An understanding of the basic idea of 'cents' is absolutely necessary, since its is used throughout this, and other, descriptions of musical tuning and temperament. However, for those who are somewhat shy of mathematics, I have presented this chapter in two parts: an initial description which tells you all you need to know to read the rest of the chapters and then a more mathematically rigorous description for those interested.

Imagine a ruler which could be used to measure intervals. It would make sense that this 'ruler' was one octave in length so that we could use it to measure all of the intervals within an octave. The ruler will need some markings on it, in the same way as we have centimeters or inches on a real ruler. Since there are 12 notes in a chromatic scale, let's go from zero, at the tonic to 1200 at the octave. In that way, we will have a note roughly corresponding to each 100 marks (which is why they are called cents).

If we look at Equal Temperament each semitone falls exactly on a 100 cent mark, which is what defines Equal Temperament.

The really important thing about 'cents', as opposed to frequencies, ratios or any other way of measuring pitch is that we can simply add and subtract them. If we consider the chromatic scale of C in the diagram, the 5th is an interval of 7 semitones or 700 cents. If we go up a further major second (200 cents), then we arrive at a 6th which is the 9th note of the chromatic scale and has a pitch of 900 cents (700+200). Come down a perfect 4th (500 cents) and we get to a major 3rd at 400 cents (900-500).

In the Introduction, it was asserted that Equal Temperament is an approximation and, in musical terms, the 5th is really to be found at a pitch ratio of 3/2 relative to the tonic (or 1.5 times the pitch if you prefer). Using the mathematics to be found in the second half of this chapter, it turns out that this is 702 cents. Given that we can safely add and subtract cents, this means that the 5th, on a keyboard tuned using Equal Temperament, is 2 cents flat. As a rule of thumb, we cannot detect differences of less than 5 cents, so it not just "close enough for jazz", its perfectly okay for classical music as well!   If we now consider Natural Tuning, which is based on naturally occurring ratios and so should be the 'most agreeable' tuning, and concentrate on the tuning key of C, the chromatic scale shows that none of the notes lie on the 100 cent marks. Unfortunately, this Natural Tuning has many other problems which make it unusable in practice. (See the chapter on Natural Tuning.) In fact, if you randomly switch between the various tunings and temperaments, you will find that the only thing that remains constant is that the tonic is always zero, and the octave is always 1200. This underlines the point made in the introduction that the octave is always fixed at twice the pitch of the tonic or 1200 cents higher, using our 'interval ruler'.

The difference or interval between two notes is always specified in terms of a ratio of their frequencies, rather than simply subtracting the higher frequency from the lower. The reason for this is that the interval of a 5th between the A at 440Hz and the E at 660Hz is the same, in musical terms as the 5th between the A at 880Hz and the E at 1320Hz. In both cases, the ration is 3/2 even though the differences in frequency are 220Hz and 440Hz respectively.

It is, of course, mathematically possible to just work with ratios. However, working with ratios has a number of problems. The first is that adding and subtracting ratios is a 'bit of a pain in the neck'. If you don't believe me, try the fairly simple task of working out the size of the major second (as a ratio) formed by going up a major 6th (5/3) and then descending a 5th (3/2) and then working out the difference between that and the equivalent major second formed by going up a 5th (3/2) and down a 4th (4/3). The answer is 1/72.

A second problem is seeing easily which is the greater of two ratios. It is fairly obvious that 3/2 is a higher pitch that 4/3 but what about 6561/4096 and 729/512, which appear in Pythagorean Tuning. (One is an F# and one a G#!)

Since ratios are the 'currency' of intervals, the way to simplify matters is to use logarithms. The key fact about logarithms is that multiplication and division of numbers is the same as addition and subtraction of their logarithms. So, if we work with the logarithms of the ratios we will be able to simply subtract the two numbers to find the interval. Since the range we are working in is an octave, and the octave is twice the tonic, base₂ logarithms are the answer because this will give us numbers between 0, log₂(1/1) and 1, log₂(2/1). As a further refinement, multiplying the logarithm by 1200 will allow us to work in integers most of the time, with the tonic at 0 and the octave at 1200. In a 12 note chromatic scale, this makes each semitone interval about 100, hence the name 'cents'

Put rather more mathematically, we define an interval in 'cents' as:

Interval = log₂(Note Pitch/Tonic Pitch) x 1200.

For some of the common intervals, we get the following for the key of C:

This agrees with our musical calculator for Natural Tuning,

To demonstrate the ease with which we can work with cents, let us return to the problem outlined earlier. The major second formed by going up a major 6th and then descending a 5th is 884-702=182. The equivalent major second formed by going up a 5th and down a 4th is 702-498=204. Therefore the difference is 204-182=22 cents, an audible difference