34zpi - Revision history

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

2025-08-20T01:07:38Z

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

2025-03-20T15:37:54Z

← Older revision		Revision as of 15:37, 20 March 2025
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	\| steps = 12.0231830072926		\| steps = 12.0231830072926
	\| step size = 99.8071807833375		\| step size = 99.8071807833375
	\| ~~tempered~~ height = 5.193290		\| height = 5.193290
	~~\| pure height = 5.084467~~
	\| integral = 1.269599		\| integral = 1.269599
	\| gap = 15.899282		\| gap = 15.899282
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	== Intervals ==		== Intervals ==

	{\| class="wikitable center-1 right-2 left-3 center-4"		{\| class="wikitable center-1 right-2 left-3 center-4"
	\|+ style="white-space:nowrap" \| Intervals in 34zpi		\|+ style="font-size: 105%; white-space: nowrap;" \| Intervals in 34zpi
	\|-		\|-
	\| colspan="3" style="text-align:left;" \| JI ratios are comprised of 16-integer-limit ratios,<br>and are stylized as follows to indicate their accuracy:		\| colspan="3" style="text-align:left;" \| JI ratios are comprised of 16-integer-limit ratios,<br>and are stylized as follows to indicate their accuracy:

2025-03-20T15:10:13Z

← Older revision		Revision as of 15:10, 20 March 2025
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	'''34 zeta peak index''' (abbreviated '''34zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 34th [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].		'''34 zeta peak index''' (abbreviated '''34zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 34th [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].

	{~~\|class="wikitable"~~		{{ZPI
	~~!colspan="3"~~\|~~Tuning~~		\| zpi = 34
	~~!colspan~~=~~"3"\|Strength~~		\| steps = 12.0231830072926
	~~!colspan="2"~~\|~~Closest EDO~~		\| step size = 99.8071807833375
	~~!colspan~~=~~"2"\|Integer limit~~		\| tempered height = 5.193290
	\|-		\| pure height = 5.084467
	~~!ZPI~~		\| integral = 1.269599
	~~!Steps per octave~~		\| gap = 15.899282
	~~!Step size (cents)~~		\| edo = 12edo
	~~!Height~~		\| octave = 1197.68616940005
	~~!Integral~~		\| consistent = 10
	~~!Gap~~		\| distinct = 6
	~~!EDO~~		}}
	~~!Octave (cents)~~
	~~!Consistent~~
	~~!Distinct~~
	\|-
	~~\|34zpi~~
	\|12.0231830072926
	\|99.8071807833375
	\|5.193290
	\|1.269599
	\|15.899282
	\|[[12edo]]
	\|1197.68616940005
	\|10
	\|6
	\|}

	== Intervals ==		== Intervals ==

2024-08-12T21:33:52Z

Approximation to JI

2024-08-12T21:30:23Z

2024-08-12T21:29:12Z

Approximation to JI

2024-08-12T21:28:12Z

← Older revision		Revision as of 21:28, 12 August 2024
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	\|}		\|}

	== ~~Intervals~~ ==		== Approximation to JI ==

	{\| class="wikitable center-1 right-2 left-3 center-4"		{\| class="wikitable center-1 right-2 left-3 center-4"
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	== Approximation to JI ==		== Approximation to JI ==

	=== Interval mappings ===		=== Interval mappings ===

	The following tables show how 16-integer-limit intervals are represented in 34zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.		The following tables show how 16-integer-limit intervals are represented in 34zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.

	{\| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"		{\| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
	\|+ style="white-space: nowrap;" \| 16-integer-limit intervals in 34zpi~~<br>~~by direct approximation		\|+ style="white-space: nowrap;" \| 16-integer-limit intervals in 34zpi (by direct approximation)
	\|-		\|-
	! Ratio		! Ratio

2024-08-12T17:45:33Z

Approximation to JI

← Older revision		Revision as of 17:45, 12 August 2024
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	{\| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"		{\| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
	\|+ style="white-space: nowrap;" \| 16-integer-limit intervals in 34zpi<br>(by direct approximation)		\|+ style="white-space: nowrap;" \| 16-integer-limit intervals in 34zpi<br>by direct approximation
	\|-		\|-
	! Ratio		! Ratio

2024-08-12T17:43:59Z

Interval mappings

← Older revision		Revision as of 17:43, 12 August 2024
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	{\| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"		{\| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
	\|+ style="white-space: nowrap;" \| 16-integer-limit intervals in 34zpi (by direct approximation)		\|+ style="white-space: nowrap;" \| 16-integer-limit intervals in 34zpi<br>(by direct approximation)
	\|-		\|-
	! Ratio		! Ratio

2024-08-12T17:21:02Z

Intervals

← Older revision		Revision as of 17:21, 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 %
	\| style="text-align:right;" \| <center>'''⟨12 19]'''</center~~><br>[[2/1\|Octave]] = 12 steps<br>[[3/2\|Fifth]] = 7 steps<br~~><br>[[9/8\|Whole tone]] = 2 steps<br>[[256/243\|Limma]] = 1 step<br>[[2187/2048\|Apotome]] = 1 step		\| style="text-align:right;" \| <center>'''⟨12 19]'''</center><br>[[9/8\|Whole tone]] = 2 steps<br>[[256/243\|Limma]] = 1 step<br>[[2187/2048\|Apotome]] = 1 step
	\|-		\|-
	! Degree		! Degree