1996 Abridged NotesKeywords: scale, music education, music theory, information theory, graph theory, set theory, visualization.
Edgar Matias, University of Toronto
James McGowan, Eastman School of Music
This is an extension of a paper written in 1993. We recommend that you read the original paper first.
Note also that while that original paper was quite basic and easy to understand, this extension is very technical and cryptic. Unless you have some knowledge or training in pitch class set theory, you're not likely to understand much of it.
Under the present theory, the following definitions apply for sets formed under M6:
component = an interval class used to form a pcset ("pitch class set")
diagonal component = ic5 ("interval class 5" or perfect fifth)
lateral component = ic6 (tritone or augmented fourth)
dangling component = ic0 for which there is no other pc within ic5 or ic6
("formed under M6" is another way of stating that the lateral component is mapped to ic6.)
Lateral and diagonal components may be referred to collectively as "non-dangling components."
This theory is based on the difference between the intervallic "have" and "need" of a pcset.
The state of music theory at the time of this writing (1996) is very good at describing the intervallic "have" of sets. For example, the interval vector is basically an itemized list of a set's intervallic "haves."
However, traditionally sets or scales were formed from the perspective of intervallic "need."
For example, the major scale (Forte/Rahn: 7-35) can be formed by stringing together perfect fifths (ic5 / diagonal components). In other words, only one interval class is "needed" to form the major scale: ic5.
Thus, the major scale is a diagonally-formed scale, since it "needs" only diagonal components (ic5 / perfect fifths), eventhough it has a lateral component (5 and B). So as not to confuse the two, we use the following NEED(have) notation:
"(l&d)" indicates that the major scale has lateral & diagonal components.
"DB" indicates that the major scale can be formed using *only* DIAGONAL components (perfect fifths) or BOTH lateral and diagonal components. It cannot be formed using *only* lateral components.
A table showing the intervallic NEED(have) component breakdown formed under M6 is available online.
Agmon, Eytan (1989). A Mathematical Model of the Diatonic System. Journal of Music Theory, 29:249-270
Clough, John and Gerald Myerson (1985). Variety and Multiplicity in Diatonic Systems. Journal of Music Theory, 29: 249-270.
Copyright 1996 Edgar Matias & James McGowan