Optimization: Difference between revisions

Roughly speaking, there are two types of tunings with diverging philosophies: ''norm-based tunings'' and ''target tunings''.  

* A norm-based tuning is optimized for not only the [[formal prime]]s, but the set of all intervals, which the formal primes are representative for. In the [[vals and tuning space|tuning space]], it minimizes the average errors on the formal primes. In the [[monzos and interval space|interval space]], it rates the [[complexity]] of all intervals through a norm, and minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals. The two perspectives are equivalent.  

* A target tuning is optimized for a particular set of intervals and considers the rest irrelevant. The irrelevant intervals are typically but further out on the [[lattice]], and thus have implicit norm values. As such, these tunings effectively correspond to all-interval damage minimizations of some sorts, though analysing them in terms of norm-based tunings may be difficult.  

This article focuses on norm-based tunings. See the dedicated page (→ [[Target tunings]]) for target tunings.  

To perform norm-based optimization, we must employ a {{w|norm (mathematics)|norm}} to rate the complexities of all intervals. 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 as we do this: 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.