Mixture Stops  

Colin Pykett

 

 " ... the mixture stops of the organ, the effect of which is traditionally admired by many people, but which in reality are an open door to the most dreadful confusion ... 

 

 ... [they] give to each note simultaneously the notes of the major chord, in other words the major third and perfect fifth. Barbaric it is nonetheless, since quite apart from the harmonic congestion produced, it necessarily introduces into the harmony the most dreadful chaos through the unavoidable inversion of chords ...

 

... the ignorance of the middle ages, groping for the laws of harmony, must probably be credited with the introduction into organs of these monstrosities which routine has preserved and bequeathed to us. It is to be hoped that they will gradually disappear ... "

 

Hector Berlioz [1]

Posted: 9 September 2014

Revised: 20 September 2014
Copyright © C E Pykett 2014

Abstract.   Shows how mixtures are constructed in terms of the number of ranks, the starting composition in the bass and how it ends up in the treble, and the various patterns of breaks in between.  All mixtures must converge to a similar composition at the top of the compass simply because it is pointless trying to make pipes smaller than a reasonable practical limit.  Therefore differences between mixtures can only be accomplished by varying their starting composition in terms of its pitches and number of ranks.  The task of the designer is then to implement a scheme of breaks designed to reach the ultimate composition in the treble while enabling the mixture to perform its functions across the compass.  These include brightening the bass and augmenting the other chorus stops elsewhere.

Quint mixtures, those containing only octave and fifth-sounding ranks, are discussed in detail.  'Full' and 'sharp' mixtures are described and contrasted and their functions explained.  The inevitable tuning problems of mixtures are illustrated which result from including the perfectly-tuned intervals in a mixture within the tempered environment of the other stops.

Tierce mixtures, including Cornets, are described which contain a rank speaking the fifth harmonic.  It is shown that their tuning problems are major and insurmountable unless one is content with a mixture in which most major thirds are grossly dissonant across the entire compass in organs tuned to equal temperament and some others.  'Harmonics' mixtures, those which contain the seventh harmonic in addition, are also discussed though their artistic rationale could not be discerned.

The article concludes with other topics including pipe scales and the problems caused by deriving mixtures by extension.

Contents

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

Introduction

Quint mixtures

Breaks

Tuning

Full and Sharp mixtures

Cornets, Tierce mixtures and Sesquialteras 

'Harmonics' mixtures

Pedal mixtures

Pipe scales

Derived mutation ranks

Concluding Remarks

Notes and References

Introduction

Judging by his purple prose quoted above, it is clear that Berlioz did not like mixture stops.  Nor did many other musicians and organ builders in the nineteenth century, which is one reason why, fifty years later, Robert Hope-Jones landed so many important contracts in so short a space of time.  Those who see him as the sole architect of their demise do not understand that he was simply the first to fully respond to customer pressure to dispose of them completely, and his view is just as interesting as that of Berlioz:

I confess myself in agreement with those who consider that the 8ft instrument, commonly called an orchestra, possesses sufficient brilliancy, and in disagreement with those who would fain add "chorus work" in the form of a few hundred piccolos playing consecutive fifths, thirds and octaves with each and all of the individual instruments comprising it. [2] 

Yet mixtures had been part of organ building for at least a millennium, indeed the earliest medieval organs in Europe were nothing but huge mixtures, so why this sudden change in the nineteenth century?  In fact there is a good and simple reason for it which is not often explained, and this concerns loudness.  Organs have always had to be loud, and until about 1930 when electronic amplifiers and loudspeakers arrived on the scene, they were the only way to achieve it in large public spaces such as Gothic cathedrals.  I think this tends to be forgotten today.  Loudness played a part in the oppressive power politics of the medieval church, complementing the dominance of the buildings themselves in a landscape populated largely by peasants who had to be kept in their place.

Acoustic power arriving at the ears is an objective measure which is different to the subjective loudness which we perceive as sentient beings, because this latter depends not only on sound pressure level (SPL) but also on the distribution of the acoustic power across the frequency spectrum.   Duplicating ranks of pipes of the same pitch is an inefficient means of increasing loudness, and this must have become obvious in the earliest days of organ building.  At best, two identical pipes will only provide 3 decibels (dB) of additional power, a factor of two, and even this only occurs when their waveforms are in phase at the listener's ears.  Doubling the power of an audio signal, though easily detectable, is not a subjectively large loudness increment in relation to the considerable additional complication and expense of doubling up on the pipework.  This is because the ear and brain process changes in stimuli logarithmically, therefore achieving a significant increase in loudness at a given frequency means that a large increase in acoustic power is required, much larger than mere doubling.  In any event, in the worst case the sound of two identical pipes will vanish altogether if the waveforms arrive at the ears in antiphase, which is quite possible.  But it must also have been observed early on that a greater increase in subjective loudness occurred if the two pipes were at different pitches.  In this case two effects are at work - one is that there can never be phase cancellation if the two frequencies are different, and the other is that subjective loudness is a strong function of how the available power is spread across the spectrum.  Sounds of a given power appear louder if they are distributed in frequency over a significant part of the audio spectrum.  When this important effect is coupled with the fact that only low blowing pressures were available to the early organ builders, and thus only low acoustic power could be generated, it must have become obvious that the greater the spectrum occupancy, the greater the loudness.  Hence the move to choruses containing many pitches higher than that of the ground tone of the notes actually keyed to spread the acoustic power across a wide frequency range, and thus arose mixtures.

But why the move to banish them in the nineteenth century?  Because it was only then that blowing pressures could be increased from a few inches of water to anything up to fifty or beyond.  This was made possible by harnessing steam, hydraulic, gas and then electrical power for organ blowing, together with pneumatic and electric actions to open the pipe valves against the higher pressures.  There was then no need to rely on mixtures to provide loudness because, for the first time, organs could be made as loud as desired in their absence.  Hence the arrival of heavy, loud principal tone backed up by batteries of powerful reeds in the Hope-Jones, Willis, Harrison and Compton types of instrument.

But now for another quotation.  Senator Emerson Richards, the designer of the largest organ ever built at the Atlantic City Convention Hall (now Boardwalk Hall), seemed humbled when he first encountered a seventeenth century Schnitger organ in the 1940's.  He wrote of this epiphany:

"Full organ fills the rather large church with a flood of pure tone - no rumble or muddiness.  Bach ... came out with an entirely new meaning.  A precise, bell-like tone, rich in harmonics, but characterized by a lightness and transparency, gave an interest to the music never achieved by the romantic organ to which we are accustomed. There is plenty of power; the Mixtures are responsible for that; but it is a different kind of power.  After becoming accustomed to it one never has the same interest in chorus reeds as instruments of power" [3]

So today we have moved full circle, from organs depending on mixtures for loudness and tone projection, to romantic instruments which either had none at all or in which they played a minor role, to today's thinking which seeks to rediscover the old secrets of organ building with the mixture work which so impressed Richards.  This article does not claim to reveal all those secrets.  However because mixtures are not universally understood, it describes what they are and how they are designed.  A basic understanding of them helps one to appreciate the instruments with which Bach and his contemporaries were familiar.

Quint mixtures

These are the commonest type.  Quint mixtures contain only ranks tuned to successive octaves above the note keyed, together with those sounding the interval of a twelfth and its octaves.  Because a twelfth is an octave plus a fifth, the old name 'quint' is used to denote this kind of mixture.  If you are unfamiliar with musical intervals, play middle C on a keyboard and count it as note number one.  Then play successive white notes until you reach number twelve, which will be G in the octave above.  This is the interval of a twelfth above C.  It is important to understand intervals, because they are used as a sort of shorthand notation to describe the construction of all mixtures - termed their composition - in a succinct manner.  As an example, if we have a 3 rank quint mixture, its composition at bottom C might be written as:

15.19.22

These numbers are the intervals which define the pitches of the three constituent pipes of the mixture for bottom C, reckoned from 8 foot pitch for the manuals and from 16 foot pitch for the pedals.  Thus '15' means a pipe which speaks the note two octaves above bottom C, because two octaves is the same as the interval of a fifteenth.  Similarly, '19' means a note one octave above the twelfth, and '22' is a note three octaves above bottom C.  To get a rough idea of what such a mixture would sound like, you could first draw an 8 foot diapason or principal stop on a digital or pipe organ.  Then play the following three notes together to simulate the mixture - middle C, the G above it and treble C.  Additionally, if you hold bottom C down with the other hand you will get the ground tone as well. 

There are many other possible mixture compositions, so this is just one example.  If there are more than three ranks, additional pitches are added such that the octave and quint ranks alternate, just as they did for the three rank mixture above.  Thus a five rank quint mixture might have a composition of 15.19.22.26.29.  The octave ranks are at intervals of 15, 22 and 29 above the note keyed and the quint ranks are at intervals of 19 and 26.

Breaks

As one ascends the keyboard from bottom C, the high-pitched mixture pipes would eventually become so small that they would be impossible to make.  A reasonable practical limit is the pipe which corresponds to the top note on a Fifteenth stop, whose speaking length is about three-quarters of an inch (19 mm) and which is correspondingly narrow.  Its tiny mouth cannot emit much acoustic power, it is difficult to voice, and its frequency (about 8.4 kHz) is inaudible to many people over sixty and to some below.  Although some organ builders do use smaller pipes, one wonders why they bother given the practical reasons just mentioned.  Therefore the pipes of a mixture break back to the octave below, usually several times as one ascends the keyboard.  

However there are other reasons for introducing breaks.  If the ranks break back before the pipes get too small, they can augment the lower-pitched chorus work by duplication.  Although this does not add much additional power per se, because duplication adds only 3 dB of power at best as we saw above, there is an enrichment effect for another reason.  This is because two pipes which are supposed to sound the same frequency will seldom be exactly in tune, therefore the slow beats which occur produce the scintillating and brilliant effect which characterises an organ chorus containing mixtures.

Breaks should only affect one rank at a time - the whole mixture should never break back at the same note on the keyboard.  Applying these guidelines to the three rank mixture discussed above might lead to the following composition:

A 5 octave keyboard with 61 notes is assumed here, with the composition in each octave shown by the rows of the table.  C1 is the bottom note.  Therefore it can be seen that the mixture breaks three times at successive C's.  Note that the final row brackets two octaves rather than just one.

Although this scheme is logical and it obeys the 'rules' set out so far, it represents a rather unexciting mixture.  If, as would be likely on a great organ division, it co-exists with a chorus consisting of 8, 4, 2 ²/₃ and 2 foot pitches (1.8.12.15 in 'interval' nomenclature), it merely duplicates those pitches from middle C upwards.  Only in the lowest two octaves does the mixture introduce additional pitches which are higher than those of the other stops, and although this is good because the mixture will brighten the bass, it does little elsewhere.  In fact it could be argued that the top three octaves of the mixture are virtually redundant.  It is a wasteful and ineffective design.  There are three ways to improve matters - one is to rearrange the breaks, the second is to start the mixture at a higher pitch, and the third is to use more ranks.  A combination of any or all of these could also be used.

Taking these options in order, breaks can be introduced anywhere across the keyboard; they need not be restricted to octave boundaries as in the above example.  In the revised scheme below, only two breaks are introduced instead of the previous three, and they occur mid-octave at F# instead of C.  

The brilliance of the initial composition (15.19.22) is therefore carried across the entire first half of the keyboard, there is then a break to 12.15.19 which continues for the next two octaves, and only for top few notes does the mixture entirely duplicate the pitches of the other stops in the chorus, which is inevitable if the pipes are not to get too small.  A possible disadvantage of this scheme is that there might be excessive reinforcement of the quint ranks in the region F#3 to F5 because there are two of them (the 12th and 19th).  If the twelfth mutation stop was also drawn this might make the resulting chorus sound too acid or 'quinty'.  This could be mitigated by careful regulation of the quint ranks so that they are of lower power compared to the octave ranks.  However the effect is often not as unsatisfactory as might be supposed at first sight because at high frequencies the ear cannot readily identify the individual notes constituting the mixture, which therefore tend to coalesce into a composite sound.

The second option, making the mixture of higher pitch, effects a further improvement.  Bearing in mind that we are assuming the presence of other stops in the chorus up to 2 foot pitch, there is no reason why the mixture needs a 2 foot (15th) rank at all low in the compass.  It would therefore be logical to start the mixture at 19.22.26 and then spread the breaks differently.  A possible scheme is shown below:  

In the middle two octaves (C2 to B3) there is now no duplication of the twelfth ranks; instead it is the octave ranks which are duplicated.  This will make the mixture instrinsically better-balanced aurally.  Overall, this scheme results in a good and serviceable mixture.

The third option is to increase the number of ranks.  Of course, this has significant cost implications so it cannot be adopted lightly.  However mixtures with an odd number of ranks are slightly less attractive than those with with an even number, because with the former there will usually be a region of the compass over which the quint ranks predominate.  It has already been pointed out that this is not necessarily a major issue, but it does not arise anyway in mixtures having an even number of ranks.  Therefore we shall now look at a possible scheme for a four rank mixture.

The pattern of breaks here is essentially the same as in the previous example.  However in the top half-octave the mixture here loses a rank because otherwise the pipes would become too small and largely inaudible if the 19th (the missing rank) were to be carried up this far.

Tuning

The individual ranks of a mixture are always tuned true to the corresponding unison note on the keyboard, that is, they are tuned exactly to the natural harmonics of the unison note.  In practice this means that a properly tuned 8 foot or 4 foot principal-toned stop is used to tune each rank of the mixture in succession so that they are perfectly tuned, which means there must be no audible beats between the two pipes.   This is always done regardless of the temperament in use for the other stops.  If it were not done, that is if the tuning of each quint pipe of the mixture were to be tempered, each note of the mixture would be out of tune with itself and it would 'beat' unpleasantly between its ranks.  It is important to realise that tuning mixtures to the natural harmonic series results in differences in tuning between the quint ranks of the mixture and corresponding notes on the other stops on the organ, because these latter are tempered (detuned) by an amount according to the temperament to which the organ is tuned whereas the quint ranks are not.

It can be difficult to get your head round this important point, so let us take an example.  In an organ tuned to A = 440 Hz (A440), a 12th rank on a mixture at A440 must be tuned to a frequency of 1320 Hz.  This is the third natural harmonic of an 8 foot stop at A (3 times 440 equals 1320), thus when the mixture rank is so tuned there will be no beat with the third harmonic of the 8 foot stop.  However the frequency of E a twelfth above A on the 8 foot stop is 1318.5 Hz assuming equal temperament is in use.  This is the E lying in the top octave of the keyboard.  This differs from the frequency of the 12th rank of the mixture at A by 1.5 Hz, thus we see that the quint ranks of the mixture are tuned slightly sharp to the slightly flattened fifths in equal temperament.  This has the practical result that there will be an easily perceptible beat frequency of 1.5 Hz (3 beats in 2 seconds) between these two pipes - middle A on the 12th rank of a mixture and top E on the 8 foot stop.  Although it is indeed perceptible it does not represent a large difference, and in fact it results in some of the richness and 'shimmer' which a quint mixture provides.  However the differences are much greater if the mixture contains 17th (tierce) or 21st ranks, and in these cases they can result in hideous dissonances.  We shall be discussing such mixtures later.

As an aside, the inclusion of mixtures on a keyboard instrument whose other stops must perforce be tempered in some way implies that one is trying to tune the instrument to two temperaments at once - these are the perfect tuning for the non-unison ranks of the mixtures and the tempered tuning used for the other stops.  Some writers have identified this as one reason why mixtures are irrational and thus why they should not be used.  One cannot disagree entirely with their logic which is perfectly correct at the arithmetical level set out above.

Full and Sharp mixtures

Towards the top of the keyboard, mixtures will typically converge to a composition whose highest-pitched rank is no higher than a 15th above unison (8 foot) pitch.  All of the mixtures discussed so far have done this, and it is for practical reasons to do with preventing the pipes getting too small.  However the starting composition at the bottom of the keyboard, the number of ranks and the number of breaks can vary widely.  These differences are seen at work in the two types of mixture termed 'Full' and 'Sharp'.

Typically, a Full Mixture will start with a composition in the bass which continues upwards the flue chorus of the other stops.  It will then break across the keyboard such that it progressively duplicates the other chorus stops, with the dual intention of adding brilliance to the bass and then generating a 'full' tone from the middle of the compass or thereabouts.  Assuming the flue chorus excluding the mixture goes up to the fifteenth, one of the schemes above might be chosen, such as the four rank mixture repeated below:

Another name for a Full Mixture is the French Fourniture.

A Sharp Mixture designed to work in conjunction with this example would start at a very high pitch to continue the chorus of the Full Mixture upwards.  It would then break at different points before converging ultimately to a similar composition at the top of the keyboard for practical reasons.  Inevitably there will be some duplication of the rank pitches between the two mixtures at some points across the compass.  A three rank Sharp Mixture could therefore be configured thus:

This is an unconventional scheme in which the breaks do not bracket key groups of the same size.  The groups are largest (8 semitones) in the bass to enable brilliance to be carried well into the second octave at F2.  After this they reduce to 7, 6 and then 5 semitones, except at the top of the keyboard where the key group returns to 7 semitones.  This tapered stagger also conveys the advantage that the mixture does not break anywhere at the same point as the Full Mixture, thereby helping to disguise the breaks of both when they are used together.  The longest pipe in this mixture corresponds to top G on the keyboard at 8 foot pitch, and the smallest is top C at 2 foot pitch.  This is a range of only two octaves plus a fourth for the entire mixture.  Consequently there is a strong sense of pitch repetition as one passes through the breaks when playing scales up and down the compass, and this type of high-pitched repeating mixture is sometimes called a Cymbale.  However some Cymbales, made for economy and usually on electric action, only contain a single octave of each rank of pipes which simply repeats five times across the compass.  There are no breaks as such.  When playing chords on such a mixture the meagre allowance of pipes (only 36 on a three rank stop) gets used up rapidly, with the result that it can throw the chorus work badly out of balance owing to the missing notes which result.  This does not happen with a properly conceived mixture such as that described here because, although the pitch range is limited, each key has its own independent set of pipes and there are never any missing notes.  With a five octave keyboard (61 keys) there will be a full complement of 183 pipes despite the limited pitch range.

A Sharp Mixture is frequently unsatisfactory if used on its own with the other chorus work because the gap between its lowest-pitched ranks and the rest of the flue chorus is too large, particularly over the first half of the compass.  Therefore it often sounds thin and badly balanced.  It is really intended only to be used with the Full Mixture when greater brilliance and loudness is required.  A pair of mixtures such as these will provide tremendous sound projection and support to a choir and large congregation without requiring recourse to high pressure reeds, which all too easily can drown and dominate the singers because they overwhelm their voices with too much power at the very notes being sung.  By contrast, the mixtures provide loudness elsewhere across the audio spectrum.

Another name for a Sharp mixture is the Scharf found frequently on organs in continental Europe.

If there is only to be one mixture on a division it should be a Full Mixture whose composition at bottom C continues the chorus upwards from its highest pitch.  Typically this will be a fifteenth stop, and the mixture should then start at the 19th.  Starting it lower is wasteful because it duplicates pitches already present, and starting it higher leaves a gap which might render the overall effect too thin.

Cornets, Tierce mixtures and Sesquialteras

So far we have discussed quint mixtures which contain only octave and fifth-speaking ranks.  Now we turn to mixtures which also include the interval of a major third in the form of a tierce rank, one which speaks the 17th interval (two octaves plus a third) above the note played.  In 'footage' nomenclature it is a 1 ³/₅ foot rank relative to 8 foot unison pitch, and it is the fifth harmonic in the natural harmonic series of any organ pipe.  These mixtures were common in Europe in the 18th century as the Cornet stop, and a glut of somewhat rambling and otherwise wearisome 'Cornet Voluntaries' characterises that era in England.  In pre-Revolutionary French organs the Cornet was highly prized and a more dignified repertoire reflected it.  In the nineteenth century Willis clung onto a tierce rank running through his chorus mixtures whereas others did not.

The Cornet is essentially a solo stop, which is why it usually existed over the treble half of the keyboard only.  The presence of the tierce rank lends it a somewhat reedy tone, not surprising in view of the importance of a strong fifth harmonic in the sound of reed pipes.  It is possible that a solo Cornet was seen as offering a way out of the tuning difficulties of organs containing both flue and reed pipes in that it would always be reasonably well in tune with the fluework whereas a reed stop often would not.  In fact Gottfried Silbermann regularly excluded manual reeds entirely from his smaller organs, relying on the tierce to project reed-like tones instead.  A Cornet mixture usually consists of five ranks having the composition 1.8.12.15.17.  Breaks are not usually introduced, and the 17th rank might simply lose a few pipes at the top of the compass if the organ builder decided they were too small to bother with.

A tierce chorus mixture includes a tierce rank running throughout its compass, though it might (with no logical foundation in acoustics) break back an octave to the 10th near the top of the keyboard.  The illogicality is because the interval of a 10th is a natural harmonic of a 16 foot stop but not an 8 foot one.  It is therefore more sensible simply to omit the tierce when the pipes get too small..  The tierce adds a much thicker, some might say unpleasant, aural texture to the sound palette compared to the transparency of a quint mixture.  But it also results in a more serious problem to do with tuning.

We have alluded already to the tuning differences between the quint ranks of a mixture and corresponding notes on the other stops on the organ, because these latter are tempered (detuned) by an amount according to the temperament to which the organ is tuned whereas the quint ranks themselves are not.  With quint ranks the problem is not severe because the associated beat frequencies are small, whereas with tierce ranks it is a major issue which has to be confronted.  Consider a tierce pipe at tenor C, tuned true to the fifth natural harmonic (as it must be).  Its fundamental frequency is the fifth harmonic of tenor C and therefore it will be 654.05 Hz in an organ tuned to A = 440 Hz and with equal temperament.  Now consider the unison note on an 8 foot stop a seventeenth above tenor C, which will be the E above treble C.  This note has a fundamental frequency of 659.26 Hz.  Therefore a beat of 5.21 Hz will heard when the two pipes sound simultaneously.  This produces a most unpleasant, rough effect when it occurs while music is being played.  Worse still, the beats occur when a major third is played on a tierce mixture almost anywhere across the keyboard as the tierce beats against the high pitched unison ranks.  Apologists for tierce mixtures point to their use in unequal temperaments where the major thirds are less sharp than in equal temperament, and this reduces the beat frequencies which are generated.  While this is true for certain keys in certain temperaments, it is not always the case for some other keys where the thirds might be even worse than in equal temperament.  In my opinion, a more satisfactory compromise would be for tierce ranks to always draw as separate stops and allow all mixtures to be quint mixtures only.  The tierce can then be added to a quint mixture if desired, and they are also available for use in synthetic tone building with other mutation stops independent of the mixtures.

The Sesquialtera was originally a two rank tierce mixture in North European organs consisting of the twelfth and tierce.  'Sesqui' is Latin for 1.5, that is a ratio of three divided by two, which is approximately the relative height of the speaking lengths of the pipes as they stand on the soundboard.  Later on additional ranks were sometimes added to form a tierce mixture complete with breaks.  The Tertian is similar except the octave quint or nineteenth is used instead of the twelfth.  Again, additional ranks were sometimes added.

'Harmonics' mixtures

In a quint mixture the 5th and 7th harmonics are missing.  A tierce mixture supplies the missing 5th harmonic, and a so-called 'harmonics' mixture also includes the 7th harmonic.  It was used by builders including Casson and Harrison towards the end of the nineteenth century and into the twentieth.

The 7th harmonic rank is present in a mixture as the interval of a flat 21st above unison (two octaves plus a seventh), though it has always mystified me quite why the adjective 'flat' (relative to the corresponding unison note) is applied only to this interval and not to the tierce which is also significantly flat.  By following this logic we should also refer to the quint ranks as slightly 'sharp'.  So here we shall refer to the 21st simply as the 21st.  Its pitch in 'footage' nomenclature is 1 ¹/₇  relative to 8 foot pitch.  Suffice to say that the shortcomings of a tierce mixture are writ large and magnified here.  Bearing in mind that a harmonics mixture is intended to be used as a chorus stop, it sounds even thicker, more opaque and unpleasant than does a tierce mixture.  And its tuning problems are so extreme that one wonders why otherwise professional and intelligent organ builders could ever have thought of using it.

To justify this iconoclasm and for the sake of completeness let us examine the tuning issues in the same way as before.  The fundamental frequency of the seventh harmonic at bottom C on the manuals is 457.84 Hz, and that of the 21st interval above it on an 8 foot stop (the B above middle A) is 493.88 Hz.  These figures apply to an organ tuned to A = 440 Hz in equal temperament.  Therefore the beat frequency between them is 36 Hz, which will of course increase as one ascends the keyboard.  The difference is in fact 131 cents (a cent is one-hundredth of a semitone), therefore the 21st is well over one semitone flat to equal temperament and consequently there is logic in calling it a 'sharp twentieth' as a few organ builders have done [4].  In this case the tuning error to the 20th is much smaller at 69 cents, though at over half a semitone it is still far worse than that of a tierce (14 cents) which is bad enough in itself.  It is astounding that such dreadful dissonances should ever have been built deliberately into a musical instrument.  Playing sevenths almost anywhere across the compass will invoke these dissonances with a harmonics mixture as the 21st rank beats with the unison and octave ones.  And let us not forget that they are in addition to those introduced through the use of the tierce.

It is sometimes argued that because the frequencies of the perfectly tuned 17th and 21st ranks are present in any musical tone, assuming it contains those harmonics to start with, there is no reason why they should not be used in a mixture which simply reflects those same frequencies.  However, supporters of this specious argument neglect to point out that the acoustic power of the separate pipes of a mixture tuned to these frequencies is much greater than that of the corresponding harmonics in natural sounds such as those of organ pipes, therefore the beats are much more pronounced as well.  This results in an abnormal amplification which can only be regarded as bizarre.  The difficulties which follow from the obligatory use of any keyboard temperament are challenging enough without magnifying the dissonances deliberately by using harmonics mixtures.

Pedal mixtures

Thus far it has been assumed implicitly that we have been discussing mixtures on the manuals only.  However they are found frequently on the pedals as well.  Most of the points made so far apply in this case, though there are some differences which are included below:

1. Normally pitches on the pedals are reckoned from 16' pitch rather than 8' as on the manuals.  For example, a pedal fifteenth will be at 4' pitch rather than 2', and this applies to the intervals and pitches of the ranks in a pedal mixture as well.

2. Because of the lower pitch together with the limited compass compared to the manuals, it is possible to compose most mixtures without breaks should that be desired, as the problem of pipes getting too small does not exist.

3. As with the manuals, it is logical to start a mixture at a pitch which continues that of the pipes in the rest of the flue chorus.  Thus if the highest pitch here is a fifteenth, then the mixture could start on the nineteenth (2 ²/₃  foot pitch).  It is preferable for a pedal mixture to have an even number of ranks to prevent it becoming dominated by the quint ranks because at these relatively low pitches the constituent tones can be perceived as separate entities by the ear, which is not always the case for the higher-pitched ranks in a manual mixture.  A mixture with an even number of ranks will (or should) always have the same number of octave and quint ranks, regardless of whether it breaks or not.

4. Because of the lower pitch reference (16') of the pedals it is arguably possible to get away with noticeably out-of-tune ranks such as the tierce (3 ¹/₅ foot in this case) which are less attractive on the manuals as we have seen, at least with equal temperament.  This is because the beat frequencies which are responsible for the dissonances with the unison and octave stops are only half those on the manuals.  Even 21st ranks have been used by builders such as Cavaillé-Coll.  He provided such a stop at 4 ⁴/₇ foot pitch on the pedal organ at Notre Dame, Paris though this was a separate mutation and not part of a mixture.  It was called the Septième and Louis Vierne is reputed to have said that it gave the pedal ensemble "the richness of a muster of double basses" [4].  However this pitch refers to a 21st at 32', therefore it would be illogical to use it with a chorus based on 16 foot stops.

A factor which can affect the design of mixtures on the pedal organ, or even whether they are used at all, concerns couplers to the manuals.  Some argue that the availability of a full coupler complement reduces the need for pedal mixtures, and it is interesting that this is seen in some Baroque organs in the 17th and 18th centuries.  For instance, Arp Schnitger built a substantial two manual instrument at St John's, Hamburg in 1680 (now at Cappel) which had a pedal organ of nine stops, including two mixtures.  There was no coupler to the manuals.  In 1742 Gottfried Silbermann built an instrument at Fraureuth, Saxony which was broadly similar as far as its manual flue choruses were concerned, yet it only had three pedal stops (16', 16', 8') and no mixture.  However there was a coupler to the Hauptwerk division.  This design was typical of Silbermann's village organs.  Because of the separation of these organs in time and distance they cannot be taken as irrefutable evidence that couplers were seen to reduce the need for a complete pedal chorus in Bach's day, but the observation is interesting nevertheless.

Pipe scales

Quint mixtures are constructed using pipes broadly of principal or diapason scale.  They should not be narrower, otherwise they inject too many upper harmonics towards the limit of audibility, and this can make them sound too shrill or shrieky.  This problem can be addressed by halving the pipe diameters more slowly across the rank than is usually the case for ordinary principals, which halve typically every 16th pipe or so.  Thus the ranks will end up somewhat wider than ordinary principals of the same pitch at the top of the compass, and thus they will sound less strident.

Cornets use pipes of wider scale similar to those of flutes, with the unison (8 foot) pipe often being stopped.  The tierce rank in a chorus mixture might also employ a wider scale than that of the other ranks.  The old German and Dutch Sesquialteras and Tertians used principal pipe scales, unless they were intended to form a bass to the Cornet in which case they would incline to a more fluty tone with correspondingly wider pipes.

With 'harmonics' mixtures anything goes.  Some builders used pipes of wider scale than principals on the grounds that, since so many harmonics were being introduced in the mixture itself, there was no need to encourage still more by using narrower pipes.  Others took the opposite view, especially when a harmonics mixture was part of a solo manual division, where one of its functions was seen to be that of reinforcing the tone of already loud solo reeds, over which the mixture had to make itself heard.

Derived mutation ranks

In theory it is possible to derive or 'pick off' the quint and other mutation notes of a mixture from a single tempered rank to which the principle of extension is applied.  In practice this works for the octave ranks from a tuning point of view, but for the other (mutation) ranks it is important to realise that a derived mixture using them is a completely different animal to those discussed so far.  Because the intervals between unison pitch and the mutations are now tempered rather than being tuned to the natural harmonic series, the mixture will always be out of tune with itself.  This is reflected in audible beats which occur for each note of the mixture.  The effects are perhaps tolerable for the quints because the tuning differences, and hence the beat rates, are small as discussed above.   However the tuning problem makes a derived tierce sound rough and unpleasant for the reasons discussed previously, even though it was used frequently in the mid-twentieth century on both 'straight' and theatre organs.  With the latter the strong tremulants disguised the problem to some extent.  But picking off a 21st is totally impractical, again for reasons of tuning.

There are also other problems intrinsic to the extension organ.  It is impossible to scale and regulate the mixture pipes independently of the other ranks of the chorus if they use the same rank, thus such mixtures can sound coarse and banal in addition to being plainly out of tune.  And of course there is the missing note problem because a pipe on an extended rank can only be used once regardless of how often it might be required.

Notwithstanding all this, Compton forged ahead and implemented derived mixtures and mutations in many of his organs, tierces and all.  Stops such as his 'harmonics of 32 foot', which was typically a derived nine rank mixture on the pedal organ, can only be regarded as weird and a travesty of artistic organ building.  In some organs he even provided spare wires in his cable harnesses which were coiled up and left dangling at the unit chests so that anyone, including the organist or anyone else capable of wielding a soldering iron, could presumably tinker with the mixture derivations!  No doubt he was not alone.

Concluding Remarks

This article has shown how mixtures are constructed in terms of the number of ranks, the starting composition in the bass and how it ends up in the treble, and the various patterns of breaks in between.  All mixtures must converge to a similar composition at the top of the compass simply because it is pointless trying to make pipes smaller than a reasonable practical limit.  Therefore differences between mixtures can only be accomplished by varying their starting composition in terms of its pitches and number of ranks.  The task of the designer is then to implement a scheme of breaks designed to reach the ultimate composition in the treble while enabling the mixture to perform its functions across the compass.  These include brightening the bass and augmenting the other chorus stops elsewhere.

Quint mixtures, those containing only octave and fifth-sounding ranks, were discussed in detail.  'Full' and 'sharp' mixtures were described and contrasted and their functions explained.  The inevitable tuning problems of mixtures were illustrated which result from including the perfectly-tuned intervals in a mixture within the tempered environment of the other stops.

Tierce mixtures, including Cornets, were described which contain a rank speaking the fifth harmonic.  It was shown that their tuning problems are major and insurmountable unless one is content with a mixture in which most major thirds are grossly dissonant in organs tuned to equal temperament and some others.  'Harmonics' mixtures, those which contain the seventh harmonic in addition, were also discussed though their artistic rationale could not be discerned.

The article concluded with other topics including pipe scales and the problems caused by deriving mixtures by extension.

Notes and References

1.  “Grand Traité d’Instrumentation et d’Orchestration Modernes”, H Berlioz, 1843.

2.  Quoted in "The Organs of Britain", John Norman, David and Charles, Devon, 1984 (p. 92).

3.  Quoted in "The Organ - its tonal structure and registration", C Clutton & G Dixon, London 1950, p.41.

4. "The Organ", W L Sumner, 3rd edition, Macdonald, 1962, p.323.