Mapping Periodicity

I’m collecting examples of regular two-dimensional mappings of linear, periodic data. If you know of any good examples, please let me know.

Here are the best examples I know about.

The earth is a three-dimensional object, but maps are two-dimensional. They have to discard an entire dimension in order to present the curved surface of the Earth in a conveniently-flat manner, and there are lots of projections than organize this discarding in a systematic manner.

But what if you want to go the other way ‘round? What if you have low-dimension data, and you want to display it at a higher dimensionality?

Chemistry provides a famous example. Each of its elements has an atomic number, corresponding to the number of protons in its nucleus. Atomic numbers form a one-dimensional continuum from 1 (Hydrogen) to at least 118 (Ununoctium). The continuum of atomic numbers has no inherent periodicity, but the physical properties of the atom impose a periodic structure on this otherwise-undifferentiated continuum of atomic numbers, as seen in the Periodic Table of the Elements.

Another example is the calendar. One can think of time as a one-dimensional count of days, such as the Julian Day Number, which has no inherent periodicity. Divide it up into 7-day weeks, however, and you get a calendar which is seven days wide and infinitely tall (usually clipped to display a single month’s days).

Music provides the third example (and the point of this article). The human ear has a hearing range that runs from about 20 Hz up to about 20,000 Hz. This one-dimensional range of frequencies has no inherent periodicity, but if two tones from a harmonic source are sounded together within that range, the coincidence of their harmonics makes some inter-frequency ratios sound more consonant than others, giving rise to the musical intervals recognized as octaves, perfect fifths, and so on. Using only the octave and tempered perfect fifth to generate all other tonal intervals, one can map linear frequency into a periodic set of intervallic relationships, as seen in isomorphic keyboards.

If linear data contains periodicity, then that periodicity can supply the information needed to provide an extra dimension.

This seems to me to be rather magical: An entire dimension of data conjured up out of thin air!

If you know of any other examples, please let me know! :-)

Tuning Invariance and the Brain

I had a great "first contact" meeting with Bob Duke yesterday. He’s the Director of UT/Austin's Center for Music Learning, and Google suggests that he's very well regarded by the music education world, with an international profile.

We're meeting again next week.

Bob wanted more information on two points I raised in my presentation, so I sent him links to two papers: the first describing Bill Sethares' work on the relationship between tuning and timbre, and the second (Burgoyne, 2005) showing the brain's perception of tonal pitch-space. This posting is an extended answer to the issues Bob raised.

Tonal Pitch Space & the ThumMusic Note-Layout
Figure 3d in Burgoyne's paper is the result of using Maximum Variance Unfolding (MVU) instead of Multi-Dimensional Scaling (MDS) to measure & display the relationships in Weber, Krumhansl, Kessle, & Lerdahl's tonal pitch space.

Why use MVS? To quote Burgoyne:

Like MDS, this algorithm produces an embedding from a matrix of pair-wise distances, but while maximizing the variance of the output embedding, it seeks to preserve only the distances between nearest neighbors. This subset of distances is locked, and a nonlinear optimization technique is used to expand the data as much as possible given these locks, analogous to stretching a ball-and-stick model in which the balls correspond to harmonies and the sticks correspond to the locked distances.

What Figure 3d shows, then, is one slice through the relationships among nearest neighbors in tonal pitch space – and along that slice, the relationships match those of the ThumMusic note-layout.

Relationship of Tuning & Timbre
The Indonesian gamelan, Thai renat, and Mandinka balafon are all traditionally tuned in an inharmonic manner. Bill’s research shows that the tuning of these instruments is closely "related" (his term) to the timbres produced by those instruments. Clearly, then, the human ear/brain/mind can accept a wide range of tunings as being "musical," as long as those tunings are "related" to the timbres in which they are played (or vice versa – same thing). The X_System's use of X_Spectra is based on this insight.

Bill's work supports the argument that the ear/brain/mind's hardware and software can process, as tonal music, a wider set of tuning relationships than has been investigated by Krumhansl, Lerdahl, etc. as above, so long as the tuning and timbre are "related."

Tuning and the Brain
Importantly, the geometry of the ThumMusic note-layout is tuning invariant – i.e., the pattern of notes is the same no matter what the tuning (with some caveats). Since the perception map shown in Burgoyne’s Figure 3d is identical to the tuning invariant ThumMusic note-layout, then it seems likely that the brain's perception of tonal relationships ought to be tuning-invariant (with related timbres), too.

I hadn't made this connection before.

Cool!

When?

I started working on Thumtronics’ innovations in September of 2003. Since then, as my long-suffering family can attest, I have been obsessed by the challenge of developing and commercializing Thumtronics’ innovations.

Shipping an affordable, expressive Thummer is Thumtronics’ one and only mission at present. Only after it reaches a high enough level of sales to make Thumtronics profitable can we consider devoting additional resources to commercializing Thumtronics’ other innovations, such as the ThumMusic System, Dynamic Tuning, or Dynamically Tempered Timbres.

Currently, Thumtronics is raising capital to fund the final design & engineering work needed to get the Thummer to market. It is expected that the Thummer will reach the market within approximately nine months of this capital becoming available.

At the moment, I’m collecting quotes from credible folks in Austin and beyond about the market potential of the Thummer. Although everyone knows that disruptive innovations can make huge profits, investors usually approach a given potentially-disruptive innovation with great skepticism. Because disruptive innovations redefine the market, exploit new channels, and attract new customers, it’s very hard to prove that the disruptive product will actually sell – until it starts selling. The quotes that I’m gathering are intended to reduce this perceived market risk, by establishing that experts in the relevant fields believe that the Thummer will sell.

I expect to start approaching potential investors in a couple of weeks. It’s hard to predict how long the capital-raising process will take. One smart guy with money, and I’m done – but more likely, I’ll need to find a half-dozen, and they’ll all debate the valuation & term sheet, so it’ll take months.

So don’t expect to see any Thummer for sale until mid-2008, at the earliest.

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).

Why?

With the help of many people, I've made what appear to be a significant scientific breakthrough which has implications to musical instrument design, music notation, electronic music synthesis, and music theory. I am attempting to bring these innovations to market through a start-up company -- Thumtronics Inc. of Austin, Texas. People keep asking me "how's it going?" This blog is the answer.