The physics of musical scales: theory and experiment
Dallin S. Durfee∗ and John S. Colton
Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602
(Dated: July 1, 2015)
The following assignments are provided as an aid to students and instructors.
Mathematical exercises
• Exercise 1. Calculate the frequency of the beats generated by the third harmonic of a middle C at 261.63
Hz and the second harmonic of a G that is a fifth (seven half steps) above the C in an equal temperament scale.
(Answer: 0.886 Hz.)
• Exercise 2. Calculate the frequency of the beats generated by the fifth harmonic of a middle C at 261.63
Hz and the fourth harmonic of an E that is a major third (four half steps) above the C in an equal temperament
scale. (Answer: 10.38 Hz.)
• Exercise 3. Calculate the number of cents in (a) an equal temperament fifth, (b) a Pythagorean just fifth,
(c) a Pythagorean wolf fifth, (d) a QC meantone tempered fifth, and (e) a QC meantone wolf fifth. (Answers:
700 cents, 701.955 cents, 678.495 cents, 696.578 cents, 737.637 cents.)
• Exercise 4. Using the QC meantone temperament, calculate the disagreement in cents between G♯ and
A♭. Recall those pitches are obtained by moving up four tempered fifths from E (and down two octaves) to
arrive at G♯, and by moving down four tempered fifths from C (and up three octaves) to arrive at A♭. (Answer:
41.059 cents.)
MIDI exercises with Temperament Studio
• Exercise 5. Exploring MIDI instruments. Play a single note, such as a middle C (marked by the letter
“q” on the virtual keyboard). See how the quality of the note changes, but not the pitch, as you select different
MIDI instruments.
• Exercise 6. Create your own instrument. To hear the mathematically-predicted beating, an instrument
that has a sustained tone with no vibrato is best. One of the best MIDI instruments for this is “21: Reed
Organ” (note: MIDI instrument names and sounds vary from computer to computer). However, creating your
own instrument may work even better, and will allow you to verify which harmonics are beating. Create your
own instrument by clicking on the drop down box labeled “Source,” and selecting “Sine + harmonics.” A control
with sliders for the first 7 harmonics will appear. Play a note such as middle C, then notice how the tone changes
as you adjust the harmonics. For example, by setting the sliders to approximately 0.75, 0.75, 1, 0.85, 0.5, 0.4,
and 0.2, a reasonably good reed organ sound can be obtained.
• Exercise 7. Equal temperament fifths. Set the tuning scheme to “equal temperament.” (a) Verify Exercise 1, namely, that when an equal temperament fifth is played with the lower note being a middle C at 261.6
Hz, a beat at 0.886 Hz is present. Time 10 beats and use the measured period to calculate the beat frequency
(or alternatively use a metronome to test the predicted beat frequency against the measured one). The beating
arises from the third harmonic of the C and the second harmonic of the G; there may also be a perceptible beat
between the fourth and sixth harmonics at twice this frequency. (b) Use the left and right arrow keys to explore
qualitatively how these fifths sound as the fifth is moved up and down the keyboard, e.g. from C-G to C♯-G♯,
etc. Verify Figure 3(a), which indicates all fifths in this temperament should be equally far from/close to the
ideal 3/2 ratio (although the beat frequency will increase/decrease as you go up/down the scale).
• Exercise 8. Equal temperament thirds. Set the tuning scheme to “equal temperament.” (a) Verify
Exercise 2, namely that when a major third is played with the lower note being a middle C at 261.6 Hz, a beat at
10.38 Hz should be apparent. That beat may be too quick for you to measure, so go down an octave to observe
a beat at 10.38/2 = 5.19 Hz. (b) Use the left and right arrow keys to explore qualitatively how these major
thirds sound as the third interval is moved up and down the keyboard. Verify Figure 3(a), which indicates that
all major thirds in this temperament should be equally far from/close to the ideal 5/4 ratio.
Durfee and Colton, p. 2
• Exercise 9. Removing the beats. Set the tuning scheme to “equal temperament,” and use the hand-made
reed organ sound from Exercise 6. (a) The just ratio of 3/2 for fifths means that when a fifth has a slightly
different ratio than 3/2, the third harmonic of the lower note of the fifth will interact with the second harmonic
of the upper note to produce beats. Play a C-G fifth interval as in Exercise 7, and listen to what happens to
the beats as you add/remove harmonics. You can remove the 0.886 Hz beating by removing either the second or
third harmonics. You can add/remove beating at twice that frequency via the fourth and sixth harmonics. (b)
Similarly, the just ratio of 5/4 for major thirds means that major third ratios which are slightly detuned from
5/4 will have beating between the fourth and fifth harmonics of the upper and lower notes, respectively. Play a
C-E third interval as in Exercise 8, and listen to what happens to the beats as you add/remove harmonics. You
can remove the 5.19 Hz beating by removing either the fourth or fifth harmonics.
• Exercise 10. Pythagorean fifths. Set the tuning scheme to “pythagorean” and the tuning root to C. (a)
Play a C-G fifth interval as in Exercise 7, and verify that, unlike equal temperament, no beats are present. (b)
Use the left and right arrow keys to explore how these fifths sound as the fifth interval is moved up and down
the keyboard. Verify Figure 3(b), which indicates that one of the fifths will be a “wolf fifth.”
• Exercise 11. QC meantone fifths. Repeat the previous exercise for QC meantone (“qc meantone”). Verify
Figure 3(c), which indicates that not only will a wolf fifth be present for QC meantone, it will howl even worse
than the Pythagorean wolf fifth.
• Exercise 12. The other major thirds and fifths. Repeat the previous exercises for five-limit fifths, and
Pythagorean, QC meantone, and five-limit major thirds. Notice how the degree of beating depends on where
the particular interval is located within the scale, and verify the patterns indicated by Figure 3.
• Exercise 13. Playing MIDI files, part 1. Check the “File Controls” box and then load Bach’s Passacaglia
In C Minor (BWV 582), a famous Baroque organ piece. Listen to it in equal temperament vs. its intended
temperament (QC meantone, root of D)1 . To the authors, the reduced beating in QC meantone makes the
piece sound noticeably more powerful: the consonant chords have less beating, and there is added contrast
between the dissonance of passing chords and the consonance of the resolution. Bach studiously avoided the wolf
intervals, but if you change the tuning root to F♯ you will hear wolf intervals are a-plenty! (It’s also interesting to
experiment with the “General Meantone” setting, where one can smoothly go between QC Meantone and equal
temperament by changing the value of the reference fifth.)
• Exercise 14. Playing MIDI files, part 2. Using File Controls, select Debussy’s Claire De Lune. This is a
famous early 20th century piano piece that employs a great deal of chromaticity and key changes, and hence
must be performed on an equal temperament instrument. Observe how pleasant it sounds in equal temperament
first (despite the small amount of beating that is naturally present in that tuning scheme), then how unpleasant
it sounds in any other tuning scheme.
∗
1
dallin durfee@byu.edu
To get the full effect, you can also change the tuning reference pitch to “Chorton” (pronounced “core-tone”), about
a half-step higher than today’s A440; this was the approximate pitch standard used by most organs during Bach’s
time in central and north Germany. See C. Wolff and M. Zepf, The Organs of J.S. Bach: A Handbook (University
of Illinois Press, Urbana, 2012), pp. 1-2.