Feeds:
Posts

## The Imperfection of the Musical Scale (Part 1)

The most surprising feature of the musical scale is its imperfection – we don’t get everything we could hope for. Allow me to explain.

The pitch of a musical note is related to its frequency, that is, the number of sound waves that pass a given point per second. Frequency is measured in hertz (Hz) – vibrations per second. The human ear can hear sounds with frequency between about 20Hz and 20,000Hz.

Within this range there are an infinite number of possible frequencies that an instrument could sound out. This is clearly impractical for musical purposes – how does one chose between an infinite number of options? The human ear can only distinguish between two frequencies if they differ by about 0.3% or more. But this still leaves over 2000 discernible frequencies within the human ear’s audible range. Imagine a piano with 2000 keys! So how should we go about making music?

To make progress, we need an insight from Pythagoras. According to legend (i.e. Wikipedia), Pythagoras was walking past a blacksmith when he noticed that two anvils would produce harmonious sounds when their sizes were simple ratios of each other i.e. when one was half the size of the other, 2/3 the size of the other etc. Thus, we would like our chosen frequencies to be simple ratios of each other, so that we can use harmonies (more than one pitch played at once).

Since it’s only the ratio of frequencies that matter, we’ll need a starting frequency. We’ll cheat and use the one used in the Western musical scale – 440 Hz. We’ll call it A. It’s the A above middle C, but that’s getting ahead of ourselves.

We’ll start with the simplest ratio 2:1. This is the octave. Thus every frequency that can be reached by doubling or halving 440Hz any number of times is one of the chosen frequencies.

We can go one step further. We can postulate that the octave will be the most important interval (i.e. ratio), in this sense: all notes separated by an octave will be “the same note”. I.e. up an octave from A, we find another A. Hence, we have reduced the problem of choosing frequencies between 20Hz and 20,000Hz to choosing frequencies within an octave. All the rest follow by halving and doubling.

As a first attempt, we could choose the simplest fractions between 1:1 and 2:1. Here’s the first seven:
a: 1:1
b: 5:4 = 1.25
c: 4:3 = 1.33
d: 3:2 = 1.5
e: 5:3 = 1.67
f: 7:4 = 1.75

Suppose we postulate a 6-note scale, with the above ratios. The problem is this: we’ve only ensured that each note sounds good when played with the note ‘a’. But if we play ‘f’ with ‘e’, then we have a ratio of 21:20. Not a very simple ratio. And adding more notes only makes it worse. Back to the drawing board.

Let’s focus on the simplest ratio after the octave, which is 3:2. This is known as a fifth. One fifth up from A is E. This immediately gives me a way of choosing the frequencies within the octave: start at A, keep going up by fifths, and if you leave the octave, just go down an octave. Going up in fifths gives the ratios

$3:2, \ (3:2)^2, \ (3:2)^3, \ (3:2)^4 \ \ldots = 3:2,\ 9:4,\ 27:8,\ 81:16 \ \ldots$

Adjusting the octaves so they all lie between 1:1 and 2:1 gives:

$3:2,\ 9:8,\ 27:16,\ 81:64,\ 243:128 \ \ldots$

These fractions aren’t very simple, and we can easily see why. The numerator is a multiple of three, the denominator is a multiple of two, and thus the fractions cannot be simplified. But the biggest problem is that this procedure of choosing the frequencies within an octave leads to an infinite number of notes. This is because, for the number of notes to be finite, we need the sequence to repeat itself. This is the same as saying that we need whole numbers $m$ and $n$ such that

m fifths = n octaves

so that after going up m fifths and then back down n octaves, we arrive back at the ratio 1:1, and the sequence repeats.

We can write the above equation as:

$(3/2)\times(3/2)\times(3/2)\times \ldots$ (m times) $= (2)\times(2)\times(2) \times \ldots$ (n times)

$(3/2)^m = 2^n$

$3^m = 2^{(n-m)}$

But this equation has no solution – the left side is a multiple of three, the right side is a multiple of two, and you cannot get a multiple of three by multiplying two by itself any number of times. To see this another way, the left side is odd, the right side is even.

So here is the problem: we can’t choose the notes of the musical scale such that, from every note, an interval of a perfect octave and a perfect fifth is available. My next post will show how modern musical scale solves this problem.