TEMPERAMENTS XXX: Pinnock ModernEntire Contents Copyright © 2010 CBH
What particular temperament might be suitable for your friends with modern instruments—not as extreme as some of the historic temperaments, but not as bland to listen to as Equal Temperament either?
Here is a practical solution devized by my English colleague Simon Neal for the 2009 North American Tour of the Emmanuel Pahud/Jonathan Manson/Trevor Pinnock trio, so let’s call it the Pinnock Modern temperament. This is probably a very authentic attitude to tuning, followed by performing musicians for centuries: If a temperament to suit your needs is unknown, make one up yourself.
Simon describes this temperament as “equal with color” or “a spread out Vallotti”. The idea is to fight for usability in every key, with none standing out as too bright or too pure.
In common with Vallotti, you can see from the diagram that there is a chain of six identically-sized fifths on the sharp side of F. Here, however, the fifths are not Vallotti’s sixth-comma, but a smaller eighth-comma—perhaps halfway towards an Equal Tempered fifth from Vallotti. Of the remaining fifths, half are pure, the other half equal-tempered, but these are not arranged evenly. In terms of key color, compared to Vallotti, you should expect the good thirds built above the notes on the sharp side of F to be not quite so good. The bad thirds elsewhere in the circle are worse than Equal Temperament, but all are better than the Pythagorean thirds of the bad Vallotti keys. The worst key here is going to be C♯.
Here is one way to realize the Pinnock temperament from scratch:
1. Tune your a' to a tuning fork, and tune a in absolute perfect tune a beatless octave below it.
2. Now we want to set the f a third below that a: Remembering that this temperament could perhaps be regarded as a Vallotti/ET compromise, tune the f perfect first of all, but then widen the interval by flattening the f until you hear almost five distinct beats per second.
3. Now your aim is to temper all the fifths in the circle between F and A by exactly the same amount: An eighth-comma narrow. All these tempered fifths are narrow, so the intervals must be squeezed. Tune middle c' pure to f, then squeeze the interval by lowering your middle c' a little until you hear a lazy beat of perhaps a bit less than once a second. If you’ve been tuning Vallotti, you are familiar with the sound of a sixth-comma fifth in this part of the keyboard, and the observation that an Equal Tempered fifth beats at half that rate. An eight-comma fifth, therefore, will be between those two extremes, a tad rougher than Equal Temperament.
4. Tune d' pure to a', then raise your d' a little, again squeezing the interval so it has a perceptible but not overly blatant wave—beating about three times every two seconds.
5. Tune g a beatless fifth below d', then squeeze the interval by raising the g. Compare f–c' to g–d': These two intervals are the same theoretical size, so they should sound pretty similar. The f–c' fifth is a tone lower, so if your intervals are indeed the same theoretical size, f–c' should beat marginally slower than g–d', but don’t fuss needlessly.
6. Tune g' up an octave from g. Compare c'–g' to d'–a': Again, these two intervals are the same size, so again they should sound pretty similar. If not, a little error may have crept in, so juggle all these fifths until you are happy with their uniform roughness. Don’t move your a, nor your f!
7. You now require the same two narrow fifths between a–e, and e–b. You know enough to check the e–b to f–c' to g–d' and a–e' to ensure all those intervals are indeed the same eighth-comma narrow, hopefully with the perceptible beats accelerating slightly as you ascend.
8. Let’s skip to the flat side of C now: It should be easy to tune up an octave from f to f' and down a perfect beatless fifth to find b♭.
9. Find your d♯ a perfect fifth below b♭ then squeezing it ever so slightly by raising the d♯ to make the first Equal Tempered fifth of this temperament: This interval will have a very slow beat, perhaps once every two seconds. Tune up an octave to d♯'. (Don’t get cross that I‘ve called this note D♯ in the diagram: I thought that was a better alternative than E♭ because there are two absolutely pure fifths on the sharp side of it, although we have determined it from the other side.)
10. While you are in Equal mode, place another Equal-Tempered fifth from b–f♯'. Always make sure you are approaching the narrow interval from perfect by squeezing it. In this case, it means flattening the f♯'.
11. Tune down the octave to f♯, another ET fifth from there to c♯', and down the octave to c♯.
12. All that remains is the g♯, which the diagram shows is perfectly located without beats between the c♯ and d♯'. Can’t get the g♯ perfect to both notes? Then an error has accumulated. It may be of little practical consequence, but it’s worth a check. Refer back to the diagram and ensure that your rendition has the gist of this temperament. Are those six eighth-comma fifths all equally lazy, if anything, with the beat speeds increasing as you ascend? Likewise, the three ET fifths: Are they in the right spot of the circle? Are you sure they are indeed narrow and not wide? Have you fulfilled Simon’s ambition for a true Equal Temperament substitute—one with more flavor?
A good tuner knows when to stop fussing: Bring the rest of the instrument in tune with your bearings area as usual, and play on.
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