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Problem set 3
Mathematics of music
Due Friday April 4th
Readings
Math exercises
1. Just tuning. Complete the table of frequency, pitch, and pitch class intervals. The intervals in bold belong to a just-tempered
scale. You might find Excel helpful so you don’t have to do everything by hand.
Frequency ratio
(round to three
decimal places
if needed)
1/1
Size of interval, in
cents (round to two
decimal places)
Name of interval
(names will appear
more than once)
Letters of pitch
classes forming this
interval, starting on C
0.00
200.00
unison
C-C
Smallest interval
between
representative pitch
classes, in cents
0.00
perfect fifth
perfect fifth (pure)
C-G
C-G
500.00
498.04
octave
C-C
0.00
9/8
5/4
400.00
4/3
1.498
3/2
5/3
500.00
700.00
701.96
900.00
15/8
2/1
1100.00
1200.00

2. Intervals in just tuning.
a. Label the lines on the scale generation template attached with fractions
representing the frequency ratios in just tuning.
b. Which interval is worst approximated (in cents) by an equally tempered interval?
Other than the octave, which is best approximated?
c. How many pure fifths are there in this scale? Draw lines on the template and
identify them by letter names. (There is more than one.)
d. How many pure major thirds are there? Draw lines on the template and identify
them by letter names.
e. How could you use this idea to find frequencies for the black notes on the piano?
Give two plausible frequency ratios for the interval C-F#. Draw lines indicating
the intervals you used.
3. Pythagorean tuning. For each of the following frequencies f = 1, 3, 9, 27, …, 312, find
the frequency  that (1) belongs in the same pitch class as f and (2) lies in the interval
1   2. Write your answers as powers of 2 and 3. What is the approximate value of
the last number in the sequence? New: Make a spreadsheet and find the pitch intervals
(in cents) that correspond to these frequency ratios.
4. The 
Bohlen-Pierce scale. The Bohlen-Pierce scale is a 13-note scale based around the
third harmonic (an octave plus a perfect fifth). Instead of using pure octaves and
approximate fifths, the BP scale has a 1:3 ratio from A4 (440 Hz) to E6 (1320 Hz). This
interval is divided into 13 units of “BP pitch.”
a. Suppose we represent A4 by 0BP and E6 by 13BP. Then D3, the note an octave
and a fifth below A4, equals -13BP. The frequency of -13BP is _______ Hz.
b. Find the frequency-to-BP-pitch conversion equation.
c. Find the BP-pitch-to-frequency conversion equation.
d. Find the exact frequency ratio that represents one unit of BP pitch (not a
decimal).
e. Make a spreadsheet that converts a BP-pitch interval to a frequency ratio, and find
the frequency ratios of 0BP, 1BP, …, 12BP, 13BP.
f. What is the best approximation of an octave by a whole number of BP pitches?
g. The idea behind the BP scale is to make music that contains only odd harmonics,
like the sound of a clarinet. Convert the 5/3, 9/7, and 7/5 frequency ratios into
units of BP pitch. Which ratio is closest to a whole number?
Music exercise
Listen to some of the BP compositions on the Bohlen-Pierce site http://members.aol.com/bpsite/
(follow links to “Early compositions in BP”) or on other web sites. Find two BP pieces by
different composers and compare them. Provide complete composer and title information so I
can listen to the pieces. Here are some questions to get you started, but you don’t have to restrict
yourself to them. You also might find the review of Charles Carpenter’s CD Splat helpful.
1. Which instruments are used? Do they sound good playing this scale?
2. Why do you think each composer chose the BP scale? Do you think he or she makes
effective use of the scale, or does it just seem like a “gimmick”?
3. Are there other things about the pieces that seem unusual?