No – not the film, but Equal Temperament, or ET for short. I have recently been reading a book on the subject. Truly fascinating. But more of that later.

Among some music I acquired from the estate of the late Maxwell Deans is a battered violin method entitled Keith’s Instructions for the violin, sixth edition, price nine shillings, published William Prouse, late Keith Prowse & Co. The inside frontispiece describes R W Keith as a pupil of the late Mr F H Barthlemon. From the elaborate and flowery design, the publication would appear to be late 19th century. Curiously, at the foot of the page, is given the following:

Where may be had Sanderson’s study of the Bow and Finger Board of the Violin. Price 8s.

Herein lies the biggest surprise of all; namely a full scale engraving of a violin and bow.

R W Keith’s Tempered VIOLIN Fingerboard SCALE

I quote capitals as in the original. Shown in detail for each string are the chromatic steps upwards with enharmonic equivalents. In addition, he shows the fractional ratios for each pitch.

I am sure, like me, most of you will be surprised by this on two levels at least. First, who would have thought it necessary to show ET at all? In the past we all learned that such a tuning system was supported by Bach and the 48. Secondly, string players tune in perfect fifths and use true intonation. This being so, why should a violin method require the student to use ET as though it was something new?

Which brings me to the point of the foregoing preamble. Some time ago I purchased a book with the provocative title of How Equal Temperament ruined Harmony (and why you should care). I can thoroughly recommend this as a good read, if you want something different to challenge the musical mind. Ross Duffin, the author, covers the history of different tuning systems in depth.

Some of you may remember acoustics as part of your studies and, in particular, the ‘Comma of Pythagoras’. As a reminder, this is where the cycle of fifths moving outwards from middle C reaches its enharmonic equivalent. (B sharp: D double flat.) If these are brought back to middle C, the upper note will be found to be sharp. This discrepancy is the ‘Comma of Pythagoras’. It was the effort to ‘correct’ this divergence in pitch that brought about Equal Temperament.

This has major implications for string players, especially when accompanied by keyboard. As a working compromise, tuning to A should always be slightly sharp to accommodate the other open strings to the piano tuning. The significance of the piano tuner tuning to middle C (and not A) as his starting point is frequently overlooked.

As a string player, I have always adjusted the leading note higher and the subdominant lower, depending on the harmonic context. This works for solo pieces, but is less satisfactory in ensemble work, where a raised leading note can make the major 3rd of the chord seem out of tune. Open strings as part of double stopped chords pose their own special problem. Ross Duffin cites as an example of this: the opening bars of Beethoven’s String Quartet in E flat Op 127 transposed to C major, where the second violin open E (double stop C, E) is sharp with the cello open C, G.

Duffin gives 1917 as the date for the adoption of ET as a universal standard. This is far later than we have been taught. But the greatest surprise of all is the variety of tunings that have been tried over the years. Extremes such as dividing the octave into 55 segments were tried; remember, this is centuries before the advent of electronic tuning aids. It has been well known that other cultures use smaller divisions of the octave – not necessarily microtones, either, which are merely halving the steps we are already used to.

For me, the most fascinating find of all was the use of the Comma to describe smaller intervals and the fact that semitones can be described as major and minor, chromatic and diatonic. Mentally, this takes some getting used to. Try the following question: Is G sharp higher or lower than A flat? Did you say higher? I would have answered so as well, that is, before I read Duffin’s book. The correct answer is lower. Lower because diatonically, A is higher than G; hence A flat is higher than G sharp. Surprised? Sceptical? Yes, I was too. Logically it makes sense, but for the performing artist, I’m not so sure.

I had a friend years ago who had some extreme views about concerts and performances. When asked about a particular concert he would say: ‘It was OK but only one of the chords was really in tune!’ In case you think I have some rather strange friends, it is worth considering the number of variables that are involved in the performance of an orchestral work. Now that I am older, I can look back and recall special moments, where all the factors come together in perfect harmony to generate a memorable performance.

As a composer, as I sit at the piano, sifting ideas and testing sonorities and harmonic spacing, I do wonder at the effect ET has on what I hear. What is excitingly evident is the thrill one has when hearing an orchestral work of one’s own, fully realised in a live performance. I am certain that tuning and not simply timbres have a lot to do with enhancing the sound.

Here is a little test you could try. If you play and repeat low B below the bass clef and add C above, you will have a minor 9th. Nothing remarkable in that. Now move the C up an octave each time. You will find the C gradually ‘slips’ downwards so that the C above the stave sounds B. This has major implications for widely spaced harmonies. One could call it the ‘shrinking octave’. A similar phenomenon seems to happen as one moves to lower notes where the lower note melodically sounds sharp.

In an article such as this, I have only touched on what is an immensely complex subject. At the very least, for all of us who have been nurtured on ET, it should stimulate some thought for possible future discussion.

Gareth Clemson BMus Auckland (NZ) BMus Edin, a composer, viola player and private teacher, belongs to the ISM’s South East Scotland Centre, of which he is currently the chairman.

How Equal Temperament ruined Harmony (and why you should care), by Ross W Duffin, is published by W.W. Norton & Co, 500 Fifth Avenue, New York, NY 10110.

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