Western music theory is notable for its rules: rules that determine which chord-pairs form a “cadence,” which chords to use to prepare for a cadence, the order in which chords should appear in a chord progression, proper voice leading, the rules of counterpoint, etc..
Some rules apply to every tuning in the syntonic temperament (Pythagorean, ¼-comma meantone, 12-tone equal temperament, etc.). For example, the I-IV-vi-ii-V-I chord progression “works” in any tuning of the syntonic temperament. “Works” means that the frequency of I is the same at the end of the progression as it was at the start (octaves aside). This same chord progression does not work in other temperaments (e.g., the schismatic) or in Just Intonation. In them, the frequency of I would drift off-key by one syntonic comma on each cycle through the chord progression.
Other rules apply only to specific tunings of the syntonic temperament. For example, Giant Steps and Central Park West both work only in 12-tone equal temperament, because that’s the only tuning in which three major thirds (or four minor thirds) precisely equal one octave. In any other tuning, each pass through the modulatory cycle would drift off-key.
Logically, then, there is a hierarchy of “rules” to be worked out:
- Rules that apply only to a single tuning in a single temperament
- Rules that apply all across a given temperament (or within a sub-range there of…but what defines the sub-range?)
- Rules that apply across all tunings of closely related temperaments (but what defines “closely related”?)
- …and so on.
It can be argued that the “research program” of the Common Practice Era was the exploration of the rules that applied to a very narrow sub-range of the syntonic temperament (the range that was most compatible with purely harmonic timbres). That research program gave us today’s “rules” of music theory. From this perspective, Musica Facta can be seen as a re-invigoration and expansion of the Common Practice Era’s research program. (To which Rameau might say, plus ça change, plus c’est la même chose.)
It can be argued that academics and musicians have been exploring alternative tunings for a very long time, and of course this is true. Musica Facta advances the state of the art in four specific ways:
- Previous research was focused almost exclusively to the tunings and temperaments that were consonant with harmonic timbres. Musica Facta can deliver similar consonance in any tuning of any temperament, which infinitely increases the set of tunings & temperaments that may be found to have emotionally-affective potential.
- Previous research approached each tuning as being a unique entity unto itself. Musica Facta approaches each tuning as being a bead on a string, with the string being the tuning continuum of the current temperament. This enables composers and theoreticians to explore dynamic changes in tuning – from one bead to another, and to another, and back again (see Composition, above).
- Musica Facta’s discovery of tuning invariance enables its new musical effects to be controlled in real time, using isomorphic keyboards.
- Controllers for alternative tunings were often tuning-specific and cumbersome. Isomorphic keyboards offer consistent fingering across an infinite number of tunings (with no wolf intervals), and can be implemented on your “mini”-tablet device, making them the very opposite of cumbersome.
These advances, taken together, enable dynamic tonality – a vast frontier of music-theoretical possibility, waiting to be explored.
Here are some examples of potential research projects in music theory.
Exploring the [Foo] temperament
Pick a temperament (perhaps from this list). Let’s call the temperament you chose the Foo temperament. Now, explore it using tuning-aligned timbres. What well-formed scales does Foo support, if any? What 2-dimensional note-layouts are isomorphic with Foo? What are the valid tuning ranges of these scales in Foo (at various prime limits)? What are the tonal resources of Foo – its scales, modes, cadences, chord progressions, tonnetz, orbifolds, etc.? Which of these are – at some useful level of abstraction – shared by other temperaments? Which are unique to Foo? Which are invariant all across Foo’s tuning continuum, and which are unique to specific tuning sub-ranges or to specific tunings? You goal, metaphorically speaking, is to produce an accurate (but high-level) map the New Musical World of Foo, so that the musicians and theorists who follow you can explore Foo in finer detail (perhaps through compositions that explore Foo’s structures’ “ability to reliably induce, in audiences, compositionally useful emotional responses;” see Composition, above). This task involves the scientific practices of reduction and abstraction: reducing each given structure down to its smallest components, and abstracting each such structure to the highest useful level of generalization. The process and results of this task, applied to many temperaments, should eventually reveal many new and compositionally-useful insights into music’s structure.
Assimilating other music theories into Musica Facta
Musica Facta already borrows (more or less) from other recent advances in music theory such as diatonic set theory, neo-Riemannian theory, topos, orbifolds, etc. For example, Musica Facta has already borrowed the concept of “well-formed scales” from diatonic set theory, in part because the rules for constructing such scales generalize well across tunings and temperaments. However, there is more to diatonic set theory than well-formed scales. How does the rest of diatonic set theory generalize across tunings and temperaments, in the presence of tuning-aligned timbres? How about topos theory? Neo-Riemannian theory? Orbifolds? Etc.? Each of these can be examined in the context of Musica Facta’s generalization of the fundamental axiom of music theory, with an eye to assimilating their core concepts into Musica Facta in a parsimonious, elegant, and useful manner.
Ultimately, any useful purpose that these other theories serve, should be careful assimilated into Musica Facta, such that Musica Facta serves the same purpose more generally and parsimoniously.
Musica Facta has opened up a vast new frontier of music theory to be explored by brave adventurers such as yourself. Let us know which research problem piques your interest, and we’ll work together to help you solve it!