186zpi - Revision history

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

2025-08-20T01:07:28Z

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
	\|-		\|-

ArrowHead294: /* Intervals and notation */

2025-03-20T15:38:11Z

Intervals and notation

← Older revision		Revision as of 15:38, 20 March 2025
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	{\| class="wikitable center-1 right-2 left-3 center-4 center-5"		{\| class="wikitable center-1 right-2 left-3 center-4 center-5"
	\|+ style="white-space:nowrap" \| Intervals in 186zpi		\|+ style="font-size: 105%; white-space: nowrap;" \| Intervals in 186zpi
	\|-		\|-
	\| 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:

ArrowHead294 at 15:37, 20 March 2025

2025-03-20T15:37:04Z

← Older revision		Revision as of 15:37, 20 March 2025
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	\| steps = 41.3438354846780		\| steps = 41.3438354846780
	\| step size = 29.0248832971658		\| step size = 29.0248832971658
	\| ~~tempered~~ height = 1.876590		\| height = 1.876590
	~~\| pure height =~~
	\| integral = 0.241233		\| integral = 0.241233
	\| gap = 11.567493		\| gap = 11.567493

ArrowHead294 at 15:23, 20 March 2025

2025-03-20T15:23:45Z

← Older revision		Revision as of 15:23, 20 March 2025
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	'''186 zeta peak index''' (abbreviated '''186zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 186st [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].		'''186 zeta peak index''' (abbreviated '''186zpi'''), is the [[Equal-step tuning\|equal-step]] [[tuning system]] obtained from the 186st [[Zeta peak index\|peak]] of the [[The Riemann zeta function and tuning\|Riemann zeta function]].

	{~~\| class="wikitable"~~		{{ZPI
	~~! colspan="3"~~ \| ~~Tuning~~		\| zpi = 186
	~~! colspan~~=~~"3" \| Strength~~		\| steps = 41.3438354846780
	~~! colspan="2"~~ \| ~~Closest EDO~~		\| step size = 29.0248832971658
	~~! colspan~~=~~"2" \| Integer limit~~		\| tempered height = 1.876590
	\|-		\| pure height =
	~~! ZPI~~		\| integral = 0.241233
	~~! Steps per octave~~		\| gap = 11.567493
	~~! Step size (cents)~~		\| edo = 41edo
	~~! Height~~		\| octave = 1190.02021518380
	~~! Integral~~		\| consistent = 2
	~~! Gap~~		\| distinct = 2
	~~! EDO~~		}}
	~~! Octave (cents)~~
	~~! Consistent~~
	~~! Distinct~~
	\|-
	~~\| [[186zpi]]~~
	\| 41.3438354846780
	\| 29.0248832971658
	\| 1.876590
	\| 0.241233
	\| 11.567493
	\| [[41edo]]
	\| 1190.02021518380
	\| 2
	\| 2
	\|}

	== Theory ==		== Theory ==

	=== Record on the Riemann zeta function with primes 2 and 3 removed ===		=== Record on the Riemann zeta function with primes 2 and 3 removed ===
	'''186zpi''' sets a height record on the Riemann zeta function with primes 2 and 3 removed. The previous record is [[125zpi]] and the next one is [[565zpi]]. It is important to highlight that the optimal equal tunings obtained by excluding the prime numbers 2 and 3 from the Riemann zeta function differs very slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.		'''186zpi''' sets a height record on the Riemann zeta function with primes 2 and 3 removed. The previous record is [[125zpi]] and the next one is [[565zpi]]. It is important to highlight that the optimal equal tunings obtained by excluding the prime numbers 2 and 3 from the Riemann zeta function differs very 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="5" \|Riemann zeta function with primes 2 and 3 removed		! colspan="6" \| Unmodified Riemann zeta function
			! colspan="5" \| Riemann zeta function with primes 2 and 3 removed
	\|-		\|-
	! colspan="3" \| Tuning		! colspan="3" \| Tuning
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	\| 30.6006474885974		\| 30.6006474885974
	\| 39.2148564976330		\| 39.2148564976330
	\|1.468164		\| 1.468164
	\| [[31edo]]		\| [[31edo]]
	\| 1215.66055142662		\| 1215.66055142662
	\| 30.5974484926723		\| 30.5974484926723
	\| 39.2189564527704		\| 39.2189564527704
	\|3.769318		\| 3.769318
	\| [[31edo]]		\| [[31edo]]
	\| 1215.78765003588		\| 1215.78765003588
	\|-		\|-
	\|[[186zpi]]		\| [[186zpi]]
	\|41.3438354846780		\| 41.3438354846780
	\|29.0248832971658		\| 29.0248832971658
	\|1.876590		\| 1.876590
	\|[[41edo]]		\| [[41edo]]
	\|1190.02021518380		\| 1190.02021518380
	\|41.3477989230936		\| 41.3477989230936
	\|29.0221010852836		\| 29.0221010852836
	\|4.469823		\| 4.469823
	\|[[41edo]]		\| [[41edo]]
	\|1189.90614449663		\| 1189.90614449663
	\|-		\|-
	\|[[565zpi]]		\| [[565zpi]]
	\|98.6209462564991		\| 98.6209462564991
	\|12.1678005084130		\| 12.1678005084130
	\|2.305330		\| 2.305330
	\|[[99edo]]		\| [[99edo]]
	\|1204.61225033289		\| 1204.61225033289
	\|98.6257548378926		\| 98.6257548378926
	\|12.1672072570942		\| 12.1672072570942
	\|4.883729		\| 4.883729
	\|[[99edo]]		\| [[99edo]]
	\|1204.55351845233		\| 1204.55351845233
	\|}		\|}

Contribution at 22:09, 12 August 2024

2024-08-12T22:09:41Z

← Older revision		Revision as of 22:09, 12 August 2024
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	\|+ style="white-space:nowrap" \| Intervals in 186zpi		\|+ style="white-space:nowrap" \| Intervals in 186zpi
	\|-		\|-
	\| 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 %
	* '''Bold:''' relative error < 16.667 %		* '''Bold:''' relative error < 16.667 %
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	=== Interval mappings ===		=== Interval mappings ===

	The following tables show how 32-integer-limit intervals are represented in 186zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.		The following tables show how [[32-integer-limit]] intervals are represented in 186zpi. 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"

Contribution at 22:00, 12 August 2024

2024-08-12T22:00:09Z

Show changes

Contribution: /* Intervals and notation */

2024-08-11T16:23:33Z

Intervals and notation

← Older revision		Revision as of 16:23, 11 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>~~⟨12 19 28]~~ '''at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps		\| colspan="2" style="text-align:right;" \| <center>'''⟨124 196] at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps
	\|-		\|-
	! Degree		! Degree

Contribution: /* Intervals and notation */

2024-08-11T16:18:53Z

Intervals and notation

← Older revision		Revision as of 16:18, 11 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>'''at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps		\| colspan="2" style="text-align:right;" \| <center>⟨12 19 28] '''at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps
	\|-		\|-
	! Degree		! Degree

Contribution: /* Intervals and notation */

2024-08-11T16:18:17Z

Intervals and notation

← Older revision		Revision as of 16:18, 11 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>'''~~<nowiki><124 196\|</nowiki>~~ at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps		\| colspan="2" style="text-align:right;" \| <center>'''at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps
	\|-		\|-
	! Degree		! Degree

Contribution: /* Intervals and notation */

2024-08-11T16:16:05Z

Intervals and notation

← Older revision		Revision as of 16:16, 11 August 2024
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	{\| class="wikitable center-1 right-2 left-3 center-4 center-5"		{\| class="wikitable center-1 right-2 left-3 center-4 center-5"
	\|-		\|-
			\|+ style="white-space:nowrap" \| Intervals in 186zpi
	\| 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 %
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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>'''~~124b~~ at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps		\| colspan="2" style="text-align:right;" \| <center>'''<nowiki><124 196\|</nowiki> at every 3 steps'''</center><br>[[9/8\|Whole tone]] = 20 steps<br>[[256/243\|Limma]] = 12 steps<br>[[2187/2048\|Apotome]] = 8 steps
	\|-		\|-
	! Degree		! Degree