Axis system

The Axis System of functional harmony was first described by Ernő Lendvai in his analysis of Béla Bartok, a somewhat xenharmonic composer. Primarily organizing chords, it describes a new complete system of chord relations / substitutions for the full system of 12 edo, rather than a meantonic approach, such that notes separated by 1\2 are primarily connected and notes separated by 1\4 are secondarily connected. It is notable for its elegance, and its popular power in analyzing complex jazz (or punk) in situations where no other system seems to work.

The purest form of Axis harmony is immediately applicable to all multiple of 12 edos, simply with additional cycles. Most conservatively, Axis Harmony is applicable to all edos which share a 1\4 tetrad, by considering the functions of all the notes in that tetrad as the same rather than individually. All multiple of 2 edos as well will be able to symmetrize functions across the tritone. In addition, all multiple of 3 edos will be able to benefit from the secondary component of the theory, the division of functions into 3 types of contrasting effect, for harmonic use. Furthermore, the principles of Axis Harmony by drawing the cycles of relationships along the lines of the edos natural symmetries will expand easily to any composite edo. It has even been postulated that the equivalences based on the intervals of 1\4 and 1\3 function even without a closed loop to the octave, allowing for unlimited expansion to ji.

Definitions

Using 12edo as an example, this is the layout and method of the Axis configuration. The notes of the scale are divided into groups, each group consisting of notes apart from each-other by the "minor third" (1\4) and the "tritone" (1\2) (constituting 1\4 diminisht tetrads) and they call each group an Axis. 12 divided by 4 results 3, so there are 3 Axis in this system. For 12edo, the Axis have been named "tonic," "dominant" and "subdominant," for the I, IV, and V chords each reside on a unique Axis; yet the meaning and effect of these terms is not always transferable with other Harmonic theories using "the same" terms.

A dyad related by 1\2 is considered a Branch, thus in 12edo each Axis sustains 2 Branches. In regards to the tonic note (if there is one) the Branches are segregated as "Primary" and "Secondary" Branches, because the motion of 1\4 loosens the connection more than 1\2 does. Further down, each member of a Branch is called a Pole or Antipole, with the tonic always a Pole. On the pole side is found the "I" "IV" chord, the "V" chord. Expanding to Primary, are found the Tritone-Substitions of all chords, and the expansion to Secondary shows the rest of the notes. For edosystems featuring more factors, more terminology may be necessary such as an addition of "Leaf" in between Branch and Pole for 1\8 edos, or the coining of more and unique names for additional Axis.

Examples

To the right, is a picture showing 16edo in a unique way.

Due to the structuced nature of the splitting from 2-factor equivalences and the weakening of the effect at each link of the chain one could consider the functions in a circumstance like this as existing only on the tonic-axis, as with each equivalence the function and color of the note drifts away, leading to a functional continuum between a single Axis and two Axis separate in theory, because the shift fails to be detected at a single point in practice.

Example of harmonic color seemlessly changing through telephone of substitution

An example of Axis Notation overlaying a 16edo circle of fifths, with little pictures replacing note-name-letters. Treating the top-most snowflake as the Tonic, the bottom-most green note is the Tonic Antipole. Additional colours yellow and blue are the secondary Pole Set for this Leaf/Branch. Addional Pole Dyad red and purple show one possible Pole Set that would addionally be treated tonically if the Axis was expanded to include 1\8.