8 The octave
8.1 The octave sound
One feature of pitch that seems to be universal to all cultures is that for musical purposes the pitch range is divided into discrete steps: for instance, the notes of a scale. This is not to say that musicians rigidly adhere to those steps when they play, but the existence of such steps is fundamental to the way music is conceived and organised. Different cultures have different ways of defining the steps in their scale of pitches, but nearly all cultures take the octave as their starting point. It has a very characteristic sound, and it corresponds precisely to a particular relationship of frequencies.
Activity 24 (Optional)
If you have access to a keyboard, play middle C (C4) and the C one octave above it (C5). Figure 23 is a reminder of the notes you need to play. Play them one after the other (in either order), Listening carefully to the sound of the two notes. You know that in musical terms these two notes are an octave apart, but can you describe the relationship between the two pitches?
Although the two pitches are clearly different (C5 has a higher pitch than middle C), most people find that there is nevertheless something very similar about them. One way to express this idea is to say that they are two versions of the same musical sound. To reinforce this idea, play a C major scale from middle C upwards to C5. When you reach C5 it feels like a return to base, although naturally you have not returned to middle C.
As you know, the two pitches played in the last activity are an octave apart − C5 is an octave above middle C (C4). The eighth note to the right of C5 is another C (C6), and is two octaves above middle C. Similarly, the eighth note to the left of middle C is an octave below middle C and is the note C3.
Sine waves whose pitches are an octave apart have frequencies in the ratio 2:1. That is, the higher pitch has a frequency that is twice that of the pitch below it. Alternatively, the lower pitch has a frequency that is half that of the pitch an octave above.
Activity 25 (Self-Assessment)
The note A4 has a frequency of 440 Hz in concert pitch. What is the frequency of the note A7?
The pitch three octaves above A4 has a frequency of 3520 Hz.
Unless you are familiar with this type of calculation, it is sensible to do it an octave at a time, as follows.
One octave above has a frequency of 2 × 440 Hz = 880 Hz (A5).
Two octaves above has a frequency of 2 × 880 Hz = 1760 Hz (A6).
Three octaves above has a frequency of 2 × 1760 Hz = 3520 Hz (A7).
The reason for doing the calculation a step at a time is to avoid a couple of traps that can easily be fallen into. First, note that a three-octave rise does not correspond to a tripling of frequency. Secondly, note that three successive doublings of frequency do not amount to a sixfold increase in frequency overall. That misapprehension would have given an answer of 2640 Hz. In fact, three successive doublings of frequency amounts to an eightfold increase. Hence the factor by which we need to multiply the original frequency is (2 × 2 × 2) or 23.
We saw earlier that doubling the frequency of a sine wave corresponds to a halving of its wavelength. This follows directly from the relationship v = f × λ. Thus a sine wave that is an octave above another sine wave has half its wavelength.