There is abundant material on the Web about the tuning problem. Brian Suits
at Michigan Tech has an excellent Physics of Music site. The Connections site at Rice
has a page on the Circle
of Fifths. If you really want to know about pianos, see
The Equal Tempered
Scale and Some Peculiarities of Piano Tuning
by Jim Campbell, on the Precision Strobe Tuners Home Page. Musical scales as psychological constructs is part of
notes to a UCLA course by Systematic Musicology Professor Roger A. Kendall.
Rayleigh's The
Theory of Sound is available on the EnviroMeasure page.
The wave
equation and continued fractions are presented rapidly in
Eric Weisstein's
World of Mathematics, courtesy of Wolfram Research. A more elementary and
attractive exposition of continued fractions is on Ron Knott's
page at Surrey. For more bird songs
you can check out my
webpage at Stony Brook and the links therein.
Sonograms and slow-motion bird songs
are produced with Martin Hairer's Amadeus software.

But along with this note, the *fundamental*, each cord or pipe
has a series of higher-frequency modes of vibration. A cord may vibrate
as two cords of half its length joined end-to-end; or three cords of
one-third the length, or four ... ; and similarly for the pipe. Since,
with other things being constant, frequency is inversely proportional to
length, these vibrations will give notes at twice, three times, four times ...
the fundamental frequency. These notes are called the *higher harmonics*
of the fundamental.

Here is an example from nature. The Hermit Thrush *Catharus guttatus,*
fairly common in eastern
North America, typically sings phrases where one long note is followed by a
sequence of more rapid ones. Examine the sonogram record of this particular
phrase from this particular bird. This record plots against time a frequency
analysis of the sound; the color encodes the power delivered at the various
frequencies. Here the first long note shows higher harmonics of order 2, 3,
4, 5, and 6 . The frequencies are 3531 (the fundamental), 6976, 10336, 13953,
17399 and 20758 cycles per second. In standard notation (A4=440)
these correspond approximately to the notes A7, A8, E9, A9, C#10, E10.

Sonogram of Hermit Thrush song. Frequency is plotted vertically from 0 to 22050 Hz. Time intervals are in seconds. |
The first note of the Hermit Thrush's song with its higher harmonics written as separate notes on a musical staff. The bird sings 4 octaves higher than scored. |

The higher harmonics give a set of pitches that are naturally related to the
fundamental. These pitches can then be heard by themselves. This is clear
in the next example, where a Song Sparrow *Melospiza melodia* (common
in eastern
North America) alternates between the second and third harmonics (the
fundamental is obscured by the second harmonic) in the beginning of its song.

Sonogram of Song Sparrow song. Frequency is plotted vertically from 0 to 11025 Hz. Time intervals are in seconds. |
The beginning of the Song Sparrow's song. The bird sings 4 octaves higher than scored. |

How to incorporate these natural harmonics
into a musical scale is the basic mathematical problem in music.
The problem already exists for the interval sung by the Song
Sparrow, between the second and third harmonics (this interval
is known in music as a *perfect fifth*). In this column
we will examine this particular case and the approximate solution, known as
Equal Temperament, which is used in tuning pianos today.

--*Tony Phillips
SUNY at Stony Brook*

- 1. Natural harmony
- 2. Frequencies and notes
- 3. The equal-tempered keyboard
- 4. The wave equation and bird song

© copyright 2000, American Mathematical Society.