Thematic Fusion
I
have developed a procedure that allows me to merge two or more musical themes
into one by averaging them. This procedure and complex of operations, which I
call the Thompson System of Thematic
Fusion and that I will refer to as Thematic
Fusion for short, can be utilized by the composer or arranger in a variety
of ways. It can be combined with traditional devices of melodic manipulation,
such as augmentation, diminution, inversion and retrograde, as well as used
independently. Thematic Fusion
furthermore allows the composer to use it either to create new thematic
material which he or she can treat as an independent dramatic event or to
create a gradual transition between two or more themes by providing a stepwise
transformation of one theme into another. The system and the procedures
involved will be described on the following pages in great detail. In
order to not only demonstrate, but also explore my procedure I have decided to
compose a set of 25 pieces for piano in all major and minor keys. The pieces
span a wide spectrum of genres and employ thematic
fusion in a variety of ways. Since the result of applying thematic fusion is always a permutation
of the original material I decided to name the collection of pieces Permutatis.
II. Purpose and Application of Thematic Fusion
Thematic Fusion was conceived by posing
the question: what if the exact middle between two musical themes was the exact
average of those very themes. By design and concept, Thematic Fusion therefore lends itself to be applied as what could
be called a transitional device, with the goal being whatever theme or thematic
event this transition leads to. Since, as will be demonstrated in detail in the
next section, Thematic Fusion by
design allows for a bias towards one theme or the other, various hybrid
versions can be created, each of them a step closer or further from the origin
or goal. With this functional aspect in mind, Thematic Fusion therefore can be utilized to create smooth thematic
transformation as well as what I like to refer to as morphing transitions. These transitions allow for one theme to
gradually become the second theme rather than having related or unrelated
material interweaved between them. The transition becomes a thematic event, the
journey becomes the destination. We can find this thematic transformation in Permutatis in no. 3, the Romanze and especially no. 8, the Minimaluett. With each reiteration theme
A becomes one step closer in sound and contour to theme B.
As
mentioned above, by using Thematic Fusion,
the transition becomes thematic material in its own right that is related to
both themes, yet new in sound and character. Keeping this in mind, Thematic Fusion can be furthermore used
to create thematic events in their own right. The hybrid of two themes can be
the climactic event of a piece, full of rhythmic and melodic activity, as it
may govern all aspects of activity of both themes. If this is taken into
consideration while using the system, the Thematic
Fusion does not need to be used to create a transition, nor does the
hybridized material need to be placed between themes. The fused theme may
appear at any time in the piece, either in the beginning, the middle or the
end. Since the new theme is a hybrid of both themes it will be a fusion of both
themes characteristics. If both themes are bubbling with activity, the activity
will likely be increased in the hybrid. If the composer chose to have two
contrasting themes, for instance an active theme with a high density of melodic
events and a legato theme with sustained notes, the hybrid of the two will
obviously be a tamed version of the first or an intensified version of the
second.
III. Procedure
At
its core Thematic Fusion uses
averaging to create hybridized themes. For demonstrative purposes this may be
as basically expressed as "H=(T1+T2)/2", thus creating the average of T1 and
T2. If one were to replace the variables T1 and T2 with musical themes then H,
the hybrid would be the exact average of the two. In mathematical terms this
could be expressed as "(6+4)/2=5." In musical and compositional praxis the
procedure however is a bit more complex and certain steps along the way have to
be taken into account. Not considering the artistic and aesthetic aspects that
must ultimately outweigh any mathematical consideration, a melody is
mechanically put, a combination of pitch and rhythm events in succession over a
period of time.
Each
pitch can be assigned a pitch class number, with the pitch C equaling zero and
the pitch B equaling eleven. The average of the pitches C and Ab for instance
then would be 4, if the pitch class numbers of C and Ab were inserted into the
formula introduced above. The mathematical expression then would be (0+8)/2=4.
In order for Thematic Fusion to work
however, the true registral position of a pitch has to be
considered.
In order to achieve this, the pitch that equals zero is the middle C. The Cs
below or above will have 12 either added or subtracted from them. This is best
demonstrated in Figure 1:
As
mentioned above, a melody is not only pitch events but also rhythmic events.
These rhythmic events need to be averaged as well to create a true hybrid of
two themes which incorporates all aspects of both themes. For the purpose of
this paper, I will use the term events
per sections or EPS. The events per sections are the attacks that occur
over a section within a theme. Should the rhythms of our themes be for instance
the ones shown below in Figure 2:
Then,
as we focus on the first beat of theme A we can find four sixteenth notes on
the first beat, while the second theme B has two eighth-notes on the first
beat. We thus have (4EPS+2EPS/2)=3EPS. These 3EPS occurring in a section of
similar length, in our case one beat, will become three eighth note triplets.
Whether this holds true to each individual case however depends on the melodic
contour of both themes, since the procedure works with a combination of both
pitch and rhythm, as the following will explain.
Since
the focus of Thematic Fusion is a
musical one, all dimensions of music and melody have to be considered. While
melodies can obviously be reduced to pitch class numbers and rhythmic attacks
over time, the aesthetic aspects of melodic material can hardly be expressed in
mere numbers. And, as I will write later, I have found that a purely
mathematical approach, an absolute one, if one so will leads to rather abstract
and detached results with both limited musical qualities and a limited range of
usability. Only two of the pieces in Permutatis
use this absolute approach, nos. 1 and 16. I will elaborate on this aspect
later in the section on limitations and problems.
I
found it necessary to create a set of rules as well as liberties, since an
uncompromising strictness leads at times to rather unpleasant and abstract
results. The first rule or guideline I established is one concerning melodic
fluidity: Always decide in favor of the melodic line. This means in practice to
avoid repeated pitches over a short period of time. If the hybrid should for
instance yield two or more repeated pitches in direct succession, then one of
them may be altered by half step towards what the contour of one of the
original themes dictates. Example 3 visualizes this:
Should
repeated pitches however be a characteristic aspect of either theme A or B,
then those should be kept intact. The rule in favor of the melodic line exists
solemnly to avoid melodically static lines. As a consequence of Rule I, Rule II
was established, which allows for chromatic alteration by one step up or down,
as long as the contour of the lines is not significantly distorted. Another
possibility to help melodic fluidity when confronted with repeated pitches is
to tie those pitches and create a longer rhythmic note value.
These
alterations become necessary when working in the diatonic system or even a
pan-diatonic system, since the hybridized theme has to conform to one
distinctive tonality, which can be challenging in the absolute approach. Should
one wish to work in an atonal aesthetic environment then these alterations may
of course be omitted. Permutatis
however, is a deliberately tonal approach to explore the system of Thematic Fusion.
Another
question that appeared was that of how to handle rests. When thus faced with
the situation of silence versus event, the rest and therefore silence is
handled like a melodic event, which it really is. As figure 4 shows, rests are
handled by fully incorporating the rhythmic value of the rest into the hybrid.
Only
considering the EPS of theme 1 and 2 on beat 2 and 3, we obtain 0 events (half
note rest) versus 5 events. Averaging here will not quite lead to obvious
results as the average would be 2.5. At this point the composer must make
melodic considerations. First off, the rest in the hybrid occurs on the same
beat as in theme 1, since this is somewhat characteristic of theme 1 and the
hybrid should contain this. Second, the 2.5 events are rounded into 3 events:
The F on the last sixteenth of beat
2 and the two G eighth notes on beat
three. The F on the second beat of
theme 2 is omitted, since the pitch F
is derived in direct succession. The remaining pitches of theme 2 which are
juxtaposed to the half-note rest in theme 1 are hybridized against the pitch
Eb, the last sounding pitch of theme 1 before the rest. This rest versus event
situation could obviously also be handled in an absolute manner, but as
mentioned before, melodic fluidity should be of the utmost importance, which
calls for musical considerations whenever necessary.
In
order to be hybridized, themes do not have to be in the same key, the same
meter or the same register, it is completely up to the composer to either
appropriate the themes first or to use them in their given state. It should
however be said, that it can contribute to the melodic usability of the hybrid
theme if the two themes are transposed into similar registers. If the two
themes are not in the same time signature then the smallest common denominator
has to be found. If this is not possible either, then one of the themes has to
be transformed into a sequence of events and then either be augmented or
diminished until the two themes are seamless in length: 
That
being said, the composer may also choose to use only a fragment of one of the
themes and hybridize it with the other after manipulating it rhythmically. Thematic Fusion is then used as a purely
artistic melodic device and not to create gradually morphing transitions.
The
effect of gradual transformation of one theme into another is created by
re-hybridizing the hybrid theme with either theme A or B, depending on which of
the themes the transformation should go towards. The resolution, and thus
number of steps that are possible until them A has completely morphed into
theme B depends on registral differences and very much on whether a
chromatic/atonal approach or a diatonic approach is chosen. If the composer
chooses the stay within one tonal system, then the resolution is reduced
considerably as the hybridized material has to fit into the tonal grid that the
tonal environment of choice dictates. In a strictly diatonic system for
example, the resolution would 7 steps as opposed to the 12 steps of a chromatic
system.
Mathematically
a hybrid theme that is closer related to theme A than to theme B would be
achieved by using the following formula: (A + H):2=Ha. The next step closer would be achieved by using the same
formula and replacing H with Ha. As mentioned above, the number of
possible transformation steps depends on the tonal system used as well as the
complexity of the themes. Figure 6 shows the right hand themes A and B of the 8th
piece in Permutatis, the Minimaluett. As one can observe here,
theme A gradually transforms into theme B, the resolution being two steps in
between hybrid an original, which amounts to a total of five steps of
transformation from theme A to B. The number of possible steps of
transformation is limited by the diatonic nature of the piece, which is written
in Eb minor.
As
mentioned earlier, Thematic Fusion
may be combined with existing methods of melodic manipulation such as
retrograde, augmentation and diminution. In Permutatis
I combined Thematic Fusion with
retrograde and different methods of rhythmic manipulation in several pieces. I
found that especially in combination with retrograde quite interesting results
were obtained.
Figure
6:
IV. Problems and Limitations
While
some of the challenges I encountered piece-specifically will be discussed in
the following section, which will provide an in-depth analysis of Permutatis, I would like to point out
some general limitations to the procedure. By design, Thematic Fusion has a limited applicability for inversion. This
results from the simple fact, that the hybrid of a theme and its direct
inversion is a static line. This can easily be expressed like this:
[5+(-5)]/2=0.
This however by no means implies that it is not possible to apply Thematic Fusion to inverted themes. One
will just have to use one of the semi-hybrid stages of transformation, which
will have some melodic qualities, while the absolute hybrid is a static line.
Example 7 shows this problem:
As
mentioned earlier, the raw hybrid can at times be of a melodic quality that can
create some challenges in terms of integrating it into an otherwise tonally
simple constructed piece. The appropriation of harmony in order to
aesthetically pleasingly integrate the hybrid theme into the piece can then
require considerable manipulation of
the
existing harmonies. This can however be avoided by following the guidelines for
melodic fluidity.
Another
aspect that may be of problematic nature is that the gradual morphing takes
place by continued restatement of the theme in one of its hybridized forms
either leading to or from one of the source themes. This repetition, while of
new material, may be unsuitable for some musical styles. The composer is thus
required to keep the piece lively by providing interesting harmonies or rhythms
to bed the hybrid themes on, so that the interest is not diminished by a sensation
of repeated material. Some musical styles have however the repeat of thematic
material as a core aesthetic aspect, for which Thematic Fusion can be a possible stylistic device. Minimalism,
with its restatements of simple, often short thematic material works quite well
in combination with the thematic morphing aspect of Thematic Fusion.
V. Benefits
The
benefits of Thematic Fusion lie
within its function. The technique not only creates morphing transitions, but
furthermore creates new musical themes. The composer is able to generate themes
that are highly related to the source material but offer new melodic traits
which both relate and differ them. In the way that the source material
prescribes the consequent melodic events, Thematic
Fusion is not unlike fugato technique, which restates the source material,
where as Thematic Fusion leads to a
re-statement that is transformed rather than transposed. This also means that a
large segment of thematic material can be generated without the need to compose
new transitional or thematic material. The reason for the composer or arranger
to use thematic fusion thus becomes the wish to add to the integrity of a piece
in terms of its structural logic.
The
flexibility in the way Thematic Fusion can
be used makes it a valuable tool for melodic manipulation. As Permutatis will show, Thematic Fusion was successfully applied
within a great number of textures and styles both poly- and homophonic and all
spectrums of harmonic complexity. Especially in polyphonic settings Thematic Fusion lead to interesting
results. No. 12, the Contrapunctus from
Permutatis, is a fugue constructed
with Thematic Fusion, a piece that
will be discussed in detail in the following
Section.
My technique thus expands the palette of traditional canonic devices such as
augmentation, diminution, retrograde and inversion, as it can either be used on
its own or add a second degree of complexity when combined with the afore
mentioned devices.
The
hybridized section furthermore has an aspect of unpredictability, which can be
refreshing in a number of compositional situations. When used with two themes
of moderate melodic activity, Thematic
Fusion will likely generate a hybrid that will be more intense and active
as it will combine the level of activity of both themes. Thematic Fusion can therefore successfully be used to generate
climactic highpoints of pieces.
VI. Permutatis
As stated earlier, Permutatis was written to thoroughly explore and test the concept
of Thematic Fusion. I chose the piano
setting as this would keep the focus on thoroughly composed transitions and
melodic turns, rather than relying on instrumental coloration and arrangement,
which would be possible in a chamber music setting. Being a pianist myself I furthermore
felt that it would be interesting to implement a pianistically challenging and
interesting aspect into each piece. With each piece being quite different in
meter, rhythm, texture and harmonic spectrum a great variety of pianistic
techniques is demanded, be it top note projection, rhythmic independence or
contrapuntal understanding.
I.
C Major:
Intrada
The
opening piece of Permuatis, the Intrada in C major is monothematic.
Merging the main theme of the piece with its retrograde generates the hybrid themes.
The Intrada begins with an initial
statement of the main theme in a chordal texture. After this initial exposition
the theme is in to top voice in m. 9 through 16, this time placed in a more
dense texture which demands a certain degree of top projection. The them is
still in its original state, it is melodically untouched.
Measures
17 through 24 contain the hybrid. As mentioned above, the 8 bar theme has been
hybridized with its own retrograde and the resulting theme features a high
amount of activity, due to containing more smaller subdivisions than before.
The sixteenth note runs that were to be found in the original theme and its
retrograde are combined in one theme now and we receive a texture that has an abundance
of melodic activity. Measures 25 through 32 contain the retrograde of the
theme, in a homophonic texture, quite similar to the one it was presented in
initially.
A
sensitivity to the dynamic nuances is very important in this piece as the
pianist has to create a clear change of mood with one and the same theme. The
opening
statement
of the theme takes place in a celebratory mezzo forte, while its consequent
restatement has to be presented rather conemplative in character.
II.
C Minor:
Passacaglia
The Passacaglia in general, is by design
based on a repeating ostinato bass. The Permutatis
Passacaglia however has two bass patterns: One which it originates from and
another one it will arrive at eventually. The metamorphosis of one ground bass
into the next yields for interesting harmonic turns and creates an further
sense of urgency and forward momentum, as the steady movement of the bass is
maintained yet harmonically intensified as it progresses.
The
bass theme of the Passacaglia is
seven bars long and is stated seven times until it transforms towards the
destination theme. The seventh statement of the theme is the hybrid of the two,
which occurs in m. 43, the middle section of the piece. The destination theme
in an unaltered form is presented for the first time in m. 64. In order to get
the stepwise transformation each theme was re-hybridized with the hybrid to
create versions that are biased to either theme 1 or 2. The resolution for this
piece is two steps of transformation. While this seems like a rather low number
of steps in transformation, one has to consider that each statement of the
theme provides new harmonic possibilities which in consequence provide a
greater spectrum of harmonic color for the thematic events occurring in the right
hand. We must thus keep in mind that in the Passacaglia, by design, the ground
bass serves as the foundation for the melodic events occurring above it.
We
can find the first hybrid theme, H2A in mm. 29 - 35, and HA in mm. 36 - 42. The
realizations of the ground bass with a bias towards its destination theme begin
in m.50 with H2B followed by HB in m. 57.
Pianistically,
the virtuoso runs, especially in the climactic section from mm. 64 to 70 are
quite challenging, as is the meter of 7/8, which may throw off the pianist who
is less experienced in the performance of compound meters.
III.
Db
Major: Romance
The
Romance in Db uses thematic fusion in combination with diminution. The voice in
the middle register represents an exact diminution of the melodic line found in
the top voice. The process of thematic fusion is here used to one hand, create
a hybrid theme and on the other hand to provide a gradual transition of
original into diminution. The hybrid theme in this case is a combination of the
original theme and its diminution. The Romance can be divided into four
sections: A, B, C, A'. Section A (mm. 1 - 8) represents the original theme and
its diminution simultaneously in the inner voice while section B (mm. 9 - 12)
is the hybrid combined with the original and the diminution placed in the left
and right hand respectively. This section functions as a transition into
section C (13-20) , which is the hybrid. Measures 21 through 28 represent a
recapitulation of section A, whereas the diminution now is in the top voice,
while the original theme is in the inner voice. A registral transformation has
thus taken place. In combining the thematic fusion with the device of
diminution a high degree of structural integrity is created.
The
pianist needs to play the Romance
with a high degree of contrapuntal sensitivity and focus on projection of inner
voices. The original theme and its diminution stand on the same level in the
hierarchy of voices.
IV.
C#
Minor: Capriccio
&nbsnbsp; The
fourth piece in Permutatis, the Capriccio in C# minor, has two themes
which are hybridized with each other. The first theme is presented in mm. 1 - 4,
directly followed by the second theme in mm. 4 - 8. The hybrid of the two
themes is found in mm. 21 - 24.
The
hybridization of the two themes created a rather interesting result, as the
contour of the second theme is clearly to be observed while the rhythmic
activity of the first theme is clearly maintained. The fact that both themes
are dominated by a melodic arch form aides as a somewhat limiting factor,
keeping the registral range of the hybrid theme similar to the two original
themes.
The
Capriccio is an unusual piece in a
number of ways. First off, the texture of the Capriccio is monophonic, the single melodic line dominates here.
This however does not mean that harmony was not taken into the highest account
when composing the piece. Each theme has a clear harmonic determination and the
features of baroque compound melody writing. Another rather unusual feature of
the piece may be its meter of 17/8. This compound meter is broken up into
groups of threes and fours. The grouping of the first theme therefore is:
3+4+3+4+3+4+4+3+3+3. The direct succession of the two groups of four at the tip
of the arch, carries the concept of arch from into the dimension of rhythmic
grouping as the moment at which the melody changes direction occurs
simultaneous to the change of
rhythmic pattern.
The
piece is most likely one of the more difficult pieces in Permutatis, as it is technically quite challenging due to the tempo
marking of allegro and the small rhythmic subdivisions. The piece is however a
great exercise for melodic fluidity and fast arpeggios.
V.
D Major:
Fantasie
The
Fantasie, the fifth piece in Permutatis is monothematic. I combined
Augmentation with Thematic Fusion to
derive the hybrid theme connecting the two statements of the original theme.
The hybrid is placed in mm. 9 through 16.
The
texture of the Fantasie is quite interesting, as it alternates
chordal sections with florid, virtuoso runs to create a quite balanced
atmosphere of activity and calmness.
The pianist will wind him or herself more
challenged by finding a way to interestingly perform the chordal sections
rather than playing the fast runs which are mostly diatonic in nature.
As
a consequence of combining the theme with its augmentation, the melodic range
of the theme was somewhat compressed, which is a logical result as the rate of
attacks is halved and fast melodic motion is then juxtaposed with a slower one.
Another feature that the hybridization provides to the generated theme is that
the new theme contains both the fast 16th note subdivision and the
calmer 8th note subdivision that its augmentation creates. These
work quite well together and add a sense of balance to the middle section.
The
hybrid in this case does not necessarily works as a transitional section: The
target theme, the augmentation of the original, is not presented in the piece,
therefore Thematic Fusion is utilized
to create new thematic material which is used as thematic event
in its own right. The Fantasie, as a
consequence has a ternary form, made up of original theme, hybrid and
restatement of the original theme.
VI.
D Minor:
Barcarole
The
Barcarole in D minor, the sixth piece
in Permutatis combines two themes of
different meters. The first theme is presented in mm. 3 - 15 and written in
9/8, while the second theme is in the meter of 4/4, spanning measures 16 - 24.
Motivically, the two themes are quite different as well, with the first themes
angular leaps and eight note subdivisions and the second themes folkloresque
dotted rhythms.
The
first theme furthermore has some structural intricacies at the micro level: The
interval at which it leaps upwards is expanding symmetrically from a fifth to a
ninth. This gives the melody a very angular mannerism, and combination with the
irregularly placed accents and its motion across the bar-lines, the first theme
is very sparkling as it reaches high into to the top registers of the piano. In
contrast to the complex first theme, the second theme is very simple and
chordal in nature, creating a well-balanced counterpart.
The
quadruplets in the left hand help to give the first theme a very light feel,
since it never quite interlocks with its accompaniment. It seems to float
effortlessly over the homophonic structure.
The
Hybrid theme is presented in m. 27. Upon closer examination is can be observed,
that the angular contour of the first theme is intact, while in a somewhat
tamer form, spanning nowhere near the range of its original form. The three
against four effect that we can find in the initial exposition of the theme is
here transformed into a two against three effect: The even and odd rhythmic
groups have traded places. This helps to maintain an interesting structure in
the piece while creating forward momentum.
The
two themes were rhythmically appropriated by turning both into sequences of
events and augmenting the second theme until its length was congruent with the
first theme. The choice of 3/4 as the meter was made simply based upon the
rhythmic feel that the hybrid created.
The
challenge to the pianist here is not a technical one, while the three against
four and two against three may be difficult for some to execute. The difficulty
in the Barcarole comes from the
complexity of the first theme and its phrases across the bar-line. The angular
melody wants rise and fall with great leaps unhindered, while the slurs stress
the need to keep it connected and smooth.
VII.
Eb
Major: Bagatelle
The
Bagatelle, utilizes Thematic Fusion to create a smooth
transition between the first and second theme. The first spans mm. 1 - 13,
while the second theme comes in at m. 15, lasting until m. 27. The hybrid is
placed in mm. 27 through 38 to help retransition back into the first theme.
The
first theme undergoes an interesting registral change as it wanders from the
left hand in to the right hand and back, while the triplet accompanying figure
is maintained throughout those registral changes. This provides the pianist
with the task to make the transition of the theme inaudible, while maintaining
its projection. This goes true especially in m.8 into m.9 where the first
registral change occurs.
The
second theme abandons the triplet feel and has a rhythmically more active
texture with sixteenth and eight notes.
The
Hybrid maintains an accompaniment similar to the second theme with an active
inner texture, while ending with a chordal accompaniment. After the hybrid
theme is stated the first theme is recapitulated. The form that is thus
generated is AABHA.
The
Bagatelle lives of the strong
contrast between the themes and this has to be expressed accordingly by the
pianist. Another aspect of performance is the drastic change in dynamics which
is encountered in the second theme, mm. 18 and 22, where a subito pp is demanded.
VIII.
Eb
Minor: Minimaluett
The
Minimaluett, introduces a stylistic
change in Permutatis, as the piece is
clearly from the genre of minimalism. The aspect of restatement in order to
created gradually morphing transitions lends itself to the genre of minimalism,
in which repetition is a vital stylistic device.
The
Minimaluett, has two themes, both
equal in length and meter. The first theme is found in mm. 1-2, the second
theme in m.13. Since the right and left hand in this piece were both
incorporated into the process of Thematic
Fusion, one could speak of four themes; however, for simplicity I want to
regard the right and left hand of each theme as a thematic unit. The resolution
of transformation is two steps, which makes the changes gradual enough to keep the
minimalist aspect. [CT1]
The
hybrid theme is placed in the exact middle of the piece in mm. 7-9. Of the
pieces in Permutatis, the Minimaluett might show the gradual
morphing the most practically and visually discernable since the themes are so
short.
The
meter of 13/8 was chosen to create unsymmetrical rhythmic groupings to aid to
keeping interest throughout the repetitions. Each thematic phase is to be
repeated at will until the first theme has morphed completely into the second
theme. The player can then choose to end the piece or reverse the form and
morph back to the first piece.
The
pianistic challenge is here no doubt the balance and precision between hands,
while realizing calmly the rising and falling dynamics of each phrase.
IX.
E Major:
Interludium
The
Interludium in E Major has two themes
that are merged to create third theme, which is used in a number of ways. The
first theme, which can be found in mm. 1 - 8 is blended with the second theme,
spanning m. 9 through 16. The hybrid in its basic form is place in mm. 17 - 24.
In m. 25 a second hybridized theme, with a bias towards the first theme
experiences some rhythmic manipulation: Measures 25 -26 contain said hybrid
(HA), rhythmically diminished by half, while m. 27 contains the rest of the
theme diminished by third. This section which now functions as a transition
leads us to a final statement of the original first theme, diminished to a
fourth of its original length.
These
rhythmic manipulations create a climactic section with fast runs, and
especially the diminution of the first theme, which creates a thirty second
note run helps to give the closing cadence in m.30 a greater sense of finality.
At
the micro level, the themes are rather simple in their design. It could be
pointed out, that the first theme harmonically moves in thirds, which helps to
create some interesting harmonies.
The
Interludium certainly provides some
technical hurdles. First off, the ostinato, which has to be maintained through
much of the piece and over which the thematic material has to be projected.
Second the runs in the end of the piece which call for a careful study of
fingerings and imagination in the area of tempo changes.
X.
E Minor:
Nocturne
The
tenth piece in Permutatis, the Nocturne, can be divided into four
sections. The first section, spanning mm. 1 - 14 is the initial statement of
the first theme. One should note the nesting chords, over which the top note
has to be projected, while letting each chord smoothly blend into the next, by
means of smart pedal work and strong legato playing with the hands.
The
second section is a re-statement of the first theme, this time in a homophonic
texture, with a small sixteenth note gimmick placed in the left hand
accompaniment figure. The sixteenth note figure is to be executed precise but
subdued, which demands considerable control, while focusing on phrasing the
melody of the first theme.
The
third section of the Nocturne
features a hybrid theme that is created out of merging the first theme with the
retrograde of the left hand. This section spans mm. 29 - 42. The resulting
hybrid has the melodic contour of the first theme, while featuring the smaller
subdivision of the left hand accompaniment figure from the second section. This
newly generated theme is placed over an accompaniment of broken chords.
The
final section of the Nocturne is a
transposed recapitulation of the first theme, this time transposed to the
subdominant. The short cadence from 57 to 61 is not part of the theme but is
rather in place to smoothly end the piece in the original key.
The
Nocturne needs to be performed in a
highly controlled manner, which allows for a clear separation of accompaniment
and thematic material, as well as a rhythmic confidence that is not thrown off
by the compound meter of 5/8.
XI.
F Major:
Valse
The Valse
in F major uses the classical Walzer
as a model, with some added intricacies that are intended to increase the
interest and of the piece as well as transporting the concept to the 21st
century.
First,
the Valse is not written in 3/4 time,
but in 9/8 which was not a deliberate attempt to break with convention, but to
ease the notation and aid to visual logic. The left hands accompaniment however
has some rhythmic anticipation, which offsets the traditional Walzer feel and gives it a more modern
touch. The extended tertian harmonies throughout the piece add to this
modernization.
Thematic Fusion is used in this piece to
create a re-transition from the second theme, starting in m. 23 back to the
first themes recapitulation in m. 45. The hybrid is therefore installed in mm.
35 - 44. The hybrid theme interestingly features the duplet motives of the
second theme along with the runs and 16th note triplet
embellishments of the first theme. These features work together quite nicely
and help to keep the hybrid section quite interesting.
The
Valse needs to be performed with a
light, effortless touch. The embellishments must be taken with a technical
proficiency that will not draw the listeners ear to the embellishments, but
keeps the focus on the line. As the hands change, this light touch must be
maintained. The player furthermore needs to bring out the contrast of duplet
versus triplet feel between the two themes. Since the Valse traditionally is a dance, keeping the main pulse and momentum
of the piece alive is paramount.
XII.
F Minor:
Contrapunctus
The
Contrapunctus in F minor, the
eleventh piece of Permutatis explores
the concept of Thematic Fusion in a
fugal setting. The Contrapunctus is
essentially a crab canon, utilizing Thematic
Fusion to create the link between the first half of the piece and the
retrograde thereof. The hybrid thus acts as transitional element, smoothly
transitioning into the retrograde.
The
piece was purposely titled Contrapunctus and
not Fuga, since the focus was to present the concept of Thematic Fusion in a contrapuntal setting. While the piece features
some of the contrapuntal intricacies of fugal writing, the focus here should be
the hybridization. In order to create the hybrid, which is found in mm. 26 -
35, every single line of the three part fugue was hybridized with its
respective retrograde. The section that was chosen for hybridization was the
episode and exposition from mm. 16 to 25. The structural aspects of the fugue
are best observed in figure 8, a formal outline of the fugue:
Figure
9:
|
M |
1 |
6 |
11 |
16 |
20 |
26 |
36 |
46 |
51 |
56 |
|
S |
|
A |
CS |
E |
S |
H |
SR |
CSR |
SR |
|
|
T |
S |
CS |
|
E |
S-Str. |
H |
SR |
|
CSR |
SR |
|
B |
|
|
S |
E |
S-Aug+Inv. |
H |
SR-Aug+Inv |
SR |
|
|
©2011 Christoph Nils Thompson
The
challenge in writing the Contrapunctus
was to feature interesting contrapuntal writing on the one hand, but to keep
the piece brief enough so that the hybridized section would not seem
proportionally insignificant to the rest of the piece. It was furthermore
important to create a working palindrome in order to be able to maintain
structural and thematic logic until the very end of the piece.
The
performance aspects of the Contrapunctus
are those of any other fugue. Contrapuntal understanding has to be exercised
and all voices must project with with same intensity, as every polyphonic
setting would prescribe.
XIII.
F#
Major: Compitum
The
piece Compitum is the only piece in Permutatis to not actively feature Thematic Fusion within the piece. This
is because the theme of the Compitum
is a hybrid itself. The theme of this piece was created by hybridizing the two
pieces in C major: The first piece Intrada
and the last piece, the Conclusio.
Since F# is the exact middle of the keys and thus the exact middle of Permutatis, I found it suitable to let
the middle piece be a hybrid of the two outermost pieces.
The
first theme of the Compitum, ranging
from mm. 3 - 10 is very simple and
diatonic, while featuring characteristics of both themes that were hybridized.
We can find the steady eight-note lines of the Conclusio in combination with the sixteenth note runs of the Intrada.
The
piece is very carefree in character and should be performed in such a fashion.
It is certainly one of the technically least challenging pieces in Permutatis, but the pianist may choose
to focus on expressing the variety of rhythms in the main melody smoothly and
with good legato technique.
XIV.
F#
Minor: Reflektion
The
expressive, elegie-esque Reflektion
features Thematic Fusion as a
thematic generator. A hybrid of the main theme, mm. 1-9 and the subordinate
theme, mm. 12 - 21 was used to create a variety of the first theme. The hybrid
can be found in the soprano line in mm. 22 through 28, followed by a
recapitulation of the original theme.
The
main contrast between the two themes lays within the textural differences: The
first themes accompaniment is dense and characterized by the pulsing sixteenth
note triplets in the inner voice, while the second theme has an airy, spacious
quality to it, seeming less urgent than the former. The hybrid was simply
implanted to the unaltered accompaniment for the first theme, to great effect.
I received a slightly altered version of the first theme, similar enough to
work within the same accompaniment, different enough to not sound repetitive.
Clearly,
the difficulty for the performer will lie within keeping the inner voice
ostinato steady while projecting the melodic notes in the top line. The Reflektion calls for neat pedaling and
good phrasing. The dramatic, ominous atmosphere of the piece will be best
expressed by not letting any temptation for virtuoso performance take over and
maintaining a conservative tempo.
XV.
G Major:
Intermezzo
The
Intermezzo in G major is monothematic
and combines Thematic Fusion with
traditional thematic devices. I used retrograde, rhythmic augmentation and
inversion. as mentioned earlier, in the section of problems in combination with
using inversion, I sidestepped those by adding retrograde to the inversion.
The
main theme of the piece is exposed in mm.
1 - 10 and repeats once. This is followed by a first hybrid theme, which
combines the first theme with its retrograde-inversion from mm. 11 - 16. These
five bars serve as a transition into the next section, spanning mm. 16 - 20,
which feature the main theme in retrograde inversion.
The
next section from mm. 21 - 35 is thematically quite dense. In the right hand in
mm. 21 - 25 we can find another hybrid version of our theme, this time blended
with its augmentation. This is followed by the theme rhythmically augmented in
mm. 36 - 35 in the right hand and the retrograde inversion of the theme in the
left hand as accompanying figure. To
close the piece the original theme finds its recapitulation in m 36.
The
Intermezzo is joyful and up-tempo.
The piece features asymmetrical rhythmic groupings of 3 and 2, which are more
or less defining the character of the piece. Throughout these rhythmic
intricacies however, the piece should be performed in a joyful, folk-dance like
fashion.
XVI.
G Minor:
Pastorale
The
Pastorale is a piece that features Thematic Fusion, not only as a
transitional device, but furthermore without the melodic considerations
discussed in the former chapters. The exact melodic material that was generated
by merging the themes was used in its unaltered state with all chromatic
features intact.
The
piece basically is based on two main themes, the first one ranging from mm. 1 -
9 and the second one, spanning mm. 27 through 34. The first hybrid theme is
found in mm. 11 - 18 and is the exact hybrid of theme 1 and two. The second
hybrid theme, placed in mm. 19 - 26 is a product of re-merging the
aforementioned hybrid with the second theme two create a bias towards the
latter.
The
results of the hybridization were highly chromatic and it took considerable
harmonic manipulation and flexibility to create an aesthetically unified piece.
Especially the chromatic runs that were created, which were at some points
almost completely made up of half-steps yielded some rather exotic sounding
potential. The chromaticism thus sped up the harmonic rhythm considerably. The
hybrid themes in the piece are nonetheless material that is, and sounds related
to both themes, and would not have been conceived without using thematic
fusion. Since the chromatic sections featuring the hybrid are rather defining
for the piece thematic fusion does play a considerable role in the structure of
the Pastorale.
The
Pastorale's main challenge to the
player may be to properly execute the thick chords of the left hand and thus
finding the right level of balance between harmony and melodic playing.
XVII.
Ab
Major: Scherzo
The
Scherzo employs Thematic Fusion as a transitional device, to create a link between
the two themes. The first and second themes are located in mm. 1 - 11, and 16 -
23. The hybrid is placed in bars 12 - 15. In this piece, I chose to use only
half of the hybrid as it seemed to support the formal structure of the piece
better. The hybrid thus functions as a four bar transition between named
themes.
The
contrast between the themes is definitely a textural one. The first theme is
quite chordal in nature while the second theme is a rapid, virtuoso succession
of attacks with both hands, creating a very active texture.
The
main challenge to the player in the Scherzo,
might be a rhythmical one, as the meter of 11/8 could require some acquainting.
The introductory phrases in the beginning call for well balanced chords and a
steady cresecendo, which climaxes into fast sixteenth note runs in the left
hand. These runs must be executed in strict time so that no momentum is lost and
the pulse of the 11/8 time signature is maintained.
XVIII.
G#
Minor: Contemplation
The
Contemplation in G# minor has three
themes and thus two different hybrid themes functioning as transitions within
the piece. The first theme ranges from mm. 3 - 11 and is followed by a hybrid
of itself and the second theme. The hybrid section ranges from mm. 12 - 20, and
as transitions into the next theme, found in mm. 21 - 33. The third theme is
placed in mm. 42 - 47. Since this theme, which was hybridized with the first theme
is shorter than the latter, only the first six bars of theme 1 were actually
used. This is followed by final statement of the first theme in m. 54.
One
of the aspects that makes this piece rather interesting is it is the only piece
in Permutatis with more than two
themes. But looking at the first theme one can find some interesting features
on the micro level. The rhythmic grouping of our 7/8 compound meter is 3+2+2,
or 3+4. Theme one begins with an 8th note triplet motive which is
tied to a long note. Comparing the accompanying unit and the thematic unit we
can therefore see that they both connect
a shorter event with a longer one: 3+4 and Triplet set followed by longer note.
The theme therefore mimics the feel of the rhythm, but does so in a compressed
form. We then receive two units in a
slightly out of focus reflection of each other.
What
will make the Contemplation
challenging for the pianist is no doubt the rhythm and the demands for rhythmic
independence that the piece makes. Another aspect that might create some
technical issues is the virtuoso run in the climactic section from mm. 45 - 47.
Despite all the possible display of technique that this piece provides, the
player must not be tempted to raise the tempo of the piece, as it has to be
performed in a contemplative, calm manner.
XIX.
A Major:
Consolation
The
Consolation in A Major is written in
the rather exotic meter of 15/16. The meter was mainly chosen because of
notational aspects since I wanted to avoid having to constantly give the reader
changing tuplets. The meter, for the first theme is divided in to five sets of
three 16th notes. The theme ranges from mm. 1 - 12. For the second
theme, which can be found in mm. 13 - 17, the 15/16 were divided into three
sets of five. This contrast keeps the piece fresh and interesting.
The
hybrid was implemented into the piece in a rather interesting way. In the Consolation, the hybrid is neither
transition nor stand-alone thematic event. I decided to weave the hybrid into
the restatement of the first theme as a contrapuntal line, meandering between
top and inner voice. Measures 18 through 29 feature the hybrid, starting on the
4th sixteenth not of measure 18. The hybrid which in contour bears
some resemblance to the first theme, therefore serves as some modified
imitation, echoing the original theme in close succession, just a dotted 8th
note away.
The
hybrid itself was generated by merging the first theme with the second theme.
Since the second theme is only half as long as the first theme, two sets of the
second theme were connected and then hybridized with the first theme.
As
with all romantic pieces, that feature a considerable amount of inner voice
movement, the pianist has to be very sensitive to which voice he chooses to
project.
XX.
A Minor:
Percusetta
The
Percusetta, the twentieth piece in Permutatis, takes a percussive approach
to the piano. The texture of the piece is dominated by a number of ostinati
which are juxtaposed and need to be performed simultaneously. The melodic part
of the piece takes place completely in the inner voices.
The
Percusetta is monothematic and uses
retrograde to create the second theme to generate the hybrid, which can be
found in mm. 17 - 24, and again in mm. 25 - 32, then with a bias toward the
retrograde, which follows in m. 33.
Since
the melodic range is very limited due to technical restrictions the main
challenge was to find a somewhat interesting theme. Creative harmonic thinking
was as important as resourcefulness in terms of ostinato placement.
The
challenge of the piece clearly lies within independence of the hands and even
fingers. This independence surpasses the rhythmic dimension, since the pianist
must further balance the volume of each of the ostinati, to not cover up the
theme taking place in the inner voice. The rhythms were somewhat inspired by
what popular music features and the patterns are organized not unlike what a
drum-set player would do.
To
clarify the analogy, one should consider the left hand bass-drum and snare,
while the higher cluster ostinato in the right hand should assume the role of
the hi-hat or ride-cymbal.
XXI.
Bb
Major: Lullaby
The
Lullaby is one of the pieces in Permutatis that uses Thematic Fusion to blend two themes that
are written in different meters. The first theme is in 4/4, spanning mm. 3 - 10,
while the secondary theme, mm. 11 - 18 is written in 3/4 time. The second theme
therefore experienced some rhythmic permutation, as can be observed in its
re-statement in m. 27, where it has been transformed to fit the meter of 4/4.
The
hybrid can be found in mm. 19 - 26 and is an interesting blend of the fast runs
of the second theme with the smooth contour of the first. The hybridized theme
works well over the accompaniment scheme used for the first theme with the
sixteenth notes in the inner voices outlining the harmony.
The
pianistic challenge of the Lullaby is
certainly the sixteenth note accompaniment in the inner voice versus the long
lines of the first theme. The pianist needs to play the inner voice with the
utmost care to maintain clarity and stabilize the harmonic structure of the
piece. The fast runs over the Lydian harmonies of the second theme will
furthermore be an opportunity to hone arpeggiation skills.
XXII.
Bb
Minor: Tango
The
Tango, along with the earlier Valse is one of the dances in Permutatis. It draws its momentum and
energy from its rhythms. The first theme (mm. 5 - 12) is placed over a left
hand ostinato pattern and is then
repeated with embellishments. The second theme which is entirely chordal
follows in mm. 21 - 31. The hybrid can be found in mm. 48 - 55; preceded another
hybrid with a bias towards the first theme in mm. 40 - 47.
Thematic Fusion was used in the Tango to create two stand alone thematic
versions. The hybridized section, mm. 40 - 55 features characteristics of both
themes and is accompanied in a suiting manner. We can find the chordal accents of
the second theme as well as a steady left hand ostinato as used to accompany
the first theme.
The
Tango demands strong rhythmic
playing, with a confident left hand that will create the necessary momentum and
steady pulse for a dance movement.
The
conflict within the piece obviously lies within the texture of the two themes,
with the second theme further more being harmonized entirely in parallel
harmony, versus the florid lines of the first theme. Interestingly, this
conflict finds resolution in the hybrid themes, where the two themes, which
really are at the opposite sides of the spectrum, form a lively symbiosis.
XXIII.
B Major:
Choral
The
Choral in B Major was modeled after
the German Hymns of the late renaissance and early baroque. It features a
simple four-part texture. The harmonic spectrum obviously has been greatly
expanded compared to the Lutheran hymns and it contains several voice leading
liberties and melodic turns which place it firmly in the twenty-first century.
The
Choral is monothematic and the hybrid
was generated by blending the each line with its direct retrograde. The hybrid
is to be found in mm. 8 - 13, followed by the retrograde of the first seven
bars in mm. 14 - 20.
In
this piece Thematic Fusion obviously
serves to link two retrograded themes together. This has been demonstrated
earlier in the Contrapunctus. In the Choral, however we do not have a
contrapuntal but purely chordal texture and we essentially created four hybrid
melodies by merging each individual voice.
The
Choral as a genre certainly offered
itself as a simple from to demonstrate Thematic
Fusion in a purely chordal setting, without dealing with expressing the
conflict of contrasting textures as in the preceding pieces.
The
piece certainly does not offer any technical challenges to the experiences
player, but does demand focus on phrasing and an even projection of all four
parts, without overpowering the top voice.
Against
my expectations, the hybrid needed very little editing and I was pleasantly
surprised by the fact that I did not end up with a sequence of harsh clusters
and dis-chords. While the hybrid does not feature the most beautiful harmonies
in the piece, it was without doubt a very interesting and unexpected outcome of
the procedure.
XXIV. Canon
The
penultimate piece of Permutatis, the Canon features my system once again in
an imitative setting. For the Canon,
I took the German term Permutationsfuge
quite literal and decided to write a canon that would morph into a different
canon with different meter and themes. The result is what one might call a
morphing double-fugue.
The
greatest challenge was to use Thematic
Fusion in the piece to an extend where it did not seem trivial, but to not
write a fugue of so generous proportions that it would seem out of place within
the collection.
First
off, the primary theme or subject of our fugue, which can be observed in mm 1 -
3 is in a different meter than the secondary theme (mm. 16 - 19, bass). The
secondary theme, when turned into a sequence of events, free of metric
categorization, is congruent in length with the first theme. The hybrid
sections, which can be found in mm. 10 - 12 and 13 - 15 were thus generated by
merging soprano, alto and bass with their respective counterpart in the other
theme. The hybrid section from 13 -15 has a bias toward the second theme, to
make the transition yet smoother.
The
effect of the transition is quite interesting and it takes place so subdued,
that the listener might be well within the second fugue before he or she grasps
the aural difference. Thematic Fusion,
is here a very successful transitional device.
What
goes for the earlier fugue, the Contrapunctus,
goes just as much for this one:
contrapuntal understanding and a patient approach to working out the different
voices is paramount.
XXV.
C Major:
Conclusio
The
final piece in Permutatis, the Conclusio closes the cycle, by returning
to the key of C major. In the Conclusio, the
hybrid theme (mm. 25 - 32,) serves as a variation on the first theme and
transition into the restatement of the second theme.
The
form of the piece is rather simple: AABAHB. The contrast between the two themes
is a textural one: The first theme (mm. 1 - 8) finds itself in a dense web of
continuously flowing 8th notes, while the second theme has a solemn
quality to it. The major/minor contrast of the two themes further intensifies
this effect.
The
first theme of the Conclusio, not
only serves as part of the hybrid but also as part of the 13th piece
in Permutatis. Together with the Intrada, the Conclusio generates the theme for the Compitum in F# major.
As
mentioned before, I found it necessary to have a 25th piece that
closes the cycle of works. While the Conclusio
does not feature Thematic Fusion in a
new way, it represents the final closure as it ends the piece on the major
chord it began on. I furthermore felt it was favorable to end the collection in
a major key and with a piece less complex than the permutation-fugue that
preceded it.
For
the experienced performer, the Conclusio
will not represent a technically difficult piece, but it might be its very
simplicity that might create difficulty as most often the simplest pieces
demand a high degree of musicality to create an interesting performance.
VII.
Conclusion
Permutatis was a valuable experience and
necessary practical test for my device. I felt it was vital to thoroughly
examine Thematic Fusion in all
possible scenarios and only the practical application of this theoretical
concept in a number of compositional approaches would provide reliable results.
Permutatis by all means is an artistic
application of Thematic Fusion, not a
theoretical display of a mathematics applied to music. The final point of
judgment in the creation of the pieces was my personal aesthetic preference as
well as a determination to strive for a high degree of compositional integrity.
This
project however demonstrates Thematic
Fusion as a valid addition to the existing methods of melodic manipulation
- be it in combination with the traditional ones or by itself. The hybrid
themes, which in some of the pieces created the most vital section, and ended
up defining the piece stylistically and aesthetically in a positive way, would
not have been written had I not applied the concept of Thematic Fusion.