View source for Parapyth

'''Parapyth''', also known as '''parapythagorean''', is the rank-3 [[temperament]] tempering out [[352/351]] and [[364/363]] in the 2.3.7.11.13 [[subgroup]]. 

Inspired by [[Secor29htt|George Secor's 29-tone high tolerance temperament]], parapyth was found by [[Margo Schulter]] in 2002, and it continued to be developed as part of her ''neoclassical tuning theory'' (NTT), although a [[regular temperament]] perspective is as viable. 

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 {{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.

== Interval lattice ==
{| class="wikitable" style="margin: 20px auto 20px auto;"
|-
| [[File:Lattice Parapyth RTT.png|1000px]]
|-
| In CTE tuning and lattice basis {{nowrap|{~2, ~3, ~7/4}<nowiki/>}}
|- style="border-top: double;"
| [[File:Lattice Parapyth NTT.png|1000px]]
|-
| In MET-24 tuning and lattice basis {{nowrap|{~2, ~3, ~33/32}<nowiki/>}}
|}

These diagrams differ by lattice bases and tunings. The first diagram is generated by {{nowrap|{~2, ~3, ~7/4}<nowiki/>}}, corresponding to the octave-reduced form of the mapping, and tuned to the 2.3.7.11.13 subgroup CTE tuning. The second diagram shows the preferred settings in Margo Schulter's neoclassical tuning theory, where it is generated by {{nowrap|{~2, ~3, ~33/32}<nowiki/>}}, and tuned to MET-24.

== Scales ==
* [[Parapyth12]] – 12-tone Fokker block in 2.3.7.11.13 TOP tuning
* [[Pepperoni7]] – 7-tone single chain of fifths in 271edo tuning
* [[Pepperoni12]] – 12-tone single chain of fifths in 271edo tuning
* [[MET-24]] – 24-tone double chain of fifths in 2048edo tuning

== Tunings ==
The most important tuning for parapyth is that given by MET-24 (''milder extended temperament''): 
* ~2/1 = 1200.000{{c}}, ~3/2 = 703.711{{c}}, ~33/32 = 57.422{{c}}. 

Another tuning derives from a 24-tone subset of George Secor's 29-HTT, thus a "24-HTT": 
* ~2/1 = 1200.000{{c}}, ~3/2 = 703.579{{c}}, ~33/32 = 58.090{{c}}. 

The fifth is in the 9th-secorian-comma tuning, which makes the augmented second of [[63/52]] pure. This fifth leads to an equal 3.247-cent error in [[9/8]] and 14/13 ({{nowrap| 63/52 {{=}} (9/8)⋅(14/13) }}) and thus a possible minimax tuning for the no-5 13-odd-limit. The minor third is extremely close to just 13/11, only off by 1/3 harmonisma. The spacer is determined such that [[7/4]] is pure. 

Yet another possible tuning is that given by [[Peppermint-24]]: 
* ~2/1 = 1200.000{{c}}, ~3/2 = 704.096{{c}}, ~33/32 = 58.680{{c}}. 

The fifth leads to [[step ratio]] φ for the [[5L 7s|chromatic scale]] and the spacer tunes the 7/6 pure. 

=== Edo tunings ===
The parapyth edos below 311 that are not contorted in 2.3.7.11.13 are {{EDOs| 17, 22, 24, 29, 41, 46, 58, 63, 65, 80, 87, 104, 109, 121, 128, 133, 145, 150, 167, 172, 184, 191, 196, 213, 230, 232, 237, 254, 259, 271, 278, 283, and 295 }}.

[[87edo]] is special for being the smallest "strict parapyth edo" (tempers out 352/351 and 364/363 and maps all of 121/120, 144/143, and 169/168 positively, meeting [[Margo Schulter]]'s criterion for "middle parapyth in the strict sense"). The following are strict parapyth edos below 311 that are not contorted in the 13-limit: {{Optimal ET sequence| 87, 104, 121, 128, 133, 145, 150, 167, 184, 191, 196, ''208'', 213, 230, 232, 237, 254, 259, 271, 278, 283, 295 }}. (Note: 208edo is contorted in 2.3.7.11.13 subgroup but not in the full 13-limit.)

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 ==
* [https://www.xenharmonikon.org/2022/07/15/met-24-a-milder-extended-temperament/ Xenharmonikon | ''MET-24: A Milder Extended Temperament''] by Margo Schulter
* [https://www.bestii.com/~mschulter/tn101812-3degrees.txt ''A Friendly Introduction to "Rank-3" Temperaments: Designing a System with Three Degrees of Freedom''] by Margo Schulter
* [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:Parapyth| ]] <!-- Main article -->
[[Category:Rank-3 temperaments]]
[[Category:Pentacircle clan]]