Parapyth: Difference between revisions

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'''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]].

{{Infobox regtemp

| Title = Parapyth

| Subgroups = 2.3.7.11, 2.3.7.11.13

| Comma basis = [[896/891]] (2.3.7.11); <br>[[352/351]], [[364/363]] (2.3.7.11.13)

| Edo join 1 = 17 | Edo join 2 = 41 | Edo join 3 = 46

| Mapping = 1; 1 0 -4 -7; 0 1 1 1

| Generators = 3/2, 7/4

| Generators tuning = 703.8, 969.2

| Optimization method = CWE

| Odd limit 1 = 2.3.7.11 11 | Mistuning 1 = 3.23 | Complexity 1 = ?

| Odd limit 2 = 2.3.7.11.13 21 | Mistuning 2 = 3.28 | Complexity 2 = ?

}}

'''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 ~~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 {{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.

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

|-

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

| 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 {~2, ~3, ~33/32}

| 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 {~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.

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 ==

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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.

=== Norm-based tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7.11-subgroup norm-based tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Tenney

| CTE: ~3/2 = 703.5763{{c}}, ~7/4 = 967.5543{{c}}

| CWE: ~3/2 = 703.7426{{c}}, ~7/4 = 969.0476{{c}}

| POTE: ~3/2 = 703.8345{{c}}, ~7/4 = 969.8722{{c}}

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7.11.13-subgroup norm-based tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Tenney

| CTE: ~3/2 = 703.7857{{c}}, ~7/4 = 967.6654{{c}}

| CWE: ~3/2 = 703.8328{{c}}, ~7/4 = 969.1612{{c}}

| POTE: ~3/2 = 703.8563{{c}}, ~7/4 = 969.9074{{c}}

|}

=== Target tunings ===

{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings

! rowspan="2" | Target

! colspan="2" | Minimax

|-

! Generator

! Eigenmonzo basis

|-

| No-5 11-odd-limit

| ~3/2 = 703.5697{{c}}, ~7/4 = 968.8259{{c}}

| 2.7.11/9

|-

| No-5 13-odd-limit

| ~3/2 = 703.5968{{c}}, ~7/4 = 968.9885{{c}}

| 2.11/9.13/9

|}

== See also ==

* [[Leapday]] – a rank-2 reduction of parapyth with additional extensions for approximating harmonics 17 and 23

== 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

~~[[Category:Temperaments]]~~

[[Category:Parapyth| ]]

[[Category:Rank-3 temperaments]]

[[Category:Pentacircle clan]]

@@ Line 1: / Line 1: @@
-'''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]].
+{{Infobox regtemp
+| Title = Parapyth
+| Subgroups = 2.3.7.11, 2.3.7.11.13
+| Comma basis = [[896/891]] (2.3.7.11); <br>[[352/351]], [[364/363]] (2.3.7.11.13)
+| Edo join 1 = 17 | Edo join 2 = 41 | Edo join 3 = 46
+| Mapping = 1; 1 0 -4 -7; 0 1 1 1
+| Generators = 3/2, 7/4
+| Generators tuning = 703.8, 969.2
+| Optimization method = CWE
+| Odd limit 1 = 2.3.7.11 11 | Mistuning 1 = 3.23 | Complexity 1 = ?
+| Odd limit 2 = 2.3.7.11.13 21 | Mistuning 2 = 3.28 | Complexity 2 = ?
+}}
+'''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 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 {{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.
@@ Line 18: / Line 29: @@
 | [[File:Lattice Parapyth RTT.png|1000px]]
 |-
-| In CTE tuning and lattice basis {~2, ~3, ~7/4}
+| 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 {~2, ~3, ~33/32}
+| 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 {~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.
+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 ==
@@ Line 53: / Line 64: @@
 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.
+=== Norm-based tunings ===
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7.11-subgroup norm-based tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Tenney
+| CTE: ~3/2 = 703.5763{{c}}, ~7/4 = 967.5543{{c}}
+| CWE: ~3/2 = 703.7426{{c}}, ~7/4 = 969.0476{{c}}
+| POTE: ~3/2 = 703.8345{{c}}, ~7/4 = 969.8722{{c}}
+|}
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7.11.13-subgroup norm-based tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Tenney
+| CTE: ~3/2 = 703.7857{{c}}, ~7/4 = 967.6654{{c}}
+| CWE: ~3/2 = 703.8328{{c}}, ~7/4 = 969.1612{{c}}
+| POTE: ~3/2 = 703.8563{{c}}, ~7/4 = 969.9074{{c}}
+|}
+=== Target tunings ===
+{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
+! rowspan="2" | Target
+! colspan="2" | Minimax
+|-
+! Generator
+! Eigenmonzo basis
+|-
+| No-5 11-odd-limit
+| ~3/2 = 703.5697{{c}}, ~7/4 = 968.8259{{c}}
+| 2.7.11/9
+|-
+| No-5 13-odd-limit
+| ~3/2 = 703.5968{{c}}, ~7/4 = 968.9885{{c}}
+| 2.11/9.13/9
+|}
+== See also ==
+* [[Leapday]] – a rank-2 reduction of parapyth with additional extensions for approximating harmonics 17 and 23
 == External links ==
@@ Line 59: / Line 124: @@
 * [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:Temperaments]]
 [[Category:Parapyth| ]] <!-- Main article -->
+[[Category:Rank-3 temperaments]]
 [[Category:Pentacircle clan]]