Parapyth - Revision history

FloraC: Linking

2026-06-05T04:49:44Z

Linking

← Older revision		Revision as of 04:49, 5 June 2026
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	\| Odd limit 2 = 2.3.7.11.13 21 \| Mistuning 2 = 3.28 \| Complexity 2 = ?		\| 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]].		'''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.		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.
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	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.		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 ==

Hkm at 22:29, 3 June 2026

2026-06-03T22:29:27Z

← Older revision		Revision as of 22:29, 3 June 2026
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	\| Edo join 1 = 17 \| Edo join 2 = 41 \| Edo join 3 = 46		\| Edo join 1 = 17 \| Edo join 2 = 41 \| Edo join 3 = 46
	\| Mapping = 1; 1 0 -4 -7; 0 1 1 1		\| Mapping = 1; 1 0 -4 -7; 0 1 1 1
	\| Generators = 3/2, 5/4		\| Generators = 3/2, 7/4
	\| Generators tuning = 703.8, 969.2		\| Generators tuning = 703.8, 969.2
	\| Optimization method = CWE		\| Optimization method = CWE

FloraC: + norm-based tunings & target tunings

2026-03-18T12:09:46Z

+ norm-based tunings & target tunings

← Older revision		Revision as of 12:09, 18 March 2026
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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.		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 ==		== See also ==

FloraC: + infobox

2026-03-18T10:51:18Z

+ infobox

← Older revision		Revision as of 10:51, 18 March 2026
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			{{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, 5/4
			\| 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]].		'''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]].

ArrowHead294 at 18:22, 3 March 2026

2026-03-03T18:22:12Z

← Older revision		Revision as of 18:22, 3 March 2026
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	\| [[File:Lattice Parapyth RTT.png\|1000px]]		\| [[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;"		\|- style="border-top: double;"
	\| [[File:Lattice Parapyth NTT.png\|1000px]]		\| [[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 ==		== Scales ==

ArrowHead294 at 14:52, 29 April 2025

2025-04-29T14:52:56Z

← Older revision		Revision as of 14:52, 29 April 2025
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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.		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 ==
			* [[Leapday]] – a rank-2 reduction of parapyth with additional extensions for approximating harmonics 17 and 23

	== External links ==		== External links ==

FloraC: - parent category

2025-04-29T09:54:10Z

- parent category

← Older revision		Revision as of 09:54, 29 April 2025
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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:Temperaments]]~~
	[[Category:Parapyth\| ]] <!-- Main article -->		[[Category:Parapyth\| ]] <!-- Main article -->
	[[Category:Rank-3 temperaments]]		[[Category:Rank-3 temperaments]]
	[[Category:Pentacircle clan]]		[[Category:Pentacircle clan]]

FloraC: "Extend" is technically incorrect as it implies keeping the rank unchanged

2025-04-25T10:08:03Z

"Extend" is technically incorrect as it implies keeping the rank unchanged

← Older revision		Revision as of 10:08, 25 April 2025
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	* 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 [[pepperoni]]. Parapyth ~~extends~~ pepperoni ~~with~~ a {{nowrap\|28/27 ~ 33/32}} "spacer" interval~~, and~~ harmonics 7, 11, and 13 are all made available simply by 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.		See [[Pentacircle clan#Parapyth]] for technical data.

ArrowHead294 at 13:32, 24 April 2025

2025-04-24T13:32:18Z

← Older revision		Revision as of 13:32, 24 April 2025
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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 (−3 fifths) is [[13/11]], tempering out 352/351;
	* 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 [[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 extends pepperoni with a {{nowrap\|28/27 ~ 33/32}} "spacer" interval, and 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 ==

Lériendil at 23:35, 19 April 2025

2025-04-19T23:35:32Z

← Older revision		Revision as of 23:35, 19 April 2025
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	[[Category:Temperaments]]		[[Category:Temperaments]]
	[[Category:Parapyth\| ]] <!-- Main article -->		[[Category:Parapyth\| ]] <!-- Main article -->
			[[Category:Rank-3 temperaments]]
	[[Category:Pentacircle clan]]		[[Category:Pentacircle clan]]