Axis system
Encyclopedia
In music
, the axis system is a system of analysis originating in the work of Ernő Lendvaï
, which he developed in his analysis of the music of Béla Bartók
.
The axis system is "concerned with harmonic and tonal substitution", and posits a novel type of functional relationship between tones and chords
. Lendvaï's analyses aim to show how chords and tones related by the interval
s of a minor third
and tritone
can function as tonal substitutes
for one another, and do so in many of Bartók's compositions.
chord ... can replace the tonic, most familiarly in an interrupted cadence." In his analyses of Bartók's music, Lendvaï identifies a novel set of tonal substitutions; substitutions that relate chords and keys in a flat mediant
relation to one another, and also those related by the tritone, a tonal relationship "normally regarded as the most remote pitch/chord/key area from the tonic." Lendvaï argued that these relationships had a naturalistic
basis (that is, were not merely an analytical or compositional contrivance), and argued that many of Bartók's compositions made essential use tonal substitutability he described. By establishing the veracity of this novel set of relationships, Lendvaï "attempts to 'explain' Bartók's chromaticism withn a tonally functional model."
into three sets; each set of tones contains those notes that are found at interval
s of a minor third
and tritone
from one another (equivalent to the notes that make up the three possible diminished seventh chord
s). Thus, there are four notes in each of the three sets, sets which Lendvaï refers to as axes
. By way of analogy with common practice harmony, the three axes are categorised as
tonic, subdominant
, and dominant
.
Tones related to each other by the interval of a tritone are said to be part of a pair or branch (e.g. C/F#, Eb/A, etc). Thus, each of the three axes contains two branches, and are further categorised as either the 'principal' or 'secondary' branch of axes in question. Continuing with the language of axes, the members of each branch are known as “pole and counterpole”. Thus, the twelve tones of the chromatic scale are identified in a two-part system of categorisation: each tone is a member of a four note axis (tonic, dominant, and subdominant); each tone is part of a branch within that axis that is either the principal or secondary branch of that axis; and each branch consists a pole and counterpole.
If 'C' is taken as tonic, then the categoristation proceeds as follows:
Essential to Lendvaï's conception of the axis system and the relationships it describes is the idea 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, E major and C minor)."
provides an illustration of axis-based substitution, where the opening is based upon the pole of the principal branch of the tonic axis (A, A-C-E-F), the middle contrasts the opening and the end of the first movement by being based on E. 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-E) and the second on the secondary (C-F).
It is interesting to note that the pattern Tonic-Super Tonic-Subdominant relationship repeats itself counter clockwise around the circle of fifths. This pattern creates the root movement of a ii/V7/I chord progression. The tonic dominant relationship is repeated around the circle in clockwise fashion.
The axis system is probably used more in analysis, and less in composition. 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.
, 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
.
in The Music of Béla Bartók (1992).
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, the axis system is a system of analysis originating in the work of Ernő Lendvaï
Erno Lendvai
Ernő Lendvai was one of the first theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in Bartók's music...
, which he developed in his analysis of the music of Béla Bartók
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...
.
The axis system is "concerned with harmonic and tonal substitution", and posits a novel type of functional relationship between tones and chords
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...
. Lendvaï's analyses aim to show how chords and tones related by the interval
Interval
Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement...
s of a minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
and tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
can function as tonal substitutes
Chord substitution
In music theory, chord substitution is the use of a chord in the place of another related chord in a chord progression. Jazz musicians often substitute chords in the original progression to create variety and add interest to a piece. The substitute chord must have some harmonic quality and degree...
for one another, and do so in many of Bartók's compositions.
Introduction
In classical and common-practice systems of harmony, certain chord substitutions are recognised and are commonly made use of by composers and arrangers: "certain chords have been able to act as substitutes for others; for example, the submediantSubmediant
In music, the submediant is the sixth scale degree of the diatonic scale, the 'lower mediant' halfway between the tonic and the subdominant or 'lower dominant'...
chord ... can replace the tonic, most familiarly in an interrupted cadence." In his analyses of Bartók's music, Lendvaï identifies a novel set of tonal substitutions; substitutions that relate chords and keys in a flat mediant
Mediant
In music, the mediant is the third scale degree of the diatonic scale, being the note halfway between the tonic and the dominant. Similarly, the submediant is halfway between the tonic and subdominant...
relation to one another, and also those related by the tritone, a tonal relationship "normally regarded as the most remote pitch/chord/key area from the tonic." Lendvaï argued that these relationships had a naturalistic
Naturalism (philosophy)
Naturalism commonly refers to the philosophical viewpoint that the natural universe and its natural laws and forces operate in the universe, and that nothing exists beyond the natural universe or, if it does, it does not affect the natural universe that we know...
basis (that is, were not merely an analytical or compositional contrivance), and argued that many of Bartók's compositions made essential use tonal substitutability he described. By establishing the veracity of this novel set of relationships, Lendvaï "attempts to 'explain' Bartók's chromaticism withn a tonally functional model."
Terms and definitions
In his analysis, Lendvaï groups the twelve tones of the chromatic scaleChromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
into three sets; each set of tones contains those notes that are found at interval
Interval
Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement...
s of a minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
and tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
from one another (equivalent to the notes that make up the three possible diminished seventh chord
Diminished seventh chord
A diminished seventh chord is a four note chord that comprises a diminished triad plus the interval of a diminished seventh above the root. Thus it is , or enharmonically , of any major scale; for example, C diminished-seventh would be , or enharmonically...
s). Thus, there are four notes in each of the three sets, sets which Lendvaï refers to as axes
Axes
Axes may refer to:* Axes, woodworking hand tools* The plural of axis* Axes , a 2005 rock album by the British band Electrelane* X and Y axes, or X, Y, and Z axes, perpendicular lines used in the Cartesian coordinate system...
. By way of analogy with common practice harmony, the three axes are categorised as
tonic, subdominant
Subdominant
In music, the subdominant is the technical name for the fourth tonal degree of the diatonic scale. It is so called because it is the same distance "below" the tonic as the dominant is above the tonic - in other words, the tonic is the dominant of the subdominant. It is also the note immediately...
, and dominant
Dominant (music)
In music, the dominant is the fifth scale degree of the diatonic scale, called "dominant" because it is next in importance to the tonic,and a dominant chord is any chord built upon that pitch, using the notes of the same diatonic scale...
.
Tones related to each other by the interval of a tritone are said to be part of a pair or branch (e.g. C/F#, Eb/A, etc). Thus, each of the three axes contains two branches, and are further categorised as either the 'principal' or 'secondary' branch of axes in question. Continuing with the language of axes, the members of each branch are known as “pole and counterpole”. Thus, the twelve tones of the chromatic scale are identified in a two-part system of categorisation: each tone is a member of a four note axis (tonic, dominant, and subdominant); each tone is part of a branch within that axis that is either the principal or secondary branch of that axis; and each branch consists a pole and counterpole.
If 'C' is taken as tonic, then the categoristation proceeds as follows:
| Tonic | Subdominant | Dominant | |||
|---|---|---|---|---|---|
| Primary | Secondary | Primary | Secondary | Primary | Secondary |
| C, F, | A, D, | F, B, | D, G, | G, C, | E, A |
Substitutability
The essential idea behind the axis system is that the axes group together "substituable key areas", and categorises key areas within a particular axis in reference to the strength and appropriateness of their inter-substitutability. Counterpoles that form one branch of an axis are more closely related that the counterpoles of the other branch of that axis: inter-substitutability within a branch is a stronger relationship than between the two branches of an axis. However, each axis possesses a "two-fold affinity", one being the relationship between pole and counterpole, the other being the relationship between the principal branch with the secondary branch. The relation between pole and counterpole, "its closest replacement" is judged to be stronger much more sensitive than the relation between the other poles of an axis, and due to this Lendvaï states that "a pole is always interchangeable with its counterpole without any change in its function."Essential to Lendvaï's conception of the axis system and the relationships it describes is the idea 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, E major and C minor)."
Analysis of Music for Strings, Percussion and Celesta
The form of the first movement of Music for Strings, Percussion and CelestaMusic for Strings, Percussion and Celesta
Music for Strings, Percussion and Celesta, Sz. 106, BB 114 is one of the best-known compositions by the Hungarian composer Béla Bartók. Commissioned by Paul Sacher to celebrate the tenth anniversary of the Basel Chamber Orchestra, the score is dated September 7, 1936...
provides an illustration of axis-based substitution, where the opening is based upon the pole of the principal branch of the tonic axis (A, A-C-E-F), the middle contrasts the opening and the end of the first movement by being based on E. 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-E) and the second on the secondary (C-F).
| Movement | Section | ||
| Beginning | Middle | End | |
| I | A | E | A |
| II | C | F | C |
| III | F | C | F |
| IV | A | E | A |
It is interesting to note that the pattern Tonic-Super Tonic-Subdominant relationship repeats itself counter clockwise around the circle of fifths. This pattern creates the root movement of a ii/V7/I chord progression. The tonic dominant relationship is repeated around the circle in clockwise fashion.
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 III of each of the principal tones: root, subdominant, and dominant.The axis system is probably used more in analysis, and less in composition. 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.
The axis system and jazz
Brazilian music, especially the music of Antonio Carlos JobimAntônio Carlos Jobim
Antônio Carlos Brasileiro de Almeida Jobim , also known as Tom Jobim , was a Brazilian songwriter, composer, arranger, singer, and pianist/guitarist. He was a primary force behind the creation of the bossa nova style, and his songs have been performed by many singers and instrumentalists within...
, 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
The Girl from Ipanema
"Garota de Ipanema" is a well-known bossa nova song, a worldwide hit in the mid-1960s that won a Grammy for Record of the Year in 1965. It was written in 1962, with music by Antonio Carlos Jobim and Portuguese lyrics by Vinicius de Moraes. English lyrics were written later by Norman Gimbel.The...
.
Further reading
Lendvaï's system is critiqued by Paul WilsonPaul Wilson (music theorist)
Paul Wilson is a music theorist and Professor of Music Theory and Composition at the University of Miami Frost School of Music, in the United States. He holds a B.A. from Harvard University, a M.A. from the University of Hawaii, and M.Phil. and Ph.D. degrees from Yale University, where he studied...
in The Music of Béla Bartók (1992).
Sources
- Wilson, Paul (1992). The Music of Béla Bartók. ISBN 0-300-05111-5.
- Cooper, David "Bartok: Concerto for Orchestra", ch. 3 Genesis and Reception, CUP