In “classical” music theory, the minor triad has always been considered subordinate to the prevailing, and more “natural”, major triad (both for major and minor tonalities).
Harmonic dualism is a principle that ascribes equal importance to the major and minor triads approached from dialectical and acoustical point of view, i.e. in terms of undertones and overtones, mediation of contrasting forces and physiological-structural understanding (following Oettingen’s (1866), Hauptmann’s (1853), and Riemann’s (1905 and 1915) studies). However, these studies do not seem to convincingly justify the existence of the minor triad; all attempts to trace the foundation of the minor triads in Oettingen’s series of undertones appears equally weak. Being the minor triad — and tonality — an indisputable piece of evidence, the David Lewin’s A Formal Theory of Generalized Tonal Functions (1982) is the only one able to demonstrate the structural-formal explanation of harmonic dualism in terms of inversion of the same intervals which belong to the major triad.
Considering that the tonal praxis is likely to come from transposition of psalm-tones in the instrumental music of the XVII century (following Barnett’s (1998), and Powers’s (2014) papers), we seek the ontological basis of harmonic dualism in the praxis of temperaments; depending on the higher or lower consonance of the two triads in meantone temperaments (for simplicity 1/3 and 1/4 of syntonic comma (s.c.) meantone temperaments), we can trace back a double dualism, in which the third in question is ontologically prevalent, and the complementary one is its dual. However, temperaments do not allow modulations without a transposition to distant keys. Conversely, Lewin’s model allows generating “dual” triads by mathematical manipulation of their constituents — i.e., inversion (“TDINV” operation) and/or conjugation (“CONJ” operation). By looking at the modal diatonic sets on which psalmodic-tones are based (the natural notes with added B ) and by arranging the pitches of these tones as thirds, one can obtain two palindromes which represent all the triadic possibilities for these modal sets, which confirms the structure theorized by Lewin (which is a palindrome as well).
Our approach contradicts the theories of “classical” harmonic dualism, which focuses only on the single constituents of the triads (regarding the topics cited above). Moreover, such approaches seek to identify how the model could be adapted to the temperament independently, i. e. without including the temperament itself into the model. All these aspects fail to provide a convincing explanation of the harmonic dualism, that wasn’t achieved until Lewin’s formulation. By considering chords as autonomous entities, different from the ordinary superposition of scalar intervals, Lewin provided for the first time a formally convincing explanation based on the equal temperament, i.e. without making comparable consonance and “naturalness” of the major and minor triad a consequence of a specific intonation (following Lewin’s 1982 paper).
My paper aims at highlighting the relevance of a historically informed genealogy of the harmonic dualism, in relation with the musical practices that led to the emergence of tonality through the use of temperaments and their interval structure.
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