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.)

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 ==

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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.)
+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 ==