159edo: Difference between revisions

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=== Mappings ===

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, despite the [[contorsion]], it still manages to have consistent mappings for the 3-limit throughout its three different [[circle of fifths|circles of fifths]] as [[Mercator's comma]], which this EDO temperes out, is less than half the size of a single step in this EDO. However, ~~the same cannot be said for the 5-limit as [[128/125]] has two different mappings. 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. ~~However, 159edo is inconsistent in the 7-limit as [[49/32]] has two different mappings. Furthermore, although~~ 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-limit, with the 19~~-limit~~ having multiple mappings.

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.

Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the harmonic 19 having multiple reasonable mappings.

=== Commas ===

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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.

== Intervals ==

@@ Line 13: / Line 13: @@
 === Mappings ===
-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, despite the [[contorsion]], it still manages to have consistent mappings for the 3-limit throughout its three different [[circle of fifths|circles of fifths]] as [[Mercator's comma]], which this EDO temperes out, is less than half the size of a single step in this EDO.  However, the same cannot be said for the 5-limit as [[128/125]] has two different mappings.  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.  However, 159edo is inconsistent in the 7-limit as [[49/32]] has two different mappings.  Furthermore, although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-limit, with the 19-limit having multiple mappings.
+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.
+Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the harmonic 19 having multiple reasonable mappings.
 === Commas ===
@@ Line 21: / Line 23: @@
 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.
 == Intervals ==