Technical Library

TEMPERAMENTS XX: Sixth-comma Meantone

Now that you are used to the sound of Vallotti’s sixth-comma fifths, you might want to try Sixth-comma Meantone. Richard Kolb, sometime lutenist at the Carmel Bach Festival, had been using this temperament successfully for some time, and persuaded me to tune it for one of the Intermezzo recital programs he devized in 2007 for his final year in Carmel. The same year, Ross Duffin’s book How Equal Temperament Ruined Harmony (and Why You Should Care) appeared: It seems this is the only temperament for those which pass through Case Western University!

I also tuned Sixth-comma Meantone on all four keyboards (Grimaldi, Neapolitan, Continuo organ and Regal) for the sole Australian appearance of Concerto Italiano under Rinaldo Alessandrini at the 2017 Adelaide Festival, performing Monteverdi’s L’Orfeo to a sold-out house in Adelaide’s historic Town Hall.

From a keyboard point of view, this temperament was supposedly favored by Silbermann and so is sometimes found described under his name. With the exception of the wolf which is greatly reduced in size from that of Quarter-comma Meantone, a moment’s contemplation of the diagram will show you that all the identically-sized fifths are here a somewhat wider sixth-comma like Vallotti. Each of the usable thirds resulting are exactly the same in size, but wider here than the absolute purity of Quarter-comma Meantone and so we sadly have no pure intervals to check.

Compared to Vallotti, though, for practical purposes we can regard all the “white” notes of the keyboard as being in exactly the same place. As with all regular meantone temperaments, all usable keys sound the same, so this temperament lacks Vallotti’s variation in key color. The “black” notes for the three sharp keys are closer to their tonics (progressively flatter than Vallotti), and the two flats are closer to their thirds (sharper than Vallotti). Yes, as in all these meantones, a G♯ is flatter than an A♭!

To change your previously-tuned Vallotti to Sixth-comma Meantone, proceed as follows:

1. Raise your b♭ so that instead of a pure fifth b♭–f', you have a narrow sixth-comma fifth sounding with the same roughness as the a–e' a semitone lower.

2. Raise your e♭ first of all to make a pure fifth with the b♭ you’ve just tuned, then raise it some more to make a sixth-comma fifth e♭–b♭. Again this can be compared in roughness to e–b which is the same size. The two flat notes have now been found.

3. The three remaining sharps must be progressively lowered from Vallotti. Work to lower your f♯' so the fifth b–f♯' sounds similar to the already tuned b♭–f'. Tune down an octave to determine the new f♯ and then lower your c♯' so that again you have the same sized narrow fifth f♯–c♯' as f–c'. Tune down an octave to c♯ and make your final narrow fifth c♯–g♯. Verify your almost-usable wolf g♯–e♭' and your tuning is complete.

4. Finally, though, you can check the accuracy of your work with a chain of ascending tempered fifths. If you’ve succeeded, each one should beat just a little more than the previous as you ascend the keyboard through c♯–g♯, d–a, e♭–b♭, e–b, f–c', f♯–c♯', g–d', a–e', b♭–f', b–f♯' & c'–g'. (Your g♯–e♭' wolf is as expected from the diagram, so that interval doesn’t occur in this check series.) And you can even pull the same trick with your major third f–a the same three beats per second as Vallotti, and each third beating slightly faster as you ascend g–b', a–c♯', b♭–d', c'–e', d'–f♯', e♭'–g' & e'–g♯'. The harsh thirds which are really diminished fourths, do not form part of this check. They are perhaps not as vulgar as in Quarter-comma Meantone, and you might find their enharmonics almost usable: f♯–b♭, g♯–c', b–e♭' & c♯'–f'. If you try to play an A♭ Major triad you’ll hear that it is almost as hideous as Quarter-comma because every note is still spelled wrong.

To tune Sixth-comma Meantone from scratch, proceed as follows:

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

2. Set the f a third below that a: Tune it pure first of all, but then widen the interval by flattening the f until you hear three distinct beats per second.

3. We are now going to squeeze all the fifths in the circle between F and A by exactly the same amount, a sixth-comma, in fact twice as narrow as an Equal Tempered fifth. Tune middle c' pure to f, then narrow the interval by lowering your middle c' a little until you hear a lazy beat of a bit more than once a second.

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. This interval will beat a tad less than twice per second, so check it against the second hand on your watch until you have a good wow-wow, wow-wow. It helps to replay the interval every second or so—listen when the notes are fresh so you don’t have to struggle to hear the beats while the sound is dying away.

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 size, so they should sound pretty similar. The f–c' fifth is a tone lower, so if your intervals are to be 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': 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. With the sole exception of the wolf, all fifths are the same size. You can therefore proceed around the circle to tune each one in turn. Work around the sharp side first. Tune e' pure to a, then lower it a little, squeezing the interval so it has a perceptible but not overly blatant wave. Compare g–d' to a–e'. Now drop down the octave from e' to e, which of course must be tuned pure. Continue similarly to determine your B, F♯, C♯ & G♯.

8. Complete the flat side with B♭ & E♭.

9. Check that all your narrow fifths c♯–g♯, d–a, e♭–b♭, e–b, f–c', f♯–c♯', g–d', a–e', b♭–f', b–f♯' & c'–g' should beat just a little more than the previous as you ascend the keyboard.

10. Likewise, the usable thirds being all the same theoretical size, f–a, g–b', a–c♯', b♭–d', c'–e', d'–f♯', e♭'–g' & e'–g♯' should increase in beat speed as you ascend.

Like we did when Transposing Quarter-comma Meantone, you can change every G♯ to A♭ if more flats are required for the piece. Raise your g♯, turning it into an a♭ so the fifth a♭–e♭' beats every so slightly slower than a–e' a halftone higher: Your wolf fifth—again really a diminished sixth—has now moved to c♯–a♭. If you‘re on a real computer, you can move your mouse over the temperament circle above to see what you have accomplished. If you require more sharps than flats, change every E♭ to D♯ instead.

Further discussion
Anonymous [Kayano, Moxzan] Dodecagon — Chi-s akt temo Tokyo 2012, p41
Duffin, Ross W How Equal Temperament Ruined Harmony (and Why You Should Care) W W Norton & Company Inc, 2007
Padgham, Charles The Well-Tempered Organ Positive Press, Oxford 1986, p59 (“Silbermann”)
Veroli, Claudio di Unequal Temperaments Artes Graficas Farro, Buenos Aires 1978, p65

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