The musical octave has been used as a way of understanding the cosmos for thousands of years. The ancient Greeks used it to represent the harmonies of the universe, and in the renaissance it was used to show hidden relationships of harmony in the music of the spheres. More recently the philosopher G I Gurdjieff used the Law of Seven to understand how processes work at all levels of existence, explaining how things can go wrong in our simplest plans, and what we can do to stop this happening. The octave can be found in the Kabbalistic Tree of Life, and Saros philosophy shows how it arises as a simple abstract pattern.
In the beginning
The story of the octave, as far as we know it, begins with Pythagoras the Greek mathematician and philosopher, who in the 6th Century BC came up with two key ideas:
- That musical harmony is based on mathematical ratio. He found that strings of different length vibrate with different sounds, and that they sound most harmonious together when the lengths are in a simple numerical ratio: 2 to 1, or 3 to 2 for example.
- That the universe is harmonious, with the planets and stars moving according to mathematical rules.
He reasoned that the motion of the planets should correspond to musical harmony, laying the foundation for the harmony of the spheres, which was used as a way of understanding our world for thousands of years.
An interesting description of the octave is contained in Plato’s dialogue on cosmology called the Timaeus. This dialogue, written around 360 BC, was Plato’s attempt at a ‘theory of everything’ describing how the world and everything in it was made . The crucial part for us is where he talks about the way ‘the divine craftsman’ created the world-soul, using a mathematical division to lay out the circles which describe the apparent motion of the fixed stars and the planets around us.
The mathematical division is based on simple numerical ratios, which define a Pythagorean octave. He starts with the simplest numerical ratios, formed from whole numbers:
2 to 1 do to do (an octave)
3 to 2 do to sol (a perfect 5th)
4 to 3 do to fa (a perfect 4th)
This gives a framework of notes: do … fa sol … do
Plato then fills out the framework, taking the ratio between fa and sol as the basic tone. The ratio of fa to sol is 4/3 to 3/2 which simplifies to 9/8, so Plato adds two notes re and mi spaced at this ratio from do, and then la and si spaced by the same ratio. He ends up with this picture:
Do-re-mi and fa-sol-la-si are all spaced by a full tone, 9/8 ratio, but there are two intervals (mi-fa) and (si-do) which are smaller, and filled by a ratio of 256/243 which is roughly a semi-tone. The white notes on a piano, CDEFGABC correspond closely to the notes of the octave. 
You can play around with the sounds of the notes to listen to the musical ratios using a musical instrument, or using an online keyboard. A good place to begin an appreciation of the musical intervals is this video.
The structure of the musical octave
Before moving on, it is worth summarising some of the structural elements of the octave:
- The octave is made up of seven notes (do-re-mi-fa-sol-la-si).
- Most of the notes are separated by a full tone, but the notes mi-fa and si-do are separated by semi tones. What does this mean? What is different about these intervals and why?
- Considering each tone as two semitones, we can see there is a total of 12 semitones in the octave.
- The perfect 4th and perfect 5th ratios are also present between other notes in the octave.
Next: Gurdieff’s Law of Seven (to come).
 Timaeus: https://en.wikipedia.org/wiki/Timaeus_(dialogue)
See: “Music and Mathematics in Plato’s Timaeus” which describes the octave process.
See: http://www.mathpages.com/home/kmath096/kmath096.htm for an interesting description of how Plato went on to describe the universe in terms of the five platonic solids.
 Plato’s semitone ratio (256/243) is approximately half of a full tone ratio (9/8), because (256/243)(256/243) ≈ 9/8. The equivalence is only approximate however. In tuning a modern piano, a slightly different set of ratios are used, with a semitone defined so that 12 equal semitones give an octave (s12 = 2), and a tone is exactly two semitones (t = s2).