159edo: Difference between revisions

Line 1:

The ''159 equal division'' divides the octave into 159 equal parts of 7.547 cents each. A salient fact about it is that 159 = 3*53, so that it shares the same 5-limit thirds and fifths with [[53edo]]. However, the patent vals differ on the ~~mapping~~ for 7. In the 7-limit it tempers out 1029/1024 and 10976/19935 in addition to the 5-limit commas 32805/32768 and 15625/15552. This makes it among other things an excellent tuning for [[Gamelismic_clan|guiron]] and [[Gamelismic_clan|tritikleismic]] temperaments. It has a very accurate 11, and in the 11-limit tempers out 385/384, 441/440 and 4000/3993. In the 13-limit it tempers out 325/324 and 364/363. It also has an accurate 17, and in the 17-limit 273/272 and 375/374. In the 19-limit it tempers out 343/342 and 361/360. It also provides the [[optimal patent val]] for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament [[Gamelismic_family|portending]].

The ''159 equal division'' divides the octave into 159 equal parts of 7.547 cents each. A salient fact about it is that 159 = 3*53, so that it shares the same 5-limit thirds and fifths with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7 and 13. In the 7-limit it tempers out 1029/1024 and 10976/19935 in addition to the 5-limit commas 32805/32768 and 15625/15552. This makes it among other things an excellent tuning for [[Gamelismic_clan|guiron]] and [[Gamelismic_clan|tritikleismic]] temperaments. It has a very accurate 11, and in the 11-limit tempers out 385/384, 441/440 and 4000/3993. In the 13-limit it tempers out 325/324 and 364/363. It also has an accurate 17, and in the 17-limit 273/272 and 375/374. In the 19-limit it tempers out 343/342 and 361/360. It also provides the [[optimal patent val]] for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament [[Gamelismic_family|portending]].

Another and notable temperament supported by 159 is [[Turkish_maqam_music_temperaments|yarman temperament]], with a generator of 2\159 which can be taken as an approximate 105/104. 159 supplies the optimal patent val for 7, 11, 13, 17 and 19-limit yarman, so they are very closely associated. Curiously, the temperament does not temper out 1029/1024, however.

Revision as of 18:37, 31 August 2020 view source Aura (talk \| contribs) autopatrolled, emailconfirmed 6,010 edits Additional information was added concerning the differing patent vals for 7 compared to 53edo, as well as the fact that 159edo has an accurate 17. Tag: Visual edit ← Older edit		Revision as of 18:40, 31 August 2020 view source Aura (talk \| contribs) autopatrolled, emailconfirmed 6,010 edits Additional information on the patent vals for 13. Tag: Visual edit Newer edit →
Line 1:		Line 1:
	The ''159 equal division'' divides the octave into 159 equal parts of 7.547 cents each. A salient fact about it is that 159 = 3*53, so that it shares the same 5-limit thirds and fifths with [[53edo]]. However, the patent vals differ on the ~~mapping~~ for 7. In the 7-limit it tempers out 1029/1024 and 10976/19935 in addition to the 5-limit commas 32805/32768 and 15625/15552. This makes it among other things an excellent tuning for [[Gamelismic_clan\|guiron]] and [[Gamelismic_clan\|tritikleismic]] temperaments. It has a very accurate 11, and in the 11-limit tempers out 385/384, 441/440 and 4000/3993. In the 13-limit it tempers out 325/324 and 364/363. It also has an accurate 17, and in the 17-limit 273/272 and 375/374. In the 19-limit it tempers out 343/342 and 361/360. It also provides the [[optimal patent val]] for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament [[Gamelismic_family\|portending]].		The ''159 equal division'' divides the octave into 159 equal parts of 7.547 cents each. A salient fact about it is that 159 = 3*53, so that it shares the same 5-limit thirds and fifths with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7 and 13. In the 7-limit it tempers out 1029/1024 and 10976/19935 in addition to the 5-limit commas 32805/32768 and 15625/15552. This makes it among other things an excellent tuning for [[Gamelismic_clan\|guiron]] and [[Gamelismic_clan\|tritikleismic]] temperaments. It has a very accurate 11, and in the 11-limit tempers out 385/384, 441/440 and 4000/3993. In the 13-limit it tempers out 325/324 and 364/363. It also has an accurate 17, and in the 17-limit 273/272 and 375/374. In the 19-limit it tempers out 343/342 and 361/360. It also provides the [[optimal patent val]] for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament [[Gamelismic_family\|portending]].

	Another and notable temperament supported by 159 is [[Turkish_maqam_music_temperaments\|yarman temperament]], with a generator of 2\159 which can be taken as an approximate 105/104. 159 supplies the optimal patent val for 7, 11, 13, 17 and 19-limit yarman, so they are very closely associated. Curiously, the temperament does not temper out 1029/1024, however.		Another and notable temperament supported by 159 is [[Turkish_maqam_music_temperaments\|yarman temperament]], with a generator of 2\159 which can be taken as an approximate 105/104. 159 supplies the optimal patent val for 7, 11, 13, 17 and 19-limit yarman, so they are very closely associated. Curiously, the temperament does not temper out 1029/1024, however.