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.

Shift Happens

Paradigm shifts – the transition from one idea or technology to a new one – are not rare. They happen all the time. Here are some examples from my personal experience.

1. Slide Rules to Pocket Calculators
In 1974, I was in the last freshman mathematics class at my high school to be taught how to use a slide rule. This brilliant device had been the standard tool for mathematical computation for 350 years (since its invention in 1630 or thereabouts). However, by 1974 – just two years after the introduction of the first pocket calculator with slide-rule-like mathematical functions (the HP-35) – pocket calculators could perform more functions, with higher accuracy, less training, and fewer errors (such as mis-placing the decimal point). While the cost of a pocket calculator was still about the same as a good slide rule in 1974, the price of calculators had already fallen by half in two years, and was clearly going to continue falling. Pocket calculators made calculating simpler, cheaper, and more powerful. Shift happened.

2. Punched Cards to Video Terminals
In 1978, I was in the last freshman computer programming class at my university to be taught how to create and use punched cards. The punched card was the de facto standard of the computing industry for almost a century (since 1889) and the basis of IBM’s computing empire. However, by 1978 – just two years after the introduction video display terminals such as DEC’s VT52 and IBM's 3270 – video terminals were clearly out-competing punched cards. Video terminals made computing simpler, cheaper, and more powerful. Shift happened.

3. Geosynclines to Plate Tectonics
Also in 1978, I was in the last freshman geology class at my university to be taught about geosynclines as the fundamental paradigm of geology. By then, plate tectonics had become widely accepted as being a superior theory, but the school’s Intro to Geology textbook wasn’t updated to reflect this change until the following year. Plate tectonics provided a simpler model, in which fewer rules explained a larger number of phenomena more accurately, reducing the frequency of failed geological predictions (such as dry holes). Plate tectonics made geology simpler, cheaper, and more powerful. Shift happened.

4. Command Line & Text Mode to Graphical User Interfaces
When I started taking classes for a Computer Science degree in 1983, the dominant input paradigm was the command line, and the standard output paradigm was text mode. By the time I got my CS degree in 1988, the command line and text mode were being superseded by graphical user interfaces (such as the Mac & Windows user interfaces), which made computing more accessible to non-professionals, reduced training costs, and enabled powerful new applications like desktop publishing, digital photo editing, and Mathematica. Graphical user interfaces made computing simpler, cheaper, and more powerful. Shift happened.

Discussion
When a new paradigm delivers a desired outcome through means that are simpler, cheaper, and more powerful, then shift happens, even if the previous paradigm has been in place for centuries.

Conclusion
Once the Thummer and ThumMusic System are commercially available, they will make learning to play and understand music simpler, cheaper, and more powerful, establishing an important new paradigm for music-making.

Why? Because…Shift Happens.

Abstraction: A Parable

One of the most powerful tools in science is abstraction, which Wikipedia defines to be “the process of generalization by reducing the information content of a concept or an observable phenomenon, typically in order to retain only information which is relevant for a particular purpose.” The story below is a parable on the power of abstraction.

1998: Redmond, WA
In 1998, Microsoft responded to a host of competitive threats and internal opportunities by undertaking the development of a language-neutral runtime engine (.NET Runtime), application framework (.NET Framework), and Integrated Development Environment (or “IDE,” Visual Studio.NET). Such a language-neutral Runtime, Framework, and IDE would have to implement all of programming functions that were shared across languages. That is, the Framework/Runtime/IDE had to provide an abstraction of programming language functionality. At this higher level of abstraction, all programming languages would look essentially the same.

However, at this time, within Microsoft, “language neutrality” meant “supporting every language whose name began with ‘Microsoft,’” such as Microsoft C++, Microsoft FoxPro, Microsoft Visual Basic, Microsoft J++, etc.

Project 7 Is Born
On a late fall afternoon in 1998 while soaking up some uncommonly-bright Redmond sun (and some uncommonly-good Pyramid Hefe Weizen) on my back deck, my brother Peter and I were discussing this strategy. I worked in Microsoft Research’s University Relations Group at the time, handing out cash to researchers at MIT and other universities, and Peter worked in Microsoft’s Developer Relations Group, helping independent companies develop great software development tools for Windows. Despite being “leaf-node” employees with no strategy-setting authority whatsoever, we decided to re-define “language neutral” to include non-Microsoft languages – specifically the dozen or so most academically-interesting or commercially-important programming languages. Then, in true Microsoft style, we gathered the support and resources we needed and Made It So. (I understand that Microsoft isn’t like this anymore, which is a terrible shame.)

Project 7….Scores!
Cutting to the chase… Project 7 was wildly successful, swatting home runs on all of its objectives and becoming a model for many subsequent technology evangelism efforts inside Microsoft. (This article mistakenly refers to my efforts as “Project 42,” but the rest of it is basically correct.) There are now over 40 programming languages for .NET. While the success of Project 7 is a testament to the power of technology evangelism, the success of .NET is a testament to the power of abstraction. By abstracting the notion of programming languages to a higher level, .NET made it possible for previously-isolated programming languages to interoperate seamlessly, providing new opportunities for discovery and innovation.

The LINQ Project
Project 7 also yielded some significant ancillary benefits. Among these, one of my personal favorites was that Microsoft was able to identify and hire some of the brightest minds in programming language design. One example of this (out of many – please forgive me, guys, for not listing you all here) is Erik Meijer, whose breakthrough work on data access has recently been incorporated into the .NET Runtime as LINQ (Language INtegrated Query).

Erik started the work that became LINQ by abstracting data access to a higher level. At this new level of abstraction, (a) all data sources look the same, (b) a small number of simple algebraic operations can perform all necessary data access tasks, and (c) these operations can be integrated into any programming language.

As a direct result of LINQ’s abstractions, LINQ is...

1. so easy to use that programming tasks that would previously have required expensive, specialized database programming skills are can now be accomplished, using LINQ, by any competent computer programmer.
2. so powerful that data access tasks that would previously have required separate, specialized code for different kinds of data sources can now be accessed with a single, simple piece of code.
3. free with Visual Studio (including its free express editions).

LINQ’s abstractions are based on ideas from the fringes of mainstream computer programming practice: lambda calculus. The vast majority of software developers working today have never even heard of lambda calculus – except for the odd programmer (often considered to be very odd indeed). LINQ uses lambda calculus to solve the perennially vexing problem of data access so elegantly – and through such a ubiquitous vehicle as the .NET Runtime – that mainstream programmers simply cannot continue to ignore it. Because all mainstream techniques had previously failed to solve the perennially-vexing problem of data access, the eventual solution had to come from the fringes.

So, what does this have to do with Thumtronics’ musical innovations?

Thumtronics and Abstraction from the Fringe
Thumtronics’ musical innovations also arise from applying the tool of abstraction to ideas from the fringes of mainstream practice. Whereas traditional approaches to the display, control, and synthesis of musical information are focused on pitch, Thumtronics’ innovations:

1. Abstract musical information to the higher levels of


1. Musical Intervals: the relationships between pitches, rather than the pitches themselves, and


2. Temperaments: the relationships among intervals, in which (for example) the major third is defined as being the same width as four tempered perfect fifths minus two octaves (syntonic temperament), or in which the diminished fourth is defined to be the same width as five octaves minus eight tempered perfect fifths (schismatic temperament);


2. Abstract the structure of the Harmonic Series and Just Intonation to the higher level of pseudo-harmonic timbres and their related tunings; and
   
3. Organize the display of musical information using the geometry of an isomorphic keyboard (also known as a “generalized keyboard”).

Thumtronics’ abstractions are based on a collection of ideas – generalized keyboards, tonic solfa, the chromatic staff, Euler’s tonnetz, tuning theory, etc. – that the vast majority of musicians know nothing about (aside from the odd microtonal musician or aficionado of alternative musical notations, considered by his peers to be very odd indeed). Thumtronics’ abstractions use ideas from the fringes of the mainstream music-making community.

However, just as LINQ’s fringy abstractions solve the perennially-vexing problem of data access, so do Thumtronics’ fringy abstractions have the potential to solve the perennially-vexing problems of (a) music education’s high failure rate, (a) the music products industry’s commoditized products, (c) the exhaustion of the resources of tonal harmony (also known as the “crisis of tonality”), and (d) the failure of any one music theory to adequately explain the many different tuning systems used around the world.

Can Thumtronics’ innovations really solve all of these problems? Maybe. I know a powerful abstraction when I see one, because I’ve seen them and driven them to success before. I’ll do the same with these.

The Moral
The moral of this story is simple: abstraction of ideas from the fringe can solve perennially-vexing problems and change the world. Never bet against an abstraction whose time has come.