Basic Analysis of Z12 Music Chords (UMAP)

This module teaches how to use the additive group Z12 and a group of musical transformations called the twelve-tone operators in analyzing any chord playable on a piano. Preliminary explanation is included of the requisite musical terms and the historical background to the music elicits Z12 analysis. Chords employed by contrasting Viennese composers Schoenberg and Schubert motivate the analysis, which begins by representing chords in Z12, followed by evaluating interval content, defining transformational equivalence, and classifying these chordal representations. Interesting geometric representations, called Krenek diagrams, complement the algebraic classification. Suggestions are given for further investigation of music with associated mathematical analyses.

Table of Contents:

INTRODUCTION

MUSICAL TERMS

HISTORICAL BACKGROUND

PITCH CLASS SETS

INTERVAL VECTORS

TWELVE-TONE OPERATORS

SET CLASSES

PRIME FORMS

FURTHER DIRECTIONS

SOLUTIONS TO SELECTED EXERCISES

APPENDIX: SUGGESTIONS FOR INSTRUCTORS

REFERENCES

ACKNOWLEDGMENTS

ABOUT THE AUTHORS