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 way preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second way preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 18edo "on the fly".

{| 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" | [[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

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== Regular temperament properties ==

=== Uniform maps ===

=== Commas ===

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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~~ ==

== 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¢ 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).

=== Keyboards ===

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

== Music ==

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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)

* [https://www.youtube.com/shorts/c26Yp_aMgDw ''Lament in 18edo''] (2025)

; [[Francium]]

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; [[Noah Jordan]]

* [https://noahdeanjordan.bandcamp.com/album/the-moon The Moon] (18edo album recorded on the 1/3 tone piano of Sonido 13 / Julian Carrillo)

* [https://www.youtube.com/watch?v=O36ZQyq6oR8 There and Back Again] (a 20-minute microtonal journey)

; [[Mandrake]]

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== See also ==

* [[~~Lumatone mapping for 18edo~~]]

* [[Fendo family]] - temperaments closely related to 18edo

[[Category:18-tone scales]]

@@ 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.
@@ Line 18: / Line 18: @@
 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 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.
-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".
 {| 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 406: / Line 404: @@
 == Regular temperament properties ==
 === Uniform maps ===
-{{Uniform map|13|17.5|18.5}}
+{{Uniform map|edo=18}}
 === Commas ===
@@ Line 568: / 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 ==
+== Instruments ==
+=== Guitar ===
 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 8-note oneirotonic 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).
+=== Keyboards ===
+[[Lumatone mapping for 18edo|Lumatone mappings for 18edo]] are available.
 == Music ==
@@ Line 586: / Line 588: @@
 ; [[Beheld]]
 * [https://www.youtube.com/watch?v=Nog2LROg8Ss Overstrung vibe]
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/-oi5eJA65Zc ''Waltz in 18edo''] (2025)
+* [https://www.youtube.com/shorts/c26Yp_aMgDw ''Lament in 18edo''] (2025)
 ; [[Francium]]
@@ Line 595: / Line 601: @@
 ; [[Noah Jordan]]
 * [https://noahdeanjordan.bandcamp.com/album/the-moon The Moon] (18edo album recorded on the 1/3 tone piano of Sonido 13 / Julian Carrillo)
+* [https://www.youtube.com/watch?v=O36ZQyq6oR8 There and Back Again] (a 20-minute microtonal journey)
 ; [[Mandrake]]
@@ Line 639: / Line 646: @@
 == See also ==
-* [[Lumatone mapping for 18edo]]
+* [[Fendo family]] - temperaments closely related to 18edo
 [[Category:18-tone scales]]