207zpi - Revision history

Pizzabagel shef at 05:53, 8 April 2026

2026-04-08T05:53:59Z

← Older revision		Revision as of 05:53, 8 April 2026
Line 4:		Line 4:
	== Theory ==		== Theory ==

	207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no harmonics (excluding powers of 2/1) below [[16/1]] exceeding 7.5¢ [[error]].		207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no harmonics (excluding powers of 2/1) below [[16/1]] exceeding 7.5¢ absolute [[error]].

	Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).		Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).

Pizzabagel shef at 05:48, 8 April 2026

2026-04-08T05:48:00Z

← Older revision		Revision as of 05:48, 8 April 2026
Line 4:		Line 4:
	== Theory ==		== Theory ==

	207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no harmonics, excluding powers of 2/1, below [[16/1]] exceeding 7.5¢ [[error]].		207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no harmonics (excluding powers of 2/1) below [[16/1]] exceeding 7.5¢ [[error]].

	Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).		Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).

Pizzabagel shef at 05:47, 8 April 2026

2026-04-08T05:47:33Z

← Older revision		Revision as of 05:47, 8 April 2026
Line 4:		Line 4:
	== Theory ==		== Theory ==

	207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no ~~non-~~powers of 2/1 below [[16/1]] exceeding 7.5¢ [[error]].		207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no harmonics, excluding powers of 2/1, below [[16/1]] exceeding 7.5¢ [[error]].

	Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).		Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).

BudjarnLambeth: categorise uncategorised page

2026-04-07T04:49:07Z

categorise uncategorised page

← Older revision		Revision as of 04:49, 7 April 2026
Line 101:		Line 101:
	1204.2886375485		1204.2886375485
	</pre>		</pre>

			[[Category:Zeta peak indexes]]

Pizzabagel shef at 04:11, 7 April 2026

2026-04-07T04:11:52Z

← Older revision		Revision as of 04:11, 7 April 2026
Line 1:		Line 1:
	207 Zeta Peak Index (abbreviated ~~207zpi~~) is the [[Equal-step tuning\|equal-step]] [[tuning system]] derived from the 207th peak of the [https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning Riemann Zeta Function].		207 Zeta Peak Index (abbreviated 207[[zpi]]) is the [[Equal-step tuning\|equal-step]] [[tuning system]] derived from the 207th peak of the [https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning Riemann Zeta Function].
	[[File:Riemann Zeta Function around 45edo, Desmos.png\|thumb\|The Riemann Zeta Function around 45edo. The highest peak, to the left of 45, corresponds to 207zpi, demonstrating its relative strength as a tuning. ]]		[[File:Riemann Zeta Function around 45edo, Desmos.png\|thumb\|The Riemann Zeta Function around 45edo. The highest peak, to the left of 45, corresponds to 207zpi, demonstrating its relative strength as a tuning. ]]

Pizzabagel shef at 04:00, 7 April 2026

2026-04-07T04:00:18Z

← Older revision		Revision as of 04:00, 7 April 2026
Line 46:		Line 46:
	\| -9.607		\| -9.607
	\|}		\|}

			== Scala file ==

	<pre>		<pre>

Yourmusic Productions: fix scala file formatting

2026-04-06T07:49:50Z

fix scala file formatting

← Older revision		Revision as of 07:49, 6 April 2026
Line 47:		Line 47:
	\|}		\|}

	~~== Scala file text ==~~		<pre>

	! 207zpi.scl		! 207zpi.scl
			!
			207 zpi
	45		45
			!
	26.7619697233		26.7619697233
	53.5239394466		53.5239394466
Line 95:		Line 97:
	1150.7646981019		1150.7646981019
	1177.5266678252		1177.5266678252
	1204.2886375485		1204.2886375485
			</pre>

Pizzabagel shef at 07:12, 6 April 2026

2026-04-06T07:12:24Z

← Older revision		Revision as of 07:12, 6 April 2026
Line 4:		Line 4:
	== Theory ==		== Theory ==

	207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It substantially ~~improves~~ on 45edo's [[harmonic]] accuracy, with no non-powers of 2/1 below [[16/1]] exceeding 7.5¢ [[error]].		207zpi is the strongest [[Zeta peak index\|zeta peak]] in the vicinity of [[45edo]], and serves as a [https://en.xen.wiki/w/Stretched_and_compressed_tuning stretched-octave] version thereof ([[2/1]] ≈ 1204.289¢). It improves substantially on 45edo's [[harmonic]] accuracy, with no non-powers of 2/1 below [[16/1]] exceeding 7.5¢ [[error]].

	Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).		Crucially, due to the octave stretch, the 207zpi [[Patent val\|patent vals]] of [[9/1]] and [[15/1]] have the same values as their "b-vals" (the second best approximation of a [[Just Intonation]] interval in a tuning system) in 45edo ([https://en.xen.wiki/w/Interval_class ''k'']=142 and ''k''=175 steps for both systems); i.e. the already-sharp direct approximations of those harmonics in 45edo (''k''=143 and ''k''=176), which are not found within its [[flattone]] [[Diatonic scale\|diatonic scale]], are "pushed out of the way" by the octave stretch within 207zpi. This means that the direct approximations of 9/1, 15/1 are now mapped to the diatonic scale, though this is not the case for their octave-reduced counterparts of [[9/8]] and [[15/8]] (''k''=8 and ''k''=41 in both systems).

Pizzabagel shef at 07:11, 6 April 2026

2026-04-06T07:11:33Z

← Older revision		Revision as of 07:11, 6 April 2026
Line 1:		Line 1:
	207 Zeta Peak Index (abbreviated 207zpi) is the [[Equal-step tuning\|equal-step]] [[tuning system]] derived from the 207th peak of the [https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning Riemann Zeta Function].		207 Zeta Peak Index (abbreviated 207zpi) is the [[Equal-step tuning\|equal-step]] [[tuning system]] derived from the 207th peak of the [https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning Riemann Zeta Function].
	[[File:Riemann Zeta Function around 45edo, Desmos.png\|thumb\|The Riemann Zeta Function around 45edo. The highest peak, ~~right~~ to the left of 45, corresponds to 207zpi, demonstrating its relative strength as a tuning. ]]		[[File:Riemann Zeta Function around 45edo, Desmos.png\|thumb\|The Riemann Zeta Function around 45edo. The highest peak, to the left of 45, corresponds to 207zpi, demonstrating its relative strength as a tuning. ]]

	== Theory ==		== Theory ==

Pizzabagel shef: The addition of the .scl file for this tuning.

2026-04-06T05:41:33Z

The addition of the .scl file for this tuning.

← Older revision		Revision as of 05:41, 6 April 2026
Line 46:		Line 46:
	\| -9.607		\| -9.607
	\|}		\|}

			== Scala file text ==

			! 207zpi.scl
			45
			26.7619697233
			53.5239394466
			80.2859091699
			107.0478788932
			133.8098486165
			160.5718183398
			187.3337880631
			214.0957577864
			240.8577275097
			267.6196972330
			294.3816669563
			321.1436366796
			347.9056064029
			374.6675761262
			401.4295458495
			428.1915155728
			454.9534852961
			481.7154550194
			508.4774247427
			535.2393944660
			562.0013641893
			588.7633339126
			615.5253036359
			642.2872733592
			669.0492430825
			695.8112128058
			722.5731825291
			749.3351522524
			776.0971219757
			802.8590916990
			829.6210614223
			856.3830311456
			883.1450008689
			909.9069705922
			936.6689403155
			963.4309100388
			990.1928797621
			1016.9548494854
			1043.7168192087
			1070.4787889320
			1097.2407586553
			1124.0027283786
			1150.7646981019
			1177.5266678252
			1204.2886375485