Pele - Revision history

FloraC: Rework

2026-07-26T06:59:46Z

Rework

← Older revision		Revision as of 06:59, 26 July 2026
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	\| Odd limit 2 = 13-limit 21 \| Mistuning 2 = 4.43 \| Complexity 2 = ?		\| Odd limit 2 = 13-limit 21 \| Mistuning 2 = 4.43 \| Complexity 2 = ?
	}}		}}
	'''~~Pele~~''' ~~is a~~ [[rank-3 temperament]] ~~generated by a perfect fifth~~ of ~~[[~]][[3/2]] and a step for~~ the [[~~81/80\|syntonic~~]]~[[~~64/63\|septimal comma~~]] ~~to reach the interval classes~~ of [[~~5/1\|5~~]], [[~~7/1~~\|7]], and ~~higher [[prime harmonic\|primes]]. Using an arrow to represent this~~ comma step~~, we have [[5/4]] at~~ the ~~down major third (C–vE),~~ [[7/4]] ~~at the down minor seventh (C–vBb),~~ and [[11/8]] ~~at the down diminished fifth (C–vGb),~~ [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].		The '''pele''' [[rank-3 temperament]] is one of the [[11-limit]] [[extension]]s of [[aberschismic]], inheriting the [[chain of fifths\|chain-of-fifths-based]] structure and the generic comma step that results from equating the [[81/80\|syntonic]] and [[64/63\|septimal commas]]. It [[tempering out\|tempers out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]], and as a consequence, [[11/8]] is mapped to the comma-down diminished fifth (C–vG♭),

	The canonical [[extension]] to the [[13-limit]] ~~finds~~ [[13/8]] at the down diminished seventh (~~C–vBbb~~)~~, tempering out~~ [[196/195]], [[352/351]] and [[847/845]], ~~and~~ a [[17-limit]] extension is available by recognizing [[17/16]] at the up minor second (C–^Db), tempering out [[256/255]].		The canonical [[extension]] to the [[13-limit]] maps [[13/11]] to the diatonic minor third, which is the exact mean of [[6/5]] and [[7/6]], and implies [[13/8]] is the comma-down diminished seventh (C–vB𝄫) and that [[196/195]], [[352/351]] and [[847/845]] are tempered out. Additionally, a [[17-limit]] extension is available by recognizing [[17/16]] at the comma-up minor second (C–^D♭), the same interval as [[16/15]], and thus tempering out [[256/255]].

	Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the up augmented second (C–^D#).		Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the quartertone-up augmented second (C–^D♯).

	See [[~~Hemifamity~~ family #Pele]] for technical ~~details~~.		See [[Aberschismic family #Pele]] for technical data.

	== Interval lattice ==		== Interval lattice ==

FloraC at 17:00, 24 July 2026

2026-07-24T17:00:34Z

← Older revision		Revision as of 17:00, 24 July 2026
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	\| Comma basis = [[441/440]], [[896/891]]; <br>[[196/195]], [[352/351]], [[364/363]]		\| Comma basis = [[441/440]], [[896/891]]; <br>[[196/195]], [[352/351]], [[364/363]]
	\| Edo join 1 = 41 \| Edo join 2 = 46 \| Edo join 3 = 58		\| Edo join 1 = 41 \| Edo join 2 = 46 \| Edo join 3 = 58
	\| Mapping = 1; 1 0 -6 -10; 0 1 1 1		\| Mapping = 1; 1 0 -6 -10 -13; 0 1 1 1 1
	\| Generators = 3/2; 5/4 \| Generators tuning = 703.4; 387.8		\| Generators = 3/2; 5/4 \| Generators tuning = 703.4; 387.8
	\| Optimization method = CWE		\| Optimization method = CWE

FloraC: + infobox

2026-07-24T09:26:14Z

+ infobox

← Older revision		Revision as of 09:26, 24 July 2026
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			{{Infobox regtemp
			\| Title = Pele
			\| Subgroups = 2.3.5.7.11; 2.3.5.7.11.13
			\| Comma basis = [[441/440]], [[896/891]]; <br>[[196/195]], [[352/351]], [[364/363]]
			\| Edo join 1 = 41 \| Edo join 2 = 46 \| Edo join 3 = 58
			\| Mapping = 1; 1 0 -6 -10; 0 1 1 1
			\| Generators = 3/2; 5/4 \| Generators tuning = 703.4; 387.8
			\| Optimization method = CWE
			\| Odd limit 1 = 11 \| Mistuning 1 = 3.51 \| Complexity 1 = ?
			\| Odd limit 2 = 13-limit 21 \| Mistuning 2 = 4.43 \| Complexity 2 = ?
			}}
	'''Pele''' is a [[rank-3 temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80\|syntonic]]~[[64/63\|septimal comma]] to reach the interval classes of [[5/1\|5]], [[7/1\|7]], and higher [[prime harmonic\|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the down diminished fifth (C–vGb), [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].		'''Pele''' is a [[rank-3 temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80\|syntonic]]~[[64/63\|septimal comma]] to reach the interval classes of [[5/1\|5]], [[7/1\|7]], and higher [[prime harmonic\|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the down diminished fifth (C–vGb), [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].

FloraC: Text replacement - "rank-3 temperament" to "rank-3 temperament"

2026-06-08T08:36:24Z

Text replacement - "rank-3 temperament" to "rank-3 temperament"

← Older revision		Revision as of 08:36, 8 June 2026
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	'''Pele''' is a [[rank-3 ~~temperament\|rank-3]] [[regular temperament\|~~temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80\|syntonic]]~[[64/63\|septimal comma]] to reach the interval classes of [[5/1\|5]], [[7/1\|7]], and higher [[prime harmonic\|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the down diminished fifth (C–vGb), [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].		'''Pele''' is a [[rank-3 temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80\|syntonic]]~[[64/63\|septimal comma]] to reach the interval classes of [[5/1\|5]], [[7/1\|7]], and higher [[prime harmonic\|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the down diminished fifth (C–vGb), [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].

	The canonical [[extension]] to the [[13-limit]] finds [[13/8]] at the down diminished seventh (C–vBbb), tempering out [[196/195]], [[352/351]] and [[847/845]], and a [[17-limit]] extension is available by recognizing [[17/16]] at the up minor second (C–^Db), tempering out [[256/255]].		The canonical [[extension]] to the [[13-limit]] finds [[13/8]] at the down diminished seventh (C–vBbb), tempering out [[196/195]], [[352/351]] and [[847/845]], and a [[17-limit]] extension is available by recognizing [[17/16]] at the up minor second (C–^Db), tempering out [[256/255]].

FloraC: Text replacement - "Category:Hemifamity family" to "Category:Aberschismic family"

2026-06-06T12:41:17Z

Text replacement - "Category:Hemifamity family" to "Category:Aberschismic family"

← Older revision		Revision as of 12:41, 6 June 2026
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	[[Category:Pele\| ]] <!-- main article -->		[[Category:Pele\| ]] <!-- main article -->
	[[Category:Rank-3 temperaments]]		[[Category:Rank-3 temperaments]]
	[[Category:~~Hemifamity~~ family]]		[[Category:Aberschismic family]]
	[[Category:Pentacircle clan]]		[[Category:Pentacircle clan]]
	[[Category:Werckismic temperaments]]		[[Category:Werckismic temperaments]]

Overthink: fixed extensions

2025-10-29T22:27:15Z

fixed extensions

← Older revision		Revision as of 22:27, 29 October 2025
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	'''Pele''' is a [[rank-3 temperament\|rank-3]] [[regular temperament\|temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80\|syntonic]]~[[64/63\|septimal comma]] to reach the interval classes of [[5/1\|5]], [[7/1\|7]], and higher [[prime harmonic\|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the down diminished fifth (C–vGb), [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].		'''Pele''' is a [[rank-3 temperament\|rank-3]] [[regular temperament\|temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80\|syntonic]]~[[64/63\|septimal comma]] to reach the interval classes of [[5/1\|5]], [[7/1\|7]], and higher [[prime harmonic\|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the down diminished fifth (C–vGb), [[tempering out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]].

	The canonical [[extension]] to the [[13-limit]] finds [[13/8]] at the ~~double-~~down diminished seventh (C–vBbb), tempering out [[196/195]], [[352/351]] and [[847/845]], and a [[17-limit]] extension is available by recognizing [[17/16]] at the minor second (~~C–Db~~), tempering out [[256/255]].		The canonical [[extension]] to the [[13-limit]] finds [[13/8]] at the down diminished seventh (C–vBbb), tempering out [[196/195]], [[352/351]] and [[847/845]], and a [[17-limit]] extension is available by recognizing [[17/16]] at the up minor second (C–^Db), tempering out [[256/255]].

	Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the up augmented second (C–^D#).		Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the up augmented second (C–^D#).

FloraC: /* Chords */ -> chords and harmony

2025-10-22T07:13:23Z

Chords: -> chords and harmony

← Older revision		Revision as of 07:13, 22 October 2025
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	This lattice shows pele as an extension of parapyth, generated by ~2, ~3/2, and ~7/4.		This lattice shows pele as an extension of parapyth, generated by ~2, ~3/2, and ~7/4.

	== Chords ==		== Chords and harmony ==
	Pele enables [[essentially tempered chord]]s of [[werckismic chords\|werckismic]] and [[pentacircle chords\|pentacircle]] in the [[11-odd-limit]], in addition to [[mynucumic chords\|mynucumic]], [[major minthmic chords\|major minthmic]], [[minor minthmic chords\|minor minthmic]] and [[cuthbert chords\|cuthbert]] in the [[13-odd-limit]].		Pele enables [[essentially tempered chord]]s of [[werckismic chords\|werckismic]] and [[pentacircle chords\|pentacircle]] in the [[11-odd-limit]], in addition to [[mynucumic chords\|mynucumic]], [[major minthmic chords\|major minthmic]], [[minor minthmic chords\|minor minthmic]] and [[cuthbert chords\|cuthbert]] in the [[13-odd-limit]].

	== Scales ==		== Scales ==

FloraC: /* Tunings */

2025-10-22T07:12:19Z

Tunings

← Older revision		Revision as of 07:12, 22 October 2025
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	\|-		\|-
	! Tenney		! Tenney
	\| CTE: ~3/2 = 703.2829, ~5/4 = 386.5647{{c}}		\| CTE: ~3/2 = 703.2829{{c}}, ~5/4 = 386.5647{{c}}
	\| CWE: ~3/2 = 703.2804, ~5/4 = 387.3911{{c}}		\| CWE: ~3/2 = 703.2804{{c}}, ~5/4 = 387.3911{{c}}
	\| POTE: ~3/2 = 703.2791, ~5/4 = 387.7906{{c}}		\| POTE: ~3/2 = 703.2791{{c}}, ~5/4 = 387.7906{{c}}
	\|}		\|}

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	\|-		\|-
	! Tenney		! Tenney
	\| CTE: ~3/2 = 703.4398, ~5/4 = 386.8933{{c}}		\| CTE: ~3/2 = 703.4398{{c}}, ~5/4 = 386.8933{{c}}
	\| CWE: ~3/2 = 703.4225, ~5/4 = 387.7761{{c}}		\| CWE: ~3/2 = 703.4225{{c}}, ~5/4 = 387.7761{{c}}
	\| POTE: ~3/2 = 703.4143, ~5/4 = 388.1971{{c}}		\| POTE: ~3/2 = 703.4143{{c}}, ~5/4 = 388.1971{{c}}
	\|}		\|}

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	\|-		\|-
	! Tenney		! Tenney
	\| CTE: ~3/2 = 703.5544, ~5/4 = 387.9654{{c}}		\| CTE: ~3/2 = 703.5544{{c}}, ~5/4 = 387.9654{{c}}
	\| CWE: ~3/2 = 703.4518, ~5/4 = 388.4909{{c}}		\| CWE: ~3/2 = 703.4518{{c}}, ~5/4 = 388.4909{{c}}
	\| POTE: ~3/2 = 703.4265, ~5/4 = 388.6202{{c}}		\| POTE: ~3/2 = 703.4265{{c}}, ~5/4 = 388.6202{{c}}
	\|}		\|}

FloraC: + basic tunings

2025-10-22T06:39:41Z

+ basic tunings

@@ Line 23: / Line 23: @@
 * [[Dekany pele]] – a transversal scale
 * The [[compdye]] scale pattern
 [[Category:Pele| ]] <!-- main article -->

FloraC: - parent category

2025-04-29T09:54:19Z

- parent category

← Older revision		Revision as of 09:54, 29 April 2025
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	* The [[compdye]] scale pattern		* The [[compdye]] scale pattern

	~~[[Category:Temperaments]]~~
	[[Category:Pele\| ]] <!-- main article -->		[[Category:Pele\| ]] <!-- main article -->
	[[Category:Rank-3 temperaments]]		[[Category:Rank-3 temperaments]]