Keyboard Temperaments with Impure Octaves
by Colin Pykett
Posted: 1 September 2008
Last revised: 23 December 2009
Copyright © C E Pykett 2008-2009
“New ideas have four stages of acceptance:
i. this is worthless nonsense;
ii. this is an interesting, but perverse, point of view;
iii. this is true, but quite unimportant;
iv. I always said so.”
J B S Haldane
Abstract. In an earlier article on this website I surveyed the historical context of tuning and temperament, concluding with some remarks about the sanctity of the octave in terms of its tuning purity. This article continues the story by asking why tempered octaves have seldom been considered in the long history of tuning keyboard instruments. Although a definite answer is elusive, a probable reason is that temperaments with impure octaves are difficult to tune by ear, and therefore it is only recently that the advent both of electronic tuning devices and digital musical instruments have made them more accessible for study.
Various temperaments with impure octaves are described, with the octaves tuned both sharp and flat from pure. The work focuses exclusively on temperaments appropriate for the organ, because a temperament suitable for this instrument might be less attractive for others, and vice versa. This is partly because of the sustained nature of organ tones, as well as the availability of stops at many pitches which other instruments do not possess. The fact that most stops constituting an organ chorus are octavely related makes the study of temperaments with impure octaves uniquely interesting for the instrument.
Three temperaments are discussed in detail, one using offset octaves and another using Cordier’s recipe where the octaves are sharpened and the fifths pure. The third temperament is called “Flat Octave 1” and it uses flattened octaves. This has the advantage that the significantly sharp thirds in conventional Equal Temperament and the even sharper ones in Cordier’s temperament can be brought closer into tune. Some mp3 sound clips are included.
Some interesting generalisations are mentioned which appear when using impure octaves, an important one being that an infinity of equal temperaments become available instead of there being just one as in the case of pure octave tuning. This fact, that impure octaves enable the exploration of more than one equal temperament, is exciting both in theory and in practice. It opens a door which has been locked for centuries. All of the temperaments with impure octaves discussed in the article are equal temperaments, which means they can be used in all keys irrespective of their different characters.
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In an earlier article on this website  I surveyed the historical context of tuning and temperament, concluding with some remarks about the sanctity of the interval of an octave in terms of its tuning purity. On the organ the octaves are tuned pure, in contrast to the other intervals such as thirds and fifths which are invariably candidates for various degrees of tempering or detuning. Thus if we follow conventional wisdom, all intervals except the octave can be tempered.
While various reasons can be proposed to explain why this cultural paradigm has become so ingrained since the dawn of human history, at least for the organ, it is by no means obvious why we still adhere to it so strongly today if we approach the issue with an open mind. The reason most often quoted is that the harmonics of two pipes an octave apart do not coincide neatly in frequency if the octave is not perfectly tuned, and in this case we will hear the pipes beating. But, and curiously, those who use this to justify pure octaves never insist that all the other intervals must also be tuned pure. Indeed they often swing the other way to actually state a preference for tempered intervals – those which generate beats - to prevent the cloying sweetness which some say would occur with too many pure intervals within the octave. This might seem to be merely making a virtue of necessity, because it is of course impossible to have all the intervals pure – if it were possible, there would be no temperament problem in the first place. But the application of different logic to insist on pure octaves on the one hand yet, on the other, to accept that any or all the other intervals may be tempered could be seen as perverse, given the difficulties of devising usable temperaments which follow as a consequence of having pure octaves.
Nevertheless, I think there is a good, down to earth, practical reason why the octaves have been tuned pure for so long, and one meets it forcefully when trying to tune an instrument by ear to a temperament using impure octaves while retaining a keyboard whose physical structure repeats every octave. Traditionally, tuning an organ by ear means that one first sets the desired temperament in the middle octave of a single stop. Next, the twelve note frequencies so defined are propagated across the compass of the chosen stop merely by tuning the octaves pure. Finally the remaining stops are tuned pure, note by note, to the rank on which the bearings have been laid. Using impure octaves would mean that one could not propagate the tuning of the middle octave across the keyboard, nor could one tune the other stops pure, just by listening for zero beats between unisons or octaves.
This would make tuning by ear a long winded and error prone procedure if the octaves were not pure, to the point where it would verge on the impossible or at least be deemed impractical for routine use. It is only in recent decades, with the availability of electronic tuning aids or with digital instruments, that this situation has changed. Consequently, given that it is now possible to tune keyboard instruments more easily today than for countless centuries past, this article examines some implications of allowing the octaves to be tempered, how they should be tempered and what might be gained.
The work described here relates specifically to the organ, because it is explained later why the characteristics of a temperament suitable for this instrument might be less attractive for others, and vice versa. This is partly because of the sustained nature of organ tones as well as the availability of stops at many pitches which other keyboard instruments do not possess. Because most stops constituting an organ chorus are octavely related, the study of temperaments with impure octaves is uniquely interesting for the instrument.
I have not presupposed that you will necessarily like the temperaments described herein. Many zealous writers on temperament, including some of the best qualified, appear to see themselves as Inquisitors by implying that anyone who questions their work is beyond redemption. Padgham’s description of the unconverted who exhibit “conservatism, fear of the unknown and ignorance”  is particularly shocking, but regrettably he is not alone and today temperament remains a subject where invective often masquerades as scholarship. For my part, I shall simply be content if you find this article interesting in some way or other. Proselytism is not its intention, as it merely attempts to open some doors which hitherto have been largely closed. Temperament is an interesting subject to some, but that does not mean that everyone has to take the medicine and I fully understand those who do not like it.
Reviewing the relevant arguments from the earlier article , as a rule students of temperament 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 mention it as an option; they proceed on the basis that pure octaves are axiomatic and always tune them true. Inevitably, this leads to a subjective tuning rigidity across the compass of a keyboard 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 octave above and below. The results are legion as will now be described.
The beat frequency of any interval on a keyboard instrument depends on the octave in which it is played. In other words, a tempered fifth played in the third octave of the keyboard 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 bears a simple relation to the octaves considered – a tempered 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 beat frequency ratios also apply to any other interval, no matter how carefully they might have been mutually adjusted within each octave by adopting a favoured temperament.
With a recently tuned organ in which all the octaves are well in tune across the whole keyboard, and with well tuned octavely-related ranks, this can lead to a hard, sterile, locked-up type of sound especially 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 as we saw above. The sterility only recedes when the tuning of the instrument drifts over time, in the course of which the octaves often become slightly impure. The subjective effect can be even more noticeable and unpleasant in digital instruments because their tuning never drifts. 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.
Such aural sterility never characterises the orchestra or other ensembles of instruments because no attempt is made by their players to maintain excessive purity of intervals to the same extent, and this includes purity of the octaves.
Not only is the effect of well tuned keyboard instruments potentially more sterile than that of the orchestra. In addition, the sometimes unattractive subjective experience due to excessively pure octaves is worse for the organ than 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, and so on. Thus all the harmonics of an organ pipe are locked in phase with each other while it sounds. Using the terms properly and rigorously, this is why the overtones in this case must 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.
Stringed keyboard instruments such as the piano and harpsichord do not usually sound as sterile, in the sense described, as the organ even when they are tuned as well as possible. The main reason is that the harmonics of a struck or plucked string (but not a bowed string) are actually not harmonics at all; they must be referred to as partials or overtones, not harmonics, because their frequencies are not exact integer multiples of each other. As the sound of a struck or plucked string dies away the overtones, which are slightly mutually sharp, exhibit beats because of their frequency independence. (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 temperaments. 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? Of course, it would not be possible to detune the octaves to such an extent that they themselves became unusable. In this case we would only have replaced one problem by another. Somewhere between the two extremes may lie a solution worth exploring.
Impure octaves are nothing new in keyboard tuning. Many piano tuners routinely sharpen or "stretch" the octaves when tuning, though the reasons quoted vary. Some maintain that the beats between the partials during the decay of the sound are 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 commonsense nature of the latter. Other reasons for stretching the octaves also exist. At the other extreme though, some tuners tune the octaves pure. So we can learn little from the piano scenario, even ignoring the physical differences in the way it and the organ produce sound as outlined above.
Yet impure octaves would without doubt have occurred from time to time if we accept the notion suggested in the earlier article  in which some “good” temperaments in the 17th and 18th centuries might have been discovered partly through serendipity because of the tuning instability of the old stringed keyboard instruments. This was a consequence of their wood frames which were inconveniently sensitive to changes in temperature, humidity and ageing . But regardless of any interesting results which happenstance might have thrown up, probably a compelling reason why impure octaves were not embraced by those who worked on temperament in those days was because of the difficulties of tuning an instrument in this way – by ear alone of course – and this has been mentioned already.
The earlier article mentioned that I planned to investigate the matter in more detail, specifically for the organ, and the results so far will be described presently. One issue foreseen at the outset has been confirmed, in that doing the work with the emphasis on arithmetic and theory which constitutes current work on temperament would almost certainly be debarred. This is because pure octaves underpin the entire concept of temperament as it is understood today, therefore 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 was considered desirable to impose a deterministic rather than an indiscriminate progression of octave tempering across the keyboard. As an example, a temperament might be set for the lowest twelve notes, say, then the successive octaves above each one would be tempered progressively according to certain rules 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 the same rules. Each octave could contain a completely different temperament in principle, though in practice it is probably better to regard the entire keyboard simply as a collection of notes upon which the notion of a distributed temperament is to be imposed.
These ideas, particularly that of a distributed temperament, are perhaps difficult to accept at first acquaintance. Therefore it might be appropriate to remind ourselves that temperament is largely a subjective matter in the last analysis, a matter of what the ear will accept. This is illustrated by the fact that among musicians there exists a spectrum of attitudes ranging from complete indifference to a neurotic interest in the subject. Put simply, this is probably related to what musicians regard as an acceptable degree of out-of-tuneness, and here again there is no single view. This will now be explored.
As night follows day, imposing a temperament on a keyboard instrument always means that some intervals will be better tuned than others. A perfectly tuned interval exhibits no beats between any of the harmonics of either of the two notes, thus we perceive no wavering, pulsating or throbbing as long as the notes sound. (In passing we might observe that the formation and meaning of beats is often misunderstood, and the phenomena are fully explained elsewhere on this website ).
Normally the octaves are tuned pure in all temperaments. However, within the octave some of the other intervals are out of tune by an amount depending on which temperament has been used. In Equal Temperament none of the intervals except the octaves are in tune. For example, all the fifths are slightly flat and the thirds are considerably sharp, whereas in the Werckmeister III temperament eight fifths within the octave are perfectly tuned. In turn, the intervals which are out of tune in any temperament govern which keys are useable and which are unusable. In the 18th century Gottfried Silbermann is said to have used a temperament in which A flat major is the worst key to an extent which most would regard as intolerably out of tune, whereas C major is very good . In Equal Temperament and some others all keys are useable.
For musical purposes the amount by which the two notes constituting an interval are out of tune is usually quantified in three ways. Firstly, if a note deviates significantly from the pitch we expect we can tell immediately that something is wrong. We form this judgement on the basis of the absolute frequency of the note, and those cursed with absolute pitch will tolerate smaller deviations than those without it. However this situation is unusual unless the temperament in use is a strongly unequal one with “wolf” notes which render certain keys unusable, or unless the instrument in question is badly out of tune anyway.
In most circumstances we use a second way of deciding whether intervals are acceptably in tune because our ears form a judgement of the frequencies of the beats when the notes sound simultaneously. In this case the judgement is based on the relative, rather than the absolute, frequencies of two notes. Thirdly, but more laboriously, we can measure or calculate their frequencies and then express the ratio in units such as cents .
However the last two measurements, involving beats and cents, are not equivalent for musical purposes, and it is therefore regrettable that many authors proceed as though they are. The number of beats per second, the beat frequency, is the same as the difference in cycles per second between two frequencies. The older frequency unit of cycles per second is properly written today as Hertz (Hz). Thus a beat of 1 cycle per second between two frequencies means that they differ by 1 Hz. This is always true regardless of the musical pitch of the notes which give rise to the beats, that is, regardless of their absolute frequencies and therefore regardless of where they lie within the keyboard compass.
On the other hand, the number of cents between the same two frequencies does depend on where they lie in the keyboard. The difference can be appreciated by taking an example. At middle C on a couple of 8 foot organ stops, two nearly in-tune pipes beating at 1 beat per second are out of tune by 6.6 cents, whereas an octave above at treble C the same beat rate would mean they are out of tune by only 3.3 cents . Extending the same beat frequency of 1 Hz to the extremes of the organ compass, at bottom C on a pair of 32 foot stops (such luxury!) the two pipes would be out of tune by 103 cents, more than a semitone. At top C on a pair of 2 foot stops the difference would be only a minute 0.2 cents, one five-hundredth of a semitone. Yet in all these cases the beat frequency perceived by the ear is the same – 1 Hz. Bear in mind that beat frequencies of 1 Hz or so over the compass would lead many to conclude that an organ was badly out of tune, and many would insist on it being significantly reduced.
The foregoing is admittedly rather laboured. However it is included to emphasise that when presented with two notes sounding simultaneously, the ear usually latches onto the beats which might exist – it does not do esoteric calculations to find the equivalent number of cents. In other words, we make a rapid aural and musical judgement as to whether intervals are adequately in tune or not on the basis of beat frequencies alone. Therefore when cents are used by modern writers on tuning and temperament to express frequency ratios between notes, it must always be borne in mind that the cent values vary dramatically across the keyboard for the same beat frequency.
This is not always obvious from the prose of many authors. For instance, Padgham in his book on organ tuning stated that “interval errors of greater than 10 cents from just values are ... significant” . This is demonstrably untrue to the point of being meaningless for the reasons just rehearsed. Taking another example, a mathematician with whom I was corresponding on this topic maintained that “cents are cents – why should it matter where they are in the keyboard?”. Clearly, specialists who allow their lives to revolve too closely around cents can lose sight of the point that the ear perceives only a beat frequency, not the difference in cents, when an interval is slightly out of tune. There is no unique correspondence between the beat frequency and its value in cents, because the latter depends on the position of the interval within the keyboard compass of the instrument in question. When we leave the comfortable confines of pure octaves and move into the realm of distributed temperaments across an entire keyboard, the issue assumes particular importance.
Even ignoring distributed temperaments, the discussion above is no mere academic nicety. One important practical consequence arises in connection with today’s synthesiser-based musical instruments such as digital pianos, harpsichords and organs. Not all of them can be tuned once purchased, but of those which can the tuning precision away from Equal Temperament is commonly limited to 1 cent. The fractions of a cent necessary for accurate tuning are unavailable in these instruments. This means that at the top of a 2 foot stop the notes cannot be tuned better than about 5 beats per second either sharp or flat from their Equally Tempered pitch. This is at least ten times worse than the accuracy a pipe organ tuner would aim for. Thus such instruments can sound rather rough and coarse when high pitched mixtures and mutations are in use, particularly when certain unequal temperaments have been set.
Because many workers on temperament today use digital instruments of one sort or another to assess their results, whether they admit it or not, this restriction is of more than passing interest and it is something I have been forced to keep constantly in mind during the work now to be described. And to forestall the obvious question, I do use digital keyboard instruments myself. They are very useful for temperament research, but only if one works consciously within their limitations.
The same remarks apply to those tuning meters or ETD’s (electronic tuning devices) where the precision available is limited to one cent. I cannot see that such items are other than a waste of money for tuning the higher notes of any instrument.
Probably the simplest way to imagine tuning an organ with impure octaves is merely to impose a frequency offset on the notes of each octave. Thus the twelve notes in each octave are shifted slightly in frequency en bloc with respect to their neighbours in adjacent octaves. Because of its conceptual simplicity this method will be described first.
There are several ways to achieve offset octave tuning, one of which is to tune the 8 foot middle octave (middle C and the 11 notes above it) to the desired temperament in the usual way and then impose the offsets while tuning the octaves above and below. This method has the advantage that the A in the middle octave can still be tuned to whichever pitch standard is desired (A = 440 Hz or whatever).
The frequencies of the remaining octaves are then adjusted so that slow beats arise between corresponding notes in adjacent octaves. For example, treble C would be adjusted so that a slow beat (say around one beat in three or four seconds) was allowed to remain when sounded with middle C. It does not matter whether treble C is adjusted to be slightly flat or slightly sharp, nor does the exact beat frequency matter either. The C above treble C would next be adjusted similarly by sounding these two notes together. In this case it is probably best to tune this octave sharp if the treble C octave was flat relative to middle C, and vice versa. This will prevent a “runaway” condition in which the extremes of the keyboard become excessively out of tune compared to the pure octaves case. This could happen if all the octaves were sharpened, or all of them were flattened, relative to their neighbours. Alternating the sharpening or flattening every other octave will prevent this. The C’s below middle C would be treated similarly, offsetting them alternately flat and sharp as before to achieve a similar beat rate across the compass.
Within each octave the notes are tuned using the same temperament for each. Provided the temperament was set accurately in the middle octave, it will usually be quickest simply to tune corresponding notes in adjacent octaves one by one such that the beat frequencies imposed on the C’s apply approximately to the other notes as well.
The question then arises how to treat stops of other pitches, because the foregoing related only to an 8 foot unison stop. A way to proceed is to consider an extension organ, and in this case the answer is simple - the impure octaves will appear automatically for all the derived pitches once they have been set across an extended rank. Therefore, for reasons of compatibility in a 'straight' (non-extended) organ, it is logical to tune each stop separately against the unison rank, setting the octave pitches slightly impure on a note by note basis as before.
Fifth-sounding ranks (such as nazards and twelfths) and third-sounding ranks (such as tierces and seventeenths) in mixtures and mutations can be tuned true to unison pitch as in normal practice. This means tuning them so there is no beat between the third harmonic of the unison and the twelfth, or between the fifth harmonic of the unison and a tierce.
The difficulty encountered in tuning an organ by this means will depend largely on the experience of the tuner. Reduced to its simplest form, it merely means that instead of tuning octavely-related pipes pure, they are all tuned so that a slow beat remains. It would be a relatively straightforward, if time consuming, matter to convert an organ normally tuned with pure octaves into one with offset octaves at its next tuning if one wished to assess the effects for oneself. If one did not like it, it could be converted back again.
In my first foray into the realm of impure octaves I applied this method of tuning some years ago to the electronic organ pictured on the home page of this site. My "Dorset Temperament" is also used in this instrument, a modest perturbation of Equal Temperament which introduces a hint of key colour while still allowing all keys to be used . It is no doubt meaningless to describe the results in words, although adjectives such as “mellow” and “warm” spring to mind. However some sound clips recorded on this instrument are usually to be found on the home page which might assist those interested to perceive a flavour of the outcome. The fact I have been able to live with this frequently-played instrument for some years is an indication that it is, at worst, not positively objectionable.
The offset octave tuning just described was an interesting exercise and it provided useful experience when taking a first step into the subject of impure octaves. However it was an end in itself – that was it. No general insights into the subject could be derived from it and in that sense it was a blind alley, an intellectual cul-de-sac.
However, if we recall that Equal Temperament is characterised by twelve equal semitone steps to the pure octave, equal in the sense that the frequency ratios of adjacent notes are the same, we can immediately get much further by realising that we can have equal semitones of any size we choose if the octave is not pure. In that case they do not need to be restricted to the frequency ratios of Equal Temperament. This insight is powerful because it enables us to develop an indefinite number of new temperaments using impure octaves in the manner to be described.
In Equal Temperament with its pure octaves, the frequency ratio for adjacent semitones anywhere within the keyboard is 21/12 or the 12th root of 2. As an example, the frequency of middle C at 8 foot pitch is 261.63 Hz, and that of the C# above it is 277.18 Hz. The ratio of these numbers is approximately 1.06, the same as 21/12. (Although the frequencies used here correspond to the usual pitch standard of A = 440 Hz, the same result will be obtained for any other standard).
The number “2” which appears in the phrase “12th root of 2” simply represents the fact we traditionally use pure octaves whose frequencies are related by a factor of exactly 2. If the octaves are impure then their frequencies can be related by any factor we like. But in that case how can we proceed to derive the corresponding semitone frequencies? The springboard to progress in this case is provided by the physical keyboard itself, that venerable collection of black and white levers of which there are 12 to the octave. The number “12” which appears in “12th root of 2” simply represents the fact we have 12 keys or semitones to the octave. It would be a brave mortal who suggested that we use a different number, stand fast the fact that some workers in temperament have dared to do so, and I certainly do not have their courage. Therefore we shall continue to assume a standard keyboard with 12 notes to the octave for the remainder of this article.
In this case, retaining 12 notes to the octave but with the octave now defined by any frequency ratio we like, the frequency ratio for the two notes constituting a semitone interval anywhere in the keyboard becomes x1/12 or the 12th root of x, where x is the frequency ratio of the impure octave. Because x can take any value in principle, we now have potentially an infinite number of temperaments in each of which the semitone frequency ratios are equal, and thus an infinite number of equal temperaments with impure octaves. This contrasts with the case for x = 2 (pure octaves) where there is only a single Equal Temperament. Infinity is an over-used word and in the present situation it has little meaning in practice, because most of the possible equal temperaments will be unusable because of gross dissonances. Qualitatively this is no different to the conventional case when trying to derive usable temperaments using pure octaves. Nevertheless, the fact that impure octaves allow us to explore more than one equal temperament is exciting both in theory and in practice. It opens a door which has been locked for centuries.
Note the deliberate use of a particular nomenclature in the foregoing. In this article Equal Temperament, spelt with upper case ‘E’ and ‘T’, means the one and only Equal Temperament which is possible if the octaves are pure. Using lower case letters, ‘e’ and ‘t’, indicates one of the many equal temperaments which are possible once the octaves become impure. The distinction is not mere pedantry as it is easy to get confused if we do not keep the differences in mind.
It is also easy to get confused if cents are used when describing temperaments with impure octaves. So beware of cents yet again! The confusion can arise if we do not remember that the cent is a measurement derived from Equal Temperament using pure octaves. Unfortunately it is often necessary to use it when tuning an instrument with impure octaves because electronic tuning meters often give readings in cents away from Equal Temperament. Similarly, most if not all digital musical instruments also require temperaments to be set in terms of cents. Whether we like it or not, conventional Equal Temperament has become the de facto standard to which all others are referred when tuning a keyboard instrument and therefore it would not be helpful to redefine the cent in what follows.
A temperament with pure fifths is an obvious idea which springs to mind when we recall one of the main problems of conventional temperaments with pure octaves. Keeping the octaves pure means that some or all of the fifths have to be flattened so they can fit into the octave. If we stretch the octaves to make them slightly sharp, this can be avoided and all the fifths can then be made pure. The idea is obvious if only because it has been cast in the guise of a fundamental problem by virtually every writer on temperament for centuries past, and the diagrams and explanations in reference  are as good as any. It is therefore less obvious, indeed it is surprising, why some of the same writers have not seen that the problem could become a virtue by developing temperaments based on pure fifths rather than on pure octaves.
Thus instead of the octaves being pure with twelve semitones tempered in one way or another, the fifths are now pure with seven, slightly larger, semitones. Twelve of these larger semitones make up the new, larger, octave. If we choose all the semitones to have the same frequency ratio one with the next, this temperament becomes one of the class of equal temperaments using impure octaves referred to above.
Numerical data for some intervals in conventional Equal Temperament (with pure octaves) compared with a temperament constructed in this way (with pure fifths) are in the table below. The numbers are given to 6 figure precision to minimise rounding errors if you use them in your own calculations.
Table 1. Some data for intervals in two equal temperaments using pure octaves and pure fifths.
The table shows that, using the nomenclature of the previous section, x in the expression x1/12 now takes the value 2.00388 instead of exactly 2 for conventional Equal Temperament. The factor x is the frequency ratio of notes separated by an octave, and in Byzantine units (which I detest) it means each octave has been sharpened by one seventh of the Pythagorean Comma. Recall that the cent still retains its usual meaning of one hundredth of a semitone in conventional Equal Temperament with pure octaves. It does not mean one hundredth of a semitone in this alternative temperament with pure fifths.
The beat frequencies in the table relate to the most obvious (and slowest) beats which the ear would usually perceive between two principal-toned stops sounding the intervals specified. In the case of the octave, this beat arises between the fundamental of the upper note and the second harmonic of the lower. For the fifths it arises between the second harmonic of the upper note and the third harmonic of the lower. For the fourths it arises between the third harmonic of the upper note and the fourth harmonic of the lower. For the major thirds it arises between the fourth harmonic of the upper note and the fifth harmonic of the lower.
A few authors have proposed temperaments using pure fifths from time to time, and that due to Serge Cordier is probably the best known . I am grateful to a French speaking correspondent  for pointing out the happy pun on his name, whose meaning can be bent to denote one who makes or mends strings. In view of this it is apposite that Cordier’s work relates primarily to the piano, though my correspondent suggested it may also have been tried on the organ . On the piano there seems little doubt that a temperament with pure fifths has merit in the opinion of some whose judgement most of us will respect if the following quotations can be taken at face value . Speaking of his own Steinway which had been tuned according to Cordier’s scheme, Lord Yehudi Menuhin was reportedly impressed after a concert during which he played the violin while his sister Hephzibar accompanied him on this piano. Apparently he said “I have never heard a piano sound so free, with such a rich tone”. And Paul Badura-Skoda apparently said “All that is great is simple. This fundamental truth applies also to the new system of tuning of Serge Cordier, who obtains astonishing results by means of just fifths and imperceptibly stretched octaves ”.
Several variants on Cordier’s scheme apparently exist and for these the literature should be consulted. This article describes the application of a pure-fifths temperament to the organ rather than the piano or any other instrument.
Putting the arithmetic aside for a while, some of the more obvious aural and musical effects of a temperament with pure fifths will now be described. The temperament is defined by the parameters in Table 1 above and its most important musical attributes are:
1. The fifths are all pure. At first sight this might be thought to have beneficial implications for the quint ranks in mixtures and mutations. However while the fifths within an octave are indeed pure, extended quint intervals such as twelfths which cross octave boundaries are not. Therefore one will not obtain quite the effects from mixtures with this temperament which one gets (in certain keys) with the same mixtures in pure-octave unequal temperaments which also contain some pure fifths. But nor will the mixtures sound quite the same as they do in conventional Equal Temperament with no pure fifths.
2. The fourths are all sharp by the same amount (3.35 cents), but the beats vary in frequency depending where they are within the keyboard compass. Table 1 shows that the beat frequency between A 440 and the fourth above is a rather uncomfortable 3.4 Hz at 8 foot pitch, nearly twice as fast as in Equal Temperament. This is a disadvantage of this temperament.
3. The octaves are all slightly sharp by the same amount (3.35 cents), but the beats between them vary in frequency depending where they are within the keyboard compass. Table 1 shows that the beat frequency between A 440 and its octave above is about 1.7 Hz at 8 foot pitch.
4. The major thirds are slightly sharper than in conventional Equal Temperament, the difference between them in the two temperaments being 1.12 cents. The difference is negligible in practice considering that the thirds are significantly out of tune in Equal Temperament in any case, as shown by Table 1. However this does not mean that the sharpened thirds should be regarded as any less objectionable than in Equal Temperament, and the sometimes unsatisfactory effect of third-sounding mutations (tierces etc) in Equal Temperament will persist with this temperament.
5. All keys can be used.
6. In theory there is no key colour or key flavour between the different keys because we are using an equal temperament in which all adjacent semitone steps are equal in terms of their frequency ratio. This attribute is the same as in conventional Equal Temperament.
7. The organ exposes the impure tuning between various stops at several octavely-related pitches in a unique manner not available with the piano or any other instrument (except maybe a large harpsichord also having stops at different pitches). The temperament is therefore well suited to the organ in this respect.
With impure octaves on the organ the question arises how to treat stops of other pitches, a problem which does not arise with the piano. The same approach is adopted here as for the offset octaves case considered previously, and it will now be repeated for convenience.
In conventional temperaments with pure octaves, octave pitches at 4 foot, 2 foot, etc are simply tuned true with the 8 foot ranks. However in this case an answer to the tuning question presents itself if we consider what would happen with an extension organ, in which an extended rank had been tuned with impure octaves. The impure octaves would then appear automatically for all the pitches derived from this rank. For reasons of compatibility when extension is not used, as in a ‘straight’ organ, it therefore seems desirable to tune the various ranks in the same way. Therefore it will be necessary to tune each stop separately against the 8 foot rank, setting the octave pitches slightly impure on a note by note basis in the same way they would appear if derived from a single extended rank.
Fifth-sounding ranks (such as nazards and twelfths) and third-sounding ranks (such as tierces and seventeenths) in mixtures and mutations can be tuned true to unison pitch as in normal practice. This means tuning them so there is no beat between the third harmonic of the unison and the twelfth, or between the fifth harmonic of the unison and a tierce.
It is not straightforward to set up this temperament by ear. At first sight it might be assumed that one would set the pitch standard of the appropriate note (e.g. A = 440 Hz), and then simply tune a principal-tone stop from it by walking up and down the keyboard using pure fifths alone. In fact the five octave compass of the organ keyboard means that one could not reach the necessary twelve fifths, for which one would need at least seven octaves. Of course, the fifths and fourths in the middle octave could be tuned in the usual way, tuning the fifths pure but tempering the fourths 3.35 cents sharp, and in fact this would be easier and quicker than tuning them to Equal Temperament in which all twelve intervals would need to be tempered. However, propagating the temperament across the range of the keyboard by tuning corresponding notes in the various octaves would then require all of them to be tempered note by note because the octaves are not pure. The whole process would then need to be repeated for stops of other footages on a stop by stop and note by note basis because these could not be tuned pure to the datum rank.
An experienced tuner might not be daunted by these problems, but others would probably find it easier to use an electronic tuning meter. To assist this process a Microsoft Excel tuning chart can be downloaded here of which an extract is shown below:
Table 2. Extract from Excel tuning chart for tuning the pure fifths temperament
The chart has five sets of columns, each set structured as in Table 2 above. This illustrates tuning data for the bottom octave of an 8 foot rank, and the complete spreadsheet continues the data downwards to cover five octaves. Together with the other four sets of data, pitches from 32 foot to 2 foot are covered for a five octave compass.
The user can type the desired pitch standard into the spreadsheet (e.g. A = 440 Hz), whereupon the data adjusts itself as necessary. For each note at each pitch, Table 2 shows that the absolute note frequencies are tabulated together with their deviations in Hz and in cents from conventional Equal Temperament. This should provide sufficient data to be compatible with most tuning meters and the frequency synthesisers in digital organs.
In the final analysis the aural and musical effect of any temperament is what matters. By far the best way to assess this temperament is to set it up on an organ, either piped or digital, and play on it and with it over a considerable period. Nothing which can be said or described here can approach its feel and capabilities, its advantages and drawbacks. However a couple of mp3 excerpts are appended which might give some slight flavour of how it sounds.
The piece in each case is an excerpt from J S Bach’s Prelude in E flat (BWV 852) from book 1 of his Well-Tempered Clavier collection, played firstly on conventional Equal Temperament with pure octaves and then on this temperament with pure fifths. This piece was chosen because it is an interplay between melodic and chordal elements from the outset and therefore it enables the temperament to be heard in both kinds of music. The excerpts were played on a small digital chamber organ with the following stop list, tuned to the two temperaments in turn:
The 8, 4, 2 and 1 foot stops were used together so that some idea can be gained of how the temperament sounds in a chorus with multiple octavely related pitches, which of course are not tuned pure one with the other in this temperament.
Prelude in E flat. BWV 852. (extract - J S Bach) - 2m 37s/2.41 MB
(played successively in two equal temperaments, first using pure octaves then pure fifths)
Some subjective remarks about the temperament include the fact that it sounds significantly warmer, richer and better in tune than I had anticipated before actually trying it. The organ throws up the attributes of this temperament differently, indeed probably better, than does the piano because the various stops at several octave pitches – which the piano does not have - are not tuned pure as they would normally be as discussed already. The temperament is therefore well suited to the organ in this respect at least. It also has some of the pleasing characteristics one experiences with certain unequal but “good” temperaments with pure octaves. For instance, from time to time it results in some unexpected but attractive mild dissonances, though these appear uniformly in all keys because this is an equal temperament in that sense. In summary, perhaps the best way to regard this temperament is to imagine a version of conventional Equal Temperament with less sterility and more warmth than the ordinary one.
There is another factor unique to the organ which has relevance when assessing how any temperament sounds, not just this one. Because virtually every organ has a different stop list, and therefore a different selection of registers having different timbres and pitches, one has to be cautious when accepting or rejecting a temperament for the instrument. Unlike those other instruments which are often tuned to different temperaments, such as the harpsichord or the clavichord, one cannot generalise so easily with the organ. Thus, what sounds well on one organ might be less attractive on another, or what is good for the clavichord or harpsichord might be intolerable for the organ and vice versa. This is because the different stops on a particular organ have different numbers and strengths of harmonics, and the huge number of beats which arise between all the harmonics of all the stops of all the notes in use at a given instant are largely responsible for the subjective effect of a given temperament. This effect will be unique for each and every organ, and even for a single instrument it will vary depending on the stops being used.
I have demonstrated many times the remarks just made, at least to my own satisfaction. With the example above, the effect of this temperament on this particular digital organ varies depending on which stops are drawn. If the 1 foot stop is put in and the same piece then played again on the 8, 4 and 2 foot stops only, the difference between Equal Temperament and this one with impure octaves is less marked. It still sounds warmer and richer, but the relatively fast beats which arose between the 1 foot stop and the others are now absent.
We have already seen that there is potentially an indefinite number of equal temperaments using impure octaves, depending on the value selected for the parameter x in the quantity x1/12 where x is the frequency ratio of two notes an octave apart. If the octaves are pure x takes a value of exactly 2, and if they are sharp (as in Cordier's temperament described above) it is greater than 2. In principle there is no reason why one should not flatten the octaves by using x with values less than 2.
x1/12 is the frequency ratio of the twelve semitones within an octave in the case where they are equal. Again, there is no reason in principle why one has to temper them equally, and by choosing frequency values empirically for the semitones (as is done in conventional unequal temperaments with pure octaves) yet further opportunities arise for an additional range of unequal temperaments with impure octaves.
Therefore some remarks follow about other temperaments which can be developed on this basis. The discussion is brief, partly to keep the length of this already long article within reasonable bounds, and partly because work in these areas has not yet progressed far enough to report it more fully.
The temperament discussed above used sharpened or stretched octaves, and it rather goes against intuition to consider the opposite situation in which the octaves might be flattened. In such cases the value of x will be less than 2, and the frequency ratio of adjacent semitones will be x1/12 for equal tempering. The counter-intuitive aspect is because the history of temperament is dominated by the battle to squeeze the intervals into a pure octave in a way that minimises the tuning problems caused by flattened fifths, and making the octaves even smaller might be thought to exacerbate the problem. Nevertheless, the flexibility of impure octaves means that one can investigate temperaments based on flattened octaves as well as sharpened ones to discover what the ear thinks of them.
A potential advantage of flattening the octave is that the thirds, considerably sharp in conventional Equal Temperament and even sharper in temperaments such as Cordier’s, can be brought better into tune. It is not possible to bring them exactly into tune while retaining equal tempering of the semitones because then the fifths and octaves will become unacceptably flat. However one approach is to choose a compromise tuning in which the frequencies of the thirds, fifths and octaves are all out of tune by about the same amount.
Considerable juggling with the numbers is possible when searching for a useable temperament with flattened octaves, but an interesting convergence appears if the octave is flattened by 11.16 cents. In this case the fifths will be 8.47 cents flat and the thirds 9.97 cents sharp. Thus the tempering of all three intervals is now approximately the same, measured in cents. In round figures they are all tempered about 10 cents away from pure. This temperament will be called “Flat Octave 1” for convenience. For comparison purposes, Table 3 below shows the tuning of these intervals in two other temperaments as well – conventional Equal Temperament (pure octaves) and the Cordier temperament discussed earlier (sharpened octaves):
Table 3. Comparison of some intervals in various equal temperaments (cents)
In Table 3 the sizes of the 3rds, 5ths and octaves are shown by the amount they deviate in cents from pure. However, as described at length earlier in this article, what matters more to the ear are the beat frequencies of these intervals, and Table 4 tabulates these for each one together with the interval of a fourth:
Table 4. Comparison of some intervals in various equal temperaments (beat frequencies in Hz, with the lower note of each interval being A = 440 Hz)
The beat frequencies vary depending where the intervals lie in the keyboard, and the values in Table 4 are for the middle octave of an 8 foot stop with the lower note of each interval being A = 440 Hz. We see for the Flat Octave 1 temperament that, although the 3rds, 5ths and octaves are detuned by approximately the same number of cents (Table 3), the beat frequencies are not the same (Table 4). Nevertheless the differences between the beat frequencies for these three intervals have been evened out to some extent for Flat Octave 1 compared to the other two temperaments. In particular, the fast beat of the thirds in Equal Temperament and Cordier’s temperament has been considerably reduced in Flat Octave 1. This is potentially important because it is the significantly sharpened thirds which make conventional Equal Temperament sound rather coarse to many ears, and Cordier’s temperament is no better (indeed slightly worse) in this regard.
In Flat Octave 1 none of the beats between the 3rds, 5ths and octaves are anything like as fast as the fastest in the other two temperaments. Moreover, and importantly, the fourths are much better in tune than the thirds, fifths and octaves, having a beat frequency between A 440 and D of only 2.74 Hz. They are therefore much better in tune than in Cordier’s temperament and they beat only slightly faster than in Equal Temperament.
The important question is – what does the ear make of Flat Octave 1? As with all temperaments, no amount of theory and arithmetic can answer this and it can only be judged by tuning an organ to it and trying it. Having done this with a digital instrument, the main subjective features include the following:
1. Playing on a single 8 foot principal-toned stop, the temperament sounds attractive on the whole - to me. It has what might be described as a somewhat quaint, olde-worlde flavour without being too out of tune over most of the keyboard (but see 4 below). The subjective difference between it and Equal Temperament (pure octaves) is more marked than for the Cordier temperament (sharpened octaves) discussed earlier.
2. The thirds are noticeably smoother than in Equal Temperament and Cordier’s temperament.
3. The fourths are much better in tune than the thirds, fifths and octaves. They are also much better in tune than in Cordier’s temperament.
4. Towards the top of the keyboard on an 8 foot stop, and with higher pitched stops lower in the keyboard, the beats between the significantly flattened fifths and octaves become excessively prominent in my opinion. This is, of course, because their beat frequencies increase with the pitch of the notes. A solution to this problem is to progressively reduce the amount of flattening of the higher octaves as the pitches of the notes increase. This will produce a genuine distributed temperament, one whose characteristics vary across the compass from octave to octave.
5. This is an equal temperament and thus all keys can be used, but for the same reason there is no key colour.
I have found the Flat Octave 1 temperament is sufficiently attractive to encourage me to play on it at length, comparing it subjectively with the other temperaments described in this article as well as more conventional ones. The excessively sharp thirds and fourths created by sharpening the octave, as in Cordier’s temperament, are less objectionable with this one where they are better in tune. A disadvantage is the noticeable beats of the fifths and octaves at higher pitches, though this could be improved by reducing the amount of octave flattening towards the top of the compass as mentioned in 4 above. On balance this temperament therefore merits further study in my view. So watch this space!
Unequal temperaments are those in which the ratio of the frequencies of adjacent semitones can take any value from note to note, unlike equal temperaments in which the ratio remains constant. These definitions of “equal” and “unequal” hold for all temperaments, regardless of whether the octaves are tuned pure or impure.
With pure octaves, unequal temperaments are often lauded in comparison to the one and only Equal Temperament which is possible in that situation. In a sense this is only to be expected because there is theoretically an infinity of unequal temperaments, and it is therefore unsurprising that there is a correspondingly inexhaustible treasure trove of experience to be discovered. One factor characterising the unequal temperaments which does not apply to Equal Temperament is that they possess key colour, a subjective flavour which heightens the effect of modulation into different keys. In fact it is an excess of key colour which makes certain keys unusable in some unequal temperaments because of the presence of “wolf” intervals which are grossly out of tune.
The same applies to equal and unequal temperaments which use impure octaves in that the equal temperaments, such as those described in this article, do not have key colour whereas the unequal ones will have. The major difference between the pure and impure octaves situation, as we have observed already, is that there is an indefinite number of both unequal and equal temperaments, unlike in the case of pure octaves in which there is only one Equal Temperament.
Therefore the obvious next step is to investigate unequal temperaments using impure octaves if only because these will possess key colour which the equal temperaments do not. Currently I have not gone this far. However the availability of impure octaves ought to give an additional degree of freedom when developing unequal temperaments which does not apply if the octaves are pure, consequently the unequal temperaments which can be developed would likely be original in the sense they had not been heard before.
There is an extra subjective dimension to the problem of devising unequal temperaments which does not apply to equal ones, and this relates again to the key colour issue. One cannot proceed until one has decided on the desired intonation of the 24 keys (12 major and 12 minor) in terms of the purities or otherwise of the intervals they contain. For instance, one might decide that certain keys must contain thirds as nearly pure as possible. Such constraints as these in the unequal temperaments using pure octaves with which we are familiar enable us to select which unequal temperament to use. However with impure octaves the problem is posed the other way round – instead of selecting a temperament which already exists because its intonation is pleasing in some way, we have to develop the temperament after having first decided which intervals and in which keys give the pleasurable effects which are sought.
Until we answer this question to our satisfaction we cannot proceed, but the answer will doubtless be different for different individuals. In this sense there cannot be a single unequal temperament using impure octaves which will please everybody, just as with the pure octaves case.
The work described in this article began by asking why tempered octaves have seldom been considered in the long history of tuning keyboard instruments. Although a definite answer is elusive, a probable reason is that temperaments with impure octaves are difficult to tune by ear, and therefore it is only recently that the advent of electronic tuning devices has made them more accessible for study. In the same timescale the appearance of digital musical instruments has facilitated experiments with these and other temperaments. It is therefore perhaps not coincidence that one of the few existing temperaments using impure octaves, that due to Cordier, did not arise until the 1970’s.
This article has described research on temperaments with impure octaves, tuned both sharp and flat. The work focused exclusively on temperaments appropriate for the organ, and it was pointed out that the characteristics of a temperament suitable for this instrument might be less attractive for others, and vice versa. This is partly because of the sustained nature of organ tones as well as the availability of stops at many pitches which other instruments do not possess. The fact that most of the stops constituting an organ chorus are octavely related makes the study of temperaments with impure octaves uniquely interesting for the instrument.
Three temperaments using impure octaves were described, one using offset octave tuning, and a second using Cordier’s recipe where the octaves are sharpened and the fifths pure. Disadvantages of the latter include the thirds which are even sharper than in Equal Temperament, and the fourths which are much sharper. The third temperament used flattened octaves. Called Flat Octave 1, this had the advantage that the significantly sharp thirds in conventional Equal Temperament and the even sharper ones in Cordier’s temperament could be brought closer into tune. The fourths were also much better in tune than in Cordier’s temperament. A disadvantage of Flat Octave 1 is the detuned fifths and octaves which become noticeable because of their beat frequencies at higher pitches. This could be overcome by progressively reducing the amount of octave flattening towards the top of the keyboard, thereby producing a distributed temperament whose characteristics vary across the compass.
Some interesting generalisations were mentioned which appear when using impure octaves, the most important being that an infinity of equal temperaments become available instead of there being just one in the case of pure octave tuning. All of the temperaments with impure octaves discussed in the article are equal temperaments, which means they can be used in all keys irrespective of their different characters.
The next step is to investigate unequal temperaments using impure octaves if only because these will possess key colour which the equal temperaments do not. The availability of impure octaves ought to give an additional degree of freedom when developing unequal temperaments which does not apply if the octaves are pure, consequently the unequal temperaments which can be developed would likely be original in the sense they had never been heard before.
1. “Temperament – a study of Anachronism”, C E Pykett, 2006. Currently on this website (read).
2. ibid, Part 4.
3. ibid, Part 2.
4. One correspondent insisted that his harpsichord and clavichord exhibited negligible tuning instability over many months and that serendipity could therefore have played no part in the development of Baroque temperaments as I had suggested. However he failed to remember that his instruments reside in a centrally heated and air conditioned room, which would have reduced environmental effects on their wooden frames to insignificance once they had acclimatised to the conditions. Werckmeister, Bach and their contemporaries had no such luxuries to offset the climatic extremes where they lived in continental Europe.
5. This has led some to state that J S Bach avoided writing his major organ works in A flat major so they could be played on Silbermann’s organs. One only has to examine the way his harmonies modulate through various keys to see the fallacy of this argument. For instance, A flat major is an important element of the harmonic texture in the ‘great’ Prelude in C major (BWV 547a). On Silbermann’s temperament this piece, while perhaps not unplayable, sounds gross in places to most ears.
6. In the same way that an Equally Tempered semitone represents a twelfth part of a pure octave, a cent represents one hundredth of an Equally Tempered semitone. Both are derived from the ratio of two frequencies, regardless of where they lie within the keyboard compass. It is important to bear in mind that the ratio of two frequencies is a dimensionless number with no units. Unlike a beat, it does not express the difference between the frequencies, which would be measured in units of Hz for example.
7. These figures assume Equal Temperament together with today’s usual tuning standard of A = 440 Hz.
8. Appendix 1 to “Temperament – a study of Anachronism”, C E Pykett, 2006. Currently on this website (read).
9. “The Well-Tempered Organ”, Charles A Padgham, Oxford, 1986.
10. “A Dorset Temperament?”, C E Pykett, Organists' Review, August 2004. Also currently on this website (read).
11. “Piano bien tempéré et justesse orchestrale”, Serge Cordier, Buchet Chastel, Paris, 1982.
12. Gilles Moreau, private communication, June 2008.
13. In the course of some extremely helpful correspondence, M Moreau also mentioned that the firm of Kleuker Orgelbau may have tuned organs to Cordier's temperament. This temperament attracted favourable reviews from Jean Guillou who wrote the Foreword to Cordier's book .
14. These quotations appear in Cordier’s book .