A musical system that is built entirely from 2:1 intervals (pure octaves

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Group Work: Pythagorean Tuning
A musical system that is built entirely from 2:1 intervals (pure octaves) and 3:2 intervals (pure
perfect fifths) is called a Pythagorean tuning system. These questions will walk you through its
construction.
(1) Start with Ab4 = 415.3047 Hz (this seems a little weird, but the reason will be clear later).
Every other Ab in the system will be related to this one by a pure octave. What are the
frequencies of Ab3 and Ab5?
(2) As we have seen, there are 12 pitches within a given octave. Your goal is now to figure out
the frequencies of the other 11 pitches in the octave. In a Pythagorean system, you can only
“reach” the 11 other pitches in the octave by stacking up a certain number of pure fifths above
the given pitch. If the resulting pitch is too high, then you need to bring it down the appropriate
number of octaves so that it is within range. ‘
Thirteen pitches are listed below, on the left column of the table. In the second column, fill in
the number of fifths you need to stack above Ab4 in order to reach the given pitch (without
regard to octave). It will help to remember that on the keyboard, a perfect fifth encompasses
seven semitones. In the third column, fill in the number of octaves you need to move the
resulting stacked-fifths pitch down in order to get it into the proper range. Based on your second
and third columns, fill in the factor by which you must multiply the frequency of Ab4 in order to
get the frequency of the pitch in the next column. Write in that frequency in the last column.
Pitch
Ab4
Number of fifths
stacked above or
below Ab4
0
Number of
octaves down to
proper register
0
Multiplicative
factor
Frequency
(3/2)0 /20 1.0000
415.3047 Hz
440.0000 Hz
A4
Bb4

B4
C5 = 72
C#5
D5
Eb5
1
(3/2)1 /20 1.5000
0
E5
F5
F#5
G5
G#4

(3) Generating the rest of the pitches in the system is now simply a matter of adding (or
subtracting) pure octaves from the 12 pitches that you’ve generated. In this system, almost any
“perfect fifth” (an interval of seven pitch classes) will be pure (i.e., will embody a frequency
ratio of 3:2). But some fifths will be out of whack. Looking at the first 12 pitches you generated
in the table, which ones won’t have a pure fifth above it in the Pythagorean system? What is the
frequency ratio between that frequency and the pseudo-fifth above it? How close is it to the pure
3:2 ratio? (Write the answer as a product of powers of 2 and 3, and convert it to a decimal.)
(4) Suppose you were not limited to just 12 pitches to the octave, but could go on stacking fifths
as long as you’d like. How far would you need to go in order to achieve a closed musical
system? In other words, how many pitches per octave would you need in order to have a system
made up completely of pure fifths and pure octaves? Justify your answer!
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