How Equal Temperament Ruined Harmony: And Why You Should Care [Paperback]

Never judge a book by its cover, they say, and it must be even more true that you should never judge a book by its title. As soon as I saw the title of this book, however, I knew I would have to make an exception in this case and read it.

Ross Duffin has written an engaging, densely argued and robust demolition of the commonly held idea that equal temperament triumphed in the time of Bach's Well Tempered Clavier and has been the one true tuning ever since. Drawing his evidence from documentary, instrumental and, for the 20th Century, recorded performances, Mr Duffin shows that the equal temperament (of 12 equally-spaced semitones to the octave) only became any form of standard much later than generally imagined, and is in many cases still more honoured in the breach than the observance - indeed the Well Tempered Clavier itself was Well Tempered, not Equal Tempered.

As to his subtitle (`And Why You Should Care') he argues that we are hearing the majority of music in a very different way than was intended by the composer - Mozart, Haydn and Beethoven, to name but three, wrote their masterpieces to be performed in temperaments other than the equal, thus fundamentally altering the way in which the very chordal progressions, and therefore the overall timbre and character within the pieces, progress.

Along the way Mr Duffin gives entertaining pen portraits of the major figures in his story, has a wealth of anecdotal asides, and writes in a generally entertaining and accessible way.

I say `generally' because there is no possible way of avoiding the mathematics, subtleties and jargon of tuning and temperament; this is a musicological work, and its readership will probably be unjustly restricted by virtue of some of the more technical sections. While being critical I would also like to have seen the aforementioned pen portraits grouped at the end of the book or at the end of chapters; the small page format of the book means that the main flow of the text is disrupted by the interjection of the portraits at the precise point in which the person or concept is first introduced. I would also have liked to have known whether Mr Duffin thought the spread of recorded music in the 20th Century affected the standardisation of tuning systems, and perhaps also seen his argument and examples straying outside the purely classical repertoire into such areas as blues and folk.

Those minor quibbles aside I would heartily recommend this book to anyone interested in the history of music, the development of musical instruments, and to anyone who, like me and also the cello student quoted in the text, wondered why the great expressive cellist Pablo Casals sounded so `out of tune' on first hearing.

Music, I was led to believe, is a supremely elegant manifestation of pure mathematics. The intervals we know as fifths (think "Twinkle - twinkle"), fourths (Auld lang syne), and octaves (Somewhere over the rainbow) correspond to simple fractional relations between the sound frequencies, of 2/3, 3/4, and 1/2, respectively. And as an illustration, one gets shown the corresponding keys on a piano keyboard.

What nobody told me in the first 44 years of my life is that the intervals you play on the piano do not correspond to the simple fractions cited above. The piano is actually tuned not in pure intervals but in a system called "equal temperament" for the simple reason that the fractions don't add up. If you add up 12 fifths, all around the circle of fifths (C - G - D - A - E - B - F# - C# - Ab - Eb - Bb - F - C) you get (3/2)^12 = 129.746. Theoretically, the first and the last C in this series should be seven octaves apart, so their frequency relation should be 2^7 = 128. And not 129.746. And there are even worse clashes with other intervals. So in fact it would be impossible to tune a piano according to the pure intervals defined by simple fractions. This is why we as a civilisation have settled for equal temperament, which means the octave is split into 12 equal semitones.

Equal temperament (ET) is so widespread today that knowledge of the alternatives has gone missing, and even many musicians are unaware of the problems that this compromise solution causes. Duffin argues that some of the "unequal" solutions favoured in renaissance music and through to the end of the 19th century (he dates the total victory of ET to 1917) would still be useful today and that the question of temperament should be considered afresh for each piece of music, taking into consideration the likely intentions of the composer, the context of its creation, and what's best for its harmonies. This will all be self-evident for practitioners of early music who use historic instruments and temperaments already, but it may be new to many people dealing with the classical repertoire from Bach to Beethoven (who, the author argues, cannot have become used to hearing ET by the time he went deaf).

This argument is all very well and convincing, but it would fill only around 30 pages, so to bulk his pamphlet up to a marketable 196 pages, the author has included lots of repetition (as you tend to do in music!) and biographical profiles of everybody who has ever voiced an opinion on temperament, from Mozart's father, via the flautist Quantz, through to the cellist Pablo Casals. And cartoons. And diagrams. But all this is redundant in principle, so if you're just after the meat of the matter, you can probably read the relevant pages within an hour, at a bookshop cafe.

What remains is the impression that music is in fact a lot less mathematically elegant than it is often claimed to be, and that it is a rather messy compromise between pure mathematical beauty and practicability. The good news is that a messy system leaves you free to mess with it, giving performers more freedom. So from now on, when I play out of tune, I can always claim I am experimenting with different temperaments.

This book is about the biggest skeleton in music's cupboard: the fact that the notes in the musical scale don't quite add up right.

A perfect fifth and a perfect fourth make an octave - that's fine. A major third and a minor third make a perfect fifth - that's fine. But though three consecutive major thirds on the piano keyboard take you up precisely an octave, three 'pure' major thirds actually make slightly less than an octave. And though four minor thirds on the piano keyboard make an exact octave, four 'pure' minor thirds actually make slightly more than an octave. So somehow, especially when tuning a keyboard instrument on which the notes are fixed (and one black note has to double as both F sharp and G flat) we have to tune the notes in such a way as to make a decent job of both the scale and the harmonies. This book is about the different methods of 'squaring the circle' that people have used over history.

In particular the author is concerned to debunk the myth that 'equal temperament', which simply divides an octave into twelve absolutely equal semitones, necessarily sounds the best and was the choice of the great composers of the 18th and 19th centuries. Bach's famous "Well-tempered Clavier" was not written (as is often thought) to demonstrate the superiority of equal temperament. Bach intended it to be played in a temperament in which (unusually for the day) every key 'worked', but yet sounded slightly different from the others. That was a revelation to me, but makes such sense.

Non-equal temperaments don't make every key sound the same. In some of them, some keys will sound wonderful and others will sound abysmal. The compromise has to come somewhere. The author's point is that simply dividing the octave into twelve equal semitones has not 'solved' the problem for all time. It brings harmonic disadvantages with it, particularly as it has to use a very wide major third. There is a cost to being able to transpose with impunity to any key you fancy.

Given that equal temperament is used almost universally on pianos and organs today, the book is far from being merely a hypothetical excursion. It might take a little bit of work in understanding, and contains a bit of elementary maths. But it is nevertheless an easy read.

I found it illuminating to fiddle around on an electronic keyboard (mine is a Roland RD-700, but I'm sure there are others) which allows the user instantly to change the temperament to one of half a dozen different options. I was amazed at what I heard.