Table of Contents

Pitch, Frequency, and Musical Scales

Pitch and Frequency

Overtones, Pitch Equivalence, and Musical Scales

The 12-Tone Equal Tempered Scale

Musical Scales within the Chromatic Scale

Logarithms

Basic Musical NotationStaff Notation, Clefs, and Note Positions

Time Signatures and Tempo

Key Signatures and the Circle of Fifths

Some Music Theory Interval and Chords

Diatonic Music

Diatonic Transformations – Scale Shifts

Diatonic Transformations – Inversions, Retrograde

Chromatic Transformation

Composing Your Own Music

Web Resources

Spectrograms and Musical TonesMusical Gestures in Spectrograms

Mathematical Model for Musical Tones

Modeling Instrumental Tones

Beating and Dissonance

Estimating Amplitude and Frequency

Windowing the Waveform: Spectrograms

A Deeper Study of Amplitude Estimation

Spectrograms and Music

Singing
Instrumentals
Compositions
Evaluating Personal Performance
Essay

Analyzing Pitch and Rhythm

Geometry of Pitch Organization and Transpositions

Geometry of Chromatic Inversions

Cyclic Rhythms

Rhythmic Inversion

A Case Study in Rhythm: Bruch’s Lok Nidrei

Construction of Scales and Cyclic Rhythms

Perfectly Balanced Rhythms, XronoMorph

XronoMoprh, Well-formed Rhythms
Comparing Musical Scales and Cyclic Rhythms

Serialism

Composing Your Own Music II

A Geometry of Harmony

Riemann’s Chromatic Inversions

A Network of Triadic Chords

Embedding Pitch Classes with the Tonnetz

Other Chordal Transformations

Tonnetz Patterns in Music

Audio Synthesis in Music

Creating New Music from Spectrograms

Phase Vocoding

How Auto-Tune Works

Time Stretching and Time Shrinking

MIDI Synthesis

Software and Other Resources

Exercise Solutions

Amplitude and Frequency Results

Complex Numbers

Autocorrelation and Periodicity

Music Software

Glossary

About the Author

James S. Walker holds a doctorate from the University of Illinois at Chicago, advised by Louis L. Pennisi. He is a Professor and teaches in the Mathematics Department at the University of Wisconsin-Eau Claire. He has published papers on topics in Fourier analysis, wavelet analysis, logic, image compression, image denoising, and mathematics & music.

Gary W. Don is a professor of music theory at the University of Wisconsin-Eau Claire. He teaches freshman and sophomore theory and aural skills, and upper-division theory courses. Additionally, he holds a doctorate in music theory from the University of Washington, and taught theory and aural skills at Skidmore College in New York before joining the UWEC faculty.