18edo: Difference between revisions
ups and downs, color names, alternate notations |
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| ja = 18平均律 | | ja = 18平均律 | ||
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== | ==Theory== | ||
18- | 18 Equal Divisions of the Octave, also known as The Third-Tone System, 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). | ||
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. 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. | ||
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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. | ||
==Notation== | ==Intervals and 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. | ||
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | | ! | [[Ratio]] | ||
! | Monzo | ! | [[Monzo]] | ||
! | Cents | ! | [[Cents]] | ||
![[Color notation/Temperament Names|Color Name]] | ![[Color notation/Temperament Names|Color Name]] | ||
! | Name 1 | ! | Name 1 | ||
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|} | |} | ||
== | ==Useful Moment-of-Symmetry Scales== | ||
Note: This list excludes scales found in 9-EDO. | Note: This list excludes scales found in 9-EDO. | ||
<span style="font-size: 1.1em;">'''<u>Pentatonic:</u>'''</span> | |||
3L2s Father Pentatonic: 4 4 3 4 3 | 3L2s Father Pentatonic: 4 4 3 4 3 | ||
<span style="font-size: 1.1em;">'''<u>Hexatonic:</u>'''</span> | |||
4L2s Bicycle: 4 4 1 4 4 1 | 4L2s Bicycle: 4 4 1 4 4 1 | ||
2L4s Rice Hexatonic: 2 5 2 2 5 2 | 2L4s Rice Hexatonic: 2 5 2 2 5 2 | ||
<span style="font-size: 1.1em;">'''<u>Heptatonic:</u>'''</span> | |||
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2 | 4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2 | ||
<span style="font-size: 1.1em;">'''<u>Octatonic:</u>'''</span> | |||
5L3s Father Octatonic: 3 1 3 3 1 3 3 1 | 5L3s Father Octatonic: 3 1 3 3 1 3 3 1 | ||
2L6s Rice Octatonic: 2 2 3 2 2 2 3 2 | 2L6s Rice Octatonic: 2 2 3 2 2 2 3 2 | ||
<span style="font-size: 1.1em;">'''<u>Decatonic:</u>'''</span> | |||
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2 | 8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2 | ||
<span style="font-size: 1.1em;">'''<u>Dodecatonic:</u>'''</span> | |||
6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1 | 6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1 | ||
== | ==Application to Guitar== | ||
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! | 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! | ||