Optimization - Revision history

Dummy index: add interwiki

2025-10-22T15:05:46Z

add interwiki

← Older revision		Revision as of 15:05, 22 October 2025
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			{{interwiki
			\| de =
			\| en = Optimization
			\| es =
			\| ja = 数理最適化
			}}
	In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments.		In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments.

FloraC: Prime-based -> norm-based. Misc. clarifications and improvements

2025-09-12T12:37:22Z

Prime-based -> norm-based. Misc. clarifications and improvements

← Older revision		Revision as of 12:37, 12 September 2025
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	== Taxonomy ==		== Taxonomy ==
	Roughly speaking, there are two types of tunings with diverging philosophies: ''~~prime~~-based tunings'' and ''target tunings''.		Roughly speaking, there are two types of tunings with diverging philosophies: ''norm-based tunings'' and ''target tunings''.
	* A ~~prime~~-based tuning is optimized for the [[formal prime]]s, but ~~they are representative for~~ the set of all intervals~~. There~~ are ~~two equivalent perspectives~~. ~~First, in~~ the [[vals and tuning space\|tuning space]], it minimizes the errors of formal primes. ~~Second, in~~ the [[monzos and interval space\|interval space]], it rates the [[complexity]] of all intervals through a norm, and it minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals.		* A norm-based tuning is optimized for not only the [[formal prime]]s, but the set of all intervals, which the formal primes are representative for. In the [[vals and tuning space\|tuning space]], it minimizes the average errors on the formal primes. In the [[monzos and interval space\|interval space]], it rates the [[complexity]] of all intervals through a norm, and minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals. The two perspectives are equivalent.
	* A target tuning is optimized for a particular set of intervals and considers the rest irrelevant. ~~However~~, ~~the interval does not get infinite complexity even if it is disregarded due to the normed nature of the interval space~~, so these tunings ~~also~~ correspond to all-interval damage minimizations of some sorts.		* A target tuning is optimized for a particular set of intervals and considers the rest irrelevant. The irrelevant intervals are typically but further out on the [[lattice]], and thus have implicit norm values. As such, these tunings effectively correspond to all-interval damage minimizations of some sorts, though analysing them in terms of norm-based tunings may be difficult.

	This article focuses on ~~prime~~-based tunings. See the dedicated page (→ [[Target tunings]]) for target tunings.		This article focuses on norm-based tunings. See the dedicated page (→ [[Target tunings]]) for target tunings.

	== Norm ==		== Norm ==
	[[File:Vector norms.svg\|thumb\|Comparison of norms on the space]]		[[File:Vector norms.svg\|thumb\|Comparison of norms on the space]]

	~~In order to~~ perform ~~prime~~-based optimization, ~~all intervals~~ must ~~be rated by complexity, so it is critical to~~ employ a {{w\|norm (mathematics)\|norm}}. Technically, this is to {{w\|embedding\|embed}} the [[just intonation subgroup\|just intonation group]] into a {{w\|normed vector space}}. There are a few aspects to consider~~. The~~ weight, which determines how important each formal prime is, and the skew, which determines how divisive ratios are more important than multiplicative ratios. They can be interpreted as transformations of either the norm or the coordinates of the space. The two views are equivalent. In addition, there is the order (or sometimes just dubbed the norm), which determines how the space can be traversed.		To perform norm-based optimization, we must employ a {{w\|norm (mathematics)\|norm}} to rate the complexities of all intervals. Technically, this is to {{w\|embedding\|embed}} the [[just intonation subgroup\|just intonation group]] into a {{w\|normed vector space}}. There are a few aspects to consider as we do this: the weight, which determines how important each formal prime is, and the skew, which determines how divisive ratios are more important than multiplicative ratios. They can be interpreted as transformations of either the norm or the coordinates of the space. The two views are equivalent. In addition, there is the order (or sometimes just dubbed the norm), which determines how the space can be traversed.

	=== Weight ===		=== Weight ===

FloraC: Improve the formulation

2025-06-12T14:35:51Z

Improve the formulation

← Older revision		Revision as of 14:35, 12 June 2025
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	The weight, represented by a diagonal transformation matrix, determines the importance of each formal prime. Since the tuning space and the interval space are {{w\|dual (mathematics)\|dual}} to each other, rating of importance in the tuning space is equivalent to rating of complexity in the interval space. The Tenney weight is the most common weight:		The weight, represented by a diagonal transformation matrix, determines the importance of each formal prime. Since the tuning space and the interval space are {{w\|dual (mathematics)\|dual}} to each other, rating of importance in the tuning space is equivalent to rating of complexity in the interval space. The Tenney weight is the most common weight:

	~~<math>\displaystyle~~ W = \operatorname {diag} (1/\log_2 (Q)) ~~</math>~~		$$ W = \operatorname {diag} (1/\log_2 (Q)) $$

	which indicates that the prime harmonic ''q'' in ''Q'' = {{val\| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Its dual states that ''q'' has the complexity of log<sub>2</sub>(''q'').		which indicates that the prime harmonic ''q'' in ''Q'' = {{val\| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Its dual states that ''q'' has the complexity of log<sub>2</sub>(''q'').
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	== General formulation ==		== General formulation ==
	In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew ~~transformation, represented by~~ transformation matrix ''X'', ~~and~~ a ''q''-norm~~. An optional~~ unit eigenmonzo list ''M''~~<sub>~~''I''~~</sub> can be added. The goal is to find~~ the generator ~~list ''G'' by~~		In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping matrix ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew transformation matrix ''X'', a ''q''-norm, and optionally a unit eigenmonzo list ''M''. Let ''G'' denote the generator tuning map, we want to

	~~Minimize~~		$$
			\begin{align}
	~~<math>~~\~~displaystyle~~ \lVert GV_X - J_X \rVert_q ~~</math>~~		& \text{find} && G \\
			& \text{that minimizes} && \lVert GV_X - J_X \rVert_q \\
	subject to		& \text{subject to} && (GV - J)M = O
			\end{align}
	~~<math>\displaystyle~~ (GV - J)~~M_I~~ = O ~~</math>~~		$$

	where (·)<sub>''X''</sub> denotes the variable in the weight–skew transformed space, found by		where (·)<sub>''X''</sub> denotes the variable in the weight–skew transformed space, found by

	~~<math>\displaystyle~~		$$
	\begin{align}		\begin{align}
	V_X &= VX \\		V_X &= VX \\
	J_X &= JX		J_X &= JX
	\end{align}		\end{align}
	~~</math>~~		$$

	== Common tunings ==		== Common tunings ==

FloraC: Oops

2025-06-12T13:50:21Z

Oops

← Older revision		Revision as of 13:50, 12 June 2025
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	== External links ==		== External links ==
	* [http://x31eq.com/temper/primerr.pdf\|''Prime Based Error and Complexity Measures''] ("primerr.pdf") by [[Graham Breed]]		* [http://x31eq.com/temper/primerr.pdf\|''Prime Based Error and Complexity Measures''] ("primerr.pdf") by [[Graham Breed]]

	~~== Notes ==~~

	[[Category:Math]]		[[Category:Math]]
	[[Category:Regular temperament tuning]]		[[Category:Regular temperament tuning]]

FloraC: Include a short discussion on these various norms

2025-06-12T12:20:15Z

Include a short discussion on these various norms

← Older revision		Revision as of 12:20, 12 June 2025
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	An orthogonal space treats divisive ratios as equally important as multiplicative ratios, yet divisive ratios are sometimes thought to be more important. For example, 5/3 is sometimes found to be more important than 15. The skew is introduced to address that.		An orthogonal space treats divisive ratios as equally important as multiplicative ratios, yet divisive ratios are sometimes thought to be more important. For example, 5/3 is sometimes found to be more important than 15. The skew is introduced to address that.

	Notably, the [[Weil ~~height~~]] is equivalent to Tenney-weighting the intervals and skewing the space such that each basis element is 60 degrees from each other.		Notably, the [[Weil norm]] is equivalent to Tenney-weighting the intervals and skewing the space such that each basis element is 60 degrees from each other.

	Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called weight–skew transformation.		Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called weight–skew transformation.
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	=== Order ===		=== Order ===
	The order of the norm determines what is a unit step in the interval space.		The order of the norm determines what is a unit step in the interval space.
			* The Euclidean norm a.k.a. ''L''<sup>2</sup> norm resembles real-world distances.
			* The Manhattan norm, taxicab norm, a.k.a. ''L''<sup>1</sup> norm resembles movement of taxicabs in Manhattan – it can only traverse horizontally or vertically, so diagonal movement counts as two steps.
			* The Chebyshevian norm a.k.a. ''L''<sup>inf</sup> norm is the opposite of the Manhattan norm – it is the maximum number of steps along any axis, so a diagonal movement is the same as a horizontal or vertical one.

	The ~~Euclidean norm aka~~ ''L''<sup>2</sup> norm ~~resembles real-world distances~~.		Note that the dual norm of ''L''<sup>1</sup> is ''L''<sup>inf</sup>, and vice versa. Thus, the ''L''<sup>1</sup> norm on the interval space corresponds to the ''L''<sup>inf</sup> tuning space, and the ''L''<sup>inf</sup> norm corresponds to the ''L''<sup>1</sup> tuning space. The dual of ''L''<sup>2</sup> norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect.

	~~The Manhattan norm or taxicab norm a.k.a. ''L''<sup>1</sup> norm resembles movement of~~ taxicabs ~~in Manhattan – it can only traverse horizontally or vertically. A diagonal movement counts~~ as ~~two steps.~~		Technically, harmony works like taxicabs, as is modeled by [[James Tenney]]'s harmonic distance, or [[Tenney norm]], so it is often believed that optimization with the Manhattan norm should result in the best tuning and that Euclidean norms are only used because of ease of computation ([[D&D's guide]] has notably taken this stance). However, Euclidean and especially Chebyshevian norms may correspond to more efficient use of optimizational resource, as they take account of how many generator steps each formal prime is mapped to in the temperament.

	~~The Chebyshevian norm a.k.a.~~ '~~'L''<sup>inf</sup>~~ norm is ~~the opposite of~~ the Manhattan norm ~~– it is~~ the ~~maximum number~~ of ~~steps along any axis, so a diagonal movement is the same as a horizontal or vertical one.~~

	~~Note that the dual norm~~ of '~~'L''<sup>1</sup> is ''L''<sup>inf</sup>, and vice versa~~. ~~Thus, the Manhattan norm corresponds to the ''L''<sup>inf</sup> tuning space~~, and ~~the~~ Chebyshevian ~~norm corresponds~~ to ~~the ''L''<sup>1</sup> tuning space. The dual~~ of ~~''L''<sup>2</sup> norm~~ is ~~itself, so~~ the ~~Euclidean norm corresponds to Euclidean tuning as one may expect~~.

	== Enforcement ==		== Enforcement ==
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	=== Destretch ===		=== Destretch ===
	Destretch is a postprocess to enforce a pure interval. The result is no longer optimal measured by the original norm, but it often works as a quick approximation to more sophisticated tunings. The most common destretched tuning is [[POTE tuning]], which approximates [[~~CTWE~~ tuning]].		Destretch is a postprocess to enforce a pure interval. The result is no longer optimal measured by the original norm, but it often works as a quick approximation to more sophisticated tunings. The most common destretched tuning is [[POTE tuning]], which approximates [[CWE tuning]].

	=== Constraint ===		=== Constraint ===
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	\|+Table of common tunings		\|+Table of common tunings
	\|-		\|-
	! ~~Weight–Skew~~\~~Order~~ !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>inf</sup> tuning)		! Weight–skew\order !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>inf</sup> tuning)
	\|-		\|-
	\| Tenney<br>Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[WE tuning]] \|\| [[TOP tuning]]<br><br>		\| Tenney<br>Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[WE tuning]] \|\| [[TOP tuning]]<br><br>
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	== See also ==		== See also ==
	* [[Generator embedding optimization]] - more information on RTT optimization techniques that follow the above concepts but give exact solutions in the form of non-JI prime-count vectors with non-integer entries (e.g. including roots)		* [[Generator embedding optimization]] – more information on RTT optimization techniques that follow the above concepts but give exact solutions in the form of non-JI prime-count vectors with non-integer entries (e.g. including roots)
	* [[:File:MiddlePath2015.pdf\|''A Middle Path between Just Intonation and the Equal Temperaments – Part 1'']] ("middle path") by [[Paul Erlich]]		* [[:File:MiddlePath2015.pdf\|''A Middle Path between Just Intonation and the Equal Temperaments – Part 1'']] ("middle path") by [[Paul Erlich]]

	== External links ==		== External links ==
	* [http://x31eq.com/temper/primerr.pdf\|''Prime Based Error and Complexity Measures''] ("primerr.pdf") by [[Graham Breed]]		* [http://x31eq.com/temper/primerr.pdf\|''Prime Based Error and Complexity Measures''] ("primerr.pdf") by [[Graham Breed]]

			== Notes ==

	[[Category:Math]]		[[Category:Math]]
	[[Category:Regular temperament tuning]]		[[Category:Regular temperament tuning]]

FloraC: /* Common tunings */ note on "E" and "CE"

2024-10-02T17:47:01Z

Common tunings: note on "E" and "CE"

← Older revision		Revision as of 17:47, 2 October 2024
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	Each has a constrained and/or destretched variant. E.g. for TE tuning there is [[CTE tuning]], and for WE tuning there is [[CWE tuning]].		Each has a constrained and/or destretched variant. E.g. for TE tuning there is [[CTE tuning]], and for WE tuning there is [[CWE tuning]].

			: '''Note''': in [https://sintel.pythonanywhere.com/ Sintel's temperament calculator], Frobenius tuning is known as ''E tuning'', and CEE tuning is known as ''CE tuning''.

	== See also ==		== See also ==

Cmloegcmluin: /* See also */

2024-06-13T01:07:34Z

FloraC: Update: it's established that Weil is a specific weight and skew scheme. Replace the otherwise weighted-skewed tunings as skewed-[weight]-[order] tuning

2024-03-20T08:38:14Z

Update: it's established that Weil is a specific weight and skew scheme. Replace the otherwise weighted-skewed tunings as skewed-[weight]-[order] tuning

← Older revision		Revision as of 08:38, 20 March 2024
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	An orthogonal space treats divisive ratios as equally important as multiplicative ratios, yet divisive ratios are sometimes thought to be more important. For example, 5/3 is sometimes found to be more important than 15. The skew is introduced to address that.		An orthogonal space treats divisive ratios as equally important as multiplicative ratios, yet divisive ratios are sometimes thought to be more important. For example, 5/3 is sometimes found to be more important than 15. The skew is introduced to address that.

	Notably, ~~adopting~~ the ~~standard~~ [[Weil height]] ~~will skew~~ the space to 60 degrees.		Notably, the [[Weil height]] is equivalent to Tenney-weighting the intervals and skewing the space such that each basis element is 60 degrees from each other.

	Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called weight–skew transformation.		Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called weight–skew transformation.
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	! Weight–Skew\Order !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>inf</sup> tuning)		! Weight–Skew\Order !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>inf</sup> tuning)
	\|-		\|-
	\| Tenney<br>~~Tenney-~~Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[~~TWE~~ tuning]] \|\| [[TOP tuning]]<br><br>		\| Tenney<br>Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[WE tuning]] \|\| [[TOP tuning]]<br><br>
	\|-		\|-
	\| Equilateral<br>~~Equilateral~~-~~Weil~~ \|\| EC tuning<br><br> \|\| [[Frobenius tuning]]<br>~~EWE~~ tuning \|\| EOP tuning<br><br>		\| Equilateral<br>Skewed-equilateral \|\| EC tuning<br><br> \|\| [[Frobenius tuning]]<br><abbr title="Skewed-equilateral-Euclidean">SEE</abbr> tuning \|\| EOP tuning<br><br>
	\|-		\|-
	\| Benedetti/Wilson<br>Benedetti/Wilson~~-Weil~~ \|\| BC tuning<br><br> \|\| [[BE tuning]]<br>~~BWE~~ tuning \|\| [[BOP tuning]]<br><br>		\| Benedetti/Wilson<br>Skewed-Benedetti/Wilson \|\| BC tuning<br><br> \|\| [[BE tuning]]<br><abbr title="Skewed-Benedetti-Euclidean">SBE</abbr> tuning \|\| [[BOP tuning]]<br><br>
	\|}		\|}

	Each has a constrained and/or destretched variant. E.g. for TE tuning there is [[CTE tuning]], and for ~~TWE~~ tuning there is [[~~CTWE~~ tuning]].		Each has a constrained and/or destretched variant. E.g. for TE tuning there is [[CTE tuning]], and for WE tuning there is [[CWE tuning]].

	== See also ==		== See also ==

FloraC: Another pass on style: matrices are italic following Wikipedia practice. Eigenmonzo -> unit eigenmonzo. Misc. cleanup

2023-12-22T12:08:06Z

Another pass on style: matrices are italic following Wikipedia practice. Eigenmonzo -> unit eigenmonzo. Misc. cleanup

← Older revision		Revision as of 12:08, 22 December 2023
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	In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments.		In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments.

	A regular temperament is defined by a [[mapping]] or a [[comma basis]]. It does not contain specific tuning information. To tune a temperament, one must define a [[tuning map]] by specifying the size of each [[~~Periods~~ and generators\|generator]]. The question is what it should be. In general, a temperament is an approximation to [[~~Just~~ intonation\|just intonation (JI)]]. Any tuning will unavoidably introduce errors on some intervals for sure. The art of tempering seems to be about compromises – to find a sweet spot where the concerning intervals have the least overall error, so that the harmonic qualities of JI are best preserved.		A regular temperament is defined by a [[mapping]] or a [[comma basis]]. It does not contain specific tuning information. To tune a temperament, one must define a [[tuning map]] by specifying the size of each [[periods and generators\|generator]]. The question is what it should be. In general, a temperament is an approximation to [[just intonation\|just intonation (JI)]]. Any tuning will unavoidably introduce errors on some intervals for sure. The art of tempering seems to be about compromises – to find a sweet spot where the concerning intervals have the least overall error, so that the harmonic qualities of JI are best preserved.

	== Taxonomy ==		== Taxonomy ==
	Roughly speaking, there are two types of tunings with diverging philosophies: ''prime-based tunings'' and ''target tunings''.		Roughly speaking, there are two types of tunings with diverging philosophies: ''prime-based tunings'' and ''target tunings''.
	* A prime-based tuning is optimized for the [[formal prime]]s, but they are representative for the set of all intervals. There are two equivalent perspectives. First, in the [[~~Vals~~ and tuning space\|tuning space]], it minimizes the errors of formal primes. Second, in the [[~~Monzos~~ and interval space\|interval space]], it rates the [[complexity]] of all intervals through a norm, and it minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals.		* A prime-based tuning is optimized for the [[formal prime]]s, but they are representative for the set of all intervals. There are two equivalent perspectives. First, in the [[vals and tuning space\|tuning space]], it minimizes the errors of formal primes. Second, in the [[monzos and interval space\|interval space]], it rates the [[complexity]] of all intervals through a norm, and it minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals.
	* A target tuning is optimized for a particular set of intervals and considers the rest irrelevant. However, the interval does not get infinite complexity even if it is disregarded due to the normed nature of the interval space, so these tunings also correspond to all-interval damage minimizations of some sorts.		* A target tuning is optimized for a particular set of intervals and considers the rest irrelevant. However, the interval does not get infinite complexity even if it is disregarded due to the normed nature of the interval space, so these tunings also correspond to all-interval damage minimizations of some sorts.

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	[[File:Vector norms.svg\|thumb\|Comparison of norms on the space]]		[[File:Vector norms.svg\|thumb\|Comparison of norms on the space]]

	In order to perform prime-based optimization, all intervals must be rated by complexity, so it is critical to employ a ~~[[Wikipedia:Norm~~ (mathematics)\|norm]]. Technically, this is to ~~[[wikipedia: Embedding~~\|embed]] the [[just intonation subgroup\|just intonation group]] into a ~~[[wikipedia: Normed vector space~~\|normed vector space]]. There are a few aspects to consider. The weight, which determines how important each formal prime is, and the skew, which determines how divisive ratios are more important than multiplicative ratios. They can be interpreted as transformations of either the norm or the coordinates of the space. The two views are equivalent. In addition, there is the order (or sometimes just dubbed the norm), which determines how the space can be traversed.		In order to perform prime-based optimization, all intervals must be rated by complexity, so it is critical to employ a {{w\|norm (mathematics)\|norm}}. Technically, this is to {{w\|embedding\|embed}} the [[just intonation subgroup\|just intonation group]] into a {{w\|normed vector space}}. There are a few aspects to consider. The weight, which determines how important each formal prime is, and the skew, which determines how divisive ratios are more important than multiplicative ratios. They can be interpreted as transformations of either the norm or the coordinates of the space. The two views are equivalent. In addition, there is the order (or sometimes just dubbed the norm), which determines how the space can be traversed.

	=== Weight ===		=== Weight ===
	The weight, represented by a diagonal transformation matrix, determines the importance of each formal prime. Since the tuning space and the interval space are ~~[[Wikipedia:Dual~~ (mathematics)\|dual]] to each other, rating of importance in the tuning space is equivalent to rating of complexity in the interval space. The Tenney weight is the most common weight:		The weight, represented by a diagonal transformation matrix, determines the importance of each formal prime. Since the tuning space and the interval space are {{w\|dual (mathematics)\|dual}} to each other, rating of importance in the tuning space is equivalent to rating of complexity in the interval space. The Tenney weight is the most common weight:

	<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) </math>		<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) </math>

	which indicates that the prime harmonic ''q'' in Q = {{val\| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Its dual states that ''q'' has the complexity of log<sub>2</sub>(''q'').		which indicates that the prime harmonic ''q'' in ''Q'' = {{val\| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Its dual states that ''q'' has the complexity of log<sub>2</sub>(''q'').

	=== Skew ===		=== Skew ===
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	=== Constraint ===		=== Constraint ===
	Constraint is a logical method to enforce one or more pure intervals. A pure interval added this way is known as an [[~~Eigenmonzo~~\|eigenmonzo a.k.a. unchanged-interval]]. It defines a feasible region for optimization, and the result measured by the original norm is feasibly optimal.		Constraint is a logical method to enforce one or more pure intervals. A pure interval added this way is known as a [[eigenmonzo\|unit eigenmonzo a.k.a. unchanged-interval]]. It defines a feasible region for optimization, and the result measured by the original norm is feasibly optimal.

	== General formulation ==		== General formulation ==
	In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping V and the [[just tuning map]] J, we specify a weight–skew transformation, represented by transformation matrix X, and a ''q''-norm. An optional eigenmonzo list M<sub>C</sub> can be added. The goal is to find the generator list G by		In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew transformation, represented by transformation matrix ''X'', and a ''q''-norm. An optional unit eigenmonzo list ''M''<sub>''I''</sub> can be added. The goal is to find the generator list ''G'' by

	Minimize		Minimize
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	subject to		subject to

	<math>\displaystyle (GV - J)~~M_{\rm C}~~ = O </math>		<math>\displaystyle (GV - J)M_I = O </math>

	where (·)<sub>X</sub> denotes the variable in the weight–skew transformed space, found by		where (·)<sub>''X''</sub> denotes the variable in the weight–skew transformed space, found by

	<math>\displaystyle		<math>\displaystyle
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	\|+Table of common tunings		\|+Table of common tunings
	\|-		\|-
	! Weight–Skew\Order !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>~~infinity~~</sup> tuning)		! Weight–Skew\Order !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>inf</sup> tuning)
	\|-		\|-
	\| Tenney<br>Tenney-Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[TWE tuning]] \|\| [[TOP tuning]]<br><br>		\| Tenney<br>Tenney-Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[TWE tuning]] \|\| [[TOP tuning]]<br><br>

FloraC: Now weight--skew transfomration is a thing and is denoted by X. p-norm -> q-norm since it's the dual norm (p is reserved for interval space norm)

2023-09-09T10:08:56Z

Now weight--skew transfomration is a thing and is denoted by X. p-norm -> q-norm since it's the dual norm (p is reserved for interval space norm)

← Older revision		Revision as of 10:08, 9 September 2023
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	In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments.		In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments.

	A regular temperament is defined by a [[mapping]] or a [[comma basis]]. It does not contain specific tuning information. To tune a temperament, one must define a [[tuning map]] by specifying the size of each [[Periods and generators\|generator]]. The question is what it should be. In general, a temperament is an approximation to [[just intonation]] (JI). Any tuning will unavoidably introduce errors on some intervals for sure. The art of tempering seems to be about compromises – to find a sweet spot where the concerning intervals have the least overall error, so that the harmonic qualities of JI are best preserved.		A regular temperament is defined by a [[mapping]] or a [[comma basis]]. It does not contain specific tuning information. To tune a temperament, one must define a [[tuning map]] by specifying the size of each [[Periods and generators\|generator]]. The question is what it should be. In general, a temperament is an approximation to [[Just intonation\|just intonation (JI)]]. Any tuning will unavoidably introduce errors on some intervals for sure. The art of tempering seems to be about compromises – to find a sweet spot where the concerning intervals have the least overall error, so that the harmonic qualities of JI are best preserved.

	== Taxonomy ==		== Taxonomy ==
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	[[File:Vector norms.svg\|thumb\|Comparison of norms on the space]]		[[File:Vector norms.svg\|thumb\|Comparison of norms on the space]]

	In order to perform prime-based optimization, all intervals must be rated by complexity, so it is critical to employ a [[Wikipedia:Norm (mathematics)\|norm]] on the space. There are a few aspects to consider. The weight, which determines how important each formal prime is, and the skew, which determines how divisive ratios are more important than multiplicative ratios. They can be interpreted as transformations of either the norm or the coordinates of the space. The two views are equivalent. In addition, there is the order (or sometimes just dubbed the norm), which determines how the space can be traversed.		In order to perform prime-based optimization, all intervals must be rated by complexity, so it is critical to employ a [[Wikipedia:Norm (mathematics)\|norm]]. Technically, this is to [[wikipedia: Embedding\|embed]] the [[just intonation subgroup\|just intonation group]] into a [[wikipedia: Normed vector space\|normed vector space]]. There are a few aspects to consider. The weight, which determines how important each formal prime is, and the skew, which determines how divisive ratios are more important than multiplicative ratios. They can be interpreted as transformations of either the norm or the coordinates of the space. The two views are equivalent. In addition, there is the order (or sometimes just dubbed the norm), which determines how the space can be traversed.

	=== Weight ===		=== Weight ===
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	Notably, adopting the standard [[Weil height]] will skew the space to 60 degrees.		Notably, adopting the standard [[Weil height]] will skew the space to 60 degrees.

	Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called ~~by either part~~.		Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called weight–skew transformation.

	=== Order ===		=== Order ===
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	The Euclidean norm aka ''L''<sup>2</sup> norm resembles real-world distances.		The Euclidean norm aka ''L''<sup>2</sup> norm resembles real-world distances.

	The Manhattan norm or taxicab norm ~~aka~~ ''L''<sup>1</sup> norm resembles movement of taxicabs in Manhattan – it can only traverse horizontally or vertically. A diagonal movement counts as two steps.		The Manhattan norm or taxicab norm a.k.a. ''L''<sup>1</sup> norm resembles movement of taxicabs in Manhattan – it can only traverse horizontally or vertically. A diagonal movement counts as two steps.

	The Chebyshevian norm ~~aka~~ ''L''<sup>inf</sup> norm is the opposite of the Manhattan norm – it is the maximum number of steps along any axis, so a diagonal movement is the same as a horizontal or vertical one.		The Chebyshevian norm a.k.a. ''L''<sup>inf</sup> norm is the opposite of the Manhattan norm – it is the maximum number of steps along any axis, so a diagonal movement is the same as a horizontal or vertical one.

	Note that the dual norm of ''L''<sup>1</sup> is ''L''<sup>inf</sup>, and vice versa. Thus, the Manhattan norm corresponds to the ''L''<sup>inf</sup> tuning space, and the Chebyshevian norm corresponds to the ''L''<sup>1</sup> tuning space. The dual of ''L''<sup>2</sup> norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect.		Note that the dual norm of ''L''<sup>1</sup> is ''L''<sup>inf</sup>, and vice versa. Thus, the Manhattan norm corresponds to the ''L''<sup>inf</sup> tuning space, and the Chebyshevian norm corresponds to the ''L''<sup>1</sup> tuning space. The dual of ''L''<sup>2</sup> norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect.
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	== General formulation ==		== General formulation ==
	In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping V and the [[just tuning map]] J, we specify a ~~weight and a skew~~, represented by transformation ~~matrices W and~~ X~~, respectively~~, and a ''p''-norm. An optional eigenmonzo list M<sub>C</sub> can be added. The goal is to find the generator list G by		In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping V and the [[just tuning map]] J, we specify a weight–skew transformation, represented by transformation matrix X, and a ''q''-norm. An optional eigenmonzo list M<sub>C</sub> can be added. The goal is to find the generator list G by

	Minimize		Minimize

	<math>\displaystyle \lVert ~~GV_{WX}~~ - ~~J_{WX}~~ \~~rVert_p~~ </math>		<math>\displaystyle \lVert GV_X - J_X \rVert_q </math>

	subject to		subject to
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	<math>\displaystyle (GV - J)M_{\rm C} = O </math>		<math>\displaystyle (GV - J)M_{\rm C} = O </math>

	where (·)<sub>WX</sub> denotes the ~~weight-skew transformation~~, found by		where (·)<sub>X</sub> denotes the variable in the weight–skew transformed space, found by

	<math>\displaystyle		<math>\displaystyle
	\begin{align}		\begin{align}
	~~V_{WX}~~ &= ~~VWX~~ \\		V_X &= VX \\
	~~J_{WX}~~ &= ~~JWX~~		J_X &= JX
	\end{align}		\end{align}
	</math>		</math>

	== Common tunings ==		== Common tunings ==
	A good number of common tuning schemes have been given names. The following table shows some of them by ~~weight-skew~~ against the order.		A good number of common tuning schemes have been given names. The following table shows some of them by weight–skew against the order.

	{\| class="wikitable"		{\| class="wikitable"
	\|+Table of common tunings		\|+Table of common tunings
	\|-		\|-
	! ~~Weight-skew~~\Order !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>infinity</sup> tuning)		! Weight–Skew\Order !! Chebyshevian<br>(''L''<sup>1</sup> tuning) !! Euclidean<br>(''L''<sup>2</sup> tuning) !! Manhattan<br>(''L''<sup>infinity</sup> tuning)
	\|-		\|-
	\| Tenney<br>Tenney-Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[TWE tuning]] \|\| [[TOP tuning]]<br><br>		\| Tenney<br>Tenney-Weil \|\| TC tuning<br><br> \|\| [[TE tuning]]<br>[[TWE tuning]] \|\| [[TOP tuning]]<br><br>

← Older revision		Revision as of 01:07, 13 June 2024
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	== See also ==		== See also ==
			* [[Generator embedding optimization]] - more information on RTT optimization techniques that follow the above concepts but give exact solutions in the form of non-JI prime-count vectors with non-integer entries (e.g. including roots)
	* [[:File:MiddlePath2015.pdf\|''A Middle Path between Just Intonation and the Equal Temperaments – Part 1'']] ("middle path") by [[Paul Erlich]]		* [[:File:MiddlePath2015.pdf\|''A Middle Path between Just Intonation and the Equal Temperaments – Part 1'']] ("middle path") by [[Paul Erlich]]