45zpi - Revision history

Fredg999: Text replacement - "Ups and Downs Notation" to "Ups and downs notation"

2025-08-20T01:07:37Z

Text replacement - "Ups and Downs Notation" to "Ups and downs notation"

2025-03-20T15:38:39Z

2025-03-20T15:17:32Z

← Older revision		Revision as of 15:17, 20 March 2025
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	'''45 zeta peak index''' (abbreviated '''45zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 45th [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].		'''45 zeta peak index''' (abbreviated '''45zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 45th [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].

	{~~\|class="wikitable"~~		{{ZPI
	~~!colspan="3"~~\|~~Tuning~~		\| zpi = 45
	~~!colspan~~=~~"3"\|Strength~~		\| steps = 14.5944346577250
	~~!colspan="2"~~\|~~Closest EDO~~		\| step size = 82.2231232756126
	~~!colspan~~=~~"2"\|Integer limit~~		\| tempered height = 2.097730
	\|-		\| pure height = 4.390681
	~~!ZPI~~		\| integral = 0.344839
	~~!Steps per octave~~		\| gap = 10.594800
	~~!Step size (cents)~~		\| edo = 15edo
	~~!Height~~		\| octave = 1233.34684913419
	~~!Integral~~		\| consistent = 2
	~~!Gap~~		\| distinct = 2
	~~!EDO~~		}}
	~~!Octave (cents)~~
	~~!Consistent~~
	~~!Distinct~~
	\|-
	~~\|[[45zpi]]~~
	\|14.5944346577250
	\|82.2231232756126
	\|2.097730
	\|0.344839
	\|10.594800
	\|[[15edo]]
	\|1233.34684913419
	\|2
	\|2
	\|}

	== Theory ==		== Theory ==

2024-08-12T21:42:10Z

2024-08-12T21:39:43Z

2024-08-11T22:55:17Z

Approximation to JI

2024-08-11T22:54:40Z

Intervals

2024-08-11T22:54:25Z

← Older revision		Revision as of 22:54, 11 August 2024
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	{{Harmonics in cet\|82.2231232756126\|columns=16\|start=16\|title=Approximation of harmonics in 45zpi}}		{{Harmonics in cet\|82.2231232756126\|columns=16\|start=16\|title=Approximation of harmonics in 45zpi}}

			== Intervals ==

			== Approximation to JI ==
	{{Stub}}		{{Stub}}

	[[Category:Zeta peak indexes]]		[[Category:Zeta peak indexes]]

2024-05-03T13:34:39Z

← Older revision		Revision as of 13:34, 3 May 2024
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	'''45 zeta peak index''' (abbreviated '''45zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 45th peak of the [[The Riemann zeta function and tuning\|Riemann zeta function]].		'''45 zeta peak index''' (abbreviated '''45zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 45th [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].

	{\|class="wikitable"		{\|class="wikitable"

2024-04-23T09:09:40Z

← Older revision		Revision as of 09:09, 23 April 2024
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	45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications.		45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications.

	No other [[Zeta peak index\|zeta peak indexes]] exhibit both a larger octave error and greater zeta ~~strength~~ than 45zpi.		No other [[Zeta peak index\|zeta peak indexes]] exhibit both a larger octave error and greater zeta height than 45zpi.

	45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities.		45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities.