1. Introduction
1.1. Introduction
pctheory is a Python module for working with atonal theory. It has classes that model chromatic pitch-classes, microtonal pitch-classes, pitches, and transformations. To make collections, you use standard Python objects, including list, set, and tuple types.
To distinguish chromatic and microtonal pitches and pitch-classes, the term “chromatic system” is used to refer to aspects of atonal theory that apply to twelve‐note equal‐temperament. The term “microtonal system” refers to aspects of atonal theory that apply to twenty‐four‐note equal‐temperament. The term “twelve‐tone system” refers specifically to Arnold Schoenberg's technique of using ordered twelve‐note aggregates; a completely different concept.
Any scale, chord, or other collection of pitches or pitch-classes can be modeled as a mathematical set, called a pitch-class set (pcset) or pitch set (pset). An ordered succession of pitches or pitch-classes is called a pitch-class segment (pcseg) or pitch segment (pseg). A partially-ordered set (poset) is (for the purposes of this module) an ordered succession of sets, lists, and/or individual pitches/pitch-classes. Any melody or musical line can be modeled as a pseg, or more abstractly as a pcseg, while chords can be modeled as either psegs, pcsegs, psets, or pcsets depending on which makes the most sense in any given situation.
1.2. Uses of pctheory
pctheory is a general-purpose platform for the use of atonal theory. There is, of course, no requirement that the music be atonal. Many of the features implemented in pctheory can be readily used in a tonal or quasi-tonal context. pctheory makes it easy to manipulate complex chords (whether “tonal”‐sounding or not), determine relationships between different kinds of chords or pitch-class collections, and perform many calculations quickly and accurately. For example, pctheory allows you to easily create and use rotational arrays (as used in Stravinsky's music). This could be useful for both composition and analysis. One interesting feature of pctheory is its inclusion of tools for working with pitch directly, rather than through the abstraction of pitch-class. This is applicable to music that uses fixed-pitch formations.
Because pctheory is a Python module, it is easy to write simple programs in Python to investigate pitch, pitch-class, and operator relations. If you want to generate all pentachordal set-classes and select only those that contain ic3, you can write a short program that uses pctheory to do this for you. If you want to study all of the subset-classes of a set-class, it is easy to generate them with pctheory. If you want to generate several different invariance matrices, this can be easily done. There is no need to work it out on paper. If you want to transform an array, pctheory can do that with a single method call. Perhaps you want to study all of the pcsets in a set-class. They can be generated with a single method call as well. If you need to know Elliott Carter's number for a particular chromatic chord, that functionality is part of the SetClass12 class, so there is no need to open a reference book. The same goes for standard properties like Forte names and ic vectors.
1.3. Prerequisites
pctheory requires Python 3.10 or newer, as well as the additional modules networkx, numpy, pandas, and pyvis.
1.4. Installation
pctheory is a Python package and can be installed with the command pip install pctheory from the terminal on your computer (Command Prompt or PowerShell on Windows, Terminal on Linux or macOS). Note that you need Python 3.10 or newer. If you are running macOS, you will need to install Python 3.x, and you can do that by following the instructions here. You need to have pip installed – if you are running Linux and need to install pip, you can find instructions here.
1.5. Sample code
Sample code to help you get started with pctheory is available here.
1.6. Copyright and license
pctheory is copyright © 2022 by Jeffrey Martin. This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. To view the GNU General Public License v.3.0, visit https://www.gnu.org/licenses/gpl-3.0.en.html.
1.7. Further reading
Following is a list of books and articles that provide helpful background for pctheory. pctheory uses terms and conventions from Robert Morris's work.
Eckardt, Jason. “Surface Elaboration of Pitch-Class Sets Using Nonpitched Musical Dimensions.” Perspectives of New Music 43 no. 1 (Winter, 2005), 120-140.
Forte, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973.
Link, John. “Harmony in Elliott Carter's Late Music.” Music Theory Online 25 no. 1 (April, 2019).
Mead, Andrew. An Introduction to the Music of Milton Babbitt. Princeton: Princeton University Press, 1994.
Morris, Robert D. Class Notes for Advanced Atonal Music Theory. 2 volumes. Lebanon, NH: Frog Peak Music, 2001.
Morris, Robert D. Class Notes for Atonal Music Theory. Lebanon, NH: Frog Peak Music, 1991.
Morris, Robert D. Composition With Pitch Classes: A Theory of Compositional Design. New Haven: Yale University Press, 1987.
Nauert, Paul. “Field Notes: A Study of Fixed-Pitch Formations.” Perspectives of New Music 41 no. 1 (Winter, 2003), 180-239.
Scott, Damon, and Eric J. Isaacson. “The Interval Angle: A Similarity Measure for Pitch-Class Sets.” Perspectives of New Music 36 no. 2 (Summer, 1998), 107-142.
Straus, Joseph. Introduction to Post-Tonal Theory. Fourth edition. New York: W. W. Norton and Co., 2016.