!

!!!!!!!!A chromatic journey through Bach’s Well-Tempered Clavier

!!!!Prepared for: Prof. Robert ShollPrepared by: Ryan Fowler, 21108189

40 Credit Independent Research ProjectBA (Hons) Music Performance with Music Technology, Level 6

!Word Count: 9544

!May 2014!

London College of Music, University of West London

Abstract !!Bach’s 48 preludes and fugues are often considered a keyboardist’s first insight into the world

of complex counterpoint and intricate hand balance. What one often misses during their initial

intensive study of the famous 48 is the foreign yet so familiar chromaticism used, some

obviously noticeable, but much more of it hidden deep inside the countered lines. This paper

aims to give a tour of some of the more interesting and developed techniques of species

counterpoint used in Bach’s fugues and compare the implications of having such interesting

harmonic dialect hidden beneath other stylistically alluring content so as to draw attention

away from the chromaticism; be it through compositional techniques, popular styles of the

time, tuning differences or hidden inner-voice swapping.

!

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Acknowledgments !!The paper has involved many others, without some of whom this project would have been a

struggle. My first acknowledgement goes to the London College of Music’s own subject

librarian, Julia Randall, who has been more than helpful and patient with me in finding rare

online sources with the journal memberships of the University of West London. I would also

like to thank my close colleague and partner Emily Myles, along with my mother, for their

continued support and motivation through the past year and proof reading abilities. I would

also like to express my gratitude towards my supervisor, Robert Sholl for his gentle nudges

and ideas which have pushed this project in a more refined direction.

!

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Contents

Prelude Introduction 4

Contextualising the WTC Origins behind the WTC and temperament considerations 8

Bach, the chromatic Chromatic appearances within the WTC 15

Dissonance caused by pedal tones 16

Bach and the 7th chord 21

Appoggiatura dissonance 23

Bach’s use of cambiata and echappée 27

Alternative solutions to a disobedient Bach 29

Fugue Conclusion 33

References Reference List 36

Bibliography Bibliography 38

Discography Discography 41

Appendices Appendix I - Bach’s temperament 42

Appendix II - Interval Tuning 43

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Prelude

Introduction !

J.S. Bach’s two volume keyboard masterpiece, the Well-tempered Clavier (hereafter referred

to as WTC) is quite often a pianist’s first insight into not only Bach but also the complex study

of fugues and counterpoint. Within these two volumes there is a heavy use of chromaticism

and dissonance which appear for the first time, historically speaking, in it’s age of

composition. This stands to great importance to comprehend the development of western

tonal harmony, though much Bach scholarship has largely ignored these aspects in favour of

a more contextual and historical view. As such, there has been no taxonomy of Bach’s

dissonance which clearly states what techniques he used and within what context; though

authors have touched delicately around the edges on what could be classed as this.

Laurence Dreyfus in ‘Bach and the patterns of invention’ begins to touch on this during his *

schenkerian analysis on the C minor figure from WTC I, though takes the narrative on a

journey of implied notes and form instead of the much more questionable dissonant phrases.

Even the essays of Christophe Wolff ignore this area all together, leaving need for a clear

explantation as to where these chromatic ideas came from and to what extend Bach used

them.

There are many different perspectives that musicologists have taken whilst studying

dissonance - one of a pedagogical, historical, contextual or analytical view; which all make for

very interesting discoveries amongst themselves. This study is therefore restricted to a

pedagogical view to show how the following selected pieces from the WTC grew and

Page ! of !4 43 See ‘Figments of the Organicist Imagination’ (Dreyfus, 1996)*

developed from works and systems before Bach. Almost all of the fugues (not so much the

preludes as these are after all just embellished chord progressions) are ripe with yielding

dissonance, of which the study into what is expected of counterpoint in Bach’s time is often

both used and ignored all within same piece. Bach was quite rightly one of the first

composers to fully utilise chromaticism within his works en masse in a tonal mediation,

certainly surpassing other composers of the period by expanding heavily on the models of

Dietrich Buxtehude and Nicolas de Grigny, of whose works Bach often copied out as a young

child and most likely influenced not only the larger structures of works but also the inner

harmonic structures (Snyder, 1987). This is not to say Bach was the first and only composer

to use dissonance in a tonal construct, but since 19th and 20th century theorists placed such

a heavy musical centre on Bach, it seems fitting that his volume of work stands consistent in

this area. What is apparent to us is that although very primitive in a structure which resembles

western tonal harmony, the works for example from de Grigny’s Livre d’Orgue often do not

expand beyond the compass of tonic, relative major/minor and the dominant in terms of

entering subjects in the contrapuntal works (Stauffer, 1993). This form is then far surpassed

by J.S. Bach whose subject entries in the WTC often pick even the most daring of intervals

such as the tritone to enter on (see Fugue no. 22 from WTC II).

The likes of Bach’s chromaticism is wildly compared to that of Don Carlo Gesualdo by the

pianist Glenn Gould. However, what Mr. Gould fails to note though, is that Bach would not

have known the works of this composer, certainly not to the extent where he would have

copied them as a child (the most promising form of composition practice), nor does the

smooth voice-leading in Bach follow the sporadic approach of Gesualdo. Further to this

argument, one must consider that Gesualdo was not composing in a tonal system, much too

astray from the strict 24 key system which Bach was so devout to, enough to create the

WTC to show off each and every one of them. It is for these reasons we must presume that

Bach’s chromatic influences came from other areas, despite the correct claim that the most

extreme examples of chromatic Bach stand up in dissonant proportion to the works of

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Gesualdo. With the structures of Bach’s works adopting those of Buxtehude and de Grigny

so devoutly, one must consider to whom was Bach’s master in chromaticism, or is it safe to

assume that Bach was the master of all chromaticism and dissonance to come in the

following 300 years? It is quite possible that Bach was not aware, and through a misreading

or misunderstanding of the rules, formed his compositional technique.

Scott Burnham’s book ‘Mozart’s Grace’ (Burnham, 2013) takes the intricate details of certain

works by Mozart and decomposes them to reveal the hidden techniques which branded

Mozart’s music with his signature. Although somewhat sporadic in approach, the jumping

and diving between pieces and musical examples makes for a very thorough overview for a

good proportion of Mozart’s catalogue. For Bach though, there is not quite a clear cut

adventure through his music, especially when studying such a broad subject as dissonance

treatment in counterpoint. There are, besides the WTC, many other Bach works which are

heavily based around chromatic motifs and encompass the dissonance techniques as set out

below in this paper; take for example the Chromatic Fantasy and Fugue BWV 903, the Art of

Fugue BWV 1080, the Musical Offering BWV 1079 or a plethora of the organ fugues such as

BWV 548 (Em) or BWV 582 (Cm). The WTC though covers many of the techniques

encompassed within these works, some of which are from entirely different periods of Bach’s

life, showing how this was a key compositional technique of his which stands consistently

throughout his works.

This paper aims to split the journey down into three main sections - first, a contextual section

which describes the motivations behind the WTC and careful temperament considerations

one must account for; secondly, a section akin to the likes of those of Scott Burnham, where

particular examples are picked from the WTC and analysed at depth to fully show the extent

of how Bach used dissonant devices in his fugues; finishing with a section which discusses

the issues with chromaticism in Bach’s work, primarily how it follows the delicate rules of

species counterpoint but often breaks all of these rules; whilst all the time still working in the

Bachian context. I hope to layout an appealing and alluring argument before you for the

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purpose of Bach delving into the chromatic language and why it was perhaps hidden

beneath other musical ideas. The suggestion of Bach being the first composer to harness

this new musical language in his works begs more questions such as why was it Bach who

was first to exploit it; is it important that Bach was the first, even more so when 19th century

theorists placed Bach at the apex of western tonal harmony ; or whether Bach was fully †

aware of whose style he had developed to create such musical developments?

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I suspect those 19th century theorists praised Bach not only for his musical greatness, but also for the reason that it was †

through Bach’s chromatic model that the later composers could compare their chromatic and dissonant passages to, cer-tainly to an higher extent than they could with other historical composers.

Contextualising the WTC

Origins behind the WTC and temperament considerations !WTC Book I which was thought to be completed in 1722 in the last period of Bach’s time at

Cöthen is believed to have began its compositional life in the late days of the Weimar period

(1708-17), quite possibly during his month in prison, though little is really known about the

true origins. David Ledbetter in his book on the 48, points out that the reason behind the

composition of book I is quite the opposite from demonstrating invention in all the keys, with

evidence of some preludes being written in the old fashioned Dorian key signatures and

others transposed (D# minor and G# minor in WTC I); as a measure of recycling from earlier

stocks, it is unlikely that Bach originally set out to compose an original set of compositions for

this purpose (Ledbetter, 2002, pp. 4-7). It is widely regarded amongst the general populace

of performers that Bach wrote the WTC to demonstrate each and everyone of the keys,

though the recycling perhaps shows how pieces were forced to fit into this framework.

The origins of WTC II are slightly more ambiguous compared to those of the first volume with

no real known date for the start of composition. It is likely, as this was reaching the peak of

Bach’s teaching career, that he composed a newer version of book I to sustain interest and

introduce new musical ideas which were more fashionable. These compositions are a little

more sustainable in origin, but more sporadically sourced than those of book I, with 13

sources linking to pieces from book II, some copies of originals have been dated back to

1720 - before Bach even began WTC I. Ledbetter continues to state how book II is compiled

in “two campaigns for covering the chromatic octave” (Ledbetter, 2002, pp. 7-9). These two

campaigns split down to the following keys, the first being: Cm, Dm, E♭, E, Em, F, F#m, G,

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Gm, A, Am and B; the simpler keys, which were the most regular keys to utilise on a

keyboard instrument. What further persuaded or enticed him to write for the ‘advanced’

remaining keys I suspect now stands as to the whole pivotal point and reason behind the

composition of not only book II but also book I of the WTC. Peter Williams argues in his text

that the key structure of the WTC is irrational with it being rare for keys to be ordered

chromatically with their minor immediately after the major; indeed the format of his colleague

at the time, Friedrichh Suppig, followed the more localised format of C-a-F-d-Bb-g-Eb-c,

etc..

…both books begin in C major, but why? For organists the classic first

key had been D minor, which traditionally also had neither flats nor

sharps… Even if Bach was following recent convention in starting with C

major as the first key for a set of secular keyboard fugues, as he was, it

was not inevitable that it would be followed by the minor version…

(Williams, 2000)

This leaves the WTC open for criticism in the fact that it is illogical in order, both in ways of

explaining temperament and tuning, along with demonstrating the ability of playing in all of its

24 keys on one well-tuned instrument (Williams, 2000, pp. 762-3). This historical form of

analysis perhaps shows the larger image of systematic chromatic programming on Bach’s

behalf which makes up one half of the chromatic narrative - that of the composition of a piece

major and minor based from all 12 keys of an instrument. There are however, many works

amongst the WTC whose chromaticism exceeds that of normal diatonic counterpoint. That is

to say, the realisation of counterpoint through the likes of Buxtehude and Froberger, which

often explore no further than standard modulations and progressions, are built upon by Bach

to exceed these general rules as confirmed by Fux.

Other considerations one must take into account whilst listening to the WTC as modern day

listeners is the matter of temperament. We often take the tuning system for granted, but this

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was not always the case as argued in Duffin’s ‘How equal temperament ruined harmony (and

why we should care)’ (Duffin, 2008). The temperament which we are now so used to was not

fully concert standardised until around 1917, some 200 years after the composition of the

WTC, which was meant to be so elegantly playable on one keyboard instrument without the

need for retuning; so how was this achieved? Many have presumed that Bach was the early

adopter of Werckmeister’s ideal equal temperament and thus composed the WTC to prove ‡

it’s worthiness; however, it is widely regarded that Bach did not allow other’s to tune his

instruments as drawn to our attention by his first biographer, Forkel.

“He [Bach] also tuned both his harpsichord… when he played from his

fancy, all the 24 keys were in his power; he did with them what he

pleased… even his transitions in the chromatic style were as soft and

flowing as if he had wholly confined himself to the diatonic scale. (…)”

(Forkel as cited in The New Bach Reader, David et al., 1998)

The ambiguous scrawl of a tuning diagram at the top of the title page of WTC I (see fig. 1.1)

was decoded in 2005 by the academic and performer Bradley Lehman. So what was this

magical keyboard tuning system that

Bach had invented and why was it not

exactly equal temperament as used

today? How did it work with all the keys

and how different did chromatic intervals and passages sound than they would in equal

temperament? All these questions are important, springing from a very much disregarded

area when analysing Bach’s harmony and chromaticism. What we as modern day 21st

century listeners hear was not the same to what Bach heard with his own ears as a result of

the temperament; granted the vague shapes stick through the centuries but even a slight

change such as a 1/6th comma will make a difference whilst listening to a chromatic

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Much to the fault of C. Wolff in my opinion, who consistently refers to both Bach and Andreas Werckmeister’s tuning as ‡

almost synonymous in his various discussions on Bach.

Figure 1.1

passage. Duffin introduces some quite astonishing facts about historical temperaments in his

book, such as string players raising the leading tone and not be restrained by the likes of

having enharmonic notes as the same pitch, ie. A♭ is not the same pitch as G#.

Another hugely argued area by Duffin using examples from the likes as Joseph Joachim with

his major and minor semitones being again another alternative to the harshness of the raised

major third in equal temperament. The WTC hosts a few examples where Bach wittingly

acknowledges the enharmonic spellings of some keys to not only allow a more acceptable

key signature but also as a way in which a major and minor semitone must be compromised

to meet in the middle and act as one note on the keyboard. The D# minor in both books is a

fine example of Bach’s acceptance to enharmonic spellings. This is further supported by

Lehman who describes the process of tuning to compensate for a compromise of syntonic

comma and lesser diesis temperament systems resulting in D# being a different note to E♭,

making it impossible for Bach to write an E♭ major prelude and fugue followed by a D# minor

prelude and fugue unless they were absolutely the same note (Lehman, feb., 2005, pg. 5).

Wohltemperirt is the term Bach writes on the title cover, but ‘equal-tempered’ does not

translate into equal temperament and to a degree this is where many musicians make a

premature assumption. This equally tempered system refers to an irregular cycling

temperament, one in which all keys can be played in but each is coloured slightly differently.

As Duffin explains this was a strong era for where keys began to develop certain

characteristics with the likes of Francesci Galeazzi describing B♭ major as “tender, soft,

sweet, effeminate, fit to express transports of love, charm and grace” in 1796 (as cited in

Duffin, 2008, pg. 44). Where it heavily links in with one’s perception of harmony and

chromaticism is how the chromatic colours are entirely different in each key with intervals

often falling short of their equally tempered equivalent (sometimes sharper too). We are now

at the luxury of having technology which allows one to do a side by side comparison of the

WTC, one in equal temperament and the other in the temperament decoded by Lehman

(from such recording artists as Egarr, Beausejour and Watchorn). When comparing selected

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movements from the WTC with particular chromatic interest such as Book I: C#, E, F and B

minor fugues and Book II: C#, D, G# and B♭ minor fugues with the A minor prelude, one only

has to sing/play along with a score to notice the truly dramatic difference in pitch. §

To what real effect does this alternate temperament have on the chromatic elements of the

fugues in question? Certainly the beginnings of key characteristics begins to shine through

here, as Lehman discusses on several occasions in various publications. In the preface to

Watchorn’s recording of the 48 he writes the following:

In the minor keys from one flat to three sharps, the dominant triad (V) is

much stronger/brighter than the tonic, creating forward motion for relaxed

resolution. In the minor keys with two or more flats, that relationship is

reversed, with a dominant that is calmer than the tonic: making a gentle

effect overall, yet leading into intense or troubling conclusions.

(Lehman in the preface to Watchorn’s WTC recordings, Nov., 2005)

Original ideas raised whilst planning this research questions just quite why all of the

chromaticism was appearing in the minor key fugues and not really the majors. This noted

unbalance certainly begins the explanation as to why modulations in the minor keys are

perhaps a little easier to the ear, this could also be the reason behind Forkel stating that

transitions of the chromatics were so “soft and flowing” in Bach’s tuning system.

With use of appendix II, one can quite easily see where some intervals which are quite harsh

sounding are now actually narrower intervals in Bach’s system. In equal temperament the

octave of 12 semitones are assigned 100 cents each (the cents is a logarithmic scale used to

compared tuning accurately in multiple octaves). Taking the C# minor fugue from book I, one

can see those first few notes of the subject which are based on two semi-tonal movements

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Perhaps this is more apparent to a well trained ear. After learning to tune my own harpsichords and pianos in the past 4 §

years my once equally tempered fatigued ears were brought into the light of what pure fifths really sound like and it is now that I find a majority of music listened to is recorded in its rightful historic temperament and thus the harshness of equally tempered music stands out even more. Please see appendix I for Lehman’s tuning chart for Bach’s temperament.

separated by a diminished fourth leap are significantly narrower than their equally tempered

equivalents (see fig 1.2). Most interesting is the narrow 6 cents on the final step from E down

to D# which is the narrowest (caused by the higher D#). This easing off in sonic difference is

really quite astonishing, yet even more so when one studies the harmonic modulations

throughout the fugue. The modulation to the dominant on various transitions further backs up

the strong tonic dominant relationship in Bach’s temperament. One only has to realise that

this fugue explores very few keys, hardly features episodes, and does not finish with a coda.

It is purely theme after theme exploring a mere 3 differing keys - C#m, G#m and E, the same

key structure as explored by Bach’s influential composer, Buxtehude.

The delay of the bitonal cadence at the end of the C# minor fugue is preceded by a

sidestepped diminished-seventh chord (see bars 107-109 in fig 1.3) as pointed out by Siglind

Bruhn in her analysis of the 48 (Bruhn, 2014), but the actual tonal makeup of a diminished

chord further adds to the argument of being a world away from the intentions of Bach. This

seventh as seen opening bar 108 totals 1200 cents as it should. However, this is split up of 3

minor thirds of 298 cents and one augmented second between A and B# at 306 cents - Page ! of !13 43

Figure 1.2

Figure 1.3

much sharper than anticipated. This wider interval which includes the leading tone (B#) harks

back to that of the discussions of Duffin in his book about instrumentalists raising the leading

tone. Furthering the debate that this diminished-seventh chord in the context of where it is,

moving towards the tonic (interrupted by a soprano E) is geared towards modulation and not

dissonance.

It is with these considerations one must look at the WTC as not only being a monumental

work which stands out from the Bach catalogue, but also one in which many ambiguities

arise with multiple solutions. We must always consider the disagreement in temperament

throughout the chromatic analysis in the following sections, but also understand that Bach’s

intentions behind the set of works was perhaps not quite as simple and clear cut as is made

out at present. Although chromaticism is more dissonant in equal temperament, the ‘relatively

dissonant’ tones in Bach’s temperament still endure the same musical subsequences - i.e. no

matter what temperament a diminished seventh chord is played in, it will still maintain the

same function. Whether or not the chromaticism is based on the temperament of the time is

also a very interesting perspective to take. Lehman states now thirds and fifths stand out in

particular keys due to the coloration of Bach’s tuning system, and it stands quite possible

that particular keys favoured particular forms of dissonance. However, one must also

consider that where the same chromatic techniques occur consistently through a larger body

of works in different keys, then the characteristic of dissonant tones in one key stand to the

same degree as they would in another.

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Bach, the chromatic

Chromatic appearances within the WTC !This following section aims to point out the uses and appearances of chromaticism and

dissonance through selected techniques within the WTC. Before delving into this form of

analysis it is important that one understands the implications of what is being used via the

rule book from which from Bach was thought to be using and building upon. The Fuxian

realisation of the contrapuntal works of Giovanni Pierluigi da Palestrina is perhaps the best

model to take when comparing Bach’s work; more often known as species counterpoint it

covers the basic rules as bulleted bellow. It is from this model in which Bach took instruction

for his own compositional works; as it was Buxtehude and de Grigny that were both heavily

influenced from Fux’s leading text: Gradus ad Parnassum; and it is their works which Bach

copied down as a child. The general rules of first and second species counterpoint are listed

below as cited from John Rothgeb (Rothgeb, 1975):

The leading tone must be raised in a minor key.

A skip or leap must be followed with step motion in the opposite direction.

The tritone should be avoided on all counts between voices.

Climax of a line must be reached on a strong beat in the bar.

In first species, avoid dissonance by major or minor 2nd, major or minor 7th, any

augmented or diminished interval along with the perfect fourth.

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In second species, the accented beats must only have consonance with the weaker beat

containing dissonance only as a passing tone.

What is most interesting to us as modern academics is the disobedience/obedience of Bach

to this general set of rules, with Bach often showing appreciation by abiding to these rules,

yet still containing movements in the WTC which break so many of them that they appear to

have stemmed from another form of contrapuntal instruction.

For the sake of not having any ambiguity, let us reflect briefly on what dissonance in

counterpoint really is. A note is dissonant if it creates an unstable interval with that of the

other note sounding at the same time. Most unstable intervals are seconds, sevenths and

diminished and augmented chords (as raised in first species counterpoint). The perfect fourth

is also classed as dissonant as it is an unstable tone - it wants to resolve down to the third or

rise to the fifth. Within the WTC there are some consistent techniques used by Bach to

integrate dissonance. To really question the meaning of its presence, one must remember

that in fugues the purpose of dissonance is to create a sense of instability so that the

following subject entry stands firm amongst the other voices. This certainly correlates

positively to the approach of a final cadence at the end of a work too in Bach. The following

sections discuss at depth a handful of these techniques, exploring origins and consequences

appropriately, creating a taxonomy of dissonance in the context of Bach.

!Dissonance caused by pedal tones !The idea of pedal tones is not really introduced until fourth species counterpoint via

suspensions but embellished further in fifth species counterpoint with the acceptance of

rhythmic augmentation on dissonant notes. Bach uses this in the two primary forms - the first

having the pedal note hold harmonic stability (almost always in approach to a cadential point)

with the tonic being held as a pedal; or the second where the pedal note is the dominant

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note (although this form is less common). Thomas Benjamin in his book on counterpoint

states:

‘[a] pedal point may occur anywhere in a fugue, but it is most often used

towards the end, settling the tonality and providing, like augmentation, a

broadening effect… The only commonly used pedal notes are tonic and

dominant, and they nearly always occur in the lowest voice.’

(Benjamin, 2003)

This idea of using a pedal tone to stabilise harmonies certainly bodes well with Bach’s body

of work, most noticeably in the fugues where an ending might seem near impossible without

the use of a pedal tone to halt the momentum. The C minor fugue from WTC I is a fine

example of this with the final two and a half bars holding a tonic pedal (see figure 2.1) above

which the upper voices come to an eventual standstill through tonal retardation. The fugue

would come to an abrupt end should it stop at the end of bar 29 on the E natural, which in

this case does not act as a tierce de picardi, but as a member of the secondary dominant to

the subdominant (F minor) which prepares the fugue for a tonal unraveling. This is an

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Figure 2.1

essential form of chromaticism for tonal music which prepares for the subdominant (on a

weak beat) and also reiterates the importance of the unflattened note.

Let us for a moment delve slightly deeper into the harmonic language contained within this

final section as it holds a fair amount of dissonance which is otherwise unseen when masked

by the repetitive subject, the hook of which is the repeated C, B-natural, C motif, acting as a

final cry at the end of the fugue. The E-natural leads on to the F minor in second inversion

(who’s dominant is the C tonic in the pedal note) which is abruptly interrupted by a diminished

seventh on F whose A♭ recoiled down a semitone to G to create a G seventh chord to return

the movement to the tonic of C minor. This is shown in figure 2.2 where the harmonies have

been vericalised. The alteration between diminished seventh and dominant seventh (only a

semitone difference) allows Bach to easily confirm that C is the home key, despite the

introduction of new harmonies in the end section. However, that analysis ignores the fact that

these changes are all suspended against a tonic pedal, to which the Cm, C major and F

major chords easily work, but the diminished chords cause an almighty clash with. After the

diminished interruption midway through bar 30, Bach could have quite easily ended on that C

minor, but he takes it a step further by introducing another discordant dominant and tonic

pattern in the final bar, this time with a tierce de picardi (as raised by Anson-Cartwright in a

text about elision of Bach’s final cadence, more often than not aided and caused by a tonic

pedal tone (Anson-Carwright, 2008)). This elision of the cadence certainly aids in reiterating

the tonic, with the weakening dissonance creating a greater contrast to the pure home key.

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Figure 2.2

This may not be fully noticeable on the piano where the pedal note naturally decays to a low

enough volume to not be fully heard by that time, but on the organ, these diminished chords

make a mighty statement in conjunction with the pedal tone (particularly when considering

temperament considerations), thus we can derive that the pedal has two purposes. The first

of these is to maintain harmonic stability with the listener firmly planting themselves in the key

of C. In contrast to that purpose, the second is to upset the harmonic apple cart, to create a

particular amount of discomfort in a listener that they strive for the tonic which sounds even

more fitting, meaning that when the upper voices do land of a harmonious note in C (major or

minor in this instance) it is like two interlocking pieces of a jigsaw, there is a release of tension,

despite all that is going on around. Thus Bach has used dissonance to for a consonant

solution via means of enhancing the consonance.

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Figure 2.3

A much grander example of this from the WTC is not far from the first, fugue number 4 in C#

minor from book I, whose pedal notes create a very similar harmonic structure is taken further

to match the scale of the this grand 5 voice work. The pedal tones start eleven bars from the

end on a dominant pedal (G#) and despite a non pedal tone for one bar (bar 109), it

continues to a tonic pedal for the final four bars but eliminates the use of the diminished

chord (as seen in figure 2.3)- Bach resolves it naturally via a plagal cadence, although this is

not heavily discussed in species counterpoint due to the lack of the leading note which

makes it difficult to confirm tonality in a cadence. The resemblance of this fugue to the older

school of fugue writing of Frecobaldi is pointed out by Ladewig although he does not treat

the pedal tone as anything of significance, an issue found quite often when comparing Bach’s

fugal writing to that of the older composers (Ladewig, 2003).

Dissonance is caused here in the dominant pedal, which, naturally by itself is an unstable

note awaiting return to the tonic, it is in essence a prolongation of the dominant seventh

chord, but Bach resolved this dominant pedal in an unusual way. The use of a deceptive

cadence is, according to Benjamin, a very dramatically effective way in which to resolve a

dominant pedal note, but is rarely used by Bach. However, it is this very cadence in bar 112

which is of the most important dissonant interest to us in this example. After a long period of

the dominant seventh being held, the chord resolves correctly to the C# chord (with a major

third) in a perfect cadence fashion, but is clashed against an A in the tenor part (the sixth

degree) and it is this clash of tonic/subdominant harmony which forces the fugue to continue

ticking over for the final section as it works its way back to C# in mostly consonant terms,

besides the third beat of bar 113 which hosts an accented passing tone in the tenor line (the

dissonant tenor) as one final chromatic outbreak before resolving harmoniously.

From these two seemingly innocent examples, one can see the plethora of dissonance

caused by pedal notes, not only in upper voice harmonies but also in the resolution of a pedal

tone. The heavy integration of secondary dominants of the subdominant often leads naturally

to dissonant passages due to the particular combinations of these alongside the tonic notes.

Page ! of !20 43

Bach and the 7th chord !The third inversion of the 7th chord is often Bach’s escape from what may otherwise appear

to be impossible fugal crosswords. The chord which has already appeared in the example of

pedal tones in the C minor fugue from WTC I is favoured heavily by Bach as a form of escape

into new and old harmonies. Benjamin states on the mater:

‘He [Bach] is especially fond of secondary fully diminished seventh chords.

In this style, any diatonic major or minor triad may be preceded by any of

its secondary dominants…Progressions around the circle of fifths may

involves series of secondary dominants, but more often alternate [with]

diatonic triads.’

(Benjamin, 2003)

The seventh chord is an easy escape route for Bach to work the harmonic direction into new

areas. The dissonance caused by this chord is perhaps why it has not occurred to earlier

composers, although the idea of modulation to particular keys is also a certain factor. The

likes of Frescobaldi, Buxtehude and de Grigny often never explored beyond tonic, dominant

and relative major/minor; but here (such as book I, C major which explores in passing the

supertonic, E♭ minor which explores the mediant G♭ etc.) we see Bach reaching further into

new keys, not necessarily for whole episode but in brief passing to help achieve tonal interest

for his listeners.

The unbalance of the diminished seventh chord makes it suitable for a magnitude of

resolutions, putting it as a key transitional and pivotal chord in contrapuntal writing. It can just

as easily return a fugue back to the subject for the final section as it could carry it off in

another key for further episodical writing. As an adaptable chord, it appears in one form or

another in a majority of works in the WTC, both preludes and fugues. The famous C major

prelude is based on diminished chords at the end to bring it back to the correct key; the D# Page ! of !21 43

minor prelude features diminished chords in the bass recitative section towards the end (on

beat one of each bar) which creates the harmonic structure, again before returning back to

the correct key; and in the penultimate bars of WTC II’s C minor fugue where it is built up over

a larger arpeggiando. So versatile is this chord that Bach seems more often than not to

reserve its use until the end of a fugue to help bring racing tonal changes to a standstill with a

chord which stands out as a dissonant sore thumb.

The G minor fugue from WTC I is a fine example of how Bach plays on the use of implied

dominant seventh and diminished seventh harmonies. Figure 3.1 shows the final three bars of

this fugue where descending voices allow for an easy interruption from the diminished

seventh chord, which in turn leads onto the dominant in the final bar. This seemingly gentle

and innocent fugue opens the door to understanding how Bach truly gets away with stark

harmonies and progressions in slower moving works, quite possibly from no real strong beat

being felt (particularly true in variation 25 of the Goldberg variations). Besides the implied

diminished sevenths throughout this piece, one chord stands out in particular - the

penultimate bar with a C diminished seventh chord. This stands as a dissonance on a strong

beat of the bar but is quickly resolved using an appoggiatura-like motif (as discussed in the

following section) to resolve the top note down to a D, and the bass C down to a B♭,

although this creates a rather unwanted chord in species counterpoint - an augmented,

caused only by the raised leading tone in the thematic motif. Bach often uses the secondary

form of dissonance to allow the voices to move in a way that suits well, often with passing

dissonance caused from a voice that is being suspended over from a main beat on to the

more dissonant weaker beat. Page ! of !22 43

Figure 3.1

Appoggiatura dissonance !Perhaps the most obvious appearances of dissonance via chromatic notes is through the use

of an appoggiatura (or ‘musical sigh’ as it is often referred to). The appoggiatura is in essence

an accented passing note which resolves to a consonant note for the second half of the

measured accompanying note. Its embellishment is in turn classed as an ornament to

introduce non-harmony notes into a work (perhaps best saved for another discussion) but in

Bach are often written out in instances like those outlined below whose purpose is not just

ornamental but makes up a strong area in his fugue subjects.

The WTC is ripe with different forms of examples; figure 4.1 shows the A minor prelude from

WTC II, which beyond its other chromatic appearances such as the tone row (as pointed out

by Alschuler and Elkies, 2009) and linear chromaticism, consists of inner part appoggiaturas

which alternate in direction. The appoggiaturas as seen in the weaving parts (treble bar 1,

bass bar 2) follow the general consensus that chromatic notes are reserved for weaker beats

as is the case in normal species counterpoint. The feeling of these appoggiaturas being on

strong beats is achieved by Bach from the delayed beginning which pushes the motif out by

a semiquaver, meaning that the rule of chromatic notes appearing only on weak beats is

obeyed, whilst still maintaining the feeling that the first note of the appoggiatura falls on the

Page ! of !23 43

Figure 4.1

stronger part of the bar despite that being where the texture is at its thinnest. Thomas

Benjamin fails to see the appoggiatura figure sustaining throughout the section, calling some

of them ‘incomplete neighbour tones’ which are what we refer to here as appoggiaturas but

then goes on to treat the B♭ in bar 1 as a passing tone when it clearly leads into the A below.

In a situation like this where the motif is clearly thematic, all must be treated equally to fully

understand what Bach has done. What is perhaps more interesting in this case is accented

chromatic notes in the alternate parts (bass m. 1, treble m. 2) whose notes fall on strong

beats, particularly the pattern of A, G#, F# and E which all fall on main beats of the bar and

via extension of the chromatic tetrachord is extending to the higher voice in the next bar.

However, this has been pointed out to be an extension of the i-V progression with a

chromatic bass line with appropriate unbalanced 1st inversion chords (see figure 4.2) as

noted in many other works of not only Bach but other Baroque composers (Didos lament

from Purcell’s Dido and Æneas springs to mind). Looking beyond these intricacies one can

see the linear intervalic progression (ignoring compound intervals) ends up as a simple

3-6-3-6-3 etc. progression (figure 4.2) which is often found in a deeper harmonic progression

in contrapuntal works showing how it is a heavily

structured work, despite all of its non-harmony

notes. The A minor is quite a peculiar work in terms

of the WTC, and is in this instance the only prelude

worth mentioning in the chromatic debate as it is

formed of so many different elements.

The form of alternating appoggiatura motif happens again in the F minor fugue from WTC I

(figure 4.3) although on a greatly augmented rhythmic scale. Again, the grouping of two notes

just a semitone apart and always in alternate directions is a recurring theme in Bach’s

adoption to appoggiaturas in thematic exploration in the following pattern: C raises up to a

D♭, followed by a C resolving down to a B, an E to an F, a B♭ to an A, etc.. The rest at the

beginning of this fugue allows the resolved note to land on the main beats of the bar, with the

Page ! of !24 43

Figure 4.2

first semi-tonal note of the appoggiatura always appearing on the weaker beat. Often when a

piece does not start on the first beat, we class it as an anacrusis and the remainder is

accounted for in the final bar, but this is not the case here for Bach. The first crotchet beat

rest is classed as part of the subject and occurs correctly in all of the proceeding subject

entries. This rhythmic offset was again used in the A minor prelude, which allowed the

chromatic non-harmony notes to fall on weak beats (as dictated in the rule book of species

counterpoint) leaving the primary harmonies to remain on strong beats. Despite this work

around on Bach’s behalf, the subject is still peculiar in terms of its harmonic structure - when

its solo first entry is heard, there is no indication of in which direction this fugue is to travel. So

ambitious is it that it is not until the last bar that it confirms the F minor shape on beats 1, 2

and 3 in bar 4.

Another example from the WTC where the appoggiaturas actually fall on the main beats is the

B minor fugue from WTC I (figure 4.4) who’s theme is entirely made up of appoggiaturas on

the main beats which all fall by one semitone, as in the F minor fugue. What one must be

careful of here is to note that the dissonant note is the note of which the appoggiatura

resolves to, in contrast to the appoggiatura beginning on the stronger beats of the bar helped

by the quaver rest at the beginning to offset the dissonance, much like in the A minor prelude

and the F minor fugue. Contrary to the F minor fugue, the B minor spells out a B minor chord Page ! of !25 43

Figure 4.3

first before going on a chromatic path which continues to spell out important harmonies

which are later to follow, such as the raised A# which helps confirm the dominant chord in

the second half of bar 1. From a species counterpoint view, this is perfectly acceptable as the

notes quickly resolve, but from a verticalised

harmonic viewpoint the implied harmonies go beyond

those of a normal simple subject. Siglind Bhruhn

removes the appoggiatura by taking only the second

note (as shown in figure 4.5), showing how the

subject hosts a dominant modulation in the middle. The implied sevenths and inversions add

to the complexity of the harmonic accompaniment, enhanced only more by the accented

chromatic passing tones.

Through rhythmic displacement Bach has managed to use this idea to integrate chromatic

and dissonant notes into his fugues through either notes the appoggiaturas resolve into, or

from chromatic notes which often resolve onto consonant notes. Although approved in

counterpoint, the adoption of this motif to create subjects is taken to the extreme by Bach,

whose most chromatic works integrate this idea.

Page ! of !26 43

Figure 4.4

Figure 4.5

(Bruhn, 2006)

Bach’s use of cambiata and echappée !Cambiata is perhaps the best way to describe some of Bach’s dissonant sections, although

does not provide a fitting solution for much of it that has been discussed thus far. In species

counterpoint, where it is introduced at the third stage, dissonant cambiata is a way of

labelling an acceptable pattern of notes which would otherwise break rules. It is approached

by step in a descending direction and allows a skip from a dissonant note by a third and

must be proceeded by a step in the opposite direction. Steven Rings points out on his book

on tonality that in Bach’s E major fugue from WTC II there is an inverted cambiata figure

which so elegantly ‘echos passages from Froberger’s fantasies and ricercares and Fux’s

Gradus’ (Rings, 2011). This figure can be seen in the bass voice in bar one both the leap

from the F# up to the A, which then resolves down to the G# as it rightly so should (see figure

5.1). Rings continues to mention that of Bach’s countersubject:

‘Bach’s countersubject, nowhere to be found in Fischer’s fugue, appears

literally in Froberger’s Fantasia II and is hinted at in his Ricercare IV, both of

which are based on the cambiata subject. Bach’s countersubject is also

present in three of Fux’s fugues…’

(Rings, 2011, pg. 151, footnote 2)

From this we can see how the greats before Bach worked these dissonant notes into the

fugues, and this is perhaps one of the few examples where Bach willingly follows the rule

book on dissonant counterpoint without developing it much further. The leaping third is a

popular motif employed by Bach, and one which seemed to fair well for Buxtehude, Fux and

Froberger - the likes of whom Bach copied and undoubtably developed this from. It was

certainly Fux who was at the forefront of developing this technique, taking the leap from not

only a dissonant note, but also from counterpoint convention.

Page ! of !27 43

Echappée or escape tone, leads on from cambiata in the form that it must move up a step

against a harmony but then leap in an opposite direction back to the next harmony note. In

her analysis of the complete 48, Siglind Bruhn points to a plethora of escape notes and how

they became much more popular in use since Bach’s adoption of them. From a performance

aspect she takes the G# minor prelude from book II and points how bar 5 G#, 6 F# and 7 E#

are often likely to be accented in performance, but care should be taken as they are merely

escape notes and act as a decoration to the music; particularly allowing more room for Bach

to work in a secondary melody such as in the C minor prelude in WTC I, without the need for

extra voices (Bruhn, 2006). Escape notes may not feel like they cause dissonance, but the

introduction of a non-tonal note to a chord is the biggest presumptions one can take for

granted in the realms of species counterpoint. Without this, much of Bach’s cadential work

would not have developed into the largely embellished sections that they are, and as such,

should not have spurred on further dissonant progressions.

As modern day listeners we do not class cambiata and echappée as dissonance worth

mentioning because they simply do not sound like dissonant passages. When comparing

Bach’s work however, to that of his predecessors, it is clear that even such minute rules

which often abide by the strict rules of species counterpoint can then be extended upon by

Page ! of !28 43

Figure 5.1

Bach to suit his own compositional style and this is one of the instances where species

counterpoint lends itself for dissonant notes to be integrated in future works.

!Alternative solutions to a disobedient Bach !Thus far we have looked at the rules of species counterpoint being a firm system of which to

compare Bach’s to. He often follows the general rules which make for a convincing fit to the

system but also further develops the methods to allow for the new stylistic harmonies and

motivic material of the fashionable Baroque period. There are, however, many instances

within the WTC where Bach not only takes the rules of species counterpoint as a form of

extended technique, but breaks all rules possibly known to the rule book when there are clear

abiding alternatives which would suit better. This causes a programmatic issue as the

consistency of this rule breaking is so irregular; often a fugue which begins by following all the

rules will end with a clash of almighty dissonance and drop all that was invested at the

beginning, with some fugues simply just following the rules with a few anomalies in small

sections. As the compositional background of both books is so ambiguous, it is impossible to

understand any factor which may have fully contributed to Bach's witting disobedience to the Gradus ad Parnassum.

Joel Lester touches ever so slightly on this area when tracing chromaticism through voice

crossing in the C minor fugue from book I (Lester, 2001) although misses the point that

chromaticism caused by this is from an apparent disobedience to the rules. Such is the Page ! of !29 43

Figure 6.1

(Lester, 2001)

extent that there is not just one solution, but multiple solutions Bach could have taken to

abide by the rules. Bar 17 from that C minor fugue is taken as an example of how Bach could

have used alternative passages to work the fugue. Figure 6.1 shows the original Bachian

solution in Lester’s diagram and points to the chromatic voice passages which track the

semi-tonal movement throughout the phrase.

To fully understand how this does not agree with species counterpoint, but does work in the

contact of Bach, one must recapitulate the rules of first and second species counterpoint as

that is all that appears in bar 17 (bar 18 is included for further contextual purposes).

Relevant rules to reconsider:

The climax of a line must be reached on a strong beat.

In first species avoid dissonance by major or minor 2nd, major or minor 7th, any

augmented or diminished interval and the perfect fourth.

In second species the accented beats must only have consonance with the weaker beat

containing dissonance only as a passing tone.

Point one suggests that the climatic point of a phrase should be reached on a strong beat -

but in Bach’s case everything is offset by one crotchet, taking the end of each soprano voice

snippet as the primary point of each phrase. Because of this, all the important climatic

phrases end up on either beats two or four, although this is very much dictated in this

instance by the pattern of the fugue’s subject which is based upon the lower mordent shape

as the second part of the phrase. The realignment of this is shown in figure 6.2 where the

phrase has been pushed back by one crotchet to allow the climax of each phrase to land on

the main beat. This delayed climax will prove importance once the following species rules are

taken into consideration.

The first species counterpoint intervals are all abided in this instance, with all intervals moving

together in thirds with the exception of the B♭ (belonging to second species counterpoint) on

Page ! of !30 43

beat two and a half in bar 17. This B♭ making up the dominant seventh in C major is

introduced on a weaker offbeat, but is still sounding on the stronger main accent of beat 3 in

bar 17, which species counterpoint forbids as it is an interval of a flattened seventh above the

lower voice’s C (this would be an acceptable suspension if it were to resolve in the following

beat). Again, this same motif happens when the C is tied over to the first beat of bar 18 to

create the dominant seventh in D, leaving an open diminished fifth between the upper voices.

It is corrected by simply resolving down to the fifth, which is acceptable in this case as the

tonic and mediant is always sounded at the same time, i.e. the interval is not left as an open

fifth.

An underlying issue that was present before the rhythmic realignment was the classing of the

D natural against the E in middle and lower voice which rapidly resolves up to the F#. This

dissonance is permitted in species counterpoint but is usually the other way round, i.e. the

consonant note falls on the beat and then the dissonance is caused after as a passing tone

to the next consonant interval. The raising of the E and the F# points that Bach is heading

towards G major but it is quickly turned around with the introduction of the C chord, and

Page ! of !31 43

Figure 6.2

Figure 6.3

further works towards D major and G major again during this sequential pattern. Rewriting

this to avoid dissonance as Bach uses results in the following possibility, shown in figure 6.4.

Other considerations to take into account is that tied notes and repeated notes are not

permitted in second species counterpoint, but these begin to take away from the rhythmic

devices Bach employs throughout this phrase. There is something very dissatisfying with the

above solution, it is not Bach. Nor is it even half way there. Such an elaborate rule breaking

episode to simply modulate from a phrase in G minor to C minor contains so much

dissonance that without, it is simply not what one expects from a Bach fugue. The flirtations

with other keys in such a short space of just a bar and a half unbalances the whole act that

when the subject does once again enter it stands out as a secure and tonal base of which

the listener can firmly plant their feet again. The rhythmic offset by a crotchet as seen in

previous works with the appoggiatura motif allows the dissonant notes to fall on weaker

beats of the bar despite them sounding like strong beats. This device is used consistently by

Bach, and is perhaps one of the many work-arounds he found to deal with rules of species

counterpoint.

This is just one of many examples in Bach’s writing where alternatives can be found, and acts

as a good exercise to rewrite them following the rules of species counterpoint to fully

appreciate and understand just how far Bach moves, say from the general consensus of

tonality and counterpoint.

Page ! of !32 43

Figure 6.4

Fugue

Conclusion !!It may seem from this distance that these distinct areas do not really add up to much in the

grand argument of Bach as the first chromatic composer (in a tonal context) who littered his

work with so much dissonance. Certainly quite a few questions arise from these areas in the

way of how they all integrate to form an argument which succinctly proves Bach as a witting

composer who grasped the power of chromaticism earlier and more confidently than his

colleagues, who’s contrapuntal works often strictly abided by Fux’s rationalisation of

Palestrina. Although the evidence strongly hints to Bach using Gradus ad Parnassum as a

rule book, there is no such real evidence that Bach willingly shares his thoughts on it,

certainly no source to hint that these were the models he adopted. The general academic

consensus that the easiest way to analyse Bach’s contrapuntal works via comparison to

species counterpoint is indeed the correct way forward. However, I feel there are often many

passages which are overlooked in Bach. The rewriting of the small passage from book I’s C

minor fugue may appear at the surface to be a tenuous link to the chromatic argument, but it

opens up for discussion many non-musical factors which could have made Bach to decide to

use his solution and not those that followed those of the species rule book. Despite need and

lust for musical expansion, my theorem is that this is the essence of Bach’s style, the one

thing an academic could pinpoint to be the signature of Bach.

This paper has looked at only a handful of works from the WTC, most of book II has not been

touched as the examples from book I stand strong for the study of the second book. As

Page ! of !33 43

discussed in the contextual section, book II was a newer and much needed more fashionable

keyboard learning book, and although new styles and forms were adopted, Bach’s

dissonance use is consistent between the two. It is this that leads me to believe that the

chromatic choices made are the signature of his style, much more than the adoption of

musical styles such as the French overture or the modern Italian Galant style (Heartz, 2003).

What is clear though, is that Bach, knowingly or unknowingly, influenced the chromatic

development of the following three centuries to such a degree that we can now appreciate

him for his capabilities. Many composers began with Bach - and the case shows clear for

young Mozart, who’s own symphony no. 40 in Gm has a section with all the chromatic notes

except the tonic; this could only be the witty interest of Bach shining through as an influence.

For Bach and his mathematical compositional technique, the time seemed to be only more

than ‘ripe for the picking’ at this moment in history as stated by Phillip Clapp:

He [Bach] happened to be born in time to epitomise the contrapuntal and

coloristic achievements of over two centuries of cumulative effort; sacred,

secular, and dramatic technique was ripe, and waited only for a great

man. So far as social influence was concerned, Bach was fortunate in

coming of a family in which music was second only to religion and in living

in communities where at least music was respected and esteemed; if his

education was not remarkably broad or far-reaching according to modern

standards… his intelligence was keen, as is shown by his preoccupation

with such mathematical problems as that of equal temperament in tuning,

which he was almost the first important composer to understand at its

true value.

(Clapp, 1916)

This paper was in no means a way to doubt the power and ability of Bach as a composer -

and although some points of this paper have been a mere brief passing, their inclusion is

Page ! of !34 43

important to understand the complete picture of Bach’s ability in various moments of his life.

It is hopefully quite clear now that throughout his career, he became more daring with his

chromatic inclusion and developed techniques from the old counterpoint school to suit

modern demand with the integration of the latest fashion. Although one can not be sure of

the reaction he received to this, it is clear that his students reacted maturely and respected

the composer to the degree that such techniques were passed down through history and

continue to influence us still to this very day.

The western world of music owes almost the entirety of itself to the work of J.S. Bach, who's

inclusion of the dissonant notes has dramatically changed the course of western tonal history

through both melody and harmony. It therefore seems fitting to end this paper with a quote

which sums up the importance of J.S. Bach in the grand scheme of western tonal music of

which the forefather has to be classed as Bach for his brave delving into newer, more exciting

and exotic musical languages. The following quote by the writer Theodor W. Adorno therefore

sums up the importance of Bach:

!What sets him apart from the practices of his age, far from being grasped

as the contradiction of his substance with them, is made a pretext for

glorifying the nimbus of provincial craftsmanship as a classical quality.

(Adorno, 1982)

Page ! of !35 43

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Anson-Cartwright, M., (2008) Elision and the embellished final cadence in J.S. Bach’s preludes, Music Analysis, 26/iii (2007 Journal Compilation), pp. 267-288

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Discography

Discography !Beauséjour, L., (2007) ‘J.S. Bach The Well-Tempered Clavier Book I (24 Preludes and Fugues)’ [CD] Canada: Naxos: 8.557625-26

Gould, G., (2013) ‘Glenn Gould Plays Bach Vol. 2: Goldberg Variations | The Well-Tempered Clavier I & II’ [CD] U.S.A: Sony Classical: 886443991972

Guillot, S., (2006) ‘BACH, J.S.: Kunst der Fuge (Die) (The Art of Fugue), BWV 1080a’, [CD] Canada: Naxos: 8.557796

Lippincott, J., (2002) ‘BACH, J.S.: Organ Music (Preludes and Fugues)’ [CD] U.S.A.: Gothic Records: G-49202

Martins, J.C., (1995) ‘BACH, J.S.: English Suites Nos. 1-3 / Chromatic Fantasy and Fugue, BWV 903’ [CD] Bulgaria: Labor Records: LAB7025

Shiina, Y., (2005) ‘BACH, J.S.: Organ Music (2+1+3+8= BACH)’, [CD] Japan: ALM Records: ALCD-1071

Stuttgart Chamber Orchestra (1955) ‘BACH, J.S.: Musical Offering, BWV 1079 (Stuttgart Chamber Orchestra, Munchinger)’, [CD] Naxos Classical Archives: 9.80766

Watchorn, P., (2006) ‘Johann Sebastian Bach (1685-1750) Das Wohltemperierte Clavier Book I (1722), BWV 846-869’, [CD] U.S.A: Musica Omnia: MO-0201

Watchorn, P., (2009) ‘Johann Sebastian Bach (1685-1750) Das Wohltemperierte Clavier Book II (1742-1744), BWV 870-893’, [CD] U.S.A: Musica Omnia: MO-0202

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Appendices

Appendix I - Bach’s temperament

(Lehman, feb., 2005) **

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A commendable table which allowed me to set up my own instrument (a flemish single manual) in Bach’s temperament **

and experience the chromatic dialect for what it was truly meant to be.

Appendix II - Interval Tuning

(Lehman, nov., 2005)

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