Optimization: Difference between revisions
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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: | ||
$$ W = \operatorname {diag} (1/\log_2 (Q)) $$ | |||
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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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, | 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 | 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. | ||
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. | |||
== 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 [[ | 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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== 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 | 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 | ||
$$ | |||
\begin{align} | |||
& \text{find} && G \\ | |||
& \text{that minimizes} && \lVert GV_X - J_X \rVert_q \\ | |||
subject to | & \text{subject to} && (GV - J)M = O | ||
\end{align} | |||
$$ | |||
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 | ||
$$ | |||
\begin{align} | \begin{align} | ||
V_X &= VX \\ | V_X &= VX \\ | ||
J_X &= JX | J_X &= JX | ||
\end{align} | \end{align} | ||
$$ | |||
== Common tunings == | == Common tunings == | ||
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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) | ||
|- | |- | ||
| Tenney<br> | | Tenney<br>Weil || TC tuning<br><br> || [[TE tuning]]<br>[[WE tuning]] || [[TOP tuning]]<br><br> | ||
|- | |- | ||
| Equilateral<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 | | 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 | 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 == | ||
* [[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]] | ||