Digital Sound & Music: Concepts, Applications, & Science, Chapter 3, last updated 6/25/2013
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1.4983, not exactly 3/2, which is 1.5. Wouldn’t the intervals be even more harmonious if they
were built upon the frequency relationships of natural harmonics?
Figure 3.45 How cycles match for pairs of frequencies
To consider how and why this might make sense, recall the definition of harmonic
frequencies from Chapter 2. For a fundamental frequency , the harmonic frequencies are ,
, , , and so forth. These are frequencies whose cycles do fit together exactly. As
depicted in Figure 3.46, the ratio of the third to the second harmonic is . This corresponds
very closely, but not precisely, to what we have called a perfect fifth in equal tempered
intonation for example, the relationship between G and C in the key of C, or between D and G
in the key of G. Similarly, the ratio of the fifth harmonic to the fourth is , which
corresponds to the interval we have referred to as a major third.
Figure 3.46 Ratio of frequencies in harmonic intervals
Just tempered intonations use frequencies that have these harmonic relationships. The
Pythagoras diatonic scale is built entirely from fifths. The just tempered scale shown in Table
3.15 is a modern variant of Pythagoras’s scale. By comparing columns three and four, you can
see how far the equal tempered tones are from the harmonic intervals. An interesting exercise
would be to play a note in equal tempered frequency and then in just tempered frequency and see
if you can notice the difference.
Interval Notes in Interval Ratio of Frequencies in Equal
Temperament
Ratio of Frequencies in Just
Temperament
perfect unison C 261.63/261.63 = 1.000 1/1 = 1.000
minor second C C# 277.18/261.63 1.059 16/15 1.067
major second C D 293.66/261.63 1.122 9/8 = 1.125
minor third
C D E♭
311.13/261.63 1.189 6/5 = 1.200
major third C D E 329.63/261.63 1.260 5/4 1.250
perfect fourth C D E F 349.23/261.63 1.335 4/3 1.333
augmented fourth C D E F# 369.99/261.63 1.414 7/5 = 1.400
perfect fifth C D E F G 392.00/261.63 1.498 3/2 = 1.500
minor sixth
C D E F G A♭
415.30/261.63 1.587 8/5 = 1.600
major sixth C D E F G A 440.00/261.63 1.682 5/3 1.667
minor seventh
C D E F G A B♭
466.16/261.63 1.782 7/4 = 1.750
major seventh C D E F G A B 493.88/261.63 1.888 15/8 = 1.875
perfect octave C C 523.26/261.63 = 2.000 2/1 = 2.000
Table 3.15 Equal tempered vs. just tempered scales
0 1 2 3 4 5 6 7
x 10
-3
-1
-0.5
0
0.5
1
C
G
f 2f 3f 4f 5f 6f
fifth
3/2
fourth
4/3
major 3rd
5/4
minor 3rd
6/5
octave
1st
harmonic
2nd
harmonic
3rd
harmonic
4th
harmonic
5th
harmonic
6th
harmonic
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