Demystifying Temperament

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James Louder

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Demystifying Temperament:

A Mixture of the Charming & Disagreeable

James Louder (© 2004; revised 2006, 2010)


I decided to write this article because of something that I heard on Canada's CBC radio network a number of years ago. The host of the show, the late Bob Kerr, had just played a recording of the beautiful old organ at Marienhafe, in Germany, which I had listened to with immense pleasure. As an organbuilder, I am always entranced by these ancient instruments. The magic of their sound has less to do with sheer antiquity than with the musical principles they embody--not least their tuning system, or temperament. But during his commentary Mr. Kerr confessed that very idea of temperament was a mystery to him.

He was certainly not the first music lover to feel this way; yet temperament is not so arcane as it is often made out to be. That it is obscure even to most educated musicians points to a gap in their training, rather than to any difficulty inherent in the subject. Far from being esoteric, temperament is a down‑to‑earth matter, concerned with a practical problem for fixed‑pitch instruments.

The problem begins with some inescapable physical laws. Musical intervals, to be truly in tune, must have the frequencies of their component pitches precisely adjusted according to the simple ratios which Pythagoras first described in the 6th century BCE. The octave must be exactly 2:1, the fifth 3:2, the major third 5:4 and so forth.[1]  For any irrational combination of frequencies the sound waves will be out of phase, giving rise to regular pulsations or "beats". The farther an interval is from "pure"[2], the faster the beat and the greater the impression of dissonance.

If music were always in one key there would be little problem. The difficulty arises when the same pipe or string of a single pitch must sound the fifth in one key, the third in another, and the octave in another still. Experience readily shows that it is impossible for this single note to be precisely in tune under all three conditions and a bit of fiddling with arithmetic proves why. Since the frequency ratios are immutable, we must find some way to beguile the listener into accepting out-of-tune intervals as just[3]. In brief, we are obliged to cheat.

Remember, this is a problem only for fixed‑pitch instruments, especially keyboard instruments. With the voice and with instruments such as the violin and the trombone, musicians can and do adjust the pitches of the notes to keep within an acceptable degree of purity. How rigorous they must be depends on the repertoire. Broadly speaking, it is choral polyphony that requires the purest intonation. The effect is ravishing when the choir is good enough to bring it off (think, for example, of the Tallis Scholars). The problem of temperament is especially acute on the organ, where the notes are sustained at an even intensity for as long as the bellows will supply wind. Consequently, any beating intervals are very apparent.

With fixed‑pitch instruments, we must decide on one frequency for each note that will give an acceptable degree of purity in a variety of musical contexts. This compromise is called "temperament" and rightly so, for like human temperament, it is always a mixture of the charming and the disagreeable. A musical temperament must create an impression of just intonation; and it is successful when it displays its consonances and hides its dissonant "dirt". 

One obvious dodge is to put the same amount of dissonance into the same intervals in all keys. In modern times, since around 1840, most fixed-pitch instruments have been tuned to just such a system, known as equal temperament, in which the degree of consonance and dissonance is identical in all keys.

However, musical expression is not so conveniently uniform. We have different expectations of different intervals, and for the same intervals in different contexts. Generally speaking, the further we find ourselves harmonically from the tonal centre of a musical passage, the more dissonance we will tolerate. On the other hand, we tend to expect purer intonation at harmonic resolutions, especially cadences. Some of our expectations arise from the fact that pure intonation reinforces the fundamental pitch of a chord, giving a sense of solidity and musical repose. Octaves must always be pure[4] (i.e. must not beat). Fifths may beat, but only slowly ‑ anything faster than about 4 per second (in the middle octave) strikes us as out of tune. Thirds may beat a good deal faster, but the so‑called Pythagorean third, which beats sixteen times per second (middle C‑E) is at the very limit of what the ear will accept as a major third. The modern equal‑tempered third, at over 10 beats per second, is only somewhat better.

Arriving at a successful temperament depends on (a) the number of notes on the keyboard and (b) the number of keys (i.e. tonalities) the player intends to use. The more notes we have and the fewer keys we play in, the better our chances of achieving something like pure intonation. Our expectations about intonation are also the result of our musical habits, which have changed over the course of history. In the Middle Ages the modes of plain-chant required absolutely pure fifths-straight "Pythagorean" temperament. In this system major thirds sound quite harsh, so mediaeval harmony simply excluded them. However, the modes were few and the polyphony fairly rudimentary.

In the Renaissance and the early Baroque periods, when the sound of a well trained choir was the ideal, and when the harmonic vocabulary had expanded to include thirds and sixths, musicians insisted on having pure thirds on their keyboard instruments. To achieve this, they adopted a tuning known as mean‑tone[5] temperament. Their mean‑tone organs and harpsichords were beautifully in tune as long as the music stayed in what we call the "near" keys (C, F, G, etc.) The "remote" keys (B, F#, Ab, etc.) simply could not be used, but to the musicians of that time it seemed an acceptable price to pay. It cannot be stated too emphatically that this was a matter of choice, not ignorance. Equal temperament was well known to theorists of the 16th century, but for keyboard instruments[6] it remained a theoretical curiosity for the next 150 years.

As the harmonic vocabulary of music grew in complexity, people began experimenting with ways to multiply the usable keys. By the first decade of the 16th century, someone had tried increasing the number of notes within the keyboard's octave. This gave rise to keyboards with two or more "split" accidentals.[7] Such keyboards had two differently‑tuned notes on the same key (really two keys sharing one space: imagine a black key that looks like a little step‑stool, each tread moving independently). At the time, these keyboards were the last word in the instrument-maker's art. Some Italian harpsichords were built with as many as 19 notes to the octave. But split accidentals proved expensive to make and difficult to play. No standard emerged, and by the end of the 17th century, no one was making them anymore.

Around this time German organbuilders began experimenting with new tunings instead. They started with expanded mean‑tone systems which mitigated the "wolf"[8] fifth. The late seventeenth century saw the emergence of so‑called "well‑tempered" systems. These were still unequal temperaments, for the degree of consonance still varied from one key to another, but the most successful ones made all keys more or less usable. These systems were more commonly employed for harpsichords than for organs. It has been suggested that Buxtehude may have had the organs of the Marienkirche in Lübeck re‑tuned in one of these temperaments[9]; and it is thought they were used for certain other German organs by the early 18th century.

This brings us to the matter of J. S. Bach's temperament. The young Bach received a reprimand from the consistory of Arnstadt for playing in outlandish keys after his return from visiting Buxtehude in Lübeck. This story is usually repeated to the discredit of the worthy elders of Arnstadt (Tut, tut! Those narrow‑minded old German burghers!). But if Bach was trying out his bold new harmonies on a mean‑tone organ[10], it is no wonder the town fathers felt their wigs curling even tighter about their outraged ears.

But what, exactly, was Bach after? Unfortunately, we cannot be sure. The myth that Bach composed the "Well‑Tempered Clavier" in order to demonstrate the advantages of equal temperament is still being retailed by many people who ought to know better. It is now accepted by most students of the period that Bach was referring to one of the well‑tempered systems mentioned above, not equal temperament. All we know for sure is that Bach tuned his harpsichord in a way of his own that allowed him to play in all twenty-four keys. The details of Bach's method have engendered much speculation, but it remains a vexed question, which may never be answered definitively.[11]

Whatever tuning Bach used for his harpsichord, we are virtually certain that no organ was ever tuned in equal temperament during Bach's lifetime. Even in Germany, where organbuilders were bolder about temperament than elsewhere, organs tuned according to well‑tempered systems were exceptional. Bach's contemporary, Gottfried Silbermann, the most famous German organbuilder of his day, clung obstinately to mean-tone temperament for his organs-even as he was making great strides in improving the newly-invented piano! His conservatism was by no means unique. Church congregations did not always welcome an organ tuned in a newfangled well‑tempered system. People missed the pure thirds. All over Europe, in fact, the old mean‑tone system continued to be the rule for organs right up until the early 19th century.

Controversy over the use of equal temperament was just heating up in Bach's later years and it continued long after his death.  Not surprisingly, the battle was waged by scholars and critics, more than by creators. Apart from Rameau, who came to champion equal temperament, the great composers have left us few thoughts on the matter. It is by no means clear, for example, what temperament Mozart preferred, but there is nothing in his piano music that strictly requires the use of equal temperament. Equal temperament for the pianoforte may not have become definitively established until after 1840 or so, coinciding with the widespread adoption of the iron‑framed piano, and the application of industrial engineering to that instrument. "Scientific" organbuilders such as Cavaillé‑Coll converted to equal temperament around this same time and for similar reasons. But it is worth noting that the English organs displayed at the Great Exhibition of 1851 were still tuned in mean‑tone.

This is also the era when the craft of the professional piano tuner emerges. This is not surprising, for of all systems of temperament, true equal temperament is by far the hardest to realize exactly. Some piano virtuosi used to keep a particular tuner on retainer and even take him on tour. This may have been because of subtle differences in key colour arising from the man's tuning technique: equal temperament, supposedly, but still some keys perhaps "more equal than others?"

Today, when electronic equipment is available to guide the tuner in his work, it is possible to arrive at a degree of exactitude that previous generations of tuners may rarely have attained (although we should not be too quick to dismiss the skills of our forebears: many of the best technicians still work by ear).  Now, anyone who wants to hear exactly equal temperament has only to walk into an electronics store at the local mall and noodle around on a cheap synthesizer. How dead this "perfect" temperament sounds -- and no better on an electronic pipe‑organ substitute, costing a thousand times more! Sadly, the advent of electronic tuning equipment may have robbed equal temperament of its last vestige of expressiveness.

But why should we care about any of this today, when equal temperament is universal, or as close as makes no difference? Just as we come to this most fundamental question, we come to the point when words begin to fail. The short answer is that temperament still makes a tremendous difference, especially on the organ. In the temperaments of the past we have re-discovered one of the expressive resources of our chosen instrument, and we have found it squarely in the notes. This, in turn, has revealed unsuspected richness in music that we thought we knew. It may also suggest new directions for organ composition as we move into the organ's twenty‑fifth century.




[1] For A=440 Hertz, its octave, A' would be 880 Hz; its fifth, E would be 660 Hz; and its major third, C# would be 550 Hz ("Hertz" means vibrations per second).

[2] For our purposes, "pure" describes perfect consonance: beat-less, exactly in tune according to the ratios of the harmonic series.

[3] The expression "just intonation" is used here in its general sense, meaning perceived as being in tune. Theorists employ a more rigorous sense, meaning an ideal system in which all intervals are pure. Just intonation in this second sense is outside the scope of this article.

[4] I deliberately pass over the piano tuner's trick of "stretching" octaves, although it is another good example of a subjective impression taking precedence over acoustical exactness.

[5] The expression "mean tone" refers to the fact that in this system, the major seconds are equal (e.g., C D = D E, where C E is a pure third) unlike the major and minor whole tones of the Pythagorean scale of medieval times. However, mean-tone temperament does have two varieties of semitone, which contribute to the flavour of each key.

[6] Equal temperament, or something close to it, was used for fretted instruments-the lute, the vihuela, and the viol-where 12 frets to the octave and enharmonic notes between strings impose it. Vincenzo Galilei flatly insisted on equal temperament for the lute (Fronimo Dialogo, Venice, 1584), writing 24 ricercare, one in each key, to prove his point-a century before Bach's birth!  But the lute's tied-on gut frets allowed some choice. We know from other sources that lutenists also used unequal temperaments.

[7] Arnolt Schlick (1511) writes of an experimental organ with extra semitones, built "within the past twelve years." He says it was a failure and derides the wasted expense. Schlick then goes on to describe his own method for tempering the octave, the earliest practical account of something like a mean-tone temperament. (Spiegel der Orgelmacher und Organisten, Chapter VIII; tr. E. B. Barber, 1980)

[8] A distasteful by product of all those pure thirds, this interval is too wide to still pass for a fifth at all. It was called the "wolf" because of its howling dissonance.  It is truly a note "in the crack between the keys" and any triad which straddles the wolf cannot be used.

[9] Dr. Kerala J. Snyder suggested this in the first edition of her magisterial biography of the great composer, Dieterich Buxtehude: Organist in Lübeck (New York, 1987).  However, her conclusions were later called into question by Dr. Ibo Ortgies, whose own research raised some serious objections. Dr. Snyder has since withdrawn her hypothesis and revised her book (Rochester, 2007).

[10] [Nov. 2004] During the 1999 restoration of the Friedrich Wender organ (1703) at St. Boniface's Church in Arnstadt, it was found that this organ was probably tuned in a well-tempered system. Nevertheless, Bach would have been the first to exploit the tonal possibilities of this brand-new instrument-hence the town fathers' consternation.

[11] [Feb. 2006] In early 2005 Bradley Lehmann of Goshen College proposed an exciting solution to this conundrum, a possible decipherment of Bach's calligraphic heading on the title page of the autograph of the WTC (1722).  (See Early Music Magazine, February & May, 2005.) Perhaps inevitably, Dr. Lehmann's findings have been vigorously rejected by certain other scholars--notably by Ibo Ortgies and Mark Lindley--and so, the debate continues.


Listen to the 1610 Esaias Compenius Organ at Fredericksborg Castle.