Bicycle: 4 4 1 4 4 1
[[4L 2s]]: 4 4 1 4 4 1


2L4s Rice Hexatonic: 2 5 2 2 5 2
[[2L 4s]]: 2 5 2 2 5 2


=== Heptatonic ===
=== Heptatonic ===


4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2
[[4L 3s]]: 3 2 3 2 3 3 2


=== Octatonic ===
=== Octatonic ===


5L3s Father Octatonic: 3 1 3 3 1 3 3 1
[[5L 3s]]: 3 1 3 3 1 3 3 1
 
[[2L 6s]]: 2 2 3 2 2 2 3 2
 
=== Enneatonic ===


2L6s Rice Octatonic: 2 2 3 2 2 2 3 2
[[3L 6s]]: 4 1 1 4 1 1 4 1 1


=== Decatonic ===
=== Decatonic ===


8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
[[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 ===
=== Dodecatonic ===


6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1
[[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
 
[[Werntz Nocturne scale]]: 2 1 1 2 2 1 1 2 2 1 1 2
 
=== Tridecatonic ===
 
[[5L 8s]]: 2 1 2 1 1 2 1 2 1 1 2 1 1
 
=== Pentadecatonic ===
 
[[3L 12s]]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1
 
== Instruments ==
=== 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!


== Application to Guitar ==
The 8-note oneirotonic scale maps very simply to a 6-string guitar tuned in "reverse-standard" tuning (tune using four 466.667{{c}} intervals, with one 533.333{{c}} 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).


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!
=== Keyboards ===
[[Julián Carrillo]] built at least one third-tone piano in 18edo.


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).
[[Lumatone mapping for 18edo|Lumatone mappings for 18edo]] are available.


== Music ==
== Music ==
=== Modern renderings ===
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=mAcBBL2lkHo ''Bluin' The Black Keys''] (1926) – rendered by Francium (2025)
=== 20th century ===
; [[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=gbPPYOygNJc ''Prélude et Etude'', Op. 48], for third-tone piano of [[Julián Carrillo]] (1966)
=== 21st century ===
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=Cp_lTUNmtd8 ''XVIII-TET Tribute to Full Moon in Virgo''] (2024)
; [[Beheld]]
* [https://www.youtube.com/watch?v=Nog2LROg8Ss ''Overstrung vibe''] (2022)
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/-oi5eJA65Zc ''Waltz in 18edo''] (2025)
* [https://www.youtube.com/watch?v=r3FypUx_iIk ''Lament in 18edo''] (2025)
* [https://www.youtube.com/shorts/hNmse4IUWL0 ''18edo improv''] (2025)
; [[Francium]]
* "excucumber", from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/2uSQv7MbMOKMLue2FMVU9y Spotify] | [https://francium223.bandcamp.com/track/excucumber Bandcamp] | [https://www.youtube.com/watch?v=dcOsIrQEsg4 YouTube]
* [https://www.youtube.com/watch?v=KgIWvxMKrlo ''Stummy Beige''] (2026)
; [[groundfault]]
* "Life and Limb", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/life-and-limb-18edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1751 YouTube (29:11–33:47)]
; [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et ''Fuga a3 in 18ET'']{{dead link}}
; [[Noah Jordan]]
* ''The Moon'' (2016) – [https://noahdeanjordan.bandcamp.com/album/the-moon BandCamp] | [https://www.youtube.com/watch?v=TunyA3gwEJw YouTube] – 7-piece album recorded on the 1/3-tone piano of Sonido 13 / Julian Carrillo
* ''There and Back Again'' (2025) – [https://noahdeanjordan.bandcamp.com/album/there-and-back-again Bandcamp] | [https://www.youtube.com/watch?v=O36ZQyq6oR8 YouTube] – 3-piece album recorded on the 1/3-tone piano of Sonido 13 / Julian Carrillo
; [[Mandrake]]
* [https://www.youtube.com/watch?v=R1uz0ok4-Zs ''Such And Flowers''] (2022)
* [https://www.youtube.com/watch?v=2AquW_cqUQc ''That Kinda Lo-Fi Feel''] (2022)
; [[Leo Matarazzo]]
* [https://www.youtube.com/watch?v=G2vrqyE0ZX4 ''Eris''] (2026)
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=vUTHZNzBwUo ''Air Triste''] (2018)
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/revealing-the-path ''Revealing the Path''] (2018)
; [[Mundoworld]]
* [https://www.youtube.com/watch?v=iIaROmh7wD0 ''I Am the Monster I Fear''] (2023)
; [[No Clue Music]]
* [https://www.youtube.com/watch?v=UHFU9-eBXBo ''WORLD PORTAL''] (2024)
; [[norokusi]]
* [https://www.youtube.com/watch?v=pEvT2oyWEuo ''3 Bagatelles''] (2021)
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=hNgI6oRYNHA ''Three Worlds Order''] (2020)
* [https://www.youtube.com/watch?v=FnYxYuukgrM "Confusion"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
* [https://www.youtube.com/watch?v=1k4rNjyaZsE ''Purgatory''] (2021)
* [https://www.youtube.com/watch?v=a7AtOuX1NAE ''The Hydrogen Atom''] (2023)
; [[Carlo Serafini]]
* ''Do Androids Dream Of 18ED2?'' (2015) – [http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog] | [http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 play]{{dead link}}
; [[TomPrice719]]
* [https://soundcloud.com/tomprice719/composition-of-june-2015 ''Composition of June 2015''] (2015)
; [[Chris Vaisvil]]
* ''Prelude in 18et'' (2009) – [https://www.chrisvaisvil.com/prelude-in-18et/ blog] | [http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 play]
* [http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3 ''Flippertronics'']
* [http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3 ''Gerbils at the Wheel of Government''] (in 9 and 18 edo simultaneously)
; [[Julia Werntz]], [[Eric Moe]] & the [[Pandelis Karayorgis Trio]]
* [https://driffrecords.bandcamp.com/album/climbing-to-sleep ''Climbing to Sleep''] (2025) – jazz album
; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=fj_AISfnFnY ''Deranged Anger''] (2021)
; [[David Zaydullin]]
* [https://www.youtube.com/watch?v=QzKNP-NwHu0 ''Phaserun''] (2024)


* [http://www.h-pi.com/mp3/18ETPrelude.mp3 18ETPrelude] by [[Aaron Andrew Hunt]] {{dead link}}
== See also ==
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et Fuga a3 in 18ET] by Aaron Andrew Hunt
* [[Fendo family]] - temperaments closely related to 18edo
* [http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 Prelude in 18et] by [http://www.chrisvaisvil.com Chris Vaisvil] &rarr; [http://chrisvaisvil.com/?p=3 composer notes]
* [http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3 Flippertronics] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3 Gerbils at the Wheel of Government] by [http://chrisvaisvil.com/?p=1402 Chris Vaisvil (in 9 and 18 edo simultaneously)]
* [http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 Do Androids Dream Of 18ED2?] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog entry])
* [https://soundcloud.com/tomprice719/composition-of-june-2015 Composition of June 2015 by TomPrice719]


[[Category:18-tone]]
[[Category:18-tone scales]]
[[Category:18edo]]
[[Category:Listen]]
[[Category:edo]]
[[Category:Teentuning]]
[[Category:listen]]
[[Category:Oneirotonic]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:teentuning]]
[[Category:theory]]
[[Category:todo:unify_precision]]
Retrieved from "https://en.xen.wiki/w/18edo"