Optimization: Difference between revisions

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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.

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.

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=== Order ===

The order of the norm determines what is a unit step in the interval space.

* The Euclidean norm a.k.a. ''L''2 norm resembles real-world distances.

* The Manhattan norm, taxicab norm, a.k.a. ''L''1 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''inf 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''2 norm ~~resembles real-world distances~~.

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

== Enforcement ==

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=== 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 ===

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|+Table of common tunings

|-

! ~~Weight–Skew~~\~~Order~~ !! Chebyshevian (''L''1 tuning) !! Euclidean (''L''2 tuning) !! Manhattan (''L''inf tuning)

! Weight–skew\order !! Chebyshevian (''L''1 tuning) !! Euclidean (''L''2 tuning) !! Manhattan (''L''inf tuning)

|-

| Tenney Weil || TC tuning || [[TE tuning]] [[WE tuning]] || [[TOP tuning]]

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== 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]]

== External links ==

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

== Notes ==

[[Category:Math]]

[[Category:Regular temperament tuning]]

@@ Line 25: / Line 25: @@
 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.
@@ Line 31: / Line 31: @@
 === Order ===
 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 ==
@@ Line 44: / Line 43: @@
 === 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 ===
@@ Line 75: / Line 74: @@
 |+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>
@@ Line 89: / Line 88: @@
 == 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]]
 == External links ==
 * [http://x31eq.com/temper/primerr.pdf|''Prime Based Error and Complexity Measures''] ("primerr.pdf") by [[Graham Breed]]
+== Notes ==
 [[Category:Math]]
 [[Category:Regular temperament tuning]]