191zpi - Revision history

Lériendil: Text replacement - "Category:Tempered_scales↵Category:Pages with mostly numerical content" to "Category:Tempered scales Category:Pages with mostly numerical content"

2025-06-24T02:53:38Z

Text replacement - "Category:Tempered_scales↵Category:Pages with mostly numerical content" to "Category:Tempered scales Category:Pages with mostly numerical content"

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	[[Category:42-tone scales]]		[[Category:42-tone scales]]
	[[Category:~~Tempered_scales~~]]		[[Category:Tempered scales]]
	[[Category:Pages with mostly numerical content]]		[[Category:Pages with mostly numerical content]]
	[[Category:Zeta peak indexes]]		[[Category:Zeta peak indexes]]
	[[Category:42edo]]		[[Category:42edo]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

Lériendil: Text replacement - "Category:Tempered scales↵Category:Pages with mostly numerical content" to "Category:Tempered_scales Category:Pages with mostly numerical content"

2025-06-24T02:17:07Z

Text replacement - "Category:Tempered scales↵Category:Pages with mostly numerical content" to "Category:Tempered_scales Category:Pages with mostly numerical content"

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	[[Category:42-tone scales]]		[[Category:42-tone scales]]
	[[Category:~~Tempered scales~~]]		[[Category:Tempered_scales]]
	[[Category:Pages with mostly numerical content]]		[[Category:Pages with mostly numerical content]]
	[[Category:Zeta peak indexes]]		[[Category:Zeta peak indexes]]
	[[Category:42edo]]		[[Category:42edo]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

Lériendil: Text replacement - "Category:Tempered scales" to "Category:Tempered scales Category:Pages with mostly numerical content"

2025-06-24T00:36:57Z

Text replacement - "Category:Tempered scales" to "Category:Tempered scales Category:Pages with mostly numerical content"

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	[[Category:42-tone scales]]		[[Category:42-tone scales]]
	[[Category:Tempered scales]]		[[Category:Tempered scales]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Zeta peak indexes]]		[[Category:Zeta peak indexes]]
	[[Category:42edo]]		[[Category:42edo]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

Lériendil: Text replacement - "Category:Pages with mostly numerical content" to ""

2025-06-24T00:24:29Z

Text replacement - "Category:Pages with mostly numerical content" to ""

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	[[Category:42edo]]		[[Category:42edo]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]
	~~[[Category:Pages with mostly numerical content]]~~

Hkm at 23:32, 23 June 2025

2025-06-23T23:32:56Z

← Older revision		Revision as of 23:32, 23 June 2025
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	[[Category:42edo]]		[[Category:42edo]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]
			[[Category:Pages with mostly numerical content]]

Contribution at 20:28, 8 June 2025

2025-06-08T20:28:22Z

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	=== Harmonic series ===		=== Harmonic series ===
	191zpi performs well in the no-7s [[13-limit]], or alternatively in the [[dual-n\|dual-7]] 13-limit.		191zpi performs well in the no-7s [[13-limit]], or alternatively in the [[dual-n\|dual-7]] 13-limit.
	{{Harmonics in cet\|28.4443828021368\|intervals=prime\|title=Approximation of harmonics in ~~186zpi~~}}		{{Harmonics in cet\|28.4443828021368\|intervals=prime\|title=Approximation of harmonics in 191zpi}}

	== Notation ==		== Notation ==

ArrowHead294 at 15:37, 20 March 2025

2025-03-20T15:37:28Z

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	\| steps = 42.187591425251		\| steps = 42.187591425251
	\| step size = 28.444382802136		\| step size = 28.444382802136
	\| ~~tempered~~ height = 3.591963		\| height = 3.591963
	~~\| pure height = 2.688894~~
	\| integral = 0.481940		\| integral = 0.481940
	\| gap = 11.612694		\| gap = 11.612694

ArrowHead294 at 14:47, 20 March 2025

2025-03-20T14:47:03Z

← Older revision		Revision as of 14:47, 20 March 2025
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	'''191 zeta peak index''' (abbreviated '''191zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 191st [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].		'''191 zeta peak index''' (abbreviated '''191zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 191st [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].

	{~~\| class="wikitable"~~		{{ZPI
	~~! colspan="3"~~ \| ~~Tuning~~		\| zpi = 191
	~~! colspan~~=~~"3" \| Strength~~		\| steps = 42.187591425251
	~~! colspan="2"~~ \| ~~Closest EDO~~		\| step size = 28.444382802136
	~~! colspan~~=~~"2" \| Integer limit~~		\| tempered height = 3.591963
	\|-		\| pure height = 2.688894
	~~! ZPI~~		\| integral = 0.481940
	~~! Steps per octave~~		\| gap = 11.612694
	~~! Step size (cents)~~		\| edo = 42edo
	~~! Height~~		\| octave = 1194.6640776897
	~~! Integral~~		\| consistent = 3
	~~! Gap~~		\| distinct = 3
	~~! EDO~~		}}
	~~! Octave (cents)~~
	~~! Consistent~~
	~~! Distinct~~
	\|-
	~~\| [[191zpi]]~~
	\| 42.187591425251
	\| 28.444382802136
	\| 3.591963
	\| 0.481940
	\| 11.612694
	\| [[42edo]]
	\| 1194.6640776897
	\| 3
	\| 3
	\|}

	== Theory ==		== Theory ==

BudjarnLambeth: /* Theory */

2024-12-12T22:58:26Z

Theory

← Older revision		Revision as of 22:58, 12 December 2024
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	== Theory ==		== Theory ==
	~~''See also:~~ [[~~Table~~ of ~~stretched~~ 42edo ~~tunings~~]]''		191zpi can be used as an [[Octave shrinking\|octave-compressed]] version of [[42edo]], with a much better [[3/1]] and [[5/1]] compared to 42edo, but at the expense of a worse [[7/1]].

	~~191zpi can be used as an~~ [[~~Octave shrinking~~\|~~octave-compressed]] version of [[42edo~~]], ~~with a much better~~ [[~~3/1~~]] ~~and [[5/1]] compared to 42edo~~, but at the ~~expense of a worse [[7/1]]~~.		It is a kind of opposite twin to the scale [[zpi\|189zpi]], as they improve 42edo’s [[JI]] approximation by about the same amount, but in opposite directions (those harmonics which are slightly sharp in one are slightly flat in the other).

	It is ~~a kind~~ of ~~opposite twin to the scale~~ [[1ed28.7c\|~~APS720jot~~]]~~. 191zpi makes 42edo into a better 2.3.5.11.13 tuning, while APS720jot makes 42edo into a better 2.3.5.7.17 tuning. They both improve 42edo but in opposite directions~~.		191zpi’s step size is very close to that of [[1ed28.5c\|APS715jot]] and [[ed255/128#42ed255/128\|42ed255/128]].

	~~191zpi is almost identical to~~ [[~~ed255/128#42ed255/128\|42ed255/128]]. 191zpi does slightly better at approximating most harmonics but 42ed255/128 does slightly better at [[3/1~~]].		See [[Table of stretched 42edo tunings]] for more.

	=== Harmonic series ===		=== Harmonic series ===

BudjarnLambeth: /* Scales */ add subsets of mos scale

2024-12-09T06:48:25Z

Scales: add subsets of mos scale

← Older revision		Revision as of 06:48, 9 December 2024
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	==Scales==		==Scales==
	* Eugene/Tritikleismic[9]: 3 8 3 3 8 3 3 8 3
	* Eugene/Tritikleismic[15]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3		;[[MOS scale]]s
	* Lemba[16]: 3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3		* Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3'''
	* Qeema/Skateboard[15]: 2 5 2 2 2 5 2 2 2 5 2 2 2 5 2		* Eugene/Tritikleismic[15]: '''3 3 2 3 3 3 3 2 3 3 3 3 2 3 3'''
	* Qeema/Skateboard[19]: 2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2		* Lemba[16]: '''3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3'''
	* Seville/Sevond[14] 1st mode: 1 5 1 5 1 5 1 5 1 5 1 5 1 5		* Qeema/Skateboard[15]: '''2 5 2 2 2 5 2 2 2 5 2 2 2 5 2'''
	* Seville/Sevond[14] 2nd mode: 5 1 5 1 5 1 5 1 5 1 5 1 5 1		* Qeema/Skateboard[19]: '''2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2'''
	* Seville/Sevond[21]: 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4		* Seville/Sevond[14] 1st mode: '''1 5 1 5 1 5 1 5 1 5 1 5 1 5'''
			* Seville/Sevond[14] 2nd mode: '''5 1 5 1 5 1 5 1 5 1 5 1 5 1'''
			* Seville/Sevond[21]: '''1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4'''


			; Subsets of MOS scales
			''(Names used are [[Template:Idiosyncratic\|idiosyncratic]].)''
			* Eugene/Tritikleismic[9]
			** Groovy aeolian pentatonic: '''11 6 8 3 14'''
			** [[Otonal]] mixolydian pentatonic: '''14 3 8 11 6'''
			** Pseudo-[[equipentatonic]]: '''11 6 8 6 11'''
			** Septimal melodic minor pentatonic: '''8 3 14 14 3'''
			** Septimal Picardy pentatonic: '''8 6 11 3 14'''
			** Undecimal lydian-aeolian pentatonic: '''8 14 3 11 6'''
			** Yokai pentatonic: '''3 14 8 3 14'''

	== Instruments ==		== Instruments ==