Vasile

Tiano 2017 Music Analysis

Bartók, Lendvai and the divided music

Ernő Lendvai was a music theorist who implemented a new form of analysing music by
the composer Béla Bartók. He wrote about this new form, in his book called “Béla
Bartók - An Analysis of His Music”1, by looking at the ‘principals’ or ‘elements’ that
contribute to the formation of Bartók‘s music. These principals include Golden Section
(or Ratio), Fibonacci series, and the Tonal Axis System as the analytical tools to use for
investigating his music. These principals are used in two ways: to show the relationship
or connection between the ‘principals’ and Bartók’s music; and the structure or
construction of Bartók’s music. The principals also integrate together to formulate and
break down how Bartók’s music work. In this written commentary, I will apply Lenvai’s
theory by analysing Bartók’s “Divided Arpeggios”2 with reference to the Golden Section
and Fibonacci series. This analysis will be divided into two sections: Firstly, I will account
the structure of “Divided Arpeggios” by using Golden Section. Secondly, I will
investigate how Fibonacci series is found in “Divided Arpeggios”.

“Divided Arpeggios” is organised in three main sections - A + B +A’. Between these

sections there are transitional sections or parts that eases into the next section. i.e. bars
26 – 29 leads into the next section. Additionally, the score has an introduction and coda,
which creates this structural plan in the score:

Intro A trans. B trans. A’ Coda

Bars/Measures: 1-5 6-25 26-9 30-45 46-9 50-67 68-80

Fig. 1.1 – Structural plan and framework

Firstly, by using Golden Section we can solidify our understanding of the form and
structure in “Divided Arpeggios”. If we understand that the ratio calculated and used in
Golden Section is 0.618 then we can start our analysis.

The total number of bars in the music is 80. 80 x 0.618 = 49.44 (roughly 49.5). That
means the height or “centre of gravity”3 (the climax of the piece) is at the halfway point

1
Ernő Lendvai, Béla Bartók - An Analysis Of His Music, 2nd ed. (Great Britain:
Redwood Burn Ltd., 1979).
2
Score provided
3
Ernő Lendvai, Béla Bartók - An Analysis Of His Music., p. 18
Vasile Tiano 2017 Music Analysis

between bars 49 and 50. The climax is at the transition point exiting section B to a return
to (a new) A section. This also creates an A and B section for our Golden Section:

Fig. 2.1: Golden section length/structure:

A+ B

GS
A B

B. 1 B. 80

We can check if this notion is correct. If we say at the beginning of bar 49 (49.00) is our
GS, then this is incorrect: 49.00/80 = 0.6125. If we say at the beginning of bar 50 (50.00)
is our GS this is also incorrect: 50.00/80 = 0.625. The true GS in “Divided Arpeggios” is
in the middle of bar 49 (49.45). This is the closest match to 0.618 – it calculates to:
49.45/80 = 0.618125.

Additionally, by listening to the music4 and following the score we can conclude that at
the GS (49.45) the music transitions into our right tempo and dynamic marking.
Previously, at bar 39 the music begins to speed up by the occurrence of diminution –
music becoming dense, tighter and smaller in a short amount of time. This can be seen
especially at bar 42 compared to the bars before. It connects and supports the structure
heard and seen on the score.

Furthermore, if we calculate the GS between the first ‘49’ bars this reveals another
achievement for our understanding of the form and structure in “Divided Arpeggios”.
The first 49 bars are our A section, seen in figure 2. If we calculate 49/0.618 = 30.282 –
bar 30 is another GS between our A section. This means at bar 30, there is a structural
point in the music, which is true to the score. By listening to the music, this can be
supported by the tonal change into a new development created by the gradual change of
tempo and dynamic marking of p – soft compared to the previous F (forte) sound and

4
Music extract given
Vasile Tiano 2017 Music Analysis

dynamic marking. The music at bar 29, on the score, is a transition to the new section B;
bar 30 is the start of section B. Similarly, if we state that bar 50 – 80 is the section B (fig
2), then our GS can be calculated as well. 30/0.618 = 18.54 – bar 68 is another GS
between B section. This would make sense as well because bar 68 is the start of the coda.

(GS) GS: (GS)

B. 30 B. 49.5 B. 68

B. 1 B. 80

Fig. 2.2 – Structure with Golden Sections

Golden Section has shown the major structural points in “Divided Arpeggios” by
Bartók. These calculations have also shown where the music changes such as the tempo,
dynamics and tonal colour. These changes support why and how the musical and
structural framework is designed in “Divided Arpeggios”.

Secondly, the Fibonacci series is another principle that can support our understanding in
“Divided Arpeggios” by Béla Bartók. It can also reveal how Bartók may have composed
by the pitches used in relation to Fibonacci numbers. Lendvai explored how the
Fibonacci series relates to Bartók’s music5. The Fibonacci series are: 0, 1, 1, 2, 3, 5, 8, 13,
21, 34 etc. By understanding the series we can start our analysis by looking at four
examples:

5
Ernő Lendvai, Béla Bartók - An Analysis Of His Music., p. 27
 Vasile Tiano 2017 Music Analysis

Example 1: The first five bars in the introduction can show if there is a melodic
intervallic relationship and connection between his music and The Fibonacci series.

Fig. 3.1: b. 1-5

In figure 3.1 we can see a relationship in the use of Fibonacci numbers such as: 1, 2, 3, 5

Fig. 3.2: The notation left to right, which includes vertical (intervallic) dyads between b.
1-5

U D E A B D E A B D E A B D E A B
♭♭♭♭ ♭♭♭ ♭ ♭♭♭ ♭ ♭♭♭♭
L B C A B D E A B D E A B D E A B C E B
♭♭ ♭ ♭ ♭♭♭♭ ♭ ♭♭ ♭ ♭ ♭♭♭ ♭
0 1 5 2 3 2 5 2 2 5 2 2 5 2 3 2 5 2 2 2 2 2 5 3 1 3

3 3 7 7 4 4 3

*(U=Upper system; L=Lover System)

In summary of this analysis, there are correlation and relationship of Fibonacci numbers
used in the opening passage bar 1-5. The use 1 (minor 2nd), 2 (Major 2nd), 3 (minor 3rd),
and 5 (perfect 4th) in the passages shows an initial chromatic interest and third
relationship. The third relationship can be seen the bars 4-5 which is an introduction to
the theme or concept of the piece called “Divided Arpeggios”.

Example 2: In section A, bars 22 – 25 shows a chordal relationship and

connection between his music and the Fibonacci series.

Fig. 3.3: b. 22-25

Vasile Tiano 2017 Music Analysis

In figure 3.3 we can see a chordal relationship in the use of Fibonacci numbers such as 1,
2, 3, 5, 8.

Fig. 3.4: chords from left to right in b. 22-25 – vertical sonority

U 2 2 8 6 5 3 1 3 5

3 3 3 3
L - - 3 - 3 3 1 - 3 6

8 5 3 5

*(U=Upper system; L=Lover System) **( “-“ symbol links into other system)

In summary of this analysis, there is a chordal framework and a relationship to Fibonacci

numbers used in the opening passage bars 22 – 25. The use 1 (minor 2nd), 2 (Major 2nd), 3
(minor 3rd), 5 (perfect 4th), and 8 (minor 6ths) show a chordal passage with the interest of
minor thirds, perfect 4ths and minor 6ths. It is also interesting to see a mirror effect or
opposite groupings in the upper and lower system. Additionally, it creates a dense and
rich harmony in “Divided Arpeggios”.

Example 3: In section B, bars 39-43 shows a liner relationship and connection

between his music and The Fibonacci series.

Fig. 3.5: lower system of b. 39-43

 Vasile Tiano 2017 Music Analysis

Fig. 3.6: chords from left to right in b. 39-43– vertical sonority

U 3 3 3 3 3

4 4 4 4 4

3 3 3 3 3
L 4 4 4 4 4

3 3 3 3 3

3 3 3 3 3

In summary of this analysis, there is a linear framework and a relationship to Fibonacci

numbers used in the passage bars 39-43. The number 3 (minor 3rd) shows a strong
interest in minor thirds, which supports the intention and musical aspects of using 3rds
in the piece. This is also seen in opening of section A’ by using thirds as a main motif
used for a variation to section B and a reference to section A.

Example 4: Other variations in the use of the Fibonacci series include tempo and
2
rhythmic groupings. The tempo marking of relates to the Fibonacci series. The music
4
is then constructed into rhythmic groupings of two – two crochet beats per bar. This
‘format’ is easy to manipulate and creates stability in the music. The only unstable
passages are when Bartók enters into three groupings against two beats. This can be seen
in bars 20 – 24. In bars 20-22, there are three groupings of semiquavers, which overflow
into the next bar (eg. 20 into 21, 21 into 22 etc.). Similarly, this can be seen in bars 23 –
24 as well. We can further explore this notion of groupings by analysing the first page.
Bars 1-5 are grouped into 3+2 – all Fibonacci numbers. Bars 6-13 are grouped into two
bars each – 8 bars in total. Similarities seen in bars 14 – 21 which shows 8 is another
Fibonacci number.

In summation, there are clear elements found by using Ernő Lendvai’s theory to analyse
the music “Divided Arpeggios” by Béla Bartók. Structure and intervals of a minor 3rds
are two main common unifying elements that make-up and form the music, which
support the meaning of the piece – ‘arpeggios’. Golden Sections show a piece can be
structured that creates a framework and plan in the music. Within the main GS, there are
Vasile Tiano 2017 Music Analysis

other Golden Sections, which supports the piece further. Additionally, our listening
experience can help us strengthen this finding by listening to tonal changes, dynamic
markings and change in tempos. Fibonacci series further add to our understanding of
this framework in four ways: intervals, chords, lineal sonorities, tempo and rhythmic
groupings. However, what lacks in this approach of analysing Bartók’s music is the
uncertainty this is the approach Bartók used. The theory seems to work with the analyst
favour (which all theories should). It lacks the discovery of finding relationships and
makes you find obvious connections. It does not show why a chord creates a certain type
of sonority or show how the use of modes used by a composer creates interest. In
essence, it is a theory that looks the mechanics and mechanisms, rather answers to the
meaning and understanding of Bartók’s music. Bartók is known that have been
influenced by folk and improvisational music. Most of his music shows connections and
references to other cultures and musical elements 6 . Lendvai’s theory does show a
different approach and view to analyse music.

Reference:

• Lendvai, Ernő. Béla Bartók - An Analysis Of His Music. 2nd ed. Great Britain:
Redwood Burn Ltd., 1979.
• Ling, Jan. A History Of European Folk Music. Rochester, N.Y.: University of
Rochester Press, 1997.
• Music extract given
• Music score given

*This analysis was a part of a unit in Music Analysis at UNSW during my

undergraduate studies*

6
Jan Ling, A History Of European Folk Music (Rochester, N.Y.: University of Rochester
Press, 1997)., p. 107 - 111