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Comment by ledauphin | Hacker News Reader

are the post and your comment not mathematically equivalent statements?

Its the use of the musical concept of Fifths that's the key here: We just derive the notes from what sounds good, not what makes sense mathematically? I'm just using Python to mirror the author's analysis -- you can derive "12 notes" pretty much just by using your ear and listening to the Fifths.

A perfect fifth is just 150% (3/2) the frequency of the root, just as an octave is 200% (2/1). "What sounds good" is in a certain sense based on the harmonic series, and in that sense it is equivalent to what makes sense mathematically.

A problem is that if you stack perfect fifths to get 12 notes then they won't sound in tune with each other across different keys. It's this issue which forms the crux of the linked post.

I think it's integral to the theorie seeing the fifth as 3 times, the octave as 4 times and the prime as 2 times an arbitrarily low root key. Trivially, the 2^n multiples form octave intervals but between C12 and C24 there's your twelve tones on a logarithmic scale. Naturally, it doesn't transpose in integer intervals if stepping down.

No, the article is essentially pointing out that the 12th root of two to the seventh power is really close to 1.5, whereas the comment you're replying to is saying that if you raise 1.5 to the 12th power, you get really close to a power of 2.

The former is more interesting when it comes to how music works psychoacoustically: the interval of a perfect fifth is fundamental to almost all music. Whereas the "circle of fifths" is more of a convenience that makes it easier to think about keys. Few songs would ever traverse the whole circle and come back to where it started, and if you stick with strict just intonation there is no circle of fifths anyways. (Maybe you could better call it a "spiral of fifths" or something.)

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