Harmonic and sequential organisation in Carl Nielsen Opus 40

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Harmonic and sequential
organisation

in Carl Nielsen Opus 40



1
)Tonality



I sat home one night a
n
d got hold of
a Theme

that begins in b
-
minor and ends in g
-
minor”.
1


With this understatement of the tonal progressions in the Theme of his Opus 40, Theme and Variations
for Piano,
Nielsen points right at a central issue of this Theme. It

s tonal ambiguity.

This could lead to considerations of whether its tonal environment

is best described through the much
used
concept

of directed tonality
, which Robert
Simpson
uses,
or whether one such try
Bailey and Krebs’

concept of paired tonality.

To my mind the variation form

of Opus 40
, which includes 15
variations, that is 15
new b
eginnings in b
-
minor
each
being directed to g
-
minor, on
ce again

lead
ing
to a new variation starting
in b
-
minor excludes
the concept of
“directed tonality” as a meaningful description.


And for paired tonality it seems that “a pair” isn’t enough. The two main tonal environments are b
-
minor and f
-
minor, wit
hin which we encounter short di
gre
s
sions to a
-
minor and g
-
minor.

We might want to speak of multilayered tonality. But I think that t
he density of this theme reaches
beyond the question of different tonalities.

I would like to present this theme as a tightly structured entity, as an example of Nielsen’s wish to “get
away from the keys”, without ending in unstructured chaos.


[
1a)
And Fo
rm
]

F
ormally
the Theme is
cast in a

bar
-
form,
with two sto
l
len and and an abgesang, which

in itself make
s

up a barform
.

[
1b)
And Motifs
]

The melodic line is constructed of
only three
motivic cells
: A rising fourth, a repeated stepwise ascend
of two notes a
nd a stepwise descend of three notes.

These elements are then c
ombined in different ways
.



However, t
he most striking feature of the Theme is
to

my view

the way it seems to make harmonic
sense within a strictly tonal


understood as
a
function harmony
bas
ed


universe
,

in spite of the
only
16
bar’s
huge tonal digressions.


2
)

Harmony

The
harmonic
analysis reflects both
the

German/Scandinavian


that is Riemaniann
-

terminology
we
use in Denmark
and the Roman Numerals
used in more or less the rest of the
world
.


While Roman Numerals designate the root of the chord, assuming that certain scaledegrees have
certain harmonic functions our Riemannean notation focus only on function, assuming that
a
function,
mainly the function of the subdominant, can be expre
ssed by
chords of
different scale degrees.


If we turn to
first part of
Opus 40
we’ll find that

t
he harmony of the
first

stolle
simply establishes the
tonality
, with a turn to its

relative

major
.

In the second stolle we move gradually away from the dominat
ion
of b
-
minor

towards


in the first place


A
-
minor.




1


Jeg sad en Aften hjemme og fik fat I et Tema, der

begynder I h
-
moll og ender I g
-
moll,..”
From an interview prior
to a concert.
Reproduced in John Fellow ed.
Carl Nielsen til sin Samtid, 220.

What we
might
want to
notice is the way Nielsen
effectuates this modulation

through consistent use of
subdominants
.

First
we see
a Neapolitan in b
-
minor continue as subdominant in G
-
major. Secondly
we
find
the

incomplete

S

o
f G
-
major
, which continues as a subdominant of the subdominant major.


This chord is
, by the way,
one of the instances of where the function of the chord overrules the actual
root of the chord. Although this chord
is
in fact
a supert
onic in first inversion

w
e interpret
it

as a
subdominant

on fourth scale degree

with an added sixth
which is
missing

the fifth.

And therefore it is seen
as “incomplete”.


[
2a)Harmony
]

As the Neapolitan might have moved back to b
-
minor through a 6
-
4
-
cadence, this incomplete S might
have moved to a 6
-
4
-
cadence in the key of G
-
major. Neither of this happens. Nielsen moves towards A
-
minor.

All these modulations are done through reinterpreta
tions of pure triads functioning as subdominants.
All are

introduc
ed
as altered subdominants
continu
ing

as un
-
altered subdominants, now pointing tonally in
new directions.


The A
-
minor reached in bar
7
is
in relation to the key of b
-
minor a
“double

subdomi
nant”, the
subdominants subdominant, SS
-
minor. Indicated by the circle,
o
, as a prefix to SS:
o
SS.

This chord is now
temporarily t
oni
c
i
zed by the
subdominant

major
,

here

acting as an applied dominant

in relation to SS
.


But when this
A
-
minor
-
chord
continue
s to F
-
minor normal tonal relations seems to break down.

We might interprete the transition as
an example of
Bifocal

voice leading
: The upper voice
repeats the
rising fourth that opened the two stollen, only now leading to F,

while the lower voice is movi
ng
back from
1. to 7.
in A
-
minor.

A more
harmonic oriented explanation cou
ld point at the relative weakness of the A
-
minor dyad,
claiming that the underlying E is still a functioning part of the harmony,

so that what we hear
in the end of
bar 8
is
not a
n

Am dyad, but
a 6
-
4
-
dominant
moving

to

a deceptive cadence
.



But both of these explanations overlook the fact, that we are dealing with third relations, and the
question of how to interpret third relations within a tonal framework.


3
)

Maegaard
’s

Terminology.

To
do so

we
will make

a short

digression into a variant of the Riemannian notation system that we use
in Denmark. It’s a variant proposed by Jan Maegaard,
the Danish Scholar/Composer who

is, I believe,
normally best known for his work on Arno
ld Schoenberg. He has however dealt intensively with Romantic
Harmony

having published a thorough book and a highly interesting article on the subject.
2

What we here will look at is the way he notates third relations. It is actually very simple. If
, for
e
xample,
the S of B
-
minor moves up a minor third it will reach its relative major indicated by the sign for
relative, which
would be

p for “parallel”. Should this chord change gender we add a “v” to indicate the
“variant
-
toneart” of Sp. That is: Spv. Follow
ing this logic another
leap a
minor third upwards will lead us to
the relative


“parallel”


of Spv, w
h
ich of course is termed Spvp. This Spvp might have had another name,
if it was reached in a different fashion. Reached
in
this way, the logic of the sys
tem dictates
the name
Spvp,
a chord which is thus heard as belonging to the subdominant area
, because of the initial “S”
.
In fact
this
naming shows the entire bar to be nothing but a prolongation of S.

The move a major third has always been a bit more comp
licated to account for within a tonal system.
However the descending major third has throughout the Romantic period been used so extensively as to



2

Maegaard , Jan og Teresa Waskowska Larsen,
Indføring i Romantisk Harmonik
,

Engstrøm og Sødring 1981, Kbh.

Maegaard, Jan, ”Harmonisk analyse af det 19 årh.s musik”. In
Musik og forskning 15, 1989
-
90
, p. 84
-
110.


have established itself as
what Leonard Meyer would call a “sound term
”, a recognizable meaningful
gesture.

W
e all know
the sound of a stable tonic that all of a sudden transforms itself into a mystic strange
place, a fairytale world, where you might dwell for a while until a mysterious but yet simple transformation


the adding of a seventh


shows that we are a
ctually very close to the dominant, and by implication:
Home.

As this move is analogue to the move from a subdominant to the Neapolitan,

f
r
om IV to
b
II,

Maegaard
calls this move for “neapolitanisation”. In romantic music we encounter neapolitanisations of
both the
tonic and the dominant and

even
o
ther regions might be neapolitanized. If we combine bar two and three
we will find a leap a major third down from Spvp, which in this terminology
then
would be
named:

Spvpn.
As this chord is identical to D, we
im
m
ed
ia
t
e
ly
reinterpret it before it moves on

-

into another
neapolitanisation.

A neapolitanized chord can namely be neapolitanized once more, resulting in the suffix “nn”, and
leading to the upper major third of the first chord neapolitanized.

For a move a
major third upwards Maegaard
in his article from 1989
suggests
the double
neapolitanisation as a describtion, while he in his 1981 book suggests
Riemanns concept of
Leittonewechselklang.
Even though Maegaard was never to fond of this concept, other scholar
s developing
new ideas around the same time as Maegaard certainly was.


4)

Neo
-
Riemannean Transformation Theory

This takes us to The Neo
-
Riemannians and their development of a transformation Theory.

The transformation theory developed by
primarily
Lewin
,

H
yers

and Cohn
3

wants to focus on the
relation between chords instead of just naming chords in relation to a tonic.
And they want to be able to do
so
even
in musical situations, where the operating tonic may be distant or temporarily shut down.

In this the
ory the c
entral
issue

is the transformation
-
operation

that makes one chord become another

chord.

In Lewin’s
1982
presentations of what he call
s

“generalized tonal functions” , he uses Riemann’s
concepts to describe the transformations possible.

A “Dominant
” operation takes you a fifth higher,
a
“Subdominant”
-
operation takes you a fifth lower
.

In 1995 Hyer suggests the use Riemann concepts for third
moves as well. A
“Relative”
-
operation
then
takes you a minor third down from a major chord and a minor
third u
p from a minor chord.
T
he leap
of
a major third
is
called
a

Leittone
-
exchange

. From a major chord
a “L”
-
operation takes you a major third up, from a minor chord the “L”
-
operation takes you a major third
down. Change of gender is called “parallel”
.


Hyer
even

argued, that all you actually need is third and parallel operations as fifth
-
operations might as
well be described as a compound operation of two thirds.
This leaves us with

three operations to describe
any possible transformation: “R”, “L”, and “P”. Applied to the example used to demonstrate

Maegaard
’s
theories
it gives this result. Note that it is progressions, not chords that are named. Therefore the signs are
written bet
ween the notes.

We see now that we have a “R”
-
operation followed by a “P”, and then again a “R”. To describe the next
move we need to perform a double operation. We first perform a “P” operation on B
-
flat major, turning it
into minor, from where we can per
form a “L” operation, that brings us down to F sharp.






3

See
David Lewin, “A Formal Theory of Generalized Tonal Functions”,
Journal of Music Theory
, Vol. 26, No.1
(1982), pp.23
-
60 and
Brian Hyer

“Reimag(in
)ing Riemann”
,
Journal of Music Theory
, Vol. 39, No.1 (1995), pp.101
-
138

and Richard Cohn, “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late
-
Romantic Triadic
Progressions”,
Music Analysis,
Vol. 15, No. 1 (1996) pp.9
-
40.


5) Neo
-
Riemann and Maegaard

If we compare the notation systems of Maegaard and Hyer we see that they are pretty much alike. The
transformation operation “R” of Hyer is in Maegaard

reflected as the adding of a “p” behind the S
indication. The “P” operation adds a “v”, the following “R” another “p”. The compound operation “LP”
shows itself as a “n”

for neapolitanisation
.

In othe
r

words
,

Maegaards system not only
shows

the
harmonic fu
nction of each chord

in
relation to
the operating tonic. It also indicates the transformation operations tak
ing

us from chord to chord.

The system is thus a combination of a nominating and a transformational
system
.


6)

The move to F
-
minor

Returning to the

transition from A
-
minor to F
-
minor
,

we could from a transformation theoretical point
of view simply state that the progression is realized through a LP
-
operation. This would tell us how the
move from A
-
minor to F
-
minor can be explained within avai
la
ble t
onal procedures.

But it does not consider an eventual harmonic relationship back to Am, or
any possible
relationship to
the B
-
minor tonality stated only a few bars earlier.

If we, as I do, feel that the progression from A
-
minor to F
-
minor relates to a well
-
known tonal
progression, then we have, I think, two options.

1) We either hear the E of the E
-
major chord as somehow still ringing beneath the A
-
C dyad, thus
making the move to F
-
minor the result of a deceptive cadence. In
the Danish
tradition notated by

setting
the original target of the dominant in question,
-

in this case
the minor SS of B
-
minor


in italics.

2) Or we could

hear the progression as a strongly colored neapolitanisation of
o
SS.
While

a
neapolitanisation normally leads to a major chord th
is move would have to be described by two suffixes:
n+v.

This actually reflects the way I hear this passage, and it

s an example of a typical Nielsen
-
handling of, or
playing with
,

the harmonic heritage of the romantic period. He uses the established chord progressions but
gives them a twist


bending them towards the flat side
, while at the same time maintaining tonality.



7) The move to G
-
minor

The following passage in F
-
minor do
es not twist or bend. It unfolds regular tonal progressions within F
-
minor, moving towards a region that could be heard as A
-
flat or D
-
flat, or simply as a flavor of the some of
the relative major tonalities within F
-
minor. The cadence which turns out to l
ead to G
-
minor actually is set
up as a potential cadence in F
-
minor
.

The only twist Nielsen
makes

is raising the note of D flat to D natural. Which still could lead to F
-
minor
.

The chord would be heard as an

incomplete Subdominant major.


So once again Nie
lsen accomplishes his modulation by a subdominantic triad, which is introduced as an
altered S and proceeds as a regular S.
I
n the case of the modulation to
G
-
minor
, this regular S in fact turns
out to be a Tonic.
I
n terms of “altered” subdominants, we not
ice that this last alteration is
the

mo
st

extravagant of the
three subdominantic modulation points.



I said that G
-
minor turns out to be the Tonic. But in fact it is not so unambiguous. The end of the theme
is a prototype of the flux between I
-
V and IV
-

I

that Reynolds notes to be a typical ambiguity in Nielsen’s
music.

And as the first variation begins with a F
-
sharp leading to b
-
minor, the G
-
minor chord might be h
e
ard as
an altered dominant

preparation, a minor
b
vi, or as we would name it: DDalt
-
minor, m
eaning the triton
e

substitution of
the supertonic,
DD
, which then is turned

minor.

No
matter what name we choose
th
is
chord
that links two tonalities, g
-
minor and b
-
minor,
is
once
again
an example

of an altered dominant
-
preparation,
that is,
a kind of alte
red S.

A short remark should be made to the notation

system, which
at
some
points

reflects the work of
another Danish Scholar/Composer, Jörgen Jersild. In Jörgen Jersilds system all chords can be replaced by
their tritone substitution
. This is
notated by t
he suffix “alt”. Furthermore Jersild extends the row of
dominants, so that he operates with not only DD, the dominant

s dominant, but also the Dominant

s
Dominant

s Dominant
-

DDD


or 3D. Actually he extends the dominant row up to 5D.
4

His main point, is

that these dominants represents
specified
positions
which
reflect
s

their distance to the tonic, represent
ing

position 1.


8) The Am


Gm analogy

But
we haven’t finished with the move to

g
-
minor.
As it is
this move

turns out to be
a structural
mirroring

of the move to a
-
minor. If we compare the two cadences, we find that they are both based on


what turns out to be


a tonic chord in first inversion moving to a Dominant by stepwise motion in the bass,
just to return to the tonic. A
-
minor moves on to F
-
m
inor by a LP
-
operation. If we should define the
transformational operation by which G
-
minor moves on to B
-
minor it would be the opposite operation of
LP. Namely PL.

For
while
Am moves a major third downwards to Fm, Gm moves a major third upwards to Bm.

There seems to be a structural thought behind these tonalities: An overall frame of a tritone pole
,

B
-
F
,

is filled out by
the notes
A and G in stepwise, contrary motion: B moves down to A. F moves up to G. The
tonalities of the first half of the Theme proc
e
eds downward: A second

(b to a)
, a major third

(a to f)
. The
tonalities of the second part of the them
e

proce
e
ds upwards: A second

(f to g)
, an
d

a major third
(g to b)
to
get back to variation I.

And all tonal areas are minor.


9) Sequential
Organisation

This tonal structure is not the only kind of
organizational

device running along with the harmonic and
motivic coherence in this theme.
A
lso sequential structures
are
running underneath the surface.

These are best shown by a jump to variation VII, where Ni
elsen reformulates his theme, while
maintaining the
inclusion of the tonal area of F
-
minor, G
-
minor and partly A
-
minor.

What is interesting here is the way
in which
he reaches G
-
minor: By a chordal progression of falling
fourths and rising seconds. A prog
ression hinted at twice before it is fully realized.


[9a) Examples from literature]

This progression reminds us of something. Of a lot of things, actually, as this progression serves as back
bone of many classical pieces. Like Pachelbels Canon in D or Bac
hs prelude in B
-
major
. Just

to name a few.

The progression is one of a very limited numbers of sequences that we meet again and again in tonal
music. This one has been called “Inganno
-
sequence”, due to its construction of pairs of falling fourths
leading to a deceptive cadence, from where the next link of the sequence starts.
And this is the reason for
the name “
Inganno


mean
ing

“d
ec
i
eve

.


The wonderful thing about sequence structures

is that they once recognized arouse an expectation:
That this sequence will run in three links, where the third link might be varied. Just as it is the case with
Pachelbel and Bach.

In Nielsen it would seem not to hold true, as it would be the first link,

that he
re was varied. But the
point is

that the motion to G
-
minor is composed as a sudden revelation of a hidden structure. A hidden
structure already present in the theme.






4

See Jörgen Je
rsild,
De funktionelle Principper i Romantikkens Harmonik belyst med udgangspunkt i Cesar Franck’s
harmoniske stil”,
EWH Kbh., 1970.

Like Maegaard’s system Jersild’s system turns out to express tranformation
operations
. Mainly D and S operations in Lewin’s

way of using them. To these operation we just need to add the “alt”
operation movin
g a chord a tritone
.


[9b) Inganno Sequence]

Already in the opening Theme
a structural inganno sequenc
e runs through the turn to F
-
minor as well
as the turn to G
-
minor
.

To
recognize

th
is

structure in the turn to F
-
minor we will have to hear the A
-
C dyad as the 6
-
4
suspension of a
n

E
-
major, turn
ing

to F in a deceptive cadence
. This would make up

the third p
air of falling
fourths in an inganno
-
sequence lying as a hidden structure

back from mm.6
.
5

Also the move to G
-
minor is
established as the third link of an inganno
-
sequence
, but this time it is
made
even more subtle.

The roots of the preceding chords make
up the pairs falling fourths, and when they reach G
-
minor the
bas
-
movement suggest a deceptive cadence. And interpreted as an altered incomplete S in F
-
minor this is
also
what follows
.

But the
incomplete S
-
chord
(IV)
is at the same time G
-
minor

(II)
,
the s
upertonic

of F
-
minor. So the
progression proceeds with the third link of the Inganno
-
structure, just as it should. But it is realized from G
to D, the real root of the chord
first
introduced as some kind of
double
altered B
-
m
inor
.


10) Structural devices



Let

s now

return to the Theme in its full length, and try to consider the
organizing

devices that keep it
together
.

It seems now that the Theme is mainly
organized
in two parts. First part contains the two sto
l
len
and
develops itself in B
-
minor with its two last bars turning to A
-
minor.

Second part consists of the abgesang, unfolded in F
-
minor, which in its two last bars
t
urns to G
-
minor.

Running across this neat division we have the two inganno
-
structures leading to F
-
minor and G
-
minor
respectively.


The melody outlines the division in its barform. However the abgesang is structured as sequence in
itself running as a bar
-
form presenting an intensified version of the two sto
l
len. This sequence runs across
not only the
two inganno
-
structures but also
across
a sequential movement in the bas
-
line, which is
developing under a changing harmonic environment.

All in all we find a

dense

intensification in the abgesang


Partly following the barform

is the rise in minor thirds from B to D, mm.4 to F mm.9.
This thirdwise
ascend could be seen as a variation of the stepwise ascending melodic sequence of the abgesang. An
extremely concentrated middleground version based on only the rising fourth.

You mi
ght
also
want to see a golden

proportion here, as the move from B to D takes three bars while it
takes five bars to reach F. Which then more or less controls the last eight bars of the theme, giving the
Fibonacci numbers 3,5,8.


11) Final Remarks: Tonalit
y

Another aspect of the theme is the foreground handling of harmony. The fact
, namely,

that Nielsen
effectuates all modulation through subdominantic triads. It generates a feeling of a constant move towards
the plagal area. And beyond.

Taken isolated we mi
ght
understand

the entrance of F
-
minor as a triton
e

substitution of T
, Talt
. Or
maybe
we should understand it as
the result of
a

middleground
series of
rising minor thirds
, making it a
Tpvp instead of a Talt
.

But considering the harmonic progressions

by wh
ich
F
-
minor is reached

-

a
s a neapolitanisation of
o
SS
-

F
-
minor
actually
rather
presents itself as belonging to a subdominantic region.




5

Already Ernst Kurth
(1923) hints at
the use of structural
sequences by describing passages where sequence is
moved from the melodi

to the harmony: “Die Melodie setzt sich aber darauf nicht mehr als Sequenz fort, hingegen
wirkt diese, wie in den Untergrund gedrängt, in der
H
armoni
si
erung

weiter” p.349.

In this perspective the turn to F
-
minor does not really represent a

mov
e

to a distant unconnected tonal
area, but to a

kind of mysterious dark subdominantic region, not previously heard of.

In this light w
e might also interpret
F
-
minor

as reached by a constant move into the subdominant area

by fifths:

from S to SS to


not SSS, but the Relative of SSS
-

turned minor.

This

distant subdominantic area is
then
brought closer to B
-
minor by turning to G
-
minor,
the
subdominant of the relative major, and in itself
a subdominantic region capable of moving directly to the
Dominant of B
-
minor.


However we would like to hear this
music the fact of a tight structural organization remains. And this
structural treating of tonal areas might in itself be seen as way for Nielsen to get away from the keys and
still work with diatonic conviction.



Lit
terature:

Cohn,

Richard,

“Maximally Sm
ooth Cycles, Hexatonic Systems, and the Analysis of Late
-
Romantic
Triadic Progressions”,
Music Analysis,
Vol. 15, No. 1 (1996) pp.9
-
40.


Hyer
, Brian

“Reimag(in)ing Riemann”,
Journal of Music Theory
, Vol. 39, No.1 (1995), pp.101
-
138


Jersild,
Jörgen,
De fun
ktionelle Principper i Romantikkens Harmonik belyst med udgangspunkt i Cesar
Franck’s harmoniske stil”,
EWH Kbh., 1970.


Jersild, Jörgen,
Dur og moll
-
harmonikkens sekvensmønstre,
EWH København, 1985


Kurth, Ernst,
Romantische Harmonik und ihre Krise in
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