Axis system

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 submediant chord ... can replace the tonic, most familiarly in an interrupted cadence." == Terms and definitions ==

In his analysis, Lendvai groups the twelve tones of the chromatic scale into three sets; each set of tones contains those notes that are found at intervals of a minor third and tritone from one another (equivalent to the notes that make up the three possible diminished seventh chords). Thus, there are four notes in each of the three sets, sets which Lendvai 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, E/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". If 'C' is taken as tonic, then the categorisation proceeds as follows: : ==Substitutability==

The essential idea behind the axis system is that the axes group together "substituable key areas", Essential to Lendvai'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 Celesta 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). : 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 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. The tritone substitution, an harmonic device common in jazz chord progressions where a dominant V chord is substituted with a bII7 chord (or a secondary dominant II7 chord with a bVI7 chord, etc.), whose common justification is the enharmonicity of the tritones of both chords (G7 has a B-F tritone whereas D♭7 has an enharmonic Cb-F tritone, and both may resolve to C-E), and which is similar to the German sixth of classical harmony, corresponds to a special case of the classification of functions as established by the Axis system. The Backdoor progression is jazz terminology for the bvii chord functioning as a dominant. It relates to the axis system because bvii is on the dominant axis. == Further reading ==