Technical Library

TEMPERAMENTS IV: Grammateus

In his Libellus de compositione regularum pro vasorum mensuratione. Deque arte ista tota theoreticae et practicae published in Vienna in 1518, Henricus Grammateus (c1495–c1525) proposed just such a temperament, with all the fifths pure except the two from B♭ to F, and B to F♯. The trick is in determining just how much half a comma is, but it is quite possible to tune it using an “equal-beating” method. Here is how:

1. Tune a chain of six pure fifths on the sharp side from C, around through to B.

2. Tune your f a pure fifth below middle c': This completes the natural keys.

3. Temporarily tune your f♯' a pure fifth above b, and then tune down the octave from both these notes to f♯ and B. (We are working in the octave below middle c', because you will find it easier to hear the beats.) We must now flatten the f♯ so that the fifth it makes with B is narrow by exactly half a comma. Lower your f♯ until it beats at the same speed with b above it, as d below it. Check your “half-wolf” (werewolf?) fifth B–f♯—is it ok?

4. Continue tuning absolutely pure fifths from F♯ to A♯. If you’ve been particularly successful, your b–f♯' fifth will beat ever so slightly faster than a♯–f, because it is just a semitone higher.

The thirds are usable, especially when you get into the sharp keys, and the two baby wolves are crude but not impossible. Many players find this temperament suitable for John Bull’s very enharmonic “Ut, re, mi, fa, sol, la” (Fitzwilliam Virginal Book LI).

Further discussion
Anonymous [Kayano, Moxzan] Dodecagon — Chi-s akt temo Tokyo 2012, p56
Barbour, J Murray Tuning and Temperament Michigan State College Press, East Lansing 1951, p140
Jorgensen, Owen The Equal-beating Temperaments The Sunbury Press, Raleigh 1981, p14

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