18edo: Difference between revisions

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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 very well, ~~and hence~~ 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.

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 very well, with the least maximum relative error out of any edos <= 100 (highest is 18.36%, in [[23/13]]). 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 <18 29 42 51|. This maps 3/2 to 733.33¢ 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!

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 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 &gt;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 very well, and hence 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.
+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 very well, with the least maximum relative error out of any edos <= 100 (highest is 18.36%, in [[23/13]]). 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¢ 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!