In Search of a Temperament

In Search of a Temperament

by John Coenraads

When our local organ builder, now retired, tunes his continuo organs for special concerts, he often gets requests for historical tunings such as Vallotti, Neidhardt, 1/4 Meantone etc., and, using today's tuning apps, these requests can usually be accommodated. But as far as the major organs in Victoria go, they all use Equal Temperament with one notable exception: Hellmuth Wolff's Opus 47 in Christ Church Cathedral. 

Not knowing what tuning scheme Hellmuth used, we resort to using  the 4 ft Principals in each division as reference ranks hoping that they are remaining stable over time. Being cone tuned, there is no sleeve to slip out of place which would affect the tuning. But this has the disadvantage that, should these pipes need tuning, it is done by gently striking the open end of the pipe with a cone shaped weight that actually deforms the end of the pipe. Over the years, cone-tuned pipes can incur damage either by splitting the metal or driving the pipe into the toe hole. On the Wolff organ, these pipes have remained basically untouched since their installation in 2005. One day I arrived an hour early for a tuning (having misread the bus schedule) so I decided to check whether our faith in these ranks was justified. Using a tuning app, I measured and recorded the frequencies of the middle octaves of the principals in the Hauptwerk, Oberwerk and Unterwerk. Scanning the numbers, it was clear that these three ranks were typically out of tune with respect to each other by a few cents. Now a cent is 1/100th of a semitone so this error is not serious and usually we are satisfied if a pipe is tuned within one or two cents of the correct pitch. But the time must come when we really need to know what Hellmuth used.

Since the cathedral organist did not know, and no one had thought to ask Hellmuth before his death in November of 2013, I decided to compare the Wolff temperament with the hundreds of temperaments available online. Early on it became clear that there was a close similarity to Vallotti with only B and F in obvious disagreement. But a closely related temperament devised by the English physicist, Thomas Young, largely corrected that. To find confirmation that I was on the right track, I contacted the organ builder John Brombaugh and he graciously responded and offered his advice. It was my luck that John had developed a sophisticated program using an Excel spreadsheet that allows one to analyze and design temperaments. Entering my numbers into the program, he proceeded to do a thorough analysis but then concluded that a temperament he had designed and used for his own Opus 29 and Opus 32 gives very similar, almost indistinguishable, results.  John shared this temperament, Brombaugh 1/6 pc WT, with me and I programmed it into the PitchLab tuning app (PitchLab is free but this requires the $3 upgraded version!). The last time we tuned the Wolff organ using the principals as reference, I compared the results with the Brombaugh Temperament and the results matched so closely as to preclude sheer coincidence. As John relates: when Brombaugh Opus 32 at Christ Church, Christiana Hundred in Wilmington, Delaware, was being finished, Hellmuth visited and seemed impressed by how well this temperament worked for music of the period the organ was made for. Subsequently he used this temperament on several of his own instruments including, apparently, Opus 47.

With my curiosity aroused, I decided to delve into the voluminous literature dealing with alternate temperaments. Starting with Wikipedia, I learned that:

"Tempering is the process of altering the size of an interval by making it narrower or wider than pure. A temperament is any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds."

So far, so good, I sort of get that, but then I was quickly overwhelmed by Pythagorean limmas, syntonic commas, just intonation and enharmonic modulation. I just couldn't get a hook on this, so I resorted to taking a look at the underlying physics.

When an organ pipe sounds, we hear not only the fundamental, or first harmonic, we also hear additional harmonics referred to as overtones. These overtones are always whole number multiples of the fundamental frequency. For example, A440 has overtones of 880 Hz, 1320 Hz etc. It is the prominence of these overtones that gives the character or colour peculiar to that stop. Thus a reed is particularly rich in the higher overtones whereas a flute will have very few overtones.

In Equal Temperament, when one plays A with E, a fifth above, one hears a slight dissonance created by the third harmonic of A being slightly out of tune with the second harmonic of E. Since this error is small, comparable to typical tuning errors, can we get away with forcing these two harmonics to be equal thus  “sweetening” the interval? Pythagoras thought so (yes, he of  “the square on the hippopotamus ...” fame) and by slightly sharpening E the dissonance is removed. In the same way, we can continue to retune all the fifths and once we have completed this circle of fifths we arrive back at A. But A is now 446.0 Hz instead of the 440 Hz with which we started. I.e., the circle of fifths is really a spiral. The accumulated error is 23.5 cents, almost a quarter of a semitone. This error is known as the  Pythagorean comma. The history of temperaments is the story of how to close the spiral. One solution involves hiding the Pythagorean comma in a rarely used interval among the black notes. But this wolf fifth will always be lying in wait, ready to howl if provoked.

The other solution, distributing the Pythagorean comma equally over all twelve intervals, results in the Equal Tempered scale. But this implicitly involves using the 12th root of 2, an irrational number which is the factor connecting successive semitones. This would have been anathema to Pythagoras and his followers who, having discovered irrational numbers (numbers that could not be written as a ratio of whole numbers) attempted to keep the existence of these monstrosities a secret. 

But there is a third possibility: sprinkle the Pythagorean comma unequally among some, but not all of the intervals. Such a scheme is said to be well tempered if it allows one to play music in most major or minor keys without sounding perceptibly out of tune. But why bother? The simple answer is that, as Rameau said, the problem with equal temperament is that it is so equal. By widening or narrowing certain intervals, one can arrange for the same interval to have a different colour or character in different keys. Some musicologists argue that Bach may have employed a well temperament, not equal, in composing the music for the Well-tempered Clavier and that he adapted each composition to exploit the particular tone colour of that key.

The Brombaugh 1/6 pc WT is quite a mild Well Temperament and many listeners may not even notice  the colour it gives to certain keys. Further analysis showed me that the following fifths are tuned pure, i.e., the third harmonic of the root note is exactly in tune with the second harmonic of the second note.

C#   -  G#

E♭ -  B♭

E     -  B

F     -  C

F#   -  C#

G#  -  E♭

The remaining fifths take up the slack by having harmonics that are tuned narrow by 1/6 of the Pythagorean comma, hence the 1/6 pc in this temperament's designation.

But the question remains, which keys does this favour? Casting around I came across a MIT research paper that suggested using the purity of the major triads as a measure of the “goodness” of a temperament. So I analyzed all twelve major triads and for each I computed the total error by which each of the three intervals within that triad deviated from the ideal ratios of 3/2, 5/4 and 6/5. For Equal Temperament the result was 2.24 % and it is the same for all keys.

When I did the same for the Brombaugh Temperament, and averaged the results, I was gratified to again get 2.24% which proves that this temperament is optimal since no Well Temperament can have an average error less than 2.24%. It was when I graphed the results that I was surprised and delighted to find that Brombaugh 1/6 pc WT now makes perfect sense. Not only is it symmetrical about the (dotted) Equal Temperament line, it is also symmetrical, although again inverted, around the keys of D and A. Furthermore, the more common keys of F and C have clearly been “sweetened” at the expense of B and F#. I'm sure John Brombaugh knows all this but I thoroughly enjoyed the challenge of discovering this for myself.

And that, dear reader, is how I entertain myself in my retirement.