Matrix

One of the coolest things about working on the Thummer has been using it to discover new things about music. Its isomorphic button-field (keyboard) is like an X-Ray lens that lets people see the deep structure of music.

Bill Sethares and Andy Milne did the heavy mathematical work to prove that what we were seeing was really there. Their proofs can be found here and here (with more papers in the pipeline). These papers, while appropriate for their purpose and venue, are mathematically impenetrable to people with tiny little heads like mine.

Therefore, I've recently posted a draft paper that presents our new musical paradigm, the Matrix, in language that is nearly math-free. You need to know what prime numbers are, and that any natural number can be factored into an unique combination of prime numbers, and that a two-dimensional array of numbers is a matrix (hence the name of the proposed paradigm), but that's about it.

Although I do not claim to be an expert in the history of science, I do know a thing or two about it, and the Matrix model of music theory has all of the hallmarks of a paradigm-shifter. For example, it solves old problems, explains previously-anomalous experimental results, makes predictions that are falsifiable, and has enabled the discovery of new properties (e.g., tuning invariance, which is the basis of Dynamic Tonality).

The Matrix paradigm accomplishes all this as a result of questioning a single key assumption of Western music theory: that musical sounds are those that follow the Harmonic Series. This assumption is embedded so deeply into Western music theory that most musicians and many theorists don't even realize that they are making it. It has been received wisdom since Pythagoras first plucked a string 2,500 years ago.

The Matrix paradigm, in brief, uses a temperament to temper both tuning and timbre in real time. It's the tempering of timbres that's new (building on Bill's previous work). This is, in effect, a generalization of the relationship between Just Intonation and the Harmonic Series that forms the core of Western music theory.

It is hard to imagine a more fundamental alteration of the theoretical basis of music than this. Hence, paradigm shift.

What?

Thumtronics’ musical innovations, taken together, abstract to a higher level both (a) the structure of musical sounds, and (b) the higher-level forms of music arising from that structure. This higher level of abstraction is both simpler and more powerful that that used in the Western musical tradition.

Thumtronics’ first breakthrough is the combination of a concertina-like keyboard with tiny thumb-operated joysticks (like on a video game controller) and motion sensors (like on Nintendo’s Wii game controller), thereby delivering the most expressive polyphonic musical instrument ever: the Thummer. This expressive power is needed to control the many new expressive opportunities enabled by Thumtronics’ other breakthroughs.

Thumtronics’ second breakthrough is the combination of the Wicki note-layout, a chromatic staff, a tonnetz, tonic solfa, and the computer keyboard, thereby producing an easily-deployable system for the display and control of musical information – the ThumMusic PLUS System – which makes music easier to teach, learn, and play.

Thumtronics’ third breakthrough is its recognition that generalized note-layouts (such as the Wicki) have the same fingering not just in every key, but also in every tuning of a given temperament. That enables Dynamic Tuning, in which the performer can change the Thummer’s tuning in a smooth continuum while retaining the same fingering. Dynamic Tuning enables tuning bends, temperament modulations, and new chord progressions, all within the time-honored framework of tonality.

Thumtronics’ fourth breakthrough is Dynamically Tempered Timbres (X_Spectra & X_Timbres), in which the partials of a given timbre are adjusted, in real time, to align with the notes of the current (dynamic) tuning, to which they are related. This can deliver perfect consonance all across a given temperament’s tuning continuum, with additional real-time effects such as dynamic dissonance, primeness, conicality, and richness. These novel musical effects can make dynamic tunings sound pleasing and familiar, while giving composers an entirely new means of creating “tension and release.”

In Thumtronics’ approach, what matters are the relationships among intervals – that is, temperaments – but not pitches. A musical composition can be specified completely, yet leave the choice of key (i.e., tonic pitch) to the needs of the performing group (to reflect its current tessitura). Computer scientists will recognize this as an example of dynamic binding.

Taken together, Thumtronics' innovations hoist the description and control of musical information to a higher level of abstraction which is both simpler and more powerful than the traditional view.

These innovations also generalize music theory beyond the Harmonic Series, to embrace a wider set of timbre-structures. This widening consequently broadens music theory beyond Just Intonation to a wider set of tunings which are related to those timbres (or vice versa -- same thing).