It is undisputed that the practical application of Just Intonation (JI) in tonal music is merely a fiction. However, JI as a theoretical concept may not be as futile as widely assumed. The first part of this paper is a historical survey, highlighting the unique perspectives on JI by lesser-known theorists Sigfrid Karg-Elert (1930) and Christoph Hohlfeld (1989). My paper focuses on their concept of "comma-free” modulation, which introduces a syntonic comma by modulating along the fifths axis, followed by a reciprocal modulation carefully "undoing" this comma.

Further developing this concept and extending it to the fifths and sevenths axes, the second part of my paper will focus on three core examples: Beethoven’s Sonata No. 20, John Coltrane’s “Giant Steps,” and Marcos Valle’s "Samba de Verão." By rendering these in strict JI and observing their Tonnetz implementation (presented as video clips), it becomes apparent that, in specific scenarios, JI may provide insights into the harmonic structures. In other words, this suggests that comma differences theoretically arising in JI may carry musical meaning. Even though the implied significance of JI would still be far from what JI apologists of the past claimed (e.g., Oettingen 1913, Vogel 1976), this perspective challenges the widely accepted ontology of today’s harmonic theories, which are either exclusively conceptualized in EQT pitch space or completely ignore aspects of the “acoustic exterior” (see Dahlhaus 1982).

In conclusion, I will briefly discuss how the given examples challenge the geometric interpretation of pitch space as a rigid geometric object, as is common in neo-Riemannian theories. As an alternative, I suggest the further exploration of elastic objects that can be dynamically permuted to be rather straight or circular to fit each passage of a composition.

In “A New Equidistant 12-Tone Temperament” (1959), the ethnomusicologist, composer, and music theorist Mieczyslaw Kolinski described a novel approach to piano tuning grounded in equal sevenfold division of the 3:2 just fifth. In support of this scheme, which employed semitones of ~100.28 cents each and therefore required that octaves be tuned slightly larger than 2:1, Kolinski invoked the deliberate use of "streched" octaves by piano tuners. But he also cited laboratory studies conducted by the psychologists and comparative musicologists Carl Stumpf, Max Meyer, Otto Abraham, and Erich von Hornbostel, who had concluded that experimental subjects preferred perfect fifths and octaves tuned slightly larger than pure.

While Kolinski’s proposal was based on a misunderstanding of the physics of non-ideal struck strings, it offers a fruitful starting point from which to examine a set of proposals made by Stumpf, Hornbostel, and their colleagues. These thinkers, whose work strongly influenced Kolinski's own, either qualified or rejected the familiar privileged role accorded to small-integer ratios and their approximations, asserting that appropriateness to melodic movement, rather than either just intonation or equal temperament, dictated the intonation used in unaccompanied singing, even in supposedly scale-bound European music (Abraham 1923); that scales or melodies might be structured by preference for equidistance between scale steps within the octave (Stumpf 1901) or by psychological universals governing the ratios between frequency ratios in melodic steps (Hornbostel 1927); and that appropriately trained research subjects could experience familiar interval qualities even if melodies were mapped onto a “micro-system” that reduced the size of all intervals to a quarter of their normal size (Werner 1926).

Considering these claims in conjunction reveals three important aspects of the Berlin School’s music-theoretical orientation. First, its laboratory practices constituted musical subjects that were intended to displace music theory’s received ideal listener. Second, its skepticism toward theories of pitch perception grounded solely in frequency ratios did not preclude a search for alternative musical universals. Third, its reduction of pitched sound to perceived fundamental frequencies at the expense of attention to the physical properties of instruments resulted in inconsistent and unworkable theoretical claims.

Solfège set theory extends previous transformational theories (Hook 2008, Rings 2011, Lam 2020) and leverages the complementary qualities of solfège and set theory to analyze recent diatonic-modal music. It shows that fixed-do, la-minor (re-dorian), and do-minor solfège map syllables to three independent dimensions of a tone (pitch, position, degree). The unification of solfège systems provides a framework for analysis and a ground truth for genuine dialogue on solfège pedagogy.

There are two orders of transpositions in solfège set theory. Movable solfège systems are first-order transpositions with intervals derived from the key or mode, but they are transpositions of syllables, not pitch. Second-order or unified solfège transpositions embed first-order transpositions and can be seen as a combination of Hook’s signature (or heptachordal) transformations and Rings’s formulation of scale degrees. The resultant three-dimensional structure is rich and complex. Unified solfège transpositions codify all permutations of solfège invariance and embed modulations. In doing so, they reaffirm the independence of the three solfège systems and redirect tactics in polarizing solfège debates to constructive use. I will focus primarily on three pivot transpositions (parallel inflection, Schubert inflection, and diatonic reinterpretation) in music by Takashi Yoshimatsu, John Williams, and others.

This study delves into the evolution of key behaviors in tonal music, charting a course from the 16th century, where keys possessed distinct identities and absolute-pitch associations, to the 20th century, where the concept of transpositional equivalence among scales became predominant. Through a blend of corpus, historical analyses, and close readings, the paper shows the shifting paradigms across European art music and popular genres. The research underscores the traditional pedagogical stance that views all major keys as interchangeable, a perspective substantiated by the uniform scale-degree distributions in 20th-century popular music as per the McGill-Billboard corpus. However, this uniformity contrasts starkly with the 16th-century Western European art music landscape, where key profiles were distinctly non-equivalent due to the modal scales' asymmetric alignment with the Renaissance gamut.

The paper tracks the gradual standardization of scales, highlighting a period around 1700 as a transitional phase wherein the twelve major keys were employed by composers, yet exhibited significant variance. This nuanced transition, marked by a gradual alignment with the circle of fifths, reveals that keys with fewer accidentals normalized sooner than those with multiple accidentals, which maintained their unique characteristics well into the 19th century.

Analytical vignettes demonstrate that certain key-specific behaviors, particularly in the 18th century, stemmed from composers' predilections for diatonic pitches, a trend that commenced in the 16th century but gradually diminished. Our investigation further explores the reasons behind the persistence of these unique key behaviors, considering various factors such as the historical development of flat and sharp signatures, cognitive biases towards familiar pitches, the ergonomic challenges posed by certain keys on specific instruments, the advent of equal temperament, and the increasing reliance on transpositional tools like the guitar capo and electronic production. Through this examination, the paper sheds light on the intricate evolution of tonal music's spatial framework, contributing valuable insights into the interplay between musical theory, practice, and cognitive perception. ​