18edo: Difference between revisions

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, 7th and 9th harmonics equally well (or equally poorly) as 12-TET 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. 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. 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 dyad 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.

However, less accurate approximations can be used, and 18edo can be treated as a 7-limit exotemperament with the mapping &lt;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-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.

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 ==

== Representations of Just Intervals ==

 

18 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 18 29 42 51 62 67 }}.)

== Useful Moment-of-Symmetry Scales ==

 

Note: This list excludes scales found in [[9-EDO]].

Pathological [[3L 9s|3L 12s]]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 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!