Temperament and Timbre - interactions between temperament and registration on the organ
Posted: 3 October 2018
Revised: 3 October 2018
Copyright © C E Pykett 2018
Abstract. The literature on musical temperament is replete with descriptions of keys which are said to be 'intolerable' and therefore 'unusable', and presumably such statements are intended to apply to all keyboard instruments. However this generalisation benefits from closer scrutiny, because not all instruments are the same where temperament is concerned. In particular, the many timbres or tone colours of the organ enable the key flavours of an unequal temperament to sound significantly different depending on which stops are used. This article uses audio examples to show that the differences can be remarkable, such as the case where even the notorious Wolf interval in quarter-comma meantone tuning becomes quite docile and usable when suitable stops are selected. These effects are unique to the organ because no other instrument has its wide range of tone colours and combinatorial possibilities.
It is shown why this interaction between temperament and timbre occurs by explaining firstly how key flavour is influenced by the beats of tempered consonant intervals, and then how the beats arise from specific pairs of harmonics in the notes defining each interval. It is then easier to see why different stops can affect key flavour, because timbre as well as the beat patterns are both strongly affected by the numbers and strengths of the harmonics in the sounds.
A consequential phenomenon is also discussed in which the subjective flavour of triads and other chords depends on how the composer constructed the harmony of a piece. In particular, the inversions in which the chords appear and whether they are written in close or open positions can result in marked changes in key flavour. This also is demonstrated with audio examples and explained as before in terms of beats and harmonics.
When unequal temperaments were more common than they are today, one can probably assume that composers would have deliberately wandered in and out of the less agreeable keys to add interest to their music. Moreover, it would not be surprising if they assumed that organists with sufficient familiarity would have learnt how to amplify or attenuate those effects by choosing their registration appropriately. Indeed, it would be more surprising if they did not, given that they would have come across the effects as part of their everyday experience. Perhaps most importantly, the article shows that keys which are routinely dismissed as 'intolerable' in the literature on temperament are not necessarily so on the organ, which has the ability to soften their effects because of the range of different timbres available.
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The subjective effects of keyboard temperaments (systems of tuning) continue to excite vigorous discussion as they have done for centuries past. In many temperaments not all of the twelve major and twelve minor keys can be used, at least in homophonic (chordal) music rather than that consisting of a purely melodic line. These unusable keys are considered so out of tune as to be intolerable for the purposes of harmony, at least that form of it used during the common practice era of European tonal music (1650 to 1900 approximately). However, aside from these extreme keys all useable temperaments have others which sound purer and more attractive (if they do not have them then a temperament cannot be considered useable). These subjective differences between the various keys make up the key colour or key flavour spectrum of a temperament, and because the flavours differ with key these temperaments are called unequal. This distinguishes them from equal temperament, almost universal today, in which all keys are useable but they have the same flavour (or lack of it, depending on one's point of view).
So far, so good - none of this is controversial. But one can go further. There is a hidden implication in the foregoing that all keyboard instruments are the same as far as temperament is concerned. In other words, it is implied that the range of key flavours experienced on a harpsichord, say, will be the same as that on a clavichord, piano or organ tuned to the same temperament. So if a given key is deemed 'good' or 'bad' on just one of these instruments, then it is left unspoken that the same will apply to all the others. The majority, if not all, literature on temperament apparently makes this assumption without pausing to consider it. However it is the purpose of this article to demonstrate, with the help of audio examples, that this is not so. Although this sameness might apply to all other keyboard instruments, it does not apply to the organ, the reason being that the organ is the only instrument which has several widely different tone colours available at several pitches selected by the various stops. It is shown that the subjective qualities of a temperament enshrined in its key flavour spectrum vary depending on how a piece is registered, firstly by reviewing how key flavour arises in the first place and then how it is affected by one's choice of stops. I pointed out the phenomenon in 2016 in two articles elsewhere on this website (, ), and it is now considered here in more detail.
The somewhat arcane arithmetical background to the subject of temperament is not covered here because the article has been deliberately structured to do without it, as it would have proved a distraction. However those wishing for an introduction to this aspect can find it in an earlier article I wrote in Organists' Review which is now available on the site , and Stephen Bicknell's online essay might also be helpful . More detailed coverage is included in Padgham's well known treatise The Well-Tempered Organ , though this must be used with caution in view of the many numerical errors it contains as I pointed out in reference .
We first consider why unequal temperaments have good and bad keys and what these terms mean in terms of tuning, intervals and harmony.
The entire edifice of musical harmony is based on the triad; there are twelve major and twelve minor triads to the octave. A root position major triad consists of a minor third sitting on top of a major third. Taking the triad C-E-G as an example, the lower interval C-E is a major third and the upper interval E-G a minor third. But in a minor triad the two thirds are reversed - it consists of a major third sitting on top of a minor third. In the minor triad C-E♭-G, the lower interval C-E♭ is now the minor third whereas E♭-G is the major one. Also, in both of these triads the bounding notes (C and G in this example) form a perfect fifth. Therefore three intervals are involved in any root position triad - major and minor thirds and a perfect fifth. These are examples of consonant intervals, those which sound purest and most acceptable when they are exactly or very nearly in tune. We also need to add other consonant intervals to the list which are the inversions of those just mentioned, because an interval and its inversion involve the same notes. Inverted intervals enable inverted triads to exist in which the notes remain the same but their ordering differs. For example, a major third (e.g. C-E) when inverted becomes the minor sixth E-C, used as the bounding notes of the inverted triad E-G-C. Thus the three inverted intervals are major and minor sixths and perfect fourths, inversions of minor and major thirds and perfect fifths respectively. Finally we have to add the octave to the list of consonant intervals for completeness.
In any temperament all of the consonant intervals must be tuned as exactly as possible, otherwise dissonances can sometimes be heard in certain keys (defined by the notes of their respective triads) which can reach intolerable proportions. Temperaments such as these place limits on the degree to which composers can modulate from one key to another during a composition, and thus on their harmonic freedom. Unfortunately, tuning perfection in all keys is impossible to achieve on the standard keyboard having only twelve notes to the octave regardless of temperament, and it is the reason why the problem of temperament exists in the first place. There are two general approaches to the predicament (other than the impractical one of increasing the number of physical notes making up each octave across the keyboard). Either the tuning purity of some intervals is sacrificed to the extent they become unusable so that the others can be nearly or exactly in tune, or all of them are slightly detuned by an amount deemed to be bearable. The first method is used, for example, in quarter-comma meantone (a grossly unequal temperament with several very poor keys but also some very good ones), and the second is used in equal temperament (all keys are useable but slightly out of tune). Neither approach can solve the problem completely, a fact which continues to fuel today's interminable debates about musical temperaments. Thus all temperaments must perforce include some deliberately mistuned or 'tempered' intervals.
What is it about a musical interval (two notes played at once) that makes it sound 'in tune' or 'out of tune'? Indeed, what do these terms actually mean? It is a defining property of all consonant intervals that they can be brought exactly into tune, or tuned pure. (This is only true when considering an interval in isolation, since we have already noted that it is impossible within the confines of any temperament to have all the intervals of all twenty four keys simultaneously in perfect tune). When a consonant interval is tuned pure on the organ its sound is perfectly stable for as long as the two notes are held down. On the other hand, if the interval is not quite in tune the aural stability vanishes, to be replaced by a regular wavering sound called a beat. The more the tuning departs from pure, the faster the beat gets. Beat frequencies of more than one or two cycles (wavers) per second become noticeable and more or less irritating to the listener if a prolonged interval or chord is sustained on the organ. (The frequency unit of cycles per second is properly written today as Hz to honour the 19th century German scientist Heinrich Hertz). Beyond a certain point, higher beat frequencies lead to a judgment of progressive out-of-tuneness which eventually becomes musically intolerable as the beats continue to get faster.
Thus the presence or absence of a beat tells the ear whether an interval is in tune or not. But what causes them? If two organ pipes of the same pitch, such a Principal and a Gamba at middle C, are not quite in tune you will hear a beat when playing on both simultaneously. The 'interval' between these pipes ought to be an exactly-tuned unison (though the unison is not generally considered to be an interval as such). Thus there should be no beat in this case, but if there is one it means the pipes are slightly mistuned as they often will be in practice. The beat frequency will equal the difference between the fundamental frequencies of the two pipes. Middle C on an 8 foot stop often has a frequency of about 262 Hz nowadays, so if the Principal pipe has exactly this frequency but the Gamba has drifted to 263 Hz, they will beat audibly at 1 Hz - one waver per second.
Now take the example of a fifth, played on a single stop this time, where things get more complicated. If the fifth is tuned pure so that there is no beat, physics tells us that the fundamental frequency of the higher note will be exactly 1.5 times that of the lower. So if the lower note is middle C at 262 Hz, the frequency of the upper one (G) will be 262 times 1.5 which equals 393 Hz. But hang on - we saw above that if the fundamental frequencies of two mistuned unisons differ by only the tiniest amount such as 1 Hz, then there will be a slow beat. But now we are being told that if the same two frequencies differ by a much greater amount (131 Hz in this example) the interval will be in perfect tune and so the beat vanishes again. What's going on? In fact the beat still exists but, as just seen, it is at a much higher frequency than before. For any fifth, the beat frequency is at half the fundamental frequency of the lower note (call this f), because 1.5f - f equals 0.5f, and this corresponds to the octave below the lower note. So the beat for the fifth between middle C and middle G will be at 131 Hz, the frequency of tenor C. We do not hear it though because it is now far too fast for the ear to discern as a beat, and neither do we hear it as a separate musical tone pitched at tenor C because there is no acoustic power in a beat. A beat is not a separate sound in the atmosphere but merely a phase interference phenomenon existing between two other sounds. These rather confusing matters are discussed in more detail in another article , but for present purposes you do not really need to understand every last nuance of beats if you are content to accept them for what they are.
But things do not stop here. If we now detune the fifth to become slightly sharp or flat from pure, a slow beat appears just as it did for the two mistuned unison pipes discussed earlier. (To be rigorous, this does not hold for some organ tones as we shall see later). Where is this slow beat coming from? The beat between the two fundamental frequencies is still there but far too fast to discern, yet we hear this new, slow, beat in addition. In fact it arises from the higher harmonics present in the two sounds. For a detuned fifth, the beat arises between the third harmonic of the lower note and the second harmonic of the upper. For a perfectly tuned fifth these harmonics coincide exactly in frequency so we hear no beat, but when the interval is not quite in tune a slow beat emerges as the difference between these two harmonic frequencies.
Beats arise between various harmonics of the two notes in any musical interval, not just fifths, and historically they were important because they enabled intervals to be tuned by ear for centuries before electronic tuning devices appeared in the 1980s. They are also partly, often largely, responsible for the subjective key flavours of a temperament. This can be appreciated by remembering that if the intervals of a particular key are perfectly tuned or nearly so, then that key will sound much purer to the ear than if it contains significantly mistuned (tempered) intervals which generate rapid and aurally-intrusive beats. The harmonic pairings responsible for generating the slowest and most noticeable beats in all the tempered consonant intervals are tabulated below. When an interval is perfectly tuned its harmonics in the table have identical frequencies and so there is no beat.
Table 1. Harmonics responsible for generating the most noticeable beats in tempered consonant intervals
We have now reached the nub of the article, which is to show how and why one's choice of stops (registration) can affect the key flavours of a temperament set up on the organ. As mentioned earlier, this sets the organ apart from other keyboard instruments because none other has the range of timbres or tone colours enjoyed by the organist.
First of all have another look at Table 1. A notable feature is that some of the harmonics are getting quite high up in the harmonic series of an organ pipe. For instance, the 8th harmonic involved in beat generation for the interval of a tempered minor 6th barely exists or is completely absent in many flute stops, whereas it will usually be stronger for principals (diapasons) and definitely so for strings and reeds. The point is that if one or both of the harmonics involved in generating the beat for a given tempered interval is weak or absent, then the beat itself will also be weak or absent. Conversely, if both harmonics are strong then the corresponding beats will be strong and therefore noticeable. Consequently the flavour of a particular key in a particular temperament will depend on the strengths of these harmonics because key flavour is strongly influenced by the shimmering aural tapestry created by the beats between the tempered intervals used by that key. Key flavour will therefore differ depending on the stop(s) used if the strengths of certain harmonics are critical to beat formation in this key, because their strengths vary with timbre.
If this explanation has been difficult to follow, the pictures in Figure 1 might help.
Figure 1. Frequency spectra of an Open Diapason and a Stop Diapason pipe
This shows the frequency spectrum of an Open Diapason and a Stop Diapason pipe at middle C on an historic Victorian organ. The harmonics are identified by the small red circles and it is obvious that the Open Diapason has far more of them than its stopped cousin. It has at least 15 harmonics, although the 18th and 19th also pop up again out of the noise towards the right hand edge of the plot. This means that all of the harmonics listed in Table 1 will easily be heard because none of these go beyond the 8th, and therefore beats between all of the consonant intervals will also be heard if they are not tuned perfectly (i.e. when they are tempered to suit a particular temperament). However the Stop Diapason is completely different, having only six harmonics. Of these, the odd-numbered ones are more prominent than the evens owing to the action of stopper in the pipe, this difference being characteristic of stopped flutes. But another aural feature of this stop is that, unlike the Open Diapason, the beats corresponding to many tempered intervals will either be weak or absent. Table 1 shows that even the beat of a tempered fifth would not be heard if it involved this particular pipe because its second harmonic barely exists. Consequently a Wolf fifth would sound quite docile if this pipe was involved, whereas this certainly would not be so for the Open Diapason. The same also goes for most of the other intervals. So this example confirms that the strengths of the beats generated by the intervals in a temperament depend intimately on the timbres of the pipes (i.e. the stops) being used. And because key flavour is so closely connected with the beats of tempered intervals, it must therefore depend on the registration chosen by the performer. As pointed out earlier, this phenomenon makes the organ unique among all other keyboard instruments. Temperament is not a one-size-fits-all subject which affects all keyboard instruments in the same way, since its effects on the organ depend on one's choice of stops.
Before proceeding to demonstrate these effects musically, a few words about the organ used to derive the spectra in Figure 1 are appropriate. This historic instrument was built originally by J W Walker in 1858 and it was described by the present organist, Paul Minchinton, in Organists' Review . It was he who stoically recorded the sound samples in a freezing cold church from which the spectra in Figure 1 were derived. When built, the organ was tuned to some unknown unequal temperament because the records show it was subsequently brought into equal tuning in 1867. Unequal temperaments were still relatively common in England at that time though exactly what they were remains something of a mystery, Nicholas Thistlethwaite writing in the making of the Victorian organ that "reliable information is hard to come by" .
For the purposes of these demonstrations I have therefore selected a grossly unequal tuning in which the sheer awfulness of the worst keys can scarcely be exceeded - this temperament is quarter-comma meantone. The most evil key is A flat major because it exposes the Wolf fifth between A flat and E flat which is hideously sharp by over one third of a semitone from pure. The rationale behind this apparently perverse choice is to show that even wolves can be tamed by a judicious selection of stops. The following demonstrations were recorded using the Prog Organ virtual pipe organ into which the complete Ponsbourne sample set was loaded, and more information about this is available in the article at reference . All the examples were recorded using the Open and Stop Diapasons on the Great Organ, the same stops whose spectra were shown earlier in Figure 1.
The first example is the hymn tune 'Charity' by Stainer played in A flat, using the Open Diapason firstly tuned to equal temperament and then to quarter-comma meantone:
There can be no doubting which is the meantone version! One does not need to listen carefully to become only too aware of the dreadful dissonance of the Wolf fifth, evidenced by its unmistakable beat, which is particularly noticeable in the sustained final chord.
The next example is of the same hymn but this time using the Stop Diapason, again played successively in the two temperaments:
This time the meantone version sounds appreciably more acceptable. The fast-beating Wolf almost seems to have disappeared, leaving only a somewhat quaint and unusual key flavour which, if not exactly attractive, one can get used to. So these two examples demonstrate the quite different subjective effects of different stops which can arise when playing on the same unequal temperament.
Why did the Wolf virtually disappear when using the Stop Diapason? Table 1 shows that for a beat to appear when playing any fifth the second and third harmonics of the notes must be present, yet Figure 1 showed that the second harmonic was very weak for the Stop Diapason. Consequently a strong beat was not generated in this case and so we did not experience the full fury of the Wolf.
The timbres of the stops used in unequal temperaments are not the beginning and end of the matter however. The music itself, in terms of how the composer distributed the notes across the page, is also important because this can modify key flavour as well. This is illustrated by the following two examples. The first is an A flat triad played in close position centred around middle C. The notes played are shown below in Figure 2(a).
Figure 2. Notation examples for chord demonstrations
The Open Diapason is heard first, followed by the Stop Diapason. Quarter-comma meantone tuning is used in both cases:
Next the same triad is played but in an open position in that the fifth (E flat) is now an octave higher, shown in Figure 2(b) above:
In this case the Wolf has returned with a vengeance even when using the Stop Diapason, which thus far has suppressed it! On first acquaintance this might seem to be an extraordinary result because the phenomenon did not arise for the same chord in close position with this stop, nor was it a major problem when playing the previous hymn tune. Thus the situation obviously merits closer attention. What is happening here is that the top E flat of the chord in Figure 2(b) has the same fundamental frequency as the suppressed second harmonic of the E flat an octave below in Figure 2(a). So, in effect, we have supplied artificially a strong second harmonic for the latter note which the Stop Diapason did not possess of itself. Consequently we now have the two necessary ingredients for the beating Wolf fifth to be heard for the Stop Diapason. As Table 1 shows, the first ingredient is the third harmonic of the root note (the A flat below middle C), and the lower spectrum in Figure 1 shows how strong this is. The other ingredient is the second harmonic of the fifth (now supplied artificially by the E flat above treble C).
We have therefore identified two features affecting how we perceive key flavour on the organ. One relates to the choice of stops and the other to the way notes are allocated to the chords which make up the harmonic texture of a piece. Note that the use of 'harmonic' here has a dual meaning - it refers both to the vertical structure of a composition in terms of those notes which sound simultaneously (i.e. conventional harmony), as well as to the harmonics present in the sound of each note. Both features obviously apply to all music other than a single melodic line, and as we are discussing temperament it is appropriate to take another example, this time from J S Bach's Das Wohltemperierte Klavier (WTK) . The Prelude in A flat major in volume 1 (BWV 862) presents as its first chord a close position triad as written in Figure 2(a). In bar 3 a similar chord occurs but in an open position. Both expose the Wolf fifth in a meantone temperament, assuming it is placed between A flat and E flat as it usually is. From the foregoing discussion, it should now be possible to understand why both chords are grossly dissonant using an Open Diapason tuned to quarter-comma meantone, whereas the opening chord of the piece sounds unexceptional using a Stop Diapason. You can try it for yourself if you have a digital organ with selectable temperaments and the appropriate tone colours. So was Bach teasing us here by writing pieces which come over quite differently when using different stops tuned to the unequal temperaments which he was often confronted with? Was one purpose of the WTK to show that even the 'worst' keys in unequal temperaments can nevertheless be used with a suitable choice of stops? We cannot know the answers to such questions, but it would be unwise to assume that they never passed through the formidable mind of the great man. Otherwise we are in danger of implying that we lesser mortals have thought of something which everyday experience would likely have thrown up yet which somehow eluded him.
The literature on musical temperament is replete with descriptions of keys which are said to be 'intolerable' and therefore 'unusable', and presumably such statements are intended to apply to all keyboard instruments. However this generalisation benefits from closer study, because not all instruments are the same where temperaments are concerned. In particular the many timbres or tone colours of the organ enable the key flavours of an unequal temperament to sound significantly different depending on which stops are used. The article used audio examples to show that the differences can be remarkable, such as the case where even the notorious Wolf interval in quarter-comma meantone tuning becomes quite docile and useable when suitable stops are selected. These effects are unique to the organ because no other instrument has its wide range of tone colours and combinatorial possibilities.
It was shown why this interaction between temperament and timbre occurs by explaining firstly how key flavour is influenced by the beats of the tempered consonant intervals, and then how the beats arise from specific pairs of harmonics in the notes defining each interval. It was then easier to see why stops with different timbres can affect key flavour, because timbre as well as the beat patterns are both strongly affected by the numbers and strengths of the harmonics in the sounds.
A consequential phenomenon was also discussed in which the subjective flavour of triads and other chords varies depending on how the composer configured the harmony of a piece. In particular, the inversions in which they appear and whether they are written in close or open positions can result in marked changes in key flavour. This also was demonstrated with audio examples and explained as before in terms of beats and harmonics.
When unequal temperaments were more common than they are today, one can probably assume on the basis of the foregoing that composers would have deliberately wandered in and out of the less agreeable keys to add interest to their music. Moreover, it would not be surprising if they assumed that most organists with sufficient experience would have learnt how to amplify or attenuate those effects by choosing their registration appropriately. Indeed, it would be more surprising if they did not. Perhaps most importantly, the article has shown that keys which are routinely dismissed as 'intolerable' in the literature on temperament are not necessarily so on the organ, which has the ability to soften their effects because of the range of different timbres available.
"Handel's Temperament revisited", an article on this website, C E
Pykett, August 2016.
https://www.albany.edu/piporg-l/tmprment.html (accessed 23 September 2018)