Singing through the syntonic comma

On a Thummer-like keyboard, from C, count up four perfect fifths (from F through C, G, and D, to A). In Just Intonation tuning—which perfectly aligns notes with the harmonic series’ partials—the perfect fifth is 701.955 cents wide, so that’s (4*701.955=) 2807.82 cents. Subtract a couple of octaves (2*1200=2400 cents) from that and you get a remainder of 407.82 cents.

Now, another way to get from F to A on a Thummer-like keyboard is to got rightward by a single major third.  In Just intonation, a major third is 386.31 cents wide. Subtracting this major third from your octave-reduced stack of four perfect fifths, you get (407.82-386.31=) 21.51 cents, which is the syntonic comma, which is the difference between a 10/9 major second (Re) and a 9/8 major second (also Re). The ratio of 10/9 over 9/8 is (9*9)/(10*8) = 81/80, which works out to this exact same 21.51 cents. (I don't want to go into more of the math here.)

21.5 cents is a LOT – more than one-fifth of a semi-tone. It is very audible, even to untrained musicians. If two singers are out of tune by a syntonic comma, you and everyone else in the audience WILL hear the difference, as a strong beating between the two singer’s voices.

This is a significant problem for vocal groups (if they actually want to sound good), especially when singing in close harmonies a capella, as (for example) barbershop quartets do.

If you sing the I-vi-ii-V-I chord progression in Just Intonation, for example, the I chord that you end on will be a syntonic comma lower than the I chord that you started on. This is called “commatic drift.” To avoid this drift, singers must learn to distinguish between two different Re’s: the 10/9 Re at the root of the ii chord and the 9/8 Re in the 5th of the V chord.  This requires singers to be able to sing the two different notes correctly, and–at least as importantly–know when to sing each one and not the other.

I don't see how the use of Do-based minor either helps or hinders a singers ability to correctly choose and sing the right Re.  I would welcome having someone explain this to me.  I can't find much discussion of this issue on the Web, which suggests that I my misunderstanding of the issues is so deep that I can't even choose the right search terms.  Either that, or writing about singing is like dancing about architecture, so none of the relevant discussion is written down.

One of the great advantages of the syntonic temperament is that it tempers out the syntonic comma (hence its name), so chord progressions like the one above “work” without either two Re’s or commatic drift. The cost of this tempering is that the notes of such a temperament are not perfectly aligned with the partials of the Harmonic Series.

There are two ways to address this. One is to adapt the pitches of the notes as they are played to align them with the proper JI intervals. There has been a ton of work on this kind of adaptive tuning. It presumes that the only timbre that’s interesting is the harmonic series, and that all tuning should be adjusted to align with harmonic partials.

Dynamic Tonality supports this. You can use the Tonality Diamond in the TransFormSynth to choose a “major JI” tuning or a “minor JI” tuning, at either vertical end of the tonality diamond. This keeps the tuning at a 5-limit JI, and adjust the 5th partial to align with the 10/9ths Re (in major) or 9/8ths Re (in minor). Of course, if you do this, then moving the tuning slider has no effect, because your use of the tonality diamond has indicated that you want to use a JI tuning, not a tempered tuning.

On the other hand, Dynamic Tonality can also address the problem by tempering the timbre to match the current tuning. Just keep the tonality diamond’s dot a near the vertical center of the tonality diamond, along the axis from “fully harmonic” timbres to “fully tempered” timbres, and you can adjust the tuning slider to your heart’s content. This option was not available to previous generations of theorists, because they didn’t have the necessary computing power. But it’s available to us. Even singers, singing into microphones, can have their amplified timbres adjusted in real time to fit the current syntonic tuning (such as 12-tet).

(I have this recurring nightmare that the only popular use of Dynamic Tonality will be to make 12-tet more consonant, thereby locking it in as the de facto standard forever.)

Point being, that Dynamic Tonality makes the problem of the syntonic comma completely disappear by tempering it out of the harmonic series, thus eliminating the syntonic comma at its source.

La-based minor revisited

I have gotten a lot of feedback on my earlier post, Why La-based Minor?

 

In brief,

• my understanding of Do-based minor was wrong.
• I am now considering a different scheme, which seems to me to combine the best of both La-based and Do-based minor.

My new thinking, which is still somewhat half-baked, goes like this.

Do-based minor emphasizes the parallel minor, while La-based minor emphasizes the relative minor. Why emphasize one over the other?  Why not clearly distinguish between the two?

Case 1: Starting in Major
Let’s assume that you’re going to notate/play a piece starts in major (diatonic Do-mode), so you transpose the pitches under the keyboard to move the desired tonic pitch to Do. Let’s say that you’re playing in C Major, so Do is C, La is A, Mi is E, and Me is Eb.

When the piece begins, in Do-mode, all is well. The tonic is on Do (C), and that mode’s third degree is Mi (E), which is where we pegged the keyboard, so those notes have those pitches.

1a) If the piece wanders from Do-mode into its parallel minor (C minor), all is well. The tonic stays on Do (C), and Do-mode’s third degree is Me (Eb), but no new transposition of the keyboard/notation is necessary, because those notes already have those pitches.

1b) If the piece wanders from Do-mode into its relative minor (A minor), all is well. The tonic moves from Do (C) to La (A), and La-mode’s third degree is Do (C), but no new transposition of the keyboard/notation is necessary, because those notes already have those pitches.

The next question is, “where do the scale dots and tonic indicator go?”

We're starting in diatonic Do-major, so the scale dots are on the usual diatonic notes, Do, Re, Mi, Fa, So, La, and Ti, with the tonic indicator on Do.

1a) Moving from Do-major to Do-minor, the scale dots change to Do, Re, Me, Fa, So, Le, and Te, with the tonic indicator staying on Do.

1b) Moving from Do-major to La-minor, the scale dots stay the same (i.e., on Do, Re, Mi, Fa, So, La, and Ti), but the tonic indicator moves to La.

The interplay of the scale dots and the tonic indicator show, in JIMS staff notation, what is happening.

• If the scale dots change, but the tonic indicator stays on the same note, then the music has moved to a parallel mode.
• If the scale dots stay the same, but the tonic indicator moves to a different note, then the music has moved to a relative mode.

This is not a factoid to be memorized, per se, but rather something which can be observed from the note-patterns on JIMS keyboard as one plays.

This system usefully distinguishes Do-major's relative minor (La-minor) from its parallel minor (Do-minor). The two would be notated using different notes on JIMS staff, and played using different buttons on JIMS keyboard.

When used strictly by “La-based minor” singers, the notation and keyboard could be transposed such that both the parallel and relative minors always used La as their tonic. However, that would NOT be the general case. the main difference between my previous proposal and this one.

Case 2: Starting in Minor
Now, let’s assume that you’re going to notate/play a piece which starts, and remains primarily, in A minor, so we map A to La.

2a) If the piece wanders from minor into its relative major (C Major), all is well. The tonic moves from La (A) to Do (C), and Do-mode’s third degree is Mi (E), but no new transposition of the keyboard is necessary, because those notes already have those pitches.

2b) If the piece wanders from minor into its parallel major (A Major), all is well. The tonic stays on La (A), and La-mode’s third degree is Do (C)…wait a minute. That’s not right. We’re talking La-MAJOR now, not La-MINOR.

Case 2b above is (I suspect) at the heart of the conflict between the La-based minorists and the Do-based minorists. Who would ever expect to find a major scale with La as its tonic?

If Case 2b's minor-to-parallel-major-and-back movement dominated a given piece, then it would make sense to start the piece in Do-minor. Then, when the mode changed from the primary minor key to its parallel major, the major mode’s tonic would be Do, as one would normally expect. Using D-minor in this way would emphasize that the core relationship in this piece was parallel, not relative.

Let's look at the scale dots and tonic indicator in Case 2.

2a) Starting in La-minor, the scale dots are on the usual diatonic Do, Re, Mi, Fa, So, La, and Ti, with the tonic indicator on La. Moving to the relative major, the scale dots remain the same, but the tonic indicator moves to Do.

2b) Starting in Do-minor, the scale dots are on Do, Re, Me, Fa, So, Le, and Te, with the tonic indicator on Do. Moving to the parallel minor, the scale dots change to Do, Re, Me, Fa, So, La, and Ti, with the tonic remaining on Do.

If the music moves farther than one relative or parallel step away from the tonic, in either direction, then it has modulated (hasn't it?). One could either keep shifting the stack of scale dots to reflect the changing set notes in the current diatonic scale (much like introducing more sharps and flats into a key signature, with all of the disadvantages thereof), or simply transpose JIMS staff and keyboard (using a transposition indicator and user-interface gesture, respectively).

Limitations
If a piece's tonal center is ambiguous, then no tonally-focused notation (like JIMS) is going to offer significant advantages over less-tonally-focused notations (like traditional notation).

However, most music played and listened to by the majority of the people in the First World is strongly tonal (and even strongly modal, if one considers the major scale to be the Ionian mode), which plays to JIMS' strengths, so I don't think that this limitation counts for much.

Advantages
The above-described approach seems to me to combine the best of both Do-based minor and La-based minor, by distinguishing unambiguously between the relative and parallel relationships. The person notating a song would need to do a significant amount of work to analyze what’s happening in a given piece, in order to notate it correctly -- but that's a GOOD thing, because once this analysis is done by the notator, it is very easily accessible by the student and/or performer.

 

Having a clear distinction, in JIMS notation, between parallel and relative intervals helps distinguish between notes that have the same name but are a comma or two apart (in Just Intonation), thereby helping singers it the right notes. Nonetheless, for purely vocal music in the La-based minor tradition, one could transpose JIMS notation (and perhaps an accompanying JIMS keyboard) to keep the scale dots constant, i.e., to use a La-based minor whether that minor was relative or parallel. But this would not be *required* under the above-proposed revisions to the JIMS system.

 

I think that this refinement is a considerable improvement to JIMS. It exposes a meaningful difference—the difference between relative and parallel keys—in a clear and unambiguous manner. This is in line with JIMS' neo-Riemannian roots.

 

Perhaps this refinement is sufficient to make JIMS useful to those who teach music using a Do-based minor system. I hope so. In my wildest dream, I imagine that JIMS, with this refinement, might be sufficient to heal the centuries-long rift between the La-minorists and the Do-minorists.

Comments and corrections welcome!  :-)

Meantone temperament

I spent most of yesterday editing Wikipedia's article on the meantone temperament.

By mid-2006, it had become clear to Bill Sethares, Andy Milne and I that our music theoretical work was focused on what have historically been called "extended meantone temperaments." We considered using that name in our own work, but we rapidly learned that the tuning community would have none of it. The term "meantone" was loaded down with an oppressive weight of historical baggage. For us to redefine the term, even slightly, or to broaden the scope of its usage, was anathema. Hence, we called our thingy the "syntonic temperament," and its valid tuning range the "syntonic tuning continuum," and so on, to avoid violating sacred historical precedent.

But it's really just a continuum of extended meantone tuning, in which we're tempering timbres in addition to notes, and thereby retaining the option of consonance across the entire tuning continuum.

The jargon of traditional tuning theory seems to me to be quite seriously muddled, especially in its failure to distinguish cleanly between a "temperament" (a set of rules, defined by a comma sequence, for mapping partials to notes) and a "tuning" (a combination of generator widths). This lack of distinction probably arises from tuning theory's obsession with the Harmonic Series (which is perfectly understandable, given the dominance of harmonic timbres in the history of Western music). If one assumes that the only timbres that matter to music are harmonic timbres, then the "mapping of partials to notes" is an irrelevant step, so tunings and temperaments become essentially the same thing...as the historical jargon-muddle reflects.

All of which made re-writing Wikipedia's Meantone temperament article harder.

NOw, the fun part will be seeing how long my edits last. Will the tuning community's Old Guard simply revert them away, hence defending tuning theory's status quo? Our will the edits be accepted, albeit perhaps with slight revisions?

Now that our theory is backed up by a slew of peer-reviewed scientific papers, it would be rather difficult to justify simply reverting the edits out of existence. Passions can run quite high in such tiny communities, however, so I am loathe to predict the outcome.

CircularSlider

Here's my first attempt at a Spark-based Flex control, using Flex's new skinning architecture. It's a circular slider.  This (uncopyrighted) file, posted by Jacob Goldstein, was a big help.  Thanks, Jacob!  :-)



There's some seriously-weird stuff happening in this control.
- Firstly, it attempts to draw a horizontal slider's track, for reasons that I haven't found yet. This "ghost track" appears to be intercepting some of the slider's events, which causes the thumb to jump around erratically during a thumb-drag.
- Secondly, it resists resizing. Apparently, I need to locate methods that faciliate resizing Spark controls and override them.

It's a start. ;-)

Custom Keyboard skins?

These days, books can be printed "on demand" ...can keyboard skins?

Keyboard skins are the ultra-thin latex sheets that fit over the keys on a computer keyboard, like the one at right.

I'd like to be able to offer a keyboard skin pre-printed with the note-pattern for JIMS Keyboard, but I can't afford to order them up front...and who knows how well they'd sell? They'd have to be printed "on demand."

If you know of any firm that offers "on demand" keyboard skins, please let me know.

Back to coding!

I've spent the last couple of months storyboarding my first batch of lessons, on Musical Sounds, the Harmonic Series, the Diatonic Scale, Modes of the Diatonic Scale, Diatonic Intervals, and the Major-Minor Axis.

Now, I'm going to start coding them up as interactive lessons, using Flex/Flash and some video. This is going to take me a while, as my coding skills are still pretty rusty. Adobe's about to release Flex 4, which I should probably use instead of Flex 3, so that my de-rustified coding skills can be as up-to-date as possible.

Once my first batch of lessons is online and gathering feedback, I expect to start storyboarding the next batch, covering Diatonic Triads and Modal Harmony, making extensive use of JIMS™ Tonnetz, which is of course aligned with JIMS™ Keyboard (see www.igetitmusic.com/papers/Perception.pdf).

The tonnetz is a great tool or exposing the relationships among triads. Consider this depiction of the neo-Riemannian PLR relationships between the C minor triad, labeled Q, and its three neighbors on a tonnetz:

This graphic shows that performing, on Q, the

• Relative operation produces Q's R-major triad (Eb-G-Bb);
• Parallel operation produces Q's P-major triad (C-E-G);
• Leading-tone exchange operation prodices Q's L-major triad (Ab-C-Eb).

Obviously, just because I'm using a tonnetz doesn't mean that I have to emphasize a neo-Riemannian approach to harmony. I can use a more neo-Rameau-ian(?), root-movement-oriented approach instead. The point is that using a tonnetz enables me to go either way, or to mix and match as appropriate.

The best thing about this is that JIMS Tonnetz is not some abstract representation of tonal space, but is, instead, a concrete aspect of JIMS Keyboard, as implemented on a computer's standard QWERTY keyboard:

Being able to relate JIMS Tonnetz directly to the sound-controlling JIMS Keyboard should make it possible for me to SHOW people how chords relate to each other, rather than trying to EXPLAIN it.

Also, the tonnetz is the dual graph of Schoenberg's chart of the regions, which is rather handy.

However, for now, I must stop thinking about harmony and start thinking about coding up the first batch of lessons.

Once upon a time...

Below is a copy of another post recently made to a discussion on Daniel Levitin's Facebook page.  It's a re-statement of the content in this older post, but I like the clarity of this new restatement.

In brief, my argument is that (a) there's a difference between "musical talent" and "the ability to handle arbitrarily high UI loads," and that (b) reducing the UI load in a given domain in can increase the success-rates that novices enjoy in that domain.

----------------------------------

Darin wrote:

> If someone has innate talent, then as that person
> practices and progresses, he or she will recognize
> the progress, will be recognized by others for the
> progress, and as a result, will develop real passion
> for the pursuit. If somone does not have innate
> talent, such person will practice, not make much
> progress, will see the lack of progress, and be told
> about it by others, and the passion will not take
> hold, but will wither, and the person will move on
> to some other endeavor.

>
> People with real talent are few.

I respectfully disagree. "Talent" has absolutely nothing to do with it. Please let me explain by example.

1. Once upon a time, the Cherokee were illiterate. The English alphabet was a poor fit with the Cherokee language, so efforts to spread literacy among the Cherokee failed. Then one Cherokee invented a writing system that fit Cherokee perfectly, enabling literacy to sweep the Cherokee almost overnight. Did the Cherokee suddenly gain a "talent" for literacy?

(http://en.wikipedia.org/wiki/Cherokee_syllabary)

2. Once upon a time, the Koreans were illiterate. The Chinese ideographic script was a poor fit with the Korean language, so the efforts to spread literacy among the Koreans failed. Then the Koreans invented Hangul, which fit Korean perfectly; now, "a bright child can become literate in a day, and a dull child in ten." Did the Koreans suddenly gain a "talent" for literacy?

(http://en.wikipedia.org/wiki/Hangul)

3. Once upon a time, physicists couldn't puzzle out the interactions of quantum mechanics, nor could students learn about them efficiently. Then, Feynman invented "Feynman diagrams," and students could understand such interactions in less than a semester. Did physics students suddenly develop a "talent" for quantum mechanics?

(http://en.wikipedia.org/wiki/Feynman_diagram)

4. Once upon a time, European mathematicians could not conceive of "x to the power of y," because Roman numerals could not notate the concept, and the Roman abacus could not calculate it. Then Fibonacci explained how to use Arabic (actually Hindu) numerals and algorithms, and the scope of European mathematical thought widened dramatically. Did Europeans suddenly develop a "talent" for mathematics?

(http://en.wikipedia.org/wiki/Liber_Abaci)

5. Once upon a time, the "value" of a church singer dependend as much on "how many songs he had memorized" as on how well he could sing them, because all songs had to be memorized by rote. Then Guido d'Arezzo invented staff notation and solfeggio, enabling novices to become valuable church singers much more rapidly. Did such novices suddenly gain a "talent" for singing?

(http://en.wikipedia.org/wiki/Guido_of_Arezzo)

6. Once upon a time, all mathematical calculations had to be executed longhand, making them expensive and error-prone. Then logarithms were invented, and many calculations could be accelerated by looking them up in tables of pre-calculated logarithms. Did this suddenly increase people's "talent" for calculation? Did the invention of the slide rule? Of the pocket calculator?
(http://en.wikipedia.org/wiki/Logarithms#History) (http://en.wikipedia.org/wiki/Calculator#Pocket_calculators)

7. Once upon a time, learning and practicing chemistry was extraordinarily difficult, with the properties of each element having to be learning individually, and its guiding principles (e.g., phlogiston) being fundamentally incorrect. Hence, few gained mastery over chemistry. Then Lavoisier discovered the combustion principle, Mendeleev invented the Periodic Table of the Elements, and Bohr deduced the planetary model of the atom, all of which reduced the investment of time necessary to master chemistry, thereby dramatically increasing the percentage of the human population that could afford to master chemistry. Did students suddenly gain a "talent" for chemistry?
(http://en.wikipedia.org/wiki/History_of_the_periodic_table)
(http://en.wikipedia.org/wiki/Phlogiston#History)
(http://en.wikipedia.org/wiki/Bohr_model)

YESTERDAY, learning and practicing music-making was extraordinarily difficult, with the patterns of each key, clef, scale, mode, tuning, instrument, timbre, etc., having to be learning individually, and its guiding principles (e.g., 12-tone equal temperament) being fundamentally incorrect. Hence, few gained mastery over music-making. Then [insert here a list of scientific discoveries and technological inventions that, arguably, have not yet been made], all of which reduced the investment of time necessary to master music-making, thereby dramatically increasing the percentage of the human population that could afford to master music-making. Did students suddenly gain a "talent" for music-making?

Of course not.

In all of the above examples, the problem was a lack of technology, not of "talent." The traditional technology of music-making—staff notation, instruments, and theory—is the problem. As with all of the above examples, fixing the technology will fix the problem.

Until we fix the technology of music-making, it hardly seems fair to blame the victims—music students—for their "lack of talent." (Why beholdest thou the mote that is in thy brother's eye, but perceivest not the beam that is in thine own eye?
(http://www.godrules.net/para/luk/parallelluk6-41.htm)

To argue otherwise is to argue that either

1. all of the above examples are wrong, or that
2. "music is different."

I would welcome the opportunity to dismember either argument. ;-)

Respectfully,

Jim Plamondon
Unaffiliated Musical Heretic

The (Isomorphic) Cortical Topography of Tonal Structures

At the request of Daniel Levitin, I added this post to his Facebook page's discussion board, which I will also paste below.

-------------------------

Gentlepersons,

Most work in music cognition assumes 12-tone equal temperament, which is a perfectly reasonable starting point. However, I suspect that the findings thereof can be easily generalized to alternative tunings, using some recent discoveries.

The first discovery is the two-dimensional syntonic temperament. Its tuning continuum includes nearly all of the tunings ever used by humankind in the real world, from the from the 7-tet ("7-tone equal temprament," hence "7-tet") tunings related to the timbres of the Thai ranat and African balafon to the 5-tet tunings related to the timbres of the Indonesian gamelan, with 17-tet, Pythagorean, 12-tet, the meantones, and an infinite number of other tunings in between.

The second discovery is the relationship between two-dimensional temperaments, such as the syntonic temperament, and two-dimensional isomorphic keyboards.

If the pattern of notes on a two-dimensional keyboard is generated by the same two intervals that generate a two-dimensional temperament (such as the syntonic temperament), then the keyboard will be "isomorphic" with that temperament. What this means is that any given interval in that temperament will have the "same shape" in every tuning of that temperament. Therefore, any given combination or sequence of intervals also has the "same shape" everywhere on an isomorphic keyboard, in every tuning of that temperament. This is tuning invariance.

For a demonstration of tuning invariance on an isomorphic keyboard (with embarrassingly-over-the-top commentary), please see this video.

This tuning invariance applies to all syntonic tunings, including tunings that are equal and non-equal, regular and irregular (such as “well-temperaments”), and also "rank-2, 5-limit Just Intonation" tunings (see proofs here or here).

Alternatively put, syntonic tunings include Western (Pythagorean, 12-tet, 1/4-comma meantone, 31-tet, “circulating”) and non-Western (Indonesian, Thai, Mandinka African) tunings, and the JI tunings used both in the West and in non-Western cultures (which rarely exceed 5-limit; the blues is, arguably, 7-limit, but that case is also well-handled by an isomorphic note-layout).

The one non-syntonic temperament which I can find to have been used by humankind in the real world is the (Turkish) schismatic temperament. Because its generators are the same as those of the syntonic temperament, it is compatible with the syntonic temperament's isomorphic keyboards, and hence with the conclusions of this posting—but it is a special case, beyond the scope of this posting, so I won't mention it again.

The third discovery—at least, we haven't been able to locate any prior art yet—is that such isomorphic keyboards include within their pattern of notes a tonnetz, as described by Euler/Oettingen/Riemann etc. (see Figures 7 and 8 in this paper.) An important point is that such a tonnetz is tempered; that is, it is not based on "ratios of small whole numbers" (i.e., Just Intonation), but rather on a mapping from these "just" intervals to intervals of the syntonic temperament.

Such a "tempered" tonnetz has the same tuning invariance as the isomorphic keyboard from which it is drawn. Hence, the relationships among the notes on such a tonnetz are tuning invariant, too.

The tonnetz is (I believe) well-known to be the dual graph of the "chart of the regions" described by Schoenberg and others (see this book, p. 105). Hence, any such tempered "chart of the regions" is likewise tuning invariant.

The map of perceptual tonal space described by Krumhansl, Janata, and other cognitive psychologists, is precisely such a tempered "chart of the regions." Hence, this map *ought* to be tuning invariant, too.

The hard-wiring of a tuning invariant map of perceptual tonal space could help explain both

1. The diversity of real-world tunings, in that an infinity of syntonic tunings are compatible with such a perceptual space, and
2. The limitations on that diversity, in that
• only the tunings of the syntonic (and perhaps schismatic) temperament fit this perceptual space, and
• a culture’s dominant instruments must produce a timbre that is closely "related" to such a tuning (wherein "related" has the meaning described here).

The latter point must not be overlooked in any related experiments. For example, using harmonic timbres for all tunings will produce invalid results.

If perceptual tonal space were indeed found to be tuning invariant, then this could would be an important scientific step towards a truly universal theory of music.

Neither I nor my collaborators have the skills or knowledge of musical cognition sufficient to execute the kinds of experiments needed to explore this issue further. We would be delighted to help, though. Ping me at jim@iGetItMusic.com.

Thanks!

Jim Plamondon
Unaffiliated Musical Heretic

The Cortical Topography of Tonal Structures

Here's an interesting experimental result, from Per Janata's article The Cortical Topography of Tonal Structures Underlying Western Music in the December 2002 edition of Science:

In contrast to distributed cortical representations of classes of complex visual objects that appear to be topographically invariant (26), we found that the mapping of specific keys to specific neural populations in the rostromedial prefrontal cortex is relative rather than absolute.

Within a reliably recruited network, the populations of neurons that represent different regions of the tonality surface are dynamically allocated from one occasion to the next. This type of dynamic topography may be explained by the properties of tonality structures. In contrast to categories of common visual objects that differ in their spatial features, musical keys are abstract constructs that share core properties. The internal relationships among the pitches defining a key are the same in each key, thereby facilitating the transposition of musical themes from one key to another.

Two observations about this:

1. The brain may recognize individual pitches using "fixed Do," but it recognizes tonal relationships using "movable Do with a La-based minor."
2. The neural topography revealed by this experiment is compatible with an isomorphic note-layout, and therefore it ought to be tuning invariant. This observation could enable the experiment's results to be generalized beyond 12-tet to include not only 12-tet, but nearly all of the pre-modern and non-Western tunings ever used by humankind.

Triad names

Aha!

Consider the traditional naming of triads (built by stacking thirds from the root upwards):
- m3, m3: diminished triad
- m3, M3: minor triad
- M3, m3: major triad
- M3, M3: augmented triad

What is "diminished" about a diminished triad? Its diminished fifth, according to the traditional interval-naming scheme.

What is "augmented" about an augmented triad? Its augmented fifth, according to the traditional interval-naming scheme.

That is, the traditional naming-rule for triads is:
- If both of the triad's thirds are the same width, name the triad after the width of the fifth.
- Else, name the triad after the width of its bottom third interval (i.e., the one between the chord's mode's 1st and 3rd degrees).

If JIMS re-names the narrow fifth "minor" and the width fifth "major," as discussed below, then the traditional triad-naming rule doesn't make sense within JIMS. That means that JIMS either needs to (a) not rename the fifths, (b) rename the triads, or (c) redefine the triad-naming rule.

I like the latter option best (re-defining the triad-naming rule). With this approach, JIMS' rule for naming triads would be:
- If both thirds are minor, the triad is "diminished."
- If the thirds differ, name the triad after the width of its bottom third.
- If both thirds are major, then the triad is "augmented."

This rule produces the same triad names as the traditional rule, but without relying on the name of the triad's fifth.

I like this approach because it recognizes that the triad is what is diminished or augmented, not the triad's fifth, which is as independently major or minor as the thirds are.

I'll have to think about the naming of seventh chords, too. There seems to be a lot of variation in seventh-chord naming anyway, between classical and jazz traditions, so a little more variation wouldn't be shocking.

Simplified solfege

One simplification which JIMS could adopt, but has not (so far), is a simplification of the solfége naming rules for sharpened and flattened notes.

The most common current rules are these:
- The vowel 'e' indicates intervals that are "flattened from the diatonic." There's a conflict with "Re," so "Ra" is used to indicate the flattened form of Re. The 'a' in "Ra" the matches the 'a' in Fa and La, of course, which confuses the rule further.
- The vowel 'i' indicates intervals that are "sharpened from the diatonic." There are two conflicts, with Mi and Ti. Instead of providing unique names for the sharpened versions of these notes, Fa and Do — which are enharmonic (in 12-tet) with the missing note-names — are used.

This traditional solfége system is not well-thought-out in two ways:
- The use of the vowels 'e' for flattened intervals and 'i' for sharpened intervals conflicts with the occurrence of those vowels in the diatonic scale, leading to exceptions and loss of obvious meaning.
- The absence of names for the sharpened versions of Mi and Ti restricts its use to 12-tone equal temperament. All other temperaments have sharpened versions of Mi and Ti that are distinct from Fa and Do, respectively. (Besides, the difference between Mi and its sharpened version is an augmented unison, not a minor second — a difference which should be exposed, not hidden.)

They followning rules would be much simpler:
- The vowel 'u' (pronounced as in moo, coo, and new) would mean "flattened from the diatonic" (i.e., diminished, using the terminology proposed here).
- The vowel 'y' (pronounced as in tie, lie, and my) would mean "sharpened from the diatonic" (i.e., augmented).

With these proposed rules, the naming of chromatic variations of diatonic notes would be entirely consistent within JIMS, and would also be consistent with the naming of diminished and augmented intervals within JIMS. Diminished intervals end in 'u'; augmented intervals end in 'y', every time.

Another reason to like the above rules is that they make the 'i' endings of Mi and Ti stand out. The only notes ending in 'i' are at the lower ends of the only two minor seconds in the diatonic scale. Unfortunately, the rest of the diatonic scale's notes' letters do not convey similar structural meanings. I've played with a lot of alternatives to the standard Do Re Mi names, and none of them are significantly better than the standard names, especially given how familiar these names are, even to non-musicians.

As to the "singability" of these vowels...

Singing 'u' is no problem. When pronounced as indicated, it's a "pure vowel" (monophthong, meaning "one vowel").

(How is it that the Greek word "phthong," meaning "vowel," has six consonants, but only one vowels? I would expect the word for "vowel" to be an anagram of the vowels — something like "youwaei." This approach does not work terribly well for consonants, I'll admit. "Bcdfghjklmnpqrstvxz" is rather hard to pronounce.)

Singing 'y' could be a problem, because it is a diphthong (i.e., two vowel sounds smooshed together, as in "eye," "you," "boy," and "cow"). To pronounce a diphthong correctly, one must slide the tongue from one location to another while speaking it. If you're singing the vowel, it's not obvious exactly when to do this tongue-slide, which makes it harder to sing (and learn to sing) diphthongs than pure vowels.

However, the diatonic scale's traditional solfa names already include diphthongs (notably Do, So, and Re). This suggests that singing 'y' would not be a major problem.

So, why doesn't JIMS use the more-complex traditional solfa-naming rules, instead of these simpler rules?

Because one of the groups that is most under-served by traditional notation is the vocal music community. Singers can transpose their voices on the fly, but transposing their notation is much harder, making JIMS attractive. Internationally-popular vocal instruction methods such as Kodály are based on "movable Do with a La-based minor." These methods would be much better-served by JIMS than by traditional notation.

If JIMS uses the solfa note-names that Kodály users are familiar with, then JIMS can slide right into their established practices, and simply work better for them than traditional notation does today.

On the other hand, if JIMS uses different solfa note-names than the Kodály standard, however, then this would be a barrier to their use of it.

So far, the benefits of being compatible with the Kodály method have, in my opinion, outweighed the advantages of shifting to the simpler system. However, Kodály is used almost exclusively in schools, and I'm not targeting schools with JIMS, because they are so incredibly resistant to change.

We'll see.

Unison and octave?

Well...darn.

In an earlier post, I explored the possibility of eliminating the "perfect/imperfect" interval class distinction. When considering the "perfect" class, I only considered the fourths and fifths, kinda sorta overlooking the (rather important) unison and octave.

Oops. My bad.

I can't see how one could have minor and major octaves or unisons. They don't have two different sizes within the diatonic scale; they only have one. Each can be augmented and diminished, but those are chromatic operations, not diatonic operations.

So, I think we're stuck with two interval classes.

Let's define "perfect" to mean "has only one interval size in the diatonic scale." That definition fits unison and octave, but no other diatonic intervals -- specifically, not the fourths or fifths.

"Imperfect," then, would mean "has two sizes in the diatonic scale." This is true for all intervals that are not unison or its octaves (i.e., 2nds, 3rds, 4ths, 5ths, 6ths, 7ths, [not 8ths], 9ths, 10ths, etc., ad infinitum). This definition of "imperfect" moves the fourths and fifths out of their traditional place in the "perfect" class, and into the "imperfect" class.

"Perfect" and "imperfect" are lousy names for these interval classes, because they are meaningless. That is, the distinguishing feature of each interval class is not encoded in the class names. Better names would be "one-width" and "two-width," for example, because these names "say what they mean."

One could still call a diatonic octave a "perfect" octave, even though its interval-class name was "one-width." The class name does not have to match the interval-name. The major third is not called the "imperfect third," after all.

Tying this all together....

There are two interval-classes:
- One-width: unison and its octaves.
- Two-width: all other diatonic intervals.

Each interval-class has its own interval-naming rule:
- One-width: diminished/perfect/augmented
- Two-width: diminished/minor/major/augmented

In the above interval-naming rules, the '/' symbol corresponds to the "augmented unison." Therefore, when the rule is read from left to right, each name denotes a note that is an augmented unison higher than the previous name. In the syntonic temperament, the augmented unison is the vector [-4, 7] (down four octaves, then up seven tempered major fifths).

The augmented unison should not be confused with the minor second, which, in the syntonic temperament, is the vector [3, -5] (up three octaves, then down five tempered major fifths). Augmented unisons separate Se, So, and Si (all of which share the same leading consonant, to show that they are all related to the same note of the diatonic scale), whereas a minor seconds separates Mi from Fa (which have different leading consonants, to show that they are different notes of the diatonic scale).

BTW, to reverse an interval-vector's direction, one reverses its signs. For example, to turn the "augmented unison" vector [-4, 7] into the "diminished unison" vector, one changes the signs of the numbers in the vector, getting [4, 7]. Likewise, to turn the "minor second up" vector [3, -5] into the "minor second down" vector, one changes the signs of the numbers in the vector, getting [-3, 5]. Same magnitude, opposite direction.

In summary, the proposed interval-naming changes are:
1. To move the fourths and fifths out of the perfect interval-class into the imperfect interval-class.
2. To rename the perfect and imperfect interval classes to "one-width" and "two-width," respectively, so that the class names "say what they mean."

I am still debating whether or not to incorporate this change in JIMS. On the one hand, this change would make interval-naming much easier to teach, and would expose many other meaningful relationships. However, these changes would also be the first in JIMS to break compatibility with the vocal music instruction methods based on "movable Do with a La-based minor," such as Kodály.

Maintaining compatibility with the Kodály method could significantly increase JIMS' rate of adoption, because its users are particularly under-served by traditional notation (and accompaniment instruments).

Nobody's "perfect"

In my previous blog post, What is a "perfect" interval, really?, I described the traditional "interval classes" as:
- Perfect: diminished/ perfect /augmented
- Imperfect: diminished/ minor/major /augmented

Then I asked, "is this distinction meaningful? And if so, how?"

The short answer is "no, it's not meaningful."

The long answer, provided by Andy Milne, is that all Moment of Symmetry (MOS) scales (also called well-formed scales), including the diatonic scale, have two interval sizes. The wider interval could always be called the "major" interval, and the narrower one, the "minor" interval. That's the interval-naming rule associated with the "imperfect" interval class.

In short, the traditional distinction between "perfect" vs. "imperfect" intervals is a distinction without a difference. One could classify all intervals as being "imperfect," in any MOS scale, without loss of information. Because all tonal intervals would therefore fall into a single class, the need for any such classification would be removed.

An Alternative to Perfection
For example, the diatonic scale includes two fourths, one smaller (e.g., F to B) and one larger (e.g., C to F). Traditionally, fourths are members of the "perfect" interval class, with the narrower interval being named the perfect fourth and the wider interval being named the augmented fourth. However, these intervals could instead be named the minor fourth and major fourth, respectively.

Likewise, the diatonic scale includes two sizes of fifths, one smaller (e.g., B to F) and one larger (e.g., C to G). Traditionally, fifths are members of the perfect class, with the narrower interval being named the diminished fifth and the wider interval being named the perfect fifth. However, these intervals could instead be named the minor fifth and major fifth, respectively.

Using only the imperfect interval class's interval-naming rule would result in the following interval names (with changed names in italics):

There are two reasons why using two naming rules, when one will suffice, is bad.

Error of the First Type
The first reason why its bad to make a distinction between perfect and imperfect intervals is that this distinction implies that there is a difference between them, when no such difference exists. The diligent student, on learning of the distinction, will naturally attempt to discover what the difference is. When the student fails to find such a difference (because there is no difference to find), this failure will tend to reinforce the student's feeling that music is Kafkaesque — that is, "marked by a senseless, disorienting, often menacing complexity." Alternatively, the student may internalize the failure, concluding that "I'm just too stupid to understand music."

Either way, this is bad pedagogy.

There's a simple rule to follow, while designing systems, that can help one avoid Kafkaesquery: Occam's Razor, which states that "entities should not be multiplied unnecessarily." In this case, the interval-naming rules of the perfect and imperfect interval classes are the "entities." If only one rule is needed, then Occam's razor says, don't create more rules.

In short, the first reason why it is bad to have two interval classes, when one will suffice, is that having two classes implies the existence of meaningful patterns that do not, in fact, exist. Exposing meaningless patterns is a false positive error (also known as a "type I error").

Error of the Second Type
The second reason why it is bad to have two interval classes, when one will suffice, is that having two classes hides the existence of meaningful patterns. Hiding meaningful patterns is a false negative error (also known as a "type II error").

Consider the relationships of interval-names to modes. Using only the one "imperfect" interval class, the relationship between interval-names and modes reveals the major-minor axis through the modes. The table below shows the diatonic modes in circle-of-fifths order, from Fa to Ti (using movable Do with a La-based minor, of course):

This table (and hence the one-class interval-naming system) exposes a number of clear patterns.
- Fa-mode (Lydian) is "the most major" mode; all of its intervals are major.
- Ti-mode (Locrian) is "the most minor" mode; all of its intervals are minor.
- As one moves down the table from Fa-mode to Ti mode (and hence around the circle of fifths), each mode introduces a new minor interval, making it "more minor" than the previous mode.
- The "new minor interval" of each mode is always the interval from that mode's tonic to Fa.

It's not that these patterns were impossible to detect when using two interval-naming rules; it's just that using a single rule makes these patterns easier to detect. Increased ease-of-detection reduces the chance of false negative errors. Alternatively put, it increases the chance that students will detect (and therefore have a chance to understand) these meaningful patterns.

Fossilized Tradition
The music establishment uses this overly-complex interval-naming system for one reason, and one reason only: because the music establishment uses this overly-complex interval-naming system. It's done because it's done. A few centuries ago, some intervals were renamed in accordance with the major/minor system, but some of the old names stuck.

Consonance
I am aware that the "perfect" moniker is applied only to "the most consonant" intervals, i.e., the unison, octave, wide fifth, and narrow fourth. At most, that's a reason
- to name the narrow fourth "perfect" while calling the wide fourth "major," and
- to name the wide fifth "perfect" while calling the narrow fifth "minor."

Diatonic and Chromatic Intervals
Another meaningful distinction that is blurred by the traditional interval-naming scheme is the difference between diatonic intervals and chromatic intervals. By calling the FaTi interval an "augmented fourth" and the TiFa interval a "diminished fifth," no distinction is made, in the naming of intervals, as to whether they are diatonic or chromatic.

Using only the "imperfect" interval-naming rule makes a clear distinction between these two different kinds of intervals: minor and major intervals occur in the diatonic scale, whereas diminished and augmented intervals do not. This is a meaningful difference, easily detected.

Communication Compatibility
Inspection of the table of interval names above shows that the likelihood of confusion, between people using the old names and those using the new, is quite low.

It's OK to have many different names that refer to the same thing, as in "Bob," "my brother Bob," "my father Bob," "Mr. Bob Johnson," etc. That's a many:1 mapping. No problem. So when a traditionalist hears someone say "minor fifth" for the first time, he'll need to learn a new name, but there's no ambiguity. Four of the five changed names fall into this many:1 category (minor fourth, major fourth, minor fifth, major fifth).

Two changed names, however, use existing names differently that they were used before (augmented fourth and diminished fifth). When a traditionalist hears someone say "diminished fifth," he'll have to wonder...is that an "old style" diminished fifth, or a "new style" diminished fifth?

That would be a problem...if it were to ever come up. However, it won't, because the "doubly altered" intervals with which new-style diminished fifth and augmented fourth may be confused are exceedingly rare. That is, someone using the new-style interval names would almost never say "diminished fifth" or "augmented fourth;" it simply wouldn't come up.

Likewise, if a person familiar with the new-style names heard a traditionalist describe an interval as a "diminished fifth," the new-stylist would say, "WTF? Why would you use a chromatic fifth there, instead of a diatonic fifth?" and after just one such interaction, the name-mapping would be clear to both parties in a very memorable way.

Hence, the ease-of-learning benefits of the proposed new names are very likely to outweigh the compatibility cost of the change.

Voronoi Expert Found!

Alan Shaw noticed my blog post Help! Voronoi expert wanted and kindly volunteered to share his 20+ years of Voronoi-specific experience.

From my blog post's description of the problem (plus a few other details), he was able to develop a Flash applet to show the Voronoi relationships among regular lattice cells as the lattice is systematically deformed. This applet is a great example of data visualization.

By playing with this applet, Bill Sethares and Andy Milne (my partners in musical heresy) were able to deduce the relevant mathematical relationships. They're now working out the details & proofs with Alan.

Data visualization is remarkably powerful, isn't it? As the Overview to Wikipedia's article on data visualization states (or, at least, as it stated when I wrote this):

"The main goal of data visualization is to communicate information clearly and effectively through graphical means...To convey ideas effectively, both aesthetic form and functionality need to go hand in hand, providing insights into a rather sparse and complex data set by communicating its key-aspects in a more intuitive way. "

Fundamentally, JIMS is nothing more than a tool for data visualization (and control), whose aim is to expose, to its users' intuition, the invariant relationships in music just as Alan's applet exposed the invariant properties of a systematically-deformed lattice.

Thanks, Alan! :-)

What Killed Thumtronics?

I killed Thumtronics, as its CEO, through my own inexperience.

Two major errors — both mine — killed Thumtronics, thus preventing the Thummer from reaching the market.

These errors were:
1) Starting Thumtronics is the wrong location.
2) Failing to observe the KISS Principle ("Keep It Simple, Stupid").

Location
I started Thumtronics in a tiny hick town (Busselton, Western Australia). Great place to semi-retire, but a lousy place to start a high-tech company. I believed that the world had become flat. However, if you're trying to get a start-up off the ground, geography still matters. Your first step must be to relocate to an appropriate start-up hub.

For Thumtronics, not relocating was fatal. Most of my other mistakes, large and small, could have been avoided simply by starting up in (for example) Austin, and taking advantage of its excellent start-up infrastructure.

KISS
I was initialy attracted to the Thummer, as an investment of my own time and money, because it was "old wine in new bottles," in which the bottle provided all of the added value. All I needed to do was wrap off-the-shelf parts in a new instrument-shape, and voila! — I'd have an inexpensive, expressive new instrument that was easy to learn, fully compatible with all existing (USB-)MIDI-based hardware and software, and patentable. Everything in the Thummer would be off-the-shelf except for its user interface, which was the only remaining source of value in the musical instrument industry's value chain (everything else having been commoditized).

Even better, every performance of the Thummer — whether live or in a video — would implicitly "endorse" the Thummer's unique abilities. Further, because the Thummer would look so unmistakably different from everything else, every performance would also be an "advertisement" for the Thummer. Our marketing expenses could be very low, because our customers would advertise the Thummer for us, simply by using it. (This approach doesn't work for guitar makers because all guitars look alike to non-guitarists. The Thummer, however, looks totally unique, even to non-musicians.)

Indeed, the Thummer was a "purple cow" — a product so different that it would attract attention effortlessly (which was later borne out by the Thummer's ability to attract press from such non-musical publications as the Wall Street Journal).

With this in mind, I should have focused exclusively on "getting version 1.0 to market ASAP, while spending as little on R&D as possible," in order to (a) keep its price low and (b) jump-start the use/endorse/advertise cycle ASAP.

Why did I not maintain this obviously-correct focus?

Because I was also aware that the Internet dramatically increased the effect of word-of-mouth communications (hence "word of mouse"). If Thummer v1 sucked, then its use/endorse/advertise cycle would never start — or, worse, an anti-use/endorse/advertise cycle would begin, "poisoning the well" for version 1 and all future versions, too.

I came to belive that, in order to ensure positive word of mouse, the Thummer v1 had to be "the most expressive instrument on the planet." It had to exceed its customers' expectations by such a wide margin that it would attract evangelically-enthusiastic word of mouse. This led me to elevate expressive potential over KISS, and therefore to invest time and money in two features that required R&D: (a) motion sensing and (b) key velocity/aftertouch.

Motion Sensing
Today, motion sensing (using accelerometers and gyroscopes) is as cheap as dirt, because it's implemented in off-the-shelf chips. Such chips are in every modern console game controller, such as Nintendo's Wii Remote and Sony's SixAxis/DualShock 3 controller.

But back then, in 2003-2005 when we were developing the Thummer, there were no cheap off-the-shelf motion-sensing solutions. Because of this, we should have written off motion-sensing as "a great feature for a later version, once motion-sensing chips became available off-the-shelf." Pretty obvious, right?

The problem was that people LOVED the motion-sensing prototype Thummers. Even skeptics became enthusiasts after seeing them demonstrated. Motion sensing made musician's expressive actions visible to the audience, which was something a tiny thumb-operated joystick could never do. Motion sensing was clearly the Thummer's killer feature.

If we could just implement motion sensing in Thummer v1 (we thought), then we'd have a hit, Hit, HIT!

However, with the crude and expensive motion-sensing chips available back then, there was no way we were going to make a motion-sensing Thummer. It took us months, and hundreds of thousands of dollars, to realize just how hard it was going to be to pull together a "complete solution" from those crude chips. Had Thumtronics been in Austin, I would have had access to people who knew that "complete solution" chips were just a couple of years away. Integrating the new chips into a Thummer would have required one-tenth the R&D effort by Thumtronics. Making the decision to wait would have been much, much easier, had I known that such chips were coming soon. (Chip advances are sporadic, so it's not easy to predict what the next year or two will bring, even if you know that "chips are getting better all the time.")

In any case, I should have stuck to my initial vision of "old wine in new bottles," and ignored motion sensing until it became "old wine," in the form of off-the-shelf chips. Deciding to spend R&D resources on motion-sensing was a mistake.

Key velocity/aftertouch
The harder you strike a piano key, the harder its strings are struck. This one extra expressive variable — "key velocity" — was enough to cause the piano to out-compete the harpsichord, pipe organ, and all other previous keyboard instruments.

"Aftertouch" goes a step further, by allowing an instrument to sense the pressure with which you continue to press a key after the initial strike.

Although there was ample off-the-shelf technology available to measure key-velocity in an electronic instrument that used a piano-like keys, there was none available for concertina-like button-field instruments (and there still is none today). The movement of a button is quite different than that of a piano-like key, so we couldn't use piano-based technology.

We figured that, if the Thummer v1 didn't implement key velocity, then it would suck, and ruin our word of mouse. Therefore, we decided to reverse-engineer the pressure-sensitive buttons of the Sony PlayStation video game controllers, which would give us both key-velocity and aftertouch. However, reverse-engineering this button-system turned out to be beyond the capabilities of our back-of-beyond, hick town company. It soaked up much more of our resources than we could afford. By the time we realized that the end of this R&D task was not in sight, the end of our capital was.

Attempting to implement key velocity/aftertouch was a mistake for three reasons. First, it required R&D, and the "old wine in new bottles" game plan was specifically designed to minimize R&D. Second, it simply wasn't necessary. An alternative feature, called "channel pressure," would have been (a) good enough, and (b) brain-dead simple/cheap/fast to implement. We were focused on key velocity/aftertouch because we listened too hard to our piano-playing beta-testers, who said it was a "must have." Third, even if the Thummer needed more expressive power to succeed, the "killer" way to get that expressive power was through motion sensing, not key velocity/aftertouch.

Bad Decisions => Lack of Cash => Death
These bad decisions cost Thumtronics time, and time is money. If I had not made these bad decisions, Thumtronics would have been able to bring v1 of the Thummer to market by Christmas 2005, at which time it still had enough capital to live cheap and market hard while sales ramped up.

Having made these errors, however, I had to attempt to raise more money. This fund-raising effort failed. Presumably, potential investors decided that if I hadn't brough the Thummer to market after a $1.5 million dollar investment — which should have been ample — then perhaps it was simply a bad idea, or I was simply a bad entrepreneur. They were probably right, on the latter point, at least (although I'd say "inexperienced" rather than "bad").

Bad Location => Bad Decsions
Had I started the company in the right location, and thus been able to assemble a board of directors (and suppliers, partners, employees, etc.) with the right experience, then they are very likely to have been able to help me (a) resist the temptation to elevate "excellence" over KISS, and (b) stick to my "old wine in new bottles" game plan, thereby getting Thummer v1 onto the market by Christmas 2005.

Purple cows don't need to be excellent, in their first version. They just need to be very, very purple...and commercially available. The Thummer would have been very bright purple indeed, even without motion sensing or key velocity/aftertouch. All it needed was to get to market, so that it could find its niche. Each subsequent version could have"sucked less," growing the niche, and climbing the Long Tail into the mainstream.

I never should have elevated "expressive potential" over KISS. Darn it.

Woulda, Coulda, Shoulda...
Let's imagine for a moment that I had not made either of these two major errors. How would the Thummer have worked out? No one can know for sure, however, but here's one possible scenario.

Thummer version 1 would have been available for sale in time for Christmas 2005, without motion sensing or key velocity/aftertouch. Although sales would not have been explosive by any means, v1 would have sold enough over the next 12 months to make Thumtronics cash-flow positive by the end of the year, with a clear growth curve. (This is especially true because we would never have employed the staff that we hired to implement motion-sensing and key velocity/aftertouch, thereby keeping our costs lower.) Sales of the open-source Monome, and of Yamaha's Tenori-On, show that demand existed for alternative instruments such as the Thummer. A history of real, proveable, black-and-white sales growth would have allowed us to attract growth capital. Growth capital is much easier to get than start-up capital, and there was no shortage of growth capital in the USA back then.

With that growth capital, we could have accelerated sales in 2006. Also, motion sensing chips became widely available in 2007, so we could have added motion-sensing to version 2 for Christmas 2007.

Thummer v2 would have been a truly excellent product, not only due to motion sensing but also due to lots of little refinements that users would have suggested after using version 1. As you can see from Ken Rushton and others, the very idea of the Thummer creates "evangelically enthusiastic" supporters. Think how much stronger this enthusiasm would be, and how much more broadly-based, if Thummers actually existed, so that one's enthusiasm could be based on experience (which Ken's now is, more or less, using his excellent DIY jammer) rather than expectation.

With motion-sensing driving the sales of Thummer v2 in 2008, we would have had the cash-flow to add channel pressure to Thummer v3, probably in time for Christmas 2008.

At least as importantly, with Thummers actually being available, the development of open-source software synths that exploited Thummer-only effects (such as dynamic tonality) would have proceeded much faster than they have in today's real time-line. By mid-2009, it is quite possible that dynamic tonality would have started showing up in pop music. (Consider this use of a Monome, or this use of a Reactable. Creative artists LOVE new gadgets.)

The more the Thummer was used by pop musicians, the more rapidly it would have ascended the Long Tail into the mainstream. That's when we would have started seeing Thummer-players being invited to join mainstream bands, or bursting into the commercial music business with Thummer-based bands. That's also when Yamaha, Fender, Roland, etc. would start considering offering their own Thummers, which Thumtronics would probably have encouraged through a patent & trademark license and reference design package.

Woulda, coulda, shoulda. Sigh.

It was all there, in the palm of my hand, but I screwed it up. By starting Thumtronics in the back of beyond, and by placing "expressive power" above KISS, I wasted my own and my investors' capital, and quite literally the opportunity of a lifetime.

A skeptic might say that Thumtronics' experience proves that "new musical instruments always fail," no matter how much "better" the new instrument might be. This is absolutely the wrong interpretation of the facts, however. The Thummer has never had the chance to succeed or fail in the marketplace. Commercially speaking, it is completely untested.

What's really frustrating is that today, motion sensing could be incorporated into the Thummer for next to nothing. Version 1 of such a Thummer, with motion sensing and channel pressure (but not key velocity/aftertouch) could be a hit product from Day 1. I've still got the key patents. It's still doable. But now I'm broke, angel investors are broke, VCs are broke, and it's just not going to happen. Argh.

Ah, well. Thumtronics is dead. Long live iGetIt Music! :-)

What is a "perfect" interval, really?

Why does tonal music include two different interval-naming classes, one ("perfect" intervals) with three variations (dim, perfect, aug) and another ("imperfect" intervals) with four (dim, min, maj, aug)? From what underlying cause does this artifact arise? How?

I’m trying to figure out how to explain the traditional interval-naming system, but it makes no sense to me, so I’m having trouble explaining it. There seems to be a pattern to the names, but I can’t quite grasp it. I have never seen any explanation of the interval-naming rules that made any sense whatsoever.

This is a typical example of an "explanation" which explains nothing. It defines perfect as follows: "The term 'perfect' explicitly indicates an interval which has not been modified, and is usually only applied to the fourth or fifth." This "explanation" raises more questions than it answers:
- Modified from what? Its width in the major scale? But...none of the intervals in the major scale are modifed from their widths in the major scale; why aren't they all called "perfect"?
- To what other intervals, besides the fourth and fifth, can the term "perfect" be applied in 'unusual' cases?

The above "explanation" doesn’t explain anything; it is incomplete; it doesn't even make sense.

Another "explanation" is that "perfect intervals are the same in major and minor." If that's the rule, then why isn't the major second called the "perfect" second? It's the same in major and minor, too.

Another "explanation" is that "those intervals that sound most consonant are called 'perfect'." But this begs the question: why do consonant intervals have only three variations (dim, perfect, aug) when imperfect intervals have four (dim, min, maj, aug)? How does consonance produce this important structural difference? Is there perhaps a deeper cause that links consonance and interval-naming classes?

Another approach to "explaining" the interval-naming system is to simply give up and say, "the following information must be memorized..." This is an egregious cop-out. It shows a failure to understand, let alone explain.

The bottom line is that tonal music seems to include two different classes of intervals:
- one with three variations (diminished, perfect, augmented) and
- one with four variations (diminished, minor, major, augmented).

Think of it this way. One could keep diminishing or augmenting either class of intervals ad infinitum, just be adding double-flats, triple-sharps, etc., so the absolute number of inter-variations isn't what matters. What matters is whether the series is centered ON ONE note (perfect), or centered BETWEEN TWO notes (minor and major).

These two interval classes do not appear to me to be an artifact of arbitrary naming rules. They seem to arise from music's deeper structure, but I can't see how.
- Does a given temperament have as many interval-classes as it has generators?
- Are the "perfect" intervals the ones that are defined by no more than one of each of the temperament' generators? That rule works for the syntonic temperament (generated by the octave and tempered perfect fifth, such that P8: [1, 0]; P5: [0, 1]; P4: [-1, 1]), but I don't know enough about other temperaments (Magic, Miracle, Hanson, etc.) to know whether it's generally true.
- Or is there some other cause?

I'm hoping to someday be able to answer more questions than I ask...