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Music: A Mathematical Oering
Dave Benson
Department of Mathematics, Meston Building, University
of Aberdeen, Aberdeen AB24 3UE, Scotland, UK
Home page: http://www.maths.abdn.ac.uk/∼bensondj/
E-mail address: \
/
\
/b
\
e/n
\
s/o
\
n/d
\
j/
\
/ (without the slashes)
at maths dot abdn dot ac dot uk
Date: 23rd August 2007
Version: Web
Dave Benson 1995–2007. Please email comments and cor-
rections to the above email address. The latest version in Adobe pdf format
can be found at
http://www.maths.abdn.ac.uk/∼bensondj/html/maths-music.html
I have noticed many people putting old versions of this text online, especially
on the usenet group alt.binaries.e-book.technical: PLEASE, PLEASE don’t
do this. The text is regularly updated, and your version is almost always out
of date, sometimes by several years. If the date you are reading this diers
by more than a few months from 23rd August 2007 then you can be sure that
you are reading an out of date copy. Go to my home page for a more up to
date copy.
This work is c
To Christine Natasha
iii
Ode to an Old Fiddle
From the Musical World of London (1834);
1
The poor fiddler’s ode to his old fiddle
Torn
Worn
Oppressed I mourn
B a d
S a d
Three-quarters mad
Money gone
Credit none
Duns at door
Half a score
Wife in lain
Twins again
Others ailing
Nurse a railing
Billy hooping
Betsy crouping
Besides poor Joe
With fester’d toe.
Come, then, my Fiddle,
Come, my time-worn friend,
With gay and brilliant sounds
Some sweet tho’ transient solace lend,
Thy polished neck in close embrace
I clasp, whilst joy illumines my face.
When o’er thy strings I draw my bow,
My drooping spirit pants to rise;
A lively strain I touch—and, lo!
I seem to mount above the skies.
There on Fancy’s wing I soar
Heedless of the duns at door;
Oblivious all, I feel my woes no more;
But skip o’er the strings,
As my old Fiddle sings,
“Cheerily oh! merrily go!
“Presto! good master,
“You very well know
“I will nd Music,
“If you will nd bow,
“From E, up in alto, to G, down below.”
Fatigued, I pause to change the time
For some Adagio, solemn and sublime.
With graceful action moves the sinuous arm;
My heart, responsive to the soothing charm,
Throbs equably; whilst every health-corroding care
Lies prostrate, vanquished by the soft melliuous air.
More and more plaintive grown, my eyes with tears o’erow,
And Resignation mild soon smooths my wrinkled brow.
Reedy Hautboy may squeak, wailing Flauto may squall,
The Serpent may grunt, and the Trombone may bawl;
But, by Poll,
∗
my old Fiddle’s the prince of them all.
Could e’en Dryden return, thy praise to rehearse,
His Ode to Cecilia would seem rugged verse.
Now to thy case, in annel warm to lie,
Till call’d again to pipe thy master’s eye.
∗
Apollo.
1
Quoted in Nicolas Slonimsky’s Book of Musical Anecdotes, reprinted by Schirmer,
1998.
Contents
Introduction
ix
Books
xii
Acknowledgements
xiii
Chapter 1. Waves and harmonics
1
1.1. What is sound?
1
1.2. The human ear
3
1.3. Limitations of the ear
8
1.4. Why sine waves?
13
1.5. Harmonic motion
14
1.6. Vibrating strings
15
1.7. Sine waves and frequency spectrum
16
1.8. Trigonometric identities and beats
18
1.9. Superposition
21
1.10. Damped harmonic motion
23
1.11. Resonance
26
Chapter 2. Fourier theory
30
2.1. Introduction
31
2.2. Fourier coe
cients
31
2.3. Even and odd functions
37
2.4. Conditions for convergence
39
2.6. Complex coe
cients
47
2.7. Proof of Fejer’s Theorem
48
2.8. Bessel functions
50
2.9. Properties of Bessel functions
54
2.10. Bessel’s equation and power series
55
2.11. Fourier series for FM feedback and planetary motion
60
2.12. Pulse streams
63
2.13. The Fourier transform
64
2.14. Proof of the inversion formula
68
2.15. Spectrum
70
2.16. The Poisson summation formula
72
2.17. The Dirac delta function
73
2.18. Convolution
77
iv
Preface
ix
2.5. The Gibbs phenomenon
43
CONTENTS
v
2.19. Cepstrum
78
2.20. The Hilbert transform and instantaneous frequency
79
2.21. Wavelets
81
Chapter 3. A mathematician’s guide to the orchestra
83
3.1. Introduction
83
3.2. The wave equation for strings
85
3.3. Initial conditions
91
3.4. The bowed string
94
3.5. Wind instruments
99
3.7. Eigenvalues of the Laplace operator
103
3.8. The horn
113
3.9. Xylophones and tubular bells
114
3.10. The mbira
122
3.11. The gong
124
3.12. The bell
129
3.13. Acoustics
133
Chapter 4. Consonance and dissonance
136
4.1. Harmonics
136
4.2. Simple integer ratios
137
4.3. History of consonance and dissonance
139
4.4. Critical bandwidth
142
4.5. Complex tones
143
4.6. Articial spectra
144
4.7. Combination tones
147
4.8. Musical paradoxes
150
Chapter 5. Scales and temperaments: the vefold way
153
5.2. Pythagorean scale
154
5.3. The cycle of fths
155
5.4. Cents
157
5.5. Just intonation
159
5.6. Major and minor
160
5.7. The dominant seventh
161
5.8. Commas and schismas
162
5.9. Eitz’s notation
164
5.10. Examples of just scales
165
5.11. Classical harmony
173
5.12. Meantone scale
176
5.13. Irregular temperaments
181
5.14. Equal temperament
189
5.15. Historical remarks
192
3.6. The drum
109
5.1. Introduction
154
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