Xenharmonic Wiki

18edo: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m removed newline at article start
Francium (talk | contribs)
4,854 edits
 
(148 intermediate revisions by 29 users not shown)
Line 1: Line 1:
{{interwiki
{{interwiki
| de =  
| de = 18-EDO
| en = 18edo
| en = 18edo
| es =  
| es =  
| ja = 18平均律
| ja = 18平均律
}}
}}
'''18 Equal Divisions of the Octave''' also known as '''The Third-Tone System'''.
{{Infobox ET}}
{{ED intro}}


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


=Basic Properties=
== Theory ==
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).
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.


==Representations of Just Intervals==
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.


{| class="wikitable"
=== Odd harmonics ===
{{Harmonics in equal|18}}
 
== Intervals ==
 
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png|thumb]]
 
{| class="wikitable center-all right-2"
! Degree
! Cents
! Nearest Ratio
! Error
! 17-Limit Ratios <ref>based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament</ref>
|-
|-
| | Degree
| 0
| | Cents
| 0.000
| | 5L3s Notation
| 1/1
| | Nearest Ratio
| 0
| | Error
| 1/1
| | 17-Limit Ratios*
|-
|-
| | 0
| 1
| | 0
| 66.667
| style="text-align:center;" | C
| 27/26
| | 1/1
| +1.329
| | 0
| 26/25, 25/24
| | 1/1
|-
|-
| | 1
| 2
| | 66.67
| 133.333
| style="text-align:center;" | Db
| 27/25
| | 27/26
| +0.096
| | +1.329
| 55/51, 14/13
| | 78/75, 75/72
|-
|-
| | 2
| 3
| | 133.33
| 200.000
| style="text-align:center;" | C#
| 9/8
| | 27/25
| -3.910
| | +0.096
| 9/8
| | 51/55, 42/39
|-
|-
| | 3
| 4
| | 200
| 266.667
| style="text-align:center;" | D
| 7/6
| | 9/8
| -0.204
| | -3.910
| 75/64
| | 9/8
|-
|-
| | 4
| 5
| | 266.67
| 333.333
| style="text-align:center;" | Eb
| 17/14 or 40/33
| | 7/6
| -2.796 +0.293
| | -0.204
| 39/32
| | 75/64
|-
|-
| | 5
| 6
| | 333.33
| 400.000
| style="text-align:center;" | D#
| 5/4 or 44/35
| | 17/14 or 40/33
| +13.686 +3.822
| | -2.796 +0.293
| 64/55
| | 39/32
|-
|-
| | 6
| 7
| | 400
| 466.667
| style="text-align:center;" | E
| 21/16
| | 5/4 or 44/35
| -4.114
| | +13.686 +3.822
| 21/16
| | 64/55
|-
|-
| | 7
| 8
| | 466.67
| 533.333
| style="text-align:center;" | F
| 15/11
| | 21/16
| -3.617
| | -4.114
| 34/25
| | 21/16
|-
|-
| | 8
| 9
| | 533.33
| 600.000
| style="text-align:center;" | Gb
| 17/12 or 24/17
| | 15/11
| -3.000 +3.000
| | -3.617
| 17/12
| | 102/75
|-
|-
| | 9
| 10
| | 600
| 666.667
| style="text-align:center;" | F#
| 22/15
| | 17/12 or 24/17
| +3.617
| | -3.000 +3.000
| 25/17
| | 17/12
|-
|-
| | 10
| 11
| | 666.67
| 733.333
| style="text-align:center;" | G
| 32/21
| | 22/15
| +4.114
| | +3.617
| 32/21
| | 75/51
|-
|-
| | 11
| 12
| | 733.33
| 800.000
| style="text-align:center;" | Hb
| 8/5 or 35/22
| | 32/21
| -13.686 -3.822
| | +4.114
| 51/32
| | 32/21
|-
|-
| | 12
| 13
| | 800
| 866.667
| style="text-align:center;" | G#
| 28/17 or 33/20
| | 8/5 or 35/22
| +2.796 -0.293
| | -13.686 -3.822
| 64/39
| | 51/32
|-
|-
| | 13
| 14
| | 866.67
| 933.333
| style="text-align:center;" | H
| 12/7
| | 28/17 or 33/20
| +0.204
| | +2.796 -0.293
| 55/32
| | 64/39
|-
|-
| | 14
| 15
| | 933.33
| 1000.000
| style="text-align:center;" | A
| 16/9
| | 12/7
| +3.910
| | +0.204
| 16/9
| | 55/32
|-
|-
| | 15
| 16
| | 1000
| 1066.667
| style="text-align:center;" | Bb
| 50/27
| | 16/9
| -0.096
| | +3.910
| 13/7
| | 16/9
|-
|-
| | 16
| 17
| | 1066.67
| 1133.333
| style="text-align:center;" | A#
| 52/27
| | 50/27
| -1.329
| | -0.096
| 25/13
| | 39/21
|-
|-
| | 17
| 18
| | 1133.33
| 1200.000
| style="text-align:center;" | B
| 2/1
| | 52/27
| 0
| | -1.329
| 2/1**
| | 75/39
|-
| | 18
| | 1200
| style="text-align:center;" | 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
<references />
 
{{Clear}}
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]]
== Notation ==
 
=== Ups and downs notation ===
=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}}
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.
{| class="wikitable center-all right-2"
 
! Degree
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.
! Cents
 
! colspan="3" | [[Ups and downs notation|Up/down notation]] using the narrow 5th of 10\18, <br> with major wider than minor
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".
! colspan="3" | Up/down notation using the narrow 5th of 10\18, <br> with major narrower than minor
 
! 5L3s Notation
{| class="wikitable"
|-
|-
! | Degree
| 0
! | Cents
| 0
! colspan="3" | [[Ups_and_Downs_Notation|Up/down notation]] using the narrow 5th of 10\18,
| perfect unison
 
| P1
with major wider than minor
| D
! colspan="3" | Up/down notation using the narrow 5th of 10\18,
| perfect unison
 
| P1
with major narrower than minor
| D
| C
|-
|-
| style="text-align:center;" | 0
| 1
| style="text-align:center;" | 0
| 67
| style="text-align:center;" | perfect unison
| up unison, downminor 2nd
| style="text-align:center;" | P1
| ^1, vm2
| style="text-align:center;" | D
| ^D, vE
| style="text-align:center;" | perfect unison
| up unison, downmajor 2nd
| style="text-align:center;" | P1
| ^1, vM2
| style="text-align:center;" | D
| ^D, vE
| Db
|-
|-
| style="text-align:center;" | 1
| 2
| style="text-align:center;" | 67
| 133
| style="text-align:center;" | up unison, downminor 2nd
| minor 2nd
| style="text-align:center;" | ^1, vm2
| m2
| style="text-align:center;" | D^, Ev
| E
| style="text-align:center;" | up unison, downmajor 2nd
| major 2nd
| style="text-align:center;" | ^1, vM2
| M2
| style="text-align:center;" | D^, Ev
| E
| C#
|-
|-
| style="text-align:center;" | 2
| 3
| style="text-align:center;" | 133
| 200
| style="text-align:center;" | minor 2nd
| mid 2nd
| style="text-align:center;" | m2
| ~2
| style="text-align:center;" | E
| ^E
| style="text-align:center;" | major 2nd
| mid 2nd
| style="text-align:center;" | M2
| ~2
| style="text-align:center;" | E
| ^E
| D
|-
|-
| style="text-align:center;" | 3
| 4
| style="text-align:center;" | 200
| 267
| style="text-align:center;" | mid 2nd
| major 2nd, minor 3rd
| style="text-align:center;" | ~2
| M2, m3
| style="text-align:center;" | E^
| E#, Fb
| style="text-align:center;" | mid 2nd
| minor 2nd, major 3rd
| style="text-align:center;" | ~2
| m2, M3
| style="text-align:center;" | E^
| Eb, F#
| Eb
|-
|-
| style="text-align:center;" | 4
| 5
| style="text-align:center;" | 267
| 333
| style="text-align:center;" | major 2nd, minor 3rd
| mid 3rd
| style="text-align:center;" | M2, m3
| ~3
| style="text-align:center;" | E#, Fb
| vF
| style="text-align:center;" | minor 2nd, major 3rd
| mid 3rd
| style="text-align:center;" | m2, M3
| ~3
| style="text-align:center;" | Eb, F#
| vF
| D#
|-
|-
| style="text-align:center;" | 5
| 6
| style="text-align:center;" | 333
| 400
| style="text-align:center;" | mid 3rd
| major 3rd
| style="text-align:center;" | ~3
| M3
| style="text-align:center;" | Fv
| F
| style="text-align:center;" | mid 3rd
| minor 3rd
| style="text-align:center;" | ~3
| m3
| style="text-align:center;" | Fv
| F
| E
|-
|-
| style="text-align:center;" | 6
| 7
| style="text-align:center;" | 400
| 467
| style="text-align:center;" | major 3rd
| upmajor 3rd, down 4th
| style="text-align:center;" | M3
| ^M3, v4
| style="text-align:center;" | F
| ^F, vG
| style="text-align:center;" | minor 3rd
| upminor 3rd, down 4th
| style="text-align:center;" | m3
| ^m3, v4
| style="text-align:center;" | F
| ^F, vG
| F
|-
|-
| style="text-align:center;" | 7
| 8
| style="text-align:center;" | 467
| 533
| style="text-align:center;" | upmajor 3rd, down 4th
| perfect 4th
| style="text-align:center;" | ^M3, v4
| P4
| style="text-align:center;" | F^, Gv
| G
| style="text-align:center;" | upminor 3rd, down 4th
| perfect 4th
| style="text-align:center;" | ^m3, v4
| P4
| style="text-align:center;" | F^, Gv
| G
| Gb
|-
|-
| style="text-align:center;" | 8
| 9
| style="text-align:center;" | 533
| 600
| style="text-align:center;" | perfect 4th
| up 4th, down 5th
| style="text-align:center;" | P4
| ^4, v5
| style="text-align:center;" | G
| ^G, vA
| style="text-align:center;" | perfect 4th
| up 4th, down 5th
| style="text-align:center;" | P4
| ^4, v5
| style="text-align:center;" | G
| ^G, vA
| F#
|-
|-
| style="text-align:center;" | 9
| 10
| style="text-align:center;" | 600
| 667
| style="text-align:center;" | up 4th, down 5th
| perfect 5th
| style="text-align:center;" | ^4, v5
| P5
| style="text-align:center;" | G^, Av
| A
| style="text-align:center;" | up 4th, down 5th
| perfect 5th
| style="text-align:center;" | ^4, v5
| P5
| style="text-align:center;" | G^, Av
| A
| G
|-
|-
| style="text-align:center;" | 10
| 11
| style="text-align:center;" | 667
| 733
| style="text-align:center;" | perfect 5th
| up 5th, downminor 6th
| style="text-align:center;" | P5
| ^5, vm6
| style="text-align:center;" | A
| ^A, vB
| style="text-align:center;" | perfect 5th
| up fifth, downmajor 6th
| style="text-align:center;" | P5
| ^5, vM6
| style="text-align:center;" | A
| ^A, vB
| Hb
|-
|-
| style="text-align:center;" | 11
| 12
| style="text-align:center;" | 733
| 800
| style="text-align:center;" | up 5th, downminor 6th
| minor 6th
| style="text-align:center;" | ^5, vm6
| m6
| style="text-align:center;" | A^, Bv
| B
| style="text-align:center;" | up fifth, downmajor 6th
| major 6th
| style="text-align:center;" | ^5, vM6
| M6
| style="text-align:center;" | A^, Bv
| B
| G#
|-
|-
| style="text-align:center;" | 12
| 13
| style="text-align:center;" | 800
| 867
| style="text-align:center;" | minor 6th
| mid 6th
| style="text-align:center;" | m6
| ~6
| style="text-align:center;" | B
| ^B
| style="text-align:center;" | major 6th
| mid 6th
| style="text-align:center;" | M6
| ~6
| style="text-align:center;" | B
| ^B
| H
|-
|-
| style="text-align:center;" | 13
| 14
| style="text-align:center;" | 867
| 933
| style="text-align:center;" | mid 6th
| major 6th, minor 7th
| style="text-align:center;" | ~6
| M6, m7
| style="text-align:center;" | B^
| B#, Cb
| style="text-align:center;" | mid 6th
| minor 6th, major 7th
| style="text-align:center;" | ~6
| m6, M7
| style="text-align:center;" | B^
| Bb, C#
| A
|-
|-
| style="text-align:center;" | 14
| 15
| style="text-align:center;" | 933
| 1000
| style="text-align:center;" | major 6th, minor 7th
| mid 7th
| style="text-align:center;" | M6, m7
| ~7
| style="text-align:center;" | B#, Cb
| vC
| style="text-align:center;" | minor 6th, major 7th
| mid 7th
| style="text-align:center;" | m6, M7
| ~7
| style="text-align:center;" | Bb, C#
| vC
| Bb
|-
|-
| style="text-align:center;" | 15
| 16
| style="text-align:center;" | 1000
| 1067
| style="text-align:center;" | mid 7th
| major 7th
| style="text-align:center;" | ~7
| M7
| style="text-align:center;" | Cv
| C
| style="text-align:center;" | mid 7th
| minor 7th
| style="text-align:center;" | ~7
| m7
| style="text-align:center;" | Cv
| C
| A#
|-
|-
| style="text-align:center;" | 16
| 17
| style="text-align:center;" | 1067
| 1133
| style="text-align:center;" | major 7th
| upmajor 7th, down 8ve
| style="text-align:center;" | M7
| ^M7, v8
| style="text-align:center;" | C
| ^C, vD
| style="text-align:center;" | minor 7th
| upminor 7th, down 8ve
| style="text-align:center;" | m7
| ^m7, v8
| style="text-align:center;" | C
| ^C, vD
| B
|-
|-
| style="text-align:center;" | 17
| 18
| style="text-align:center;" | 1133
| 1200
| style="text-align:center;" | upmajor 7th, down 8ve
| perfect 8ve
| style="text-align:center;" | ^M7, v8
| P8
| style="text-align:center;" | C^, Dv
| D
| style="text-align:center;" | upminor 7th, down 8ve
| perfect 8ve
| style="text-align:center;" | ^m7, v8
| P8
| style="text-align:center;" | C^, Dv
| D
|-
| C
| 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
|}
|}


For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).
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)


==<span style="font-size: 1.3em;">Useful Moment-of-Symmetry Scales</span>==
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
Note: This list excludes scales found in 9-EDO.


===<span style="font-size: 1.1em;">Pentatonic:</span>===
'''<u>heptatonic 3rd-generated:</u> D * * E * F * * G * A * * B * C * * D''' (generator = 5\18 = perfect 3rd)
3L2s Father Pentatonic: 4 4 3 4 3


===<span style="font-size: 1.1em;">Hexatonic:</span>===
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
4L2s Bicycle: 4 4 1 4 4 1


2L4s Rice Hexatonic: 2 5 2 2 5 2
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


===<span style="font-size: 1.1em;">Heptatonic:</span>===
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")
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2


===<span style="font-size: 1.1em;">Octatonic:</span>===
genchain of thirds: ...A4 - A6 - A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 - d3 - d5...
5L3s Father Octatonic: 3 1 3 3 1 3 3 1


2L6s Rice Octatonic: 2 2 3 2 2 2 3 2
===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====


===<span style="font-size: 1.1em;">Decatonic:</span>===
{{Sagittal chart|Evo}}
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2


===<span style="font-size: 1.1em;">Dodecatonic:</span>===
====Revo flavor====
6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1


==<span style="font-size: 1.3em;">Application to Guitar</span>==
{{Sagittal chart}}
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).
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=18}}


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


{| class="wikitable"
{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name(s)
|-
|-
! | Comma
| 3
! | Monzo
| [[536870912/387420489|(18 digits)]]
! | Value (Cents)
| {{monzo| 29 -18 }}
! | Name 1
| 564.81
! | Name 2
| Wa-18
| 18-comma
|-
|-
| style="text-align:center;" | 128/125
| 5
| | | 7 0 -3 &gt;
| [[128/125]]
| style="text-align:right;" | 41.06
| {{monzo| 7 0 -3 }}
| style="text-align:center;" | Diesis
| 41.06
| style="text-align:center;" | Augmented Comma
| Trigu
| Augmented comma, diesis
|-
|-
| style="text-align:center;" |  
| 5
| | | 23 6 -14 &gt;
| [[6115295232/6103515625|(20 digits)]]
| style="text-align:right;" | 3.34
| {{monzo| 23 6 -14 }}
| style="text-align:center;" | Vishnuzma
| 3.34
| style="text-align:center;" | Semisuper
| Sasa-sepbigu
| [[Vishnuzma]], Semisuper comma
|-
|-
| style="text-align:center;" | 50/49
| 7
| | | 1 0 2 -2 &gt;
| [[50/49]]
| style="text-align:right;" | 34.98
| {{monzo| 1 0 2 -2 }}
| style="text-align:center;" | Tritonic Diesis
| 34.98
| style="text-align:center;" | Jubilisma
| Biruyo
| Jubilisma, tritonic diesis
|-
|-
| style="text-align:center;" | 686/675
| 7
| | | 1 -3 -2 3 &gt;
| [[686/675]]
| style="text-align:right;" | 27.99
| {{monzo| 1 -3 -2 3 }}
| style="text-align:center;" | Senga
| 27.99
| style="text-align:center;" |
| Trizo-agugu
| Senga
|-
|-
| style="text-align:center;" | 875/864
| 7
| | | -5 -3 3 1 &gt;
| [[875/864]]
| style="text-align:right;" | 21.90
| {{monzo| -5 -3 3 1 }}
| style="text-align:center;" | Keema
| 21.90
| style="text-align:center;" |
| Zotriyo
| Keema
|-
|-
| style="text-align:center;" | 1728/1715
| 7
| | | 6 3 -1 -3 &gt;
| [[1728/1715]]
| style="text-align:right;" | 13.07
| {{monzo| 6 3 -1 -3 }}
| style="text-align:center;" | Orwellisma
| 13.07
| style="text-align:center;" | Orwell Comma
| Triru-agu
| Orwellisma
|-
|-
| style="text-align:center;" | 16875/16807
| 7
| | | 0 3 4 -5 &gt;
| [[16875/16807]]
| style="text-align:right;" | 6.99
| {{monzo| 0 3 4 -5 }}
| style="text-align:center;" | Mirkwai
| 6.99
| style="text-align:center;" |
| Quinru-aquadyo
| Mirkwai comma
|-
|-
| style="text-align:center;" | 3136/3125
| 7
| | | 6 0 -5 2 &gt;
| [[3136/3125]]
| style="text-align:right;" | 6.08
| {{monzo| 6 0 -5 2 }}
| style="text-align:center;" | Hemimean
| 6.08
| style="text-align:center;" |
| Zozoquingu
| Hemimean comma
|-
|-
| style="text-align:center;" | 99/98
| 11
| | | -1 2 0 -2 1 &gt;
| [[99/98]]
| style="text-align:right;" | 17.58
| {{monzo| -1 2 0 -2 1 }}
| style="text-align:center;" | Mothwellsma
| 17.58
| style="text-align:center;" |
| Loruru
| Mothwellsma
|-
|-
| style="text-align:center;" | 100/99
| 11
| | | 2 -2 2 0 -1 &gt;
| [[100/99]]
| style="text-align:right;" | 17.40
| {{monzo| 2 -2 2 0 -1 }}
| style="text-align:center;" | Ptolemisma
| 17.40
| style="text-align:center;" |
| Luyoyo
| Ptolemisma
|-
|-
| style="text-align:center;" | 65536/65219
| 11
| | | 16 0 0 -2 -3 &gt;
| [[65536/65219]]
| style="text-align:right;" | 8.39
| {{monzo| 16 0 0 -2 -3 }}
| style="text-align:center;" | Orgonisma
| 8.39
| style="text-align:center;" |
| Satrilu-aruru
| Orgonisma
|-
|-
| style="text-align:center;" | 385/384
| 11
| | | -7 -1 1 1 1 &gt;
| [[385/384]]
| style="text-align:right;" | 4.50
| {{monzo| -7 -1 1 1 1 }}
| style="text-align:center;" | Keenanisma
| 4.50
| style="text-align:center;" |
| Lozoyo
| Keenanisma
|-
|-
| style="text-align:center;" | 9801/9800
| 11
| | | -3 4 -2 -2 2 &gt;
| [[9801/9800]]
| style="text-align:right;" | 0.18
| {{monzo| -3 4 -2 -2 2 }}
| style="text-align:center;" | Kalisma
| 0.18
| style="text-align:center;" | Gauss' Comma
| Bilorugu
| Kalisma
|-
|-
| style="text-align:center;" | 91/90
| 13
| | | -1 -2 -1 1 1 &gt;
| [[91/90]]
| style="text-align:right;" | 19.13
| {{monzo| -1 -2 -1 1 0 1 }}
| style="text-align:center;" | Superleap
| 19.13
| style="text-align:center;" |
| Thozogu
| Superleap comma, biome comma
|}
|}
<references/>
== 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]].
== Scales ==
Note: This list excludes scales found in [[9edo]].
=== Pentatonic ===
[[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
[[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!
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).
=== Keyboards ===
[[Julián Carrillo]] built at least one third-tone piano in 18edo.
[[Lumatone mapping for 18edo|Lumatone mappings for 18edo]] are available.
== 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)


=Music=
== See also ==
<ul><li>[http://www.h-pi.com/mp3/18ETPrelude.mp3 18ETPrelude] by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]</li><li>[http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 Prelude in 18et] by [http://www.chrisvaisvil.com Chris Vaisvil] =&gt; [http://chrisvaisvil.com/?p=3 composer notes]</li><li>[http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3 Flippertronics] by Chris Vaisvil</li><li>[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)]</li><li>[http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 Do Androids Dream Of 18ED2?] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog entry])</li><li>[https://soundcloud.com/tomprice719/composition-of-june-2015 Composition of June 2015 by TomPrice719]</li></ul>
* [[Fendo family]] - temperaments closely related to 18edo


[[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"