Technical Library

TEMPERAMENTS IV: Grammateus

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Grammateus temperament ©2012 CBH 4K gif By splitting the wolf into two or more parts and moving it around the circle, we can reduce its severity.

Henricus Grammateus proposed just such a temperament in 1518, 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 naturals.

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 Bf—is it ok?

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

The thirds are usable, especially when you get into the sharp keys, and the two baby wolves not impossible. Many players use this temperament for the Fitzwilliam Virginal Book.


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


Temperament index:


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