18edo: Difference between revisions
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==Theory== | == Theory == | ||
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). | 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). | ||
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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. | ||
==Intervals and 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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genchain of thirds: ...A4 - A6 - A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 - d3 - d5... | genchain of thirds: ...A4 - A6 - A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 - d3 - d5... | ||
==Representations of Just Intervals== | == Representations of Just Intervals == | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]] | [[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]] < 18 29 42 51 62 67 |.) | 18 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes the [[val|val]] < 18 29 42 51 62 67 |.) | ||
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|} | |} | ||
==Useful Moment-of-Symmetry Scales== | == Useful Moment-of-Symmetry Scales == | ||
Note: This list excludes scales found in 9-EDO. | Note: This list excludes scales found in 9-EDO. | ||
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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== | == 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! | ||
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). | 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 == | ||
* [http://www.h-pi.com/mp3/18ETPrelude.mp3 18ETPrelude] by [[Aaron Andrew Hunt]] {{dead link}} | |||
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et Fuga a3 in 18ET] by Aaron Andrew Hunt | |||
* [http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 Prelude in 18et] by [http://www.chrisvaisvil.com Chris Vaisvil] => [http://chrisvaisvil.com/?p=3 composer notes] | |||
* [http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3 Flippertronics] by Chris Vaisvil | |||
* [http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3 Gerbils at the Wheel of Government] by [http://chrisvaisvil.com/?p=1402 Chris Vaisvil (in 9 and 18 edo simultaneously)] | |||
* [http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 Do Androids Dream Of 18ED2?] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog entry]) | |||
* [https://soundcloud.com/tomprice719/composition-of-june-2015 Composition of June 2015 by TomPrice719] | |||
[[Category:18-tone]] | [[Category:18-tone]] | ||