Music Algebra: Harmonic Progressions Analysis and CAT (Cataldo Advanced Transformations)

In this article we formally introduce an original method, the purpose of which fundamentally lies in providing musicians with a reliable instrument that may effectively assist them in carrying out, as simply and intuitively as possible, the analysis of whatever chord progression, without resorting to the so-called "modal interchange". Net of a single exception (a routine named "structure reduction"), the whole method is based on a series of harmonic transformations. The above-mentioned transformations, named CAT (the acronym stands for Cataldo Advanced Transformations), turn out to be nothing but inverse chord substitutions characterized by specific conditions and restrictions. The method arises from the analysis of a considerable number of chord progressions, devoting particular (although not exclusive) attention to traditional jazz compositions: in this regard, it is worth highlighting how a significant improvement of CAT has been achieved by conducting an extremely thorough analysis of the so-called LEGO Bricks (public domain harmonic patterns).


I. SHORT INTRODUCTION
The purpose of the method fundamentally lies in providing musicians with a reliable instrument that may effectively assist them in carrying out the harmonic progressions analysis. The method is primarily based upon the application, carried out by following a specific order, of a series of transformations, named CAT (Cataldo Advanced Transformations), by means of which whatever harmonic progression may be converted, within certain limits, into a mere sequence of Plagal and Perfect Cadences [1]. As far as jazz is concerned, a significant improvement of the method has been achieved by conducting an extremely thorough analysis of the so called LEGO Bricks (public domain harmonic patterns) [2] [3].

II.
LIMITATIONS OF THE METHOD The method is characterized by the following limitations: The Key of any song must be considered as being major. Consequently, if the key of a song is manifestly minor, the analysis must be carried out by referring to the relative major key (for example, C Major instead of A Minor). It is worth specifying how a direct analysis of the songs written in minor key is obviously feasible: however, the procedure would require slight modifications concerning the conditions related to some transformations, herein not addressed in order not to weigh down the discussion.
Each Minor Major Seventh chord must be instantly replaced by a Minor Seventh one; similarly, each Augmented Major Seventh Chord must be instantly replaced by a Major Seventh. In other terms, the analysis is carried out by taking into consideration, exclusively, the first five kinds of Seventh Chords.
In the light of their extreme subjectivity, the (inverse) substitutions based on the so-called "Modal Interchange" are herein intentionally ignored. In fact, the outcomes usually obtained by resorting to the modal interchange can be alternatively deduced by exploiting the Quality (Dominant to Major) and Similitude Substitutions. [1] The time signature must always be imagined as being equal to 4/4. For example, even if we deal with a 3/4, we have to consider four pulses per measure (four beats per bar): each beat, in this case, will be characterized by a duration equivalent to a dotted quaver (see fig.1)

III.
DESCRIPTION OF THE METHOD The method consists of ten consecutive phases: 1. Selection of the Key (bearing in mind that the global tonal centre is herein regarded as necessarily major). For example, if we set X = C, we can banally write: For example, by setting X = C, we obtain: 4. Structure Reduction (net of which a correct application of CAT would be de facto impossible). Very simply, the number of bars, as well as the duration of the chords, must be iteratively halved. The procedure is stopped the moment in which even a single chord characterized by a duration equal to a beat appears. Actually, the structure reduction should be applied every time it is possible, so as to obtain the highest simplification level.
In order to explain how to interpret the notation we have been resorting to, the last relation is equivalent to the following assertion: if a Diminished Chord, denoted by an, is followed by a Dominant Seventh Chord, denoted by Z7, and if an, concurrently, belongs to the set of the Diminished Chords that can be obtained by applying a Diminished Substitution to the Dominant Seventh Chord distant an ascending perfect fifth from Z7, an must be replaced exactly by this chord (an must be regarded as deriving from a Diminished Substitution applied exactly to this chord).

Diminished Chords followed by Minor Seventh Chords
According to CAT, with obvious meaning of the notation, we have to consider the following transformations:

Diminished Chords followed by Half-Diminished Chords
Albeit the case has never occurred during the analysis of more than 300 jazz harmonic progressions, we admit the possibility that a Diminished Chord may be followed by a Half-Diminished one. If this happens, the Diminished Chord cannot be immediately replaced by a Dominant Seventh: in this case, in fact, we have to necessarily wait for the Half-Diminished Chord to be subjected to an inverse substitution, so returning the analysis to one of the cases previously considered. [1]

Minor Seventh Chords and Half-Diminished Chords deriving from Expansion Substitutions
According to CAT, denoting with Y a generic note, with bark the k-th bar, with T(chord) and beat(chord), respectively, the duration and the metric placement of the chord in round brackets, we have: