Sunday, January 03, 2010

Temperament 2

I showed some scepticism in an earlier post about mean-tone tuning on the lute.

This is reinforced by the wide range of keys in which lute music is written. Tuning systems other than equal temperament have a 'home' key in which they sound best. The further away you get from the home key, the less good they sound. So non-equal tempered systems work best in a narrow range of keys.

Assuming a lute tuned in G, a good starting point is the key of G major or minor. The instrument sounds good, the fingering works well, and as a result there are plenty of pieces in those keys. Yet the repertoire stretches a long way on either side. On the sharp side there's a lot of music in D major and minor, A minor (think Lachrimae), and even in E minor (Dowland's Mignarda). On the flat side we find pieces in C major and minor, F major and minor, B flat major and minor, E flat major, and A flat major (think Walsingham). In modern terms, that's a range of key signatures from two sharps to five flats. The music isn't kept strictly separated into different keys, either: for example, the Fantasie by Laurencini in Varietie of Lute Lessons (Robert Dowland, 1610) finishes in D major and is immediately followed by Ferrabosco's Fantasie in B flat minor.

Within pieces, the keys range still further. The first example below is from Sir John Langton's Pavan by John Dowland, which appears in Varietie. The piece has started in D major but at this point we're on a chord of F sharp major, the dominant of B minor. In the second example, from a Pavan in F minor by John Johnson in Dd 2.11 (f. 44v), we've hit G flat major at the beginning of the second bar. The same chord? Whichever way you approach it, it's going to be problematic in a non-equal tempered tuning system.

The argument is sometimes made that the different sound quality of different keys is one of the positive characteristics of non-equal tempered systems. Up to a point, I would say, but examples like this go well beyond that point for me. It's also an aesthetic that would apply only to instruments, since singers don't generally choose to sing out of tune in more remote keys. That makes it unconvincing for me.

Interestingly, John Dowland gives detailed instructions in Varietie on how to position the frets on a lute. He recounts the anecdote about how Pythagoras discovered musical proportions by hearing smiths in a forge beating iron with different sized hammers, and he tries to follow these simple Pythagorean ratios in his fretting system. The 12th fret is positioned at half the length of the string, the 7th at one-third the length, the 5th at one-quarter, and the 2nd at one-ninth the length. The positioning of the lower frets is more complex: the first fret at 2/33 of the string length, the third at 5/33, and the fourth at 53/264 which is almost, but not quite, a pure Pythagorean major third, which would be at 1/5 the length. The 8th, 9th and 10th frets are positioned at one third of the sounding length of the string stopped at the 1st, 2nd or 3rd fret: in other words, a pure Pythagorean fifth higher.

Phew. That would seem to be definitive, coming from the master himself. But the problem is that this tuning system is as skewed as any other. In particular, the semitones are of very different sizes, with the widest being 38% bigger than the narrowest, and as a result various octaves and other intervals are out of tune. So it really can't be said to give a more satisfactory result than any other system. Incidentally, there's no suggestion from Dowland that frets might be moved for different keys.

So what's the answer? Well, as I said before, every system is a flawed compromise. But for me this is one further factor in favour of equal temperament on the lute.

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