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, ~~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.

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

|-

| Tenney ~~Tenney-~~Weil || TC tuning || [[TE tuning]] [[~~TWE~~ tuning]] || [[TOP tuning]]

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

|-

| Equilateral ~~Equilateral~~-~~Weil~~ || EC tuning || [[Frobenius tuning]] ~~EWE~~ tuning || EOP tuning

| Equilateral Skewed-equilateral || EC tuning || [[Frobenius tuning]] <abbr title="Skewed-equilateral-Euclidean">SEE</abbr> tuning || EOP tuning

|-

| Benedetti/Wilson Benedetti/Wilson~~-Weil~~ || BC tuning || [[BE tuning]] ~~BWE~~ tuning || [[BOP tuning]]

| Benedetti/Wilson Skewed-Benedetti/Wilson || BC tuning || [[BE tuning]] <abbr title="Skewed-Benedetti-Euclidean">SBE</abbr> tuning || [[BOP tuning]]

|}

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 ==

@@ 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, 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.
@@ Line 77: / Line 77: @@
 ! 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 ==