iGetIt! Music blog

This blog will receive no new entries after this one.

I've started a new blog devoted to iGetIt! Music. See you there! :-)

It's over

Thumtronics is dead.  :-(

For years now, I've been trying to raise money to finish the Thummer's  design and to manufacture its first production run. In that time, I received many promises, but no checks. Now, the global financial crisis has dried up all funding for early-stage companies. Thumtronics is now in bankruptcy.  It's over.

To Thumtronics' investors, I offer my sincerest apologies. I did my best. I put everything I had into it -- every penny, every hour, every effort.  I'm sorry. I hope that you will forgive me, and more importantly, that you won't hold my failure against the next guy who comes to you with a great idea.

I've tried diligently for years to license Thumtronics' patents to other companies, with no success. Most of Thumtronics' (pending) patents will fall into the public domain due to non-payment of fees.

Once Thumtronics' bankruptcy is final, I'll place the design documents for the Thummer prototypes on the web, as the basis of an open-source hardware project. Completing the Thummer's design would be a great team project for an electronics/mechanical engineering class. Once an open-source reference design was completed, anyone who wanted to make Thummers could do so.

For example, a firm like Hong Kong's Medeli to work with HK UST's students to finish the Thummer's design, then to collect partially pre-paid orders online until receiving enough orders to justify making the initial production run. If nothing else, this would be a great way to identify (and hire) the university's best students.

Hopefully, through an open-source approach, Thummers will someday become available.  I hope so, because I want one!  :-)

In the meantime, I've started a new company, iGetIt! Music. It has no website yet (although I've registered the domain). iGetIt! is just me and an Internet-connected computer in my bedroom -- a classic micro-ISV. iGetIt! is focused on developing online music education courseware using the computer keyboard (and possibly iPhone) as its input device. iGetIt! doesn't have the potential to rake in the big bucks that Thumtronics had, but if I'm lucky I ought to be able to make a living out of it. In these troubled economic times that's a whole lot better than nothing.

My biggest challenge now is getting back into the swing of computer programming. It's been 22 years since I got my Computer Science degree, and 17 years since I last programmed for a living. I was on the cutting edge of object-oriented programming back then, and mostly I'm finding that today's tools make it very easy to do things that required lots of hand-coding back then.  I'll start a new blog shortly to document my progress and share what I've learned.

So...Thumtronics is dead. Long live iGetIt! Music!  :-)

Tonnetz

Since Euler, and especially since Hugo Riemann, the tonnetz has been thought of as being generated by a combination of major thirds and perfect fifths. Reading an excellent paper by Yale's Richard Cohn, I have suddenly realized how this traditional approach could be generalized using the Matrix/ThumMusic paradigm.

It is much more general to think of the tonnetz as being generated by octaves and (tempered) perfect fifths, just like everything else in the Matrix/ThumMusic paradigm.

Here's a portion of the Matrix's two-dimensional note-space expressed in the ThumMusic System's isomorphic note-layout:



Each note is of the form [alpha, beta] where alpha is the number of octaves (each of width P8) and beta is the number of perfect fifths (each of width P5) which, when added together, give the width of the indicated interval. For example:

• the origin note[0, 0] is zero octaves and zero perfect fifths away from itself, (0 * P8) + (0 * P5);
• note[1, 0] is one octave higher in pitch than the origin, (1 * P8) + (0 * P5);
• note[0, 1] is one P5 higher than in pitch the origin, (0 * P8) + (1 * P5);
• note[-2, 4] is two octaves lower, but four P5's higher, than the origin, (-2 * P8) + (4 * P5).

Assuming that the P8 is 1200 cents wide and the P5 is 700 cents wide, the notes of the note-matrix would have these widths:
Now, let's build a portion of the tonnetz on the note-matrix, following Cohn's paper:


The minor triad Q is surrounded by three major triads P, L, and R.

• P: Parallel;
• L: Leading-Tone Exchange;
• R: Relative.

The above construction of the Q, P, L, & R triads from octaves and tempered perfect fifths is much more general than the traditional construction, because these intervals are the generators of the syntonic temperament, so the tonnetz's properties are invariant across the syntonic tuning continuum, no matter what the specific width of the P5 (within the range 686-720). This continuum includes an infinite number of individual tunings, not just the small number of N-edo tunings (in which N mod 3 = 0) over which Cohn's paper generalizes the tonnetz' traditional construction.

Cohn's paper makes much of the toroidal topology of such equally-tempered tunings (as do many neo-Riemann theoreticians). This emphasis overlooks the syntonic temperament's general topology, which is cylindrical. The tonnetz' octave axis forms a closed loop around the cylinder; its axis of major thirds runs parallel to the cylinder's inifintely-long axis; and its axes of minor thirds and perfect fifths form spirals around the cylinder's inifintely-long axis. Many common chord progressions, such as the IV-vi-ii-V-I, require only the syntonic temperament's cylindrical topology (without which the ii below the vi would differ from the ii above the V by a syntonic comma).

At those points along the tuning continuum that correspond to an equal division of the octave, such as 12-edo, 17-edo, 19-edo, 31-edo, etc., the cylinder snaps into a torus. Each n-edo's toroidal tonnetz has (a) all of the properties of the cylindrical tonnetz, (b) all of the properties shared by all toroidal tonnetzs, and (b) the properties specific to that unique n-edo's tonnetz. These points of equal temperament are like beads on a string -- but what's really interesting is not the beads, but the string.

From Thumtronics' perspective, the potential of the neo-Riemannian PLR operations to provide an invariant basis for music theory across the whole syntonic tuning continuum is very exciting (I think). Or, to express the same thought from the neo-Riemannian perspective, the Matrix/ThumMusic paradigm may give neo-Riemannian theory the opportunity to expand its scope to embrace the entire syntonic tuning continuum, and perhaps also the tuning continua (and tonnetz') of other rank-2 temperaments (e.g., magic, hanson, schismatic, etc.). These other temperaments temper out different commas, so their tonnetz' will be different from the syntonic tonnetz, but the same general principles ought to apply (at some level of abstraction, anyway).

Cohn's paper (like Riemann himself) makes a number of statements regarding the relationship between the tonnetz and "acoustics" that are only true if one assumes that "acoustics" means "the Harmonic Series." Yet the Matrix/ThumMusic paradigm generalizes "acoustics" -- by dynamically aligning a timbre's partials with a tuning's notes, as specified by a temperament's defining intervals -- such that the relationship between the tonnetz and "acoustics" is 1:1. The Matrix/ThumMusic tonnetz is a direct embodiment of generalized musical reality.

I think I'd read something about the PLR approach to chord relationships, chord progressions, and the like before reading Cohn's paper, but it hadn't clicked. Now, it has definately clicked. I suspect that the PLR approach to chord relationships will prove to be a very powerful tool in the Matrix/ThumMusic System.

Cool bananas! :-)

[Update, Thur Sep 25th: A couple of prominent neo-Riemannians have (very) informally agreed (a) that the proposed application of neo-Riemannain theory to the syntonic tuning continuum appears to be both novel and interesting, and (b) that they would read the relevant Matrix/ThumMusic papers and get back to me.]

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.

Amdahl's Law

Amdahl's Law describes the efficiency gains that come from paralellizing part of a sequential process. I suspect that it will be useful in estimating the increase in efficiency that can be gained in music education from the use of the ThumMusic System, but I'm not mathematically astute enough to figure out exactly how.

The ThumMusic System has the potential to speed up many aspects of music education -- i.e., to increase their efficiency -- but not all aspects. What is the balance? It is possible to optimize the wrong thing, increasing its efficiency enormously without significantly improving the efficiency of the whole process. I don't think that this is the case with the ThumMusic System, because it improves the efficiency of everything from theory to practice by reducing the symbol, concept, and gesture sets, exposing the relationships among the set-members geometrically, and exposing the consistency of those relationships to more senses (touch and sight in addition to hearing). But...how do I measure this? Perhaps Amdahl's Law can help.

Increasing the Efficiency of Music Education

I've recently posted to the Web a draft paper that describes the ThumMusic System and its potential to increase the labor efficiency of music education.

The paper will remain there until it is submitted for publication to an appropriate journal, at which point I may have to take it down. In the meantime, comments are welcome.

Transposing Tradition

In a letter to the Austin Chronicle dated July 17, 2008 and titled The Question: How Good are the Musicians?, author Tom Bowman asks

Who wants to go down to Red River and get their ears assaulted by band after band of “musicians” who know about 10 chords, write insipid lyrics, and can’t even transpose their own songs from one key to another? If the level of talent were higher, the established clubs wouldn’t have to bring in so many touring acts.

How fascinating that Mr. Bowman equates the ability to “transpose songs from one key to another” with the bands’ “level of talent”! This is by no means an isolated example, however; the ability to transpose is often equated with musical skill and/or talent. A Google search for the keywords +transpose +sight +key +talent turns up almost 3,000 hits.

Yet the difficulty of transposition has no relevance whatsoever to musical knowledge, skill, or talent. It is merely an artifact of the pitch-focused design of traditional musical instruments and notation. At most, it is an artificial barrier placed in the path of aspiring musicians.

Using the ThumMusic System, transposition is a complete non-issue.

I am reminded of upper-class English schoolchildren, who were compelled to learn Latin as recently as the 1980’s. Why? Because Latin was the universal language of scholarly discourse…200 years previously. To be recognized as being One of Us, one had to speak the upper class' secret language, for purely exclusionary reasons. Anyone who could not afford to waste time and money learning an utterly useless secret language was excluded from the upper-class club.

I don’t think that Americas' taxpayers can afford to waste their time and money attempting – with a low success rate – to teach its children the secret language of music’s upper classes. By using the ThumMusic System, students can concentrate on learning about music.

Which is the point, really, isn’t it?