Harmonic and sequential organisation in Carl Nielsen Opus 40
1)Tonality
“I sat home one night and 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 beginnings in bminor each being directed to g-minor, once again leading 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 bminor and f-minor, within which we encounter short digressions to a-minor and g-minor.
We might want to speak of multilayered tonality. But I think that the 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 Form]
Formally the Theme is cast in a bar-form, with two stollen and and an abgesang, which in itself makes
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 and a stepwise descend of three notes. These elements are then combined in different ways.
However, the 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 based – 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 expressed by chords of different scale degrees.
If we turn to first part of Opus 40 we’ll find that the 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 domination
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 of 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 supertonic in first inversion we 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 Aminor.
All these modulations are done through reinterpretations of pure triads functioning as subdominants.
All are introduced as altered subdominants continuing 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 subdominant”, the
subdominants subdominant, SS-minor. Indicated by the circle,o, as a prefix to SS: oSS. This chord is now
temporarily tonicized by the subdominant major, here acting as an applied dominant in relation to SS.
But when this A-minor-chord continues 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 moving back from
1. to 7. in A-minor.
A more harmonic oriented explanation could 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 an 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 Arnold 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
example, 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. Following this logic another leap a minor third upwards will lead us to
the relative – “parallel” – of Spv, which 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 system 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 complicated 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.
We 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 actually very close to the dominant, and by implication:
Home.
As this move is analogue to the move from a subdominant to the Neapolitan, from IV to bII, Maegaard
calls this move for “neapolitanisation”. In romantic music we encounter neapolitanisations of both the
tonic and the dominant and even other 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 immediately 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 scholars 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, Hyers and Cohn3 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 theory the central issue is the transformation-operation that makes one chord become another
chord. In Lewin’s 1982 presentations of what he calls “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 up from a minor chord. The 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 between 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 perform 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 other words, Maegaards system not only shows the harmonic function of each chord in relation to
the operating tonic. It also indicates the transformation operations taking 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 available tonal 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 oSS. While a
neapolitanisation normally leads to a major chord this 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 does not twist or bend. It unfolds regular tonal progressions within Fminor, 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 lead 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 Nielsen accomplishes his modulation by a subdominantic triad, which is introduced as an
altered S and proceeds as a regular S. In the case of the modulation to G-minor, this regular S in fact turns
out to be a Tonic. In terms of “altered” subdominants, we notice that this last alteration is the most
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 heard as
an altered dominant preparation, a minor bvi, or as we would name it: DDalt-minor, meaning the tritone
substitution of the supertonic, DD, which then is turned minor.
No matter what name we choose this 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 altered 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 the 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 reflects their distance to the tonic, representing
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-minor 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 proceeds downward: A second (b to a), a major third (a to f). The
tonalities of the second part of the theme proceeds upwards: A second (f to g), and 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. Also sequential structures are running underneath the surface.
These are best shown by a jump to variation VII, where Nielsen 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 progression 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 Bachs 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” meaning “decieve”.
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 here 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 Jersild, 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 moving a chord a tritone.
[9b) Inganno Sequence]
Already in the opening Theme a structural inganno sequence runs through the turn to F-minor as well
as the turn to G-minor.
To recognize this structure in the turn to F-minor we will have to hear the A-C dyad as the 6-4
suspension of an E-major, turning to F in a deceptive cadence. This would make up the third pair 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 supertonic 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-minor.
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 stollen 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 turns 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 stollen. 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 might 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: Tonality
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 might understand the entrance of F-minor as a tritone 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 which F-minor is reached - as a neapolitanisation of oSS 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 Harmonisierung weiter” p.349.
In this perspective the turn to F-minor does not really represent a move to a distant unconnected tonal
area, but to a kind of mysterious dark subdominantic region, not previously heard of.
In this light we 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.
Litterature:
Cohn, Richard, “Maximally Smooth 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 funktionelle 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 Wagners ”Tristan”, Berlin, 1923.
Lewin, David “A Formal Theory of Generalized Tonal Functions”, Journal of Music Theory, Vol. 26, No.1
(1982), pp.23-60
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. 84110.