Ratios Summary thus far

We have the duple relationship, such as 4:1 and 2:1, which are known commonly as octaves and are basically the same note. We have unison, 1:1, which is actually the same note. We have the triple relationship, 3:2 and 4:3 which are the perfect fifth and the perfect fourth respectively.
Pythagorus derived from the triple ratio the circle of fifths F,C,G,D,A,E,B. C is F times three, G is F times three times three, D is the triple triple triple of F, etc... These pitches, when translated to the same 2:1 or octave becomes our diatonic scale: C,D,E,F,G,A,B. Around the 15th and 16th century the use of sharps and flats came into practice and the idea of the chromatic 12 note scale surfaced, followed soon after with Equal Temperament. For the last 400 years we in the western world have been infatuated with, immersed in, and frustrated by the equally tempered twelve tone scale.

The problem is this:
There are other ratios to be heard. Our twelve tones are only an approximation of the richness that could be possible if we could break out of our cocoon. I wouldn't dream of saying Pythagorus is to blame. I think the biggest factors that have led to our situation is our constraining notation system and the composers who have failed to think outside of its box.

next time:
the drama of fives

Roll over Pythagorus

The ancient Greeks, around 550 BC, believed that there was something mystical about the number three, that it must be godly somehow. Since music was also important, it must be related to three. (btw, this also seems to be a reason why our Christian Church fathers were so adamant about defining to the smallest iota the theology of the Trinity. The Greeks were neurotic about anything related to threes, especially religion.)

Pythagorus utilized a monochord, a crude, one-stringed instrument useful for taking measurements of the precise ratio of the vibrating string. When he halved it, he got the octave above the original plucked string sound. When he divided the string in thirds, he got the perfect fifth above. Okay, that's the end of his experiment, he decided to deduce everything we know about music from that. Yeah, those Greeks were thorough. So he built a scale based on fifths which is our way of saying the ratio 3/2. So, as the ages went by, western music ended up with a scale based on the notes of the circle of fifths: F, C, G, D, A, E, B and the chromatics went beyond that. Of course we added those names to the pitches much later, like 2000 years later. The important part is that if Pythagorus had decided to maybe try the five ratio instead of just threesies, we would very likely have H, I, J, K, and L added to our keyboards today.

Up to around 1600AD western musicians were infatuated with the perfect fifth and perfect fourth, both of which are ratios involving 3 (3/2, and 4/3 respectively). It may also be interesting to note that our current popular music revolves around the I, IV, V chords, the roots of which are the same as above.
Hmm..
Now you can say, "That band is sooo... 6th century... BC."