User:Mike Battaglia/Generated Tone Systems

Unmapped regular tuning systems

Sometimes, when working with scales, it can be useful to consider those scale-theoretic properties which are independent of the tempered JI intervals the scale is considered to represent, and which are independent of the absolute tuning of a scale. Such considerations arise when working with MODMOS's, for instance, when looking at the Graham complexity of an arbitrary MODMOS, when taking the Product word of two existing patterns of scale steps, and when attempting to explore higher-dimensional generalizations of MOS and MODMOS.

This perspective becomes particularly useful when working with scales which are completely generated by taking Z-linear combinations of an (ideally smaller) set of intervals. When working with such scales, it can be useful for us to conceive of them as existing within a larger unmapped Abstract Regular Tuning System (ARTS). To define an abstract regular tuning system, we first define a Regular Tuning System (RTS) as a set of real numbers, representing logarithmic musical intervals, which also has the structure of being a free abelian group under addition. To consider the general structure of such scales independently of their tuning, we then define the ARTS as an equivalence class of regular tuning systems which are related by isomorphism.

The subsets of an ARTS are called abstract scales.

Motivation

In plain English, an ARTS is much like the JI lattice, except the "monzos" now refer to abstract unmapped generators rather than primes, and the "vals" now refer to mappings of these unmapped generators.

Mathematically, we note that there is exactly one ARTS for each free abelian group, and that all of the RTS's of the same rank correspond to the same ARTS. This is useful, not because we intend to abandon harmonic mappings or regular temperaments, but because when we prove theorems about scales which depend on nothing but their properties as a subset of the ARTS, these theorems will still apply to all of the mappings and tunings of the same scale.

Properties:

There are a number of important properties that exist to classify the various scales of an ARTS, which we list below.

Periodicity: an abstract scale of an ARTS can be said to be an abstract periodic scale if there exists some element p in S such that, for all elements e in S, e+n*p is also in S for every natural number n.

Rank: the rank of an abstract periodic scale in an ARTS is the rank of the smallest subgroup containing that scale.

Epimorphicity: A subset S of an ARTS A is called "abstractly epimorphic" if there exists an element h in Hom(A,Z) such that the restriction of h to S is a bijection.

Unmapped Fokker Block: The