A DORSET TEMPERAMENT?

by Colin Pykett

Posted: 3 December 2002

Last revised: 18 December 2009

Copyright © C E Pykett

(This article was also published in Organists' Review, August 2004.   In February 2008 the version posted here was revised to include the absolute frequencies of the Dorset Temperament at the request of a number of people involved with digital music, rather than just giving a table of beats.  The deviations in cents of each note from Equal Temperament have also been included.  See Appendix 2).

Dorset is an equable county. Its climate, people and landscape make it so, particularly when one considers that it is still possible to travel from the coast to the Wiltshire border without encountering even a dual carriageway, let alone a motorway. Its famous son Thomas Hardy immortalised the place, and in Under the Greenwood Tree he wrote an entire novel in which church music played a central role. So it was in this frame of mind when I played an unremarkable organ in an unremarkable village church a few years ago, when everything that sunny rural afternoon was redolent of the nineteenth century.

Yet after a while I felt that the organ was not, perhaps, so unremarkable after all. It was normally tuned to equal temperament; indeed, a few weeks after my visit the tuner descended and proved that. But at the time it had drifted slightly, not enough to be unpleasantly out of tune, but enough to endow certain keys with a definite flavour. This phenomenon cannot occur, by definition, in an instrument tuned exactly to equal temperament. The common keys such as C and G major were tolerably well in tune, not much better or worse than in equal temperament. But some of the more remote keys seemed curiously attractive. The tune Ewing (to Jerusalem the golden) in D flat fitted perfectly with my mood and the occasion, but also with the "temperament" that chance had conferred on the instrument.

So struck was I by the situation that I investigated the tuning in more detail. Playing across the compass on the great 4 foot Principal revealed that in the middle octave two pure fifths existed. These were adjacent, between D and A and between E flat and B flat. In equal temperament there are no pure fifths; all of them exhibit slow beats. I therefore took a note of the beat rates of the remaining fifths and fourths in that octave, although being without a watch I was unable to time them accurately. What unfolded subsequently, by fits and starts as time permitted, now forms the basis of the rest of this narrative. Obviously, that organ was not tuned to this temperament throughout. Because it had not received attention for some time the tuning within several of the octaves had become different, thus whatever attractions the instrument possessed must have been related to a distributed temperament in which even some octaves were slightly impure. Nevertheless the tuning mentioned above may have been partly responsible for the subjective effect of the instrument, and I considered it worth following up.

Before going further be assured I shall not be delving into the arcane arithmetic and acoustical theory of tuning and temperament. That has been done by countless other authors since the time of the ancient Greeks. One of the best known references as far as organs are concerned is that of Dr Charles Padgham [1]. Yet even his scholarly treatise contains a sprinkling of numerical errors, which are doubly unfortunate given the frequency with which they have been propagated by some uncritical authors. Instead you might like to do an experiment if you want to gain a better understanding of temperament. Sit at the piano and, starting at the lowest C, ascend the keyboard by fifths. Thus you will first play C and G, then G and D, then D and A, and so on. Presently you will encounter a black note when you reach B and F# and thereafter all subsequent fifths will be on the black notes, until at the top of the keyboard you regain the naturals with B flat to F. The next and final interval returns you to the root note, top C, after a journey of seven octaves. During this journey you will have played twelve fifths.

So what, you may ask? It is obvious that twelve fifths make seven octaves. But the point is that none of these fifths is pure, whereas the octaves are (or nearly so on the piano, and exactly so on the organ). If the fifths were pure it would be impossible to fit them into the seven octaves, and the topmost C would be about a quarter-tone sharp by the time you reached it in the exercise above. A piano tuned this way would certainly deserve the honky-tonk epithet [but see note 2 under Postscript] . To avoid this, each fifth is flattened slightly or "tempered". You can hear this, for example in the middle of the keyboard, where the slow beats which are characteristic of equal temperament will be heard clearly as the sound dies away. They are even more obvious on a diapason stop on the organ. In equal temperament each fifth is flattened by the same amount.

Detractors of equal temperament point not only to the slightly impure fifths but, with more justification, to the considerably sharpened thirds which are an unfortunate by-product of the process. It is this which makes equal temperament sound rough compared to certain keys played in some other temperaments. This factor also renders the use of third-sounding mutation and mixture ranks (e.g. tierces) highly questionable in ET because they beat hideously with the corresponding actual notes in the scale. Moreover, from a musical rather than an aural perspective, the fact that all keys are out of tune by the same amount means that much of the point of modulation vanishes, because all keys have the same subjective flavour (or lack of it).

All is not lost however. It should be reasonably obvious that there is an infinite number of ways by which the twelve fifths can be stretched or shrunk so that they will still fit into seven octaves. They do not all need to be flattened by the same amount as in equal temperament. For example some need not be flattened at all, so that they then remain pure. But others will then need to be flattened more than in equal temperament to compensate. This flexibility is the basis of the various temperaments which have been developed over the last few centuries, bearing names such as Werckmeister, Vallotti and Kirnberger. In practice, though, the amount of flexibility to play around with the fifths is not infinite at all; many arrangements will produce unusable dissonances in many keys. Therefore the reason why some arrangements are dignified by eponymous titles is because these individuals hit on tuning recipes in which most, if not all, keys remained useable whereas some were better in tune than in equal temperament. These keys sound pure and attractive to most modern ears, though some regard them as insipid. It is this variation in tuning purity between keys in an unequal temperament which endows them with subjective key flavour.

We can now return to my Dorset experience. What struck me at first was that I was unaware of any well-known temperament which has as few as two pure fifths. Some temperaments have none, but if they are present most of them have around 8. However some French temperaments have only 3 or 4, and another by Neidhardt also has 4. The latter is well regarded nowadays, partly because it has some characteristics of equal temperament and of some earlier ones. Therefore the question was: did this chance discovery in Dorset suggest yet another way of tuning which might have attractions? To investigate the matter further I did calculations and experiments in which the tuning of the impure fifths was varied by different amounts (recall that two remain pure). It is only when one tries to do this that the mind-numbing difficulty and tedium of it becomes all too clear. Therefore at this point it is worth paying tribute to the pioneers of yesteryear who surmounted these difficulties without the benefit of computers and electronic keyboards to make things as painless as possible. All of their calculations were done by hand with long multiplication and division, and they could only test their results by laboriously re-tuning hundreds of pipes. Much of their work would have been done with quill pen and candle light after a long day building organs!

What resulted from these experiments was a method of tuning, a temperament, retaining the two pure fifths but with the others flattened by certain amounts. Of these flat intervals, 6 were flattened slightly more than in equal temperament and the remainder by slightly less. Qualitatively, the advantage of equal temperament had been retained as far as the ability to play in all keys was concerned. None was seriously out of tune, indeed the differences from ET were barely noticeable in this respect. On the other hand none was as pure (in the sense of having pure thirds for example) as those in more extreme temperaments. However the gain was a return to a definite range of key flavours, and this deserves further discussion.

In most unequal temperaments it seems that the designer of the temperament was concerned mainly to ensure that as many of the commonly used keys as possible were as well tuned as possible. The out-of-tuneness was lumped towards the more remote, less frequently used, keys. Thus C major is invariably in excellent tune, usually far better than in equal temperament, whereas D flat often verges on the unusable. What a pity that we cannot play Jerusalem the golden on such a temperament and enjoy the different colour of D flat! To avoid this, I tempered the ten flattened fifths in such a way that all the commonly used keys remained in good tune, whereas keys with the "sharp" (black) keynotes were at least as good if not slightly better. This was possible for all major and all but one of the minor "sharp" keys. This unusual choice has resulted in the noticeable key flavours associated with the temperament, while still allowing all keys to be used.

The table below shows how each key compares with equal temperament. This is an intonation table which is actually based on how well the tuning errors in each key compare in an arithmetical sense with acoustically pure intervals. By deriving a similar set of numbers for equal temperament, the two temperaments can be directly compared. The method is similar to that described fully in Padgham’s book already referred to, which will enable the qualitative descriptions in the table to be better understood for those who require a more detailed insight.

The main features of this comparison are:

1. 12 out of the 24 keys have better intonation than equal temperament

2. All of the "sharp" major keys are better than in equal temperament.

3. All but one of the "sharp" minor keys are better than in equal temperament.

4. Some keys are much better, in some cases as good as the best of other well known temperaments.

5. The poorest keys are entirely useable and only a keen ear will detect their departure from equal temperament. They are all better than the poorest keys in what are considered to be "good" temperaments today.

This comparison of temperaments needs to be put into an aural context however, in which the judgement of the ear is more important. For example, in concluding that the intonation of C sharp/D flat major in this temperament is "much better" than in equal temperament, I merely made arithmetical comparisons of frequencies on a note-by-note basis throughout that scale. It does not necessarily mean that the scale sounds much purer in the sense that the thirds and sixths are much better in tune (in fact they are not). It merely sounds subjectively different to some other keys. For the same reason, the "poorest" keys are only poorest in a relative numerical sense; they are by no means as subjectively out of tune as the poorest keys can be in some other temperaments. It must be remembered that this temperament is a relatively minor perturbation of equal temperament, with the main object of enabling different key flavours to be enjoyed.

For the same reasons the intonation of mixtures with this temperament is, subjectively, not much different to equal temperament. Quint mixtures (those containing only octave and fifth-sounding ranks) produce beats with the corresponding notes of the scale which are broadly acceptable, just as in equal temperament. Third-sounding ranks such as seventeenths generate more noticeable beats which in some cases are slower than in equal temperament but in others faster. In neither temperament do these ranks sound attractive to my ears in all circumstances, though it does depend to some extent on the type of music being played. These conclusions were drawn from experiments in which the temperament was set up on an electronic organ for convenience. (And speaking of electronic organs, it is likely that many old analogue organs of 20 years ago or so would have had the benefit of a temperament similar to this. It is improbable that the designers or anyone else were aware of it though. This possibility is discussed further in Appendix 1).

Tuning details will now be provided for those who might want to try this temperament for themselves, although I would not encourage the inexperienced to retune an organ which does not belong to them. Such matters must be left to the professional. Nevertheless a significant number of people possess house organs and are capable of tuning them, and it is to them as well as the professional tuner that the following notes are addressed. The data are equally applicable to electronic organs of course, provided they allow the owner to adjust the tuning (many do not).

Being a physicist, my first thought was merely to tabulate the required frequencies of the twelve semitones of the octave. However I have been associated with organ tuners for long enough to accept and respect the fact that they much prefer a table of beats! [Appendix 2 has been added subsequently at the request of some correspondents involved with digital music which does contain the absolute frequencies, and the deviations of each semitone interval in cents from ET].   In fact it is remarkable how accurately and quickly an expert tuner can set up a temperament on an organ only by counting beats. Therefore the table below contains tuning data in this form. Note that the table follows convention by referring only to a single octave; the twelve fifths of our earlier example which were spread across seven octaves are now compressed into one. This is done simply by inverting those fifths which would lie outside the octave into fourths.

The reference pitch of the table is A=440.00 Hz, giving the C below (middle C) as 261.43 Hz. This is 0.08% flat compared to its frequency in equal temperament. It is assumed this note is first tuned to a fork or other standard, and the use of an equal temperament fork would probably be accurate enough for most purposes in view of the miniscule deviation mentioned. The table refers to the middle octave of an 8 foot stop. A stop with reasonable but not excessive harmonic development is necessary for tuning, and a diapason type of tone is usually used. If there are insufficient harmonics the beats will be difficult or impossible to hear; if there are too many there will be multiple beats which can confuse the ear.

(C first tuned to middle C at 8 foot pitch)

The "equivalent beat rate" data may be preferred, and they can with experience be interpreted practically. For example, that relating to the final check interval is just over 7 beats in 5 seconds. Provided the final check interval is not too far out, it can be concluded the temperament has been set up properly.

The stop used for the exercise above is then brought into tune by tuning all the octaves true, and this stop is then used to tune the others.

The temperament discussed here was suggested by serendipity. It has most of the characteristics of equal temperament, both good and bad, but it has key flavours which equal temperament does not. Since it arose from an experience in a part of the world with an equable temperament in other respects, perhaps it might be called The Dorset Temperament!

There is an amusing twist to this story which might interest those with a liking for technical history. In the early days of electronic music the semitones of the equally tempered scale were sometimes derived from a single oscillator driving twelve frequency divider circuits. In those days it was important to simplify the hardware as far as possible, a constraint which no longer applies. This was done by keeping the division ratios as small as possible, and one of the simplest arrangements resulted in an approximation to equal temperament in which the two pure fifths discussed in this article sometimes appeared. I have investigated the scale produced in this way, and not surprisingly it turns out to be very similar to the temperament described here. However it can scarcely be claimed that the approximation was used because of subtleties of temperament when in fact things were driven by the necessity for simplicity and cheapness!

Reference

1.  The Well-Tempered Organ, C A Padgham, Positif Press, Oxford 1986.

oooOOooo

Postscript

1.  (March 2006): The electronic organ pictured on the home page is tuned to the Dorset Temperament, together with some stretched octaves as proposed in Temperament - a study of Anachronism

2.  (December 2009): The statement that "a piano tuned this way [i.e. with perfect rather than tempered fifths] would certainly deserve the honky-tonk epithet" was made without having tried it.  Since writing the article I have tested Cordier's method of tuning using perfect fifths, which results in stretched octaves across the entire compass.  Perhaps surprisingly, it results in some interesting sounds on the organ, and it is certainly not "honky-tonk"!  An article elsewhere on this website describes the work in detail and contains some sound examples (read).

Appendix 2 (February 2008)

This table has been added in view of a number of requests from people in the digital music field who wanted the absolute frequencies of the notes in the Dorset Temperament, rather than the beat rates given in the article itself.  The table did not form part of the original article as published in Organists' Review.

The table also shows how the Dorset Temperament deviates from Equal Temperament in terms of the deviations in cents for each note.  This is probably the most useful representation for digital synthesisers and digital organs because most if not all of these use ET as their default tuning standard, which is tuned precisely for all practical purposes in these instruments.  However, few of these instruments are able to specify note deviations from ET with a precision better than one cent, which is not really good enough for tuning unequal temperaments accurately.

As the table below shows, it is necessary to specify deviations to a fraction of a cent, and this is also true for any other non-ET temperament if it is to be accurately tuned.  Merely tuning the notes of the octave to the nearest cent becomes progressively less satisfactory as the absolute frequency increases.  In other words, in the upper octaves the note frequencies will deviate from their proper values by progressively larger amounts, and this leads to beat rates between intervals which are noticeably incorrect in these octaves.  In the case of the Dorset Temperament, it means that the two perfect fifths (D - A and D# - A#) will not in fact be perfect at all - there will be residual beats which will become faster the higher the octave in which they are tested.  The same will apply to the intervals which ought to be perfect in any other unequal temperament.  This is an inevitable and unsatisfactory consequence of commercial digital music technology.

However if you do set up the frequency deviations to the nearest cent using the data in the table below, you will find that in the octave starting at middle C at unison (8 foot) pitch, the two notionally perfect intervals will have only a very slow beat rate which you might therefore deem satisfactory for practical purposes - they will be "nearly perfect" in this octave.  Indeed, this should be used as a test to check that the data have been entered into the synthesiser correctly.