EECS20N: Signals and Systems

Musical scale

The frequencies 440Hz and 880Hz both correspond to the musical note A, but one octave apart. The next higher A in the musical scale would have the frequency 1760Hz, twice 880Hz. In the western musical scale, there are 12 notes in every octave. These notes are evenly distributed (geometrically), so the next note above A, which is B flat, has frequency 440 × β where β is the twelfth root of two, or approximately 1.0595. The next note above B flat, which is B, has frequency 440 × β 2.

 A 440 B flat 466 B 494 C 523 C sharp 554 D 587 D sharp 622 E 659 F 698 F sharp 740 G 784 A flat 831 A 880
Left is a table of the complete musical scale between middle A and A-880. Each frequency is β times the frequency above it. The applet on the previous page has a button that you can use to play the musical scale.

The psychoacoustic properties of the musical scale are fascinating. The musical scale is based on our perception of frequency, and harmonic relationships between frequencies. The choice of 12 evenly spaced notes is based on the so-called circle of fifths.

Frequencies that are harmonically related tend to sound good together. In the following applet, you can combine any set of frequencies in the scale.

If you were able to run applets, you would have a Scale demo here.

The blue waveform is the sound you hear, which is a combination of the other (pure tone) waveforms. Try combining A, C sharp, and E. If you have a musical ear, you will recognize this as a major triad. What is special about this frequency combination? Notice that the following frequencies all correspond to the note A:

440,
440 × 2 = 880,
880 × 2 = 1760,
1760 × 2 = 3520.

What about 440 × 3 = 1320? Notice that 1320/2 = 660, which is almost exactly the E in the scale at the left. Thus, 440 × 3 is the note E, one octave above the E immediately above A-440. E and A are harmonically related, and to most people, they sound good together. It is because

440 × 3 ≈ 659 × 2

For somewhat more arcane reasons, the interval between A and E, which is a frequency rise of 3/2, is called a fifth. The note 3/2 above E has frequency 988, which is an octave above B-494. Another 3/2 above that is approximately F sharp (740 Hz). Continuing in this fashion, multiplying frequencies by 3/2, and then possibly dividing by two, you can approximately trace the twelve notes of the scale. This progression is called the circle of fifths. The notion of key in music and a scale are based on this circle of fifths.

Where does the C sharp come from in the major triad? Notice that

440 × 5 ≈ 554 × 4.

Among all the harmonic relationships in the scale, A, C sharp, and E have among the simplest. This possibly accounts for the predominance of the major triad in western music.

A major triad can be written as a sum of sinusoids

s(t) = sin(440× 2π t) + sin(554 × 2π t) + sin(659× 2π t).

The human ear hears frequencies. Musical sounds such as chords can be characterized as sums of pure tones. Of course, truly musical sounds are much more complex. For one thing, pure tones are not particularly appealing sounds. Musical instruments produce notes that are more complex than pure tones. The characteristic sound of an instrument is its timbre, and as we shall, some aspects of timbre can also be characterized as sums of sinusoids.