Technical Library TEMPERAMENTS II: Pythagorean Entire Contents Copyright ©2005 CBH

The accomplishment of any tuning system requires you to take account of the impossibility of compressing the spiral into a true circle. Don’t fret too much about this—it has driven philosophers, mathematicians, and musicians crazy for centuries, and will continue to do so.

If you can hear a perfectly tuned fifth, you can tune your instrument Pythagorean. This system—I’m avoiding calling it a “temperament”, because no intervals are actually tempered—is useful for the earliest keyboard music, from the Robertsbridge Codex c1320. At this time of mediæval organum, the fifth was a consonance, and so long as you don’t try to play major thirds, you may find yourself overcome with the beauty of sound.

To tune Pythagorean simply:

1. Tune a chain of perfect (ie absolutely beatless) fifths from C around the flat side of the circle to E♭.

2. Tune another chain of perfect fifths around the sharp side of C until you reach G♯.

The interval from G♯ to e♭ sounds hideous, and well-deserving of its name, the wolf. As a coincidence, you actually have four perfect major thirds, but these are in keys which are probably unused at this time in history. These perfect thirds are denoted by the straight lines in the diagram. Unfortunately, all the other thirds are very wide—more out of tune, in fact, than the Equal Temperament we have become used to on the piano.

When you are happy with your middle octave, you must bring the rest of your instrument into tune with what you have done. Tune down in octaves, and up in octaves, then tune the other choirs in tune with your first register by proceeding from bass to treble, checking as you go. Those fifths that are perfect in the middle of the keyboard, should also be perfect everywhere else.

Wanna explore some more?

	Tuning Bibliography
	Technical Library overview
	Harpsichords Australia Home Page