Parapyth: Difference between revisions
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[[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.) | [[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 don't deal with any conceptual issues arising from [[15/13]] not being present 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, tempering ([[23/16]])/[[729/512|(9/8)<sup>3</sup>]] = [[736/729]]. Alternatively, if you want a more accurate [[9/7]], [[7/6]], [[13/11]], [[25/16]] and the symmetric [[19/16]] from [[4edo]] (building on that with [[25/21]] and [[12/11]] place appropriately in [[8edo]]), [[104edo]] is an excellent rank 1 19-limit etypyth tuning that supports [[Magic_family#Septimal_magic|Magic]] by patent val and [[srutal archagall]] using the 104c val (which affords more consistency on intervals of 5 overall) - both valuable temperaments, reflecting 104edo as a "dual-5" system so that the [[25/16]] is very accurate (and even distinguished from [[11/7]] and [[14/9]] simultaneously!). The 104c may be preferable as a default due to it observing [[100/99|S10]] and [[121/120|S11]] by equating them, thus tempering [[8019/8000|S9/S10]] = ([[11/8]])/([[10/9]])<sup>3</sup>, although the patent val does still observe S11 (but tempers S10). | |||
== See also == | == See also == | ||
Revision as of 21:24, 18 February 2024
Parapyth is the rank-3 temperament tempering out 352/351 and 364/363 in the 2.3.7.11.13 subgroup.
Inspired by 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 a little sharp such that:
- the major sixth (+3 fifths) hits 22/13, 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 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.
See Pentacircle clan #Parapyth for technical data.
Interval lattice
These diagrams differ by lattice bases and tunings. The first diagram is generated by {~2, ~3, ~7/4}, 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 {~2, ~3, ~33/32}, 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
Tunings
The most important tuning for parapyth is that given by MET-24 (milder extended temperament): ~2/1 = 1\1, ~3/2 = 703.711, ~33/32 = 57.422. Another tuning is taking a 24-tone subset of George Secor's 29-HTT, thus a "24-HTT". Yet another possible tuning is that given by Peppermint-24.
Edo tunings
The parapyth edos below 311 that are not contorted in 2.3.7.11.13 are 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: 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 S10 and S11) - then the minimal strict etypyth (a.k.a. 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 don't deal with any conceptual issues arising from 15/13 not being present 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, tempering (23/16)/(9/8)3 = 736/729. Alternatively, if you want a more accurate 9/7, 7/6, 13/11, 25/16 and the symmetric 19/16 from 4edo (building on that with 25/21 and 12/11 place appropriately in 8edo), 104edo is an excellent rank 1 19-limit etypyth tuning that supports Magic by patent val and srutal archagall using the 104c val (which affords more consistency on intervals of 5 overall) - both valuable temperaments, reflecting 104edo as a "dual-5" system so that the 25/16 is very accurate (and even distinguished from 11/7 and 14/9 simultaneously!). The 104c may be preferable as a default due to it observing S10 and S11 by equating them, thus tempering S9/S10 = (11/8)/(10/9)3, although the patent val does still observe S11 (but tempers S10).