2:3:4 Harmony within the tritave
In the Pythagorean tuning system, the fifth is used to generate a scale of 12 notes per octave. In this paper, we use the octave to generate a scale of 19 notes per tritave; one can play this scale on a traditional piano. In this system, the octave becomes a proper interval and the 2:3:4 chord a proper chord. We study harmonic properties obtained from the 2:3:4 chord, in particular composition elements using dominants, inversions, major and minor chords, and diminished chords. The Tonnetz (array notation) turns out to be an effective tool to visualize the harmonic development in a composition based on these elements. 2:3:4 harmony may sound pure, yet sparse, as we illustrate in a short piece.
Musimathics: The Mathematical Foundations of Music
Volume 2 of Musimathics continues the story of music engineering begun in Volume 1, focusing on the digital and computational domain. Loy goes deeper into the mathematics of music and sound, beginning with digital audio, sampling, and binary numbers, as well as complex numbers and how they simplify representation of musical signals. Chapters cover the Fourier transform, convolution, filtering, resonance, the wave equation, acoustical systems, sound synthesis, the short-time Fourier transform, and the wavelet transform. These subjects provide the theoretical underpinnings of today's music technology. The examples given are all practical problems in music and audio. Additional material can be found at http://www.musimathics.com.
A Guide to Musical Temperament
Part 1 Foreword Part 2 Preface Part 3 Part 1: Main Material Chapter 4 1. The Basis for Temperament Chapter 5 2. Selected Temperaments Chapter 6 3. Musical Aspects Chapter 7 4. Generating Tuning Instructions Chapter 8 5. The Tuning Process Chapter 9 6. Instructions for Setting Selected Temperaments Chapter 10 7. A Historical Overview Part 11 Part 2: Supplementary Material Chapter 12 8. Starting Notes, Pitch References, and Transposition Chapter 13 9. Theoretical and Equal-Beating Versions Chapter 14 10. Near-Equal Temperament Chapter 15 11. Derivation of the Cent Equation Chapter 16 12. Temperament Data Chapter 17 13. Temperament Spreadsheets Chapter 18 14. Instructions in Musical Notation Part 19 Glossary Part 20 Bibliography Part 21 Index Part 22 About the Author