2. Atonal theory
2.1. Introduction
Atonal theory is a branch of music theory that deals with atonal music, much of it written using the twelve-tone method. The theory uses mathematical concepts, most notably from set theory and group theory, to model collections of pitches and pitch-classes and transformations of these collections. It is a flexible model, and offers compositional and analytical insight to many different kinds of music, including some scalar and tonal music. It is particularly helpful when working with nonstandard harmonies in the 12‐EDO system, whether the context is tonal, quasi‐tonal, or atonal.
A thorough introduction to atonal theory is beyond the scope of this documentation. Instead, this page explains the conventions used in the pctheory package. These are based on the work of Robert Morris, so readers familiar with Composition with Pitch‐Classes: A Theory of Compositional Design should already be familiar with them.
2.2. Pitch and pitch‐class
We begin with pitch‐classes in twelve‐tone equal‐temperament. A pitch‐class is often represented with a name like C, D, E, etc. In atonal theory, it is often useful to assign a number to each pitch‐class. So C becomes 0, C♯ becomes 1, D becomes 2, D♯ becomes 3, etc. Often theorists use T for B♭ and E for B♮ instead of the numbers 10 and 11, but pctheory follows the convention of using A and B instead of T and E. This avoids confusion with the use of T to represent the transposition operator, and also follows the standard for hexadecimal digits.
It is also possible to work directly with individual pitches. In pitch‐space, 0 is C4, 1 is C♯4, 2 is D4, -1 is B3, -2 is B♭3, etc. Positive numbers represent notes above C4 and negative numbers represent notes below C4. This is in contrast to the standard MIDI numbering, where 60 is C4. It is then possible to make sets and permutations of pitches, just as we would with pitch-classes. This is useful when working with Xenakis's sieves and fixed‐pitch formations.
2.3. Sets
Any collection of pitch-classes, whether a chord, scale, or something else, can be modeled with a mathematical set. A set is written with curly braces {} and contains 0 or more elements. Sets cannot contain duplicate elements, and the elements are not ordered with respect to each other. In atonal theory, we call sets of pitch‐classes pcsets. We could represent the diatonic scale as the pcset {4, 9, 0, B, 7, 2, 5}. Often it would be notated {0, 2, 4, 5, 7, 9, B}, which is exactly the same set. I show both orderings to demonstrate that order does not matter in mathematical sets.
2.4. Unordered operations
Because the pitch-classes are represented as numbers, it is possible to perform mathematical operations on them. All pitch-class operations are done mod 12. Transposition (T) is addition: Tn(x) = x + n mod 12. Inversion (I) is often defined as I(x) = 0 - x mod 12, but it is also identical to 11x mod 12. We will use the latter definition. The operation of multiplication (M) is defined as Mm(x) = mx mod 12. Inversion is M11. Other useful M operators are M5 and M7. M1 is an identity operator; it leaves pitch-classes unchanged. The combined transposition and inversion operator, TnI is redefined as TnM11(x) = 11x + n mod 12. The general TnMm operator is defined as TnMm(x) = mx + n mod 12. Any Tn or TnMm operator can be represented using the UTO class.
2.5. Set-classes
Any two pcsets that can be mapped into each other under the operations of transposition and/or inversion can be said to belong to the same set‐class. Any set‐class can be represented by its prime form, a special set in the class. The calculation of prime forms is beyond the scope of this documentation, and is done automatically in pctheory using the SetClass12 class. There are 224 set‐classes in the chromatic system.
2.6. Rows, permutations, and segments
Twelve‐tone rows are nonduplicative orderings of the twelve pitch‐classes. This means they are mathematical permutations. In atonal theory, permutations of pitch‐classes are often called pitch‐class segments, or pcsegs. A pcseg may or may not have duplicate pitch‐classes, and can be of any length. A row is a 12‐note nonduplicative pcseg. The pctheory package has multiple functions for generating random pcsegs. Any melody can be modeled as a pcseg, or more precisely as a pitch segment (pseg). Any chord may also be modeled as a pcseg or pseg – whether the ordering is from top to bottom or from bottom to top is arbitrary.
2.7. Microtonal pitch and pitch-classes
The pctheory package contains functionality for working with the 24-tone equally‐tempered microtonal scale. Microtonal pitch‐classes are represented with the numbers 0 to 23, written as 00, 01, 02, ... 21, 22, 23. Microtonal pitches follow the same convention as chromatic pitches, but now we move up a quarter‐step with each number instead of a half‐step. Pcsets (and pcsegs) can be made of microtonal pitch‐classes. The functionality for microtonal pcsets is somewhat more limited because of the comparatively enormous number of microtonal set‐classes – 352,698 set‐classes total.