Tuning and temperaments

When I was tuning my bass viol the other day, I wondered if I was getting the best out of it?  And this started me thinking about tuning, and temperaments: the best way to tune notes in a scale.

As you know, in Western music, the smallest pitch difference commonly used is a semitone, of which 12 make an octave.  We speak of an interval as the difference between two pitches, and intervals played sequentially form a melody; in parallel a chord.  It turns out that two or more notes played simultaneously sound best when the ratio of their frequencies is a fraction with small integer values.  For example, two notes with frequencies 300Hz and 450Hz would sound good together, as the ratio of their frequencies is 3/2 (or 2/3; doesn’t matter).  We can see this by plotting

    \[ \sin(x)+\sin\left(\frac{3}{2}x\right) \]

which produces something like this:

figure_1

As you see, it’s a nice, regular shape, with all internal maxima and minima lining up. If we play the notes first singly and then together:

They “meld” together well – they sound like they go together, and make a nice concordant sound.

If instead we plotted

    \[ \sin(x)+\sin(\sqrt{3}\,x) \]

we would have this messy graph:

figure_2

As a sound this would come across as discordant:

As you hear, it sounds like two pitches with no relationship between them.

A difference in pitch between two notes is called an interval, of which the most fundamental is the octave: two notes whose frequency ratio is 2:

figure_3

Describing pitches and intervals

A single pitch can be described in terms of cycles per second, or Hertz; the greater the Hertz, the higher the pitch.  This value is called the frequency of the sound. Multiply the frequency by 2 and you have the same note an octave higher.  For example, modern concert pitch A is defined as 440Hz, so the A one octave higher has a frequency of 880Hz, and one octave lower 220Hz.  The range of sounds perceptible to the human ear is about 20Hz to 20,000Hz; the range of a modern concert piano is 27.5Hz to 4186.01Hz.

Frequency is an exponential scale, so the ratio of two similarly named notes will be equal no matter their individual frequencies.  Thus if A=440Hz, and the E above A has frequency E=660Hz, their ratio is 660/440=3/2=1.5.  If you go three octaves higher, you will have \mbox{A}=440\times 2^3\mbox{Hz}=3520\mbox{Hz} and \mbox{E}=660\times 2^3 \mbox{Hz}=5280\mbox{Hz}.  The frequency ratio is 5280/3520=1.5.

To turn an exponential scale into an additive scale, we use cents, of which 100 make a semitone (hence the name “cents”), and so 1200 make an octave.  This means that a single cent c must satisfy

c^{1200}=2

and so

c=2^{1/1200}.

So given a ratio r between two frequencies, the number of cents c satisfies

\left(2^{1/1200}\right)^c=r

or more simply

c=1200\log_2(r).

So a ratio of 1.5, for example, corresponds to 1200\log_2(1.5)\approx 701.955 cents.  The use of cents is very convenient, as to consider the frequencies between a set of notes, we can simply add (or subtract) the number of cents.

The problem with a scale

In (Western) music, an octave is equal to 12 semitones.  The pitch ratio 1.5, which we heard above as a concord, sits at seven semitones.  So every time we go up seven semitones, we multiply the pitch by 1.5.  Since 7 and 12 are relatively prime, we would hope that 1.5^{12}=2^7, which it can’t, as this would imply (3/2)^{12}=2^7 or 3^{12}=2^{19}, thus contradicting the fundamental theorem of arithmetic.  The pitch ratio in cents of 12 lots of 1.5 is

12\times 1200\log_2(1.5)\approx 8423.46

and 7 octaves is of course

7\times 1200=8400.

The difference of these two pitches is 23.46 cents, and this value is called the Pythagorean comma.  In general, in discussions of tuning and pitch, a comma is the difference in pitches between a note obtained using different tunings.  This means that the basic ratios of 2 (for the octave) and 1.5 (called a perfect fifth) are fundamentally incompatible.  The business of trying to manage this incompatibility has occupied theorists and players for hundreds of years, and continues to do so.  The result is that it is impossible to have a perfect scale where every ratio (of low values) is fully concordant.

We will look at some different approaches to scale building in future posts.

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