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18edo: Difference between revisions

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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.
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 (with 3s) exotemperament with the mapping {{val| 18 29 42 51 }}. This maps 3/2 to 733.33¢, 5/4 to 400¢ 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-odd-limit tonality diamond, but hey, whatever floats your boat! This 7-limit mapping [[support]]s 7-limit [[sixix]] thus is strongly associated with 18edo's [[4L 3s]] [[mos]].  
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]].  


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 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.
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{{Harmonics in equal|18}}
{{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>
|-
| 0
| 0.000
| 1/1
| 0
| 1/1
|-
| 1
| 66.667
| 27/26
| +1.329
| 26/25, 25/24
|-
| 2
| 133.333
| 27/25
| +0.096
| 55/51, 14/13
|-
| 3
| 200.000
| 9/8
| -3.910
| 9/8
|-
| 4
| 266.667
| 7/6
| -0.204
| 75/64
|-
| 5
| 333.333
| 17/14 or 40/33
| -2.796 +0.293
| 39/32
|-
| 6
| 400.000
| 5/4 or 44/35
| +13.686 +3.822
| 64/55
|-
| 7
| 466.667
| 21/16
| -4.114
| 21/16
|-
| 8
| 533.333
| 15/11
| -3.617
| 34/25
|-
| 9
| 600.000
| 17/12 or 24/17
| -3.000 +3.000
| 17/12
|-
| 10
| 666.667
| 22/15
| +3.617
| 25/17
|-
| 11
| 733.333
| 32/21
| +4.114
| 32/21
|-
| 12
| 800.000
| 8/5 or 35/22
| -13.686 -3.822
| 51/32
|-
| 13
| 866.667
| 28/17 or 33/20
| +2.796 -0.293
| 64/39
|-
| 14
| 933.333
| 12/7
| +0.204
| 55/32
|-
| 15
| 1000.000
| 16/9
| +3.910
| 16/9
|-
| 16
| 1066.667
| 50/27
| -0.096
| 13/7
|-
| 17
| 1133.333
| 52/27
| -1.329
| 25/13
|-
| 18
| 1200.000
| 2/1
| 0
| 2/1**
|}
<references />
{{Clear}}
== Notation ==
== Notation ==
=== Ups and downs 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.  
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}}
{{Mavila}}
{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
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{{Sagittal chart}}
{{Sagittal chart}}
== Representations of JI intervals ==
{| 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>
|-
| 0
| 0.000
| 1/1
| 0
| 1/1
|-
| 1
| 66.667
| 27/26
| +1.329
| 78/75, 75/72
|-
| 2
| 133.333
| 27/25
| +0.096
| 51/55, 42/39
|-
| 3
| 200.000
| 9/8
| -3.910
| 9/8
|-
| 4
| 266.667
| 7/6
| -0.204
| 75/64
|-
| 5
| 333.333
| 17/14 or 40/33
| -2.796 +0.293
| 39/32
|-
| 6
| 400.000
| 5/4 or 44/35
| +13.686 +3.822
| 64/55
|-
| 7
| 466.667
| 21/16
| -4.114
| 21/16
|-
| 8
| 533.333
| 15/11
| -3.617
| 102/75
|-
| 9
| 600.000
| 17/12 or 24/17
| -3.000 +3.000
| 17/12
|-
| 10
| 666.667
| 22/15
| +3.617
| 75/51
|-
| 11
| 733.333
| 32/21
| +4.114
| 32/21
|-
| 12
| 800.000
| 8/5 or 35/22
| -13.686 -3.822
| 51/32
|-
| 13
| 866.667
| 28/17 or 33/20
| +2.796 -0.293
| 64/39
|-
| 14
| 933.333
| 12/7
| +0.204
| 55/32
|-
| 15
| 1000.000
| 16/9
| +3.910
| 16/9
|-
| 16
| 1066.667
| 50/27
| -0.096
| 39/21
|-
| 17
| 1133.333
| 52/27
| -1.329
| 75/39
|-
| 18
| 1200.000
| 2/1
| 0
| 2/1**
|}
<references />
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]]


== Regular temperament properties ==
== Regular temperament properties ==
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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!
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).
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 ===
=== Keyboards ===
[[Julián Carrillo]] built at least one third-tone piano in 18edo.
[[Lumatone mapping for 18edo|Lumatone mappings for 18edo]] are available.
[[Lumatone mapping for 18edo|Lumatone mappings for 18edo]] are available.


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=== 20th century ===
=== 20th century ===
; [[Ivan Wyschnegradsky]]
; [[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=gbPPYOygNJc] ''Prélude et Etude, Op.48''], for third-tone piano of [[Julián Carrillo]] (1966)
* [https://www.youtube.com/watch?v=gbPPYOygNJc ''Prélude et Etude'', Op. 48], for third-tone piano of [[Julián Carrillo]] (1966)


=== 21st century ===
=== 21st century ===
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; [[Beheld]]
; [[Beheld]]
* [https://www.youtube.com/watch?v=Nog2LROg8Ss Overstrung vibe]
* [https://www.youtube.com/watch?v=Nog2LROg8Ss ''Overstrung vibe''] (2022)


; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/-oi5eJA65Zc ''Waltz in 18edo''] (2025)
* [https://www.youtube.com/shorts/-oi5eJA65Zc ''Waltz in 18edo''] (2025)
* ''Lament in 18edo'' (2025)
* [https://www.youtube.com/watch?v=r3FypUx_iIk ''Lament in 18edo''] (2025)
** [https://www.youtube.com/shorts/c26Yp_aMgDw Short version with demo of Lumatone layout)]
** [https://www.youtube.com/watch?v=r3FypUx_iIk Full version]
* [https://www.youtube.com/shorts/hNmse4IUWL0 ''18edo improv''] (2025)
* [https://www.youtube.com/shorts/hNmse4IUWL0 ''18edo improv''] (2025)


; [[Francium]]
; [[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]
* "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]
 
; [[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]]
; [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et Fuga a3 in 18ET]
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et ''Fuga a3 in 18ET'']{{dead link}}


; [[Noah Jordan]]
; [[Noah Jordan]]
* ''The Moon'' (18edo album recorded in 2016 on the 1/3 tone piano of Sonido 13 / Julian Carrillo) – [https://noahdeanjordan.bandcamp.com/album/the-moon BandCamp] | [https://www.youtube.com/watch?v=TunyA3gwEJw YouTube]
* ''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
* [https://www.youtube.com/watch?v=O36ZQyq6oR8 There and Back Again] (a 20-minute microtonal journey)
* ''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]]
; [[Mandrake]]
* [https://www.youtube.com/watch?v=R1uz0ok4-Zs Such And Flowers]
* [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]
* [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]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=vUTHZNzBwUo Air Triste]
* [https://www.youtube.com/watch?v=vUTHZNzBwUo ''Air Triste''] (2018)


; [[Herman Miller]]
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/revealing-the-path Revealing the Path] (2018)
* [https://soundcloud.com/morphosyntax-1/revealing-the-path ''Revealing the Path''] (2018)


; [[Mundoworld]]
; [[Mundoworld]]
* [https://www.youtube.com/watch?v=iIaROmh7wD0 I Am the Monster I Fear]
* [https://www.youtube.com/watch?v=iIaROmh7wD0 ''I Am the Monster I Fear''] (2023)


; [[No Clue Music]]
; [[No Clue Music]]
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; [[norokusi]]
; [[norokusi]]
* [https://www.youtube.com/watch?v=pEvT2oyWEuo 3 Bagatelles]
* [https://www.youtube.com/watch?v=pEvT2oyWEuo ''3 Bagatelles''] (2021)


; [[NullPointerException Music]]
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=hNgI6oRYNHA ''Three Worlds Order''] (2020)
* [https://www.youtube.com/watch?v=hNgI6oRYNHA ''Three Worlds Order''] (2020)
* [https://www.youtube.com/watch?v=FnYxYuukgrM ''Edolian - Confusion''] (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=1k4rNjyaZsE ''Purgatory''] (2021)
* [https://www.youtube.com/watch?v=a7AtOuX1NAE ''The Hydrogen Atom''] (2023)
* [https://www.youtube.com/watch?v=a7AtOuX1NAE ''The Hydrogen Atom''] (2023)


; [[Carlo Serafini]]
; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 Do Androids Dream Of 18ED2?] ([http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog entry])
* ''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]]
; [[TomPrice719]]
* [https://soundcloud.com/tomprice719/composition-of-june-2015 Composition of June 2015]
* [https://soundcloud.com/tomprice719/composition-of-june-2015 ''Composition of June 2015''] (2015)


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 Prelude in 18et], [http://chrisvaisvil.com/?p=3 composer notes]
* ''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/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)
* [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]]
; [[Julia Werntz]], [[Eric Moe]] & the [[Pandelis Karayorgis Trio]]
* ''[https://driffrecords.bandcamp.com/album/climbing-to-sleep Climbing to Sleep]'' - jazz album (2025)
* [https://driffrecords.bandcamp.com/album/climbing-to-sleep ''Climbing to Sleep''] (2025) – jazz album


; [[Xeno*n*]]
; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=fj_AISfnFnY Deranged Anger]
* [https://www.youtube.com/watch?v=fj_AISfnFnY ''Deranged Anger''] (2021)


; [[David Zaydullin]]
; [[David Zaydullin]]
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