The axis system involves harmonic and tonal substitution, and creates a functional relationship between tones and chords. This system shows how chords and tones relate by intervals of a minor third and a tritone, which function as tonal substitutes for one another. The axis system was created by Ernő Lendvaí, who was one of the first music theorists to explain the music of Béla Bartók in terms of the golden rule and Fibonacci numbers.
In classical harmony certain chord substitutions are rather common, such as the submediant (sixth scale degree) chord being replaced with the tonic (first scale degree), such as Am chord being replaced with a CM. However, Lendvaí discovered a novel set of tonal substitutions that relates chords and notes in a flat mediant (third scale degree) manner, and also chorus and notes related to the tritone, this tonal characteristic is normally regarded as being the farthest from the tonic.
To explain the axis system, group the 12 chromatic tones into three sets, where each set contains notes found at intervals of a minor third and a tritone (i.e. the notes that make up a diminished seventh chord). These sets are called axes, the three axes are referred to as the tonic, subdominant, and the dominant. The tones in the axis system relate to one another by the interval of a tritone as part of a brach (C/F#, A/Eb), so each of the axes contains two branches; the principal and secondary branch. The members of each branch are referred to as either poles or counter poles. In other words, each tone is a member of a four note axis, and each branch consists of a pole and a counter pole.
The axes group together substitutable key areas, and assign strength and appropriateness of their inter-substitutability. The counter poles that form on the branch of an axis are more closely related than the counter poles of the other branch of that axis. Intersubstitutability 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 a relationship between pole and counter pole the other being between principal branch and secondary branch. It is important to remember that a pole is always interchangeable with its counter pole without any change in its function.
The particular axes should not be considered as chords of the diminished seventh, but as the functional relationships of four different tonalities. A good comparison to this would be the major-minor relations of classical music (i.e. CM and Am, EbM and Cm).
For example, the axis system is none other than recognizing the fact that the tonic A and Eb, not only have C as a common relative, but also the F#. Likewise the common relative between the subdominant D and Ab is not only F, but it is also B. The following diagram illustrate this concept:
The opposite poles-counter poles (C and F#) are more directly attached than the relative keys of classical harmony. The following diagram shows how each of the poles are related to their respective counter pole:
The counter pole tension becomes the most important and fundamental constructional principle in Bartók’s work.
In the axis system, the same function can be attained by a major second step in one direction or a minor second step in the other direction:
An extension to this rule is that by moving a major second, fourth, minor sixth, or major seventh interval in one direction, or by a minor second, major third, fifth, or minor seventh in the other direction, you will arrive at the same function in the axis system.
Relationship of Three Functions
Dividing the circle of fifths into three equal parts, for example, C-E-Ab, in the sense of the tonic, dominant, and subdominant, it turns out that the tonal system created by the equal division of the circle of fifths matches the model of the axis system. The following diagram represents this relationship:
The following represents the relationship between the pole and counter pole with respect to dimension:
where T = Tonic, D = Dominant, and S = Subdominant. Now that we have covered the basics of the axis system, the next step is to compose a piece of music. Lets start with a progression that looks like this: ii – V – I, where ii = supertonic V = dominant, and I = tonic. We will let I = C major, then V = G major, and ii = D minor. However, from what we have learned above, we can substitute G major for Db major (tritone substitution) and retain the same functionality. We now have a progression that looks like this: Dm – GM – CM, and along the way we will substitute the GM7 with DbM7, turning the progression into Dm – DbM – C. As can be observed, this create a downward movement of thirds and sevenths, which you will distinctly hear in the piece we have composed. We are also going to give this piece the minimalistic feel and with a little added flair of a retrograde-inversion (highlighted in green below) which is set in slightly before going into the final ii – V – I progression and tritone substitution (highlighted in blue below). The following represents our piece, appropriately named, Cartesian V:
Please enjoy listening to Cartesian V.