This review by Peter Bavington first appeared in British Clavichord Society Newsletter 41 (June 2008)

Ross W. Duffin: How Equal Temperament Ruined Harmony (and Why You Should Care), New York and London, W. W. Norton & Co., 2007. ISBN 978-0-393-06227-4. Price: £17.99

A. C. N. Mackenzie of Ord: The Temperament of Keyboard Music: Its Character; its Musicality; and its History, Bristol, published by the author, 2007. ISBN 978-0-9556030-0-6. Price: £75; available from bookshops or direct from the author at 24 Kingsdown Parade, Bristol, BS6 5UF

Temperament might seem a dry subject, but over the centuries it has frequently stirred strong passions. These two books are in that tradition: they both vigorously promote particular points of view, and seek to bring about a change in musical practice. Both direct much of their fire at the system of tuning – equal temperament – which for many years has been in almost universal use in Western music. But there the similarity ends: in style and approach, the two could hardly be more different.

Ross Duffin’s How Equal Temperament Ruined Harmony is written in the same easy-going style as Stuart Isacoff’s Temperament,[1] to which it is in part a riposte. Every effort is made to seem unstuffy and approachable, even to the extent of including some remarkably unfunny cartoons. Pages are broken up with boxes containing short essays on various subjects: some of these cover, with admirable clarity, such points as ‘What is the Harmonic Series?’, but most are short biographies which, while interesting in themselves, are of little direct relevance to the subject-matter of the book. It is almost as if the author is desperate to get his message across, but afraid that readers will desert him at the first trace of formality or a difficult idea.

This approach has its advantages. The first chapters, explaining the basic facts which give rise to the need for temperament on an instrument with only twelve fixed pitches per octave, are so clearly written that anyone new to the subject should find them easy to follow. Pure concords, the syntonic and Pythagorean commas, just intonation, and the difference between mean-tone and irregular temperaments are among the concepts which are painlessly introduced. Painlessly, that is, if you are at ease with numbers. Unfortunately, when explaining the phenomenon of consonance, Ross Duffin, like so many others, falls into the trap of starting with mathematics – an approach which he tries rather unconvincingly to ‘sell’ to the reader with the remark ‘Note to Mathophobes: This is not math, it’s arithmetic’. In my view, it is better to start from the musical experience itself: a concord such as a perfect fifth, for example, is pure when we hear it to be so (essentially, when it is free of that roughness or wavering which we call beats). It is not pure because the frequencies of the two notes are in the ratio 3:2; that fact is incidental.

This fascination with numbers and ratios can easily lead down irrelevant by-ways. It is true, for example, as Duffin says in more than one place, that in sixth-comma mean-tone temperament the augmented fourth (e.g. C–F#) is ‘acoustically pure’ in the sense that it has the ratio 32:45; but this is of no musical significance, since we do not use this interval musically as a concord, and there is no need for it to sound consonant – rather the reverse.

Much of the argument of the book centres on Duffin’s answer to the question posed at the start of the first chapter: ‘Shouldn’t leading notes lead?’ The leading note in a scale is normally the major third in the dominant chord: in the key of A major, for example, the G# in the chord whose bass is E. In a normal cadential progression, this G# would lead on to the tonic note A. Should the third E–G# be made as consonant as possible, which implies a comparatively low pitch for the G#; or should the G# be raised so that it leads more smoothly and ‘expressively’ on to the A, creating in the process a wide and possibly dissonant third E–G#?

This is discussed primarily in terms of non-keyboard intonation. Duffin shows that in the seventeenth and early eighteenth centuries string players were taught to pitch the leading note quite low – the G# nearer to G than to A – but that nowadays they are taught to raise its pitch close to the note on to which it resolves. He maintains that this change is comparatively recent – he attributes it to the great cellist Pablo Casals (1876–1973), though he implies that the rot set in with the attempt by string players to match equal temperament (with its wide thirds) when playing with the piano. He also maintains that it ‘spoils’ the harmony.

Both these points are open to question. Patrizio Barbieri[2] showed that ‘Pythagorean–expressive’ intonation – in terms of our example, putting G# closer to A than to G – was already beginning to be taught in the 1760s and 70s. And I would contend that it does not necessarily ‘spoil’ the harmony. Harmony is not merely a succession of sweet sounds: it is the art of creating and resolving the tension produced by dissonance. A comparatively dissonant third in a cadential dominant chord can increase the feeling of tension in that chord, enhancing the sense of fulfilment when it resolves on to the following tonic.

Incidentally, one of the genuine problems with equal temperament that Duffin, in his quest for mathematical ratios, misses altogether is that too often dissonances are smoothed away. Thus, for example, the diminished fourth (e.g. C#-F), which should be an agonized discord, sounds the same in equal temperament as a major third. For strings and other non-keyboard instruments, Duffin is a strong advocate – at any rate, for all but ‘modern’ music – of a ‘harmonic intonation’ which puts each sharp note lower than its enharmonic flat (thus, in our example, G# below A flat and closer to G than to A). Specifically, he favours an ‘extended sixth-comma mean-tone’. Now a characteristic of mean-tone is that it is a ‘regular’ system, meaning that all the fifths are the same size. Equal temperament is also a regular system; for keyboards, however, Duffin argues strongly against equal temperament and in favour of an irregular circulating temperament, with varying sizes of fifths and thirds, for music before Debussy. This is advocated partly as a compromise, since extended mean-tone is not possible on a keyboard with only twelve notes to the octave, but mainly because the varying thirds create ‘key-colour’ by making each chord sound subtly different. It is true that some eighteenth- and nineteenth-century writers objected to the introduction of equal temperament on the grounds that it would destroy key-colour. However, this idea that the character of the different keys depends on temperament is open to several objections. Firstly, nobody has been able to link the alleged key-colours consistently with the degree of consonance produced in different chords. How is it, for example, that Beethoven and Chopin thought of A flat major as calm and restful, producing sublime cantabile melodies in that key, when the tonic major chord A flat–C–E flat is dreadfully discordant in most irregular systems? And above all, if key-colour depends on temperament, how is it to be produced in non-keyboard music, where nobody suggests that adherence to an irregular temperament is practical or desirable?

The discussion of these matters is spirited and stimulating but, I suggest, best read critically. To the author’s credit, he does include evidence which might seem to contradict his thesis, such as Louis Spohr’s plain and unambiguous advocacy of equal-temperament intonation on the violin: his attempts to undermine the significance of this and similar sources seem a little strained.

Before I leave Ross Duffin’s book, I must express my disappointment that he has perpetuated the idea that Bradley Lehman has ‘discovered’ J. S. Bach’s keyboard tuning, encoded in the decorative device on the title page of the 1722 autograph manuscript of the first book of the Well-Tempered Clavier.[3] All that Lehman, and his predecessor Andreas Sparschuh, have shown is that it is possible to interpret the ‘squiggles’ as a key to a practical keyboard temperament: there is no evidence to suggest that this was really Bach’s intention. Readers will be left with the impression that the question of Bach’s keyboard tuning has been settled without doubt, which is very far from being the case.

With Alexander Mackenzie of Ord we encounter an author who, far from making things easy and entertaining for the reader, seems determined to place obstacles in his way. Much space – far too much, in my view – is devoted to attacking the terms used by other writers. For example, previous authors such as Murray Barbour[4] and Mark Lindley[5] have used the term ‘regular’ to describe systems in which the fifths are all the same size, and ‘irregular’ those in which they vary. Mackenzie objects to this, perhaps because ‘irregular’ suggests to him something untidy or even slightly disreputable. He proposes ‘equal’ for ‘regular’ and ‘unequal’ for ‘irregular’: but this cannot help creating confusion, since ‘equal temperament’ is the universally accepted name for the modern system, which in Mackenzie’s terminology would be only one type of equal temperament. He is forced into the cumbersome expedient of distinguishing that specific meaning by using capital E and T.

Nor does the potential for confusion end there. Mackenzie insists on calling the syntonic comma the ‘diatonic’ comma, a usage which I have not come across before, and which is bound to cause confusion since the alternative name of the other principal comma – the comma of Pythagoras – is the ‘ditonic’ comma. A glossary would have helped readers understand these unfamiliar usages as they meet them in the book; but, alas, there is none. Neither is there an index, which in a book on such a complex subject – and costing £75 – is inexcusable.

As with Ross Duffin, the approach is rigorously mathematical. When he needs to convey the size of musical intervals in numbers, instead of the comparatively familiar ‘cents’ (100 cents make up one equal-temperament semitone), Mackenzie insists on using ‘savarts’. I read with some incredulity the claim that

the basically very simple numerical relationship [of the mean-tone chromatic and diatonic semitones] between 29.25 and 19.05 in Savarts, is very much clearer than it is in terms of Cents, when blown up to 117.11 and 76.05.

I think any non-mathematical reader is going to be discouraged by all these decimals – but there is no need for them! One cent is well below the amount of pitch difference that the ear can hear, and the whole point of using cents is that intervals can be denoted quite accurately enough using whole numbers: 117 and 76 in this case, which I find a lot easier to grasp than 29.25 and 19.05. This is another example of the perils of approaching the subject of temperament through mathematics rather than through music.

Nonetheless, readers who are interested in the subject should persevere with this book. The positive counterpart to the author’s infuriating idiosyncrasies is that he is able to examine historical evidence with a fresh eye and without preconceptions. The most outstanding example of this – and probably the most valuable contribution of the book – is his exploration of what was meant by ‘the common method of tuning’. This phrase, or something similar, recurs in many English sources from the late seventeenth to the mid-nineteenth centuries, often with a partial description or comment but never with sufficient detail to enable us to deduce a particular temperament. Having established that it cannot normally refer either to mean-tone or equal temperament, the author, like a careful detective, collates the scraps of evidence in the sources to see whether, by eliminating other possibilities, it is possible to arrive at a specific practical tuning that fits all the evidence.

From phrases such as ‘variety without sudden change’ and ‘pleasing and orderly variety’ he draws the new and perhaps controversial conclusion that the differences between the sizes of the fifths and major thirds in closely related keys were small, so that there was a gradual progression in size going round the circle of fifths, rather than sudden jumps as in such temperaments as Werckmeister III. Often the most revealing evidence is incidental, as when Tiberius Cavallo (1778) is describing (with approval) his experiment with equal temperament:

. . . the harmony was perfectly equal throughout, and the effect was the same as if one played in the key of E natural on a harpsichord tuned in the usual manner
– which tells us the size of the third E–G# in ‘the usual manner’ of tuning, and also confirms that there must have been some difference between the effect of different keys.

It is from reasoning like this that Mackenzie derives his two varieties of ‘Ord’ temperament, the name referring wittily both to the French ‘tempérament ordinaire’ and his own surname. He is so fond of ‘Ord’ that he comes close to recommending it from first principles for all types of music, old and new. He is ingenious and, for eighteenth- and early nineteenth-century music, probably on the right lines; but, once again, I am cautious about the mathematical precision of his approach. Is it really possible as ‘Ord’ requires – would it have been possible for eighteenth-century tuners – to produce accurately and reliably, by ear, fifths narrowed by one-sixteenth of a Pythagorean comma, and to distinguish them from others narrowed by one-twelfth of a comma? I wonder.

Readers who look for practical tuning instructions in this book – as in Ross Duffin’s – will be disappointed. It comes with two CDs of organ music: one is essentially a temperament demonstration, but the other is an enjoyable and stylish recital by David Ponsford of mostly English music, from Byrd to Stanley, played on the organ of Dingestow Court, which certainly demonstrates the suitability of ‘Ord’ for music of this period.

1. New York, Alfred A. Knopf, 2001; London, Faber and Faber, 2002; reviewed in BCS Newsletter 27.[back]
2. Patrizio Barbieri, ‘Violin Intonation: a Historical Survey’, Early Music, Vol. XIX, No. 1 (February 1991).[back]
3. See Bradley Lehman, ‘Bach’s Extraordinary Temperament: our Rosetta Stone’ Parts 1 and 2, Early Music, Vol. XXXIII, Nos. 1 and 2 (February and May 2005).[back]
4. James Murray Barbour, Tuning and Temperament – a Historical Survey, East Lansing, Michigan, 1951; reprinted by Dover Publications Inc., 2004.[back]
5. Mark Lindley: article ‘Temperaments’ in The New Grove Dictionary of Music and Musicians, London, Washington and Hong Kong, Macmillan, 1980.[back]

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