A Formalization Musical "Sets"

Someone2841

For those of you who have taken Music Theory IV (or upper division or even graduate courses on 20th Century Music Analysis), musical "set theory" should be a familiar concept. I use quotation marks because, as those who are familiar with mathematical set theory, musical group theory would be a much more applicable term. Professionally, I am a music theorist, but I also enjoy studying mathematics on the side. It is disappointing that does not find mathematical definitions in music textbooks, but I know many concepts and terminologies from mathematics are applicable to the study of music (especially but not limited to the post-tonal variety).

Anyway, for those who know what I am talking about (those with a musical and mathematical background), I wanted to put some thoughts out there for a few rudimentary mathematical definitions for musical set theory. While one could write a book on these, I was just wondering what people thought about these definitions for pitch-class set classes and pitch-class set congruence.

Where T_n(X) is the transposition of set X up n semitones and I(X) is the mirror of set X, given pitch-class set class S and pitch-class A:

A∈S, S = { X | ∃n, A=T_n(X) ∨ A=T_n(I(X)) }

That is, given that A is a member of S, S must include all transpositions of A and the transpositions of its inversion as well.


Given pitch-class sets A and B and pitch-class set class S:

A∈S ^ B∈S → A ≅ B

That is, if A and B are both an element of S, A and B are congruent (belong to the same pitch-class set class).

I can't say I'm an expert as this, so feel free to propose better definitions or just criticize mine. If anyone has other definitions they would like to discuss, that would be cool too.

Stephen Tashi

I suggest that you explain the concepts involved and then perhaps someone can comment on whether your translation to set notation is correct.

Apparently the population of forum members who took this type of music theory is quite small. (I took the old fashioned kind, where you analyze Bach chorales.)

BWV

this is a core concept in contemporary music theory (http://en.wikipedia.org/wiki/Set_theory_(music) )

its needed to analyze non-tonal music, but tends to be too general for tonal works (for example major and minor chords are equivalent as a minor triad is a mirror inversion of a major triad)

Not sure about the definition - I read it as saying that a pc-set includes every possible transposition of an individual member the set, rather than of the set as a whole, i.e. (0,1) and (0,2) would be equivalent since (0,2) transposes the second member of the set by a semitone