Musical Intuition

We denote this primary wisdom as Intuition, whilst all later teachings are tuitions.
(Ralph Waldo Emerson)

Intuition is a funny thing. Mostly, intuition means that “something new corresponds with your expectations,” which means that it corresponds with your experience.

But what if your experience was misleading?

Consider, for example, a caveman observing the Sun. It is "intuitively obvious" to this caveman that the Sun is moving around a fixed Earth, because that’s what he experiences every day. Or consider the incidence of infectious diseases. In a unsanitary city of foul water, tainted food, and ubiquitous disease-vectors like mosquitoes, fleas, lice, and cockroaches – that is, in almost any city in the world, until very recently – it would have been “intuitively obvious” that illness, health, death, and survival were all essentially random, or in the hands of the Gods. The underlying patterns were hidden by the experience of randomness.

So it is with music. Most people’s experience with music-making misleads them into thinking that music is about pitch, because pitches are what’s notated, pitches are what are controlled by traditional instruments’ interfaces, and pitches are what musicians talk about among themselves. It seems intuitively obvious that music is about pitch.

However, this experience is misleading. Music is not about pitch. It’s about intervals – i.e., the gaps between pitches. At this level of abstraction, any given musical structure – an interval, a melody, a chord, a chord progression, or even an entire musical piece – is the same in any octave or key. This is not at all obvious to most instrumental musicians, for whom the ability to transpose “on sight” is rare and awe-inspiring. Those who’ve learned music by singing using tonic solfa are more likely to recognize this higher-level abstraction, because their key-independent experience prepares their intuition to recognize the “invariance” of musical structures across keys and octaves.

Likewise, the experience of most musicians is misled by their implicit assumption that musical timbres are, and must be, harmonic – i.e., follow the spectral pattern defined by the Harmonic Series. This assumption is so deeply ingrained in Western music theory – dating at least from Pythagoras, 2,500 years ago – that most music theorists assume it without even recognizing that an assumption has been made. When the music of some indigenous cultures – in Indonesia, Thailand, and Mandinka Africa – was discovered to be inharmonic, this physical basis for music theory was challenged. Many people just threw up their hands and said that musical structure had to be “just cultural; just experience” – i.e., intuition.

However, if you abstract music to the next higher level – i.e., to patterns of relationships among intervals, as defined by a comma sequence – then it becomes clear that the music of the above-listed cultures and that of the West all share the same deep structure, and that the sonic spectra (timbres) of the instruments used by each culture bears an invariant relationship to its characteristic tuning within that deep structure. Yet this “tuning invariance” – first described just last year (2007) – is so non-intuitive that it had been overlooked by generations of music theorists, arguably because their experience was so firmly grounded in the Harmonic Series that their intuition misled them.

It is remarkable that so many of the world’s musical cultures use combinations of tuning & timbre that share the same deep, invariant structure. Why this one structure, and not others?

It is entirely possible (but entirely speculative at present) that the human brain contains a hard-wired isomorphic note-layout which reflects this deep structure. Such a note-layout presents any given musical interval, chord, chord progression, etc., with invariant geometry in all tunings of such a deep structure. The findings of many recent studies in music cognition can be interpreted as supporting this hypothesis. Like everything else in Western music theory, those studies have tended to be pitch-based, and to assume the use of 12-tone “equal temperament” tuning, but Occam’s Razor suggests that this one entity – a hard-wired isomorphic note-layout of interval-detecting brain cells – can explain their findings very simply. No studies have yet been performed to determine whether such a hard-wired note-layout exists, in part because the discovery of tuning invariance is so recent, and was made by relative outsiders to the music cognition community (as is so often the case).

Which brings me back to the tonic of this piece: musical intuition. The only possible source of “intuition” that’s deeper than personal experience is the hard-wired physical reality of the human brain. If the brain did indeed contain a hard-wired isomorphic note-layout, then that note-layout would be the ultimate source of musical intuition – invariant across octaves, keys, tunings, and cultures.

For the experience of music-making to be deeply and truly intuitive, the tools of music-making – music notation, music control interfaces, music synthesizers, etc. – would need to reflect this hard-wired geometry of music. This hasn’t been technically feasible until recently, nor commercially feasible until even more recently, but it is entirely feasible today.

If music-making were to be made truly and deeply intuitive, in a culturally-invariant way, then the percentage of the world’s population that could afford to successfully gain a self-sustaining level of musical competence could increase dramatically. Furthermore, it would elevate music to being truly a single universal language, with lots of interesting regional dialects.

I submit that this would be a maifestly Good Thing.

An Exceptionally Simple Theory of Music

I’ve watched with interest the reaction to the publication of Garret Lisi’s paper, An Exceptionally Simple Theory of Everything. He is being revered and reviled simultaneously.

When something happens that people think might be important, but they don’t really understand it, they tend to look around for an expert. If the experts disagree, then those who don’t understand the details are left with an ink-blot test, from which they divine meaning by faith alone. On the one hand, the decision to revere or revile tends to be based largely on internal factors – one’s faith in “progress,” for example. On the other hand, trivial and extraneous details of the ink-blot can become disproportionately influential – such as one’s feelings about surfing.

My collaborators and I are pursuing a similarly-simple Grand Unified Music Theory (which underlies the ThumMusic System). I suppose that we can expect it to receive a similarly split reaction…assuming anyone even notices. We have the advantage that our work’s foundations have been accepted for publication in peer-reviewed scientific journals (Computer Music Journal, Winter 2007, and Journal of Mathematics and Music, Spring 2008), which Lisi’s paper was not.

Oh, well. There’s no such thing as bad publicity, right?

sus4

Here’s an interesting effect in Dynamic Tuning.

Using the Max/MSP implementation of Dynamic Tuning found here, first

• Slide the slider on the lower left of the screen rightward to “fully tempered”
• Set the tuning slider to 696.6 cents (1/4-comma meantone or 31-tet)
• Play a major triad on your QWERTY keyboard (e.g., the buttons labeled H-K-U)
• While the triad is sounding, slide the tuning slider up to its maximum (720 cents, 5-tet)
• Hold the slider there a moment, and then slide it back to where it started (696.6 cents)

What did you hear?

What I hear is:

• A nice, pure-sounding major triad, then
• A sus4, then
• A major triad again.

How’d that happen?

Well, when you push the slider up from 696.6 cents to 720 cents, you’re widening the tempered perfect fifth accordingly. The pitch of the major third (and the placement of the fifth harmonic of the tempered timbre) is higher than that of the root by four tempered perfect fifths minus two octaves. That means that the major third is widening from

• ((4 * 696.6) - (2 * 1200) = (2786.4 - 2400) =) 386.4 cents, which is a nearly-just major third, to
• ((4 * 720.0) - (2 * 1200) = (2880.0 - 2400) =) 480.0 cents, which is 18 cents flat from a just perfect fourth.

Just by wiggling the tuning slider, you can go from a very restful major triad to a tense sus4 – with the sharpened P5 adding to the tension – and back again.

Cool! :-)

Non-Western Cultures

The endpoints of the syntonic tuning continuum are 7-tone equal temperament (7-tet, P5=686 cents) and 5-tone equal temperament (5-tet, P5=720 cents).

This is particularly interesting because some non-Western cultures use these tunings (or tunings very similar to these). For example,
- The traditional Indonesian slendro scale is similar to 5-tet.
- The traditional scale of the Thai renat is similar to 7-tet.
- The traditional scale of the Mandinka African balafon is similar to 7-tet.

It has been suggested that these cultures' instruments emit sound spectra which (in isolation or when crossed with a harmonic timbre such as a human voice) are maximally consonant when played in these tunings.

This is not to say that these cultures necessarily use other musical structures from the syntonic temperament -- scales, chords, etc. But...who knows?

If the human mind categorizes tonal relationships in a tuning invariant manner, then perhaps tuning invariance can provide the foundation for a unified theory of music that generalizes music theory beyond the Harmonic Series to embrace a wide range of pseudo-harmonic tunings/spectra, from 7-tet to 5-tet and everything in between, including the ubiquitous Western 12-tet.

Research Projects

I am occasionally asked if Thumtronics can propose research projects associated with its innovations. Please find a list below. I regret that I do not have the time to supervise such projects. If you undertake any research project related to Thumtronics' innovations, I would be happy to learn know how it turns out! :-)

Ease of Learning: Test the efficiency with which human subjects learn musical concepts using the piano and traditional notation vs. the ThumMusic PLUS System, and relate the human subjects’ differences in learning outcomes to differences in the systems’ respective Kolmogorov complexity.

Ergonomic Risk: What criteria are relevant to the ergonomic risk posed by playing musical instruments, what metrics are appropriate to these criteria, and how can all musical instruments’ ergonomic risk be normalized to a single common metric, such that the ergonomic risk of a given novel instrument can be benchmarked against the ergonomic risk posed by various traditional instruments?

Expressive Potential: What culture-independent criteria are relevant to the expressive potential of musical instruments, what metrics are appropriate to these criteria, and how can all musical instruments’ expressive potential be normalized to a single common metric, such that the expressive potential of a given novel instrument can be benchmarked against the expressive potential of traditional instruments?

ThumLine: Implement a ThumLine plug-in for Finale! or Sibelius. Add ThumLine support to Calliope, Lime, LilyPad, or any other open-source music notation editor.

ThumMusic Pedagogy: How should ThumMusic-based music pedagogy be different from traditional music pedagogy, to leverage the strengths of the ThumMusic System? What concepts should be introduced sooner, later, or differently, relative to the traditional system?

ThumMusic-based Music Education Materials: What materials should be developed to make the ThumMusic System’s pedagogical approach simple to deploy, use, and assess? How can modern digital media be leveraged to increase the cost efficiency of ThumMusic-based music education – that is, to maximize the positive learning outcomes while minimizing the cost of deployment, use, and assessment? How can these materials best support traditional approaches to music education?

Pressure-Sensitive Keyboard: Design a pressure-sensitive 57-button Thummer keyboard that uses a button-pressure sensing technology similar to that used by the Sony PlayStation 3 SixAxis game controller.

Motion Sensing: Design a motion-sensing module that uses a motion-sensing technology similar to that used by the Sony PlayStation 3 SixAxis game controller.

QWERTY Thummer: Implement the ThumMusic note-pattern on a standard alphanumeric (QWERTY) computer keyboard such that it emits standard MIDI and/or OSC, thereby allowing electronic musicians to use their laptop keyboards to control musical data using the ThumMusic note-pattern.

Web Thummer: Implement a Web-based applet that implements the ThumMusic note-pattern on a standard alphanumeric (QWERTY) computer keyboard such that the Web page responds to keyboard button-presses by (a) sounding the pressed note, and (b) indicating, on an interactive web page, the buttons/notes currently being pressed/sounded.

ThumTone Synth: Implement an electronic music synthesizer that implements some or all features of the X_System, e.g., (a) Dynamic Tuning, (b) tuning-aligned timbres, and (c) primeness, richness, dissonance, etc..

Dynamic Tuning: Compose music that creates and releases tension using the unique musical effects of Dynamic Tuning (tuning bends, tuning modulations, temperament modulations, new chord progressions, etc.). Induce or deduce the rules governing the effective use of these effects

Commas: Commas are ratios of small whole numbers that arise from the structure of the Harmonic Series to plague traditional music theory. Examples include the Pythagorean comma, the syntonic comma, and the schisma. Tunings such as 12-tone equal temperament "temper out" commas...but they're still in the timbre of harmonic sounds. Tempering the partials to match the tuning could eliminate the commas from the timbre, too. This suggests that pesky commas can be truly eliminated from the music theory of the X_System. Prove that this is or is not so, and if so, demonstrate the musical consequences of the result.

Ethnomusicology: Examine the tunings, timbres, and musical structures associated with the indigenous gamelan, renat, and balafon, to see if they can or cannot be explained by the X_System's pseudo-harmonic approach in a manner identical to the approach's treatment of the Western 12-tet. What do these results say about the X_System's generality?

Music Perception: For the human ear/brain/mind to accept a continuum of pseudo-harmonic tunings and timbres as being tonal, it would need to categorize pitch relationships in a tuning invariant manner. There is a hint of evidence that this is exactly what happens. Perform experiments to explore the perception of tonal structures when using a wide range of pseudo-harmonic tunings, timbres, and temperaments. What does these results say about the tuning invariance of pitch perception?

Musical Paradoxes and Illusions: Explore, using pseudo-harmonic timbres/tunings, a variety of musical paradoxes that are known to exist in harmonic/just music, such as the Missing Fundamental, Combination Tones, Shepard Tones, and Diana Deutsch's paradoxes and illusions. In what ways (if at all) does the perception of these paradoxes and illusions differ (a) among different pseudo-harmonic timbres/tunings, and (b) between harmonic/just timbres and pseudo-harmonic timbres/tunings?