pc-set

[Joe Monzo]

The term "pc-set" is an abbreviation for "pitch-class set", and refers to a set of distinct integers (i.e., no duplicates) belonging to some EDO, which represent the pitch-classes in a collection of musical notes. It is written in square brackets, with each integer separated by commas. The integers are calculated modulo the cardinality of the EDO, and listed in order from smallest to largest. By convention, "0" is equated with the note "C". Here are some examples in 12-edo (i.e., the integer set 0 to 11):

 C-minor triad [0,3,7] C-major triad [0,4,7] D-minor triad [2,5,9] G-dominant-7th chord [2,5,7,11] C-major pentatonic [0,2,4,7,9] A-minor pentatonic [0,2,4,7,9] C-major scale [0,2,4,5,7,9,11] A-harmonic-minor scale [0,2,4,5,8,9,11] whole-tone scale 1 [0,2,4,6,8,10] whole-tone scale 2 [1,3,5,7,9,11] octatonic diminished scale 1 [0,1,3,4,6,7,9,10] octatonic diminished scale 2 [0,2,3,5,6,8,9,11] chromatic scale [0,1,2,3,4,5,6,7,8,9,10,11] 6-note 4th-chord built on D [0,2,3,5,7,10]

(This last example is that used by Schoenberg for the main theme of his Kammersymphonie: D:G:C:F:Bb:Eb/D#.)

The idea of pc-set was originally introduced in the 1960s in reference to atonal musical practice which assumed 12-edo as the tuning, however, the concept is extendable to any EDO. The reader must be aware of the cardinality of the EDO in order to understand the meaning of the integers in the pc-set; if no cardinality is mentioned, then it must be assumed that cardinality = 12.

The standard reference for 12-edo pc-sets is Forte 1973, which contains an appendix listing all pc-sets of cardinalities 3 thru 9, along with their interval-vectors. Forte's naming scheme is to provide the cardinality of the pc-set, a hyphen, and then simply an integer which designates that set's order within the cardinality as ranked by size of the integers in the pc-set, starting with the leftmost and continuing to the right.

Larry Solomon has extended Forte's nomenclature by separating Forte's prime-forms from their inversions, arguing that musical usage does not usually consider sets and their inversions to be equivalent, and using a suffix "B" after the Forte set-name to indicate the inversion. Solomon also argues that the sets's interval strings are a better nomenclature for the sets, giving a unique identifier to each set which is not dependent on Forte's more-or-less arbitrary catalog order.

REFERENCES

Babbitt, Milton. 1960.

"Twelve-Tone Invariants as Compositional Determinants,"
The Musical Quarterly 46/2, 246-259.

Forte, Allen. 1964.

"A Theory of Set-Complexes for Music,"
Journal of Music Theory 8 [Winter], 136-184.

Forte, Allen. 1973.

The Structure of Atonal Music
Yale University Press, New Haven and London
Library of Congress catalog card number: 72-91295
ISBN-10: 0-300-02120-8, ISBN-13: 978-0300021202

Solomon, Larry. Music-Theory webpages:

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