A Mixture of the Charming & Disagreeable
James Louder (revised; © 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 very practical problem for fixed‑pitch
The problem begins with some inescapable physical laws. Musical intervals, to be truly
in tune, must have the frequencies of their component sounds precisely adjusted according to the simple ratios which Pythagoras
first described in the sixth century BC. The octave must be exactly 2:1, the fifth 3:2, the major third 5:4 and so forth. 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", 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. 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".
obvious dodge is to put the same amount of dissonance into the same intervals in all keys. In modern times, since around 1800,
most music has been construed according 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 (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
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 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 remarked too emphatically that this was a
matter of choice, not ignorance. Equal temperament was first described in the sixteenth century, but 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. Early in the seventeenth century, someone had the bright idea of increasing
the number of notes within the keyboard's octave. This gave rise to keyboards with two or more "split" accidentals.
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 century, no one was making them
Around this time German organbuilders began experimenting with new tunings instead. They started
with expanded mean‑tone systems which mitigated the "wolf" fifth. The late seventeenth century saw the emergence of the 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; and it is thought they were used for certain other German organs by the early 18th century.
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, 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.
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 Sibermann, the most
distinguished 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, first championed by Rameau, 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.
Few of the great composers have left us their 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; which is not surprising, for
of all the systems of temperament, true equal temperament is by far the hardest to realize. 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
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.
 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).
 For our purposes, "pure" describes perfect consonance: beat-less, exactly in tune according to the ratios of
the harmonic series.
 The expression "just intonation" is used here in its general sense, meaning perceived as being in tune.
Theorists used the expression in 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.
 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.
 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 medieval church modes. However, mean-tone temperament
has two varieties of semitone, which contribute to the flavour of each key.
 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.
 Dr. Kerala J. Snyder suggested this in 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.
 [Nov. 2004] Since I wrote this article, further study of the Wender organ at St. Boniface's Church in Arnstadt has revealed
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-whence the town fathers' consternation.
 [Feb. 2006] In early 2005 by Bradley Lehmann of Goshen College proposed an exciting solution to this conundrum,
based on a decipherment of the Bach's calligraphic heading on the title page of autograph of the WTC (1722). See
Early Music Magazine, February & May, 2005. Perhaps inevitably, Dr. Lehmann's findings have been vigorously questioned
by certain other scholars-notably by Ibo Ortgies and Mark Lindley-- and the debate goes on.
Listen to the 1610 Esaias Compenius Organ at Fredericksborg Castle.