Axis system
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In music the axis system, proposed by Ernő Lendvai (1971, p.1-16) in his analysis of the use of tonality in the music of Béla Bartók, is an assignment of harmonic function to all twelve pitch classes in relation to an assigned tonic, determined by that pitch class's interval from the tonic. The three functions used are tonic, subdominant, and dominant, and each contains four notes (forming three diminished seventh chords). Each group is then divided into principal and secondary groups a tritone apart. Thus there are six pairs of notes in three categories.
It is essential that the particular axes should not be considered as chords of the diminished seventh, but as the functional relationships of four different tonalities, which may best be compared to the major-minor relations of classical music (e.g. C major and A minor, Eb major and C minor).
—(Lendvai 1979, p.3)
If the tonic is C, then the functions are as follows:
| Tonic | Subdominant | Dominant | |||
|---|---|---|---|---|---|
| Primary | Secondary | Primary | Secondary | Primary | Secondary |
| C, A | F♯, D♯ | F, D | B, G♯ | G, E | C♯, A♯ |
Functions are assigned based on position within the circle of fifths, starting with F as subdominant: F (s), C (t), G (d), D (s), etc.
The system is critiqued by Paul Wilson (1992, p.203-208).
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[edit] Harmonic properties
The axis system shares the "essential properties of classical harmony, i.e."
- the functional affinities of the fourth and fifth degrees
- the relationship of relative major and minor keys
- the overtone relations
- the role of leading notes
- the opposite tension of the dominant and the subdominant
- the duality of tonal and distance principles
- (Lendvai 1979, p.1)
[edit] Components
The axis system consists of the tonic, dominant, and subdominant axes, each of which consist of a principal and secondary branch, each of which consist of a pole and counterpole (Lendvai 1979, p.5).
If the pole of the principal branch is C then its counterpole is F#, together they comprise the principal branch while the secondary branch of the same axis consists of the pole A and the counterpole Eb. The system is completed by the subdominant and dominant axes, F-D-B-Ab and G-E-C#-A# (ibid, p.3). The relation between counterpoles is much more sensitive than the relation between the other poles of an axis, so much so that "a pole is always interchangeable with its counterpole without any change in its function." (ibid, p.4)
"Thus each axis has a two-fold affinity depending on whether we oppose the pole with the counterpole, or the principal branch with the secondary branch." (ibid, p.5) The form of the first movement of Music for Strings, Percussion and Celesta illustrates this, where the opening is based upon the pole of the principal branch of the tonic axis (A, A-C-Eb-F#), the middle contrasts the opening and the end of the first movement by being based on Eb. In the second movement the beginning and ending based on C are contrasted with the middle based on its pole F#. The two movements themselves are contrasted through the first being based on the principal branch (A-Eb) and the second on the secondary (C-F#).
| Movement | Section | ||
| Beginning | Middle | End | |
| I | A | Eb | A |
| II | C | F# | C |
| III | F# | C | F# |
| IV | A | Eb | A |
- (ibid, p.5)
It is interesting to note that the pattern D-T-S repeats itself around the circle of fifths (ibid, p.2):
c g d a e b f# db ab eb bb f T D S T D S T D S T D S
[edit] Composition and analysis with the axis system
It may be noted that each of the above notes forms a descending diminished chord. In the case of I or C, C A F♯ D♯, or an E♭ fully diminished chord. This connects the axis system not only with diminished chords, which often form the basis for the movement in a piece based on the axis system, but also links the axis system with the diminished scales formed on bIII of each of the principal tones: root, subdominant, and dominant.
The axis system is probably used more in analysis, and less in composition[citation needed]. Uses of the system have a characteristic sound, but that sound is similar to that which can be found in uses of tritone Dominant Substitutions, and Deceptive Cadences using the ♭VII dominant chord. Older systems of harmonic theory segue very easily into the axis system. It might be noted that many composers, like Bartók, are extremely "mathematical" while many are not.
[edit] The axis system and jazz
Brazilian music, especially the music of Antonio Carlos Jobim, uses harmonies that are easily analyzed by the axis system and not easily analyzed in other ways. A great example is the bridge of The Girl from Ipanema, which uses a series of ♭III dominant and ♭VII dominant chords that looks like a real mess according to normal jazz analysis[citation needed].
[edit] See also
[edit] Source
- Wilson, Paul (1992). The Music of Béla Bartók. ISBN 0-300-05111-5.
- Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill. Revised 1979: ISBN 0-900707-04-6.
