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]] (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.  

* 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 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.  

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.  

The weight determines the importance of each formal prime. For instance, the Tenney weight is represented by the transformation matrix

<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) </math>

which indicates that the prime harmonic ''q'' in Q = {{val| 2 3 5 … }} has the importance of 1/log₂(''q''). Since the tuning space and the interval space are [[Wikipedia:Dual (mathematics)|dual]] to each other, such a rating of importance in the tuning space has the dual effect in the interval space: the prime harmonic ''q'' has the complexity log₂(''q''). The more complex an interval is, the more error will be allowed for it.  

Notably, adopting the standard [[Weil height]] will skew the space to 60 degrees.  

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 by either part.  

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

The Euclidean norm aka ''L''² norm resembles real-world distances.  

The Manhattan norm or taxicab norm aka ''L''¹ norm resembles movement of taxicabs in Manhattan – it can only traverse horizontally or vertically. A diagonal movement counts as two steps.

 

The Chebyshevian norm aka ''L''^infinity norm is the opposite of the Manhattan 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''¹ is ''L''^infinity, and vice versa. Thus, the Manhattan norm corresponds to the ''L''^infinity 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.

Destretch is a postprocess to enforce a pure interval. The most common destretched tuning is [[POTE tuning]], which works as a quick approximation to the more sophisticated CTWE tuning.  

Constraint is a logical method to enforce one or more pure intervals. A pure interval added this way is known as an [[eigenmonzo]]. It defines a feasible region for optimization, and the result measured by the original norm remains optimal.

In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping V and the [[just intonation point]] (JIP) J, we specify a weight and a skew, represented by transformation matrices W and X, respectively, and a ''p''-norm. An optional eigenmonzo list M_C can be added. The goal is to find the generator list G by

 

<math>\displaystyle \lVert GV_{WX} - J_{WX} \rVert_p </math>

 

subject to

 

<math>\displaystyle (GV - J)M_{\rm C} = O </math>

where (·)_WX denotes the weight-skew transformation, found by

V_{WX} &= VWX \\

J_{WX} &= JWX

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

! Weight-skew\Order !! Chebyshevian<br>(''L''¹ tuning) !! Euclidean<br>(''L''² tuning) !! Manhattan<br>(''L''^infinity tuning)

| Tenney<br>Tenney-Weil || || [[TE tuning]]<br>[[TWE tuning]] || [[TOP tuning]]<br>

| Equilateral<br>Equilateral-Weil || || [[Frobenius tuning]]<br>EWE tuning ||  

| Benedetti/Wilson<br>Benedetti/Wilson-Weil || || [[BE tuning]]<br>BWE tuning || [[BOP tuning]]<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]].

[[Category:Regular temperament theory]]

[[Category:Tuning]]

[[Category:Overview]]