Musimathics: The Mathematical Foundations of Music 

Bradley Lehman

Physics Today 61 (12), 54–55 (2008);

https://doi.org/10.1063/1.3047685

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30 June 2025 16:46:49

 Sound that doesn’t
books sound so simple
Musimathics they describe complicated phenomena.
The systematic sequence of material
A website gives a detailed outline of
both volumes and includes a list of er-
The Mathematical will serve not only experts as an in- rata to be corrected in later printings
Foundations of Music valuable desk reference but also in- (http://www.musimathics.com); Loy
structors for a university course—I’d es- responds cordially and quickly to sub-
Gareth Loy timate about two semesters per volume. mitted suggestions. Both volumes and
Volume 1
Loy’s approach, as presented in the another website (http://www.musimat
MIT Press, Cambridge, MA, 2006.
preface of volume 2, is refresh- .com) present Loy’s free
$52.00 (482 pp.).
ing: “I believe that enlightened MUSIMAT programming lan-
ISBN 978-0-262-12282-5
common sense and inference guage, which resembles C++
Volume 2 are the whole of mathematics or C# and contains some built-
MIT Press, Cambridge, MA, 2007. and that inference itself flows in data types and methods for
$52.00 (562 pp.). from enlightened common musical applications. Such a
ISBN 978-0-262-12285-6 sense.” Accordingly, he uses language for experimentation
Reviewed by Bradley Lehman enlightening analogies and ex- is welcome, and I feel ungra-
In the preface of volume 1 of Musimath- cellent illustrations. He uses cious in rejecting a free tool,
ics: The Mathematical Foundations of the measuring of tides via but I disagree with too many
Music, Gareth Loy assures readers that water moving through pipes as of the MUSIMAT details. The
the two-volume set presumes no ad- a model for digital sampling. programming language re-
vanced background in calculus or The algebra of complex num- flects the first volume’s perva-
trigonometry. The text of volume 1 can bers gives us a gumdrop on a sive misuse of the term
serve as a good general primer of New- rotating turntable, which rep- “rhythm.” I would rename its

30 June 2025 16:46:49

tonian physics, and Loy develops the resents a phasor function. “Rhythm” data type either
mathematical principles with lucid ex- Flashlights project the gum- “Duration” or “Meter,” be-
planations. The second and more chal- drop’s shadows onto two per- cause the main purpose of
lenging volume presents a comprehen- pendicular screens; the traces that type is apparently to
sive development of mathematical give the shapes of sine and co- subdivide a regular bar.
models that apply to music, sound, sine waves. A device resembling a pan- “Rhythm” broadly includes repeated
recording, synthesis, and acoustics; it tograph shows another way to draw si- or varied patterns, articulation, accen-
completes an excellent set. Every chap- nusoid graphs with a pen, while e raised tuation, rests, and more. To play some-
ter of Musimathics has a welcome sum- to imaginary powers spins us through thing rhythmically can be profoundly
mary at the end that bypasses the cal- transforms. A piston demonstrates re- expressive; to play merely metrically,
culations to reiterate the concepts. The sistance and inductance. A 1913 snap- as machines do, is usually dull and too
material in both volumes is well fo- shot of a race car, visually distorted by predictable.
cused, interrelated, and admirably con- the era’s photographic technology, illus- There are also some inaccurate and
cise throughout more than 1000 pages. trates convolution. unfortunately misleading points in
Loy is a musician, composer, and What would I put on my wish list for Loy’s coverage of pitch and intonation.
president of Gareth Inc, a company that Loy’s intellectual voyage? At the top For example, in both volumes Loy de-
provides software engineering and con- would be a CD or an extensive set of web fines “glissando” and “portamento” in-
sulting services internationally. He is an downloads to demonstrate exactly what correctly as each other. The biggest con-
interdisciplinary thinker with fine ped- those interesting-looking waves and fil- ceptual predicament is providing
agogical instincts. Some of the material ters do for us. Listening to the sound appropriate names for all the notes.
in volume 2 is so abstruse that it would would allow me to understand, as a Historically, correct diatonic spelling
take a year or two of expert, professo- practicing non-electronic musician, why has been vital in tonal music because it
rial guidance and commitment to fol- I should care about or memorize any for- affects both sound and musical expres-
low through all the derivations. It is not mulas from the books. The second vol- sion. Twentieth-century theory and
the author’s fault that the modeling ume illustrates triangular waves (page practice have oversimplified about
equations really are that complicated; 375), phase modulation (page 399), 28 common-practice notes—naturals,
flanging (page 429), and the Indian in- sharps, flats, double sharps, and double
Bradley Lehman works for Angel.com, strument rudra vina (page 442). What do flats—down to the “Western scale”
based in McLean, Virginia, where he the formulations and instrument sound (Loy’s term) of only 12 frequencies per
designs and produces automated tele- octave, with glibly interchangeable
like, beyond the calculations and visual
phone systems. A performing musician and
aids? Loy mentions white noise (page pitch-class names. According to 18th-
composer, he holds a doctoral degree in
34), but I could not find an explanation century textbooks, however, sensitive
harpsichord. His current research involves
in either volume for the “pink noise” cal- musicians were alert to fine nuances in
an unequal tuning system he deduced
ibration button on my equalizer or for everyday practice: for example, per-
from Johann Sebastian Bach’s music and
pedagogy (http://www.larips.com). other colors of noise, such as blue, forming B-flat about one-ninth of a tone
brown, red, and gray. higher than A-sharp. The instruments

54 December 2008 Physics Today © 2008 American Institute of Physics, S-0031-9228-0812-240-2

sometimes had separate holes, keys, or
fingering charts to play the different
recognized and enjoyed first for non-
intellectual reasons. Whether the math- Radiation
notes accurately. MUSIMAT says those
several notes are simply identical by
ematical structure is the foundation, the Oncology
framework, or only the external scaf-
definition (page 440, volume 1). The folding, Loy demonstrates that it has its A Physicist’s-Eye View
“Pitch” data type does not concern ei- own abstract beauty. That beauty, as he Michael Goitein
ther intonation or pitch perception, as states in volume 1, was a gift he en- Springer, New York, 2008. $129.00
its name suggests. Perhaps it should be countered by his own precocious in- (330 pp.). ISBN 978-0-387-72644-1
called “PianoKey” instead, as the data quisitiveness at age 11. Thus Loy sug- The title of Michael Goitein’s book Ra-
type evidently only assigns ordinal gests an alternative subtitle for diation Oncology: A Physicist’s-Eye View
numbers to the 88 available notes on a Musimathics: “Everything I wanted to is a play on words. The concept of
piano. MUSIMAT has a “PitchList” know about music when I was eleven.” “beam’s-eye view” in radiation-therapy
data type to offer some melodic trans- If I were 11 years old again, these well- treatment planning was developed by
position features, but it naively obliter- organized volumes would seem totally Goitein himself to describe the radi-
ates the musical distinction between awesome and inspiring. ographic view of a patient’s anatomy, as
chromatic versus diatonic semitones.
Therefore, according to the printed ex-
ample of the musical couplet “Shave
and a Haircut” (page 442, volume 1), it
yields enharmonically wrong notes.
Loy’s musical assumptions as a com-
poser and writer are understandably ori-
ented toward synthesizer players and
digital technology mavens, not music
historians or expert players of conven-
tional instruments. Musimathics will
draw in physicists and engineers at the
risk of alienating ordinary musicians
who seek straightforward answers to
practical questions. The indexing of both
volumes and the website’s outline do not

30 June 2025 16:46:49

serve, for example, the likely interests of
a cellist—finding sections about string
vibrations, tone production, the friction
and velocity of the bow, and so forth.
Why are the higher notes closer together
on the fingerboard? What are the com-
parative acoustical properties of steel,
gut, or wound strings? I could not find
anything about the double reeds of
oboes or bassoons. A harpsichord
builder studying plucking points or
soundboard resonances would have to
page carefully through both volumes
side by side, seeking the topics of nodes
and modes. My own favorite “musi-
mathic” topic, the history and theory of
unequal tuning methods, gets only a
thin and long-outdated treatment in vol-
ume 1. But, to keep everything in
broader perspective, any of those topics
slighted in musical mathematics could
by themselves require 1000 pages. Loy’s
generalist approach is excellent for
sparking curiosity.
Musimathics and its subtitle suggest
that everything springs from “mathe-
matical foundations,” through intense
calculation and careful modeling. Yet I
lose the sense that most of those musi-
cal and acoustical ideas were already
commonly practiced before the mathe-
matics came along to measure them.
The mathematical modeling is not nec-
essarily a foundation; rather, it de-
scribes accurately some of the things we

December 2008 Physics Today 55 See www.pt.ims.ca/16307-23