18edo: Difference between revisions

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== Theory ==

18edo does not ~~approximate~~ the 3rd ~~harmonic at all, unless an error of >30{{c}} is considered acceptable~~, and ~~it approximates~~ the ~~5th, 7th and 9th harmonics equally well (or equally poorly)~~ 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).

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. 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 2.9.5.7 subgroup temperament.

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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.

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.

=== Odd harmonics ===

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== 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¢ 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.

The first defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{~~nowrap|D + M2}~~} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Chord names don't follow diatonic nominals because {{dash|C, E, G|med}~~} is not {{dash|P1, M3, P5|med}}.~~

The second approach is to essentially pretend 18edo's antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. This allows music notated in 12edo or another diatonic system to be directly translated to 18edo "on the fly", and it carries over the way interval arithmetic and chord names work from diatonic notation.

{| 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" | [[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

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=== Pentadecatonic ===

~~Pathological~~ [[3L 12s]]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1

[[3L 12s]]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1

== Application to guitar ==

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; [[Beheld]]

* [https://www.youtube.com/watch?v=Nog2LROg8Ss Overstrung vibe]

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/-oi5eJA65Zc ''Waltz in 18edo''] (2025)

; [[Francium]]

@@ Line 11: / Line 11: @@
 == Theory ==
-edo does not approximate the 3rd harmonic at all, unless an error of &gt;30{{c}} is considered acceptable, and it approximates the 5th, 7th and 9th harmonics equally well (or equally poorly) 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).
+edo 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. 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 2.9.5.7 subgroup temperament.
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 edo 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.
+edo 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.
 === Odd harmonics ===
@@ Line 24: / Line 26: @@
 == Notation ==
 === Ups and downs notation ===
-edo 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:
+edo 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.
+{{Mavila}}
-The first defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Chord names don't follow diatonic nominals because {{dash|C, E, G|med}} is not {{dash|P1, M3, P5|med}}.
-The second approach is to essentially pretend 18edo's antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. This allows music notated in 12edo or another diatonic system to be directly translated to 18edo "on the fly", and it carries over the way interval arithmetic and chord names work from diatonic notation.
 {| 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" | [[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
@@ Line 567: / Line 566: @@
 === Pentadecatonic ===
-Pathological [[3L 12s]]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1
+[[3L 12s]]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1
 == Application to guitar ==
@@ Line 585: / Line 584: @@
 ; [[Beheld]]
 * [https://www.youtube.com/watch?v=Nog2LROg8Ss Overstrung vibe]
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/-oi5eJA65Zc ''Waltz in 18edo''] (2025)
 ; [[Francium]]