Technical Library TEMPERAMENTS III: Pythagorean Entire Contents Copyright © 2016 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.

Pythagorean is simple to tune:

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

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

Pythagorean tuning is represented in the circle of keys at the right. The “0” denotes all the fifths tuned absolutely pure. The comma occurs in the untuned interval from G♯ to E♭, denoted by the “-1” because it is narrow by one whole Pythagorean comma. This interval sounds hideous, and well-deserving of its name, the wolf. To reduce its severity, we may divide the comma and move it to different locations around the circle in subsequent temperaments.

As a coincidence, you actually have four aurally-in-tune major thirds, but these oocur in keys which are probably unused at this time in musical history. The notes forming these pure thirds are connected 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 so used to on the piano. These wide thirds are not a problem if you choose the appropriate music.

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 bring the other choirs into tune with your first register by proceeding from bass to treble, checking as you go. Those fifths that are pure in the middle of the keyboard where you set the bearings, should also be pure throughout the entire compass..

Wanna explore some more? Let’s see what Grammateus did to improve the wolf.

Further discussion
Anonymous [Kayano, Moxzan] Dodecagon — Chi-s akt temo Tokyo 2012, p52
Barbour, J Murray Tuning and Temperament Michigan State College Press, East Lansing 1951, p1
Klop, G C Harpsichord Tuning Werkplaats voor Clavecimbelbouw, Garderen 1974, p10
Kottick, Edward L Harpsichord Owner’s Guide University of North Carolina Press, Chapel Hill & London 1987, p151
Veroli, Claudio di Unequal Temperaments Artes Graficas Farro, Buenos Aires 1978, p20
Zuckermann, Wolfgang Joachim The Modern Harpsichord October House, New York 1969, p242

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