BAIN MUSC 525 Post-Tonal Music Theory

Maximally-Even Collections

TERMS & CONCEPTS

Ch. 1: Spatial Relations and Musical Structures diatonicism pandiatonicism spatial relations and pc sets Johnson circle diagram

- crossing lines - dots

specific intervals or c distances general intervals or d distances maximally even property dinner table analogy deep scale property (unique multiplicity of ic)

TWO TYPES OF INTERVALLIC DISTANCES

c-distances d-distances

The clockwise distance between dots, measured in crossing lines

The clockwise distance between dots, measured in dots

Same as opci A measurement of distance within a collection.

Ex. The c distance from C-D is always 2; The c distance from C-E is always 4; etc.

Ex. The d distance from C-D within the C-major collection is 1, whereas the d distance from C-E is 2; etc.

MAXIMALLY-EVEN COLLECTIONS

c1 Collection name Set Class c Collection name Set Class 3 Augmented triad 3-12 (048) 9 Messaien’s Mode 3 9-12 (01245689T) 4 Fully-diminished

seventh chord 4-28 (0369) 8 Octatonic scale 8-28 (0134679T)

5 Major pentatonic scale

5-35 (02479) 7 Diatonic scale 7-35 (013568T)

6 Whole-tone scale 6-35 (02468T) 6 Whole-tone scale 6-35 (02468T)

Quotable

The Dinner Table Analogy “Suppose you have a round dinner table surrounded by twelve evenly dispersed chairs, none of which can be removed. You have invited seven guests to dinner and want guests to be spread out as much as possible around the table.”

The Definition of Maximally Even “…a circle diagram is defined as maximally even if for each d distance there are only one or two possible c distances. And if there are two c distances for a particular d distance, then the c distances are consecutive numbers (such as 2 and 3, 3 and 4, and so forth).” For example, for the major pentatonic collection (C,D,E,G,A):

d distance c distance C-D, D-E, E-G, G-A, A-C 1 2, 3 C-E, D-G, E-A, G-C, A-D 2 4, 5 C-G, D-A, E-C, G-D, A-E 3 7, 8 C-A, D-C, E-D, G-E, A-G 4 9, 10

Source: Johnson, Timothy. Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals.

Emeryville, CA: Key College Publishing, 2003.

1 c – cardinality.