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__FORCETOC__
__FORCETOC__


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


Line 18: Line 18:
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.
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.


==Representations of Just Intervals==
==Notation==
 
{| class="wikitable"
|-
| | Degree
| | Cents
| | 5L3s Notation
| | Nearest Ratio
| | Error
| | 17-Limit Ratios*
|-
| 0
|0.00
| style="text-align:center;" | C
| | 1/1
| | 0
| | 1/1
|-
| | 1
| | 66.67
| style="text-align:center;" | Db
| | 27/26
| | +1.329
| | 78/75, 75/72
|-
| | 2
| | 133.33
| style="text-align:center;" | C#
| | 27/25
| | +0.096
| | 51/55, 42/39
|-
| | 3
| | 200
| style="text-align:center;" | D
| | 9/8
| | -3.910
| | 9/8
|-
| | 4
| | 266.67
| style="text-align:center;" | Eb
| | 7/6
| | -0.204
| | 75/64
|-
| | 5
| | 333.33
| style="text-align:center;" | D#
| | 17/14 or 40/33
| | -2.796 +0.293
| | 39/32
|-
| | 6
| | 400
| style="text-align:center;" | E
| | 5/4 or 44/35
| | +13.686 +3.822
| | 64/55
|-
| | 7
| | 466.67
| style="text-align:center;" | F
| | 21/16
| | -4.114
| | 21/16
|-
| | 8
| | 533.33
| style="text-align:center;" | Gb
| | 15/11
| | -3.617
| | 102/75
|-
| | 9
| | 600
| style="text-align:center;" | F#
| | 17/12 or 24/17
| | -3.000 +3.000
| | 17/12
|-
| | 10
| | 666.67
| style="text-align:center;" | G
| | 22/15
| | +3.617
| | 75/51
|-
| | 11
| | 733.33
| style="text-align:center;" | Hb
| | 32/21
| | +4.114
| | 32/21
|-
| | 12
| | 800
| style="text-align:center;" | G#
| | 8/5 or 35/22
| | -13.686 -3.822
| | 51/32
|-
| | 13
| | 866.67
| style="text-align:center;" | H
| | 28/17 or 33/20
| | +2.796 -0.293
| | 64/39
|-
| | 14
| | 933.33
| style="text-align:center;" | A
| | 12/7
| | +0.204
| | 55/32
|-
| | 15
| | 1000
| style="text-align:center;" | Bb
| | 16/9
| | +3.910
| | 16/9
|-
| | 16
| | 1066.67
| style="text-align:center;" | A#
| | 50/27
| | -0.096
| | 39/21
|-
| | 17
| | 1133.33
| style="text-align:center;" | B
| | 52/27
| | -1.329
| | 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
 
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]]
 
=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.
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.
Line 184: Line 36:


with major narrower than minor
with major narrower than minor
!5L3s Notation
|-
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0
Line 193: Line 46:
| style="text-align:center;" | P1
| style="text-align:center;" | P1
| style="text-align:center;" | D
| style="text-align:center;" | D
|C
|-
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 1
Line 198: Line 52:
| style="text-align:center;" | up unison, downminor 2nd
| style="text-align:center;" | up unison, downminor 2nd
| style="text-align:center;" | ^1, vm2
| style="text-align:center;" | ^1, vm2
| style="text-align:center;" | D^, Ev
| style="text-align:center;" | ^D, vE
| style="text-align:center;" | up unison, downmajor 2nd
| style="text-align:center;" | up unison, downmajor 2nd
| style="text-align:center;" | ^1, vM2
| style="text-align:center;" | ^1, vM2
| style="text-align:center;" | D^, Ev
| style="text-align:center;" | ^D, vE
|Db
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
Line 211: Line 66:
| style="text-align:center;" | M2
| style="text-align:center;" | M2
| style="text-align:center;" | E
| style="text-align:center;" | E
|C#
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
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| style="text-align:center;" | mid 2nd
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ~2
| style="text-align:center;" | E^
| style="text-align:center;" | ^E
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ~2
| style="text-align:center;" | E^
| style="text-align:center;" | ^E
|D
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
Line 229: Line 86:
| style="text-align:center;" | m2, M3
| style="text-align:center;" | m2, M3
| style="text-align:center;" | Eb, F#
| style="text-align:center;" | Eb, F#
|Eb
|-
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 5
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| style="text-align:center;" | mid 3rd
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ~3
| style="text-align:center;" | Fv
| style="text-align:center;" | vF
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ~3
| style="text-align:center;" | Fv
| style="text-align:center;" | vF
|D#
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
Line 247: Line 106:
| style="text-align:center;" | m3
| style="text-align:center;" | m3
| style="text-align:center;" | F
| style="text-align:center;" | F
|E
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
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| style="text-align:center;" | upmajor 3rd, down 4th
| style="text-align:center;" | upmajor 3rd, down 4th
| style="text-align:center;" | ^M3, v4
| style="text-align:center;" | ^M3, v4
| style="text-align:center;" | F^, Gv
| style="text-align:center;" | ^F, vG
| style="text-align:center;" | upminor 3rd, down 4th
| style="text-align:center;" | upminor 3rd, down 4th
| style="text-align:center;" | ^m3, v4
| style="text-align:center;" | ^m3, v4
| style="text-align:center;" | F^, Gv
| style="text-align:center;" | ^F, vG
|F
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
Line 265: Line 126:
| style="text-align:center;" | P4
| style="text-align:center;" | P4
| style="text-align:center;" | G
| style="text-align:center;" | G
|Gb
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 9
Line 270: Line 132:
| style="text-align:center;" | up 4th, down 5th
| style="text-align:center;" | up 4th, down 5th
| style="text-align:center;" | ^4, v5
| style="text-align:center;" | ^4, v5
| style="text-align:center;" | G^, Av
| style="text-align:center;" | ^G, vA
| style="text-align:center;" | up 4th, down 5th
| style="text-align:center;" | up 4th, down 5th
| style="text-align:center;" | ^4, v5
| style="text-align:center;" | ^4, v5
| style="text-align:center;" | G^, Av
| style="text-align:center;" | ^G, vA
|F#
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
Line 283: Line 146:
| style="text-align:center;" | P5
| style="text-align:center;" | P5
| style="text-align:center;" | A
| style="text-align:center;" | A
|G
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
Line 288: Line 152:
| style="text-align:center;" | up 5th, downminor 6th
| style="text-align:center;" | up 5th, downminor 6th
| style="text-align:center;" | ^5, vm6
| style="text-align:center;" | ^5, vm6
| style="text-align:center;" | A^, Bv
| style="text-align:center;" | ^A, vB
| style="text-align:center;" | up fifth, downmajor 6th
| style="text-align:center;" | up fifth, downmajor 6th
| style="text-align:center;" | ^5, vM6
| style="text-align:center;" | ^5, vM6
| style="text-align:center;" | A^, Bv
| style="text-align:center;" | ^A, vB
|Hb
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
Line 301: Line 166:
| style="text-align:center;" | M6
| style="text-align:center;" | M6
| style="text-align:center;" | B
| style="text-align:center;" | B
|G#
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 13
Line 306: Line 172:
| style="text-align:center;" | mid 6th
| style="text-align:center;" | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ~6
| style="text-align:center;" | B^
| style="text-align:center;" | ^B
| style="text-align:center;" | mid 6th
| style="text-align:center;" | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ~6
| style="text-align:center;" | B^
| style="text-align:center;" | ^B
|H
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
Line 319: Line 186:
| style="text-align:center;" | m6, M7
| style="text-align:center;" | m6, M7
| style="text-align:center;" | Bb, C#
| style="text-align:center;" | Bb, C#
|A
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 15
Line 324: Line 192:
| style="text-align:center;" | mid 7th
| style="text-align:center;" | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ~7
| style="text-align:center;" | Cv
| style="text-align:center;" | vC
| style="text-align:center;" | mid 7th
| style="text-align:center;" | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ~7
| style="text-align:center;" | Cv
| style="text-align:center;" | vC
|Bb
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
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| style="text-align:center;" | m7
| style="text-align:center;" | m7
| style="text-align:center;" | C
| style="text-align:center;" | C
|A#
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 17
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| style="text-align:center;" | upmajor 7th, down 8ve
| style="text-align:center;" | upmajor 7th, down 8ve
| style="text-align:center;" | ^M7, v8
| style="text-align:center;" | ^M7, v8
| style="text-align:center;" | C^, Dv
| style="text-align:center;" | ^C, vD
| style="text-align:center;" | upminor 7th, down 8ve
| style="text-align:center;" | upminor 7th, down 8ve
| style="text-align:center;" | ^m7, v8
| style="text-align:center;" | ^m7, v8
| style="text-align:center;" | C^, Dv
| style="text-align:center;" | ^C, vD
|B
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 18
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| style="text-align:center;" | P8
| style="text-align:center;" | P8
| style="text-align:center;" | D
| style="text-align:center;" | D
|C
|}
|}


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)  


==<span style="font-size: 1.3em;">Useful Moment-of-Symmetry Scales</span>==
pentatonic genchain of fifths: ...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...
Note: This list excludes scales found in 9-EDO.


===<span style="font-size: 1.1em;">Pentatonic:</span>===
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)
3L2s Father Pentatonic: 4 4 3 4 3


===<span style="font-size: 1.1em;">Hexatonic:</span>===
'''<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)
4L2s Bicycle: 4 4 1 4 4 1


2L4s Rice Hexatonic: 2 5 2 2 5 2
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


===<span style="font-size: 1.1em;">Heptatonic:</span>===
'''<u>heptatonic 3rd-generated:</u> D * * E * F * * G * A * * B * C * * D''' (generator = 5\18 = perfect 3rd)
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2


===<span style="font-size: 1.1em;">Octatonic:</span>===
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
5L3s Father Octatonic: 3 1 3 3 1 3 3 1


2L6s Rice Octatonic: 2 2 3 2 2 2 3 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;">Decatonic:</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")
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2


===<span style="font-size: 1.1em;">Dodecatonic:</span>===
genchain of thirds: ...A4 - A6 - A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 - d3 - d5...
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>==
==Representations of Just Intervals==
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!
{| class="wikitable"
|-
! | Degree
! | Cents
! | Nearest Ratio
! | Error
! | 17-Limit Ratios*
|-
| 0
|0.00
| | 1/1
| | 0
| | 1/1
|-
| | 1
| | 66.67
| | 27/26
| | +1.329
| | 78/75, 75/72
|-
| | 2
| | 133.33
| | 27/25
| | +0.096
| | 51/55, 42/39
|-
| | 3
| | 200
| | 9/8
| | -3.910
| | 9/8
|-
| | 4
| | 266.67
| | 7/6
| | -0.204
| | 75/64
|-
| | 5
| | 333.33
| | 17/14 or 40/33
| | -2.796 +0.293
| | 39/32
|-
| | 6
| | 400
| | 5/4 or 44/35
| | +13.686 +3.822
| | 64/55
|-
| | 7
| | 466.67
| | 21/16
| | -4.114
| | 21/16
|-
| | 8
| | 533.33
| | 15/11
| | -3.617
| | 102/75
|-
| | 9
| | 600
| | 17/12 or 24/17
| | -3.000 +3.000
| | 17/12
|-
| | 10
| | 666.67
| | 22/15
| | +3.617
| | 75/51
|-
| | 11
| | 733.33
| | 32/21
| | +4.114
| | 32/21
|-
| | 12
| | 800
| | 8/5 or 35/22
| | -13.686 -3.822
| | 51/32
|-
| | 13
| | 866.67
| | 28/17 or 33/20
| | +2.796 -0.293
| | 64/39
|-
| | 14
| | 933.33
| | 12/7
| | +0.204
| | 55/32
|-
| | 15
| | 1000
| | 16/9
| | +3.910
| | 16/9
|-
| | 16
| | 1066.67
| | 50/27
| | -0.096
| | 39/21
|-
| | 17
| | 1133.33
| | 52/27
| | -1.329
| | 75/39
|-
| | 18
| | 1200
| | 2/1
| | 0
| | 2/1**
|}
*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament


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).
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]]


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


Line 396: Line 389:
! | Comma
! | Comma
! | Monzo
! | Monzo
! | Value (Cents)
! | Cents
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 1
! | Name 2
! | Name 2
Line 403: Line 397:
| |<nowiki> | 7 0 -3 </nowiki>&gt;
| |<nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:right;" | 41.06
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | Augmented Comma
Line 409: Line 404:
| |<nowiki> | 23 6 -14 </nowiki>&gt;
| |<nowiki> | 23 6 -14 </nowiki>&gt;
| style="text-align:right;" | 3.34
| style="text-align:right;" | 3.34
| style="text-align:center;" |Sasa-sepbigu
| style="text-align:center;" | Vishnuzma
| style="text-align:center;" | Vishnuzma
| style="text-align:center;" | Semisuper
| style="text-align:center;" | Semisuper
Line 415: Line 411:
| |<nowiki> | 1 0 2 -2 </nowiki>&gt;
| |<nowiki> | 1 0 2 -2 </nowiki>&gt;
| style="text-align:right;" | 34.98
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | Jubilisma
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| |<nowiki> | 1 -3 -2 3 </nowiki>&gt;
| |<nowiki> | 1 -3 -2 3 </nowiki>&gt;
| style="text-align:right;" | 27.99
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" | Senga
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| style="text-align:right;" | 21.90
| style="text-align:right;" | 21.90
| style="text-align:center;" |Zotriyo
| style="text-align:center;" | Keema
| style="text-align:center;" | Keema
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| style="text-align:right;" | 13.07
| style="text-align:right;" | 13.07
| style="text-align:center;" |Triru-agu
| style="text-align:center;" | Orwellisma
| style="text-align:center;" | Orwellisma
| style="text-align:center;" | Orwell Comma
| style="text-align:center;" | Orwell Comma
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| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| style="text-align:right;" | 6.99
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 6 0 -5 2 </nowiki>&gt;
| |<nowiki> | 6 0 -5 2 </nowiki>&gt;
| style="text-align:right;" | 6.08
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | Hemimean
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| style="text-align:right;" | 17.58
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
| |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
| style="text-align:right;" | 17.40
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
| style="text-align:right;" | 8.39
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | Orgonisma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| style="text-align:right;" | 4.50
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 475: Line 481:
| |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
| |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
| style="text-align:right;" | 0.18
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | Gauss' Comma
|-
|-
| style="text-align:center;" | 91/90
| style="text-align:center;" | 91/90
| |<nowiki> | -1 -2 -1 1 1 </nowiki>&gt;
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| style="text-align:right;" | 19.13
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | Superleap
| style="text-align:center;" |  
| style="text-align:center;" |  
|}
|}
==<span style="font-size: 1.3em;">Useful Moment-of-Symmetry Scales</span>==
Note: This list excludes scales found in 9-EDO.
===<span style="font-size: 1.1em;">Pentatonic:</span>===
3L2s Father Pentatonic: 4 4 3 4 3
===<span style="font-size: 1.1em;">Hexatonic:</span>===
4L2s Bicycle: 4 4 1 4 4 1
2L4s Rice Hexatonic: 2 5 2 2 5 2
===<span style="font-size: 1.1em;">Heptatonic:</span>===
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2
===<span style="font-size: 1.1em;">Octatonic:</span>===
5L3s Father Octatonic: 3 1 3 3 1 3 3 1
2L6s Rice Octatonic: 2 2 3 2 2 2 3 2
===<span style="font-size: 1.1em;">Decatonic:</span>===
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
===<span style="font-size: 1.1em;">Dodecatonic:</span>===
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>==
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=
=Music=
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