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