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?

Keiretsubator

An interesting new business model is emerging here in Austin: the keiretsubator, a combination of an incubator and a keiretsu.

Keiretsu is the Japanese word for a set of companies with interlocking business relationships and shareholdings, usually headed by a bank which finances the group’s companies. Examples include Mitsubishi, Mitsui (Toyota), and Fuyo (Canon, Hitachi, Nissan, Yamaha).

An incubator is an organization – usually not-for-profit – which provides start-up companies with shared infrastructure, basic business services and equipment, technology support services, and assistance in obtaining outside financing from angel groups and VCs. Examples include the Austin Technology Incubator, Georgia’s Advanced Technology Development Center, and Washington State’s William M. Factory Small Business Incubator.

In the keiretsubator model, investors invest directly in the keiretsubator, which then invests in a small number of firms which share the same technology & industry focus (e.g., electronic music technology). In addition to the usual incubator services, the keiretsubator hires senior start-up veterans to fill the senior management roles – CFO, CTO, COO, VP Marketing, etc. – of its member companies on a shared basis. For example, one top-flight Chief Financial Officer will be hired by the keiretsubator, whose services are then shared by the member firms, none of whom attract or afford such a top-flight CFO on their own.

It's a pretty good deal from the CxOs' perspective, because they get to work on a number of interesting projects while being compensated with cash and equity in the keiretsubator – and thus indirectly with equity in the keiretsubator's member companies.

The keiretsubator model thus combines the hands-on management and financial services of a keiretsu with the early-stage focus of an incubator.

The keiretsubator model appears to be ideal for tiny start-up companies with great ideas, no money, and no management other than their founder. Such tiny firms can rarely attract the top-flight management needed to steer the company to success. By banding together, they can pool their opportunities.

Generally speaking, only one in six start-ups ever goes public, and only one in three is acquired by another firm, so incubation – in any model – needs to spread its risk among a number of different member firms to minimize its overall risk. On the other hand, venture capital firms that focus on early-stage companies have earned more than 20% per year over the last twenty years – so if well-managed, a keiretsubator could be a very good investment.

Will the keiretsubator model work? If it attracts great people, great ideas, and sufficient capital, then the odds are in its favor.

ClarisWorks, OLE vs. OpenDoc, and Microsoft

An online history of ClarisWorks contains the following passage:

Before ClarisWorks 5.0 was finished, the rift between Claris and the remaining development team grew too wide. Scott Holdaway, Tom Hoke, Scott Lindsey, Bruce Hammond, and Carl Grice left Claris, and formed what would later become Gobe Software. These were the core engineers, the ones who had been on the project the longest. Several newer engineers remained. Three in particular were responsible for OpenDoc integration. At Apple WWDC '96, they demoed a version of ClarisWorks with OpenDoc support. Apple was particularly eager to demonstrate to developers that a flagship product was supporting the new component technology - ergo, everyone else should too. Not surprisingly (to me, anyway), the following week the three engineers were recruited by Microsoft. Result: no OpenDoc support in ClarisWorks 5.0. That maneuver is just one typical example of the way Microsoft stifles competition [emphasis added]. Ironically, in this case it hit a product conceived in defiance of the Microsoft way.

This is a complete mis-characterization of Microsoft’s role.

With Windows sales booming in 1992-1993, Claris’ customers – who had previously used only Apple’s Macintosh computers – had acquired Windows-based computers too. They needed versions of Claris’ applications that ran on both Mac and Windows. These customers made it clear that if they didn’t get cross-platform versions of Claris’ applications soon, they would have to switch to some other company’s products. This put Claris’ management under the gun to get Windows versions of Claris’ applications onto the market ASAP. Producing Windows software was major change for Claris, because it was a wholly-owned subsidiary of Apple Computer, which made the Macintosh, which competed against Microsoft’s Windows.

As Microsoft’s Technology Evangelist to Claris, I facilitated Claris’ exploration of Windows’ technologies. I worked mostly with Larry Slotnick, Claris’ VP of Product Development, and Don Bradford, Claris’ Senior Director of Integrated Product Development.

As I understand it, Don hired outside contractors to implement two different test versions of ClarisWorks, one using OpenDoc (backed by Apple, Novell, and IBM)and one using OLE (backed by Microsoft). Based on the results of this experiment, Don concluded that OpenDoc objects were too slow and heavy to meet ClarisWorks’ need for a fast, lightweight, cross-platform object model. Don and Larry decided that ClarisWorks 4.0 would use OLE instead of OpenDoc (as would FileMaker, BTW).

When informed of this technical decision, Apple’s management was furious, insisting that ClarisWorks WOULD support OpenDoc, emphasizing that “we OWN you, and you WILL do as you’re told,” with the clear implication that any Claris employee who persisted in championing OLE would be fired, irrespective of the prevailing technological realities.

In response to Apple’s threat, the entire ClarisWorks team had an all-day off-site meeting (at Vancouver’s Red Lion Inn at the Quay) to discuss its options. At the end of the meeting, Don called me at my office in Redmond, with his entire team participating by speakerphone.

As I recall, the conversation went something like this:

DON: James, I’ve got the whole ClarisWorks team together here, and we’ve been talking all day about Apple’s decision to ram OpenDoc down our throats.

TEAM: [boos, hisses, etc.]

DON: We’ve decided – and we’re all amazed at this decision…

TEAM: [laughter]

DON: …we’ve decided that we all want to join Microsoft as a team.

JAMES: [pause] Excuse me?

TEAM: [laughter]

DON: Microsoft is the only company that’s big enough to keep our whole team together so we want to join Microsoft, don’t we, guys?

TEAM: [cheers, laughter]

JAMES: Cool! I’ll see what I can do.

I called a senior staffer in Microsoft’s HR group (I don’t remember who) and suggested that she send a recruiter right away to hire the entire ClarisWorks team. She nearly had a coronary! She explained that because Microsoft had a competing product (Microsoft Works), hiring the ClarisWorks team would be seen as “predatory hiring” – a violation of anti-trust rules – despite the team’s “invitation to hire” (which I would have thought to be a water-tight defense). However, if INDIVIDUAL Claris employees wanted to explore career opportunities at Microsoft, then…fine. So Microsoft ended up cherry-picking the best people out of the ClarisWorks team.

Microsoft’s first hire was Don, who formed Microsoft’s new Macintosh Internet Team in Silicon Valley, into which he hired many other Claris folks. His team went on to produce Internet Explorer 2.1 for the Mac, which won every comparative review and was subsequently licensed by Apple as its default browser.

Within months, Apple realized that Don and Larry had been right – that is, that OLE really was the only viable technology for ClarisWorks 4.0 – and aligned its development efforts accordingly. After much delay, it released ClarisWorks 4.0 in late 1994 with extensive OLE support.

About the same time, Novell released PerfectOffice 3.0, also with extensive OLE support. The release of these OLE-supporting products from two of OpenDoc’s leading backers told the whole world that OpenDoc was dead, and it happened in late 1994. By the time Microsoft hired Claris’ last three OpenDoc-savvy engineers – in 1996, according to the above – OpenDoc was already deader than Babbage’s Difference Engine.

In summary, the ClarisWorks story is not, in any way, “an example of the way Microsoft stifles competition.” It is, instead, a story of Apple’s early-1990’s management being critically out of touch with technological reality. Claris’ technologists tried to speak truth to power, but those in power at Apple refused to listen until it was too late. All Microsoft did was provide a better technology option, and pick up the pieces when Apple shattered its own ClarisWorks team.