Cadential sets, quantized modulations, PLRQ group, neo-Riemannian theory, seventh chords, jazz harmony, coslice category, Charlie Parker

1 Introduction↩︎

The problem of modulation between relative major and minor keys (such as C major and A minor) has occupied theorists since long ago. For example, Kirnberger [1] suggests a progression \(\mathrm{C}\to \mathrm{D} \to \mathrm{G} \to \mathrm{E}7 \to \mathrm{Am}\), while Rimsky-Korsakov [2] uses \(\mathrm{C} \to \mathrm{Dm}\to \mathrm{E}7 \to \mathrm{Am}\) to effect this modulation in a rather straightforward fashion. The principle seems to be rooted in the notion of standard cadence from the fifth degree towards the tonic.

Guerino Mazzola’s theory of global tonality [3], on the other hand, introduced minimal cadential sets or cadences: minimal collections of chords that uniquely identify a scale (or, more precisely, an interpretation of a scale), as the mechanism for asserting tonality. In order to define them, first let \(E=\{x_{0},\ldots,x_{6}\}\subseteq \mathbb{Z}_{12}\) be a major scale, ordered by the circular order induced by octave equivalence. Its triadic and tetradic interpretations are obtained by stacking generic thirds, namely: \[E^{(3)}=\{\{x_{i},x_{i+2\bmod{7}},x_{i+4\bmod{7}}\}:i\in\mathbb{Z}_{7}\},\] and \[E^{(4)}=\{\{x_{i},x_{i+2\bmod{7}},x_{i+4\bmod{7}},x_{i+6\bmod{7}}\}:i\in\mathbb{Z}_{7}\},\] respectively. The Roman numerals I, II, \(\ldots\), VII denote¹ the starting scale degrees \(x_{i}\).

For the major scale in the triadic interpretation, the cadences are: \[k_1 = \{\mathrm{II}, \mathrm{V}\}, k_2 = \{\mathrm{II}, \mathrm{III}\}, k_3 = \{\mathrm{III}, \mathrm{IV}\}, k_4 = \{\mathrm{IV}, \mathrm{V}\}, k_5 = \{\mathrm{VII}\}.\]

It is important to remark that, throughout this paper, the term cadence is used in Mazzola’s technical sense of a cadential set: a minimal chordal signature that identifies an \(n\)-adic interpretation, i.e., the set of diatonic \(n\)-ads over a scale. It is not used in the common-practice sense of an ordered closing syntagm. In any case, the present analysis is not incompatible with syntactic jazz-harmonic analyses like those of ii-V motion, blues form, or turnarounds. Hence, once all the members of a cadence in Mazzola’s sense have been displayed, the corresponding \(n\)-adic interpretation (i.e. the tonality in Mazzola’s terminology) is identified. This does not mean that centricity or formal closure has been established in the common-practice syntagmatic sense.

Mazzola developed a so-called quantum theory of modulation for the transit between one tonality and another, but it does not properly model the modulation between minor and major tonalities, particularly between a major tonality and its relative minor (for they are constructed over the same scale) thus rendering them as non-quantized. Also del Pozo and Gómez-Martin propose a modulation theory based on discrete optimization [4], but it is trivial for a major tonality and its relative minor, since their scales coincide.

In order to manage this modulation we resort to the PLR group², acting on the set of major and minor triads, for it provides morphisms connecting these cadential sets. In particular, the relative \(R\) satisfies \(R \circ T^5 = T^5 \circ R\) and \(R \circ T^7 = T^7 \circ R\), yielding the commutative diagram \[\label{E:RforTriads} \begin{tikzcd} \mathrm{F} & \mathrm{C} \arrow[l,swap,"T^{5}"] \arrow[r,"T^{7}"] & \mathrm{G} \\ \mathrm{Dm} \arrow[<->,u, "R"] & \mathrm{Am} \arrow[l, "T^{5}"] \arrow[r,swap,"T^{7}"] \arrow[<->,u, "R"] & \mathrm{Em} \arrow[<->,u, "R"] \end{tikzcd}\tag{1}\]

Thus \(R\) sends the cadence \(k_4 = \{\mathrm{IV}, \mathrm{V}\}\) to \(k_{2} = \{\mathrm{II}, \mathrm{III}\}\) (which lies within the relative minor), which is similar to Rimsky-Korsakov’s solution. The relative also transforms \(k_{1} = \{\mathrm{II}, \mathrm{V}\}\) into \(k_{3} = \{\mathrm{III}, \mathrm{IV}\}\) and back, but only as sets, for there is no commutative diagram that clarifies the relation.

This paper extends this framework to tetradic (seventh chord) harmony, where the relevant symmetry group is PLRQ [6], and discovers a conglomerate structure connecting six cadential sets.

2 Tetradic Cadential Sets↩︎

Regarding the tetradic interpretation of the major scale, the cadential sets are [7]: \[\begin{gather} J_1 = \{\mathrm{I}^7, \mathrm{II}^7\}, J_2 = \{\mathrm{I}^7, \mathrm{IV}^7\},\\ J_3 = \{\mathrm{II}^7, \mathrm{III}^7\}, J_4 = \{\mathrm{III}^7, \mathrm{IV}^7\},\\ J_5 = \{\mathrm{V}^7\}, J_6 = \{\mathrm{VII}^7\}. \end{gather}\]

Note that we include the superscript to indicate that these belong to the tetradic interpretation of the major scale.

3 The PLRQ Group↩︎

The PLRQ group [6] generalizes PLR to seventh chords. We will enlist the morphisms that we require from it, each with an illustrative example.

Define \[L_{42}: [\underline{x},x+4,x+7,x+11]\leftrightarrow [x+2,\underline{x+4},x+7,x+11].\]

Then, in Cannas’ notation we have \[T^{8}\circ L_{42} = P_{42},\] for \[\begin{align} T^{8}\circ L_{42}([\underline{x},x+4,x+7,x+11])&= T^{8}([\underline{x+4},x+7,x+11,x+2])\\ &=[\underline{x},x+3,x+7,x+10], \end{align}\] as claimed.

4 The Coslice Category Construction↩︎

We need to define the underlying category for the main constructions of the paper. Let \(\mathcal{X}\) be the set of dominant, minor, half-diminished, major, and diminished seventh chords in \(\mathbb{Z}/12\mathbb{Z}\). Let \(PLRQ\) be the PLRQ group of transformations of seventh chords as described by [6]; its generators exchange specified pairs of chord types and fix the remaining types. We set \[G=\langle PLRQ,T^{s}\mid s\in\mathbb{Z}_{12}\rangle,\] where \(T^s\) denotes transposition by \(s\). We take \(\mathcal{C}\) to be the action groupoid \(\mathcal{X}\mathbin{/\mkern-5.5mu/}G\). More explicitly, the objects of \(\mathcal{C}\) are the chords in \(\mathcal{X}\), and a morphism \(X\to Y\) is a pair \((g,X)\), where \(g\in G\) and \(g(X)=Y\). Composition is given by \[(h,g(X))\circ(g,X)=(hg,X)\] and, when no confusion is possible, we will write the morphism \((g,X)\) simply as \[g:X\to g(X).\]

Thus transformations such as \(R_{42}\), \(L_{13}\), \(P_{42}\), and the relevant transpositions determine morphisms between chord objects.

The pairs of cadential sets of interest to us have common members: \(J_1\) and \(J_2\) share \(I^7\), while \(J_3\) and \(J_4\) share \(III^7\). Hence, if we were to generalize 1 to tetrads, the difficulty is that the transformations connecting the non-common members do not generally stabilize these common members. For instance, \(R_{42}\) sends \(IV^7\) to \(II^7\), but it does not fix \(I^7\). Thus the relation between paired cadential sets is not best described as a transformation of two-element sets fixing their intersection. We resolve this using the coslice category construction.

Taking the common member of each paired cadential set as the distinguished object, we ask: do morphisms exist between the various seventh chords appearing in cadential sets?

This yields commutative triangles in \(\mathcal{X} \mathbin{/\mkern-5.5mu/}G\) representing morphisms in the relevant coslice categories: \[\begin{tikzcd} \mathrm{I}^7 \arrow[r, "T^5"] \arrow[dr, "R_{42}\circ T^5"'] & \mathrm{IV}^7 \arrow[d, "R_{42}"] \\ & \mathrm{II}^7 \end{tikzcd} \qquad \begin{tikzcd} \mathrm{III}^7 \arrow[r, "T^{10}"] \arrow[dr, "R_{42}\circ T^{10}"'] & \mathrm{II}^7 \arrow[d, "R_{42}"] \\ & \mathrm{IV}^7 \end{tikzcd}\]

In these coslice triangles, the object under which the coslice is taken is the common member of a pair of cadential sets. Thus the triangle with distinguished vertex I\(^{7}\) relates \(J_{1}=\{\text{I}^{7},\text{II}^{7}\}\) and \(J_{2}=\{\text{I}^{7},\text{IV}^{7}\}\). The outgoing arrows have the non-common members II\(^{7}\) and IV\(^{7}\) as codomains and \(R_{42}\) connects them. Similarly, the triangle with distinguished vertex III\(^{7}\) relates \(J_{3}=\{\text{III}^{7},\text{II}^{7}\}\) and \(J_{4}=\{\text{III}^{7},\text{IV}^{7}\}\). Thus the morphisms in Proposition 3 make explicit the relations between pairs of cadences by representing each cadence through the corresponding object of the appropriate coslice category.

5 The ABC Conglomerate Structure↩︎

Now we see the six cadential sets organize as follows.

6 Analysis: Blues for Alice↩︎

Charlie Parker’s Blues for Alice (1951) [8] manifests the prism structure explicitly. The chord progression is: \[\begin{gather} \boxed{\text{F6}} \mid \text{Em7 A7\flat9} \mid \text{Dm7} \boxed{\text{G7}} \mid \\ \text{Cm7} \boxed{\text{F7}} \mid \boxed{\text{B\flat7}} \mid \boxed{\text{B\flatm7}} \text{E\flat7} \mid \\ \boxed{\text{Am7}} \text{D7} \mid \text{A\flatm7 D\flat7} \mid \boxed{\text{Gm7}} \mid \\ \text{C7} \mid \boxed{\text{Am7}} \mid \text{Dm7} \mid \boxed{\text{Gm7}} \text{C7} \mid \end{gather}\]

Before we proceed to the harmonic analysis, it is important to remark that the tetradic interpretation and cadence model entail a rigidity that, nevertheless, allows us to understand the overall structure of the tonality phenomenon. Yet, when it comes to analysis and performance, and particularly in jazz, there is certain flexibility in the interpretation of the chords. Here, the piece opens with F6, but it is clearly signaling the departing tonic chord. As Mark Levine writes, for instance: “A C major chord can be notated as Cmaj7, CM7, C6, C\(\substack{9\\6}\), or C\(\Delta\), and they all mean pretty much the same thing” [9]. Teriete [10] states, which is quite to the point in our case: “›Lead Sheets‹ sind so spezifisch wie nötig, aber so abstrakt wie möglich.”³

For the purposes of the following analysis, “articulates” means that the lead-sheet harmony projects the corresponding root-degree skeleton of the cadential set, possibly through standard jazz procedures such as sixth chords, dominantization, modal mixture, or local ii–V syntax; literal diatonic tetrads will be identified when they occur.

Now we can see that the harmonic structure of Blues for Alice traces a path through the BA prism:

1. Measures 1–3 articulate \(J_1 = \{\mathrm{I}^7, \mathrm{II}^7\}\) with continuous melody; F6 realizes the tonic member, while the G-root sonority in the local ii-V chain projects the II degree in altered dominant form.

2. Measures 4-5 articulate \(J_2 = \{\mathrm{I}^7, \mathrm{IV}^7\}\) while the first melodic silence occurs; F7 and B\(\flat\)​7 project the skeletons of I and IV degrees, dominantized by blues syntax.

3. Measures 6-7 supply B\(\flat\)m7 as an altered IV degree and Am7 gives the literal III\(^{7}\), articulating \(J_{4}=\{\text{III}^{7},\text{IV}^{7}\}\).

4. Measures 9-12 display \(J_{3}=\{\text{II}^{7},\text{III}^{7}\}\) literally through Gm7 and Am7, embedded in the final turnaround.

The passage through the prism is not equally transparent at every stage. In the opening measures, \(J_{1}\) and \(J_{2}\) are articulated via lead-sheet realization, blues dominantization, and local harmonic syntax rather than by bare exhibition of the diatonic tetrads. The trajectory becomes progressively clearer as the piece approaches the Alice side of the prism: \(J_{4}\) is articulated with an explicit III\(^{7}\) and an altered sonority on degree IV, and \(J_{3}\) is finally literal with respect to the occurrence of its two members.

7 Analysis: Cherokee and Quantized Modulation↩︎

Ray Noble’s Cherokee (1938) [8] demonstrates two types of modulation.

7.1 Section A: Quantized modulations↩︎

The very opening measures feature a modulation from B\(\flat\) to E\(\flat\): \[\text{B\flatmaj7} \to \text{F}^{+}7 \to \text{Fm7} \to \text{B\flat7} \to \text{E\flatmaj7} \to \text{A\flat9} \to \text{B\flat6}\]

The tonality of B\(\flat\) is established with cadential set \(J_5 = \{\mathrm{V}^7\}\) (represented by F\(^+\)​7)⁴. The modulation to E\(\flat\) uses pivot chords \(\mathrm{II}^7\) (Fm7) and \(\mathrm{VII}^7\); here \(\mathrm{VII}_{\text{E}\flat}^{7}=\text{Dm}7\flat 5=\{\text{D},\text{F},\text{A}\flat,\text{C}\}\), which is a subset⁵ of A\(\flat\)​9\(\cup\)B\(\flat\)​6\(=\{\text{A}\flat,\text{C},\text{E}\flat,\text{G}\flat,\text{B}\flat,\text{D},\text{F},\text{G}\}\). This is quantized because full cadential sets establish each tonality with the corresponding pivots [7].

The modulation back from E\(\flat\) to B\(\flat\) is as follows: \[\text{Dm}7 \to \text{C}7 \to \text{Cm}7 \to \text{Dm}7 \to \text{G}7\flat 9 \to \text{Cm}7 \to \text{F}^{+}7 \to \text{B\flatmaj}7.\]

Here the pivots are \(\mathrm{III}^{7}\) (Dm7) and \(\mathrm{V}^{7}\) (represented by F\(^+\)​7); see [7]. It is remarkable that these correspond to the only quantized modulations available for two tonalities that are separated by a fifth (the only other case with a similar property corresponds to tonalities separated by one major third).

7.2 Bridge: Non-quantized modulations via \(P_{42}\)↩︎

Here we have the famous bridge \[\begin{gather} \text{C\sharpm7} \to \text{F\sharp7} \to \text{Bmaj7} \mid \\ \text{Bm7} \to \text{E7} \to \text{Amaj7} \mid \\ \text{Am7} \to \text{D7} \to \text{Gmaj7} \mid \cdots. \end{gather}\]

These modulations are not quantized, but the \(P_{42}\) morphism serves as a bridge \[P_{42}(\mathrm{I}^7_B) = \text{Bm7} = \mathrm{II}^7_A,\] for the set \(\{\mathrm{II}^{7},\mathrm{V}^{7}\}\) is cadential, but not minimal (i.e., \(J_{5}=\{\mathrm{V}^{7}\}\) suffices). Here the subscripts A and B denote the corresponding keys, not the regions of the conglomerate.

8 Comparison with Syntactic Approaches↩︎

Pachet [11] analyzed Blues for Alice using a syntactic/AI approach with ontological “shapes”. His system produces parse trees. In a subsequent work, Rousseaux and Pachet [12] acknowledge a fundamental limitation: they cannot capture the vécu (lived experience) of transitioning between régions de complexité.

Pachet’s analysis is useful precisely because it is successful at the syntactic level. His grammar parses a chord sequence by means of hierarchical objects such as TwoFive, ResolvingSeventh, ChordSubstitution, and BluesShape. Thus it explains how local harmonic objects are combined into a larger blues form. The present approach asks a complementary question: once such syntactic objects have been recognized, which cadential regions are being traversed, and which morphisms relate them? The contrast is summarized in Table 1.

This comparison should not be read as a rejection of Pachet’s syntactic analysis, but as an overlay of two perspectives. The syntactic layer describes sequential construction; the categorical layer describes paradigmatic relations between cadential sets. In this sense, the morphism \(R_{42}\) offers a formal correlate of the vécu of the transition between regions of complexity discussed by Rousseaux and Pachet. Musicians may describe this transition informally as “going to the relative”; the categorical apparatus identifies the corresponding morphism.

9 Stylistic Signatures in the Conglomerate↩︎

The conglomerate structure illuminates how different jazz styles inhabit different regions of cadential space. Each style can be characterized by which pairs of cadential sets it primarily exploits.

• Standard bebop blues: Primarily navigates within the Blues triangle (\(J_1 \leftrightarrow J_2\)).

• Parker’s innovations: Traverses the full prism, including the Alice triangle.

• Cherokee’s uniqueness: Exploits the C edge (\(J_5 \leftrightarrow J_6\)) for quick quantized modulations.

This explains Parker’s own description of his breakthrough while practicing Cherokee [13]:

“Now I’d been getting bored with the stereotyped changes that were being used at the time, and I kept thinking there’s bound to be something else. I could hear it sometimes but I couldn’t play it. Well, that night I was working over ‘Cherokee,’ and as I did I found that by using the higher intervals of a chord as a melody line and backing them with appropriately related changes, I could play the thing I’d been hearing. I came alive.”

In the present model, Parker’s description can be read as a discovery of harmonic paths that traverse the conglomerate beyond the conventional bebop-blues region.

10 Conclusion↩︎

The extension of Mazzola’s cadential set theory to tetradic harmony via the PLRQ group reveals a conglomerate that organizes the six cadential sets available. This structure:

1. Unifies the Mazzola (global tonality) and neo-Riemannian (local voice-leading) traditions through the coslice category construction.

2. Explains the phenomenological difference between cadential sets through categorical morphisms.

3. Models a formal correlate of what syntactic approaches leave implicit: the transformational experience musicians have when navigating harmonic space.

4. Illuminates Parker’s bebop innovations as an exploration of the full prism.

5. Distinguishes quantized modulation (full cadential establishment) from non-quantized modulation (\(R_{42}\) or \(P_{42}\) as direct bridges).

The naming convention: Alice (\(J_3 \leftrightarrow J_4\)), Blues (\(J_1 \leftrightarrow J_2\)), Cherokee (\(J_5 \leftrightarrow J_6\)), anchors the abstract structure in the jazz repertoire that first revealed it.

11 Source Code↩︎

The following code in Octave 9.2.0 (using the signal package) was used to calculate the mean absolute melodic intervals and the root-mean-square intervals of the opening segment (blues) and the continuation after the first rest (alice).

pkg load signal
blues=[17 12 9 16 12 9 14 16 11 14 13 10 7 9 5 2 7 9 5 4 3 7 10 14 13];
alice=[5 7 5 12 10 5 8 -2 7 15 13 10 8 12 5 7 9 4 0 2 13 11 3 10];
alice=[alice 8 7 17 14 10 2 9 7 12 10 15 12 9 4 7 14 10 2 9];
mean(abs(diff(blues)))
rms(diff(blues))
mean(abs(diff(alice)))
rms(diff(alice))

References↩︎