Temperament: A Beginner's Guide
Stephen Bicknell
oneskull@dircon.co.uk


April 25, 1997
Revised May 14, 1997

Temperament is one of those subjects that attracts a) buffs and b) incomprehensible jargon. The usual explanations start with an analysis of the dreaded comma — whether Pythagorean or Syntonic — and most of us never really get past the problem of trying to work out what on earth such a thing might be.

I hope, kind readers, that you will allow me to come to your rescue, and attempt to explain in simple terms what the temperament enthusiasts are on about.

There is a problem to be solved in tuning any musical instrument: the notes cannot be made to fit into the octave, and some have to be de-tuned to make sense. If you tune a circle of pure fifths —

c - g - d - a - e - b - f# - c# - g# - d# - a# - f - c

— the c you end up at is not exactly in tune with the one you started with — a mathematical anomaly known as a comma.

In equal temperament all the notes in the scale are shifted by the same amount in order to resolve the problem.

In all other temperaments the notes in the scale are shifted by differing amounts, giving each temperament a certain character. These can be arranged in more or less chronological order:

The earliest is Pythagorean temperament, which seems to have been in use up to the end of the 16th century. Almost all the fourths and fifths are dead in tune, and the entire comma is 'dumped' on one interval (according to Arnaut de Zwolle between F and Bb), which is therefore unusable. This temperament is easy to explain and to tune, but it leaves a lot of the notes of the scale in quite odd positions. It is quite satisfactory for music written in the old 'modes' that preceded the major and minor scales, provided there is no modulation whatever.

By the early 17th century meantone temperament was the norm. In this temperament the major thirds are perfectly in tune and the fourths and fifths slightly compromised — except for one hideously catastrophic fifth, usually between G# and Eb, the famous 'wolf.' However, this is now a 'regular' temperament, for in keys with less than four accidentals the notes of the major scale are in the same relative positions, the thirds all pure. This, for the first time, allows the composer freedom to include harmonic modulation in one direction or another, and to choose a key that mirrors his thoughts. However, during the course of modulatory passages there will be an audible 'shift' of tonality, rather like changing gear. The appearance of a black note that is technically 'unavailable' in music of the 17th century (they are A flat, A sharp, D flat, D sharp and G flat) is a sure indication that a sudden clash was intended — rather like the deliberate use of false relations. The more extreme accidentals (C flat and onwards) barely ever appear. The occasional appearance in the mean-tone era of keys like F minor suggests the dawning of an awareness of the possibilities of key-colour: with four flats it has a very strange minor third (G sharp, not A flat) and if the G flat is called for there is further trouble in store. Mean-tone (Also known as quarter-comma meantone)

The wolf in meantone tuning is so horrible and such an obstacle that, by the later seventeenth century, it was being modified substantially in practice. Modified meantone is probably the most appropriate temperament for most of the 'early' organ music we now hear — even though Buxtehude and Bach were clearly among those exploring new tuning systems, their compositional technique remains informed by the meantone system. Simply put, the pure thirds of meantone are de-tuned a little bit in order to try and lessen the wolf. Modified meantone temperament was still being used by English organ builders, including Willis, as late as the 1850s.(includes fifth-comma meantone and sixth-comma meantone, the latter sometimes also known as Silbermann temperament). Naturally it allows the composer to modulate a little more freely and frequently, perhaps permitting an occasional excursion into five sharps or flats before returning to a more reasonable home key.

Late in the seventeenth century theorists started to experiment with various well-tempered systems, or circulating temperaments. The object was to finally hide the wolf, making all keys usable. It is perfectly obvious that this could be done by distributing the intervals equally across the scale, but this was not the path they took (except as an academic exercise). Why? The answer lies in the fact that these circulating (i.e. no-wolf) temperaments are those which allow the widest exploration of key colour. There is every indication that musicians of the 18th century were very happy with the expressive possibilities offered by writing in different keys, and sought to exploit the quite different character of each in their writing. Temperaments of this type include the various tunings by Werckmeister (organ expert, 1691), Kirnberger (Bach pupil, early 18thC), Neidhardt (1724) and Vallotti (c1730). Of these systems, Werckmeister III is notable for its purity in the best keys and its suitability for organs with large quint mixtures (many of the fourths and fifths are in tune); but it is irregular and bumpy in the way it deals with modulation and key colour. Vallotti is smooth and regular, but the key colour is generally rather mild . In all these systems it is possible to play in any key, though the more remote keys may sound unpleasant, and enharmonic modulation is not always happy. Other circulating temperaments have been devised in modern times, almost all of them suffering from the grave defect that they are difficult to commit to memory and therefore difficult to use in practice (you can't tune an organ with a book in one hand).

Finally, equal temperament. This very obvious solution has been known since 350 BC (!), but did not become widespread until the late 18th century (50-100 years later in the English speaking world). The advantages are obvious — all keys are usable without fear or favour, and full enharmonic modulation is possible. The disadvantages are also clear: not one interval is dead in tune (indeed in any major scale the thirds and leading notes are extremely sharp), and there is no key-colour whatever. In organs, reeds sound grittier and tierce mixtures begin to scream.

A few words on Bach and temperament. Bach did not at any time advocate the use of equal temperament. He wrote two sets of pieces called 'Das Wohltemperierte Klavier' ('The Well-tempered Keyboard'), avoiding the German term for equal temperament, which is 'Gleich-Schwebende Temperatur.' These 48 pieces are designed to exhibit the full range of key-colour available from a circulating temperament, and careful examination of the texts shows that Bach varied his compositional technique according to the key he was writing in.

All the rest of Bach's music falls into the more conventional patterns of the day, most of it being quite well suited to modified mean-tone temperament (even if it continually pushes at the boundaries of this system). It is particularly noticeable in the organ music that Bach uses modulation as an expressive device, and dares to use the highly coloured remote keys for periods of tension. Today we often wonder how 17th and 18th century players could stand playing long pieces without changing the stops from time to time. The answer is that the modulation from one key to another during the course of the music was an audible 'event', a noticeable change — and is one of the factors that renders registrational changes unnecessary. When Bach plunges from a passage in the home key into a section of wild dissonance peppered with suspensions and discords, he is deliberately invoking the colourful effects obtainable only from unequal temperament.

Why, then, did Bach write such works as the Mass in B minor, in four sharps? In this instance it is possible that the instrumentalists were playing instruments tuned to chamber pitch, a whole tone flatter than organ pitch. The organ continuo would be transposed by the player, who would be playing in serene, no-accidentals, A minor. A quick check of the score will show that, if this transposition is assumed to have taken place, then the continuo player will have been able to avoid the G# — Eb wolf at all times.

Today's instrument makers and theorists have always erred on the side of caution and have avoided large scale use of meantone or modified meantone, preferring to offer us more 'usable' and 'versatile' circulating temperaments. My own feeling is that this is a great shame, for, despite Bach's famous jokes at Silbermann's expense (he is supposed to have played Gottfried Silbermann's organs in outlandish keys, until the builder was forced to retire, 'his wolf howling in his ears'), his music is surely centered round the meantone tradition and the many colourful possibilities it offers.

Those who wish to hear large scale demonstrations of major works — Bach and others — played on old unequally tempered organs, are recommended to try Harald Vogel's recordings of the Schnitger at Norden and Ton Koopman's recordings of Bach, especially on the G. Silbermann at Freiberg. Amongst instrumental recordings I would recommend those of the music of Monteverdi (Vespers et al.) made by Andrew Parot and the Taverner Choir and Consort (for which I tuned the Mander continuo organ in quarter-comma meantone).

Mention of Italian music brings us naturally to the question of chromaticism in early music. This is an area in which the Italians excelled, but there are examples from many other countries, especially in the seventeenth century, where the chromatic scale is used for special effect. On equal temperament this is uniformly dull, and music of this type fails dismally to engage the listener. However, in meantone tuning a quite different picture emerges. The half-steps of the scale are all of differing sizes, and the chromatic scale, far from being smooth and oily, becomes a bumpy, eccentric and challenging affair. The harmony it brings with it displays rapid, even kaleidoscopic, changes of key colour through constant modulation, and the repeated build-up and release of harmonic tension is characteristic and fascinating.

I would also mention one other genre especially suited to meantone — the English Trumpet Voluntary. The acid whining of thirds and sixths in equal temperament makes these pieces less than harmonious to the modern ear. Even in a circulating temperament they fail to 'come across.' In meantone they present a quite different picture, one that I tried to hint at in my imaginary conversation with John Stanley, posted to the list a few months ago.

I hope that this introduction will encourage some of you to listen out for the many positive features of unequal temperament, even though the wide range of the modern repertoire means that most new organs will (and probably should) be tuned to equal temperament.



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Last modified January 28, 1998.