A recent discussion with a colleague reminded me of an idea I had some time back: composing non-equal-tempered music. This is not a new idea, it’s just something I’ve never delved into much before.

### Quick Intro to the Physics of Sound

Sound is caused by vibrations. Whenever you make a sound, some object or part of an object has to vibrate. This could be, for example, a metal saucepan, a guitar string, or your vocal chords. For you to be able to hear a sound, the vibrations have to reach your ear drum. The vibrations travel as a wave from the source of the sound to your ear. Sound waves travel through air, but also travel through just about any other medium you can think of (ok, so a perfect vacuum doesn’t count as a medium ;-).

### Quick Intro to Musical Notes

Musical notes are periodic sound waves. The volume of the note is related to the amplitude of the wave and the pitch of the note is related to the frequency of the wave. A greater frequency corresponds to a higher pitch and vice versa. The interval between any two musical notes is related to the ratio of the note frequencies. For example, the interval of an octave corresponds to a frequency ratio of 1:2. The note which many orchestras use for tuning, known as concert A, has a frequency of 440 Hz. So the A an octave higher has a frequency of 880 Hz and the A an octave lower 220 Hz.

### The Equal-Tempered Scale

Most instruments used to play western music today are tuned to a scale known as the equal-tempered scale. Basically, this means that every octave is divided into 12 equal semitones. (For the mathematically minded out there, the interval of one semitone corresponds to multiplying a frequency by the twelfth root of two, or by about 1.059.) This method of tuning instruments results in a very versatile scale, where a piece of music may be transposed to a different key, and still sound essentially the same. The only drawback is that music doesn’t sound quite as *nice* as it could. Which doesn’t really have to be a drawback, unless you’re a purists.

### Just Intonation

Two notes generally sound nice to the human ear if they are separated by an interval called a “just interval”. This means that the ratio of the frequencies of the two notes can be expressed using small numbers. For instance, the frequencies 440 Hz and 660 Hz are related by a ratio of 2:3. These notes sound nice together. In fact this interval has a name: a perfect fifth.

If you’ve followed everything so far, and you’re both mathematical and musical, you might have discovered a problem here. Musicians would be aware that a perfect fifth is equivalent to seven semitones. And mathematicians would be aware that if you multiply 440 Hz by 1.059 seven times you don’t get exactly 660 Hz. You’re off by about 2%. What’s going on?

The truth here is that the even-tempered scale sacrifices these just intervals in order to have the nice property of every semitone being equal. So when you play a perfect fifth on a piano, the interval you’re hearing is not quite a 2:3 ratio. But there are methods for tuning instruments which ignore the modern conventional scale in order to make things sound nice. And even putting tuning aside for a minute, I bet many musicians subconsciously adjust their pitch to fit with just intervals when singing or playing an instrument where small pitch adjustments are possible, such as a string instrument or the trombone.

### “Impossible” Music

Which brings us back to where we started. Other musicians have already tried their hands at this. Basically, I want to compose music where I care more about how the piece sounds than about following a conventional scale. A computer will help me immensely by synthesising notes that are not constrained to the even-tempered scale. And when I’m done, musicians will probably find it easiest to listen to their part and play by ear. If I come up with anything decent, I shall post it on this site.