Achttaktigkeit. Analogous to his idealized notions of harmonic progression, Riemann posits an idealized metric configuration comprising eight-measure units.
Auftaktigkeit. Refers to Riemann's supposition of a universal iambic configuration in the metric domain of musical phrasing. Even where it is not notated, Riemann understands musical phrases to begin with an implicit upbeat.
Cadence. Riemann posited two idealized successions of functions: in major, T–S–D–T; in minor, ˚T–˚D–˚S–˚T (see Dualism).
Dualism. At a most basic level, dualism reflects the structural equality of major and minor systems: in a dualistic framework, the minor triad is recognized as the mirror-symmetric image of the major. The concept became problematic in the nineteenth century when theorists sought acoustical justification for dualism. For example, Riemann sought to locate the source of minor triadic generation in a supposed undertone series, the mirror-symmetric dual of the overtone series, which historically was considered the source of generation for the major triad. Even though Riemann's specific explanation for dualism changed over the course of his writings, dualism remained an essential and foundational component of his harmonic thought.
Function. Riemann distinguishes three functions: tonic (T), dominant (D), and subdominant (S). However, it remains unclear whether Riemann construed functions as chords or categories. Understood as chords, each function is manifest by primary forms (I, V, and IV) and modified forms (see Scheinkonsonanz). Understood as categories, the functions reify stages in idealized harmonic successions (see Cadence).
Harmonieschritte. Riemann's dualist system to categorize particular harmonic successions between two chords according to the directed intervals between chordal Haupttöne (see Hauptton). Riemann divided Harmonieschritte into Schritte that relate like-mode triads, and Wechsel that relate mode-opposed triads. Riemann further modifies Schritte and Wechsel according to the direction of the interval between Haupttöne relative to their mode: Schlichte (simple) relations progress in the direction of generation of chord components relative to the Hauptton of the first (outgoing) chord, Gegen (contrary) relations progress in the direction opposite to that of chord components. For example, the schlichter Quintschritt relates any major triad to the major triad a perfect fifth higher (e.g., c+–g+, the directed relation from C, the Hauptton of the C-major triad, upward to G, the Hauptton of the G-major triad) or any minor triad to the minor triad a perfect fifth below (e.g., ˚e–˚a, the directed (p. 580) relation from E, the Hauptton of an A-minor triad, to A, the Hauptton of a D-minor triad). For Riemann, “schlicht” is an unmarked term, understood when generally omitted: “Schlichter Quintschritt” is more simply referred to as “Quintschritt.” The Gegenquintschritt relates any major triad to the major triad a perfect fifth lower (e.g., c+–f+) or any minor triad to the minor triad a perfect fifth higher (e.g., ˚a–˚e). The (schlichter) Terzwechsel relates any major triad to the minor triad whose Hauptton lies a major third higher (e.g., c+–˚e) or any minor triad to the major triad whose Hauptton lies a major third lower (e.g., ˚e–c+). The Gegenterzwechsel relates any major triad to the minor triad whose Hauptton lies a major third lower (e.g., c+–˚af) or any minor triad to the major triad whose Hauptton lies a major third higher (e.g., ˚a♭–c+).
Hauptton (principal or referential tone). In his dualist harmonic conception, Riemann posits a Hauptton, distinct from a triad root, specifically that tone in a triad to which the other tones refer as major third and perfect fifth. In major, the Hauptton corresponds to the triadic root (C in a C-major triad); in minor the Hauptton corresponds to the triadic fifth (E in an A-minor triad) (see Klangschlüssel, Dualism, Klang).
Klang. Klang is multifaceted concept in Riemann's writings. At the most general level, Klang refers to the acoustical signal of a sounding body. Riemann, however, posits a more specific meaning, distinct from, but not always clearly differentiated from, the triad. In line with his dualistic conception, a Klang is an abstraction comprising a Hauptton surrounded by its upper and lower triadic components (major third and perfect fifth). A consonant, sounding Klang (i.e., a triad), however, manifests only one side of the idealized abstraction (Hauptton and upper components in major, Hauptton and lower components in minor).
Klangschlüssel. Riemann's system of notation, in which triads are labeled according to their Haupttöne, with symbols affixed to indicate mode. For example, c+ represents a C-major triad (Hauptton C with affixed “+” for major); ˚e represents A-minor (Hauptton E with affixed “˚” for minor). Modifications to any triad can be indicated with numerals and symbols affixed to the letter names. Arabic numerals represent tones at intervals above a major-mode Hauptton; Roman numerals represent tones at intervals below a minor Hauptton. Thus, c6 indicates a C-major triad with added sixth (C–E–G–A); aVII indicates a d-minor triad with added under-seventh (B–D–F–A); d5〉 indicates a D-major triad with a chromatically lowered fifth (A♭–D–Fs), and so on.
Klangvertretung. Refers to the ability of individual tones and of dyads to project triadic and functional identities. For example, the tone C can assume the identity of the root of a C-major or C-minor triad, the third of an A♭-major or A-minor triad, or fifth of an F-major or F-minor triad. Similarly, an A–C dyad can represent either the upper third of an F-major triad or the lower third of an A-minor triad.
Scheinkonsonanz (apparent consonance). A collection that appears to be a triad but that for Riemann represents an elliptical manifestation of a dissonant collection. (p. 581) For example, Riemann could construe an A-minor triad in the context of C major either as an elision of a dissonant configuration, F–A–C–E, in which the Hauptton F is suppressed and replaced by the E, or the dissonant configuration C–E–G–A, in which the chordal fifth, G, is suppressed and replaced by the A. In Riemann's function theory (see Function), apparent consonances form the basis for functional substitution. In the given example, the A-minor triad as ellipsis of F–A–C–E would be construed as a subdominant Leittonwechselklang (i.e., an apparent triad that results from the substitution of the triadic Hauptton by the leading tone); alternately, the A-minor triad as ellipsis of C–E–G–A, would be construed as a tonic Parallelklang (i.e., an apparent triad that results from substitution of the triadic fifth with sixth).
Tonvorstellung. The imagination or mental representation of a tone. Tonvorstellung became a central concept in Riemann's late writing when he located the foundation of tonal relations within psychology. This relieved him of the obligation to argue from an acoustical perspective.
Transformation. As a contemporary music-theoretical concept, transformation has a twofold meaning. As a general term, transformation refers to a perspective that places emphasis on the relationships between musical objects, rather than upon the objects themselves (in contrast to a set-theoretical perspective). More specifically, transformations refer to mathematical functions that map elements of a set to themselves in a manner that is both one-to-one and onto. Musically, the set may contain chords, tones, keys, and so on.
Voice-leading parsimony. Refers to relations between triads or other pitch collections—in the literal or underlying voice-leading—that minimize moving voices and voice-leading distances between them. For example, a C-major harmony can be voice-led parsimoniously to an E-minor harmony: the tones E and G can remain fixed and C can move by semitone to B to complete the transformation. The Tristan progression is an example of a parsimonious tetrachordal relationship: B and Gs remain fixed; F and Ds move by semitone to E and D♮, respectively.