The Music & Writings of Graham Jackson

How is it tuned?

 For the usual, equal-tempered tuning, one takes the interval of the perfect 5th, found in the overtone series, and starting from any note--say F--sets one 5th on top of another, continuing for twelve steps until one circles back to the original note. The resulting twelve notes constitute the 5 black and 7 white notes of the piano keyboard.

Unfortunately, they do not come back exactly to the same note, but overshoot slightly. Hence, since Bach's day, tuners make each of those 5ths slightly smaller--out-of-tune, one could say--to round off the circle neatly.

The difference from the natural 5th in this equal-tempered tuning is so slight that most people do not notice it consciously, although subconsciously they do, as it causes a slight dulling of the resonance. Hence string quartets, choirs, etc. often prefer slightly adjusting their pitches to the more natural consonances.

For the "twelve true-5ths tuning": you first set C at 256 Hz. Then you tune the 7 "white keys" by the circle of 5ths, using however natural 5ths. Then you divide the octave at C exactly in half (which can be done handily with a special tuning fork), and tune the 5 "black keys" by natural 5ths to that F#.

You end up with two series of natural 5ths--one of 7 notes and one of 5 notes, linked by an "unnatural" interval of an augmented 4th (which is actually the same augmented 4th found in the equal-tempered system).