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Comment by haberman | Hacker News Reader

Most musicians aren't aware of this, but an interesting corollary of the circle of fifths is the Pythagorean Comma: http://en.wikipedia.org/wiki/Pythagorean_comma#Circle_of_fif...

We generally treat Gb and F# as the same note, but if you actually count perfect Pythagorean fifths (frequency ratios of 3:2) in both directions from C, Gb and F# don't actually perfectly meet in the middle!

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That's the cause of that famous piece being called the Wohl Temperierte Klavier, those little rounding errors add up, once you have settled on a system that works for everybody by convention Gb and F# have to be the same notes (since there is only the one key for them...).

http://www.phy.mtu.edu/~suits/notefreqs.html

The ratios between subsequent notes differ, but not by much.

Edit: a quickie spreadsheet gives me 0.009502 for the largest difference between two tones.

But that again is just a convention, and before western music converged on some kind of well tempered or chromatic tunings, there were quite some experiments with keyboards featuring both flat and sharp keys, see http://de.wikipedia.org/w/index.php?title=Datei:Cembalo_univ... or http://en.wikipedia.org/wiki/Archicembalo

The circle is a Procrustean bed - the real way to derive the scale is to use a line of fifths that never returns to the starting pitch. Circle any group of seven consecutive fifths, and the group's major scale tonic is the one second from the left. This line goes into double and triple sharps/flats, etc.

"Perfect" pitch isn't necessary, as slight deviations produce interesting sounds related to the "flanger" effect. It's the reason why pianos have more than one string per key: it is the intention that they are slightly detuned, or else you wouldn't hear that there's more than one string.

The deviation should preferably be less than, say, 1Hz, because you hear the difference in frequency as the beat frequency.

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