User:CompactStar/Super-pitch - Revision history

Fredg999: Fredg999 moved page Super-pitch to User:CompactStar/Super-pitch over redirect: OR-3 (idiosyncratic framework and terminology)

2025-03-26T19:22:11Z

Fredg999 moved page Super-pitch to User:CompactStar/Super-pitch over redirect: OR-3 (idiosyncratic framework and terminology)

← Older revision	Revision as of 19:22, 26 March 2025
(No difference)

BudjarnLambeth: Niche -> mathematical interest, because I think “mathematical interest” is more accurate, but feel free to revert if you disagree, I’m not confident

2025-03-26T06:32:32Z

Niche -> mathematical interest, because I think “mathematical interest” is more accurate, but feel free to revert if you disagree, I’m not confident

← Older revision		Revision as of 06:32, 26 March 2025
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	{{~~niche~~}}		{{Mathematical interest}}
	'''Super-pitch'''{{idiosyncratic}} is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.		'''Super-pitch'''{{idiosyncratic}} is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.

BudjarnLambeth: Niche

2025-03-18T03:01:33Z

Niche

← Older revision		Revision as of 03:01, 18 March 2025
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	{{~~Novelty~~}}		{{niche}}
	'''Super-pitch'''{{idiosyncratic}} is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.		'''Super-pitch'''{{idiosyncratic}} is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.

Fredg999: Credit CompactStar for name proposal

2023-12-12T19:16:08Z

Credit CompactStar for name proposal

← Older revision		Revision as of 19:16, 12 December 2023
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	The super-logarithm is traditionally defined the number of times a logarithm must be iterated to get to 1. For example, slog<sub>b</sub>(1) = 0, slog<sub>b</sub>(b) = 1, slog<sub>b</sub>(b<sup>b</sup>) = 2, slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3, and so on. This definition only allows for inputs of the form 1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, etc., although there are various continuous extensions of it for other outputs (most commonly the linear and quadratic approximations, as mentioned on the Wikipedia article) which have differing definitions.		The super-logarithm is traditionally defined the number of times a logarithm must be iterated to get to 1. For example, slog<sub>b</sub>(1) = 0, slog<sub>b</sub>(b) = 1, slog<sub>b</sub>(b<sup>b</sup>) = 2, slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3, and so on. This definition only allows for inputs of the form 1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, etc., although there are various continuous extensions of it for other outputs (most commonly the linear and quadratic approximations, as mentioned on the Wikipedia article) which have differing definitions.

	There is, notably, one extension for complex numbers developed by Kneser, which so far seems the best when implemented to the reals.		There is, notably, one extension for complex numbers developed by Kneser, which so far seems the best when implemented to the reals.

			The term "super-pitch" was proposed by [[User:CompactStar\|CompactStar]].

	== "Super-pitch equivalents" of different concepts ==		== "Super-pitch equivalents" of different concepts ==

Fredg999: Mark term as i

2023-12-12T18:57:49Z

Mark term as i

← Older revision		Revision as of 18:57, 12 December 2023
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	{{Novelty}}		{{Novelty}}
	'''Super-pitch''' is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.		'''Super-pitch'''{{idiosyncratic}} is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.

	The super-logarithm is traditionally defined the number of times a logarithm must be iterated to get to 1. For example, slog<sub>b</sub>(1) = 0, slog<sub>b</sub>(b) = 1, slog<sub>b</sub>(b<sup>b</sup>) = 2, slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3, and so on. This definition only allows for inputs of the form 1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, etc., although there are various continuous extensions of it for other outputs (most commonly the linear and quadratic approximations, as mentioned on the Wikipedia article) which have differing definitions.		The super-logarithm is traditionally defined the number of times a logarithm must be iterated to get to 1. For example, slog<sub>b</sub>(1) = 0, slog<sub>b</sub>(b) = 1, slog<sub>b</sub>(b<sup>b</sup>) = 2, slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3, and so on. This definition only allows for inputs of the form 1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, etc., although there are various continuous extensions of it for other outputs (most commonly the linear and quadratic approximations, as mentioned on the Wikipedia article) which have differing definitions.

Fredg999: Marked as novelty (at least for as long as super-pitch doesn't have evidence of specific psychoacoustic effects)

2023-07-25T05:45:16Z

Marked as novelty (at least for as long as super-pitch doesn't have evidence of specific psychoacoustic effects)

← Older revision		Revision as of 05:45, 25 July 2023
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			{{Novelty}}
	'''Super-pitch''' is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.		'''Super-pitch''' is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.

CompactStar at 03:36, 22 July 2023

2023-07-22T03:36:20Z

← Older revision		Revision as of 03:36, 22 July 2023
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	=== Individual pages for EDSO ===		=== Individual pages for EDSO ===
	{{Main\|[[EDSO]]}}		{{Main\|EDSO}}

	== References ==		== References ==

CompactStar at 03:36, 22 July 2023

2023-07-22T03:36:14Z

← Older revision		Revision as of 03:36, 22 July 2023
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	=== Individual pages for EDSO ===		=== Individual pages for EDSO ===
			{{Main\|[[EDSO]]}}
	* [[~~8edso~~]]

	== References ==		== References ==

Eliora: /* "Super-pitch equivalents" of different concepts */

2023-07-01T11:03:16Z

"Super-pitch equivalents" of different concepts

← Older revision		Revision as of 11:03, 1 July 2023
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	The super-pitch equivalent of [[just intonation]] is intervals of the form log<sub>b</sub>(x) for positive integers b and x. This includes all of just intonation, since all just intervals can be described as logarithms (e.g. [[3/2]] = log<sub>4</sub>(8)), in addition to some irrational numbers such as log<sub>2</sub>(3).		The super-pitch equivalent of [[just intonation]] is intervals of the form log<sub>b</sub>(x) for positive integers b and x. This includes all of just intonation, since all just intervals can be described as logarithms (e.g. [[3/2]] = log<sub>4</sub>(8)), in addition to some irrational numbers such as log<sub>2</sub>(3).

	It is possible to construct super-pitch equivalents of most concepts in [[regular temperament theory]]. There exists a super-pitch equivalent of prime factorization–every integer greater than 2 can be uniquely expressed as a power tower of numbers in the sequence OEIS [https://oeis.org/A007916 A007916] (non-perfect powers). For example, 8 = 2<sup>3</sup>, 16 = 2<sup>2<sup>2</sup></sup>, 25 = 5<sup>2</sup>, 27 = 3<sup>3</sup>, 36 = 6<sup>2</sup>, and 81 = 3<sup>2<sup>2</sup></sup>. From this, it is straightforward to define the super-pitch equivalent of [[monzo]]s, or "super-monzos" (just substitute prime factorization for this power tower representation). Super-[[vals]], super-[[mapping]]s, and even super-[[temperament]]s can be derived by using super-monzos instead of regular monzos.		It is possible to construct super-pitch equivalents of most concepts in [[regular temperament theory]]. There exists a super-pitch equivalent of prime factorization–every integer greater than 2 can be uniquely expressed as a power tower of numbers in the sequence OEIS [https://oeis.org/A007916 A007916] (non-perfect powers). For example, 8 = 2<sup>3</sup>, 16 = 2<sup>2<sup>2</sup></sup>, 25 = 5<sup>2</sup>, 27 = 3<sup>3</sup>, 36 = 6<sup>2</sup>, and 81 = 3<sup>2<sup>2</sup></sup>. From this, it is straightforward to define the super-pitch equivalent of [[monzo]]s, or "super-monzos" (just substitute prime factorization for this power tower representation). Super-[[vals]], super-[[mapping]]s, and even super-[[temperament]]s can be derived by using super-monzos instead of regular monzos. This means that subgroups in super-pitch theory are made of non-perfect powers, like 2.3.5.6.7.10 for example.

	== Super-pitch division ==		== Super-pitch division ==

Eliora: /* Super-pitch division */

2023-06-30T15:21:33Z

Super-pitch division

← Older revision		Revision as of 15:21, 30 June 2023
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	This is arguably the way to make super-pitch divisions of a given interval. For example, the following table shows 10 super-pitch divisions of the octave <ref name=":0" />:		This is arguably the way to make super-pitch divisions of a given interval. For example, the following table shows 10 super-pitch divisions of the octave <ref name=":0" />:
	{\| class="wikitable"		{\| class="wikitable"
	! colspan="2" \|10 ~~tetrative~~ divisions of the ~~octave~~		! colspan="2" \|10 equal divisions of the superoctave
	<small>per Kneser's solution</small>		<small>per Kneser's solution</small>
	\|-		\|-
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	\|}		\|}
	Since exponentiation is not commutative and has two inverses - root and logarithm which breed two distinct numbers, likewise tetration has similar inverses which are too sets of disjoint numbers - solution to x^x = 2 is not the same number as solution to slog2(x) = 0.5. A pure interpolative function does not cancel out, for example the interval step 5, 1.458782..., when raised to the power of itself does not yield 2. Likewise, the first step, 1.089118, power-tower-ated 10 times does not yield 2 either.		Since exponentiation is not commutative and has two inverses - root and logarithm which breed two distinct numbers, likewise tetration has similar inverses which are too sets of disjoint numbers - solution to x^x = 2 is not the same number as solution to slog2(x) = 0.5. A pure interpolative function does not cancel out, for example the interval step 5, 1.458782..., when raised to the power of itself does not yield 2. Likewise, the first step, 1.089118, power-tower-ated 10 times does not yield 2 either.

			=== Individual pages for EDSO ===

			* [[8edso]]

	== References ==		== References ==
	<references />		<references />