Temperament - a study of Anachronism
by Colin Pykett
Posted: March 2006
Last revised: 12 February 2013
Copyright © C E Pykett
"a cul-de-sac of a subject"
Peter Williams 
" ... 1200 cents x 2 + 386.14 = 2786.314 cents ... "
Christopher Kent 
Abstract The subject of tuning and temperament continues to provide a never ending source of interest and income for a constant stream of academics. Because it requires just that little extra effort to comprehend the necessary simple arithmetic (it is wrong to dignify it as mathematics), it is easy for those with the inclination to wrap up their work in a cloak of mystery and authority which is actually largely spurious. It disguises the fact that many of the claims made about the temperaments favoured by Bach, say, are completely unproveable. In reality, they fall into the same category as the story that he once found some coins in fish heads thrown out of the window of an inn, and they are about as useless.
This article shows that an uncomfortable proportion of recent work on temperament is unscholarly in that it projects today's understanding, values and culture several centuries backwards as though these things have never changed. Thus the authors of such material are merely wallowing, apparently unconsciously, in a sea of reverse anachronism. They are literally out of time. Some are also apparently unconscious of the errors in their work. By looking at the realities of musical life in the 17th and 18th centuries it is suggested that some if not many contemporary temperaments can be traced to the fact that stringed keyboard instruments had wood frames, with the consequential tuning instability this implies. Also the role possibly played by impure octaves in these temperaments is examined.
on the headings below to access the appropriate section)
Because a correspondent drew to my attention an article on temperament of such utter fatuousness, carried by a journal with learned pretensions which has clearly abandoned any attempt at peer review, I now rest easy in the knowledge that anything I might write on the subject could not possibly be any worse. I am grateful to him.
To study temperament is to read the entire story of music itself across the world since the dawn of time. Melodic as opposed to percussive musical instruments arose firstly because of the aural observation of the first three natural harmonics in the sounds of pipes. These were then given a separate existence by tuning additional strings or pipes to the same frequencies. The intervals between these would have been octaves and fifths because of their genesis from the natural harmonic series. As time went on it was found that the fifth alone had the magical property of generating eleven notes which could be used to fill the gap between two octavely related ones, thereby giving us the chromatic scale still used today. Simultaneously, it became obvious that the frequencies of these notes had to be adjusted from what was regarded as their “pure” values so they could fit into the octave. Thus arose an appreciation of the root problem of temperament together with the concepts of scale and key, and the basic architectures of all melodic instruments. In turn, these tools have given us all of music.
Yet the overlay of history often applied to this simple story is perverse and skewed. It distorts the beauty of a narrative which is nothing less than transcendental. Today’s practitioners of temperament, the Temperamentalists, sometimes make a particularly crude blunder by translating today’s understanding of the physics of music, together with other aspects of our culture and values, several centuries backwards as though they have never changed. This has facilitated the generation of the quantity of material we have today on subjects such as “Bach’s Temperaments”. When assimilating such material it is therefore essential to understand this background before reaching a true verdict of its worth. This self-indulgent reverse anachronism on the part of some Temperamentalists is reflected in the title of this article.
The article is divided into four parts. Part 1 shows that, although the ancient Greeks are generally credited with much of today’s understanding of temperament, most of it was in fact well in place by the time of Pythagoras. Part 2 looks at some of the realities of musical life in Europe in the 17th and 18th centuries, avoiding the anachronistic translation referred to above. It relates the genesis of contemporary temperaments to the single fact that stringed keyboard instruments then had wood frames, with the consequential tuning instability. Part 3 next reviews some current work in temperament, demonstrating among other things a disgraceful shoddiness in much of it which is reflected in the level of numerical error. Finally, Part 4 considers whether impure octaves should form part of modern studies of early temperaments rather than being virtually ignored, and if so, how they should be approached.
Part 1 - History, Pythagoras and All
There is a breed of scholar turned out by university departments of the humanities which has apparently been trained to believe that it is OK for them to ignore things which do not interest them or about which they know little. Some historians are a case in point. It is incredible that a recent three-volume 1500-page “History of Britain” could have appeared which virtually overlooks the contribution of our scientists and engineers to national and world culture , just like the TV series which spawned it. I thought that history seen only in terms of dates, kings and battlefields is regarded as juvenile and superficial today, as well as boring and passé. In similar vein is the “Cambridge Biographical Encyclopedia” which devotes less space to Michael Faraday than to James Curley . And who was he? He was a mayor of Boston (Massachusetts). Hands up those who have heard of him. Both these examples illustrate a perverse view of the relative importance of past events and characters, indicating a need to look carefully and independently at codified history when forming judgements about earlier times.
Therefore, what role have historians played in the story of temperament? More to the point, what role did people such as Pythagoras play? Most people think they know how pivotal his contributions were – if indeed they were. But if historians get things wrong how, if at all, do they attempt to put them right? Indeed, what is history, and how should we regard it? How should we regard historians, who seem to be able to get away with scholarly blue murder when academics in most other disciplines would be rightly ridiculed by their peers? All these questions are pertinent to the consideration of the ancient subject of temperament, because I am going to tell the story in a way which leaves much of its conventional history looking a bit skewed. I am going to adopt the straightforward, though apparently unusual, approach of looking at the facts rather than myths, legends and received wisdom to see what they really tell us about temperament.
Pythagoras is widely quoted by Temperamentalists, among whom are a good number of historians, to the extent that people might be forgiven for believing he invented the whole subject of tuning and temperament. After all, what paper on temperament would be complete without a mention of the Pythagorean Comma (perhaps better called the Pythagorean Coma in view of the boredom it induces among the much-lectured). This article is no exception, but I shall mention it only once more a little later on – promise (although you might find that promises can sometimes be broken, even by me). But when we consider that Pythagoras probably had little or no connection even with the origins of his eponymous triangle Theorem, let alone with temperament, we have to consider what history itself means and how we should approach it. Without an understanding of how historians approach their calling, we cannot understand what temperament really means today, why so many people remain interested in it, and how to understand what it is really all about.
Pythagoras lived in the 6th century BCE (I use the term CE - Common Era - here to avoid a potential conflict with an unnecessary overlay of Christianity). Most will remember his Theorem from their schooldays, which states that triangles with sides of lengths 3, 4 and 5 units (or 5, 12 and 13, as well as others) are right-angled. These values also satisfy the equation which says that the square of the largest number equals the sum of the squares of the other two. It is important to realise that you do not have to know about the equation before you can understand empirically the right angled property of the triangles (some historians do not seem to have twigged this - Pythagoras’s Theorem is actually two theorems in one). In other words, the Theorem could have existed in an empirical and practical form long before algebra had been invented, and it almost certainly did.
How Pythagoras has come to be known as the discoverer of this truism eludes me. A mere couple of centuries later, Euclid does not mention him by name when treating the subject . One reason for this was probably the extreme secrecy and isolation in which Pythagoras worked. Outside a carefully-chosen few known as the Pythagorean Brotherhood, nobody knew of his discoveries and achievements. Members of the Brotherhood were sworn to secrecy, so much so that at least one of them was executed (by drowning) for breaking his oath. Charming feller, wasn't he, dear old Pythagoras? So how, more than two millennia after Euclid, can we associate Pythagoras so definitely with “his” Theorem today? Even if we were to grant, on the basis of non-existent data, that Pythagoras was one of several who discovered independently this property of three particular integers rather than merely retailing it to his disciples, there is now evidence that others had beaten him to it by a long chalk. For example, it is described on a Babylonian tablet of c. 1800 BCE, and other evidence suggests that Chinese and Indian cultures of about the same time were also aware of it. In fact, significant involvement with the problems of constructing reasonable-looking rectangular buildings, pyramids, etc must have led to an empirical understanding of the rule almost as night follows day – there is a high probability that it would have dawned in the mind of an experienced architect in any civilisation. If you can’t lay out right angles on your building site you can’t really call yourself a builder.
So if Pythagoras was not the first to discover the Theorem, or maybe not even among those who did, are we entitled to enquire why he is also credited with so much to do with temperament? Yes, we are, because the reasons have as much to do with myth and legend as anything else. One of the stories is that he, and only he, discovered the natural harmonic series merely because he happened to hear a blacksmith hammering bits of metal. Well, I ask you ..... The fact is that, like the Theorem, the harmonic structure of musical sounds was known to mankind long before Pythagoras (we know this from artefacts such as ancient Chinese flutes). But even ignoring the flutes, it’s pretty insulting to our forbears, when you think about it, to believe that they knew nothing about right-angled triangles and harmonics in music for the untold millennia before Pythagoras just happened along. Just like triangles, the notion of harmonics in musical sounds is something which any number of even half-perceptive individuals must have pondered on since man first walked the earth.
Why is this? Because you can hear them! Maybe historians can’t – I could not say - but most other people can. If you blow a pipe you can hear the octave above the fundamental note. The twelfth is even more obvious. As we are organists on this website, try wedging down a note around the middle of an organ keyboard with a pencil (carefully!) and then draw the Open Diapason. Walk slowly round the building. If you can’t hear the twelfth strongly in certain positions (where it is augmented by its own standing waves) I would be surprised. By blowing a pipe harder you can hear the harmonics even more clearly. Indeed, an open pipe will eventually overblow to the octave, and a closed one to the twelfth. A child could, and probably often did, do this, sitting in that incomparable Mediterranean sunshine all those centuries ago when there was so little else to do all day. A few of those children might have come to realise they had heard something remarkable in later life. That’s how many discoveries have happened.
Beyond the twelfth the harmonics become more difficult to detect by ear. But from the fundamental and these two harmonics above it, just these three notes, the entire structure of music and (particularly) Western harmony can be derived. It is a tremendous vista, a wonderful thing, and the reason why any musician worth the name needs to understand it. From just three notes has sprung our entire human musical experience, maybe mankind’s prime example of cultural unity in diversity. The two harmonics are the octave and an octave-plus-a-fifth (i.e. a twelfth) above the fundamental. In fact the fifth is so dominant both in its sound and in the theory of tuning that it became known as the dominant, and so it remains in today’s nomenclature. It was no great step beyond this to realise that exactly the same frequencies could be reproduced at will by giving an instrument several strings or pipes, tuned in octaves and fifths. When correctly tuned, the music they produced would sound the most euphonious and concordant – in fact, “in tune”. What does “tuning” actually mean? In those days it meant adjusting the tensions of two strings or the lengths of two pipes until there was no beat or wavering when they were sounded simultaneously. This was true whether the two notes in question were octaves or fifths (see Appendix 1 for an explanation of tuning using beats).
As time moved on it was realised that the gap between two notes an octave apart could be filled with eleven other notes, twelve if you count the bottom note as well. This too was probably discovered largely by serendipity, rather than being seen as a remarkable intellectual leap as some historians claim. Pythagoras is again credited by some of them with much of the clever thinking here and (as a deservedly famous mathematician) it is possible he did contribute to the outcome, or reinvent some of it. But he was not the first, because those Chinese had almost certainly beaten him to it again. It could have arisen as a result of adding more and more strings to an instrument, all tuned in octaves and (particularly) fifths of each other. Be that as it may, why this magic number twelve? Because the twelve notes are all fifths of each other. Fifths again!
We still use these self-same twelve notes in the octave today. To see this for yourself, sit at the piano and, starting at bottom C, ascend the keyboard by fifths. Thus you will play the following sequence of notes before you reach another C :
C, G, D, A, E, B, F#, C#, G#, D#, A#, F, and then top C
Rearranging them you will find – surprise, surprise - that you have played all twelve semitones of the octave:
C (twice), C#, D, D#, E, F, F#, G, G#, A, A#, B
You will get the same result by starting on any other note and ascending by fifths until you get back to it again – the starting note does not have to be C.
No wonder the fifth was regarded as dominant! It can generate all the notes of the chromatic scale we use today, including the octave, from any starting note. Thus the fifth is the generating interval which has created all melodic musical instruments and therefore all of music itself. In a sense, if you play a fifth you have played all music ever written. In recent times a mathematician was so intrigued by the properties of Mandelbrot’s fractals that he used them to regenerate an entire Beethoven symphony from its fractal representation. What a pity he apparently didn’t know he could have done this just by playing a fifth.
This spine-tingling, amazing discovery (and it is amazing still when it first dawns on you, however it might have arisen), this seemingly magical property of the fifth to generate all other notes, is one of those transcendental, golden achievements of mankind’s collective intellect which reinforces our belief that the universe is fundamentally simple in the sense it can be understood. In the case of the Greeks it also led to all that Byzantine mythical mumbo-jumbo which usually clouds discussions of temperament, such as the divine ratio of 2 to 3, the music of the spheres, the purity of integers, the relationship of music to the gods, etc. Although all of this did indeed exist at the time, it can be largely swept aside for present purposes and left to the historians and the classicists to pick at. But one problem could not, and cannot today, be ignored. This concerns tuning purity.
If all the fifths are tuned true or pure, in other words if you tune them so the two notes in each fifth do not create a wavering beat, the topmost C in the piano experiment above will be significantly out of tune with the bottom C you started from. It will be painfully sharp. To get it back into tune, you have to flatten some or all of the intervening fifths just a little, so that when you reach top C it is exactly in tune – exactly seven octaves higher in fact - with bottom C. Therefore, those fifths which were flattened are now impure – you can hear a beat between the two notes. The difference between the out-of-tune top C and the in-tune top C is called the Pythagorean Comma. Having arrived once again at this orgasmic climax of the discussion I promise not mention it again (maybe), together with other unhelpful terms such as Lesser Diesis, Syntonic Comma, Schisma and Greater Diesis. Together with somebody else called Didymus whom we’ve never heard of and don’t want to either. Those Temperamentalists who delight in immersing themselves in such argot today can nevertheless fall prey to the most elementary mistakes in arithmetic, some examples of which I will give later on. Although it’s funny to see them tripping themselves up, it’s actually pretty serious.
The ancients strongly disliked this tuning impurity of the fifths because it went against their mystic ideas. If you believed that your favourite god was up there on the nearest hill listening to the music you were playing, you were going to be jolly careful that you didn’t offend him or her. The last thing you would want to do would be to deliberately choose some hideous irrational number to represent the frequency ratio of a flattened fifth, rather than the primitive integral beauty of 3 to 2 for a perfect one (the fundamental frequencies of the notes forming a purely tuned fifth – no beats - are in the exact ratio of 3 to 2). So they decided to make music using fewer than twelve notes to the octave, even though they knew all about the twelve note scale. Hence those boring and funereal old modes which characterised music until about a millennium ago, at least in Europe. The restriction to 6 or 8 notes to the octave made modal music-making pretty simple during all that time.
But for some reason the Europeans, uniquely and awkwardly among all musical cultures, suddenly started to invent the rules of harmony – they wanted to be able to play several different notes simultaneously, and they wanted less and less restrictions to be placed on the notes which could be used. So they dusted off the old idea of a twelve note scale. Instead of modes, they also invented the concept of key along with that of harmony, and this meant that they wanted to be able to play chords in any key they chose. This brought them up against the old problem of how to tune the twelve notes of the octave so that all keys could be used, which is the root problem of temperament. “Tempering” the fifths, or flattening some or all of them so they will all fit into the octave, is what temperament is all about. And having approached the subject in forward time, we shall shortly jump to the present day and look backwards. But before doing that it is worth saying a few words about the truly great gift which the ancient Greeks, Pythagoras included, gave to us.
We have already touched on the mad antics of their gods and the notions of purity and correctness which pervaded their culture. Yet at the same time the gods did not seem to mind the Greeks indulging in any amount of philosophy about life, the world and everything. This laudably liberal attitude on the part of their otherwise idiosyncratic deities was remarkably unlike the attitude of the Christian Church and some other religions closer to our time. So we find that the notion of philosophy itself was born around the time Pythagoras lived. It could be said that philosophy was invented by Thales of Miletus in Greece. He taught that the Earth was a flat disc floating in an infinite sea, and that everything arose from this watery medium. What is significant about this is that it was a theory which offered an explanation, however bizarre we might think it today, for the otherwise mysterious and inexplicable things which we all perceive as part of our being and of living.
Thales’s idea was not a religion, relying on divinely revealed wisdom. It was purely an intellectual construct of the mind which could be tested and, if found to be wanting, it could be discarded. In parallel, the work of mathematicians such as Pythagoras and (later) those with a more experimental bent, such as Archimedes in nearby Sicily, allowed this and other theories to be tested against observable facts, against discovered truth as that polymathic thespian Stephen Fry called it in a memorable lecture to young minds a few years ago . The Greeks realised that if a theory was disproved by just one observation, it was disproved for ever. It could no longer exist. Conversely, a theory could never be proved either. It would remain on the stocks just as long as it wasn’t disproved. These fundamental tenets of philosophy and science have remained unchanged ever since, and they have stood us in good stead.
Thus, during a golden age lasting only some 500 years, the Greeks and their immediate successors laid the foundations of our modern systems of thought which are still used unchanged today, a tremendous legacy of intellectual freedom which later threatened to undermine those who saw their future in a different direction. Among these was the Church. For about the first 1500 years CE it displaced the notion of discovery by priestly revelation, and it was not until this stranglehold began to disintegrate a few centuries ago that the next phase in the story of temperament really began.
Before proceeding, what have we covered so far?
1. A single musical note has a natural harmonic structure which can be heard.
2. The first two harmonics above the fundamental note, the octave and the twelfth, can be given a separate existence by tuning separate strings or pipes to the same frequencies.
3. The twelfth is particularly important because it is a fifth above the octave.
4. Using only the interval of a fifth, the octave can be spanned by twelve different notes, all generated by fifths.
5. But if derived from purely-tuned fifths, these notes do not quite fit exactly into an octave.
6. Therefore some or all of them have to be squeezed or flattened.
7. The choice of which fifths are flattened, and by how much, is the study of temperament.
Part 2 - Looking Backwards
In the previous section we got roughly as far as the 16th century by travelling rapidly forwards in time from the era of the ancient Greeks (nothing much happened for the first 1500 years or so after BCE became CE, so we skipped over it). Now we shall look back towards the same period from today’s vantage point.
Why look backwards? Because the study of temperament today is almost invariably backward-looking. What Temperamentalist writes about anything other than the temperaments allegedly used by Bach or his contemporaries, for example? More than a few Temperamentalists seem to want to make a name for themselves by propounding increasingly fantastic hypotheses about subjects such as the temperaments favoured by Bach. It is fertile ground for those with the inclination, simply because much of what they claim is, and will forever remain, unproveable. The big problem in temperamental retrospection of this kind is that it almost invariably overlooks the differences between life today and life three hundred or so years ago. Thus the work is by definition anachronistic. It is out of time, studies not so much of temperament but examples of reverse anachronism. This sort of Temperamentalist transplants today’s understanding, culture, values and capabilities several centuries backwards as though these things have not changed in the intervening period. Reading some of this material suggests to me that the authors imagine that 17th or 18th century theorists, musicians and composers understood the physics of music just as we do today, that they had electronic tuning aids, and that they had no difficulty in performing calculations of the most involved sort as though electronic calculators were the norm. Such assumptions are of course grossly false. These authors either do not know their history at all, or they have only absorbed that regrettably skewed version of it which ignores the awesome leaps made in human understanding over this period. So let us have a peek at what life in Europe was really like at the end of the 17th century when Bach was becoming active as a musician and composer.
A big issue was that life was still pretty much dominated by the Church. In Bach’s Germany and in Britain that meant Protestantism. And it wasn’t the mellow experience we get today when we go to church, either. The Protestants had not long stopped burning Roman Catholics in village squares, leaving those who lived in nearby houses to scrape the human grease off their walls before they became alive with maggots. Also witchcraft was an ever-present fear among the population, with the last documented witch execution still to take place in Germany in the 1770’s, over twenty years after Bach’s death. Officially-sanctioned torture of suspected witches was not abolished in Bavaria until the early1800’s.
In such times you would therefore have been well advised not to get on the wrong side of the Church or the civic dignitaries. Although the political clout of the Church at national level was on the wane, the local clergy could still wield a lot of influence when a young man wanted a job, particularly a job as a church or Court musician. Physically trapped within the insularities of village life in an age without means of transport beyond your own legs or the horse, you would generally not want to attract attention by doing anything out of the ordinary, such as being seen to take an excessive interest in anything at all. Made wary by the religious wars which raged in Europe in the 17th century, even Descartes said that "to live you must live unseen". Therefore, dabbling in science (an anachronistic term in itself because the word did not gain its current meaning until the 1830’s) would have been a definite no-no. By and large, the ancient Greek concepts of unfettered intellectual freedom had been forgotten for more than a millennium because of religious suppression of the most vicious kind. Isaac Newton did not go up to Cambridge until 1661, only 24 years before Bach was born, and even he had trouble with the university because he was reluctant to accept the Doctrine of the Trinity. What was not to be found in the Bible was potentially dangerous, associated only with reactionaries.
So are the Temperamentalists really expecting us to believe that organ builders, organists and composers routinely sat up nights at a window by the light of a guttering candle doing sums about tuning? Would that have fitted with the contemporary notions of good citizenship? Was there a danger that it could all too easily have been misinterpreted, maybe as witchcraft, if somebody else caught sight of all that spidery scrawl and countless numbers? Page after page after page of it would have been necessary to develop a new temperament. If floods or other pestilences descended on a village, would not such people have been regarded as architects of these misfortunes? I do not know the answers to these questions, though there is a certain plausibility about a scenario which suggests that on the whole they would not have done these things. But I do know that, apart from a minority, they could not and would not have done them for other, very good, reasons. That minority would either have been extraordinarily determined, or it would have possessed knowledge and capabilities well beyond the average, or both. Let us look first at what was known about what we would call the physics of music (though they would certainly not have called it that, again because our term ‘physics’ was not defined until the 1830’s).
At the most basic level, we can ask whether there was even a general awareness in those days of the concept of frequency, in the sense that a string vibrating a certain number of times per second would emit a note of a certain pitch (we ignore here the difference between objective frequency and subjective pitch). Of course, it would have been obvious that a string did vibrate transversely while it was emitting sound, because of the tizz it made if you brought your finger nail up against it. But was there any realisation that a string vibrating at middle C vibrated with exactly twice as many oscillations per second as when it was sounding tenor C? If so, how would they have known this for sure? This is not mere pedantry, because exactitude necessarily lies at the heart of tuning and temperament as we shall see. There was no possible means of rendering the detailed motion of the string visible, nor of measuring frequency. The numerical exactitude required in temperament studies is around 1 part in 100,000 or 0.001% (see Appendix 2).
Note that I said "general awareness of the concept of frequency" above, meaning that the majority of musicians would probably not have given these matters a second thought. This does not mean that the knowledge did not exist at all, because some texts from the mid-17th century mention it  and I am grateful to the harpsichord maker J-P Baconnet for suggesting I point this out . But my point remains : it stretches credulity to believe that these matters would have been taught routinely as part of the general education of the day, whatever that might mean. Even in Britain before the second world war hardly any science was taught in the State school system as my late father often reminded me, and he had won a scholarship to a relatively élite school!
Notwithstanding all this, musical instruments still had to be tuned, and tuning at that time would have been carried out by counting beats as it often is today. No problem there then, provided you had a source of portable time. By this I signify some means of measuring beat rates at the keyboard, expressed as (for example) 5 beats in 7 seconds. But what form would such a source of portable time have taken? A wristwatch? Hardly, considering they did not become available until the end of the 19th century. What we would call pocket watches had been around in some form since the 1520's but they were generally crude, expensive and relatively rare. 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 quite exquisitely beautiful pocket chronometer until the 1760's, by which date Bach had died.
At the time we are considering it is more likely that most clocks used pendulums and were therefore not portable. The timekeeping property of a pendulum had been discovered by Galileo around 1580, but it was not until Newton had explained fully how it worked that the pendulum escapement became a domestic commonplace a century later - around the time Bach was born - in timepieces such as the long case (Grandfather) clock in Britain. Some 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. However, my favourite conjecture is that musicians 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 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. My first teacher in the 1960's used one frequently in the organ loft). Such a musician's pendulum in the 17th century or so 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 rates of the fifths of your favourite temperament .
There is also another aspect of the matter. If measuring beat rates was in fact regarded as more difficult than I have assumed here, then, obviously, the more pure fifths in a temperament the better from the point of view of ease of tuning. It is much easier to tune a fifth pure than to temper it, because when in tune it has no beat. The unequal temperaments which sprang from the period we are discussing frequently had many pure fifths, such as Werckmeister III (c. 1690 - 8 pure fifths out of 12), Vallotti (c. 1730 - 6 pure fifths out of 12), etc. Setting such temperaments by ear, especially for an amateur tuner, is much easier than setting Equal Temperament, because ET contains no perfect fifths and all twelve have to be accurately tempered. Anyone who disagrees will need to explain why professional tuners today frequently do not tune by ear but instead use electronic devices when tuning organs or pianos. Maybe this humdrum matter contributed to a preference for unequal temperaments until instruments with better tuning stability were developed in the 19th century, an issue to which I shall return presently.
But returning to the physics of music, was it understood what a “beat” was, in the quantitative sense of it being a measure of frequency difference? And a difference of which frequencies exactly? How many Temperamentalists even today understand why the interval of a fifth creates a beat when it is not exactly pure? The fundamental frequencies are grossly different even when the interval has been tuned pure so that there remains no beat – the frequencies of a pure fifth are in the exact ratio 3 : 2. Why is this? How can there be no beat when there remains such a large frequency difference, yet why does a beat appear when this massive difference is made a tiny amount smaller or larger? In a recent paper Lehman attempted an explanation by stating that the “upper harmonics of both notes have a frequency that is almost identical, at some point several octaves above the fundamental” . This statement is incorrect. Clearly, then, these are matters which tax the understanding even of some specialist writers today, so are Temperamentalists telling us that such knowledge was widespread among the general musical population in Bach’s day? (The answers to the questions just posed are in Appendix 1).
Then there was the sheer problem of doing arithmetic. We have already noted that the arithmetic involved in tuning and temperament calculations has to be done to a precision of at least 1 part in 100,000. The necessity for this is explained in Appendix 2. In other words, the calculations have to be done using six significant figures. How would multiplication and division of two 6-figure numbers have been done in those days? By long multiplication and division of course. It hardly seems necessary to add that calculators were unavailable! Logarithms had already been invented, and these could in principle have eased the task somewhat. Although log and antilog tables were available (though how widely available in your average village I do not know) they were full of errors, literally thousand upon thousand of errors, which is one reason why ships often had no idea where they were (the other main reason was the impossibility of keeping accurate time on board). Were boys such as the young Werckmeister and the young Bach taught logarithmic methods of doing arithmetic? Even if they were, with the errors in the tables it would have been a pretty fruitless endeavour as far as temperament was concerned. Various types of slide rule had recently (c. 1660) been introduced, based on logarithmic methods of doing arithmetic, but the precision offered was not sufficient for the necessary calculations involved in nuances of tuning. A vast variety of “ready reckoner” tables was later published, but not until the 18th century was well advanced and by then Bach had died. Even then, they were riddled with errors just like the earlier log tables. In fact the problem of error-ridden typography was the main driving force behind Charles Babbage’s unsuccessful endeavours in the 19th century to produce a mechanical computer – one of its key features, on which Babbage insisted, was that it had to be able to print its results automatically and thus not rely on a human type setter and proof reader.
Let’s look at an example of Baroque arithmetical problems. The Pythagorean Comma (oops – I’ve broken my promise – here it is again) can be represented as the ratio of the two repellent numbers 531441 and 524288. Notice they both have those six figures which we mentioned earlier. Although these numbers would have been known to the clever minority we spoke of above, how would this quotient – the answer to the division - have been arrived at in the 17th or 18th century? Such a calculation typifies those involved in the arithmetical study of temperament. Still, having done such arithmetic at junior school in the 1950’s, I know it can be done and how to do it. I also know how error prone the results can be, and a rap on the knuckles with a ruler was not unknown in those days if you got your sums wrong too often. (How things have changed, and for the better!). The method of long division by repeated subtraction is demonstrated in Appendix 3 for the benefit of those who are curious to see how sums were done in pre-calculator days. I do, however, question whether doing such sums was a favourite way of whiling away the evenings in the 17th century. Even with the benefits of today’s computers, calculating aids and computerised typesetting, modern Temperamentalist literature is (disgracefully) still riddled with numerical errors, examples of which I shall demonstrate presently. What it was like then can well be imagined.
Yet the extraordinary paradox is that musicians in those days probably had a better intuitive feel for tuning and temperament than their brethren do today. This is largely due to the single fact that the stringed keyboard instruments used for general music making, such as the harpsichord, had wood frames. In fact the harpsichord case itself had to withstand the combined pull of the many iron and brass strings – there was no frame at all as such - and the fact it sometimes did not succeed is shown by some surviving examples where the case has virtually imploded. Today’s piano has a massive cast iron or welded steel frame (early ones did not) to resist the tension of the strings, which is why it is so heavy. The rigidity of the frame endows it with a tuning stability that people would have given their eye teeth for three hundred years ago. Music can be played quite happily on a modern piano, even a cheap upright, which is tuned only once or twice a year, whereas the tuning of instruments with wood frames was much less stable. Tuning might have been required several times a week in some cases, and the only way to do it was for virtually every village player to learn how to tune as part of his or her music lessons, just as today’s string players still have to do.
I cannot believe that tuning would have been regarded as anything other than a chore, to be undertaken before the day’s music making started perhaps. It would have been carried out as quickly as possible, and doubtless it was often slapdash to some extent. Often I imagine it would have been skipped altogether. Thus tuning errors would have abounded, the only criterion of whether a tuning exercise had been successful being whether the latter state was better in some sense than the former. But, surely, herein lies the clue we need to understand why so many temperaments discussed today sprang from that period. The argument has two strands: firstly, we do not need to waste time discussing here whether “good” temperaments were sought because they obviously were. By a “good” temperament I mean one in which most if not all keys can be used without gross dissonances. Bach’s ‘48’ is probably the best known example demonstrating the hunger for instruments which can be played in all keys. But please, please bear in mind that there are lots of “good” temperaments. The term does not solely mean Equal Temperament, and in any case some today would deny that ET is “good” anyway. Maybe that was true then as well. I do not intend to get dragged down this side alley here. Consult the writings of almost any Temperamentalist for more on this.
The second strand to the argument is that, because of the probably slapdash tuning or its complete omission, occasional happy coincidences would arise in which a particularly euphonious result was achieved in a particular piece of music. Yesterday it might have sounded terrible, but today it sounds much better because of the hurried tuning done this morning, or because we didn’t bother to tune the damn thing at all and it just happened to have been very cold last night. Noting the felicitous result with the piece we have just been practicing, a few other pieces in other keys would quickly be played. If the results remained good, someone with the sharp wit and intellectual mightiness of Bach could well have paused to write the tuning down. And how would he have done this? Just like today, he would probably have run through all the fifths and fourths (fourths are inverted fifths) over the middle octave or two or three, counting the number of beats in a given time interval such as ten seconds, and writing them down. Or he might have used some form of shorthand to indicate whether they were flatter or sharper than those which had actually been intended when he last tuned the instrument (though having counted the beats anyway, I suspect he would have simply have written the numbers down unless there was a need for covertness).
I have had the good fortune to discover a “good” temperament this way, merely by playing a village organ which by chance had not been tuned for a while. I called it “The Dorset Temperament”, and both it and the experience which led to it are described elsewhere in the Temperamentalist literature and on this website . As a physicist, I hope you will not regard me as immodest if I claim some familiarity with mathematics and number manipulation. Yet, like most other people I suspect, the thought of trying to sit down and design a new temperament from scratch by doing sums seems not only repulsive, but completely the wrong way round to do the job from an intuitive, musical, point of view. I suggest most erstwhile Temperamentalists three hundred years ago would have felt the same, but even more so.
If this argument is accepted it means that a number, maybe the majority, or even all “good” temperaments would likely have arisen largely through serendipity. It is just too implausible to believe that the inventor of a temperament first sat down and did lots of sums before trying it out for real. In reality it would have been the other way round – the temperament appeared one day as if by magic, and it was then analysed retrospectively by those with the inclination. This does not mean that an interesting tuning arrangement which arose by chance would not have been tweaked subsequently after doing a few sums. But lest this seems to denigrate the capabilities of those who have donated their names to their eponymous temperaments, bear in mind that much of today’s science is still done this way. Something is observed, maybe by chance (such as the discovery of the bactericidal action of a mould which turned out to be Penicillium), and the explanation is sought later.
The same thing happens routinely in the natural sciences such as physics – experimentalists do an experiment which occasionally might have a completely unexpected outcome, and then the theoreticians get going to explain why. Seldom do the processes of science take place the other way round (one of the few counter-examples being the theory of black holes in cosmology, where the theory was developed in detail over about two hundred years before the observational evidence of recent times was obtained). Why should this not have been the way in which “good” temperaments were discovered – serendipity followed by theory? Alexander Fleming, the discoverer of penicillin, said some memorable things when looking back to that day in 1928 when some mould or other just floated through the window of his laboratory. Two of them are “if my mind had not been in a reasonably perceptive state I would not have paid any attention to it”, and “before you can notice any strange happening you have got to be a good workman, you have got to be a master of your craft” . Nothing better epitomises my view of the serendipitous way “good” temperaments arose than these words: they could just as easily have been written by Bach himself.
There is an amusing sequel to this story which might have relevance for historians studying temperament. When writing up the results of their work, some scientists find it difficult to admit that they stumbled across their results by accident, so they write their paper as though they were terribly clever and worked up the theory first, testing it by experiment afterwards. Some modern historians seem to take much 17th and 18th century work on temperament at face value – I feel they do not analyse the situation deeply enough, and with sufficient understanding of the real process of discovery. It is generally more plausible to accept the serendipity-followed-by-theory sequence rather than the reverse, despite what the old texts might imply.
Interestingly, there can be no argument that an exact reversal of the hypothesis (that instruments with unstable tuning led to the discovery of "good" temperaments) occurred later on. In the 19th century the universality of Equal Temperament came about precisely because of the dominance of keyboard instruments whose tuning was extremely stable. These were the free reed instruments such as the concertina, the accordion, the harmonium and the American Organ. Unlike organ pipes, the tuning of free reeds is virtually unaffected by temperature and wind pressure, indeed they were often used as frequency standards in Victorian laboratories. The rapid spread of such instruments throughout the world was facilitated by the various European empires which then existed, and therefore the instruments took Equal Temperament along with them. Both examples, that is, instruments whose tuning is either stable or unstable, resulted in the propagation of different sorts of temperaments quite independently of what the theorists of the day might have said or thought. On the whole musicians will simply play and compose for whatever instruments happen to be around, and their music reflects this. They seem to take little notice of the Temperamentalists wittering away in the background. Whatever our views about Equal Temperament, we can be grateful for the music it has given us from the 19th century.
However, we digress. Reverting to unstable tuning and going a bit further, a by-product of the serendipity process might have been that some “good” temperaments included impure octaves. It is unclear to me why the octaves should be any more sacrosanct in the regime of controlled detuning which is temperament than any other interval. Yet what well known temperament today actually uses impure octaves? If the scenario of frequent and slapdash tunings which I have postulated is correct, it is likely that euphonious results were occasionally obtained in which at least some of the octaves were impure – and most likely sharp. I shall return to this possibility later.
So at the end of part two, what are the main conclusions?
1. Modern temperament studies are necessarily retrospective, but often with insufficient heed paid to the circumstances of life at the time studied. In this sense they can be anachronistic.
2. The influence of the Church would have made people more wary of indulging in free thought and research than we take for granted today.
3. In earlier times there were not the arithmetical tools available, nor the depth of knowledge about the physics of music, to enable the quantitative analysis of temperaments to be made as readily as today.
4. Nevertheless there was probably a better and more widespread qualitative feel for temperament and tuning issues than today, largely because of the notorious tuning instability of stringed keyboard instruments which required most players to be able to tune them.
5. The need for frequent retuning might have led to the serendipitous discovery of some or all of the temperaments we associate with that era.
6. Routinely developing a new temperament by the reverse process of first doing the theoretical work would probably have been rare, despite the implications contained in some contemporary publications from the period.
7. Some temperaments might have contained impure (probably sharpened) octaves.
Part 3 – Temperament Studies Today
Apart from the issue of reverse anachronism, it is worth taking a wider look at some modern work on temperament. It seems to be one of those subjects which attracts more than its fair share of armchair experts if the contents of Internet chat lists is anything to go by. These have done no actual work in the subject at all in the sense of having published anything. This means they do not distinguish between erudition and scholarship, and a large proportion of them also believe that invective is an acceptable substitute for either.
But of the work which is published, probably the most widespread and least excusable matter is that too much of it is peppered with arithmetical errors. For example, Kent and his publisher (CUP) apparently believe that 1200 x 2 + 386.14 equals 2786.314 . Like many other authors he also draws heavily on data from Padgham’s treatise on organ tuning , and in view of the errors in the latter it is to be hoped that he verified them independently. For example, there is no way that the Werckmeister III temperament could be set up using Padgham’s numbers. Similarly, Padgham’s table on Just Intonation contains a misprint that should have been obvious to even the most casual proof reader. Throughout his book are innumerable examples of varying arithmetical precision, revealed by unexplained (indeed, inexplicable) differences in the numbers of significant digits. You cannot do meaningful temperament research unless you work to at least 6 significant figures throughout (see Appendix 2 to this article).
Continuing, Lehman  states that “two consecutive 5ths of equal size ... have beat rates in a 3:2 ratio”. He quotes the example of the interval G-D which includes middle C and the interval D-A above it, though "adjacent" might have been a better adjective here than "consecutive" (which has confusing associations with harmony). The meaning of the terms "beat rate" and "equal size" is also unclear. For the former, perhaps Lehman meant "beat frequency" and we shall assume here that he did. As for the latter, if these two "equal size" intervals were tuned to Equal Temperament (in which they are both equally flattened from pure) the ratio of their beat frequencies would be 1.49849, which differs from his value by slightly more than 0.1%. Such differences are highly significant in the context of tuning and temperament, where we have already seen that the difference between two numbers can only be ignored if it is less than 0.001%, one hundred times smaller than the percentage error here. If both “equal size” intervals were to be tuned pure, which they might be in a number of actual temperaments, there are no beats at all. This would lead to a ratio of 0 divided by 0 which is a mathematically indeterminate quantity, very different to Lehman’s 3:2. Thus we have been badly cast adrift here in an arithmetical sense.
These are just a few examples. I do not have the time nor the inclination to act as honorary proof reader for all of the myriad publications I consult when doing theoretical work on temperament, thus they are merely some of the examples I have come across by chance. More than once I have wasted hours tracking down arithmetical problems which turned out not to be mine (my first thought) but due to errors in published data, including one of the cases above. The examples quoted are all relatively recent, and two of them are scarcely from what one would normally regard as twopenny-halfpenny publishers. So if the quality of the data is as bad as it obviously is, what is the point of publishing it? Do the authors themselves actually use their own numbers for anything beyond conflating their publication tally, or is it perhaps a case of “see how clever I am”? If the latter, it is in reality a case of “see how good I am at digging traps for myself” I’m afraid.
I have mentioned earlier the effort which goes into trying to prove which was Bach’s favourite temperament. One of the most memorable (to my mind) papers was published quite recently. Entitled “Bach’s extraordinary temperament – our Rosetta Stone” , I have already cited two erroneous statements from it.
But the main thrust of the paper is that the cypher used by Bach on the title page of his Das Wohltemperirte Clavier (above) is no mere squiggle but encodes the temperament he wanted to be used when playing these pieces. Despite the inconvenient facts that the squiggle apparently has to be turned upside down before it can be analysed, and that no comparable method of encoding temperament information was apparently used again by Bach or anybody else, the author claims that its meaning was so obvious that it was used by the Leipzig authorities when assessing Bach in his absence for a teaching position as a successor to Kuhnau. So obvious, in fact, that they would have tuned a harpsichord to this temperament before trying out his pieces in all keys which were part of his submitted CV for the job. Unfortunately not a shred of collateral factual evidence is presented to support the hypothesis though. Surely if Bach wanted to encourage the use of a particular temperament when playing these pieces he would have set out its characteristics in a less equivocal manner? I wonder if the author was conscious of the inevitable comparison he invites by his choice of title between his capabilities in decryption and those of Thomas Young and J-F Champollion?
Taking a more charitable view of the idea there is, however, a further opportunity for determining the hidden meaning of the cypher which the author of this paper missed, and that concerns the terminating flourish at the right hand side. It was added deliberately in the sense the pen was first lifted from the paper before drawing it. If there really is the amount of information to be extracted that he maintained, perhaps he might have regarded that flourish as evidence that Bach stretched the octaves of his temperaments, a possibility not discussed in his paper and one to which I shall now turn.
Part 4 – Impure Octaves
As a rule Temperamentalists ignore or dismiss the possibility that the octaves might play a greater role in their subject than merely marking the boundaries between successive sets of twelve tempered semitones. Most of them never even mention it as an option; they proceed as though pure octaves are axiomatic and always have them tuned true. Inevitably, this leads to a subjective tuning rigidity across the compass of the keyboard of an instrument, regardless of the temperament to which it is tuned. The tempering of the intervals in every octave is the same, and every note is tuned true with its octaves above and below. The results are legion.
For example, the beat rate of any interval played depends on the octave in which the interval resides. In other words, a fifth played in the third octave will beat faster than if it is played in the second octave, but slower than if it were to be played in the fourth octave. With any temperament which uses pure octaves, the ratio of these beat frequencies has a simple numerical relationship to the octaves considered – a fifth in the third octave beats exactly twice as fast as when it is played in the second octave, four times as fast as in the first octave, and so on. These exact and simple integral beat frequency ratios also apply to any other interval, no matter how finely they might have been mutually adjusted within each octave by adopting a particular temperament. With a recently tuned instrument in which all the octaves are well in tune across the whole keyboard, this can lead to a hard, sterile locked-up type of sound when chords are played which span a significant part of the compass. Not only are there no beats at all between the octaves, but the beat rates between similar intervals in different octaves are related by exact integer ratios. The sterility only recedes when the tuning of the instrument drifts over time. The subjective effect can be even more noticeable and unpleasant in electronic instruments if unintelligently tuned, because their tuning never drifts once set. Again, remember that the effects we are discussing are independent of the temperament actually used; they follow purely because the octaves are locked in frequency.
Subjectively, the effect is worst for the organ; less so for stringed keyboard instruments. This is because of the different overtone (partial) structures in the two cases. The overtone frequencies of organ pipes have an exact integer relationship with each other: the second harmonic of any pipe, sounding the octave above the fundamental, is at exactly twice the frequency of the fundamental, the third harmonic (the pure twelfth) is at exactly three times the fundamental, etc. All the harmonics of an organ pipe are rigidly locked in phase with each other while it sounds. Using the terms properly and rigorously, this is why the overtones in this case can be called harmonics. The situation described only pertains when the pipes are sounding in their steady state speaking regime after the attack transients have died away, which of course occurs relatively quickly. Therefore, if an organ is well in tune so that the octaves are as exact as possible, the sterility both of the octaves and of the beat ratios between octaves is amplified subjectively by the lack of numerical freedom in the frequency ratios among the harmonics of the pipes themselves. The mere fact that the sounds of organ pipes do not die away until the keys are released adds yet further to the potential subjective hardness of the overall effect of an organ with well tuned pure octaves.
Subjectively, the effect of stringed keyboard instruments does not suffer as badly as that of the organ even when they are tuned as well as possible. The main reason is that the harmonics of a struck or plucked string (not a bowed string) are actually not harmonics at all; they are overtones because their frequencies are not exact integer multiples of each other. As the sound of a struck or plucked string dies away, the overtones beat with each other because the higher ones are slightly sharp to those lower in frequency. (This also happens during the attack and release transient phases of organ pipe speech, but because these phases are of such short relative duration the effect is dominated by the steady state phase in which the overtones are true phase-locked harmonics as described above).
Another disadvantage following from the use of pure octaves is that an opportunity is missed to ease the tight straitjacket of conventional temperament work. The root problem of temperament is to squeeze that uncomfortable set of bedfellows called the semitones into an octave in such a way that none of the intervals between them is grossly out of tune. This is done by making small adjustments to their frequencies (e.g. making the fifths flat, the thirds sharp, etc). Why not ease this problem a little by making the octaves themselves adjustable as well? Sharpened or “stretched” octaves are nothing new in keyboard music, of course. Many piano tuners routinely stretch the octaves when tuning, though the reasons quoted vary. Some maintain that the change in frequency of the partials during the decay of the sound is less dissonant (when several notes have been keyed) if the octaves are tuned slightly sharp towards the top of the keyboard. Others say that it is better to tune the octaves sharp so that they will in time come better into tune as the string tensions relax slightly. Of the two reasons I incline preferentially to the common sense nature of the latter. Other reasons for stretching the octaves also exist. At the other extreme though, some tuners will have nothing to do with it and one well known book on piano tuning does not mention it at all . So it seems we can learn little from the piano scenario.
Yet impure octaves would without doubt have occurred if we accept the notion described earlier in which “good” Baroque temperaments arose largely through serendipity because of the tuning instability of the old stringed keyboard instruments. Therefore I plan to investigate the matter in more detail, specifically for the organ. Currently I have yet to decide on a definite road map for the study though. Doing it with the degree of emphasis on arithmetic and theory which constitutes current work on temperament is almost certainly debarred. It is debarred because pure octaves underpin the entire concept of temperament as it is understood today, and removing them will also remove the relative arithmetical simplicity of the subject. If the octaves are no longer pure, the subject could easily become theoretically anarchic and entirely experiential. Any note on the keyboard could in principle take any frequency value, and the frequencies actually chosen would then arise solely through empiricism – trial and error.
To prevent this unpleasing prospect developing, it will probably be necessary to impose a deterministic progression of octave sharpening across the keyboard. As an example, a temperament would be set for the lowest twelve notes, say, then the successive octaves above each one could be sharpened progressively to generate its upper brethren. Or the generating temperament could be set in an octave closer to the middle of the compass and then propagated in both directions, up and down, according to certain rules. Each octave could contain a completely different temperament in principle, though in practice it will probably be better to regard the entire keyboard as a collection of notes upon which the notion of a distributed temperament is to be imposed. The availability of computers with electronic keyboards connected to them makes the execution of such a study not only precise but rapid and interesting, as the results of any algorithm for distributing the note frequencies can be immediately assessed by ear. And that, at the end of the day, is what it is all about.
Currently, I have inserted some impure octaves into the tuning of the organ pictured on the home page to see whether I can live easily with them. Otherwise this instrument is tuned to my Dorset Temperament.
Since this article was written I have now posted another which reports in detail the results of further research into temperaments with impure octaves, tuned both sharp and flat from pure .
Appendix 1 – Tuning Unisons, Octaves and Fifths by Beats
Why do we perceive a beat – a wavering effect – when we are trying to bring a pair of pipes or strings into tune? It is more complicated than most people think, and I have yet to see a proper explanation of it in print. The following is that proper explanation.
To begin the explanation, consider a case where two pure tones – sine waves – are sounding simultaneously. The combined waveform is merely the sum of the two individual sine waves, so it can be written as:
sin 2π f1 t + sin 2π f2 t
where f1 and f2 are the frequencies of the two tones in cycles per second or Hertz (abbreviated as Hz), and t is time in seconds. π (pi) is the Greek symbol having the value 3.141593 (to a precision of 7 significant figures).
The diagram below is a plot of this equation for frequency values 588.00 and 587.32 Hz. The term SPL stands for Sound Pressure Level for the sound wave in the air. I’ll explain presently why these apparently outlandish values of frequency were chosen.
You can see immediately that there is a slow modulation of the strength of the sound, which rises to a maximum before falling momentarily to zero. This cycle is repeated, leading to the phenomenon of what we call beats. The beat frequency equals the difference between the two frequencies f1 and f2 , so in this case the beat frequency is 0.68 Hz. Hertz is another name for cycles per second so the reciprocal of this figure, 1.47 seconds per cycle, shows that each beat cycle takes place over 1.47 seconds which is a little under 1 ½ seconds. This is a very slow beat, characteristic of the beat rates of the slightly impure intervals used in temperament work. The pitch of the note heard is that of treble D (an octave plus a tone above middle C).
But in fact this beat waveform is not at all like the beats you actually hear when tuning organ pipes, and there are two reasons for this. Firstly, the sound of two pipes beating does not actually go through zero once each beat cycle as it does in the example we see here. This is because the sounds of the two pipes are mixed in the listening room, and this mixing process is far more complex than simply adding the two sine waves together which we did here to generate the waveform. The natural mixing process involves a huge number of additions due to the huge number of reflections in the room, and this results in a less pronounced beat. Although there would still be a beat, it would not go through zero because the multiple reflected sounds would prevent complete cancellation at any point in the beat cycle.
The other reason why real beats do not sound like this example is because the sound of any organ pipe does not consist of just one harmonic – a single sine wave – but of several harmonics. This is illustrated below for a Stopped Diapason pipe.
The Frequency Spectrum of a Stopped Diapason Pipe
The vertical lines in this diagram indicate the strength in decibels of each harmonic (you can ignore the fact that decibels are used, and that the harmonic lines are not equally spaced along the horizontal axis). The fundamental frequency of the pipe is the same as the frequency of its first harmonic. Successive harmonics lie at successive exact integer multiples of this frequency i.e. at exactly twice, three times, four times the fundamental frequency, etc. When two such pipes are sounded together, you get a separate beat generated between each harmonic of one pipe and every harmonic of the other, leading potentially to a large number of beats. However, because the strengths of the harmonics fall off rapidly, only the beats generated by the first few harmonics are heard in practice for this type of pipe.
When tuning two pipes which are intended to have the same pitch (e.g. middle C on an 8 foot Open Diapason and an 8 foot Cornopean) you adjust their fundamental frequencies to be the same. By adjusting their fundamental frequencies, you are also adjusting all their harmonics at the same time. You simply adjust the frequency of one of the pipes until no beats are heard, and at that point they are then in tune.
A different situation pertains if you want to tune, say, an 8 foot Open Diapason to a 4 foot Principal. The two are an octave apart, so even when they are in tune there will still be a large frequency difference between their fundamental frequencies. At middle C this difference frequency will be about 262 Hz. Therefore, how can you use beats to tune them when there will still remain this large frequency difference? Does this not result in a very fast beat even when the two pipes have been brought into perfect tune?
Because in this case of two perfectly tuned pipes, the beat frequency between the two fundamentals is exactly the same as the fundamental frequency of the 8 foot pipe, and the concept of a beat then becomes meaningless. There can be no beat at a frequency which is the same as one of the generating frequencies. But if this is so, can beats still be used in the tuning process, as they were for the more obvious case of the unisons? Yes they can, and this is how the beats arise. When the pipes are very nearly in tune, there will be only a small difference between the fundamental frequency of the Principal and the second harmonic of the Open Diapason, and it is between these two frequencies that the main beat will be heard. (There will also be subsidiary beats at different, faster, beat frequencies between other pairs of harmonics, but because these are of lower amplitudes, the main and slowest beat just referred to is the one the ear will latch onto). By adjusting one of the pipes, this main beat will eventually disappear together with all the others, and the two are then in tune.
The situation gets a further step more complicated when tuning two pipes which are an interval of a fifth apart. It is necessary to do this to “lay the bearings” or “set the temperament” when tuning an organ. How can the necessary slow, noticeable beats arise in this case, when the fundamental frequencies of the two pipes are not the same (as they were in the ‘unison’ case), or not exactly an octave apart (as they were in the ‘octave’ case)?
The main beat in this case, which becomes more and more noticeable (and slower and slower) as the pipes are pulled progressively more closely into tune, arises between the third harmonic of the lower note and the second harmonic of the upper note. Take the example of the fifth formed by playing the G below middle C and the D above middle C. For an instrument tuned to the usual modern standard where the A above middle C has a fundamental frequency of 440.00 Hz, and assuming Equal Temperament, then the fundamental frequency of the G is 196.00 Hz and that of the D is 293.66 Hz.
The third harmonic of the lower note, G, is therefore at 196.00 x 3 = 588.00 Hz. The second harmonic of the upper note, D, is at 293.66 x 2 = 587.32 Hz. See how close these two frequencies are! That is why the ear is easily able to detect the beating between them. The frequency difference is minute, 0.68 Hz, the same as in the sine wave example we opened the discussion with. That is why I chose these two apparently peculiar numbers for that example. All the fifths in equal temperament have a slowish beat (none are tuned pure), although the actual beat rate for this G-D interval varies depending on which octave you are playing it in. If this fifth was to be tuned pure, you would tune it until no beat could be heard at all, and in that case there would then be a multiplier of exactly 1.5 between the two fundamental frequencies – in other words, they would have a frequency ratio of exactly 3:2. Hence the love of the ancient Greeks for pure intervals, which are represented by ratios of exact integers. They discovered this by observing the lengths of pipes and strings tuned to these intervals.
The reason why the frequency of the sine waves in the earlier example corresponded to treble D, an octave above the upper note of the fifth (middle D) we are now discussing, is because it is the frequency of the second harmonic of that upper note. This is the same as the fundamental frequency of the note an octave above the upper note, i.e. treble D.
We can now see why Lehman’s explanation  for how the interval of a fifth creates a beat was incorrect. He said that “upper harmonics of both notes have a frequency that is almost identical, at some point several octaves above the fundamental frequency”. We have shown that it is not “several octaves” above the fundamental at all, but only one octave above the fundamental in the case of the upper note, and one and a half octaves in the case of the lower.
Unlike the octave, the interval of a fifth also generates a beat between the fundamental frequencies as well as between the upper harmonics. This is the case even when the interval is exactly tuned (pure). The reason lies in the arithmetic, because the difference between the two fundamental frequencies of a fifth equals half the frequency of the lower note, rather than coinciding with one of those frequencies as in the case of the octave. However the beat is far too fast to be perceived except when the two fundamental frequencies are very low, as in the case of fifths played on 16 foot pedal stops in the lowest octave. Then the beat frequency of the fifth is the same as that of a 32 foot stop, hence the use of this “quinting” technique to derive an acoustic bass.
However this brings us onto yet another aspect of beats which creates much confusion. There is absolutely no acoustic energy in the air in any beat regardless of its frequency, including at the frequencies generated by a quinted acoustic bass, and that is the main reason why an acoustic bass is so unsatisfactory. It is also the reason why you cannot usually hear a beat as a separate sound at any frequencies other than very low beat frequencies, when the ear merely becomes able to follow the time pattern of the envelope of the sound as in the picture above. Energy would only appear at the beat frequencies if the propagation of sound in the atmosphere was nonlinear, which it is not. In those circumstances the mechanism would become one of amplitude modulation of one frequency by another rather than beat formation. In amplitude modulation true frequency sidebands are generated whose energy has been robbed from the generating waves. The two processes are very different.
Even so, but rarely, you can hear the actual beat frequencies, and this occurs when the generating sounds are very loud. Under these conditions the small nonlinearities in the ear itself cause energy to appear at the beat frequencies within our hearing mechanism, both the sum and difference frequencies, and you can hear them as faint tones even though they do not actually exist in the sound wave impinging on your ear. It is interesting that it is musicians and musical instrument technicians who usually speak as though these spurious frequencies are more common than they actually are. If you play an instrument for your living, you live close to the sound generating mechanism much of the time, and its sheer loudness is such that you will often hear the sum and difference frequencies. The same applies to an organ tuner or voicer, who hears the beats as separate tonal entities between, say, the ranks of a mixture when close to the pipes. I think this factor is responsible for such people in the trade assuming that their audiences will always hear them also, though generally they do not because they are much further away from the sound source and therefore the sum and difference tones vanish into inaudibility. This also means that oft-quoted statements such as "many difference and addition tones is the thing that makes a fine organ chorus more than equal to the sum of its parts"  apply more to the tuner, voicer and possibly the player than to Joe Public away down in the body of the church.
Can you tune using beats when there are no upper harmonics?
No. You could not tune two pure tones – two sine waves – using beats except when tuning them to the same frequency. Trying to tune them in octaves or fifths is impossible using beats because the beats are not formed. There are no harmonics to form them with. Fortunately, pure sine waves do not arise in nature as they can only be generated electrically.
However, the converse problem can also occur if there are too many harmonics. Reed stops have large numbers of harmonics whose strengths diminish only slowly against harmonic number. If you try to tune two reed stops in octaves and fifths, there is a cacophony of strong multiple beats between all these harmonics which can confuse the ear. This is why organs are first tuned using a diapason type of stop, whose harmonics are well suited both in number and relative strength for beat formation. Having got the flue work in tune, the reeds are then tuned in unison with the flues, note by note and stop by stop.
The End - (well, the end of this Appendix)
At the end of this rather exhausting and surprisingly long Appendix on beats, do you now agree that the average organ builder, organist or composer in Bach’s time would have been unlikely to understand all about them? Admittedly, there is nothing particularly difficult provided we keep a clear head and (more importantly) grant that terms such as sine waves, harmonics, spectra, difference frequencies and nonlinear processes were known and widely understood. But that would not and could not have been the case. They would have known about tuning by beats at a practical level in those days, but I suspect the reasons why beats occurred would have seemed rather mysterious to most of those who thought about it at all. Therefore, for modern Temperamentalists to imagine that they can transplant today’s level of understanding backwards two or three centuries can only be a nonsense.
Appendix 2 – Arithmetical Precision required in Temperament Studies
The degree of precision required in numbers and arithmetic operations to do with temperament arises as follows.
Consider the process of tuning by beats. Let there be two flue pipes, one of which is already in tune and one which is to be tuned to it. Tuning becomes progressively more critical and difficult the higher the frequency because the beats, which are frequency differences, become faster as those frequencies increase for a given change in length of the pipes.
A verdict of “pretty well in tune” would probably be given if there was, say, one beat in around ten seconds for any pair of pipes. Although stricter criteria could be adopted there is no point in making things too difficult, partly because of the tuning drift which occurs naturally due to temperature variations etc after an organ has been carefully tuned. Therefore, using this criterion, the two pipes would have to be tuned until their fundamental frequencies did not differ by more than about 0.1 Hz, because one beat in ten seconds implies a frequency difference of 0.1 Hz.
Remembering that tuning is most critical in the upper reaches of the compass, consider the fifth C on the keyboard, i.e. the C below top C on a 61-note organ keyboard. Even higher notes could be chosen, but again we have to adopt reasonable parameters if the discussion is to remain sensible and practical. The fundamental frequency of this note on an 8 foot stop is 1046.5 Hz for an organ tuned to A = 440.00 Hz in Equal Temperament. For simplicity we shall use the approximate figure of 1000 Hz.
A frequency tolerance of about 0.1 Hz at a frequency of about 1000 Hz implies a tuning accuracy of the order of 0.0001 or 0.01%. This is therefore also the precision required in temperament calculations which have to deliver the frequencies of the notes in a particular temperament. But because there are usually several steps in the calculation of each frequency, it is necessary that the numerical precision of the numbers used in each step is greater than that required in the final answer, otherwise the answer will not be accurate enough owing to truncation or rounding errors. Therefore at least one more significant figure is required throughout the calculations, meaning that numbers must be represented to at least a precision of 0.00001 or 0.001%. This is the same as a precision of 1 part in 100,000, or 6 significant figures, as stated in the main body of this article.
3 – Long Division
The example below shows how the two numbers 531441 (the numerator) and 524288 (the denominator) can be divided using the age-old method of long division to give the answer 1.01364. This is the minimum precision, 6 significant digits, required to represent numbers when doing temperament arithmetic (Appendix 2). Desirably it would be better to work to at least one further digit so that the 6 digit answer could be rounded rather than truncated. No problem with an electronic calculator, though I am sure you will agree that the enthusiasm to do this by hand would rapidly wane.
The method works by first subtracting the denominator repeatedly from the numerator until no further subtractions can be done. The denominator is then divided by 10 and subtracted repeatedly from the remainder left after the first stage. The whole process is then repeated, dividing the denominator by a further 10 each time, until the desired degree of precision has been reached. (A computer or calculator still does exactly the same when you ask it to divide two numbers, except that the numbers are represented in binary form rather than decimal, therefore the intermediate divisions are by two rather than ten).
Bach and his unfortunate contemporaries would almost certainly have been taught how to do this because there was no alternative method at the time. Even so, I doubt it would have developed into one of their favourite pastimes and therefore I contend that studies of temperament would have been largely qualitative in those days rather than quantitative as today, with serendipity playing a more dominant role than many current Temperamentalists grant. Even for those with the determination, the opportunities for error would have been so great that, again, fully quantitative numerical studies of temperament would have been the exception rather than the rule.
The method was still being taught and examined when I was at junior school in the 1950’s. A few years later I was taught the use of logarithms to do the same job, although the business of looking up the numbers in log tables, doing the necessary arithmetic on them, and then reconverting the answer using antilog tables was so involved that I doubt it was really very much easier. There was also still far too much opportunity for error. The arrival of electronic hand held calculators in the 1970’s finally swept away all this centuries-old arithmetical baggage the human race had carried for so long.
Notes and References
1. Elements, Proposition 47, Book I, Euclid.
2. “Here’s to Plato”, Rectorial Address to the University of Dundee, Stephen Fry, 1 November 1995.
3. “Bach’s extraordinary temperament : our Rosetta Stone”, Bradley Lehman, Early Music, February 2005. (Lehman, and presumably his publisher, were apparently unaware of previous material by Sparschuh and Zapf which had covered much of the same ground. For more on this, see subsequent correspondence such as the letter by Mobbs and Mackenzie of Ord, Early Music, August 2005).
4. “A Dorset Temperament?”, C E Pykett, Organists’ Review, August 2004. Also available on this website (read).
5. “Temperament and Pitch”, Christopher Kent, in The Cambridge Companion to the Organ, Cambridge University Press 1998.
6. “The Well-Tempered Organ”, Charles A Padgham, Positif Press 1986.
7. “History of Britain”, Simon Schama, BBC Publications 2000 – 2002.
8. “Cambridge Biographical Encyclopedia”, D Crystal, Cambridge University Press 1998.
9. “Piano Tuning”, J C Fischer, 1907 (reprinted Dover, 1975).
10. “Fleming – Discoverer of Penicillin”, L J Ludovici, Andrew Dakers, London 1952.
11. Some might object to intervals such as A# to F being called a fifth. However, because of the equivalence of the notes played in the cycle of twelve fifths to the twelve semitones of a chromatic scale with any starting note, no concept of key was implied in this discussion and none should be assumed. It is only necessary here to identify the physical notes on the keyboard in some convenient manner, and the tuner's convention of referring to the white notes as naturals and the black ones as sharps seems as good as any. The point is simply that a fifth consists of seven adjacent semitones (eight if counting the starting note). Using MIDI note numbers would have served just as well, but it would have caused unnecessary difficulties for the uninitiated.
12. With his laws of motion and his calculus, Newton proved in the mid-17th century that the time period of a simple plumb-line type of pendulum is proportional to the square root of its length. This statement is true provided the mass of the string is negligible compared to that of the bob it supports, that the dimensions of the bob are negligible compared to the length of the string, and that the arc of swing is small (typically ten degrees or so). In these circumstances a "seconds pendulum", which takes one second to swing from one extremity to the other, is almost exactly one metre long, hence the size of long case clocks which appeared shortly afterwards. Such a pendulum could easily have been carried around for tuning and other musical purposes. It is difficult to see how else beat rates could have been measured accurately at the keyboard at the time we are considering in this article, until experience had maybe conferred a sufficiently accurate mental notion of time on an experienced tuner.
13. In "The Mixture - to be taken as before?", S Bicknell, number 3 of 6 articles published in 'Choir and Organ' in 1998-9 under the heading 'Spit and Polish'.
14. See, for example, "L'Harmonie Universelle, livre troisième : Des instruments à chordes", M Mersenne, Paris, 1636.
Probably stimulated by proposals put forward earlier by Galileo, Mersenne made the first documented, but rather crude, determination of the absolute frequency of a musical note. This was at 84 Hz, which lies between bottom E and F on an 8 foot stop of a modern organ tuned to A = 440 Hz in ET. He first timed the motion of a long heavy wire that moved slowly enough to be followed by eye. From theoretical considerations based on his own laws describing the pitch of a taut string, he could then calculate - but not measure directly - the approximate frequency of a shorter and lighter wire, one that produced an audible sound. Note that this was a rather shaky application of logic in that his "laws" themselves remained hypothetical until such time as any frequency could be measured to verify them. However, even neglecting this problem, the result was a long way from the accuracy required for actual studies in temperament as we pursue them today, and therefore it does not conflict with my conjecture that the concept of frequency and the part played by it in musical acoustics was not widespread in the 17th and 18th centuries. How could it have been, when the foregoing shows it was so difficult to measure and when education - the dissemination of such knowledge - was so limited both in terms of availability and content?
15. J-P Bacconet, private communication, December 2006.
16. "Keyboard Temperaments with Impure Octaves", C E Pykett, 2008. Currently available on this website (read).
17. "J. S. Bach's Well-tempered clavier : a new approach", Peter Williams, Early Music, November 1983.