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.
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.
Labels: harmonic series, just intonation, Matrix, music theory, paradigm
posted by ThumMeister | 7:38 AM
3 Comments:
I often get cranky with the idea that Thomas Kuhns ideas about scientific revolutions are current philosophy of science -- the field has moved on since the early 1960s when Kuhn was writing, and the idea of a paradigm shift is no longer really believed to characterise scientific advancement by most in the field (personally, I like Imre Lakatos and his idea of research programmes, but I am nowhere near current in the field himself).
Anyway, that is just quibbling - the problem with Kuhn is his ideas apply better to non-scientific fields - and music, ultimately a field based on aesthetics, is one such. So paradigm shift away.
I appreciate your taking the time to comment! :-)
I am a bit confused by your statement that "the problem with Kuhn is his ideas apply better to non-scientific fields." It is my understanding that the opposite is true; that his ideas (and Popper’s, for that matter) apply better to areas of study that are predominantly provable/falsifiable than those that are predominantly aesthetic.
Either way, your subsequent comment that "music is ultimately based on aesthetics" appears to beg the question, i.e., does music have a physical, scientific basis, or is it "just cultural"? Without knowing the answer to this question, we can’t apply ANYONE’s theories of scientific change – Kuhn’s, Popper’s, Lakatos’, Mayo’s, or anyone else’s.
I find Sethares’ work on the cultural independence of an acoustical basis of harmonic structures to be convincing. There is ample ground for disagreement here, however. As I state in the draft Matrix paper, "Some concluded that the perception of music was merely cultural, with no acoustic basis whatsoever," citing Cazden (1945) and Lundin (1947) as references.
Hopefully, this ground will be plowed by many researchers in the years to come, producing a raft of papers that will, collectively and asymptotically, approach The Truth.
In the meantime, I offer the Matrix paradigm as a framework around which a research program can be constructed, and invite other researchers to join in exploring some of the questions it raises.
David's thoughtful comment above prompted me to search for more-recent scholarship on this topic.
The most interesting stuff I found was that of Jesús Zamora Bonilla. I like his application of economic theory to the practice of science, which he accomplishes by considering scientists to be seeking recognition instead of profit.
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