Towards a Music Algebra: Fundamental Harmonic Substitutions in Jazz

In this paper the most common harmonic substitutions, at least as far as jazz music is concerned, are unconventionally addressed. The novelty consists in introducing a new method finalized to formally defining and logically applyingall the fundamental harmonic substitutions, by exploiting anunusually rigorous notation. After defining the substitutions and discussing their applicability, we resort to them in order to modify some simple harmonic progressions substantially based upon a banal major turnaround. As explicitly suggested by the title, the modifications are carried out by following an extremely formal line of reasoning: all the logic passages are accurately described by resorting to a notation so similar to the one commonly employed in mathematics and physics, that the harmonic analysis of a song turns out to be de facto comparable to the demonstration of a theorem.


I. INTRODUCTION
In this paper, all the fundamental harmonic substitutions [1] [2] are discussed by resorting to a notation, generally employed in fields such as mathematics and physics, extremely formal and rigorous. We herein exclusively refer to a single harmonization, the Ionian one: consequently, for the sake of clarity, we reveal in advance that the socalled "Modal Interchange" is not addressed in this article. Obviously, the line of reasoning we exploit can be equally followed starting from other scales, such as the Natural Minor Scale (that can be obtained from the Ionian Scale by means of a banal translation), the Ipoionian Scale (Bach Minor Scale), the Harmonic Scales (Major and Minor).

II.
DIATONIC SUBSTITUTIONS Two chords that arise from the harmonization of the same scale are interchangeable if the distance between them (between the roots) is equal to a diatonic third (both ascending and descending). [1] [2] Exclusively referring to the Ionian Harmonization, the Diatonic Substitutions in From (28) we immediately deduce that the dominant seventh chord obtained from the fourth degree (F7, in the specific case) leads towards a note, to extremely simplify, that does not belong to the C Ionian Scale (Bb, in the specific case). Consequently, the Secondary Dominant that arises from the forth degree is to be regarded as nonfunctioning (or non-functional).

IV. TRITONE SUBSTITUTION
If we set, for example, X = C, from the foregoing relations we immediately obtain the following: V. DIMINISHED SUBSTITUTION Any Dominant Seventh Chord, especially if it is provided with the flat ninth and even if it arises from a previous harmonic substitution, can be replaced by a Diminished Chord distant a major third, a perfect fifth, a minor seventh or a flat ninth from the initial chord.
All the above-mentioned intervals are implicitly regarded as ascending. In other terms, we can concisely write, with obvious meaning of the notation, as follows: More explicitly, we have: By setting, for example, X = C, from (37) we obtain:

VI. EXPANSION
The Dominant 9sus4 and b9sus4 Chords can be expressed by resorting to the so-called Slash Chords. In very formal terms, with obvious meaning of the notation, we can write: By virtue of (39) and (40), we can state that, in a certain measure, any Dominant Seventh Chord can be imagined as being preceded by a Minor Seventh Chord or a Half-Diminished Chord distant a descending perfect fourth. [3] Consequently, employing a vertical line to separate two consecutive beats or bars, we have: If we set, for example, X = C, from (39) and (40) we obtain: Coherently with the setting (X = C), from (41) we have:

Case 1: Rhythm Changes (Bridge)
Let's consider the following simple harmonic progression: Harmonic Progression 1.1 If we set I = B b , we immediately obtain: Harmonic Progression 1.2 Taking into account the previous progression, let's carry out the following substitutions:  46) and (47), we finally obtain: Harmonic Progression 1.3

Case 2: I'm Getting Sentimental Over You (first 8 bars)
Let's now consider the following harmonic progression:

Harmonic Progression 3.3
The chord obtained in the eight bar (A b 7) leads towards B b maj7 (ninth bar, not displayed in the Harmonic Progression 3.3). The progression b VII7 | Imaj7 is commonly named "Back-Door Solution". [5]

Case 4: Easy Living (first 8 bars)
Let's consider the following banal harmonic progression: Harmonic Progression 4.1 If we set I = A b , from the foregoing progression we obtain:

Harmonic Progression 4.2
Taking into account the previous harmonic progression, let's carry out the following substitutions: The second chord in the fourth bar (G b 7) leads towards A b maj7: once again, a typical "Back-Door Solution".

Case 5: Giant Steps
Let's now consider the following harmonic progression: Harmonic Progression 5.1 By setting I = B, we immediately obtain:

Harmonic Progression 5.2
Taking into account the previous progression, let's carry out the following substitutions: Taking into account (74) and (75) we finally obtain: Harmonic Progression 5.4

VIII. FINAL REMARKS
Let's start from a banal sequence of four bars (or beats, or sets composed of an equal number of beats) characterized by the same chord (in our case, a Major Seventh Chord): Imaj7 Imaj7 Imaj7 Imaj7 Harmonic Progression 6.1 Although it cannot be formally regarded as a harmonic substitution, we are clearly allowed to resort to the socalled "Tonicization" [ It is evident that, starting from a single Major Seventh Chord, we can easily obtain a Major Turnaround. In other terms, we are implicitly stating that all the harmonic progressions we have obtained in the previous section may be imagined as arising from a single chord. In a certain sense, we could even state that, net of some progressions explicitly based upon the so-called "Pure Plagal Cadence" (IVmaj7 | Imaj7), all the Popular Jazz Songs, as far as the harmonic structure is concerned, may be considered as originating from a single chord (a Major Seventh Chord or, exploiting the harmonization of whatever minor scale, a Minor Seventh). It is worth underlining that we have explicitly referred to the "Pure Plagal Cadence", since the "Authentic" or "Extended" one (IVmaj7 | V7 | Imaj7) can be immediately deduced, by simply resorting to a banal Diatonic Substitution, from a IIm7 | V7 | Imaj7 harmonic progression which, in turn, can be obtained from a single Major Seventh Chord by exploiting a "Tonicization" and, taking into account (76), an Expansion.