an Unequal Temperament

The clavichord needs to be tuned every two or three weeks, and it doesn’t take long to do it. I have taken this as opportunity to experiment with unequal temperaments.

Back in piano tuning school, we were taught to always tune in equal temperament, which has many advantages. It makes all of the keys equally usable, and is almost universal nowadays for everything except early music (up through J. S. Bach, maybe a little further – some extend the use of unequal temperaments on into the Classical era of Haydn and Mozart.)

Hermann Helmholz in his nineteenth-century book “On the sensations of tone” speaks ill of equal temperament, especially for the training of singers (see chapter 16 for an extended discussion of tempered intonation). For example: “The singer who practices to a tempered instrument has no principle at all for exactly and certainly determining the pitch of his voice.” (p. 326)

So when I tuned the choir room piano about six weeks ago, I decided to apply one of my clavichord tunings and see how it sounds. So far, so good: none of the choristers have yet complained, and I think that I prefer it to equal temperament. The “good” keys (C major, F and G major, etc.) are cleaner than they are in equal temperament, and the keys with two sharps or flats are almost as good. The keys with three and four flats and sharps are about like they are in equal temperament. The others? Not so much. The crunchiest is G flat major, as described below; it is barely usable, with D flat major about as bad. Is that too great a price for the improvements in the other keys? That is an open question.

For those who might be interested, here is my tuning plan, such as it is. On the clavichord, I don’t follow a plan, beyond making the fifths and thirds increasingly “noisy” as they move around toward the back side of the circle of fifths (that is, the G flat to D flat fifth, or maybe the D flat to A flat). The details differ every time I tune it, depending on whether I am in a mood that day to favor the sharp keys or the flat keys.

Tuning a piano is a bigger undertaking, so I was more careful:

The Foundation

Tune the A-440 from the tuning fork, then down an octave to the next A, the one on which you will base the tuning.

Tune the following intervals:
F-A pure (no beats. This is a considerably smaller major third than equal temperament)
A-E pure
C-E pure (that is, tune the C based on the E that you have just tuned)
Check F-C; should be about one beat
Check F-A-C triad; should be almost pure
Check A-C-E minor triad: should be pure or almost pure

Tune C-G pure
Check C-E-G triad; should be pure

Tune G-D, about 1 beat narrow
Check D-A (down a fourth from the D), should be about 1 beat wide; adjust D to balance the D-A fourth and the G-D fifth

Tune G-B pure
Check B-E, should be about 2 beats
Check G-B-D triad, should be almost pure
Check G-B-E minor triad (first inversion), should be clean but not pure.

Tune the F to F octave.

This completes the white notes for the temperament octave.

The Balancing

Now for the fun part: the black notes, which are a matter of tradeoffs.

Tune the following intervals:
F up to B flat, about 2 beats wide
B down to F sharp, about 2 beats wide
A up to C sharp, about 1 or two beats wide

Check the intervals from F sharp to A sharp, which will be fairly clean, and F sharp to C sharp, which will be noisy, 3 or 4 beats. Check the F# - A# - C# triad. This will be the worst in the temperament because of the fifth.

Adjust these three pitches (F#, A#, C#) to your satisfaction, checking the major and minor triads (most of them in one or another inversion) that they complete: B flat major, B flat minor, B minor, F# minor, A major, F# major. They will have varying degrees of cleanliness; which of them are better is up to you.

Take a deep breath.

Tune the B flat up to E flat (interval of a fourth), about 1 beat wide.
Check B-D#, should be fairly clean. Check B major, likewise fairly clean.

Tune the E flat down to A flat (interval of a fifth), about 1 beat narrow.
Check the A flat-C-E flat triad. It should be fairly clean. Adjust E flat and A flat to your preference.

Check A flat-D flat, compare with the F# - C# fifth, adjust the C#/D flat so that these two intervals beat at about the same speed, about three beats, checking the thirds from A to C# and A flat to C so that they are both acceptably clean (they will not be pure). This is a point where you could choose to favor the sharp keys over the flat keys by making A-C# cleaner than A flat – C, or vice versa if you would prefer to favor the flat keys. That may depend on what music you plan to play on the instrument before its next tuning. Check the other triads that include A flat and D flat.

Check all major and minor triads. The best way to do this is by going around the circle of fifths, starting with the C major triad and moving either direction (C-G-D-A etc. or C-F-B flat-E flat etc.). The triads should get increasingly “noisy” in a smooth progression up to the G flat major triad, then gradually smoother as you go on back down the other side toward C major. The minor triads should do likewise, starting with A minor.

It is up to you as to how extreme you make the temperament, and you can favor the sharp keys over the flats or vice-versa; the above is only a guideline, especially the latter part of it on the black notes. It is all a matter of tradeoffs.

The Rest of the Piano

From here, tune pure octaves as usual, up to the top and then down to the bottom. The one wrinkle for me was that the octave check intervals I normally use in equal temperament – chains of chromatic tenths in the treble, which should gradually increase in beat speed as one ascends – do not work at all. The chromatic thirds even in the temperament octave will have radically different beat speeds. Same for the low bass, where I likewise check with the interval of the tenth, also the minor seventh (or rather, minor fourteenth – the seventh plus an octave) which in the low bass beats slowly enough to be useful). But with the unequal temperament, about the only way to check the octaves is with the fourth and fifth that lie within the octave that you are tuning; are they what one would expect, given the degree of beating in those intervals in the temperament octave? Remember that the beat speeds will be faster as one goes up and slower as one goes down.

For further study

If these matters are of interest, you might look in a library for Owen Jorgensen’s book “Tuning the historic temperaments by ear” (Michigan State Univ. Press, 1991) which appears to be out of print.

Here is a discussion of the book in the excellent forum “Piano World.” Notice in particular the remarks by Bill Bremmer, an experienced piano technician. He gives a very different “recipe” for an unequal tuning, based on the Thomas Young Well Temperament (1799), which leaves the F-A and C-E thirds somewhat impure, but would make the more distant triads (F# major, C# major) significantly cleaner than the plan outlined above.

Also, here is a website that discusses historical temperaments, giving a good overview of the subject. The author (Kyle Gann) gives the pitches in cents, which shows the difference of each pitch from its equally-tempered equivalent (in equal temperament, each half step is 100 cents, with 1200 cents to the octave). This would be more useful than tuning-by-ear accounts such as mine if one is tuning with the use of an electronic tuner.

But I leave you with the lesson I have learned from all this: the precise details of an unequal temperament are of little importance, especially on instruments such as the clavichord and harpsichord which are easily re-tuned. I have tried to suggest some basic principles, and encourage those interested to experiment freely on their own instrument.

The details matter somewhat more on a piano, and quite a lot on a pipe organ, for which a change of temperament is a multi-day operation and definitely not to be undertaken lightly or as an experiment.

[Edited 12/18/18 to add: Here is an article from the excellent (and free!) online journal "Vox humana," describing one of the oldest extant pipe organs in the world. It is in the village of Rysum in North Germany, built some time around 1450 and little modified since then. Of interest for the topic of my essay is a playing of the triads of the circle of fifths in the organ's temperament, quarter-comma meantone. It is a brief sound file, 32 seconds, and a good demonstration of unequal temperament on an organ; the "distant" keys at the backside of the circle of fifths are quite dissonant. There are weeks when my off-the-cuff clavichord temperament is almost this extreme, but I would not do this to the choir room pianoforte. The whole idea of temperament is to make these "distant" triads and keys either equal to all the others (equal temperament) or cleaner in varying degrees than they are in meantone (unequal "well" temperaments).]