TEMPERAMENTS IV: GrammateusEntire Contents Copyright © 2012 CBH
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 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 not impossible. Many players use this temperament for the Fitzwilliam
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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