Riemann and Helmholtz — Functional Theory and Acoustics

The Maturing of Harmonic Theory

Rameau (Article 4) gave chords a “root” and “the germ of function,” and equal temperament (Article 5) made all-key playing possible. In the 19th century, as Romantic music perfected expression with chromaticism and bold modulation, theory reached a mature period that organized it. Two giants recaptured music from two directions — Riemann (structure) and Helmholtz (physics).

Riemann — the Completion of Functional Harmony

Hugo Riemann (1849-1919) — a German musicologist. He shaped Rameau’s “germ of function” into a lucid system that reduces all chords to three functions:

• T (Tonic) — stability, point of arrival (the tonic chord I)
• S (Subdominant) — departure, preparation (the subdominant chord IV)
• D (Dominant) — tension, wants to resolve to T (the dominant chord V)

The basic progression is T → S → D → T (e.g., I-IV-V-I). Riemann explained even complex chords as “a substitute for some function (a substitute chord).” For example, the VI chord is a substitute for T, the II chord for S, and so on.

This idea of function is the direct ancestor of modern harmony teaching (Roman-numeral analysis, understanding chord progressions). The analysis “this chord is the nth degree in what key, in what function” is Riemann’s legacy. The understanding of “two-five (II-V)” in jazz and pop is on this same line.

Helmholtz — Explaining Consonance with Physics

Hermann von Helmholtz (1821-1894) — a physicist and physiologist. He brought science (acoustics and physiology) into music theory. His main work, On the Sensations of Tone (1863):

• sound is composed of a fundamental + overtones
• consonance and dissonance are determined by how much “beating” the overtones of two tones produce — if overtones align well, consonance (smooth); if they are close and interfere, dissonance (rough)
• he backed Pythagoras’s “simple integer ratio = consonance” with the physics of overtones

That is, Helmholtz integrated the “number vs. ear” opposition that had run since Article 1. He explained “why simple ratios sound pleasant” from both the physical (alignment of overtones) and the physiological (the workings of the ear). Number (ratio) was right — but that it “sounds beautiful” is due to the mechanisms of ear and brain. The birth of psychoacoustics.

The Character of the Two Theories

Riemann and Helmholtz answered different questions:

	Riemann	Helmholtz
Question	how do chords progress	why do chords sound consonant
Method	analysis of musical structure	experiments in physics/physiology
Legacy	functional harmony, Roman-numeral analysis	psychoacoustics, acoustic engineering

The former is “the grammar of music,” the latter “the science of sound.” Both matured music theory in the 19th century, each in its own domain.

Relation to Romantic Music

This theoretical maturing ran alongside the practice of Romantic music. The chromatic harmony represented by the “Tristan chord” of Wagner’s Tristan und Isolde stretched functional harmony to its limit, weakening “the gravity of tonality.” Riemann’s functional theory was both a tool for organizing the Romantics’ complex harmony and a foreshadowing of the collapse of tonality beyond it (Article 8).

“The gravity of tonality (the force wanting to return to the tonic)” was exploited to the fullest, and stretched to the limit — the end of the 19th century was both the apogee of functional harmony and the prelude to its dissolution.

From This Article to the Next

We have traced the 19th century — Riemann’s functional theory and Helmholtz’s acoustics. In the 20th century, theory unfolds greatly in two directions: “how to analyze works” and “how to overcome tonality.” The next article turns to Schenker, who established an analysis that reduces a work to a few background structures (Ursatz) — a new paradigm of analysis, “a masterpiece is the ornamentation of a skeleton.”