Technical Library

TEMPERAMENTS XXX: Eighth-comma well temperament

We’ve seen many possibilities on these pages of splitting the comma amongst the twelve fifths of our circle: Kirnberger III’s four narrow fifths and eight pure, Kellner’s five narrow fifths and seven pure, and perhaps you’ve managed to get your head around the ever-popular Vallotti temperament, with half of its fifths narrow and the other half pure. Within each of these temperaments, the narrowed fifths are all the same size: Kirnberger III fifths are quarter-comma narrow, Kellner fifth-comma, and Vallotti sixth-comma. The fewer pure fifths which remain on our circle, the more tempered fifths, and if there are more of them, the tempered fifths must be relatively wider. (Make sure you understand that last sentence before proceeding!)

There is nothing stopping us playing around a bit to see what happens when we might have eight narrow fifths—and four corresponding pure fifths.

I suggested this eighth-comma well temperament—something a little less harsh in the remote keys than Vallotti—for Australian Haydn Ensemble’s Beethoven’s Piano series in June 2015. We needed to consider the wide-ranging tonalities of Beethoven’s Third Piano Concerto in c minor, with its slow movement in the unexpected key of E Major. It was very ably tuned, as always, by Colin van der Lecq.

Looking at the diagram at the right, you can see we have really just extended the area covered by Vallotti’s six narrow fifths, adding a narrow fifth on both the sharp and flat sides. With eight narrow fifths, we have four remaining pure. (If you’re on a real computer rather than reading this on one of those newfangled mobile devices which lack mouse-over commands, move your mouse over the diagram to compare this temperament with Vallotti.)

Here’s how you can tune this Eighth-comma well temperament, using a very similar procedure to Vallotti by defining an initial Major third, and dividing that third into four equal fifths:

1. Tune a' to your pitch source and tune a in absolute perfect tune an octave below it.

2. Find the f a third below that a: Tune it pure first of all, and then widen the interval by flattening the f until you hear five distinct beats per second. Vallotti was three beats per second, but here we want five. The resultant third is wider than Vallotti, but not as wide as Equal Temperament.

3. Look at the diagram and you’ll see all the fifths in the circle between F and A must be narrowed, like Vallotti. Because we tuned our f somewhat lower than it was in Vallotti, the fifths between F and A must each be a little wider, only an eighth-comma narrow instead of Vallotti’s sixth-comma narrow, if that makes sense. Divide these fifths evenly. As a guide, the g–d' fifth beats exactly once per second at A440; f–c' a fraction slower because it is a tone lower.

4. Once you are used to the slow beat of eighth-comma fifths, you can place them in the other positions they are required around the circle. Tune b♭ pure to f', then raise the b♭ so it has a nice slow beat, similar to the c'–g' a tone higher.

5. Tune four pure fifths around the circle from the flat side of B♭ until you hit G♭ (which will serve nicely as our F♯).

6. You now have three remaining fifths to temper between A and F♯: Make them equally narrow.

7. As a final check, remember that each of your eight tempered fifths should seem to beat slightly faster as you ascend the keyboard through e–b, f–c', g–d', a–e', b♭–f', b–f♯, c'–g' & d'–a'. All four others are pure: e♭–b♭', f♯–c♯', a♭–e♭', d♭–a♭'.

8. Bring the rest of your instrument into tune with your bearings area, and you are ready to play.

Further discussion
Asselin, Pierre-Yves Musique et Tempérament Éditions Costallat, Paris 1985, p152
Bavington, Peter Clavichord Tuning and Maintenance Keyword Press, London 2007, p165

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