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: | 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 minor third ( | |||
* The major third (+4 fifths) hits [[14/11]], tempering out [[896/891]]; | * The major third (+4 fifths) hits [[14/11]], tempering out [[896/891]]; | ||
* The augmented unison (+7 fifths) hits [[14/13]], tempering out [[28672/28431]]. | * The augmented unison (+7 fifths) hits [[14/13]], tempering out [[28672/28431]]. | ||
This temperament is now known as [[ | 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 == | == Interval lattice == | ||
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|- | |- | ||
| In CTE tuning and lattice basis {~2, ~3, ~7/4} | | In CTE tuning and lattice basis {~2, ~3, ~7/4} | ||
|- | |- style="border-top: double;" | ||
| [[File:Lattice Parapyth NTT.png|1000px]] | | [[File:Lattice Parapyth NTT.png|1000px]] | ||
|- | |- | ||
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== Tunings == | == Tunings == | ||
The most important tuning for parapyth is that given by MET-24 (''milder extended temperament''): | The most important tuning for parapyth is that given by MET-24 (''milder extended temperament''): | ||
* ~2/1 = | * ~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": | Another tuning derives from a 24-tone subset of George Secor's 29-HTT, thus a "24-HTT": | ||
* ~2/1 = | * ~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 (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. | 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]]: | Yet another possible tuning is that given by [[Peppermint-24]]: | ||
* ~2/1 = | * ~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. | The fifth leads to [[step ratio]] φ for the [[5L 7s|chromatic scale]] and the spacer tunes the 7/6 pure. | ||
=== Edo tunings === | === Edo tunings === | ||
The parapyth edos below 311 that are not contorted in 2.3.7.11.13 are {{ | 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.) | [[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]] | 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 == | == See also == | ||
* [[ | * [[Leapday]] – a rank-2 reduction of parapyth with additional extensions for approximating harmonics 17 and 23 | ||
== External links == | == 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 | * [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:Parapyth| ]] <!-- Main article --> | ||
[[Category:Rank-3 temperaments]] | |||
[[Category:Pentacircle clan]] | [[Category:Pentacircle clan]] | ||