71zpi - Revision history

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

2025-08-20T01:07:29Z

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

← Older revision		Revision as of 01:07, 20 August 2025
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	! Cents		! Cents
	! Ratios		! Ratios
	! Ups and ~~Downs Notation~~		! Ups and downs notation
	! Step		! Step
	\|-		\|-
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	! Cents		! Cents
	! Ratios		! Ratios
	! Ups and ~~Downs Notation~~		! Ups and downs notation
	! Step		! Step
	\|-		\|-

ArrowHead294 at 15:36, 20 March 2025

2025-03-20T15:36:48Z

← Older revision		Revision as of 15:36, 20 March 2025
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	'''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 71st [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].		'''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 71st [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].

	{~~\| class="wikitable"~~		{{ZPI
	~~! colspan="3"~~ \| ~~Tuning~~		\| zpi = 71
	~~! colspan~~=~~"3" \| Strength~~		\| steps = 20.2248393119540
	~~! colspan="2"~~ \| ~~Closest EDO~~		\| step size = 59.3329806724710
	~~! colspan~~=~~"2" \| Integer limit~~		\| height = 3.531097
	\|-		\| integral = 0.613581
	~~! ZPI~~		\| gap = 12.986080
	~~! Steps per octave~~		\| octave = 1186.65961344942
	~~! Step size (cents)~~		\| consistent = 6
	~~! Height~~		\| distinct = 6
	~~! Integral~~		}}
	~~! Gap~~
	~~! EDO~~
	~~! Octave (cents)~~
	~~! Consistent~~
	~~! Distinct~~
	\|-
	~~\| [[71zpi]]~~
	\| 20.2248393119540
	\| 59.3329806724710
	\| 3.531097
	\| 0.613581
	\| 12.986080
	\| ~~[[20edo]]~~
	\| 1186.65961344942
	\| 6
	\| 6
	\|}

	[[File:71zpi.png\|thumb\|right\|The Riemann zeta function around 71zpi]]		[[File:71zpi.png\|thumb\|right\|The Riemann zeta function around 71zpi]]

Recentlymaterialized: reflecting convention to lower the integer-limit to the highest integer that is supported by the subgroup and is not a multiple of the equave, when measuring consistency of an ET that equally divides a just equave

2024-09-12T01:29:08Z

reflecting convention to lower the integer-limit to the highest integer that is supported by the subgroup and is not a multiple of the equave, when measuring consistency of an ET that equally divides a just equave

← Older revision		Revision as of 01:29, 12 September 2024
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	== Theory ==		== Theory ==
	'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes\|no-2s zeta peak EDT]] (consistent in the ~~no-2s~~ [[Odd_limit#Nonoctave_equaves\|21-~~throdd~~-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed\|the no-2s peak]] at 59.271105 cents.		'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes\|no-2s zeta peak EDT]] (consistent in the [[Odd_limit#Nonoctave_equaves\|no-2s 19-integer-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed\|the no-2s peak]] at 59.271105 cents.

	71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of [[20edo]].		71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of [[20edo]].

Contribution at 21:54, 12 August 2024

2024-08-12T21:54:04Z

Show changes

Contribution: /* Record on the Riemann zeta function with prime 2 removed */

2024-08-12T13:49:45Z

Record on the Riemann zeta function with prime 2 removed

← Older revision		Revision as of 13:49, 12 August 2024
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	{{Harmonics in cet\|59.2711049295348\|columns=18\|start=16\|title=Approximation of harmonics in 71zpi with prime 2 removed }}		{{Harmonics in cet\|59.2711049295348\|columns=18\|start=16\|title=Approximation of harmonics in 71zpi with prime 2 removed }}

	{\| class="wikitable center-1 right-2 left-3 center-4 center-5"		{\| class="wikitable center-1 right-2 left-3 center-4 center-5 mw-collapsible mw-collapsed"
			\|+ style="white-space:nowrap" \| Intervals in 71zpi with prime 2 removed
	\|-		\|-
	~~\|+ style="white-space:nowrap" \| Intervals in 71zpi with prime 2 removed~~
	\| colspan="3" style="text-align:left;" \| JI ratios are comprised of 32-integer limit ratios,<br>and are stylized as follows to indicate their accuracy:		\| colspan="3" style="text-align:left;" \| JI ratios are comprised of 32-integer limit ratios,<br>and are stylized as follows to indicate their accuracy:
	* '''<u>Bold Underlined:</u>''' relative error < 8.333 %		* '''<u>Bold Underlined:</u>''' relative error < 8.333 %

Contribution: /* Record on the Riemann zeta function with prime 2 removed */

2024-08-12T11:11:47Z

Record on the Riemann zeta function with prime 2 removed

← Older revision		Revision as of 11:11, 12 August 2024
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	== Record on the Riemann zeta function with prime 2 removed ==		== Record on the Riemann zeta function with prime 2 removed ==
	'''[[71zpi]]''' sets a height record on the Riemann zeta function with prime 2 removed. The previous record is [[53zpi]] and the next one is [[93zpi]]. It is important to highlight that the optimal equal tunings obtained by excluding the prime ~~numbers~~ 2 from the Riemann zeta function differs slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.		'''[[71zpi]]''' sets a height record on the Riemann zeta function with prime 2 removed. The previous record is [[53zpi]] and the next one is [[93zpi]]. It is important to highlight that the optimal equal tunings obtained by excluding the prime number 2 from the Riemann zeta function differs slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.
	{\| class="wikitable"		{\| class="wikitable"
	! colspan="6" \|Unmodified Riemann zeta function		! colspan="6" \|Unmodified Riemann zeta function

Contribution: /* 71zpi with prime 2 removed */

2024-08-12T11:09:27Z

71zpi with prime 2 removed

← Older revision		Revision as of 11:09, 12 August 2024
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	* <small><small>Small Small:</small></small> relative error < 41.667 %		* <small><small>Small Small:</small></small> relative error < 41.667 %
	* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %		* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
	\| colspan="2" style="text-align:right;" \| <center>'''⟨81 128] at every 4 steps'''</center><br>[[2/1\|Octave]] = 81<br>[[3/2\|Fifth]] = 47<br>[[9/8\|Whole tone]] = 13 steps<br>[[256/243\|Limma]] = 8 steps<br>[[2187/2048\|Apotome]] = 5 steps		\| colspan="2" style="text-align:right;" \| <center>'''⟨81 128] at every 4 steps'''</center><br>[[2/1\|Octave]] = 81 steps<br>[[3/2\|Fifth]] = 47 steps<br>[[9/8\|Whole tone]] = 13 steps<br>[[256/243\|Limma]] = 8 steps<br>[[2187/2048\|Apotome]] = 5 steps
	\|-		\|-
	! Degree		! Degree

Contribution: /* 71zpi with prime 2 removed */

2024-08-12T11:05:33Z

71zpi with prime 2 removed

Show changes

Contribution: /* Record on the Riemann zeta function with prime 2 removed */

2024-08-12T10:57:44Z

Record on the Riemann zeta function with prime 2 removed

Show changes

Contribution at 10:48, 12 August 2024

2024-08-12T10:48:54Z

← Older revision		Revision as of 10:48, 12 August 2024
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	== Theory ==		== Theory ==
	'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes\|no-2s zeta peak EDT]] (consistent in the no-2s [[Odd_limit#Nonoctave_equaves\|21-throdd-limit]]), but with less extreme stretch than [[#~~No-2s_zeta_peak~~\|the no-2s peak]] at 59.271105 cents.		'''71zpi''' marks the most prominent [[zeta peak index]] in the [[vicinity]] of [[20edo]]. While [[70zpi]] is the nearest peak to [[20edo]] and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength. 71zpi may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes\|no-2s zeta peak EDT]] (consistent in the no-2s [[Odd_limit#Nonoctave_equaves\|21-throdd-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed\|the no-2s peak]] at 59.271105 cents.

	71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of [[20edo]].		71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of [[20edo]].