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Pythagorean Music Theory: Musical Ratios that Give Rise to Too-sharp Notes
Pythagorean music theory has almost given us the major diatonic scale. But one note’s missing, namely F.
So, continuing with Pythagorean music theory, why not try to get that last note by playing the next note, a fifth interval (seven semitones) up from B, which was the last note you played in the series?
Try it.
What’s the note you get?
Alas, it’s F♯, not plain old F.
Worse, the fifth above F♯ is C♯, not C.
Dang.
Worse still, suppose you go away from the piano and instead decide to derive the series of notes using a calculator. You start with the frequency 261.6 (Middle C) and use your calculator to derive the series of fifth intervals as exact ratios of 3:2. Then you compare your list of calculated frequencies with the actual frequencies of the corresponding piano notes (available on Roedy Black’s Musical Instruments Poster).
What you discover is that all the theoretical notes you calculated are slightly but noticeably sharper than the notes on the piano!
Dang again.
In any case, the fact that you can almost get a complete major diatonic scale simply by using notes derived from consecutive overtone frequencies with the single simple frequency ratio 3:2 (the perfect fifth) illustrates the central role of simple frequency ratios in scale building.