Handy guide to choosing temperaments
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  A handy guide to choosing temperaments for the practical musician  

Colin Pykett

Posted: 29 January 2020
Revised: 29 January 2020
Copyright © C E Pykett

Abstract.   Most writers on musical temperament concentrate almost exclusively on the theoretical background to the subject, indeed the numerology becomes an end in itself in many cases, so consequently the interests of the practical musician are not well served by much of the literature.  This article takes a different approach by completely ignoring the mathematics and physics in favour of two straightforward but important musical aspects from which others then follow.  For 11 unequal temperaments the variations in key flavour are emphasised, together with a clear indication of whether any keys are unusable.  This is done by a simple tabular representation showing at a glance how the 24 major and minor keys of each temperament fall into categories labelled 'good', 'reasonable', 'poor' and 'awful'.  From the tables it is also sometimes possible to discern the hand of the temperament designer, showing that some were undoubtedly better than others.  Werckmeister and Neidhardt were particularly good at it in that they succeeded in distributing the major and minor keys reasonably evenly across the spectrum of key flavours, whereas some others were less successful.  It is singular that the latter usually failed to achieve enough variation within the minor keys compared with the major ones.  The tables also identify two temperaments in which the worst keys comprise the largest category, which is scarcely a recommendation to use them!  Kirnberger seemed to be one of the less competent sources in these respects.  The tabular format also confirms how temperaments evolved from the early and rather unsubtle meantones, through a progression of later ones with no Wolf intervals and more useable keys, to those incorporating distinct echoes of equal temperament yet which retain a pleasing range of key flavours.

 

 

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

 

Introduction

 

Equal temperament

 

Quarter-comma meantone

 

Fifth-comma meantone

 

Sixth-comma meantone (Silbermann's temperament)

 

Werckmeister III

 

Kirnberger II (modified)

 

Kirnberger III

 

Neidhardt I

 

Tempérament ordinaire I

 

Tempérament ordinaire II

 

Vallotti

 

Young II

 

Summary

 

Concluding remarks

 

Notes and references

 

 

Dedication

 

"Over the weekend I was playing with different temperaments on my digital organ, playing classical French music with Dom Bédos, Corette etc. As I don't know what I'm doing, it's a bit hit and miss, but I'm at least beginning to get a feel for the colours these temperaments provide"

Recently I came across these words on the internet, written by an organist in the Netherlands who just wanted to get to grips with the subject of temperament without being hassled by the arcane theory which usually accompanies it. So this article is dedicated to him and others like him. 

 

 

 

Introduction

 

Musical temperament is a vast subject which is difficult to assimilate, partly because it straddles the boundaries between music, physics, mathematics and the history of all three going back many centuries. So unless one feels comfortable within all of these disciplines it can repel efforts to understand it. However, speaking as a physicist, to my mind the practical musician is the most important focus because the ultimate aim of the whole business is (or should be) to make music rather than to regard numerological nicety as an end in itself. Consequently I have written this article in the hope that musicians will be able to take away the important features of temperaments in common use without needing to grapple with arithmetic or anything else not of direct relevance to actually performing the music. Not every temperament you might come across is included here, indeed some have been excluded deliberately. The main omissions are those which are claimed to be the temperaments favoured by J S Bach, for which there is still (2020) not the slightest historical evidence. Thus modern temperaments such as those invented by Kellner (1978), Barnes (1979) and Lehman (2005) are not considered here. Other omissions include certain temperaments which figure mainly in the context of the early English organ, which are similarly speculative as well as being of rather narrow interest.

 

Features of a temperament important to a musician probably include key flavour, and whether there are keys which are so out of tune as to be intolerable. Respectively, these govern the overlay of different colours which a given temperament adds to a performance, and whether all keys can be used. Both these aspects are of course strongly subjective, thus I have made no attempt to define more precisely what might be meant by terms such as 'colour' or 'intolerable'. Moreover, although in this article I shall be classifying the keys of a given temperament using adjectives such as 'good', 'poor' or 'awful', these names are mine and they will doubtless provoke argument [1]. However the reason for this approach is precisely to encourage this sort of discussion because today it is so easy to assess temperaments using digital keyboard instruments which offer the choice of several. Thus one no longer needs to visit particular instruments such as specially-tuned pipe organs in order to evaluate a given temperament, nor does one need to know anything about tuning them. I therefore encourage readers to experiment with as many different temperaments as possible when the opportunity arises, perhaps using this article as a springboard.

 

Temperaments are listed here in an approximate historical order and following a loose classification scheme beginning with equal temperament. This is followed by some examples of meantone tunings and then by some later temperaments whose structure is more irregular in certain theoretical respects.


Equal temperament

Because of its near-universal use today, equal temperament is sometimes thought to be relatively new whereas its roots in codified history in fact go back at least to the 15th century when it was used to determine the fret spacings on guitars, and for those on lutes not long afterwards.  It was discussed by 17th century theorists including Werckmeister and it was in use across Europe during Bach's lifetime, though his views on it remain unknown (as of 2020).  After 1750 it became increasingly widely used until it almost completely displaced other methods of tuning.  It is the most mathematically perfect temperament, all 24 keys (12 major and 12 minor) can be used without restriction, they all sound the same apart from pitch and thus there is no variation in key flavour, but no interval other than the octave is exactly in tune.  However there is considerable variation in the amount by which the other intervals are out of tune.  The fifths, for example, are pretty well-tuned and unobjectionable in that they only beat slowly in the middle of the keyboard for a stop of 8 foot pitch.  On the other hand the major thirds are badly out, being wide or sharp of pure, and it is probably only because we are so habituated to equal temperament that their rapid beats are not regarded more widely as disagreeable.  The out-of-tune thirds are highlighted starkly when one plays major thirds on Fifteenth and Seventeenth (Tierce) stops drawn simultaneously, because these ranks are themselves separated by the interval of a major third.  A similar rough effect occurs with a solo mixture stop such as a Cornet which incorporates the same ranks.  Thus the 'Cornet voluntaries' by English composers such as Maurice Greene and John Stanley are not well served by equal temperament, nor are many compositions from the classical (pre-Revolutionary) French school which exploited the synthetic tone building possibilities of mutation stops including the Tierce.  For such as these one needs to look at the unequal temperaments now to be described, where the choice will be partly dictated by the key(s) one wishes to use.  But by the same token, if it is thought appropriate to use unequal temperaments for such music, it must surely also be deemed proper to perform romantic works from the 19th century and later on the sort of equally tuned instruments which composers such as Liszt and Reger knew?  It seems perverse to apply different logic to the two situations in a contrived attempt to reject equal temperament out of hand, as so many commentators find it fashionable to do today.

 

 

Quarter-comma meantone

 

Quarter-comma meantone is a grossly unequal temperament in which the intonations of keys range from good to unusable (hence the adjective 'unequal').  This occurs because the temperament contains a Wolf fifth which is so out of tune that it 'howls' horribly in those keys which call on it.  The Wolf, conventionally placed between A flat and E flat, is more than one third of a semitone sharp or wide from pure.  However the important upside is that the Wolf soaks up the out-of-tuneness which would otherwise corrupt some other keys, enabling the temperament to boast 8 major thirds and 8 minor sixths which are perfectly tuned in that no beats are heard when they are played in isolation.  In turn, this leads to 6 major and 3 minor keys with very good intonation giving a very 'pure' sound to modern ears, though some find the effect insipid.  The temperament is of considerable antiquity, probably having evolved to meet the demand for usable thirds and sixths in early polyphony which was not satisfied by the simpler tuning systems used even earlier for plainsong.  Having said that, it continued to survive for a surprisingly long time given its disadvantages, which included heroic efforts on the part of organ builders to mitigate its worst shortcomings by providing split keys giving differently-tuned notes on their keyboards in the A flat and E flat positions (i.e. the notes defining the Wolf).  Quarter-comma meantone is a good example of the universal problem common to all temperaments in that one cannot devise one in which all keys are perfectly intonated with all intervals in tune.  The nearest one can get to that ideal is the approximation of equal temperament whose shortcomings have already been mentioned.  All other temperaments such as this one pride themselves on having some keys which are very good, better than in equal temperament, at the expense of having to accept others which are worse. 

 

Presently we will discuss some other types of meantone tunings since quarter-comma temperament is but one of a class of many.  However its historical importance usually means that when the term 'meantone' is used without qualification, it is quarter-comma meantone which is implied (though it is always advisable to check this point).

 

Musicians might find the tables below helpful when deciding whether this temperament will meet their needs.  They summarise some attributes of the consonant intervals and keys in terms of their euphony, assuming that the Wolf is placed as per conventional usage between G# and D#.  All of the raised notes on the keyboard are denoted in the tables as 'sharps' following the usual tuner's convention, regardless of their musical context in terms of scale or key.  The first table below illustrates the distribution of the pure (perfectly tuned) consonant intervals, excluding the octave which is always perfectly tuned in all the temperaments discussed in this article:

 

Major

Thirds

Minor

Sixths

C - E

C# - A

D - F# D - A#
D# - G E - C
E - G# F# - D
F - A G - D#
G - B G# - E
A - C# A - F
A# - D B - G
 (8 in all)   (8 in all) 

 

Quarter-comma meantone: the pure major thirds and minor sixths

 

It can be seen that all three major thirds lying within the C major diatonic scale (C - E, F - A and G - B) are tuned pure, together with their inversions as minor sixths (E - C, A - F and B - G respectively).  This suggests that the key of C major in quarter-comma meantone is likely to sound very 'good' and attractive in a subjective sense, far better than in equal temperament where all major thirds are badly out of tune.  The other pure intervals are similarly associated with other 'good' keys.  Conversely, other intervals which are badly out of tune render their associated keys less desirable.  The second table below confirms this, showing how the subjective intonation varies between all 24 keys, which have been classified as 'good', 'poor' or 'awful':

 

Good

Poor Awful

C maj

D# maj

C# maj

D maj E maj F# maj
F maj C min G# maj
G maj C# min B maj
A maj E min D# min
A# maj F# min F min
D min B min G# min
G min   A# min
A min    
 (9 in all)   (7 in all)   (8 in all) 

 

Quarter-comma meantone: relative intonations of all 24 keys (major keys in red)

 

These tables should therefore provide some guidance in deciding whether quarter-comma meantone is a promising candidate for the piece you wish to perform.  It is stating the obvious that the written key signature of a composition gives no indication as to how it might modulate into different keys.  For example, J S Bach's Prelude in C major for organ (BWV 547) would be very well catered for by this temperament if it merely stuck to this key, but of course it does not.  Instead it wanders in and out of several of the less attractive keys before it concludes.  Similar remarks apply to the Fugue, especially 8 bars or so before the end!  One therefore wonders whether Bach was trying to make a point here about the use of grossly unequal temperaments such as this one.  If so, was he encouraging us to use them to add a generous helping of spice at certain points?  Or, on the other hand, was he perhaps demonstrating how restrictive they can be by constraining a more timid composer's freedom to modulate?  One cannot know.

 

Unlike equal temperament, quarter-comma meantone obviously possesses marked differences in key flavour between its groups of  'good', 'poor' and 'awful' keys as these names themselves imply.  However it is important to note that this does not apply within the 9 keys comprising the 'good' category.  The mathematical relationship between the keys in this group represents a type of equal temperament, and we have already remarked that equal temperament itself exhibits no variation in key flavour across its keys.  This is no mere academic nicety since the point appears to be not widely appreciated or understood.  It means that, if composers restrict themselves only to the 'good' keys, quarter-comma meantone offers nothing at all beyond equal temperament in terms of key flavour.  The unfortunate upshot of this misapprehension is that the temperament is frequently assumed to have a wider range of finely-graded key flavours than is in fact the case.  The practical consequence is that the marked variations in key flavour are not finely-graded at all.  On the contrary, key flavour tends to change in a jerky fashion as one moves from key to key, and particularly when crossing the boundaries between 'good', 'poor' and 'awful'.

 

 

Fifth-comma meantone

 

This is another member of the class of historically important meantone temperaments, in this case with roots going back to the mid-17th century.  It is not quite as extreme as quarter-comma meantone in that its Wolf interval is not as badly out of tune, though it still 'howls' unpleasantly in those keys which call on it and therefore the temperament is again a grossly unequal one with identifiable groups of  keys ranging from 'good' to 'awful' as before, and these groups give rise to marked variations in key flavour.  But unlike quarter-comma meantone, this temperament does not have the benefit of any pure consonant intervals at all.  It has gained some pure dissonant intervals though, in the shape of major sevenths or leading notes and their inversions as pure semitones.  While of small value to harmony these have some relevance to melody.  However it does possess 8 major thirds and 8 minor sixths which, although not pure, nevertheless remain in better tune than those in equal temperament.  These relatively pure consonant intervals sit in the same positions as the purely-tuned ones shown above for quarter-comma meantone, repeated for convenience in the table below:

 

 

Major

Thirds

Minor

Sixths

C - E

C# - A

D - F# D - A#
D# - G E - C
E - G# F# - D
F - A G - D#
G - B G# - E
A - C# A - F
A# - D B - G
 (8 in all)   (8 in all) 

 

Fifth-comma meantone: the nearly-pure major thirds and minor sixths

 

Because all these intervals are now slightly impure, the mathematics shows that some other previously badly out of tune intervals in quarter-comma meantone become better-tuned in compensation.  In turn this means that some of the formerly 'poor' keys can be promoted into a new category labelled 'reasonable' with 6 entries.  For the same reason some of the formerly 'awful' keys have improved to 'poor'.  Overall there has been a useful reduction in the combined number of 'poor' and 'awful' keys from 15 to 9.  All this is shown in the following table:

 

Good

 Reasonable  Poor Awful

C maj

D# maj

B maj

C# maj

D maj E maj C# min F# maj
F maj C min F min G# maj
G maj E min G# min D# min
A maj F# min A# min  
A# maj B min    
D min      
G min      
A min      
 (9 in all)  (6 in all)  (5 in all)   (4 in all) 

 

Fifth-comma meantone: relative intonations of all 24 keys (major keys in red)

 

Therefore, on the whole fifth-comma meantone has taken some of the rough edges off its quarter-comma cousin, thereby enabling more keys to be deemed usable.  There is a corresponding expansion in the range of key flavours associated with the temperament in that 6 keys now fall into the new 'reasonable' category.  However there remains no variation of key flavour among the 'good' keys, as with quarter-comma meantone.

 

 

Sixth-comma meantone (Silbermann's temperament)

 

The mathematical differences between the fifth and sixth-comma temperaments are small.  It is therefore arguable whether their relative attributes would have been scarcely noticeable in practice at a time (the 18th century) when tuning practices were approximate and the tuning stability of organs was questionable owing to their unstable wind supplies.  However there is some evidence that Gottfried Silbermann tuned his instruments to this temperament, and that J S Bach was familiar with it having played on some of them.  Because of the similarities with fifth-comma meantone, nothing further will be said since the characteristics of both temperaments are the same at the qualitative level of this article.  Therefore everything said above, including the two data tables, applies here so it will not be repeated.

 

 

Werckmeister III

 

Andreas Werckmeister was primarily a practical musician who taught himself enough mathematics and acoustical theory to understand the theoretical background to temperament, and he designed a number of them by doing painstaking confirmatory experiments with a monochord.  He wrote several important treatises in the late 1600s, initially inclining towards unequal temperaments such as his well known Werckmeister III [2] but later expressing dissatisfaction with them in favour of equal temperament which had by then been known for a long time.  Bach was presumably acquainted with his work because he owned at least one of Werckmeister's books.  This temperament is quite different to any of those discussed above.  It has no Wolf interval, and an important feature is its many (8) pure fifths and their inversions as fourths, shown in the table below.  These allow quint mutations such as Twelfths and Larigots to be exactly in tune with many of the corresponding notes on the keyboard, which adds a sparkling purity to the mixtures which of course also contain these ranks (provided they do not include third-sounding ranks such as the Tierce which muddy the effect somewhat).  There are no pure thirds, though five are better-tuned than in equal temperament.

 

 Fourths   Fifths 
C - F C# - G#
C# - F# D# - A#
D# - G# E - B
E - A F - C
F - A# F# - C#
G# - C# G# - D#
A# - D# A - E
B - E A# - F
 (8 in all)   (8 in all) 

 

Werckmeister III: the pure fourths and fifths

 

There are no longer any 'awful' keys as there were with the meantone temperaments, and many keys are better-intonated than in equal temperament with some being very good and none positively unusable.  There is consequently a generally pleasing spectrum of key flavours associated with the temperament.  Other than the happy absence of the 'awful' group, keys have been allocated to the same range of qualitative categories used earlier ('good', 'reasonable' and 'poor') as in the table below.  The variations in key flavour tend to be smoother and not as jerky as those encountered with the meantone temperaments.

 

Good

 Reasonable  Poor

C maj

D maj

C# maj

F maj D# maj F# maj
A# maj E maj C min
E min G maj  
A min G# maj  
  A maj  
  B maj  
  C# min  
  D min  
  D# min  
  F min  
  F# min  
  G min  
  G# min  
  A# min  
  B min  
 (5 in all)  (16 in all)  (3 in all) 

 

Werckmeister III: relative intonations of all 24 keys (major keys in red)

 

Werckmeister III is widely reckoned to be one of the best temperaments for the organ, at least for the repertoire up to the romantic period when equal temperament arguably becomes a more compelling choice.  It has a broad range of key flavours, mixtures and mutations sound very well, and it is clear that Werckmeister was an expert in the subject of temperament, not only at the theoretical level but as a practising musician and tuner as well.  Many pipe organs tuned to Werckmeister III exist in the UK, that in Eton College School Hall having been widely recorded [3].

 

 

Kirnberger II (modified)

 

J P Kirnberger was one of J S Bach's better-known pupils, though the temptation to therefore regard his work as revealing the temperament(s) favoured by his master should be resisted unless evidence were to be discovered in the future.  On the contrary, this one is so poor that one cannot easily imagine a musician of Bach's discernment being satisfied with it.  Kirnberger is sometimes regarded as an expert on temperament, though when one compares his efforts with those of Werckmeister nearly a century earlier it is less easy to identify the intuitive and happy blend of theory and practice which is so obvious in the work of the latter.  On the contrary, his work demonstrates more of a suck-it-and-see approach and this is why we will not consider his earliest crude attempts at designing temperaments here.  His Kirnberger II temperament was modified (by himself) to become more subtle than its predecessor, hence its name, and in my estimation it is the first of his which merits discussion.

 

The temperament has many pure consonant intervals, mainly fourths and fifths, distributed as in the table below.  There are also some pure dissonant intervals (semitones, whole tones, augmented fourths, minor sevenths and major sevenths)  but they are not listed here.  The large number of pure fourths and fifths can add sparkle to quint mutations and mixture work in certain keys, as with Werckmeister III and for the same reason.

 

Major

Thirds

 Fourths   Fifths 

Minor

Sixths

C - E

C - F C# - G# E - C
G - B C# - F# D# - A# B - G
  D# - G# F - C  
  F - A# F# - C#  
  F# - B G# - D#  
  G# - C# A# - F  
  A# - D# B - F#  
 (2 in all)   (7 in all)   (7 in all)   (2 in all) 

 

Kirnberger II (modified): the pure consonant intervals

 

The relative intonations of the various keys are indicated in the table below.  However the preponderance of 'poor' keys should be noted, and in fact the situation is arguably worse than the table implies because some of them (particularly E major) verge on the unpleasant.  The marked shift away from 'reasonable' to 'poor' compared with Werckmeister.III is unfortunate, and it detracts from the slightly greater number of 'good' keys (7 versus 5).  Although there is a wide range of key flavours, the differences between some of them seem clunky as one modulates, an experience reminiscent of the crudeness of the meantone temperaments  And a temperament which has more 'poor' keys than any other kind would seem to be the work of an amateurish dilettante, with little merit on the face of it.  Why would anyone prefer a temperament in which most of the keys were badly intonated?

 

Good

 Reasonable  Poor

C maj

D# maj

C# maj

D maj A# maj E maj
F maj C min F# maj
G maj C# min G# maj
E min F min A maj
F# min G min B maj
B min A min D min
  A# min D# min
    G# min
 (7 in all)  (8 in all)  (9 in all) 

 

Kirnberger II (modified): relative intonations of all 24 keys (major keys in red)

 

 

Kirnberger III

 

Taking a more positive view, Kirnberger continued to refine his temperaments by developing this one which is a considerable improvement on its predecessor.  In terms of intervals it now has only one pure major third and minor sixth instead of two each as before, and several pure fourths and fifths remain though they are not quite the same ones as previously.  These latter continue to add clarity to quint mutation stops and mixtures in some keys.  A sprinkling of pure dissonant intervals exists, also as before.  The pure consonant intervals are tabulated below:

 

Major

Thirds

 Fourths   Fifths 

Minor

Sixths

C - E

C - F C# - G# E - C
  D# - G# D# - A#  
  F - A# E - B  
  F# - B F - C  
  G# - C# G# - D#  
  A# - D# A# - F  
  B - E B - F#  
 (1 in all)   (7 in all)   (7 in all)   (1 in all) 

 

Kirnberger III: the pure consonant intervals

 

These changes result in a considerable and advantageous shift to the distribution of keys among the 'good', 'reasonable' and 'poor' categories, which is evident from the table below.  The most striking improvement is in the large number of keys which now fall into the 'reasonable' category, thereby removing the major shortcoming of the previous temperament which had far too many 'poor' keys.  In fact, comparing the table to the corresponding one for Werckmeister III demonstrates immediately that the distributions are similar in that both have a marked preponderance of 'reasonable' keys.  However Werckmeister's temperament still retains the edge in that it is better balanced - there are arguably too many 'reasonable' minor keys in Kirnberger III compared to only three major ones.  Nevertheless the visual correspondence between the two tables is striking, and it makes one wonder whether Kirnberger might have been trying to follow Werckmeister's lead here.  The temperament is used quite frequently, perhaps for the reasons just mentioned, though perhaps also because it has the aura of one who was so close to Bach.

 

Good

 Reasonable  Poor

C maj

D maj

C# maj

F maj D# maj E maj
G maj G# maj F# maj
A# maj C min A maj
E min C# min B maj
  D min  
  D# min  
  F min  
  F# min  
  G min  
  G# min  
  A min  
  A# min  
  B min  
 (5 in all)  (14 in all)  (5 in all) 

 

Kirnberger III: relative intonations of all 24 keys (major keys in red)

 

 

Neidhardt I

 

J G Neidhardt was a German musician with a very good theoretical and practical understanding of temperament, and in these respects he was comparable to the older Werckmeister whose published work he would probably have been familiar with.  Born the same year as Bach, he worked indefatigably at the subject, going so far as to recommend certain ones as suitable for different venues such as small, medium and large buildings.  This one can be regarded as forming a bridge between the excessive crudeness of the meantone tunings and the excessive blandness of equal temperament.  All keys can be used and there is only one 'poor' key (E major) which is nevertheless useable.  There are no pure thirds, 6 of them being slightly worse (even sharper) than in equal temperament though, counterbalancing this, 4 are slightly better.  The remaining 2 are the same.  It has 4 pure fourths and fifths in addition to 4 others which are tuned the same as in equal temperament.    It therefore incorporates many numerical echoes of equal temperament with which Neidhardt would undoubtedly have been familiar, hence its ability to smooth out the roughnesses of the more extreme unequal tunings while not going too far in the direction of a fully equal solution, though some might disagree.  It retains a pleasing range of key flavours which equal temperament does not.  Despite this it is sometimes criticised today for a lacklustre character, presumably when compared with the less subtle qualities of the unequal temperaments discussed previously.  However it has only one 'poor' but nevertheless usable key compared with the several or many in other temperaments which are not only poor but downright intolerable.  One cannot have it both ways when dealing with temperaments since there is no ideal solution.  It is rather like moving sand around in a tray - piling it up at one end means that it gets thinner elsewhere.

 

The relative intonations of all keys are in the table below, which illustrates the move towards equal temperament very clearly.  The vast majority (20) of keys are 'reasonable' whereas in equal tuning this category would embrace all 24.  Therefore it is perhaps best considered as a mildly unequal temperament which nevertheless retains an attractive range of subtle key flavours.

 

Good

 Reasonable  Poor

C maj

C# maj E maj
F maj D maj  
G maj D# maj  
  F# maj  
  G# maj  
  A maj  
  A# maj  
  B maj  
  C min  
  C# min  
  D min  
  D# min  
  E min  
  F min  
  F# min  
  G min  
  G# min  
  A min  
  A# min  
  B min  
 (3 in all)  (20 in all)  (1 in all) 

 

Neidhardt I: relative intonations of all 24 keys (major keys in red)

 

 

Tempérament ordinaire I

 

Like knowledge more generally, temperaments (as with pitch standards and tuning methods) could not have propagated hundreds of years ago quickly, uniformly and widely across national boundaries in the facile though anachronistic manner assumed by some writers today who are accustomed to the instant global dissemination of information across the internet.  But where they did take root there was a tendency for local modifications to become embedded as part of their characters, and this is probably true of the two 18th century French temperaments described here.  This one has echoes of the old quarter-comma meantone tuning but with some (probably trial and error) adjustments which removed the howling Wolf fifth.  The only remaining pure consonant intervals are a few fourths and fifths, thus the perfectly tuned quarter-comma thirds and minor sixths no longer appear.  However, the sand-in-the-box analogy referred to above means that the suppression of the Wolf results in the banishment of all the 'awful' keys of which quarter-comma tuning had 8.  The resulting relative intonations of all keys are shown in the table below.  Unfortunately, when displayed in this way one sees immediately that the 'poor' keys now predominate as they did in Kirnbeger II, and I do not regard such temperaments as having much to recommend them in terms of good design.  They convey an impression of having evolved through mere tinkering rather than being created more intelligently from the bottom up.  On the other hand the temperament is undoubtedly a considerable improvement on quarter-comma meantone, and the range of key flavours was presumably one which French composers of the day would have probably exploited.

 

Good

 Reasonable  Poor

C maj

D maj C# maj
F maj A maj D# maj
G maj C# min E maj
A# maj D min F# maj
B min E min G# maj
  F# min B maj
  G min C min
  G# min D# min
  A min F min
    A# min
 (5 in all)  (9 in all)  (10 in all) 

 

Tempérament ordinaire I: relative intonations of all 24 keys (major keys in red)

 

 

Tempérament ordinaire II

 

This is probably a modification of another meantone temperament, in this case sixth-comma meantone.  In the same way that sixth-comma was an improvement on quarter-comma, this temperament is an improvement on tempérament ordinaire I.  As in that case the Wolf interval has again been suppressed, enabling all the previously 'awful' keys to vanish as before.  But there is now a considerably augmented 'reasonable' category which removes the shortcoming of ordinaire I in which the 'poor' keys were predominant, though the distribution is still somewhat unbalanced in that there are too many 'reasonable' minor keys as in Kirnberger III.  This results in more noticeable variations in key flavour across the major keys than for the minor ones.  Although none are pure, the twelve major thirds fluctuate both slightly sharp and slightly flat about the equal temperament mean, and these variations lead to corresponding key flavours which are of value when playing Tierce en Taille and similar music from the French pre-Revolutionary period.

 

There is an interesting analogy here with the differences between Kirnberger II (modified) and Kirnberger III in contemporary Germany.  In both pairs of temperaments (i.e. the two Kirnbergers and the two tempéraments ordinaires) the earlier one suffers from an unfortunate preponderance of 'poor' keys, whereas in the later this has been remedied by augmenting the number of 'reasonable' keys instead.

 

Good

 Reasonable  Poor

C maj

E maj C# maj
D maj A# maj D# maj
F maj C# min F# maj
G maj D min G# maj
A maj D# min B maj
  E min C min
  F# min F min
  G min  
  G# min  
  A min  
  A# min  
  B min  
 (5 in all)  (12 in all)  (7 in all) 

 

Tempérament ordinaire II: relative intonations of all 24 keys (major keys in red)

 

 

Vallotti

 

Vallotti was an Italian composer and a contemporary of Bach and Handel, and this eponymous temperament arose around 1730.  All keys can be used and there is a range of subtle key flavours, though it is not as pronounced as with some other temperaments.  Today some find this attractive and safely unadventurous whereas others, who prefer a more noticeable variation with less perception of a pull towards equal temperament, do not.  These characteristics can be predicted simply by looking at the table below listing the relative intonations of all keys.  Like some of the temperaments discussed already, the preponderance of 'reasonable' keys is obvious.  These would total 24 for equal temperament in which all keys would be classed as 'reasonable', so Vallotti can be viewed as a step towards equal tuning which was by then well known, and away from the roughnesses of the meantone ones.  However, like Kirnberger III and  tempérament ordinaire II, Vallotti's distribution is lopsided in that too many minor keys (in fact all of them) fall into the 'reasonable' category.  Werckmeister III also has a dominant 'reasonable' category of the same size but its balance of major and minor keys is much better.  The first asymmetry results in a consequential lopsided aspect in that the major keys are spread more evenly across all three categories, meaning that the corresponding variations in major key flavour are much more pronounced than for the minor keys.  Whether this was intended by Vallotti is impossible to say, though it is a feature which needs to be borne in mind.   The temperament is employed quite widely.

 

Good

 Reasonable  Poor

C maj

D maj C# maj
F maj D# maj E maj
G maj G# maj F# maj
A# maj A maj B maj
  C min  
  C# min  
  D min  
  D# min  
  E min  
  F min  
  F# min  
  G min  
  G# min  
  A min  
  A# min  
  B min  
 (4 in all)  (16 in all)  (4 in all) 

 

Vallotti: relative intonations of all 24 keys (major keys in red)

 

 

Young II

 

Thomas Young was a practising physician and polymath who also distinguished himself in the fields of physics, physiology and Egyptology, and this was his second temperament, proposed in 1800.  It is identical to that of Vallotti except for a small theoretical modification (the circle of fifths has been rotated one place clockwise).  So it cannot claim to be original in any major respect because Vallotti had beaten him to it 70 years earlier.  It is therefore curious that his work merited a paper to the Royal Society, except perhaps on the basis that almost anything from the pen of so august a member would have been accorded a respectful hearing.   Thus, although he was one of the first in Britain to enter the field rather late as a temperament theorist, he had little to contribute on this occasion compared with the achievements of his peers going back several centuries in continental Europe.

 

The table below lists the intonation of each key across the same three categories used for Vallotti, whence it can be seen that a few of the major keys have been redistributed.  This is the only difference between the two temperaments.  It results in their intonations becoming progressively poorer as the number of sharps and flats increase.  This is an advantage if that is what one wants but of little account otherwise, since it results in the temperament moving even closer to the mathematical predictability of equal temperament.  The situation regarding the minor keys remains the same as in the case of Vallotti in that all of them appear in the 'reasonable' category.  This results in the same twofold asymmetry of Vallotti reappearing here - the 'reasonable' category is unbalanced because of the preponderance of minor keys, whereas the variations in major key flavour are again more pronounced than for the minor keys because they are distributed across all three categories.  Redressing these matters means we would have to go back to Werckmeister which is a better-balanced temperament in these respects, though doing so would also introduce a wider range of variations in key flavour.  Staying with Young (or indeed Vallotti) means we would be playing it safer with a temperament closer to equal and therefore less likely to offend more conservative musicians and their audiences, though today some regard this as one of its weaknesses.

 

Good

 Reasonable  Poor

C maj

D# maj C# maj
D maj E maj F# maj
F maj A maj G# maj
G maj A# maj B maj
  C min  
  C# min  
  D min  
  D# min  
  E min  
  F min  
  F# min  
  G min  
  G# min  
  A min  
  A# min  
  B min  
 (4 in all)  (16 in all)  (4 in all) 

 

Young II: relative intonations of all 24 keys (major keys in red)

 

 

Summary

 

A lot of information has been presented above so an attempt will now be made to summarise it.

 

1. All of the meantone temperaments discussed here (quarter, fifth and sixth-comma) have a Wolf fifth which results in several or many keys (depending on the temperament) falling into an 'awful' category of key flavour, but they all possess 'good' and 'poor' keys as well.  However the fifth and sixth-comma meantones have a less objectionable Wolf so they can therefore offer an additional less extreme 'reasonable' category, though there still remain strong variations in key flavour.  Within the 'good' category of each temperament there is no variation in key flavour, therefore the differences tend to be more noticeable when modulating across the category boundaries rather than within categories, and the variations are consequently somewhat jerky rather than finely graded.  At the qualitative level of this article there is no difference between the fifth and sixth-comma temperaments.

 

2. Werckmeister III was the first temperament he published despite the confusing numbering system.  It is very carefully designed and has no Wolf interval and therefore no 'awful' category.  So all keys can be classified as 'good', reasonable' and 'poor', though the 'reasonable' category predominates.  There is a well-balanced distribution of major and minor keys across the categories, unlike some other temperaments discussed here in which the 'reasonable' minor keys heavily outnumber the major ones.  The large number of pure fifths results in sparkling and attractive quint mutation and mixture stops  in many keys which are well in tune with the rest of the  fluework on the organ.

 

3. Kirnberger II (modifed) suffers from the considerable shortcoming that most of its keys fall into the 'poor' category.  It is difficult to see how such a temperament can be regarded as other than badly designed, and therefore it is also difficult to understand why it would be a temperament of choice.

 

4. Kirnberger III does not suffer from this problem and in other respects it is also an improvement on the previous one.  It has a qualitative distribution of keys across the categories similar to that of Werckmeister III, except for an excessive preponderance of minor keys in the 'reasonable' category compared with the major ones.  Also like Werckmeister III its many pure fifths result in a range of attractive and sparkling quint mutation and mixture stops.

 

5. Neidhardt I is another carefully designed temperament which is an evolutionary waypoint between the rawness of the earlier meantones and the blandness of equal temperament, though it nevertheless retains an attractive range of key colours.  Its distribution of major and minor keys across the categories is well balanced.

 

6. Tempérament ordinaire I is probably based on quarter-comma meantone but with some empirical adjustments which suppress the Wolf fifth.  Like Kirnberger II (modified), it suffers from the significant shortcoming that most of its keys fall into the 'poor' category.  It is difficult to see how such a temperament can be regarded as other than badly designed, and therefore it is also difficult to understand why it would be a temperament of choice.

 

7. Tempérament ordinaire II is probably based on sixth-comma meantone but with some empirical adjustments which suppress the Wolf fifth  Its distribution of keys across the various categories is much better balanced than for tempérament ordinaire I, though it still has too many minor keys in the 'reasonable' category.  In this respect it is similar to Kirnberger III.

 

8. Vallotti is an obvious move towards equal temperament and away from the inequalities of the other temperaments discussed above.  It shares with tempérament ordinaire II and Kirnberger III an unbalanced predominance of minor keys in its 'reasonable' category, resulting in the variations of major key flavours being more noticeable than for the minor ones.

 

9. Young II is very similar to Vallotti, indeed practically a copy of it, so the remarks above also apply here.

 

 

Concluding remarks

 

Most writers on temperament concentrate almost exclusively on the theoretical background to the subject, indeed the numerology becomes an end in itself in many cases, so consequently the interests of the practical musician are not well served by much of the literature.  This article has taken a different approach by completely ignoring the mathematics and physics in favour of two straightforward but important musical aspects from which others then followed.  For 11 unequal temperaments the variations in key flavour have been emphasised, together with a clear indication of whether any keys are unusable.  This has been done by a simple tabular representation showing at a glance how the 24 major and minor keys of each temperament fall into categories labelled 'good', 'reasonable', 'poor' and 'awful'.  From the tables it was also sometimes possible to discern the hand of the temperament designer, showing that some were undoubtedly better than others.  Werckmeister and Neidhardt were particularly good at it in that they succeeded in distributing the major and minor keys reasonably evenly across the spectrum of key flavours, whereas some others were less successful.  The latter usually failed to achieve enough variation within the minor keys compared with the major ones.  The tables also identified two temperaments in which the worst keys comprised the largest category, which was scarcely a recommendation to use them!  Kirnberger seemed to be one of the less competent sources in these respects.  The tabular format also confirmed how temperaments evolved from the early and rather unsubtle meantones, through a progression of later ones with no Wolf intervals and more useable keys, to those incorporating distinct echoes of equal temperament yet which retained a pleasing range of key flavours.

 

 

Notes and references

 

1.  The intonation classification of keys described in this article by adjectives such as 'good', 'poor' and 'awful' follows to some extent the method used by Charles Padgham in his book 'The Well-Tempered Organ' (Positif Press, Oxford, 1986).  However care should be exercised in using the book because it contains a large number of numerical errors, many of which I have identified and corrected in another article on this website.

 

2.  The numbering system of Werckmeister's temperaments  is confusing.  Werckmeister III refers to the first temperament he invented, or at least described, so occasionally it is denoted as Werckmeister I.

 

3.  The organ in the school hall at Eton is not as 'authentic' as some writers imply, but it is nevertheless a most pleasing instrument to listen to.  Of the recordings available, Thomas Trotter's rendition of C P E Bach's organ works is suggested as it not only demonstrates the organ itself but serves as a good example of what Werckmeister III actually sounds like when used to render the mid-to-late 18th century repertoire (Regent REGCD 314).