159edo: Difference between revisions

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== Theory ==

As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having so many steps as to have individual steps blend completely into one another~~. As a bonus, 159edo is [[consistent]] up to the 17-limit – though it proves to be inconsistent in the 19-limit~~.

As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having so many steps as to have individual steps blend completely into one another.

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]].

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 [[contortion]], 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 multiple 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.

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]] 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.

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 == Theory ==
-As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having so many steps as to have individual steps blend completely into one another.  As a bonus, 159edo is [[consistent]] up to the 17-limit – though it proves to be inconsistent in the 19-limit.
+As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having so many steps as to have individual steps blend completely into one another.
-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]].
+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 [[contortion]], 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 multiple 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.
+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]] 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.

159edo: Difference between revisions

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