Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- Overture: Some Issues Facing the Contemporary American Composer
- Part One Essays on Composers
- Part Two Talks on My Music
- Part Three Essays on Criticism and Aesthetics
- A Some Serial Music Terms
- B Set-Class Table
- C Hexachordal Combinatoriality
- D Two-Row Combinatoriality
- Notes
- Bibliography
- Index
C - Hexachordal Combinatoriality
Published online by Cambridge University Press: 17 February 2023
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- Overture: Some Issues Facing the Contemporary American Composer
- Part One Essays on Composers
- Part Two Talks on My Music
- Part Three Essays on Criticism and Aesthetics
- A Some Serial Music Terms
- B Set-Class Table
- C Hexachordal Combinatoriality
- D Two-Row Combinatoriality
- Notes
- Bibliography
- Index
Summary
Hexachordal combinatoriality depends the properties of the set A, the first six notes of a row:
1. If A can be transposed and inverted so the result has no tones in common with A, I-type combinatoriality is possible.
2. If A can be transposed so the result has no tones in common with A, T-type combinatoriality is possible.
3. If A can be transposed and inverted so the result is the same set as A, RI-type combinatoriality is possible.
4. If A can be transposed so the result is the same set as A, R-type combinatoriality is possible. Here the transposition can always be by 0 semitones, so all rows can produce this type. However, there are some A sets that can produce identity under a nonzero transposition.
We can write the four cases algebraically, where B is the complement of A; that is, A and B have no tones in common, and A and B comprise the aggregate:
1. I-type: B = TnIA for at least one value of n.
2. T-type: B = TqA for at least one value of q.
3. RI-type: A = TsIA for at least one value of s.
4. R-type: B = TuA for at least one value of u.
An all-(hexachordal)-combinatorial row has a set A that fulfills each case.
- Type
- Chapter
- Information
- The Whistling BlackbirdEssays and Talks on New Music, pp. 367Publisher: Boydell & BrewerPrint publication year: 2010