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.