159edo: Difference between revisions

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A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7, 11 and 17– in fact, this EDO has a very accurate 11 and an only slightly less accurate 17. In the 7-limit it tempers out 1029/1024, 10976/10935, 117649/116640, 235298/234375 and [[Meter comma|703125/702464]] in addition to the 5-limit commas [[32805/32768]], [[15625/15552]], [[amity comma|1600000/1594323]], [[semicomma|2109375/2097152]], and [[vulture comma|10485760000/10460353203]]. This makes it among other things an excellent tuning for [[Schismatic_family#Guiron|guiron]] and [[Gamelismic_clan #Tritikleismic|tritikleismic]] temperaments, as well as a possible tuning for metric temperament. In the 11-limit, it tempers out not only [[385/384]], [[441/440]], [[3025/3024]], [[4000/3993]], [[4375/4356]], and [[6250/6237]], but– in a first for EDOs that are multiples of 53– both [[Nexuma|1771561/1769472]] and [[Quartisma|117440512/117406179]], which, in turn means that [[Symbiosma|19712/19683]] is tempered out as well. In the 13-limit it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[cantonisma|10985/10976]] and 13720/13689. In the 17-limit tempers out 273/272, 375/374, 595/594, 715/714, 936/935, [[1089/1088]], 8624/8619 and 15379/15300. In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas. 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|portending]].

In addition to the above, 159edo actually tempers out the 7-limit [[termite comma]], as well as the 17-limit [[sparkisma]], the latter of which is also tempered out by 53edo despite it having a different mapping for 17.

In addition to the above, 159edo actually tempers out the 7-limit [[termite comma]], and the 13-limit [[chalmersia]], as well as the 17-limit [[sparkisma]], the latter of which is also tempered out by 53edo despite it having a different mapping for 17.

Another 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. Yarman temperament has [[MOS]] of 79 and 80 notes to the octave, and the 79-note MOS has been proposed by Ozan Yarman as a tuning standard for [[Arabic,_Turkish,_Persian|arabic/turkish/persian]] music.

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 A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7, 11 and 17– in fact, this EDO has a very accurate 11 and an only slightly less accurate 17.  In the 7-limit it tempers out 1029/1024, 10976/10935, 117649/116640, 235298/234375 and [[Meter comma|703125/702464]] in addition to the 5-limit commas [[32805/32768]], [[15625/15552]], [[amity comma|1600000/1594323]], [[semicomma|2109375/2097152]], and [[vulture comma|10485760000/10460353203]]. This makes it among other things an excellent tuning for [[Schismatic_family#Guiron|guiron]] and [[Gamelismic_clan #Tritikleismic|tritikleismic]] temperaments, as well as a possible tuning for metric temperament.  In the 11-limit, it tempers out not only [[385/384]], [[441/440]], [[3025/3024]], [[4000/3993]], [[4375/4356]], and [[6250/6237]], but– in a first for EDOs that are multiples of 53– both [[Nexuma|1771561/1769472]] and [[Quartisma|117440512/117406179]], which, in turn means that [[Symbiosma|19712/19683]] is tempered out as well. In the 13-limit it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[cantonisma|10985/10976]] and 13720/13689. In the 17-limit tempers out 273/272, 375/374, 595/594, 715/714, 936/935, [[1089/1088]], 8624/8619 and 15379/15300. In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas. 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|portending]].
-In addition to the above, 159edo actually tempers out the 7-limit [[termite comma]], as well as the 17-limit [[sparkisma]], the latter of which is also tempered out by 53edo despite it having a different mapping for 17.
+In addition to the above, 159edo actually tempers out the 7-limit [[termite comma]], and the 13-limit [[chalmersia]], as well as the 17-limit [[sparkisma]], the latter of which is also tempered out by 53edo despite it having a different mapping for 17.
 Another 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.  Yarman temperament has [[MOS]] of 79 and 80 notes to the octave, and the 79-note MOS has been proposed by Ozan Yarman as a tuning standard for [[Arabic,_Turkish,_Persian|arabic/turkish/persian]] music.