Early Keyboard Temperaments
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  Early keyboard temperaments - their development and cultural environment  


Colin Pykett

 

"every single note of our musical repertoire is a monstrous compromise"

Howard Goodall [15]

 

Posted: 29 December 2016

Last revised: 3 February 2017

Copyright © C E Pykett 2016-2017

 

Abstract.  This article discusses the tendency of some modern authors to reflect today's norms of fashion, musical culture and understanding of physics into their writings about early temperaments, when these matters were in fact very different several centuries ago. A prime example concerns the difficulty of tuning intervals accurately until relatively recently when electronic tuning devices appeared in the late twentieth century. Until then tuners of keyboard instruments had to time beats using various less precise methods, and even this only became routine from about 1800. Prior to that tuning was done for centuries using the vaguest of instructions which appear ludicrous to modern eyes. The reasons for this are that beats and the harmonics which generate them were for long imperfectly understood, and practical means for timing them accurately did not exist. The plethora of different pitch standards made things worse because beat frequencies depend on absolute frequency. There were also problems due to the slow dissemination of temperament theory, together with widespread educational narrowness which meant that most musicians and tuners would not have understood it anyway. In addition, hand blowing and poorly-designed winding systems meant that the tuning stability of organs was badly controlled regardless of how tuners might have struggled to set a temperament accurately. The upshot of all these factors is that the sharp focus applied to subjects such as key colour today can be anachronistic if it is assumed that our predecessors several centuries ago viewed them as we do. Obviously, the key colours of a temperament become elusive if it cannot be set up accurately in the first place on an instrument with stable tuning, and for centuries this would have been the case with the organ.

Therefore several important features of musical temperament have been placed in an historical context, including an understanding of frequency, harmonics, beats and how these influenced contemporary tuning practices. Another factor unique to the organ concerns the subjective colours of temperaments, not only in different keys, but also with different stops. In addition, differences among early scholars between a philosophical and empirical approach to knowledge are discussed to suggest how material before the mid-eighteenth century might be better interpreted. In particular, some results presented by theoreticians might well not have been tried for real because of an absence of how to translate them into practice.

 

 

Contents

 

(click on the headings below to access the desired section)

 

 

Introduction

 

The temperament problem and when it arose

 

Philosophy versus Empiricism

 

Frequency

 

Harmonics

 

Beats

 

Tuning practices

 

Key colour

 

Concluding remarks

 

Notes and References

 


Introduction


The environment in which early keyboard temperaments arose was utterly different to that we are familiar with today. Now, tuning is routinely executed to high precision using electronic devices; tuners and musicians are well-educated in their respective arts and more broadly; international travel and networking with peers is part of everyday life; and information on anything and everything is available virtually instantaneously. None of this applied in the seventeenth century, say, when theorists such as Mersenne and Werckmeister were active. Although their achievements were impressive, there was an almost unbridged divide between them and the average tuner, teacher or performer across Europe (often each was all three). Most of these lived in villages and small towns and they worked within a cultural vacuum unimaginable today, with little routine contact beyond their peers in nearby communities - "nearby" in this context being measured by a few hours staring at a plodding horse's backside. Among other things, this isolation explains why musical pitch was a random function of geography for so long, and why some evolutions affecting temperament arrived later in Britain than across the Channel in continental Europe. Thus the theoretical advances pioneered by a few would have been little known to most practical musicians, and most would have been unable to understand them in any case.

So how did temperaments develop and propagate, depending as they do on an elaborate theoretical framework of physics and mathematics which would have seemed little different to witchcraft to the ordinary practitioner of music? But how well was this framework understood even by the contemporary experts? Did they understand phenomena like beats, harmonics or even the concept of frequency itself? Did favourite temperaments spring up and spread Europe-wide in the neat and facile way implied by some of today's writers? How accurately were practical tuners able to set a temperament, and did they care? In fact, what were their tuning procedures anyway - did they listen for beats for example? These and similar questions are addressed in this article, whose raison d'être is to highlight the danger, seen in some published material, of assuming that today's culture with its understanding of musical acoustics can be back-projected onto life hundreds of years ago. It has led to a skewed version of temperament history rewritten by certain authors, or sometimes just conjured up, to fit a sparse set of facts padded out by an overlay of dubious assumptions. Thus an account of the main factors influencing the development of temperament is given, together with their places along a historical timeline where possible.



The temperament problem and when it arose


No attempt is made in this article to explain the problem of temperament in detail as it can be found in numerous other places both online and in print. However an outline of the main issues and when they arose follows for completeness.

European church music in the medieval (pre-Renaissance) era was based on organum, in which singers harmonised a plainchant melody by adding parts moving mainly in parallel octaves, fifths and fourths. They could easily adjust their pitch intonations so that all these intervals were pure (in perfect tune), and so could the earliest organs because the octave was divided into only the seven diatonic ('white') notes of a modern keyboard. An extra B flat gave two diatonic major scales with their tonic notes (F and C) separated by a fifth, thus two sets of seven Gregorian modes could be sung or played. No problems of temperament arose because there were only six fifths to each diatonic octave (three of them inverted as fourths), and this enabled all of them to be tuned pure. The tuning system would often have been the Pythagorean temperament in which all fifths and fourths are pure, except for one fifth which is called the Wolf on account of its being grossly narrowed or flattened by nearly a quarter-tone. However this problem only appears when imposing the temperament over a fully chromatic octave with its twelve semitones. Within a diatonic octave with only six fifths all can be tuned pure; there is no Wolf because it can be notionally placed where there is no physical note on the keyboard (such as the interval B-F#) so it can never be played in practice. In modal music the Wolf is a virtual interval which never appears, a considerable advantage. However in the Pythagorean system most major thirds become very sharp and thus they would have been little used. 

By the time the Renaissance was underway around 1400, composers were already demanding that the interval of a useable (reasonably well tuned) third be added to the mix. Thus they were now writing music based on triads rather than being limited to fifths and octaves. What we recognise as modern harmony developed, and keyboard instruments had to be physically augmented to encompass twelve semitones to the octave instead of only two in the diatonic scale - this change had already begun in the thirteenth century. It was necessary so that thirds and all other intervals could be physically played starting from any note. The new keyboard also provided composers with a choice of twenty four keys (diatonic scales), twelve major and twelve minor. Another new concept was the possibility of modulating at will between keys because the expanded keyboard now offered a wide choice of accidentals (sharps, flats and naturals). Skilled singers and instrumentalists whose notes could be 'bent' slightly in pitch in real time as the music progressed could continue to produce purely-tuned intervals in any key as in the days of organum, but players of the new keyboards could not, nor could those using fretted stringed instruments. Thus the instruments could be out of tune with the singers unless the latter were particularly expert in intonation and sensitive to the problem. This was because there were now twelve fifths and fourths crowding into the octave rather than only six, and some or all of this expanded population had to be narrowed or flattened so they would fit into a pure octave. Thus arose the problem of choosing a suitable temperament which is still with us today, in which the twelve note frequencies of the octave are a compromise to suit intervals played in as many keys as possible. There exists a large number of different compromises and thus different temperaments, ranging from some (such as equal temperament) in which all intervals are somewhat out of tune but useable, to others in which some intervals are perfectly tuned at the expense of making others intolerable.

With twelve semitones, octaves could now be spanned by three major thirds starting from any note. As just mentioned, the octave could also be spanned by twelve fifths (seven of them being inverted as fourths so they would all fall into a single octave). In terms of arithmetic, the problem of temperament arises because the frequency ratio of the two notes of a purely-tuned octave is 2:1, that of a pure fifth 3:2 and of a pure major third 5:4. The octave was considered sacrosanct and it always remained pure [1], but this imposed a straitjacket on how the thirds could be tuned. Three pure thirds such as C-E, E-G# and G#-C [2] represent a total frequency span of 125:64 (5:4 cubed), narrower by nearly half a semitone than the 2:1 ratio of the octave C-C, thus one or more of them has to be stretched or 'tempered' to fit. Hence the major thirds, or some of them, have to be sharpened quite a lot in any temperament. At the same time some or all of the fifths have to be narrowed or flattened from pure so they (and their inversions as fourths) will all fit into an octave. The problem is compounded by other interactions between these three intervals (thirds, fifths and octaves) in that a third is in effect the sum of four fifths. Starting at C the successive fifths are C-G, G-D, D-A and A-E, resulting in a major third between the first and last note of this sequence if the intervening octaves are ignored, and this third is again considerably sharp if the fifths are pure. So the fifths cannot be tuned without disturbing the thirds, and vice versa. Some or all of them have to be tempered in some way, or adjusted in frequency away from pure, while allowing the octaves to remain pure.

Note that only octaves, fifths and thirds were discussed above, but to encompass more of the intervals which should be consonant, the major sixth also needs to be included. Then, with their inversions (fourths, minor sixths and minor thirds respectively) we have all the quasi-consonant intervals whose distances from pure tuning largely determine the subjective acceptability of a temperament. Because no temperament exists in which they can be tuned pure simultaneously across all twenty four keys, one is reduced basically to a process of empirical fiddling with nuances of intonation which has still to reach a conclusion after several centuries of trying. So why bother to write this article - what is its rationale? It is hardly likely to move such an ancient and indefinite free-for-all towards closure. I can only refer you to what was said at the outset, which suggests that if people want to continue picking over these old bones, they might at least do so with an eye to historical fact rather than filling in the gaps with baseless assumptions such as "Bach would have used this temperament". As we shall see, the issue of historicity applies particularly to how practical tuners actually tuned and how accurately they could have done it on instruments whose frequency stability was poor. Accurate and stable tuning have always been vital if the concept of temperament is to have meaning in practice rather than just remaining a theoretical nicety for the delectation of numerologists.



Philosophy versus Empiricism

 

"the sole remaining task for philosophy is the analysis of language"

Ludwig Wittgenstein

 

"philosophy is dead"

Stephen Hawking


It should be remembered that philosophy alone still ruled the mindset of many educated people in Europe when the earliest temperaments were being developed. The scientific method familiar today, in which empiricism (practical experimentation) is key, was anathema to philosophers who believed that solutions to all problems could be obtained just by sitting and thinking about them. The minority of scholars in those days who saw the power of doing experiments were persecuted, reviled or at least regarded as a lower species. This was because a single experiment could in principle deliver results capable of sweeping away millennia of accepted wisdom, and this notion had long caused fear within academia and the church. Think of what the Roman church did to experimentalists such as Copernicus or Galileo. Therefore we have to read the early pre-Enlightenment texts on temperament in this context. Much that is presented in them was never tried for real, which again illuminates the then-contemporary chasm between theory and practice. Although a few authors were astonishingly gifted in mathematics, their theoretical insights were often not translated into practical tuning instructions at all, or the instructions which were given were so vague as to be almost useless. Most philosophers of the day believed that their job was done when their thoughts had been written down. Trying them out in practice was simply not part of the contemporary zeitgeist and it would seldom have occurred to many of them. Therefore it can be a mistake for today's writers to take what they find in early texts to be a reflection of what was actually implemented by practical tuners, or even by the scholars themselves.

This dichotomy persisted into the eighteenth century. Rameau's 'New System of Music Theory' [3] sets out an impressive arithmetical framework for a class of French temperaments, but his tuning instructions are disappointingly vague and incomplete. Other examples will be discussed in more detail later. However, by this time other, more practical, problems were rearing their heads. One was that accurate tuning means the frequency ratios between the two notes of intervals such as thirds and fifths have to be set with some precision. At the time this could only have been done by counting and timing beats and we shall see later that this was surprisingly difficult, partly because beats do not measure ratios directly even when timed accurately (they measure frequency differences, which is quite a different thing). Even ignoring this practical problem, the mere idea of timing beats, regardless of its difficulty, did not seem to percolate down to the average tuner for centuries after its importance had been realised by at least some theorists.



Frequency


We now turn to a series of concepts which are fundamentally important to the subject of temperament, and the first of these is frequency. The ancient Greeks discovered simple numerical relationships between the lengths of strings and the frequency ratios of the notes they emitted when plucked. For example, strings sounding notes an octave apart had lengths in the ratio 2:1, for a pure fifth it was 3:2, and so on [4]. However it is important not to assume that the Greeks therefore understood all about frequency, because measuring frequency ratios of intervals is quite different to being able to measure the absolute frequency of a given note. Although it would have been obvious that plucked strings vibrated transversely when emitting sound, they would have had no means of measuring the number of vibrations in a given time period for sounds in the audible range, a procedure which is required to define their frequencies. Nor could they have measured time periods accurately enough in any case. It is conceivable they might have set a long, heavy string into oscillation so that its motion was slow enough to be observed by eye and therefore timed approximately. But for the reasons given previously, this would probably have been classed as one of those undesirable experiments which were not allowed to taint their philosophical approach to an understanding of life's problems.

However just such an experiment was eventually done by Mersenne more than two thousand years later in the early seventeenth century [5]. Mersenne was one of a few emerging thinkers following Galileo who realised the importance of an empirical approach - they were called natural philosophers. He also derived, or at least verified experimentally, an equation relating frequency to the length, mass per unit length and tension of the string. Using it, he then concluded that a note lying between the E and F above our modern 8 foot C had a frequency of about 84 Hz [6]. The measurements he made could not have been accurate enough for use in temperament studies if only because of the difficulty of measuring time with sufficient precision. Nevertheless this was a considerable achievement for its day because what he had done was to illuminate a causal link between the periodic motion of a vibrating source, the consequential air movement and the perception of musical pitch by our ears and brain. This is the essence of what is meant by frequency or pitch in music. It is even more remarkable when one considers that Mersenne lived and worked during the appalling upheaval of the Thirty Years' War in Europe during which millions died, yet he also managed to travel and maintain contact with other prominent intellectuals of his day such as Descartes. (The latter, no doubt wisely, spent much of his time keeping his head well down - he wrote that "to survive one has to live unseen").  Mersenne's lead was taken up by others, including Robert Hooke who demonstrated his technique of measuring frequency to the Royal Society in the late seventeenth century.  He showed that a mechanical siren, a piece of stout paper applied to a rotating toothed wheel, would emit a musical sound - even this seemingly simple experiment was thought to be startlingly novel at the time. Using a modified clock mechanism, he was also able to maintain the speed of the wheel constant and measure its rotation rate. This provided another way of measuring musical pitch or absolute frequency.  Techniques such as this were used to calibrate the first tuning forks which appeared shortly afterwards in the early eighteenth century, and sirens remained in laboratory use until the electronic age dawned. Although the concept of frequency is well engrained today in musical acoustics, a universal definition of it did not finally enter common usage until the twentieth century. According to Audsley [7], as late as the nineteenth century the French organ builder, Aristide Cavaillé-Coll, was one of those who quoted frequencies having twice the numerical values of those used today (he counted two air impulses in organ pipes for each complete cycle of oscillation). This can cause considerable confusion when reading some material from that era unless one is aware of it.

Having said all this, why is a knowledge of absolute frequency (the frequency of an individual string or pipe) as opposed to relative frequency (the difference between the frequencies of the two notes comprising a musical interval) important in a temperament context? After all, temperaments can be set up independently of pitch, can't they? Well, the answer is both yes and no. The theoretical basis of temperament relies on setting the frequency ratios (not the arithmetical differences) between intervals such as fifths and thirds according to some particular prescription, and pitch or absolute frequency does not enter into this at all. However a problem arises when trying to tune a temperament in practice because, when tuning by ear, there is no means of determining a frequency ratio. Only frequency differences can be detected, and that is done by listening for beats. And to temper an interval accurately, the beats have to be timed as well. Obviously, some frequency standard also has to be adopted (A = 440 Hz or whatever) before any other notes can be tuned. The cusp of the problem is then encountered because the 'goodness' of the various intervals depends on listening to beats, if not explicitly timing them, and beat frequencies vary with the pitch standard chosen. Therefore a tuner would first need to tune a reference pipe or string to the correct frequency before proceeding, and this would then define all his subsequent beat frequencies. So how would this initial tuning have been done?

The recent literature of musical pitch is replete with confident statements about the pitches of ancient organs going back at least as far as the sixteenth century. But how do these authors know for sure what the actual frequencies were in numerical terms (i.e. expressed in Hertz)? Obviously they cannot, because the builders and tuners of the day did not know either. Frequency was not a word in their vocabulary. The foregoing discussion has shown that the mere concept of frequency, so familiar today, could not have been properly understood until at least the time of Mersenne in the seventeenth century. Galileo before him had declared that measuring frequency in the audible range was an impossibility and would remain so. Even post-Mersenne, in all probability the concept of frequency did not become the property of the average tuner until much later. And how were pitch standards defined anyway, meaning how did tuners approach an organ or stringed instrument and proceed to set its pitch? Some sort of pitch pipe could have been carried around, probably a wooden one without a stopper because stoppers can move around unpredictably. The pipe would probably have been wooden because metal ones are notoriously sensitive to temperature changes arising from handling or even the proximity of the tuner's body, and the issue here is the danger of transferring that heat to the internal air column rather than the negligible effect it has on dimensions. Should there be any doubt, both wood and metal flue pipes are very sensitive to air temperature - they go sharp by almost a semitone for a temperature increase of 30 degrees Celsius. Thus the need to blow the pipe when tuning a stringed instrument would have made even a wooden one sharpen quickly as the sound speed within increased - blowing it in a cold building around freezing point would have raised the air temperature rapidly inside the pipe by nearly 40 degrees. However when tuning an organ the pipe could have been inserted into the soundboard rather than being blown with warmer air from human lungs, so this problem would be lessened. And when the tuning fork arrived in the eighteenth century the problems would have been reduced further. It is also from surviving pipes in a few organs from this later period that we can be more confident of what pitches they were tuned to, as well as their temperaments in some cases. But on the whole such certainties are rare in the long history of tuning and temperament.



Harmonics


Harmonics in musical sounds are inextricably bound up with the subject of temperament, because it is they which generate the beats that were used exclusively to determine whether intervals were properly tempered until electronic tuning devices arrived in the late twentieth century [8]. The presence of harmonics, frequencies above the fundamental or ground tone of a note, must have been apparent from the earliest times because they can easily be heard by an educated ear. This is particularly true for a few prominent early harmonics such as the octave (the second harmonic) and twelfth (the third).  These observations must also have been correlated with the same frequencies which can be heard individually by stopping a vibrating string at the appropriate points with a finger. Therefore it must have been clear for a very long time that a string (and an organ pipe) is capable of vibrating at many frequencies simultaneously, though no satisfactory explanation of the phenomenon was forthcoming until much nearer our own era. It was not until the early nineteenth century that harmonic formation was explained theoretically, though Fourier's role in this is often misunderstood. It is true that he showed mathematically that any periodic (cyclically repeating) variation could be represented as a harmonic series, but in fact this work was not inspired by anything to do with acoustics or music because it was originally an attempt to explain how heat propagated in solids [9]. Some years later Helmholtz concluded that harmonics strongly influence the subjective tone colour or timbre of musical sounds which we perceive consciously, and this is of some interest in the context of temperament. But far more important is the intimate link between harmonics and beats and this will now be explored.



Beats


Having mentioned beats several times already we now need to discuss them in more detail. The subject is many-facetted in terms of acoustical physics, perhaps surprisingly so, and it is expanded in another article elsewhere on this website [10]. They are so important in the context of temperament that they must be treated at some length, particularly as regards their place in history.  For millennia it had been observed that beats disappear when two unison strings or pipes are drawn into perfect tune. Although accepted simply as the way things are, this effortless observation was for long imperfectly understood because, as we have noted already, the concept of frequency remained vague until the seventeenth century.  But it was even less clear why the same phenomenon also applied to the consonant intervals such as octaves, fifths and thirds, because when perfectly tuned (no beats) there remains a large frequency difference between the two notes, and it was not known why this did not continue to generate a beat. Many people who have not thought about it do not understand it to this day, so the subject obviously needs to be covered at some length here.

Beats are heard when sounds at two nearly-coincident frequencies reach the ear simultaneously. They are not formed within our auditory system as sometimes supposed because they exist independently in the atmosphere as an objective phase-interference phenomenon. The subjective effect is a relatively slow periodic loudness variation having a frequency of its own, the beat frequency, typically in the range from several cycles per second to less than one cycle. In the former case it might be described as a throb or purr, and in the latter as a slow waver. The beat frequency is the arithmetical difference between the two generating frequencies, thus when these become identical the beat vanishes [16]. 

Beats arise between different harmonics of the two notes defining intervals other than unisons, where the harmonics involved are the same (the fundamental in both cases). In the case of an octave the harmonics are the first harmonic (fundamental) of the upper note and the second harmonic of the lower. For a fifth they are the second harmonic of the upper note and the third of the lower. For a fourth they are the third harmonic of the upper note and the fourth harmonic of the lower. For a major third they are the fourth harmonic of the upper note and the fifth harmonic of the lower. For a minor third they are the fifth harmonic of the upper note and the sixth harmonic of the lower - and so on [11]. It is largely the beat frequencies which determine whether the tempered intervals of a particular temperament are subjectively acceptable and musically useable in harmony, unless the individual notes are so out of tune as to be intolerable for melody as well as harmony, and in this event they will be flat or sharp by a significant fraction of a semitone. When any interval is tuned pure the beat between their two notes vanishes, in fact the absence of a beat is the practical tuner's definition of interval purity. 

It is less well appreciated that the strengths or amplitudes of the generating harmonics affect how strong and noticeable the corresponding beat is. Intervals with strong beats are intrusive and subjectively unpleasant and they dominate our subjective responses to some temperaments in terms of attributes such as key colour. But if the harmonics are weak the beat will not be as noticeable as when the harmonics are stronger. In the most extreme case no beats at all (other than between mistuned unisons) will be heard if there are no harmonics above the fundamental, but this only occurs for sine waves, pure tones. Real acoustic musical instruments never produce pure tones, although some electronic ones can such as the Hammond tonewheel organ. Some intervals played on pipe organs produce weak or absent beats because the generating harmonics are also weak or absent. Regardless of temperament, this often applies to major and minor thirds played on a flute stop, for two reasons. Firstly the harmonics involved in beating thirds are getting quite high up the harmonics series - they are higher order harmonics than those for octaves and fifths. In the case of the major third the fourth and fifth harmonics are relevant, and for a minor third the pertinent harmonics are the fifth and sixth. The second reason that beats can be weak for thirds is that these higher harmonics are weak or even nonexistent in flute tones because their strengths fall off very quickly above the fundamental. An important conclusion follows from this - the tuning purity of intervals, particularly thirds, in a temperament produces subjective effects on the organ which vary for different stops. If tempered thirds sound objectionable on a principal stop it is quite possible they might be acceptable when played on a flute. This is because principals have a more extensive and stronger harmonic series than flutes, and therefore they generate stronger beats. This property makes the organ unique among keyboard instruments in that the effect of a given temperament can vary considerably depending on which stops are drawn. No other instrument has this attribute, yet it is a subject seldom appreciated or discussed. When appraising old texts on temperament it is important to read them with this point in mind when relating the material to the organ because not all keyboard instruments are the same when it comes to tuning. The subjective effect of a temperament is not independent of the instrument it is set up on.

How have beats been treated throughout temperament history? The plain fact is that people have always listened to them even though they might have done so with little thought. You cannot avoid it when trying to draw a pair of notes into tune, because when the interval or unison is pure the beats disappear. As already mentioned, the absence of a beat is the practical definition of purity to a high degree of precision in terms of the frequency equality of the generating harmonics. But at what point did it become known what a beat was in the terms of the foregoing discussion, that is in terms of physics? Like beats themselves, the harmonics which generate them are equally obvious to the trained ears of tuners and musicians and therefore it does not seem unreasonable to conclude that harmonics have been recognised (though probably not well understood) from antiquity. But it is unclear when the causal connection between harmonics and the beats they generate was made and the associated arithmetic put in place. From Werckmeister's writings in the late 1600s it is obvious that he listened for beats and used them in tuning, but whether he understood that harmonics generate the beats is less clear. A related issue is whether he timed beats to temper intervals accurately. He certainly distinguished between slow and fast beats, but it is less obvious whether he went beyond this to quantify his descriptions numerically and describe how they should be timed in practice.

 

The stature rightly accorded to Andreas Werckmeister sometimes obscures an objective view of him. Unlike many of his peers he was essentially a highly-gifted and self-taught amateur in the sense he had not the academic background which they enjoyed, and this may have reflected the legacy of the Thirty Years' War which had wrecked the infrastructure and culture of his country rather than implying a lack of financial means. His writings suggest limited interactions with other theorists because there is no mention of other great names such as Joseph Sauveur, Christiaan Huygens, Robert Hooke, Thomas Salmon and even Mersenne, Descartes and Galileo. All of these were members of an influential European cohort who transformed the understanding of musical acoustics in the sixteenth and seventeenth centuries. For their part, his contemporaries seemed mostly unaware of Werckmeister and he did not influence them. His arithmetic was performed heroically without the assistance of Napier's newly-invented logarithms to calculate ratios, and he was probably only hazily acquainted with fundamental concepts such as that of frequency at the level of physics. It is therefore unsurprising that one looks in vain for detailed explanations of phenomena such as beats in his work although he certainly made use of them. This too is not surprising because he came to the physics of music from music itself, acquainting himself with enough mathematics to develop his theories but no more. He was a pragmatist who carried out experiments when he needed to, using devices like monochords to develop temperaments experimentally as much as theoretically. It was therefore predictable that other practical musicians, including J S Bach, found his work approachable and understandable (Bach owned at least one of Werckmeister's books during his lifetime) and it is at this level that we are best able to understand his influence on tuning and temperament.  In fact his work was probably assimilated more quickly into the portfolio of knowledge and techniques of the average tuner than that of the more well known names mentioned above - they could understand it and put it to use immediately.  He was one of them.

Fifty years or so later the uncertainties concerning beats outlined earlier were comprehensively removed by the appearance in England of a remarkable book by mathematician Robert Smith, Master of Trinity College, Cambridge [12]. Among many other attributes, it describes various experiments such as one in which beats arise or disappear when two cello strings of varying lengths are sounded. In effect he was using the instrument as an elaborate monochord to demonstrate harmonics and how they produce beats (a non sequitur on my part because using two strings on a device means it cannot strictly be called a monochord). Together with similar treatises such as that of Rameau already alluded to [3], it is therefore possible to say that the status of beats in the physics of music was pretty much understood by the mid-eighteenth century at the latest. It is therefore unfortunate that tuning practices were in nothing like so advanced a state, as we shall now see.



Tuning practices


It is reasonable to enquire why tuners did not move eagerly to time beats once they had become better understood, yet the fact is that as a rule they did not. As late as 1780, well after Smith's book [12] had appeared, typical tuning instructions in Britain were still entirely qualitative and of the form "tune the Fifth pretty flat and the Third considerably too sharp". Or "let the Fifth be nearer perfect than the last tho' not quite". Such statements are laughable today and even at this relatively late date, almost into the nineteenth century, they show categorically that the unavoidably haphazard result of a tuner's intervention might well not have been that which was hoped for - perhaps even by the tuner himself. These extracts are from a complete set explaining how to set up the so-called 'Handel Temperament' (which probably had nothing to do with Handel, incidentally) [13].

Some obvious reasons explain the lag between understanding beats in theory and beat timing edging later into the tuner's craft. One is the slowness with which information propagated in those days and another is the generally narrow confines of contemporary education. Musicians and tuners would not have rushed to buy books such as that by Smith even if they were aware of their existence, because most could not have understood (or afforded) them. Today his book remains an impressive though difficult read - he addresses a mathematically expert and classically-educated audience from the outset, interspersing Latin, French and Greek freely among the physics and mathematics, much of which is far from trivial. Just as important, capitalising on the theoretical understanding of beats outlined in the previous section implies that tuners would have had to develop a convenient means of timing them. Obviously they could count beats fairly readily, but they also needed to know how many beats were occurring within five seconds, say. This was the practical nub of the matter, as it had been for centuries past. Another important topic concerns pitch. The pitch standard to which an instrument is tuned (A = 440 Hz or whatever) defines the beat frequencies of all intervals across the keyboard, therefore there would be no point in a tuner timing beats unless he knew the beating frequencies which must be generated by the pitch standard he used. It involves a lot of awkward arithmetic to calculate these for a given pitch standard. Although by the mid-eighteenth century books such as those by Smith would have eased the problem in principle, in practice only those of higher than average numeracy would have found the material comprehensible and useable. It seems likely, therefore, that scraps of paper containing the beat frequencies of favourite temperaments would have circulated informally among musicians and tuners, much as they did until the late twentieth century when electronic tuning devices became common. Many tuners would have had little idea how the numbers arose, and most would probably not have cared provided they were thereby enabled to set temperaments reasonably accurately.

Consider the problems of tuning the various meantone temperaments in which eleven out of the twelve fifths are narrowed or flattened by the same frequency ratio. The actual ratio depends on which version of meantone is used (quarter-comma, etc). To enable the octaves to remain pure the remaining (twelfth) fifth then has to be sharpened by an intolerable amount, and this fifth is therefore a Wolf in all meantone tuning systems. A non-tuner might think that because eleven fifths are all flattened equally this means that their beat frequencies will be the same, but this is not so. Although the frequency ratios of these fifths are identical, their beat frequencies will vary because they occur at different positions in the key compass, a practical tuning issue which Werckmeister had observed and perhaps understood. It arises because beats reflect frequency differences, not ratios, therefore they get faster towards the treble where frequencies are higher and slower in the bass. In contrast, ratios remain the same across the keyboard regardless of absolute frequency. This variation in the beat frequencies makes meantone temperaments (like some others) tricky to tune because many fifths have to be tempered accurately [14]. However for the particular case of quarter-Syntonic comma meantone, the several pure thirds and sixths make things easier because some of these can be used as 'check' intervals for the tempering of the fifths and fourths - if the latter are set incorrectly the check intervals will not be pure as required. Fifth and sixth-comma meantone temperaments do not have these convenient pure intervals to aid the tuner. Therefore relative ease of tuning might partly explain why this version of meantone with its several unsatisfactory attributes persisted for so long. But in general these and other temperaments (including equal temperament) must have often been set up rather roughly because of the sheer difficulty of achieving repeatable results in an era when beats were not timed. For this reason I find the opinion of some authors, who maintain that performers would have thought nothing of retuning a few notes on an instrument to suit the music being played, to have little merit. Maybe they should try it themselves, tuning by ear of course, and then ask whether their positions should be revised. In any event, whether it was done or not for stringed instruments, it certainly could not have been undertaken lightly on the organ for obvious reasons, so we can ignore the possibility here. As Howard Goodall has said, "the complexity of the 'meantone' temperaments meant that they ... varied absurdly from town to town, country to country. Everyone had their version of it, and what worked for one piece of music on one instrument in one place almost certainly wouldn't in another" [15].

So if tuners were to time beats they would have had to do two quite difficult things, the difficulties arising from practical considerations rather than necessarily a lack of understanding. The first would be to set one note on the instrument to be tuned to the pitch standard of choice, and this has been discussed already. The second requirement raises the question as to whether they would have carried an accurate source of time around with them and if so, what form would it have taken? A wristwatch perhaps? Well, not until the end of the nineteenth century when they first became available. What we would call pocket watches had been around in some form since the 1520s but they were large, generally crude and relatively rare. They were also expensive, which explains why that conspicuous consumer Henry VIII was once portrayed wearing one hanging round his neck. By no means all of them had the necessary sweep seconds hand, or indeed any form of seconds hand. That sort of pocket timepiece had to await the mind-blowing work of Harrison who did not produce his exquisite pocket chronometer until the 1760s, and by that time Bach and Handel had died. It is more likely that most clocks used pendulums and were therefore not easily portable. The timekeeping property of a pendulum had been discovered by Galileo around 1580, but it was not until Huygens and Newton had explained fully how it worked that the pendulum escapement became a domestic commonplace a century later - around the time Bach, Handel and Scarlatti were born - in timepieces such as the long case ('Grandfather') clock in Britain. Many of these would have measured the passage of time audibly with a 'seconds pendulum' whose ponderous tick-tocks were a second apart, and there is a chance you might have developed an innate mental capability to estimate seconds reasonably accurately if you had grown up in a house containing such a clock. It is also conceivable that musicians and tuners after about 1700 or so might have carried an elementary pendulum around with them, thereby solving the problem of a source of portable time for tuning and maybe for setting the pace of music as well. (Maelzel's more convenient metronome did not appear until 1815. Until the advent of today's electronic devices, cheap plastic pocket metronomes using the pendulum principle were readily available from music shops almost up to the present day. They were rather like extending tape measures and my first teacher used one frequently). Such a musician's pendulum in the eighteenth century could have been identical to the plumb line which had been used by architects and builders for countless millennia, consisting merely of a piece of string with a weight at one end. The string could easily have been calibrated with marks indicating the beat periods of a favourite temperament. It is interesting that in the eighteenth century Smith wrote "times of beating may be measured by a watch that shews seconds or a simple pendulum of any given length" [12], thus my conjectures above are clearly reasonable.

Failing this, tuners would have had little option but to time beats using some instinctive mental sense of how time passes. There is also the strong possibility that some simply used their pulse rate, a practice which was commonly resorted to by workers across science and industry until well into the nineteenth century. So what does all this say about the accuracy with which tuners could set up a temperament? Quite a lot is the trite answer, because inaccuracy might be a better word. If you could not or did not want to time beats, how on earth could you have set up a temperament properly and repeatably in those days? The question of tuning accuracy illuminates a particular issue which still exercises many writers on temperament today, and this concerns the relative attributes of fifth (Syntonic) comma and sixth (Pythagorean) comma meantone tunings. In terms of the tempering (beat rates) of the fifths, these are almost identical and very careful beat timing is necessary to set either of them up properly by ear. Eleven fifths have to be tempered in both cases, and this fact alone exacerbates the difficulties. It is necessary to time the beats over an extended period, perhaps up to one minute, if the necessary accuracy is to be achieved. While this would have been straightforward, if tedious, on the organ it would obviously have been impossible on stringed instruments with their transient sounds. Therefore one is led to the assumption that on the latter the difference in practice between these temperaments was quite possibly more notional than real. It is true that several authors from the eighteenth century onwards wrote learnedly about the relative attributes of the two methods of tuning, but this does not imply that tuners were as a rule capable of setting them up correctly. It is just as likely that such material simply reflects the disconnect between theory and practice which has been remarked on already.

Accuracy aside, another important conclusion relates to ease of tuning. Temperaments in which all the fifths need to be tempered, including equal temperament, are difficult and time consuming to tune. However those which contain pure fifths are more straightforward, so the more such fifths, the merrier. It is much easier to tune a fifth pure than to temper it, simply because when in tune it does not beat. Some well known temperaments had several pure fifths, such as Werckmeister III (c. 1690 - 8 pure fifths out of the 12), though having expounded its theoretical basis he subsequently admitted to disliking it in practice. Another is that by Vallotti (c. 1730 - 6 pure fifths). Maybe this humdrum matter as much as anything else contributed to a preference for unequal temperaments with their pure fifths at a time when tuning itself was such a hit and miss affair.

One can reasonably reach a threefold conclusion from the foregoing. Firstly, achieving accurate tuning from the earliest times presupposes, without much evidence, that tuners have always been numerically aware of the beat frequencies they were aiming for. But these were the output of an arcane theoretical framework, and it is difficult to be sure how many of them possessed knowledge of either the theory or the frequencies. And if the pitch standard of an instrument changed, so did all the beat frequencies, which made things even more difficult. Secondly, the difficulty of timing beats means that accurate tuning was at least elusive and, more likely, next to impossible throughout much of temperament history. The ludicrously imprecise instructions for the 'Handel' temperament [13] in the late eighteenth century have already been discussed, and similar examples grace the literature going back hundreds of years earlier. Yet it was obviously such instructions which tuners were still relying on into the nineteenth century in some cases, because the 'Handel' instructions were published as late as 1780. Therefore accurate tuning must have been the exception rather than the rule for hundreds of years past, until well into the nineteenth century in fact. Thirdly, unequal temperaments containing multiple pure fifths might have been attractive simply because they were easier to tune, rather than because of the sometimes excessive enthusiasm for their musical attributes which is common today.



Key colour


Key colour or key flavour is one of those musical attributes just mentioned relating to temperaments. It concerns how a temperament sounds in different keys, and this is strongly influenced by differences in the beat rates of intervals. It is difficult to be certain how key colour was regarded in earlier times, particularly since the emphasis placed on it today can sometimes lead to a lack of objectivity when looking back into temperament history. Many early writings and positive anecdotes relating to the properties of temperaments can just as easily be interpreted as expressing simple relief that at least some keys were reasonably well in tune, rather than reflecting a sharp focus on key colour for its own sake. Given the widespread tuning inaccuracies which must have dominated the scene for centuries while beat timing was not practised, a disproportionate spotlight on key colour seems difficult to justify and probably anachronistic. Accurate tuning would have been even more elusive when one recalls the uneven winding of organs before the emergence of steam, hydraulic or electric blowing in the nineteenth century. 

This view is supported by the fact that several well known and widely used temperaments for hundreds of years do not have key colour, at least in the sense that what colouration there is varies among their keys. Equal temperament is of course one such, and its history goes at least as far back as the fifteenth century when it was used for fretting guitars. The fret positions could be defined accurately and repeatably because they could be established once and for all by careful measurements derived from monochords kept in the workshop. Such accurate tuning was obviously more difficult and therefore less common with stringed keyboard instruments and the organ; it is unlikely that tuners routinely carted monochords around with them when tuning those instruments. Hazier evidence suggests equal temperament might have been known to the ancient Greeks and possibly to the Chinese before them. It is frequently claimed that its sharp thirds caused it to fall out of favour until its almost universal adoption by the mid-nineteenth century, though another reason might have to do with the fact that it is difficult to tune. Any temperament having twelve tempered fifths, all with different beat rates, might not have been popular until beat timing became better accepted as part of the tuner's craft in the late eighteenth and nineteenth centuries. This coincides neatly with the resurrection of interest in equal temperament at that time.

However another temperament, or class of temperaments, which do not have key colour is the important meantone temperaments which also persisted in some form for centuries. This seems not to be universally appreciated, sometimes by those who sing their praises loudest. Meantone is in fact a form of equal temperament except for the single Wolf fifth because all the others are tuned with identical tempering. Therefore keys which do not involve the Wolf have the same subjective colour, or lack of it depending on one's point of view. Another correspondence between meantone and equal tunings is that the fifths in sixth-Pythagorean comma meantone are tempered exactly twice as fast in terms of beat frequency as they are in equal temperament. Perhaps this is one reason why Gottfried Silbermann, who often tuned in sixth-comma meantone, was sometimes persuaded to offer equal temperament (the 'sharp' tuning which he apparently disliked) because it is easy for a tuner with experience in the one to switch to the other simply by tuning an octave adjacent to the one usually used.

Another factor which is specific to the organ has already been mentioned and this concerns the subjective colour of temperaments, not only in different keys, but also with different stops. The beat amplitudes (strengths) of major and minor thirds are particularly sensitive to the amplitudes of the fourth, fifth and sixth harmonics in the sounds of the individual pipes. These are strong and clearly audible for principals but less so for flutes, where they are almost always weaker and in some cases absent altogether. Therefore key colour on the organ varies both with key and the registration employed, because a beat which might be strong and objectionable on a principal stop can sound considerably tamer on a flute. This aspect of the matter is seldom discussed but it is stretching a point to assume that it was unknown in earlier times.



Concluding remarks


There is sometimes a tendency for modern authors to reflect today's norms of fashion, musical culture and understanding of physics into their writings about early temperaments, when these matters were in fact very different several centuries ago. A prime example concerns the difficulty of tuning intervals accurately until relatively recently when electronic tuning devices appeared in the late twentieth century. Until then tuners or owners of keyboard instruments had to time beats using various less precise methods, and even this only became routine from about 1800. Prior to that tuning was done for centuries using the vaguest of instructions which appear absurd to modern eyes. The reasons for this are that beats and the harmonics which generate them were for long imperfectly understood, and practical means for timing them accurately did not exist. The plethora of different pitch standards made things worse because beat frequencies depend on absolute frequency. There were also problems due to the slow dissemination of temperament theory, together with widespread educational narrowness which meant that most musicians and tuners would not have understood it anyway. In addition, hand blowing and poorly-designed winding systems meant that the tuning stability of organs was badly controlled regardless of how tuners might have struggled to set a temperament accurately. The upshot of all these factors is that the sharp focus applied to subjects such as key colour today can be anachronistic if it is assumed that our predecessors several centuries ago viewed them as we do. Obviously, the key colours of a temperament become elusive if it cannot be set up accurately in the first place on an instrument with stable tuning, and for centuries this would have been the case with the organ.

Therefore several important features of musical temperament have been placed in an historical context in this article, including an understanding of frequency, harmonics, beats and how these influenced contemporary tuning practices. Another factor unique to the organ was also mentioned and this concerns the subjective colours of temperaments, not only in different keys, but also with different stops. In addition, differences among early scholars between a philosophical and empirical approach to knowledge were also discussed to suggest how material before the mid-eighteenth century might be better interpreted. In particular, some results presented by theoreticians might well not have been tried for real because of an absence of how to translate them into practice.



Notes and References


1. In principle there is no reason why the octaves should remain pure. They can be tempered just like the other intervals. An example is Cordier's temperament in which the octaves are widened to enable all fifths to become pure. This temperament was highly regarded by musicians such as Yehudi Menuhin, Paul Badura-Skoda and Jean Gillou. I have discussed this and other temperaments with impure octaves in another article on this site at:


Keyboard Temperaments with Impure Octaves


2. In this article the physical notes of the keyboard are sometimes denoted by the tuner's nomenclature of 'naturals' and 'sharps'. This means that some intervals written using this convention do not make sense musically. An example is the major third between G sharp and C which, musically, should properly be written today as A flat to C. However it is assumed readers will understand and accept this.

3. "Nouveau Système de musique théorique", J-P Rameau, Paris, 1726

4. The ancient Greek length ratios of plucked strings only apply if the strings are made of identical material and thickness so that their masses per unit length are the same. Their tensions must also be identical.

5. "L'Harmonie Universelle, livre troisième : Des instruments à chordes", M Mersenne, Paris, 1636. 

6. This statement assumes equal temperament with middle A tuned to 440 Hz.

7. "The Art of Organ-Building", G A Audsley, New York, 1905.

8. In this article no distinction is made between the exact harmonics of a driven sustained sound (a forced oscillation), such as that from an organ pipe or bowed string, and the transient anharmonic partials arising from plucked or struck strings as their sound decays.

9. "Mémoire sur la propagation de la chaleur dans les corps solides, présenté le 21 Décembre 1807 à l'Institut National, Paris", Nouveau Bulletin des Sciences par la Société Philomatique de Paris, J B Fourier, March 1808. 

10. "Resultant Bass, Beats and Difference Tones - the facts", an article on this website, C E Pykett, 2011.

11. Beats also arise from other pairs of higher-order harmonics if the harmonics are strong enough in the sounds of their corresponding notes. The harmonics for the intervals mentioned here are usually those which generate the most important (strongest and slowest) beats and it is these which the ear latches onto, though some of the others can often be heard singing away in the background as well.

12. "Harmonics, or the Philosophy of Musical Sounds", Robert Smith, London, 1748.

13. "Twelve Voluntaries and Fugues for the Organ or Harpsichord with rules for tuning by the celebrated Mr Handel", Book IV, Longman & Broderip, London, c. 1780.

I have discussed this temperament in detail in an article elsewhere on this site. See  'Handel's Temperament revisited', C E Pykett, 2016.

14. For Silbermann's sixth-Pythagorean comma meantone tuning, the beat frequencies of the eleven fifths and fourths in the octave starting at middle C lie in the range from 107 to 200 beats per minute, excluding the Wolf between G# and D# which beats extremely quickly and is not tuned separately. These figures assume an 8 foot stop with A set to 440 Hz. Yet all fifths are nevertheless flattened by the same ratio. This example illustrates the numerical divergence between the frequency ratio of the notes of an interval and their beat frequencies which are actually heard - although ratios might be the same, beat frequencies are not. It is the variation in beat frequencies which makes temperaments difficult to tune when many fifths have to be tempered rather than tuned pure. Moreover, all the beat frequencies just mentioned would change if a different pitch standard were to be adopted.

15. "Big Bangs - the story of five discoveries that changed musical history", Howard Goodall, London, 2000.

 

16. Treating the matter rigorously, the mathematics shows that the beat frequency equals half the difference between the two generating frequencies, not the difference itself.  Yet it is the difference (not the half-difference) which is actually heard as the beat frequency.  The apparent paradox is because the ear perceives each half-cycle of the beat as a separate and perceptually-identical entity, in other words it hears two pulsations per beat-cycle rather than just one.  The matter is explained in greater detail in Appendix 1 to reference [10].  The point is emphasised to illustrate that one needs a reasonable grasp of mathematics to properly understand beat formation, and it is unlikely that this was widespread at a time when even such basic concepts as frequency were not well understood.  The literature shows clearly that it was not until the eighteenth century was well advanced that correct explanations of the phenomenon appeared.  Even so, one still has to consider whether other than a small minority of practical musicians and tuners were capable of grasping what was happening in beat formation until some time later, probably not until the nineteenth century in fact.  This helps to explain why tuners did not routinely time beats until then.