Parapyth: Difference between revisions

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In the early prototype, there was only a single chain of fifths, tuned slightly sharp such that:

* The minor third (−3 fifths) is [[13/11]], tempering out 352/351;

* The major third (+4 fifths) hits [[14/11]], tempering out [[896/891]];

* The augmented unison (+7 fifths) hits [[14/13]], tempering out [[28672/28431]].

This temperament is now known as [[pepperoni]]. Parapyth encapsulates pepperoni, and adds a ~~spacer representing~~ 28/27~33/32~~. Prime~~ harmonics 7, 11, and 13 are all made available simply using two chains of fifths.

This temperament is now known as [[pepperoni]]. Parapyth encapsulates pepperoni and adds a {{nowrap| 28/27 ~ 33/32 }} spacer interval such that harmonics 7, 11, and 13 are all made available simply by using two chains of fifths.

See [[Pentacircle clan #Parapyth]] for technical data.

See [[Pentacircle clan#Parapyth]] for technical data.

== Interval lattice ==

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If we instead mean "parapyth" to refer to [[etypyth]] – its most elegant extension to the no-5's 17-limit (so we ignore [[100/99|S10]] and [[121/120|S11]]) – then the minimal strict etypyth (a.k.a. [[etypyth|17-limit parapyth]]) is [[46edo]], although this requires accepting its [[21/17]] as standing in for ~[[16/13]] and ~[[26/21]], corresponding roughly to (the [[octave complement]] of) [[acoustic phi]] so that stacking this interval gives a ~17:21:26:32 chord. The benefit of taking this no-5's interpretation is you do not deal with any conceptual issues arising from an out-of-tune [[15/13]] in 46edo, but you could deal with this alternately by interpreting simply only in the [[13-odd-limit]] adding odds 17, 21 and 23, which highlights that a benefit of 46edo is a fairly accurate [[23/16]] in the usual parapyth mapping of a tritone (C–F♯), tempering out {{nowrap| ([[23/16]])/[[729/512|(9/8)<sup>3</sup>]] {{=}} [[736/729]] }}. Alternatively, if you want a more accurate [[9/7]], [[7/6]], [[13/11]], [[104edo]] is an excellent etypyth tuning. 104edo is a dual-5 system that supports both the [[sensamagic]] (104) and [[pele]] (104c) mappings of 5, so that the combined [[25/16]] is very accurate (tempered together with the 81/52 (C–vG♯), distinguished from [[11/7]] (C–A♭) and [[14/9]] (C–^G) simultaneously). Pele may be preferable as a default due to it observing [[100/99|S10]] and [[121/120|S11]]. Sensamagic has the capacity to observe them too, but in the specific case of 104edo it tempers out S10.

== See also ==

* [[Leapday]] – a rank-2 reduction of parapyth with additional extensions for approximating harmonics 17 and 23

== External links ==

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* [https://www.bestii.com/~mschulter/met24-partage.txt ''The MET-24 temperament for Maqam music: Partitions or divisions of the apotome in context''] by Margo Schulter

~~[[Category:Temperaments]]~~

[[Category:Parapyth| ]]

[[Category:Rank-3 temperaments]]

[[Category:Pentacircle clan]]

@@ Line 4: / Line 4: @@
 In the early prototype, there was only a single chain of fifths, tuned slightly sharp such that:
 * The minor third (−3 fifths) is [[13/11]], tempering out 352/351;
 * The major third (+4 fifths) hits [[14/11]], tempering out [[896/891]];
 * The augmented unison (+7 fifths) hits [[14/13]], tempering out [[28672/28431]].
-This temperament is now known as [[pepperoni]]. Parapyth encapsulates pepperoni, and adds a spacer representing 28/27~33/32. Prime harmonics 7, 11, and 13 are all made available simply using two chains of fifths.
+This temperament is now known as [[pepperoni]]. Parapyth encapsulates pepperoni and adds a {{nowrap| 28/27 ~ 33/32 }} spacer interval such that harmonics 7, 11, and 13 are all made available simply by using two chains of fifths.
-See [[Pentacircle clan #Parapyth]] for technical data.
+See [[Pentacircle clan#Parapyth]] for technical data.
 == Interval lattice ==
@@ Line 53: / Line 52: @@
 If we instead mean "parapyth" to refer to [[etypyth]] – its most elegant extension to the no-5's 17-limit (so we ignore [[100/99|S10]] and [[121/120|S11]]) – then the minimal strict etypyth (a.k.a. [[etypyth|17-limit parapyth]]) is [[46edo]], although this requires accepting its [[21/17]] as standing in for ~[[16/13]] and ~[[26/21]], corresponding roughly to (the [[octave complement]] of) [[acoustic phi]] so that stacking this interval gives a ~17:21:26:32 chord. The benefit of taking this no-5's interpretation is you do not deal with any conceptual issues arising from an out-of-tune [[15/13]] in 46edo, but you could deal with this alternately by interpreting simply only in the [[13-odd-limit]] adding odds 17, 21 and 23, which highlights that a benefit of 46edo is a fairly accurate [[23/16]] in the usual parapyth mapping of a tritone (C–F♯), tempering out {{nowrap| ([[23/16]])/[[729/512|(9/8)<sup>3</sup>]] {{=}} [[736/729]] }}. Alternatively, if you want a more accurate [[9/7]], [[7/6]], [[13/11]], [[104edo]] is an excellent etypyth tuning. 104edo is a dual-5 system that supports both the [[sensamagic]] (104) and [[pele]] (104c) mappings of 5, so that the combined [[25/16]] is very accurate (tempered together with the 81/52 (C–vG♯), distinguished from [[11/7]] (C–A♭) and [[14/9]] (C–^G) simultaneously). Pele may be preferable as a default due to it observing [[100/99|S10]] and [[121/120|S11]]. Sensamagic has the capacity to observe them too, but in the specific case of 104edo it tempers out S10.
+== See also ==
+* [[Leapday]] – a rank-2 reduction of parapyth with additional extensions for approximating harmonics 17 and 23
 == External links ==
@@ Line 59: / Line 61: @@
 * [https://www.bestii.com/~mschulter/met24-partage.txt ''The MET-24 temperament for Maqam music: Partitions or divisions of the apotome in context''] by Margo Schulter
-[[Category:Temperaments]]
 [[Category:Parapyth| ]] <!-- Main article -->
+[[Category:Rank-3 temperaments]]
 [[Category:Pentacircle clan]]