Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Introduction
- 1 Waves and harmonics
- 2 Fourier theory
- 3 A mathematician's guide to the orchestra
- 4 Consonance and dissonance
- 5 Scales and temperaments: the fivefold way
- 6 More scales and temperaments
- 7 Digital music
- 8 Synthesis
- 9 Symmetry in music
- Appendix A Bessel functions
- Appendix B Equal tempered scales
- Appendix C Frequency and MIDI chart
- Appendix D Intervals
- Appendix E Just, equal and meantone scales compared
- Appendix F Music theory
- Appendix G Recordings
- References
- Bibliography
- Index
4 - Consonance and dissonance
Published online by Cambridge University Press: 05 March 2013
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Introduction
- 1 Waves and harmonics
- 2 Fourier theory
- 3 A mathematician's guide to the orchestra
- 4 Consonance and dissonance
- 5 Scales and temperaments: the fivefold way
- 6 More scales and temperaments
- 7 Digital music
- 8 Synthesis
- 9 Symmetry in music
- Appendix A Bessel functions
- Appendix B Equal tempered scales
- Appendix C Frequency and MIDI chart
- Appendix D Intervals
- Appendix E Just, equal and meantone scales compared
- Appendix F Music theory
- Appendix G Recordings
- References
- Bibliography
- Index
Summary
In this chapter, we investigate the relationship between consonance and dissonance, and simple integer ratios of frequencies.
Harmonics
We saw in Sections 3.2 and 3.5 that when a note on a stringed instrument or a wind instrument sounds at a certain pitch, say with frequency ν, sound is essentially periodic with that frequency. The theory of Fourier series shows that such a sound can be decomposed as a sum of sine waves with various phases, at integer multiples of the frequency ν, as in Bernoulli's solution (3.2.7) to the wave equation. The component of the sound with frequency ν is called the fundamental. The component with frequency mν is called the mth harmonic, or the (m – 1)st overtone. So, for example, if m = 3 we obtain the third harmonic, or the second overtone.
Figure 4.1 represents the series of harmonics based on a fundamental at the C below middle C. The seventh harmonic is actually somewhat flatter than the B♭ above the treble clef. In the modern equally tempered scale, even the third and fifth harmonics are very slightly different from the notes G and E shown above – this is more extensively discussed in Chapter 5.
There is another word which we have been using in this context: the mth partial of a sound is the mth frequency component, counted from the bottom. So, for example, on a clarinet, where only the odd harmonics are present, the first partial is the fundamental, or first harmonic, and the second partial is the third harmonic.
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- Information
- Music: A Mathematical Offering , pp. 144 - 160Publisher: Cambridge University PressPrint publication year: 2006