Optimization: Difference between revisions
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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. | ||
An abstract regular temperament does not contain specific tuning information. To tune a temperament, one must specify the size of each [[generator]]. The question is what it should be. | An abstract regular temperament is defined by a [[mapping]] or a [[comma basis]]. It does not contain specific tuning information. To tune a temperament, one must specify the size of each [[Periods and generators|generator]]. The question is what it should be. In general, a temperament is an approximation to [[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. | ||
== Basic concepts == | == Basic concepts == | ||
* [[Mapping]] and [[comma basis]] | |||
* [[Periods and generators]] | * [[Periods and generators]] | ||
* [[Tuning map]] | |||
* [[Just intonation point]] (JIP) | * [[Just intonation point]] (JIP) | ||
== Taxonomy == | == Taxonomy == | ||
Roughly speaking, there are two types of tunings with diverging philosophies: | 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 | * A prime-based tuning is optimized for the [[formal prime]]s, but they are representative for the set of all intervals. There are two 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 all intervals through a norm, which serves as a complexity measure, and it minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals. | ||
* A target tuning is only optimized for a particular set of intervals and considers the rest irrelevant and/or infinitely complex. | * A target tuning is only optimized for a particular set of intervals and considers the rest irrelevant and/or infinitely complex. | ||
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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 | 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 space or the norm. 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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The order of the norm determines what is a unit step in the space. | The order of the norm determines what is a unit step in the space. | ||
The | The Euclidean norm aka ''L''<sup>2</sup> norm resembles real-world distances. | ||
The | The Minkowskian norm, 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 mounts to two steps. | ||
The | The Chebyshevian norm aka ''L''<sup>inifinity</sup> norm is the opposite of the Minkowskian norm – it is the maximum number of steps in any direction, so a diagonal movement is the same as a horizontal or vertical one. | ||
It should be noted that the dual norm of ''L''<sup>1</sup> is ''L''<sup>infinity</sup>, and vice versa. Thus, the Minkowskian norm corresponds to the ''L''<sup>infinity</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. | It should be noted that the dual norm of ''L''<sup>1</sup> is ''L''<sup>infinity</sup>, and vice versa. Thus, the Minkowskian norm corresponds to the ''L''<sup>infinity</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 == | ||
Enforcement is the technique where certain intervals are forced to be tuned as desired. The [[octave]] is the most common interval subject to enforcement, and is often assumed to be ''the'' enforced interval, but other intervals are possible. The two common methods are destretch and constraint. | |||
=== Destretch === | === Destretch === | ||