Cardinality invariance

All isomorphic note-layouts, by definition, have the property of transpositional invariance: the same fingering in every key.

Non-trivial isomorphic keyboards also have the property of tuning invariance: the same fingering in every tuning (of those temperaments with the same generators as the note-layout).

I've blogged before about the fact that the Wicki note-layout has another invariant property, not yet named: its fingering patterns are the same for well-formed scales of any cardinality (again, assuming that the layout and temperament use the same generators). However, that property has not yet been assigned a name.

I hereby define cardinality invariance as "the same fingering in every well-formed scale, regardless of cardinality" (for a given generator-pair).

JIMS' (Wicki) note-layout has this property. The Wesley note-layout has it, too. Most other isomorphic note-layouts don't have it.  I don't yet know what mathematical characteristics confer it. But now, at least, it has a name: cardinality invariance.

Isomorphism & diatonic set theory

There are lots of isomorphic note-layouts -- for example, the Bosanquet, Fokker, Janko, Wesley, Chromatic Button Accordion (B-system and C-system), and Wicki.

JIMS uses the Wicki note-layout for a variety of reasons that are beyond the scope of this post.

The Wicki note-layout is proving to have some interesting mathematical properties. For example, consider any well-formed scale constructed by stacking N tempered perfect fifths and subtracting octaves (an "alpha-reduced beta-chain," where alpha is the octave and beta is the tempered perfect fifth), and N is the "cardinality" of the scale (that is, the number of notes in the scale).

The Wicki note-layout appears to be unique in that such well-formed scales are always tightly packed together on the keyboard, with no "holes" between the notes of the scale.

For example, consider the well-formed scale of cardinality 5 (pentatonic). It's notes [Do Re Mi So La] form a single tight group that (a) has no "holes" in it, and (b) is symmetrical around Re.

The well-formed scale of cardinality 7 (diatonic) is likewise tightly grouped and centered.

So is the well-formed scale of cardinality 12 (chromatic). Notice that both Le and Si are included, which is redundant; they represent the same note in the 12-tone well-formed scale, whether in 12-tone equal temperament tuning or not. I've just included both in the drawing for symmetry. The chromatic scale is the only well-formed scale with even cardinality (well, among those scales with cardinality less than or equal to 19, anyway), which is kinda messing with my head a bit.

And so on, for the well-formed syntonic scale of cardinality 17:

...and 19:

...and 21:

...and so on, ad infinitum.

To put it another way, the Wicki note-layout appears to be unique in that, to increase the cardinality of the syntonic scales playable on a Wicki note-layout, all one needs to do is add more notes to the left & right edges of the note-layout.

The other isomorphic note-layouts do not share this property. Their design intermingles scale notes and non-scale notes. As a result, they do not present the same pattern of notes for well-formed scales of all cardinalities.

By way of comparison, consider the Chromatic Button Accordion's C system note-layout (CBA-C), shown at right.

The CBA-C layout works fine for the chromatic scale, but if you wanted to use it exclusively for the pentatonic or diatonic scales, the note-layout would be full of holes. Alternatively put, neither the pentatonic nor diatonic note-sets map to compact, contiguous button-sets in the CBA-C note-layout.

Likewise, look at the line of "semi-tones" running up-and-rightwardly from C on the CBA-C note-layout. If one wanted to put the Db and C# on separate buttons there's no room. There's only one button-space between C and D; if has to serve for both Db and C#. The CBA-C note-layout does not have a clean "edge" to which the Gb could be added, as the Wicki note-layout does. As a rule of thumb, any note-layout with a contiguous line of "semitone"-controlling buttons has the chromatic scale "baked in," because the "semitone" is only a meaningful concept in chromatic scale (i.e., in the well-formed scale of cardinality 12). In scales of cardinality higher than 12, there is no "semitone." There are augmented unisons and there are minor seconds, but there are no semitones.

Now, look back at the patterns that well-formed scales make on the Wicki note-layout. These patterns all share three characteristics:

(a) They have no "holes" between the notes of a scale of given cardinality.

(b) They are symmetrical around Re.

(c)  All of their notes fall on adjacent rows, with one row being one button/note wider than the other (including the chromatic/12, because I included both Le and Si, which is cheating, just a little).

On the other hand, one can see (using the scale chooser on the interactive keyboard below) that non-well-formed scales, such as the Neapolitan, Melodic, Harmonic Major, Marmonic Minor, and Double Harmonic Minor, do not share all of these characteristics.

This suggests that there is some common element that is shared by (a) the definition of well-formedness and (b) the definition of the Wicki note-layout. I do not yet know what that common element is, but it's pretty obvious that it's in there somewhere.  (I think that it has something to do with the fact that on the Wicki note-layout, the "beta-stack" corresponds directly to one hexagonal line of note-controlling buttons, and the "alpha-stack" corresponds directly to a second, semi-perpendicular line.  But I'm not sure.)

If you can shed any light on this common element, please don't hesitate to let me know.  :-)

Help! Voronoi expert wanted

If you know anything about the mathematics of Voronoi diagrams, I need your help.

The Isomorphic Conspiracy (Andy Milne, Bill Sethares, and myself, with various single-paper collaborators) has published three peer-reviewed journal papers so far, and a bunch of conference papers, on isomorphic keyboards and their unique musical capabilities. In that work, we have, so far, used an approach that tied the notions of "note-layout" and "button-arrangement" inextricably together.

For example, consider the Wicki-note-layout, mapped to a perfectly-regular hexagonal button-arrangement. If you squish the rows of the button-arrangement to be a smidgeon closer together (as the Thummer does), then the result wouldn't be the Wicki note-layout anymore, under our original definition, because the button-arrangement was no longer strictly hexagonal. It would be something else, albeit something closely related, but we had way to describe the relationship mathematically.

We're hot on the trail of a new approach that separates note-layouts from button-arrangements. The mathematics of Voronoi diagrams are central to the new approach, because they can help us define the conditions under which one button is adjacent to another, as a regular lattice of buttons is systematically deformed away from perfect regularity. We're making pretty good progress, but the work would procede much faster if we could bring someone into the Conspiracy, for this one paper at least, who knew Voronoi mathematics inside-out.

If you know a Voronoi-math expert, preferably one who also has an interest music, please let me know.

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.