Béla Bartók’s Axis System One of the most prominent characteristics of Bartók’s music i s the symmetry contained in his works. Sometimes subtle, sometimes obvious, it could cover everything from melodies, structures, tonal centers, rhythms, rhythms, to forms and chord relations. During the course of his professional life, Bartók never revealed or spoke openly about his compositional compositional techniques (nor did he publish anything on the topic), he always claimed to be an instinctive composer. It was the Hungarian musicologist Ernö Lendvai who dedicated a big portion of his life to investigate Bartók’s compositional style, and revealed to the world important techniques that he considered Bartók used in his compositions. The list includes, for example, the use of the Fibonacci series, mi-pentatony, sixfour structures, hypermajor and hyperminor chords, alpha h armonies, golden section, etc. Among those techniques one represents the culmination of Lendvai's research, the “axis system.” The axis system as described by Ernö Lendvai, is a method used to divide the chromatic scale symmetrically considering the harmonic functions of subdominant, tonic and dominant. The aim of this technique is to create chord successions using the twelve pitches from the chromatic scale and still maintain
169
a sonority reminiscent of the tonal system. The explanation Lendvai gives to this theory can easily be visualized from the circle of fifths.
60
Figure 71. Circle of Fifths. C F
G
Bb
D
Eb
A
Ab
E Db
B F#
Lendvai starts by organizing the circle of fifths according to the harmonic functions of subdominant, tonic and dominant major triads in the key of C major.
Figure 72. Harmonic Functions. Subdominant
Tonic
Dominant
F
C
G
60
Karen Anne Bates, “The Fifth String Quartet of Béla Bartók: An Analysis Based on the Theories of Ernö Lendvai,” PhD diss., University of Arizona, 1986. In ProQuest Dissertations and Theses, http://wwwlib.umi.com/dxweb/ details?doc_no=1186293 (accessed February 21, 2008), 44-49.
170
Lendvai then draws a symmetric axis using C (tonic) as the starting point, creating a link between opposing poles C, A ,F#, and Eb, named as the “tonic axis”.
Figure 73. Tonic Axis. C
Eb
A
F#
Using the same concept, Lendvai then draws the “dominant axis”, which links opposing poles G, E, Db, and Bb.
171
Figure 74. Dominant Axis. G Bb
E Db
In the same manner the “subdominant axis” links the opposing poles F, D, B, and Ab.
Figure 75. Subdominant Axis. F D
Ab B
When superimposed, the three axis (tonic, dominant and subdominant) complete the “axis system” diagram.
172
Figure 76. Axis System Complete Diagram. T
C
SD
F D
T
D
G
Bb
D
Eb
SD
SD
AT
Ab
E Db D
D
B F#
SD
T
What this symmetry creates according to Lendvai is an equivalent-function relationship between all the pitches contained in each axis. 61
Figure 77. Function Symmetry. Subdominant
Tonic
Dominant
F, D, B, Ab.
C, A, F#, Eb.
G, E, Db, Bb.
Once the axis pitches are organized in this way, Lendvai creates a secondary subdivision of the pitch-classes contained in each function into primary and secondary “branches” in clockwise motion, as shown in the next figure. 61
Ibid.
173
Figure 78. Primary and Secondary Branches. Subdominant
Tonic
Dominant
Primary
Secondary
Primary
Secondary
Primary
Secondary
F, D.
B, Ab.
C, A.
F#, Eb.
G, E.
Db, Bb.
The categorization into primary and secondary branches is derived from the relative minor relationship. The starting point of each axis ( IV, I, V) is related to the degree of its relative minor and grouped as primary, and the opposing poles are the secondary branches. Considering this, Lendvai points out that the symmetry founded in the relation of perfect fifths between F, C, G (creating the functions of subdominant, tonic, and dominant), can also be found in other symmetric relations. For example the symmetrical relationship of major thirds Ab, C, and E, creates, in an atonal manner, a sonority similar to the tonal functions of subdominant, tonic, and dominant.
Figure 79. Major Third Relationship. Subdominant
Tonic
Dominant
Ab
C
E
174
Lendvai also points out that regardless of symmetrical relations, any chord can be substituted for another from its axis, creating in an atonal manner, a sonority similar to tonal functions.
Figure 80. Chord Substitution Table 1. Subdominant
Tonic
Dominant
F
A
Bb
Subdominant
Tonic
Dominant
D
C
Db
Subdominant
Tonic
Dominant
B
Eb
G
Subdominant
Tonic
Dominant
Ab
F#
E
The final step in Lendvai’s analysis is to add to every axis major triad its parallel minor.
175
Figure 81. Parallel Minor Axis. T SD
C-
FD
T
D
G-
Bb-
D-
Eb-
SD
SD
A- T
Ab-
EDbD
D
BF#-
SD
T
Lastly, any of the major triads can be substituted for its parallel minor.
Figure 82. Chord Substitution Table 2. Subdominant
Tonic
Dominant
F-
A
Bb-
Subdominant
Tonic
Dominant
D
C-
Db
Subdominant
Tonic
Dominant
B
Eb-
G-
Subdominant
Tonic
Dominant
Ab
F#-
E
176
Since the very first moment the axis system appeared in Ernö Lendvai’s book entitled Bartók’s Style in 1955, there have been many defenders of Lendvai’s theories like András Szentkirályi. At the same time, some very sharp detractors like Peter Petersen, János Kárpáti, Malcolm Gillies and Paul W ilson, argued that the axis system lacks valid fundamentals and is pure fantasy. 62 Regardless of what position any musician assumes in this debate, the axis system is still a great tool for composing music. Whether this was one of the means Bartók used to achieve his compositional goals, it is something that may never be established. Nevertheless, it remains as a very useful tool to analyze and understand Bartók’s music. In order to apply this system to the style of jazz, the first step is to choose a root movement derived from the three axis. In this case six roots have been chosen, as shown in the next figure.
Figure 83. Axis System Application 1 Subdominant
Tonic
Dominant
Ab
F#
E
B
Eb
G
62
János Kárpáti, “Axis Tonality and Golden Section Theory Reconsidered,” Studia Musicologica 36 (1995): 365-380, http://iimp.chadwyck.com/marketing.do (accessed February 18, 2008).
177
The next step is to determinate the modes that are going to be assigned to each root. In order to do that, i t is important to consider what types of chords are conventionally used in a subdominant, dominant, tonic progression in jazz. Since most of the time this type of cadence is achieved through a ii-7, V7, Imaj7 progression, the following example maintains a minor chord for the subdominant, a dominant 7th for the dominant, and a major 7 th for the tonic roots. It is also important to point out that the same modes (from chapter 3) have been maintained for the two subdominant, two dominant, and two tonic roots, in order to continue with the symmetry.
Figure 84. Axis System Application 2.
Ab Dorian Augmented #11
E Lydian b7#9
F# Lydian #9#13
B G Dorian Lydian b7#9 Augmented #11
Eb Lydian #9#13
Finally, the resulting chords and modes are assigned to an electric piano, and an electric bass in a style of jazz-rock, similar to the music of Mike Stern.
178
Figure 85. Axis System Application 3.
179
