Klangfarbenakkord and Klangfarbenharmonien Metric Space Models for Music on Informational Geometry 1

In this article, we present a method of spectral serialization envisioned by G. Mahler and A. Schoenberg between the late 19th and early 20th centuries that enables the selection of timbres rooted in musical structure. Based on information geometry, we evaluate the Wasserstein distance between polyphonic timbre spectra, as demonstrated in our previous paper, in terms of the second moment of spectral chords comprising three or more notes, expressed in squared hertz. By embedding these quantities in a Banach space, we analyze specific passages from works by R. Strauss, Mahler, A. Berg, and A. Webern from the perspective of timbre, showing how the same chord exhibits different affinities and divergences depending on orchestration. Taking the wah wah mute as an example, we propose a model that treats instrumental timbre and speech within the same Banach space, and we attempt to quantify the "vowel triangle" in Hz. These efforts constitute a partial but integrated solution to the historical three problems of Schoenberg. Serialism, Klangfarbenmelodie, and Sprechgesang which neither Mahler nor Schoenberg in their lifetimes, nor the postwar generation of composers including Nono, Boulez, and Stockhausen, was able to resolve fully. We aim to reconsider musical thought as a whole from the perspective of timbre and to open new possibilities for interpretation, performance, and composition. All tools are publicly available on GitHub, so that anyone can reproduce and extend these analyses.

This paper presents a web-based application for navigating and composing microtonal harmony, built on the Harmonic Eigenspace, a four-dimensional psychoacoustically grounded space in which tetrad chord types are located by their spectral dissonance profiles, computed with Sethares's roughness/dissonance model. The coordinate system is transposition-invariant: a coordinate triple (α, \b{eta}, γ) locates the three upper notes in relation to the root, so a chord quality corresponds to a direction in the space, the invariant ray along which transposition acts, while the root frequency sets the scale. The dissonance field over these coordinates can be computed at any register; the locations of its local minima are register-invariant, as they arise from partial-coincidence ratio conditions. The dissonance volume contains 100 local minima that align with just-intonation intervals and act as landmarks, organising the space into basins around the most consonant tetrads. We embed tetrads from three tonal equal temperaments as discrete lattices within this continuous volume. The application presents this space through two components: the Harmonic Eigenspace as a navigable 3D visualisation of the dissonance volume in which all nodes are playable, and a Modal Studio that extends modal interchange logic to the ten-gradation interval vocabulary of 53-TET. The application also functions as a MIDI controller with MIDI Polyphonic Expression support, usable in any digital audio workstation that supports this format. A listening study with 31 participants used both scenes of the application: listeners first rated isolated 53-TET chords alongside chords familiar from Western practice, such as the maj7 and the m7; they then rated chord progressions composed in the Modal Studio, measuring their acceptance or rejection of microtonal progressions heard for the first time.

We propose a general approach to enharmonicism within syntactic music theory; that is, we formalize several abstract structures of enharmonicism without any acoustic considerations. The structures of concern to this paper are constructed from diatonics, chromatics, and enharmonics, and we seek to present a general enharmonic theory that builds upon Hook's enharmonic theory published in "Enharmonic Systems: A Theory of Key Signatures, Enharmonic Equivalence and Diatonicism" (2007). We also propose that the two criteria of reflection and maximal evenness can be used to determine whether an arbitrary enharmonic system is musically practical. Moreover, we argue that preferential treatment of the standard enharmonic system (SES) can be motivated from an abstract mathematical perspective divorced from pitch consideration and conventional acoustic constructions of SES.