When studying musical analysis, traditional pedagogy often relies on intuitive, qualitative metaphors to describe how a piece moves over time. In Dougal Green’s foundational text Form in Tonal Music, terms like "shape," "curve," and "tension" are introduced to describe the surface contour of a composition.

While these heuristic models provide a fantastic mental picture, they lack mathematical precision. What does it actually mean for a phrase to have a "shape"? Can we define a musical curve rigorously enough to compute its rate of change?

In this post, we will summarize Green’s introductory concepts from my notebook, map out his baseline definitions, and then bridge the gap into pure mathematics—constructing a topological and metric framework using Fuxian ratios to calculate a true harmonic derivative.

Part 1: Green's Foundations — Form, Shape, and Genre

To understand how a piece organizes its materials, we first have to separate the overarching structural architecture into distinct components.

1. Key Definitions & Terminology

  • Genre: Categories of musical composition (e.g., Minuet, Symphony, Concerto).
  • Form: The dual organization of a composition’s design and its tonal structure.
    • Design: The organization of surface elements including melody, rhythm, cadences, timbre, texture, and tempo.
    • Tonal Structure: The harmonic progression and tonal centers governing the piece (e.g., $I \to V \to I$).
  • Phrase: The fundamental unit of music—a musical thought that comes to a point of relative repose, signaling "completeness." Major phrases typically conclude with a consonant triad.
  • Cadences & Caesuras:
    • Cadence: Chords bringing a phrase to a close (e.g., Authentic Cadence, Plagal Cadence).
    • Caesura: A light break in the flow of the music preceding or accompanying structural divisions.

2. The Qualitative "Musical Shape"

Green defines shape as the surface contour of a piece, depending heavily on the interaction between tension and relaxation. He visualizes this as an arched "curve": an accumulation of intension that rises to a climax and subsequently subsides.

According to Green, this shape is influenced by six primary factors:

  1. Rise and fall of melodic lines (especially in outer voices).
  2. Rhythmic activity (e.g., accelerating note values).
  3. Dynamics.
  4. Texture (density of voices).
  5. Instrumentation.
  6. Relative amount/degree of consonance and dissonance.

3. Scale Degrees and Extended Intervals

Harmonic tension is anchored by the natural hierarchy of scale degrees within a key. Mapped across their octave extensions, these are:

  • 1: Root / Tonic
  • 2: Supertonic ($9^{\text{th}}$)
  • 3: Mediant ($10^{\text{th}}$)
  • 4: Subdominant ($11^{\text{th}}$)
  • 5: Dominant ($12^{\text{th}}$)
  • 6: Submediant ($13^{\text{th}}$)
  • 7: Leading Tone / Subtonic ($14^{\text{th}}$ / $15^{\text{th}}$ framework)

Part 2: Rigorizing the "Curve" — Topology and Metrics

While Green uses a drawing of a curve as an intuitive diagram, we can convert this metaphor into a formal mathematical function by defining the underlying space on which music operates.

1. Fuxian Ratios as the Base Space ($X$)

To move beyond arbitrary note names, we turn to Johann Joseph Fux’s Gradus ad Parnassum. Fux establishes intervals via frequency proportions, categorizing them into:

  • Superparticular Ratios: $\frac{n+1}{n}$ (e.g., $2:1$ octave, $3:2$ fifth, $4:3$ fourth, $5:4$ major third).
  • Superpartient Ratios: $\frac{n+k}{n}$ where $k > 1$ (e.g., $5:3$ major sixth).

Let our space $X$ be the collection of chord states defined by these pitch ratios relative to a central tonic origin.

2. Defining Open Sets (Topology)

We define a topology $\tau$ on $X$ using open metric balls:

$$B_\epsilon(x) = \{y \in X \mid d(x, y) < \epsilon\}$$

An open set $U \in \tau$ represents a harmonic field or structural neighborhood (such as a local prolongational zone around the tonic). A continuous musical phrase is then modeled as a continuous mapping:

$$\gamma: [0, t_{\text{end}}] \to (X, \tau)$$

3. Constructing the Tension Metric ($d$)

To compute distance between the "null state" (the tonic triad identity $r_{\text{tonic}}$) and any active chord state $c$, we construct a metric combining logarithmic frequency distance and a Fuxian ratio complexity penalty:

$$T(t) = d(r_{\text{tonic}}, c) = \sum_{k} \left( \vert{}\log_2(r_k)\vert{} \times (p_k + q_k) \right)$$

Where $r_k$ is the frequency ratio of the $k$-th interval and $\frac{p_k}{q_k}$ represents its reduced fraction terms. Because pitch space is cyclical modulo the octave ($\log_2(2) = 1$), register shifts scale naturally without breaking harmonic equivalence classes.

Part 3: Computing the Harmonic Derivative (Case Study: Mozart's Minuet)

With a metric space $(X, d)$ established, we can define the instantaneous velocity of tension (the harmonic derivative) as:

$$\frac{dT}{dt} = \lim_{\Delta t \to 0} \frac{d(r_{\text{tonic}}, \gamma(t + \Delta t)) - d(r_{\text{tonic}}, \gamma(t))}{\Delta t}$$

Applying this calculus to Mozart’s Minuet from Don Giovanni (Ex. 1-1 in Green) highlights the power of this formalization:

1. Measures 1–8 ($\vert{}\vert{}: I \to V :\vert{}\vert{}$)

  • m. 1 ($t = 1$): Begins on the $G$ major triad—our "null" identity state ($T(1) = 0$). Velocity is flat ($\frac{dT}{dt} = 0$).
  • mm. 2–4 ($1 < t \le 4$): As the line moves away from the tonic root, metric distance increases smoothly ($\frac{dT}{dt} > 0$).
  • mm. 5–7 ($4 < t \le 7$): Secondary dominants introduce complex superpartient ratios. Tension accelerates ($\frac{d^2T}{dt^2} > 0$) toward the melodic zenith at m. 7.
  • m. 8: The phrase cadences on $V$. Velocity drops ($\frac{dT}{dt} < 0$), but because it lands on the dominant rather than $I$, $T(t)$ does not return to zero, mathematically preserving the "incomplete feeling" noted by Green.

2. Measures 9–16 ($\vert{}\vert{}: V_7 \to I :\vert{}\vert{}$)

  • m. 9: Phrase 2 resets on $V_7$, creating a discontinuous jump to a high baseline tension coordinates.
  • mm. 10–14: Sixteenth-note accompaniment replaces the previous quarter/eighth pulse. Because the rate of interval state changes per unit time ($\Delta t$) drastically increases, the magnitude of velocity ($\frac{dT}{dt}$) and acceleration ($\frac{d^2T}{dt^2}$) is vastly higher than in Phrase 1.
  • mm. 15–16: Climaxes at a higher zenith (m. 15) before executing a steep negative velocity drop ($\frac{dT}{dt} \ll 0$), collapsing metric distance all the way back to $0$ as it resolves to the final tonic cadence.

Conclusion

Green's qualitative description gave us the visual bracket—a curve that rises and falls. But by introducing a topological space, a metric derived from Fuxian superparticular ratios, and differential calculus, we transform that qualitative sketch into a quantitative graph.

We are no longer just saying music "feels more intense"; we can explicitly measure the velocity and acceleration of harmonic tension as a curve traverses a formal metric manifold.