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Lewin used mathematics to abstract 2 models: the Generalized Interval System (GIS) andthe Transformational network. The former was supposed to define a set of musical objects,generalize intervals of pitch and time, and infer mathematical functions that mapped all Page 6 of 16possible pairing of musical objects. “Transformations” were supposed to model possibleactions upon the musical objects. Such modeling might be useful for automatic computer-based analysis of music and for the design of psychoacoustic experiments, but it hasnothing to do with the pragmasis of musical communication. This is not that different fromthe notorious theory of generative grammar by Chomsky, which Chomsky himself had topronounce wrong, but which nevertheless proved to be very useful for computerized studyof languages. The dichotomy between the nomothetic math-based prescriptive and empirical psychology-based descriptive approaches to music theory dates back to Classic Antiquity, to thefamous criticism of Pythagorianism by Aristoxenus (the Aristotle's pupil). A brilliant old-timeAmerican music theorist and ethnomusicologist, Norman Cazden, succinctly summarizedtheir arguments in his essay: Cazden, Norman. 1958. “Pythagoras and Aristoxenos Reconciled.” Journal of theAmerican Musicological Society 11 (2/3): 97–105.https://doi.org/10/gmnfj5.Riemann's main idea was that chords that emerge from the counterpoint of multiple partsend up forming a finite number of harmonic "classes". These classes are consistentlyreproduced in different keys and in different compositions by different composers in avariety of implementations (e.g., inversions) and can be identified by their harmonicfunctions - i.e., relative fluctuations in harmonic tension and relaxation, which determinedthe positioning of these classes within a musical form. Unlike the Neo-Riemannian"transformations", these functions reflected the conventions of musical communicationbetween composers, performers, and listeners - the common practice of engaging specificharmonies to initiate, terminate, and culminate musical phrases and sentences. Importantly, Riemann made a few critical errors in defining the functional characteristics ofminor keys and what he called iambic meters. However, these shortcomings werecorrected by his followers - especially in Russia, where the "Golden Age" of classical musicpromoted the boom of music theory, and the widespread of modal folk music, very differentfrom Western music, directed music theorists towards greater attention and deeperunderstanding of principles of tonal organization that guide creators of very diverse musiccultures. As a result, Riemann's theory of functional harmony merged with the theory ofmusical mode. In contrast, in the West, modal theory was largely abandoned after the 17th century. Thedifference between musical mode and musical scale (a principal structural unit of theSchenkerian and Neo-Riemannian theorists) is that mode incorporates not only a set ofpitch classes but also the functional relations between them as revealed in typicalprogressions of harmonies within a given musical composition. From this point of view, notall compositions nominally created in the same key share the same functionalcharacteristics. A C-Major prelude from volume I of the Well-Tempered Clavier by Bach, C-Major "Simple" sonata by Mozart, “Locus iste” by Bruckner, Bolero by Ravel, "DoctorGradus at Parnassum" by Debussy, Valse des fleurs by Stravinsky, and "Juliet - the girl",No.10 from Prokofiev's ballet "Romeo and Juliet" - all these compositions noticeably differ Page 7 of 16No.10 from Prokofiev's ballet "Romeo and Juliet" - all these compositions noticeably differin the use of harmonic progressions and therefore present different modal implementationsof the same key of C Major. Neo-Riemannian analysis fails to capture their differences(that are quite obvious from mere comparative listening to these pieces), while RussianRiemannian-based analysis easily spells out their specificities.