Listening to

In Our Time - Mathematics And Music (downloadable until Wednesday, maybe) made me remember I once promised to start this thread, so here it is.

Note: the maths is hopefully accurate, the rest of it may be made up.

**Pythagoras, Intervals, and Rational Numbers**Pythagoras liked triangles. He noticed that triangles of different sizes made sounds of different pitches when you bashed them. A triangle of half the size of the first triangle made a sound that was pleasing in combination to the sound made by a triangle that was the size of the first triangle. And a triangle 2/3 the size of the first triangle was also pleasing (although a little less so), similarly a triangle 3/2 the size of the first triangle created a pleasant effect.

Further investigation revealed that

**ratios of small integers** made pleasant sounds together.

Here are some ratios of small integers, and the corresponding name of the interval in Western-European parlance, and the note-letter if the base note was C:

1::1 unison (C)

2::1 octave (C)

3::2 perfect fifth (G)

4::3 perfect fourth (F)

5::4 major third (E)

6::5 minor third (Eb)

5::3 major sixth (A)

8::5 minor sixth (Ab)

Extending this further gets you into all sorts of trouble as the link above describes:

Say you want to find the ratio for a major second (D). Then you can start from unison, go up a fifth and down a fourth, which gives you:

(method A) 1::1 * 3::2 / 4::3 = 9::8

Or, you can go up a fourth and down a minor third, which gives you:

(method B) 1::1 * 4::3 / 6::5 = 10::9

Two methods for reaching the same note give different results, which gives big problems if are making music with triangles of fixed sizes, because you'd effectively need an infinite number of triangles to cope depending on your melody (how you reach the note in question), and harmony would be even more problematic.

That's enough typing for one evening, next time I think I'll write a load of nonsense about strings and harmonics and beat frequencies and equal temperament, unless someone has any better ideas or questions or anything.