1. Musical Set Theory See also:WACM Reader, pp. 51–54 Suggested Reading in Paul’s Handouts Modeling Pitch Pitch Classes and Pitch Class Sets Transposition, Inversion, Equivalence Classes Transposition, InversionEquivalence Classes Interval Vector, Common-Tones and Complements Interval VectorCommon-Tones and Complements

2. Modeling Pitch Continuous and Discrete Models Continuous pitch in Tenney, For Ann (Rising)For Ann (Rising) Temperament Blackwood Examples: 14ET, 23ET14ET23ET Methods: Handle everything @ receiving end (rescale synth response to usual MIDI note messages) PBend on note-by-note basis Channelize and use pbend to offset tuning of each ch Height, Chroma, Octave Equivalence Pitch Classes Intervals

3. Pitch Classes and Pitch Class Sets A Pitch class set (pcset) is a collection of pcs In 12ET there are 4,095 distinct pcsets (plus the “empty” pcset) Pitch-space realizations of pcset (many to one)

4. Transposition, Inversion Transposition pitches pcs — mod-12 arithmetic pcsets Inversion pitches pcs pcsets

5. Equivalence classes of pcsets t-setclass, ti-setclass TI set classes (12ET) enumerated in Allen Forte 1973 351 t-setclasses, 224 ti-setclasses (brute force method, group theory techniques — Hook 2007) Prime forms, algorithms (see Reader) Prime forms as a compact database, “reconstituting” the entire space Unifying harmonic language Example — Nauert, SpiritSpirit

6. Interval vector Example: pcset (2 4 5 8 9) Z-relation (distinct set classes, same interval vector) — e.g. all-interval tetrachord classes Interval class123456 # of occurrences 212221

7. Using interval vectors Common-tone theorem Complement theorem An application in the domain of rhythm (Reich, Clapping Music)

8. The Twelve-Tone Branch Early development of musical set theory Forte: “free atonal” music Babbitt: twelve-tone principles Worth exploring: the pc array (Babbitt, Morris, others) Robert Morris, Out and Out (1989) Mel Powell, String Quartet (1982)