Have you ever been singing along to the radio and one of your friends moans how painfully ‘out of tune’ you sound? Well, it turns out that it may not be your singing that is to blame, but in fact maths!
The western musical scale is made up of 12 notes1, and each note corresponds to a different frequency. We hear frequencies on a logarithmic scale – this means that doubling the frequency produces the same note, but one octave higher. For example, if the frequency 440Hz corresponds to a note ‘A’, then 880Hz will also correspond to ‘A’, and in between these two ‘A’s will be 11 other notes. Now, loosely speaking, two notes will sound ‘nice’ together if the ratio between their frequencies is a simple rational number2. We’ve already seen the simplest ratio, 2:1, corresponds to an octave, and the next simplest ratio is 3:2, called a perfect fifth. If the ratio of a perfect fifth is not exactly 3:2, the notes are out of tune. In the 12 note scale, a perfect fifth is 7 notes above the starting note (confusing, I know!).
As a consequence of this, stacking up successive fifths means you cycle through all of the 12 notes, and this is called the circle of fifths.
From a maths point of view, the reason this happens is because 12 and 7 share no common factors, so the smallest positive integer solution of 7a = 12b is a = 12, b=7. In fact, 12 and 5 also share no common factors, and so we also get a circle of fourths, which is precisely the circle of fifths in the clockwise direction. The only other possible circle will correspond to 12 and 1, or equivalently 12 and 11, but this is just the 12 note scale in order3.
We said previously that a perfect fifth corresponds to a ratio of 3:2. So if we wanted to tune our keyboard, one sensible way to do so would be to start with the frequency of the note A (the choice of A here is arbitrary), and multiply/divide this frequency by 3/2 to generate frequencies of other notes. For example, starting again with A = 440Hz, we would get the frequency of E, a perfect fifth up, to be 440*3/2 Hz = 660Hz. From the circle of fifths, if you go up by a perfect fifth exactly 12 times, you end up at the same note, except 7 octaves higher. This means that the following should be true:
However, this is in fact false! For instance, 312 is an odd number, and 219 is an even number, so they cannot be equal. It turns out that (3/2)12 = 129.746…, and 27 = 128, so under this tuning system, going once around the circle of fifths leaves you approximately one quarter of a note out of tune.
This is a disaster! We have just proven that there’s no way to tune music such that every single perfect fifth has the simple ratio of 3:2, and every octave has the ratio 2:1, which is a complicated way of saying that any tuning system must contain some ratios which are out of tune!
The tuning described above is closely related to Pythagorean tuning, invented by the Ancient Greek mathematician Pythagoras. One solution here is to have 11 of the fifths perfectly in tune, and tune the remaining fifth such that the circle of fifths does ‘close up’ as we previously wanted. The one out of tune fifth is called the Wolf interval, the name deriving from its resemblance to the howling of a wolf!
Below is a great real world demonstration – can you spot the wolf? As a consequence, Pythagorean tuning is almost never used for actual music making.
Historically, there have been three tuning systems competing against each other – Meantone temperament, Well temperament, and Equal temperament.
Meantone temperament is similar in construction to Pythagorean temperament, except the ratio 3:2 is slightly narrowed to 5(¼):1, or approximately 1.495:1. This means that stacking 4 fifths together, we get the simple ratio 5:1, so that the interval of a third (for example, A to C) has a simple ratio of 5:4, which makes some chords sound better! Like with Pythagorean tuning, a Wolf interval is needed to make the circle of fifths close up. Below is a demonstration – you might hear that the tuning progressively gets less stable, culminating in a dramatic wolf interval at 1:28! In contrast, Well temperament is a compromise to “tame the wolf”, that is to make the wolf interval more in tune, at the cost of other intervals being slightly less out of tune.
However, the clear winner is Equal temperament, which not only tames the wolf, but in fact kills it! In Equal temperament, tuning is based on stacking semitones (for example, the interval between C and C♯ is a semitone). In particular, the ratio of the nth note of any 12 note scale is given by 2(n/12). This means that all octaves are perfectly in tune (2(12/12) = 2), but every other interval, such as the fifth or third, are slightly out of tune. For example, in Equal temperament, a fifth has a ratio of 2(7/12):1, or 1.498:1, which is close, but not equal to 3:2. Listen below for a comparison of the same piece from before, now in Equal rather than Meantone temperament. The benefit of Equal temperament is that the tuning remains no matter how complicated the harmony of the music is. But this is also a potential drawback, since in other tuning systems, such as Well temperament, each key has it’s own unique sound and personality, and this is completely lost in Equal temperament. Either way, given that you’ve probably never noticed the music you listen to every day is slightly out of tune, Equal temperament clearly does a pretty good job!
1Other cultures around the world use a different number of notes.
2This a controversial and complicated topic in itself – culture and upbringing also makes a big difference in what intervals are perceived as ‘nice’.
3Any group theory enthusiast will see that this is equivalent to saying the cyclic group Z12 is generated by elements of order 1,5,7,11.