Optimization: Difference between revisions

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.  

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 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.  

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.  

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/log₂(''q''). Its dual states that ''q'' has the complexity of log₂(''q'').  

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.  

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

* 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 ''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.  

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 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 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 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.

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

where (·)_''X'' denotes the variable in the weight–skew transformed space, found by

\begin{align}

V_X &= VX \\

J_X &= JX

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

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

| Tenney<br>Weil || TC tuning<br><br> || [[TE tuning]]<br>[[WE tuning]] || [[TOP 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>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 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''.