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 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 [[~~subgroup temperaments #Pepperoni|'''~~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 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.

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| In CTE tuning and lattice basis {~2, ~3, ~7/4}

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|- style="border-top: double;"

| style="~~background~~-~~color~~: ~~#000000; font-size: 3px~~;" ~~|  ~~

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| [[File:Lattice Parapyth NTT.png|1000px]]

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=== Edo tunings ===

The parapyth edos below 311 that are not contorted in 2.3.7.11.13 are {{~~Optimal ET sequence~~| 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 }}.

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 {{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.

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 ==~~

* [[Peppermint-24]]

== External links ==

@@ Line 5: / Line 5: @@
 In the early prototype, there was only a single chain of fifths, tuned slightly sharp such that:
-* The minor third (&minus;3 fifths) is [[13/11]], tempering out 352/351;
+* 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 [[subgroup temperaments #Pepperoni|'''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 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.
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 |-
 | In CTE tuning and lattice basis {~2, ~3, ~7/4}
-|-
+|- style="border-top: double;"
-| style="background-color: #000000; font-size: 3px;" | &nbsp;
-|-
 | [[File:Lattice Parapyth NTT.png|1000px]]
 |-
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 === Edo tunings ===
-The parapyth edos below 311 that are not contorted in 2.3.7.11.13 are {{Optimal ET sequence| 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 }}.
+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]]&mdash;its most elegant extension to the no-5's 17-limit (so we ignore [[100/99|S10]] and [[121/120|S11]])&mdash;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&ndash;F&#x266F;), tempering {{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&ndash;vG&#x266F;), distinguished from [[11/7]] (C&ndash;A&#x266D;) and [[14/9]] (C&ndash;^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.
+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 ==
-* [[Peppermint-24]]
 == External links ==