Intro to Pythagorean tuning
Math and Music
Pitch: Pitch is frequency measured on a logarithmic scale. It accords with our perception than
octaves are equally spaced. The unit of pitch is normally a semitone, the smallest step on a
piano; there are twelve semitones to the octave. A semitone is the smallest distance recognized
in most Western music (music that uses intervals smaller than a semitone is called microtonal).
Pitch can also be measured in cents; one semitone equals 100 cents. Most people can’t hear
intervals smaller than six cents. The frequency-to-pitch conversion equation is
p  2112log 2 ( /27.5) , where p is measured in semitones and  is measured in Hertz.
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Warmup questions
1. What is the pitch of A440?
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2. Explain why the numbers 2, 21, 12, and 27.5 appear in the conversion equation.
3. Use the rules of logarithms to rewrite the equation in the form p  A12log 2 ( ) .
4. What effect does multiplying a frequency by a factor r have on its corresponding pitch?
5. Find the pitch-to-frequency conversion equation. How would it change if pitch were
measured in cents?
 frequency of middle C? This C is
6. “Middle C” corresponds to the pitch “60.” What is the
also called C4.
Pitch class: The pitch class of any given pitch is the set of all pitches separated from it by a
whole number of octaves. Pitch classes are labeled by letter names. For example, the pitch class
“A” consists of the pitches {…, 9, 21, 33, 45, 57, 69, …}, also known as {…, A0, A1, A2, A3,
A4, …}. Humans—and even cats—naturally identify pitches that are separated by an octave as
“the same” in some sense.
7. What letter name is attached to the pitch class of “0”?
8. Explain why every pitch class has exactly one representative x in the interval 0  x  12 .
Pitch classes are normally identified by this number.
9. Which frequencies are present in the pitch class of the pitch with frequency  ?
10. Every pitch class has exactly one representative frequency  where1   
(fill in
the blank).
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Octaves and fifths: Octaves and fifths are the most consonant
intervals
played
by a stringed or
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wind instrument. (Why?)
11. For each of the following frequencies f = 1, 2, 3, 4, 5, 10, 33, find the frequency  that
(1) belongs in the same pitch class as f and (2) lies in the interval you identified in
question (10) above.
12. For each of the following frequencies f = 1, 3, 9, 27, …, 312, find the frequency  that (1)
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belongs in the same pitch class as f and (2) lies in the interval you identified
in question
(10) above. Write your answers as powers of 2 and 3. What is the approximate value of
the last number in the sequence?
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