iGetIt! Music

ExploreTuning1

2010-02-27T00:47:00.002-06:00

One of the cool things about JiMS iGetIt! note-layout (also used on the now-defunct Thummer) is that it has the same fingering in every tuning of the syntonic temperament.

This is kinda hard to explain, so I wrote a little Flash app to help. Here it is (source code here):

The slider on the left controls the frequency of Re⁰; all of the other notes' frequencies are determined by their geometric relationship to Re⁰, as a combination of octaves and fifths (as described here and here).

The slider on the right changes the width of the tempered major fifth (traditionally, "perfect fifth"), thereby changing the widths of all non-octave intervals -- that is, changing the tuning. A few notable tunings are labeled along the slider's track.

This chart shows what's happening:

(The colors in the chart do NOT correspond to the colors of the keyboard buttons in the applet above.)

On the keyboard app above,
1. Every note in a given note-class (such as all of the Re's) has the same color.
2. Two dfferent note-classes' notes have the same color if their frequencies, in the chart above, intersect in the current tuning.

For example, in 7-tet, a given diatonic note and all of its chromatic variations (a) control the same frequency, and hence (b) are drawn with the same color. Example: Ra, Re, and Ri are all red in 7-tet. Hence, there are only 7 "frequency classes" in 7-tet. That is, only 7 frequencies, and their octaves, occur in it.

BUT THERE ARE STLL 19 NOTES PER OCTAVE. Many of them just share the same frequency-classes. For example, Ra, Re, and Ri are still different NOTES; they just happen to control the same frequencies when tuned to 7-tet.

Likewise, if one moves the right-hand slider all the way down to 5-tet, then only the 5 notes of the pentatonic scale have unique frequency-classes, all of the diatonic, chromatic, and enharmonic notes (i.e., all of the notes of well-formed scales of cardinality higher than the pentatonic) share/duplicate these pentatonic notes' frequency-classes.

If one slides the slider up to 12-tet, only the chromatic notes have unique frequency-classes; the enharmonic notes (that is, the notes of those well-formed scale with cardinality higher than the chromatic) share/duplicate these chromatic frequency-classes.

In 19-tet, or 31-tet, or in most other tunings, each note-class of the enharmonic scale controls a different frequency-class.

(One of the strangest tunings is 17-tet, in which the pairs De-Li and Se-My are enharmonic. Set the slider to 17-tet, and play Se⁰ and My⁰, in the upper-left and lower-right corners of the keyboard, respectively. Different notes, same frequencies.)

This makes me wonder about the relationship between "scales" (that is, subsets of the enharmonic scale's note-classes) and "tunings" (is the pentatonic scale "really" the pentatonic scale all across the tuning range? Why or why not? How about the diatonic scale...in 5-tet?).

Now, the tunings that are far from 12-tet sound like crap when played using harmonic timbres (try it!), such as the timbre produced by the keyboard applet above. That's because the applet is only tempering the tuning, not the timbre, too. Tunings sound best when played using a "related" timbre -- that is, a timbre in which the partials align with the tuning's notes. Indonesian gamelan orchestras, playing in slendro's 5-tet scale, are playing instruments that emit timbres that (when crossed with a harmonic timbre) fit 5-tet. Tradtitional Thai and African music, played in 7-tet, is played on instruments that emit timbres that fit 7-tet...just as Western timbres fit the tunings near 12-tet.

With electronic sound synthesis, one can temper the timbres to match the tuning in real time -- by shoving a timbres' partials around -- so that voila! You get to have (or choose not to have) consonance in any tuning.

Which bring us to Dynamic Tonality.

Here's a simple example of dynamic tonality, using the above keyboard applet:
1. Slide the tuning to 19-tet (using the tuning slider at the right).
2. Play the ReFiLa triad. Very nice; very restful.
3. Slide the tuning to 5-tet (at the top of the slider).
3. Play the ReFiLa triad again. Too much tension! Must release!
4. Slider the tuning back to 19-tet, and play the ReFiLa triad again. Aha...sweet relief.

What you're experiencing is a novel means of creating tension and relief -- that is, of controlling emotional affect -- in tonal music.
A. In 19-tet, the ReFiLa triad is your basic major triad, which fits well with the harmonic series, and sounds restful.
B. Widening the fifth from 19-tet to 5-tet widens the triad's major third (Re-Fi) by so much that it begins to sound like a sus4 instead. That's one form of tension.
C. Also, widening the fifth from 19-tet to 5-tet pulls the tuning's notes out of alignment with the timbre's (harmonic) partials, creating another form of tension. The notes are "out of timbre."
D. Tuning back to 19-tet relieves the tension of the pseudo-sus4, and also brings the notes back "into timbre."

If one can temper one's timbres in addition to tempering one's tunings, then one can introduce "out of timbre" tension to any triad, including the tonic major triad.

The above experiment would be more compelling if the underlying synth could alter the frequency of a note being played after it started playing (i.e., pitch bend), but, alas, it cannot (so far as I can tell).

You can explore Dynamic Tonality more deeply with the Max/MPS-based TransFormSynth, described here.

P.S.: Why the ReFiLa triad, instead of the DoMiSo triad? Because Re⁰ -- being the center of symmetry (more or less) of all well-formed scales -- is the "origin note" from which the frequencies of all all other notes are determined. As such, Re's frequency doesn't change when the tuning changes, but the frequencies of all other notes do change. Clearly, the applet need to be extended to support the ability to specify a "tonic note-class," which would make the tonic note-class' members (e.g., Do) stable instead of Re. Always more work to do. ;-)

Guthman Musical Instrument Competition

2010-02-24T18:33:00.000-06:00

The Thummer is a contestant in Georgia Tech's Guthman Musical Instrument Competition, being held later this week.

Dr. Monty Cole, a high school friend of mine, happens to work just down the road at Mercer University, and has kindly offered to present the Thummer there on my behalf.

Unfortunately, we haven't been able to get one of the (rapidly aging) Thummer prototypes working, so the presentation will rely on videos of other people performing on it, rather than a live performance.

Here's the presentation, as a compressed PowerPoint file: Guthman.zip

It's super-short, relying primarily on three videos. I'm not sure that the PowerPoint file will be able to locate the videos properly after one downloads, unzips, and moves them to some other file folder. One may need to open the presentation in PowerPoint, go to the slides that contain the videos, double-click on the video graphics, and update the video-link to reference the video files' new location.

Why do the videos have a "Wondershare" logo across their upper-left corner? Because I used a trial version of Wondershare's Video Converter for Mac to convert the video files from WMV to MOV format.

Marek Zabka: Let's Talk

2010-02-23T16:11:00.000-06:00

Marek Zabka, a Lecturer at Slovakia's Comenius University, is hot on our heels.

His paper Generalized Tonnetz and Well-Formed GTS: A Scale Theory Inspired by the Neo-Riemannians shows that he's investigating the same generalized approach to music theory that Andy Milne, Bill Sethares, and myself are pursuing (our references here), on which JiMS iGetIt! Music System (JiMS) is based.

Interestingly, Dr. Zabka does not cite any of our papers, which I presume means that he's unuaware of them.

He has not yet connected his approach to isomorphic keyboards or -- more importantly -- to a generalization of timbre, so we're still ahead of the pack.

Clearly, the foundations of our mutual approach are "in the air," much as infinitesimal calculus was in the 1660's and natural selection was in the 1850's.

I don't have Dr. Zabka's contact information, and can't find it on the web. If you, kind Reader, know how to contact him, or can forward this to him, I would welcome the opportunity to welcome him to into our growing collaboration.

Lesson 005.0

2010-02-20T01:02:00.001-06:00

Here's my first draft of Lesson 5 in JiMS iGetIt! Music System (source code here):

Same crummy state-controlling button-bar at the bottom, for now. I really must fix that.

This lesson is 640x480, rather than the much smaller dimensions of the previous lessons. The larger size doesn't fit this blog very well, but it makes the lesson's text easier to read -- especially the note-button labels.

In this lesson, we build the "Fundamental Scales" -- that is, music's "well-formed scales." I'm not using the "well-formed scale" phrase yet, because to do so, I also need to introduce Myhill's property, and we're still a few lessons away from that.

In Lesson 6, I expect to introduce the notion of tuning, to show how the world's different musical cultures are related, and to establish the argument that to learn music using JiMS is to use a very general approach -- not limited to traditional Western music, for example. I had hoped to put that into Lesson 5, but it was just too much information. It needed its own lesson.

As of this lesson, my courseware has not just drifted, but positively galloped away from mainstream approaches to music education. Yet one can see that the concepts it introduces are quite simple, when shown using JiMS isomorphic keyboard and on-screen animations.

This lesson is late because I spent a week doing the final packing, cleaning, etc. to get our Austin house on the market. That's done; the coast is clear. More lessons! (More cowbell!)

Cardinality invariance

2010-02-07T01:23:00.000-06:00

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.

Languages, Frameworks, and Idioms

2010-02-06T23:01:00.001-06:00

I've recently realized that I'm not "re-learning how to do computer programming," as I had intended. I'm just learning
- new languages (ActionScript, [M]XML),
- a new framework (Flex),
- new development tools (Eclipse), and
- their relevant idioms.

The data structures, algorithms, and fundamental abstractions are all pretty much the same as they were 17 years ago. There are a few new concepts, but basically, it's old wine in new bottles.

It could be, that a more recently-experienced programmer would look at my code and ask "why aren't you using glorpization here?" -- and the answer would be, I haven't a clue what glorpization is, because its use became widespread during my programming hiatus, so I never encountered it before.

One example of this, I suspect, is iterators. The use of iterators for traversing collections had only recently come into vogue went I left programming in the early 1990's. So when I returned to programming, using Adobe's Flex, I tended to traverse its ArrayCollections using indexing (that is, "for i=0 to foo.length") rather than iterators, whether implicit ("for each element in foo") or explicit ("cursor = foo.getCursor; result = cursor.findAny(key)...").

Another thing I missed was the introduction of associative arrays as an underlying mechanism of dynamic programming. "Static typing be damned -- just slam another property on that instance!" This was anathema, back then. Now, you can hardly turn around without bumping into associative arrays. XML appears to one big nested associative array. Flex depends on them.

It's quite odd to see a practice that was formerly considered to be a hanging offense -- "Egad! Self-modifying code! Run away, run away!" -- find its way into the core of modern programming.

I am still finding XML to be impenetrable. It's so simple that I can't understand it, because its (assumed) simplicity is shrouded in impenetrable layers of complexity.

My current confusion on the topic reminds me very much of how confused I was when I was first learning object-oriented programming (OOP). Everyone kept describing OOP in anthropomorphic/magical terms that clouded the issue. They'd say "message passing" when what they really meant was a function call. They'd say "overriding a method" when what they meant was a function call. Only after I discovered C++'s vtables did OOP become clear to me. At its heart, OOP was just a little bit of compiler (and runtime) code that maintained tables of pointers to functions. It was simple. Why did everyone insist on hiding this simplicity with layers of complexity?

I'm pretty sure that XML is the same -- a very simple idea layered in complex crap.

For example, my online music education lessons rely heavily on the use of the computer keyboard as a musical input device. I am quite aware that much of the world does not use the English language's "standard" QWERTY keyboard layout, using other layouts like QWERTZ, AZERTY, etc. instead. I would like to write my code such that I can add new layouts without recompiling all of my lessons.

I've got all of the relevant data structures working just fine in my code, as in-line ArrayCollections of Key objects (which define a name, code, and physical position for each key) -- but I can't figure out how to implement them in XML. It ought to be brain-dead simple. It probably IS brain-dead simple. But I just wasted an entire day on this issue without making any progress whatsoever, as far as I can tell. Too many layers of complexity around XML's simple core.

So, screw it. For now, I will continue to keep XML in the "too hard" file, along with fly fishing, calculus, preparing beef Wellington, and serialism.

Lesson 004.0

2010-02-04T23:48:00.000-06:00

Here's my first draft of Lesson 4 (source code here):

It uses the same crummy state-controlling button-bar at the bottom, for now. I'm going to write a new control for progressing-through-the-lesson-control soon, probably this weekend.

Moodle
I spent some time earlier this week looking at Moodle, the open-source learning management system. I'd like to package my lessons in a Moodle wrapper, because it has excellent support for all sorts of things (like gradebook databases) that I don't want to have to think about. Unfortunately, Moodle's Flash/Flex support is seriously deficient. Fortunately, an effort appears to be underway to address this deficiency. Therefore, I will proceed as if Moodle will have excellent support for Flash within the near-enough future.

I met the Moodle guys when I was living near their home base in Perth, Western Australia. I knew at the time that they had a good chance of beating their commerical competition. The signs were there, even then. Moodle hasn't been gaining market share as rapidly as I had expected, though. It needs some professional help with its evangelism, I suspect, to accelerate its rate of growth. If Moodle doesn't pick up the pace, it could be the next MySpace. It's "do or die" time.

Music & Pedagogy
Lesson 4 is the first lesson to introduce JIMS keyboard. The keyboard is introduced by deriving the pentatonic scale from an octave-reduced stack of (tempered) (major) fifths. Notice that the lesson never qualifies the term "fifth" -- that is, it doesn't call it a "perfect" fifth or a "major" fifth. I don't want to, or need to, open that can of worms quite yet. All in good time.

The next lesson, Lesson 5, will state that JIMS' unique approach gives its students the power and flexibility to understand and describe the music of many cultures. It will suppor this statement by extending the Stack of Fifths to produce the diatonic, chromatic, and enharmonic scales, and by showing that -- using a tuning slider -- the student can change the tuning to match that of many different non-Western cultures and Western eras, while retaining the simplicity and consistency of JIMS' keyboard's pattern.

I think that it's important to make this point early on, because immediately after making it, the lessons will shift their focus to the diatonic scale, and spend a LOT of time in the diatonic world thereafter. If the flexibility of the JIMS keyboard isn't demonstrated early on, a knowledgeable music teacher, reviewing JIMS' early lessons, might reasonably conclude that JIMS teaches concepts that are applicable only to the diatonic scale. I need to plant the seed of JIMS' power early on, even if I don't water it until much later.

Programming
I'm becoming more comfortable with the architecture that I'm using for these lessons, in which the lesson's content is implemented in the transitions between Flex's application states. If I choose the states wisely, then the architecture works well -- even if this architecture is, as I suspect, an unanticipated application of Flex's "state" feature.

Flex's 4 Beta 2's implementation of stateGroups seems to be a bit buggy (as one might expect from a new feature in a beta-version framework). It seems to clobber properties that aren't set by the relevant states. For example, you'll notice that in Lesson 4 above, the note's octave numbers disappear partway through the lesson. That appears to be a manifestation of the stateGroup bug. I spent a couple of hours trying to work around it, before deciding that the lack of octave numbers, in those states, was not a big enough bug to worry about. Also, don't use stateGroups to affect the setting of the properties of a slider, because the max/min/value will be set to NaN under conditions that I haven't spent the time to rigorously quantify.

Now, if I were a really serious beta-tester, I'd dig into Adobe's online bug reports and open-source nightly builds of Flex, to track down the bug and try to identify a fix. However, I'm confident that the bug is severe enough that others will have done this work, so it will be fixed in the release version. Although my use of the state feature in my application's architecture is, as I've suggested above, likely to be unusual, the use of stateGroups is not, so other people should be encountering this bug. If it persists in the next release, I will become more actively concerned.

Schedule
I've decided to try to post a new lesson each week. That would give me fifty lessons in a year. Assuming that I'll make the first dozen free, on a "try before you buy" basis, then those who subscribe to the paid lessons will get an additional 38 lessons (because 50 - 12 = 38). Thirty-eight is more than three dozen, and hence is three times the number of free lessons -- which ought to make the paying customer feel like they are getting enough to make the $29.95 purchase worthwhile. Of course, I'll be adding new lessons constantly thereafter, but with 50, I ought to have enough to "go live" and start selling subscriptions.

I'll have to pause my output while writing a "notation" component, but since JIMS' sequencer-like notation is so much simpler than traditional notation, it shouldn't slow me down by too much.

Lookin' good. ;-)

Lesson 003.0

2010-01-29T20:00:00.001-06:00

Here's my first draft of Lesson 3 (source code here):

Like the first two lessons, it uses a gross-looking and non-intuitive button-bar (along the bottom) to move from state to state. I need to replace that with a simpler/better "Next" button that only appears when one can proceed, and along with "Quit" and "Previous" buttons. The button bar is better for my development purposes, though, because it allows me to jump around non-sequentially.

Musically, this lesson shows JIMS starting to diverge from traditional representations of musical information. There is no international standard way of indicating the octave to which a note belongs. Some musicians indicate the octave of the piano keyboard; some musicians use MIDI numbers; some musicians use apostrophes -- it varies across the globe. So one more variation can't hurt, and might help.

In JIMS, octaves are numbered relative to the octave of the "origin note." In Lesson 3, Fred takes the note Bob sings as his origin, and numbers all octaves from it. Octaves are numbered along a number line, with higher octaves being positive and lower octaves being negative, as described in Lesson 3.

This system is entirely relative. The note Xx⁰ is in the same octave as the origin note, irrespective of the frequency associated with the origin note. As my father used to say, "Everything is relative (but relatives aren't everything)."

Develping Lesson 3 took much longer than it should have, in part because I spent a week (or more) rewriting my QWERTY/Wicki keyboard code to Flex 4...which I then decided not to use in this lesson after all. I'll use it soon enough, though.

StateGroup bug in Flex 4 Beta 2

2010-01-29T19:09:00.000-06:00

I hit an annoying bug in Flex 4 Beta 2 today: using stateGroups to set the min/value/max of a slider control SOMETIMES sets those values to NaN instead of the specified values. It can be worked around by not using state groups...which of course makes state groups considerably less useful.

Here's a Flex app that reproduces the bug 100% of the time (source code here):

I've edited the code and walked Flex's source code in the debugger, trying to track down the cause of the failure, and also tried to find a mention of the bug in Adobe's bug database...but it's already eaten up my whole day, and I've got lessons to finish, so I'll just stop using stateGroups. That's life in beta-land.

Bummer. :-(

wordWrapping a Flex Panel's title?

2010-01-10T19:12:00.000-06:00

How does one turn on word wrap in the title of a Flex Panel?

I'm using Panels to present questions, and the question statements can occupy more than one line. However, the Panel clips them to just one line. There must be a way to make the Panel use as many lines as necessary to display the title's text in full, preferably from MXML.

How?

I suspect that the answer involves using "Cascading Style Sheets," which is on my steadily-growing "must-learn" list. That list seems to be growing from the bottom faster than I can scratch newly-learned things off the top, which is a bit worrisome. The problem isn't that cascading style sheets are rocket science; I'm sure that they are not. It's just one more thing on the list.

Lesson 2

2010-01-09T16:50:00.000-06:00

Here's my first cut at Lesson 2 in JiMS iGetIt! Music System (source code here):

No radical departures from mainstream theory or pedagogy, so far. I'm not super-happy with the state-based architecture that I'm using, and there are some bugs (unimplemented events, actually) in the Flex 4 beta that I had to work around, but...so far, so good.

Interval width changes across the syntonic tuning continuum

2010-01-03T13:03:00.002-06:00

If we stack nine tempered major fifths (traditionally called "perfect fifths") above Re, and nine below it, we get the following generated collection:
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
De-Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li-My

Plotting these intervals' relationships across the syntonic temperament's tuning continuum produces this chart:

(You might want to open this chart into its own window, so that you can look at it, without scrolling, while reading the text below.)

This chart follows the following JIMS conventions:
- Interval names are traditional, except for
+ 4ths and 5ths: wider is "major," narrower is "minor"
+ (that way, 4ths and 5ths follow the same naming-pattern as all of the other non-octave intervals)

- All intervals follow the standard JIMS color-code:
+ major intervals in blue
+ augmented intervals in cyan (an "extreme blue")
+ minor intervals in red
+ diminished intervals in magenta (an "extreme red")

- All chromatic variations of a given diatonic interval share the same note-line symbol. For example,
+ All the unisons (Ra, Re, Ri) are marked with x's.
+ All of the seconds (Me, Mi, My) are marked with squares.
+ All of the thirds (Fa, Fi) are marked with vertical lines.
+ etc.

The legend, at the right of the chart, displays the generated collection of notes, in the same order (bottom to top) as they appear in the list at the top of this blog post. Each note's name is followed, after a colon (':'), by its interval-from-Re. Observe that the follow a pattern: augmented intervals at the top, then major intervals, then unison (Re), then minor intervals, then diminished intervals at the bottom of the list.

The vertical scale, on the left, indicates the width of a given note from Re.

The horizontal scale, on the bottom, indicates the width of the tempered major fifth (M5), that is, of the generator of the generated collection. The scale includes the valid tuning range of the syntonic temperament, which can be thought of an an extended meantone tuning system.

The widths of the intervals between Re and every other (non-octave) note is controlled by the width of the generator, M5. As the width of the M5 increases, from left to right across the chart, the widths of all of the non-octave intervals change:
- The intervals below Re in the legend, representing minor and diminished intervals, slope downwards as the M5 increases, indicating that they narrow.
- The intervals above Re in the legend, representing major and augmented intervals, slope upwards as M5 increases, indicating that they widen.
- The farther a note is from Re in the legend, the steeper its slope.

Consider, for example, the widths of the unisons. As the generator (M5) increases in width:
- Re (unison) is unchanged at 0, because it is the basis from which all other intervals are measured. Its note-line is shown at the very bottom of the chart area, as a series of black x's.
- Ra (diminished unison, d1), shown with magenta x's, decreases in width. It's note-line drops from 0 cents below Re (i.e., 1200 cents above Re), on the left edge of the chart, to 240 cents below Re (i.e., 960 cents above Re) at the right edge.
- Ri (augmented unison, A1), shown width cyan x's, increases in width, from 0 cents above Re on the left to 240 cents above Re on the right.

All of the unisons start, on the left, at 0, and separate as the width of the generator increases.

Likewise, consider the widths of the seconds-from-Re:
- Me (minor second, m2) drops rapidly from 171 cents to 0.
- Mi (major second, M2) rises slowly from 171 cents to 240.
- My (augmented second, A2) rises sharply from 171 cents to 480.

Just as with the unisons, all of the seconds start together (at 171 cents) and separate as the width of the generator increases. Generally, all of the chromatic variations of a given diatonic degree start at the same point on the left-hand edge of the chart, and diverge as the M5's width increases rightwards across the chart. (Note that 1200 and 0 are the same octave-reduced interval, so that Ra, which intersects the left edge at 1200, intersects it at the same interval as Re and Ri, which intersect it at 0.)

7-edo
The seven left-edge-intersection-points divide the octave into 7 equally-wide intervals, forming a 7-note "equal division of the octave," abbreviated "7-edo."

(The phrase "N-tone equal temperament" and its abbreviation "N-TET," used in Wikipedia and elsewhere, is avoided in JIMS, because it confuses the important distinction between tunings and temperaments...an explanation of which is beyond the scope of this blog post.)

5-edo
Likewise, the right-hand edge of the chart, at M5=720, shows that a completely different combination of notes intersect to divide the octave into five equally-wide intervals: 5-edo. (Again, note that 1200 and 0 are the same octave-reduced interval, so Di, intersecting the right edge at 1200, and Me, intersecting the right edge at 0, are intersecting it at the same interval.)

12-edo
Near the middle of the chart, at M5=700, you can see that seven pairs of note-lines cross. From top to bottom, the crossing pairs are:
1100 - Ra and Di (d1 and M7)
900 - De and Ti (d7 and M6)
800 - Te and Li (m6 and A5)
600 - Le and Si (m5 and M4, traditionally named d5 and A4)
400 - Se and Fi (d4 and M3)
300 - Fa and Mi (m3 and A2)
100 - Me and Ri (m2 and A1)

The notes in the crossing pair are always 12 notes apart in the 19-note stack of M5's (check for yourself, using the chart's legend).

The crossing note-pairs are said to be "enharmonic" (i.e., have the same pitch) in 12-edo. This is the "equal temperament" tuning familiar to most modern musicians -- so familiar, in fact, that many such musicians do not realize that other tunings exist, or that there is such a thing as a tuning (let alone a temperament).

17-edo
Slightly to the right of 12-edo, at M5-706 cents, two other note-lines cross:
352 - Se and Mi (d4 and A2)
847 - De and Li (d7 and A5)

All of the note-lines intersect the vertical line labeled "17-edo" at 17 equally-spaced intervals, so M5=706 is 17-edo tuning.

In 17-edo, the major second is subdivided into three equally-wide intervals by the augmented second and minor second. For example, see how the gap between Re (black x's, at the bottom) and Mi (blue squares, near the 200 cent horizontal line) is evenly divided by Ri (A1, cyan x's) an Me (m2, red squares). Note that at this point along the horizontal axis (M5=706), Me is closer to Re (i.e., lower in pitch) than Ri is.

In 17-edo -- and indeed everywhere rightward of 12-edo -- minor/diminished intervals are lower in pitch than the augmented/major intervals with which they are enharmonic in 12-edo.

19-edo
Likewise, the vertical line labeled "19-edo" marks the spot, at M5=695, where the note-lines subdivide the octave into 19 equally-wide intervals: 19-edo tuning. At this tuning, a major second (for example, Re-Mi) is divided into three equally-wide intervals by and augmented unison (Ri) and a minor second (Me).

In 19-edo -- and indeed everywhere leftward of 12-edo -- minor/diminished intervals are higher in pitch than the augmented/major intervals with which they are enharmonic in 12-edo.

Dynamic Tonality
Despite the changes among the relationships between intervals across the syntonic temperament's tuning continuum, the sound of tonal harmony's basic structure survives, as shown in this video (with over-the-top narration, for which I apologize):

This dynamic flexibility of tuning, combined with the consistent fingering of the Wicki/JIMS keyboard, can be used to create musical effects that are truly new, such as the tuning progression in this piece, C to Shining Sea, by William Sethares. We call the result Dynamic Tonality.

Well-formed scales beyond the chromatic

2010-01-02T18:21:00.044-06:00

First, let's review the construction of the chromatic scale.

Stacking 13 tempered perfect fifths (P5's) one atop the other, centered on Re, produces the following 13-note generated collection:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si

The Le-to-Di and Me-to-Si 12-note subsets of this generated collection are both just transposition of each other, so either can be used to represent a 12-note contiguous subset of the above 13-note generated collection. In the following discussion, the Me-to-Si subset will be used.

The Me-Si generated collection's notes can be adjusted so that they all fall within a single octave. We will arbitrarily define the octave to start on Do. The result is a "well-formed scale," in this case the chromatic scale.

The chromatic scale has the following note sequence and interval sequence:
note sequence: Do-Di-Re-Me-Mi-Fa-Fi-So-Si-La-Te-Ti-[Do2]
interval sequence: A1-m2-m2-A1-m2-A1-m2-A1-m2-m2-A1-m2

...where:
A1: augmented unison
m2: minor second

In the syntnonic temperament's valid tuning range -- that is, when the width of the P5 is anywhere between 686 and 720 cents wide -- the m2 is wider than the A1. Hence, in the syntonic temperament, the chromatic scale has the following width sequence:
width sequence: S L L S L S L S L L S L

That's 7 L's and 5 S's.

With that review, we can now go...

Beyond the Chromatic
In the syntonic temperament, then, the well-formed scale with the next-highest cardinality after the chromatic's 12 will have the cardinality:
Cardinality' = 2L + S = (2 * 7) + 5 = (14) + 5 = 19.

Stacking 19 tempered P5's one atop the other, centered on Re, produces the following generated set:
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
De-Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li-My

...with the extra notes (relative to the chromatic scale) appearing the ends and shown in boldface.

Octave-reducing this generated set, and arbitrarily defining the octave to being on Do, gives the following 19-note note sequence and interval sequence:
Do-Di-Ra-Re-Ri-Me-Mi-My-Fa-Fi-Se-So-Si-La-Li-Te-Ti-De-[Do2]
A1-d2-A1-A1-d2-A1-A1-d2-A1-d2-A1-A1-d2-A1-d2-A1-d2-A1

...where:
A1: augmented unison
d2: diminished second

Clearly, as we sub-divide the octave into more pieces (i.e., into higher-cardinality scales), those pieces must get smaller.
Scale Cardinality Large Small
Pentatonic 5 m3 M2
Diatonic 7 M2 m2
Chromatic 12 m2 A1
Enharmonic_19 19 A1 d2

At each successively-higher cardinality, the formerly-small interval width becomes the new large width, and a new small width is introduced.

On a 19-note-per-octave Wicki/JIMS note-layout, and played in 19-tone equal temperament (P5=695, at which the A1 and d2 are both 1200/19=63.16 cents wide), this scale looks/sounds like this (source code here):

Now, let's explore the alternative cardinality-successor to the chromatic scale.
Stacking 17 tempered P5's one atop the other, centered on Re, produces the following generated set:
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li

...with the extra notes, relative to the chromatic, added to either end, and shown in boldface.

Octave-reducing this generated set, and arbitrarily starting defining the octave to being on Do, gives the following 17-note note sequence and interval sequence:
Do-Ra-Di-Re-Me-Ri-Mi-Fa-Se-Fi-So-Le-Si-La-Te-Li-Ti-[Do2]
m2-d2-m2-m2-d2-m2-m2-m2-d2-m2-m2-d2-m2-m2-d2-m2-m2

On a 17-note-per-octave Wicki/JIMS note-layout, played in 17-tone equal temperament (P5=706), this scale looks/sounds like this (source code here):

And there you have it: the next-higher-cardinality scales after the chromatic are 17 and 19.

Syntonic and Mavila

2010-01-02T13:34:00.003-06:00

In an earlier post, I calculated the cardinalities of successive well-formed scales -- pentatonic (5), diatonic (7), and chromatic (12) -- and animated their interval-patterns on the Wicki/JIMS note-layout.

What we saw -- with some interpretative help from Andy Milne -- was that:

each successive well-formed scale came in two versions: one with X large intervals and Y small intervals, and one that was vice versa (Y large and X small); and that
the sequence of intervals that defined both versions was the same; the only difference between the two versions was the tuning (that is, the width of the tempered perfect fifth, since that is the generator of the generated set that defines a well-formed scale).

For example, the diatonic "generated set" is Fa-Do-So-Re-La-Mi-Ti, which produces the note-sequence (in Do-mode) Do-Re-Mi-Fa-So-La-Ti-[Do2], which has the inter-note interval sequence M2-M2-m2-M2-M2-M2-m2.

In the syntonic temperament's valid tuning range (P5=(686, 720)), the M2 is wider than the m2, so this sequence can be written as the width sequence L-L-S-L-L-L-S, which is 5 large (L) and 2 small (S) intervals.

However, as P5's width shrinks towards 686, the m2 widens and the M2 shrinks, such that they become equal at around P5=686 cents, producing 7-tone equal temperament tuning.

If one narrows the P5 even further, one leaves the syntonic temperament and enters what Erv Wilson called the Mavila temperament, in which the m2 is wider than the M2. There, this same pattern (note sequence: Do-Re-Mi-Fa-So-La-Ti-[Do2] == interval sequence: M2-M2-m2-M2-M2-M2-m2 ) has the width sequence S-S-L-S-S-S-L, because in the Mavila temperament's valid tuning range, m2 > M2.

Alternatively put, the diatonic note note sequence and (hence) interval sequence are unchanged from syntonic to Mavila; the only thing that's changed is the relationships among the interval-widths, in that syntonic's m2 < M2 becomes Mavila's m2 > M2.

The same meta-pattern applies to the chromatic scale (all from Do):
note sequence: Do-Di-Re-Me-Mi-Fa-Fi-So-Si-La-Te-Ti-[Do2].
interval sequence: A1-m2-m2-A1-m2-A1-m2-A1-m2-m2-A1-m2

Within the syntonic temperament's valid tuning range, the m2 is wider than the A1 (i.e., m2 > A1), so the above chromatic note/interval sequence produces the following width sequence:
width sequence: S L L S L S L S L L S L

However, if the P5's width is narrowed so that it crosses out of the syntonic temperament's valid range into the Mavila temeprament's valid tuning range, then the width-relationship of the m2 and A1 is reversed, such that m2 < A1 -- producing a chromatic width sequence in Mavila that's the opposite of that in the syntonic:

width sequence: L S S L S L S L S S L S

Apparently, Andy's algorithm for calculating the sequence of cardinalities for successive well-formed scales, and the count of large & small intervals in each, produces a single scale, of which there is a syntonic variant and a Mavila variant. Let's see, in my next post, if that pattern continues, as we explore well-formed scales beyond the chromatic.

Kodály, Wicki, and iSlate

2009-12-31T13:43:00.003-06:00

The Kodály music education Methodstarts young students with pentatonic songs, then slowly introduces them to the "extra" diatonic intervals, and then eventually to the "extra" chromatic intervals.

Now, imagine that the youngest students were presented with a "pentatonic keyboard" in the Wicki/JIMS note-layout, like the one shown in the (non-interactive) image at right.

For the pentatonic songs used initially by the Kodály Method, this pentatonic-only keyboard would be ideal. It would contain only the notes (intervals) that the students were currently learning. (Other keyboard controls, not shown, would be used to indicate the tonic and define its pitch.) Also, it would give students a visual, tangible metaphor for tonal space, hence (potentially) accelerating their development of audiation skills.

Then, when they were introduced to the "extra" intervals of the diatonic scale, they could get a new diatonic keyboard.

As you can see, it's the same as the pentatonic keyboard, with the addition of Fa and Ti along the left and right edges, respectively. In effect, the diatonic keyboard's extra notes expand the "tonal space" to which the student is exposed.

Again, by containing only the notes in the scale currently being studied, such a keyboard has the potential to sharpen student's focus.

Later, as the student progressed to learning about chords, they could be presented with a two-handed diatonic keyboard, suitable for self-accompaniment. (The note-layouts are mirrored for cognitive convenience, and angled for ergonomic convenience.)

...which would, in turn, be superseded by a two-handed chromatic keyboard:

...and eventually, a two-handed enharmonic keyboard, featuring all 19 intervals of the enharmonic scale:

The latter keyboard looks rather overwhelming, and it probably would be, if it were the first keyboard a student encountered. However, after starting with the simple pentatonic keyboard and working progressively up through the diatonic an chromatic keyboards, the enharmonic keyboard wouldn't seem like such a big deal. It just adds a few extra notes at the outer edges of the keyboard, leaving its pentatonic/diatonic/chromatic core unchanged.

The main advantage of this approach is that the student always uses a keyboard that has precisely enough note-controlling buttons to achieve the required pedagogical goals, thus encouraging proper focus and minimizing distraction/confusion. Of all of the isomorphic note-layouts, the Wicki note-layout is best for this purpose. Each successively-wider Wicki keyboard enables the student to see farther into tonal space, literally expanding their tonal horizons.

The main disadvantage is that the student must trade-up keyboards rather frequently.

Perhaps this disadvantage could be ameliorated by using a virtual multi-touch keyboard, such as the much-rumored Apple iSlate (see article here):

Such a multi-touch sensitive display would perhaps lack the tactile feedback needed in a true performance instrument...but that's not the point. The Kodály Method stresses the use of one's voice as one's performance instrument. Hence, in a Kodály context, the Wicki note-layout keyboard would be used not for performance (absent the Thummer [sigh]), but rather for pedagogy -- i.e., in helping students apply additional senses (sight, touch) to the development of proper audiation skills.

Using a virtual keyboard would enable new intervals to be introduced not just one scale at a time, but one note at a time -- first just So and Mi, then also Do, then Re, then La, etc. -- following the standard sequence of the Kodály Method.

Apple's iSlate is likely to be to expensive for K-12 music instruction. Give it 10 years, however -- maybe less -- and iSlate clones will be cheaper than traditional band instruments.

Cardinality of well-formed scales

2009-12-31T06:24:00.004-06:00

Let’s apply Andy's next-highest-cardinality-MOS-scale-calculating algorithm, starting with the pentatonic scale.

First, here's an animation of the pentatonic scale on a Wicki note-layout (source code here):

As you can see, the pentatonic scale has two large steps (m3’s) and three small steps (M2's).
Name         L   S   Cardinality
Pentatonic   2   3       5

So far, so good.

Remember, a well-formed scale is drawn from a stack of tempered perfect fifths; the stack has the same number of notes in it as the scale does: the cardinality of the scale. The pentatonic scale's stack of P5's (laid on its side) looks like this:
-2 -1 0 1 2
Do-So-Re-La-Mi

Now, let’s apply Andy's algorithm to these values of L and S to get the L and S of the next-higher-cardinality MOS scale, i.e., L’ and S’ respectively.

Cardinality’ = 2L + S = (2 * 2) + 3 = 4 + 3 = 7, which agrees with our expectations for the diatonic scale, which is a god sign. ;-)

X = L + S = 2 + 3 = 5
Y = L = 2

Therefore, the next-higher-cardinality-than-pentatonic MOS scale will have either 5 large steps and 2 small steps, or vice versa.

The diatonic scale has five large steps and two small steps, with cardinality 7, so that seems to be the “right choice.”

However, I am confused. Does this mean that there ALSO exists some “Bizarro-Diatonic” scale of cardinality 7 which has five small steps and two large ones? If so, what is that scale? If not, why not?

This process gives us the following result:

Name              L   S   Cardinality
Diatonic          5   2        7
Bizarro-Diatonic 2   5        7

Hmmmm, it’s a little weird to have the Bizarro-Diatonic scale as a result of this algorithm, but what the heck, let’s press on.

[Edit: Andy Milne was kind enough to point out that the Bizarro-Diatonic scale is more properly named the Mavila scale, following Erv Wilson. I haven't been able to find out much about it on the web. When I understand it better, I'll put up an appropriate animation of its interval pattern.]

The diatonic scale described above looks like this (source code here):

The diatonic scale's stack of tempered P5's is just like the pentatonic's, but it has one additional note at each end (Fa and Ti):
-3 -2 -1 0 1 2 3
Fa-Do-So-Re-La-Mi-Ti

Now, let’s apply the stated algorithm to the diatonic scale’s L and S values, to find the next-higher-cardinality MOS scale (which OUGHT to be the chromatic scale, if all goes well).

Taking the values L' = 2 and S' = 5 from the diatonic scale...

Cardinality’’ = 2L’ + S’ = (2 * 5) + 2 = 10 + 2 = 12, which is the cardinality of the chromatic scale, which is encouraging.

X’ = L’ + S’ = 5 + 2 = 7
Y’ = L = 5

Therefore, the next-higher-than-diatonic MOS scale will have either 7 large steps and 5 small steps, or vice versa.

Hmmmm, that's odd. Can there be two different versions of the chromatic scale?

Yes, it turns out that there can, and the existence of the two different versions answers a question that's been puzzling me for the last couple of weeks.

Here's an animation of the first version of the chromatic scale (source here):

There are 12 intervals in the chromatic scale, so any drawing of them is going to look complicated, and this animation's drawing is no exception. But if you look closely, you can see a lot of structure in its pattern of intervals.

Firstly, the scale has only two interval sizes, as predicted: minor seconds (m2's, in red) and augmented unisons (A1's, in pink).

Second, there are five A1's (smaller intervals) and seven m2's (larger intervals). In 12-tone equal temperament ("12-tet"), the A1 and m2 happen to be equally wide (at 100 cents), but they are still different intervals, so they have different shapes on the Wicki note-layout.

Third, all the interval lines are parallel to each other. Among them, they outline a chromatic staff (well, kinda sorta).

One of the note-buttons, Le, isn't used in the animation above, however -- none of the interval-arrows ever reach it. You might well ask, "why did Jim include Le in the animation, then?"

To answer this leading question, let's look at an animation of the other version of the chromatic scale (source here):

It looks very much the same, as you would expect. The only difference is that, after the scale goes up an m2 from Fi to So, it goes up another m2 from So to Le, rather than turning back towards Si with an A1 as the previous version of the chromatic scale did.

We can call the first version "Chromatic Si," and the second version "Chromatic Le." In 12-tone equal temperament, there is no difference between them, but there would be a difference in (say) 1/4-comma meantone tuning, which had a leading role in Western music for many centuries (and which appears to have been used in the tuning of ancient Chinese bells).

In the stack of tempered perfect fifths that forms the chromatic scale, Le and Si are on opposite ends:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si

...which is the same as the diatonic stack, but with extra notes added at the ends (in boldface).

This chromatic stack has 13 notes, but only 12 can be included in the chromatic scale. One must choose whether to include Le or Si; you can't have both, because then you'd have a scale of cardinality 13, not 12.

If you choose to include Le, you get the Chromatic_Le scale.
If you choose to include Si, you get the Chromatic_Si scale.

[Edit: I got all excited when I first saw this, because, after about 4am, my counting skills decline precipitously -- so I thought that Chromatic_Le had 5 small and 7 large intervals, and Chromatic_Si, vice versa. They are just transpositions of each other, and nothing to get excited about. One should never trust (or at least, never post) late-night epiphanies.

As further evidence of the decline in my late-night counting abilities, I also declared the down-and-right-pointing intervals (e.g., from Do to Di) to be "diminished seconds," when they were clearly augmented unisons (how "clearly"? In both chromatic animations, the A1 arrows ALWAYS connect notes that begin with the same initial consonant -- that is, chromatically-altered versions of the same note. Hence, the interval MUST be a variation on unison. What an idjit I am!). I have relabeled them accordingly, and updated this blog post's text accordingly. Thanks to Andy Milne for giving me the heads-up on these errors; see his comment below.

I realy should have changed the color of the A1 arrows to be cyan, while I was relabelling them, to follow my convention that all augmented intervals are colored cyan...but I forgot, and now I'm too tired again. Later, perhaps.]

I've been up all night working on this blog posting (and its animations), so now, at 6am, I'm off to bed. Soon, I'll put up another post that continues walking up the cardinality chain to 17-tone scales, 19-tone scales, and beyond.

Cardinality sequence of MOS scales

2009-12-30T19:59:00.001-06:00

It is well-known (among mathematically-inclined music theorists) that there are well-formed scales with cardinality 5 (pentatonic), 7 (diatonic), and 12 (chromatic). ("Cardinality" is the number of notes in the scale.)

But, why these cardinalities, and not others? Why are there not well-formed scales of cardinality 6, 8, 9, etc.? Also, what cardinalities come after 12, that are well-formed?

I asked this of Andrew Milne, who gave a great answer, which I have appended below (with hyperlinks and occasional [editorial comments] added for your convenience...).

The quick answer is that the next couple of well-formed scales fter the chromatic have cardinalities 17 and 19.
------------------------------------------------------------

From: Andrew Milne [mailto:andymilne@tonalcentre.org]
Sent: Wednesday, December 30, 2009 11:10 AM
To: 'Jim Plamondon'; 'Bill Sethares'
Subject: RE: well formed scales, cardinality: next after 12?

Hi Jim and Bill

A lot of these concepts – well-formedness, Myhill’s, etc. are mathematically related and their definitions are hard to separate. But this is the method I use to determine the cardinality and tuning range of MOS/well-formed scales.

First you need a Farey sequence of order N. Then any Farey triple (three consecutive terms of the sequence), whose middle member has a denominator of N (i.e., j/k, M/N, p/q) gives the tuning range for an MOS scale of N tones. For example, Farey(12) has the triple 4/7, 7/12, 3/5, which tells us that there is a 12-tone MOS (the chromatic scale) which occurs for tunings where 4/7 < beta/alpha < 3/5. We know this method works, but Bill and I still haven’t worked out quite why.

By generating successively higher-order Farey sequences, you can find triples with higher denominators, and hence the tuning ranges of higher cardinality MOS scales. Note that as the cardinality goes up, the tuning range gets smaller. (This is all illustrated in the MOS labyrinth picture [for which, see Figure 5 in this paper]).

For your example, the next well-formed scale after the 12-tone chromatic is 17 if 7/12 < beta/alpha < 3/5, and 19 if 4/7 < beta/alpha < 7/12. You can either calculate this yourself or, more easily, just read it directly off the MOS labyrinth diagram.

Actually, I’ve just remembered, there’s another easy way to do this. If your MOS has L large steps, and S small, the next higher MOS has cardinality 2L + S with L’ large steps and S’ small where EITHER

L’=L+S large steps and S’=L small OR
S’=L+S small steps and L’=L large; and so on.

For example, the 7-tone diatonic has L=5, S=2; the next higher MOS has 2L + S = 12 tones.

If this 12-tone MOS has L’ = L+S and S’ = L, then the next higher MOS is 2L’+S’ = 2(L+S)+L = 3L+2S = 19 tones.
If this 12-tone MOS has S’ = L+S and L’ = L, then the next higher MOS is 2L’+S’=2L+L+S = 3L+S = 17; and so on.

I presume that this rule is a direct result of well-known properties of the Farey sequence.

Every MOS/well-formed scale has a tuning range over which it is also proper (Rothenberg’s term, or “coherent” in Balzano’s terminology). There is a method to find this using the Farey sequence of a higher order (or Stern Brocot tree – which is the same thing but arranged into layers), which was explained by Thomas Noll, in one of the emails he sent when reviewing our JMM paper [see draft here]. I can’t quite remember what he said (I do have a copy, so I can check), but it amounted to something like stepping up to the next higher MOS scale and using that tuning range, or similar. For example, the diatonic scale has a tuning range of 4/7 to 3/5, but is proper only over 4/7 to 7/12.

Presumably there are also ways to generalise these things for 3-D tunings producing “pairwise well-formed” scales, and higher-D tunings. This would certainly form the basis of a groundbreaking paper...

Andy

------------------------------------
More to follow.

Lesson 001.0

2009-12-21T22:00:00.002-06:00

Here's the first lesson in JiMS iGetIt! Music System (source code here):

It's not terribly impressive, of course, but one must start somewhere, both as a student of music and as a student of coursware development.

Well, it doesn't display correctly in IE/Windows, just as my last couple of test projects didn't. Works fine on Safari/Mac. Clearly, I can't continue to ignore this IE/Windows problem.

States in Flex 4 (entry 3)

2009-12-16T09:33:00.000-06:00

Here's a minor iteration to my recent state-controlling button-bar (source code here):

The main difference is that, in this version, whenever the application's state changes, the StateButtonBar's currently-selected state is updated to reflect that change.

Say, for example, that the user presses the StateButtonBar's State3 button, thus setting the application's currentState to "State3". Then, the user presses the stand-alone foo button, beneath the StateButtonBar. This sets the application's currentState to "foo", and because the StateButtonBar now listens for application state changes, it is notified of the state change and can update its own state (by selecting the ButtonBarButton labeled "foo") accordingly.

US Patent up for review

2009-12-14T12:56:00.000-06:00

The US version of one of my WIPO patent applications, Musical Button-Field Layout for Alphanumeric Keyboards, has recently been assigned for examination.

I'm too broke to pay a patent attorney to "prosecute" (drive forward) this patent application -- heck, Thumtronics already owes patent attorneys more than it (or I) can repay. However, if I can prosecute it myself for a couple of hundred bucks and a couple of dozen hours, I can probably do that, on behalf of iGetIt! Music.

The twist claimed by this patent application is not just the Wicki-to-QWERTY mapping, but rather, mapping each note of tonic solfa (in the Wicki pattern) to the QWERTY keyboard, and relying on the QWERTY keyboard's underlying computational power to map these intervals to the correct pitches, given a common reference pitch. Hence, "La" always maps to the same button on the QWERTY keyboard, but that button does not always map to the same pitch. It's a "movable Do keyboard," right there on your computer.

If I'd realized, before starting the Thummer project, that I could map Wicki's note-layout to an alphanumeric keyboard as described in this patent application, then I might have focused on a software-only product from the outset, and saved myself a fortune in hardware-development costs.

Based on my experience with Thumtronics, here's my advice to budding inventors:

One language, one market. If you're a native speaker of English, file only in the USA (although perhaps via WIPO's international process, because it gives you more time). If Japanese, in Japan only, and if Chinese, in China only. If you're European, file in the USA only, until Europe adopts a single unified patent system that enables prosecuting (i.e., "driving forward") a single pan-Euopean patent in a single language.
Don't hire a patent attorney. Instead, (a) carefully read dozens of patents related to the one you expect to file, paying particular attention to the specialized language used, (b) carefully compose your own patent description, claims, and drawings, based on what you learned from that study, and finally (c) buy the latest version of Patent It Yourself, and follow its instructions to the letter.

Thumtronics broke both of these rules, and the resulting expenses were an unsustainable drain on its finances. According to LegalZoom, "thousands of inventions are patented each year but only a minuscule amount actually generate substantial, if any, profits." Too true.

With such low odds of returning any value, the cost of patenting must be equally low for patents to be worth filing at all.

All of the patent attorneys in the world will tell you the same lie: that they cannot possibly estimate the cost of prosecuting your patent; that every patent is different; and that the best thing to do is to go ahead and get the process started. This is complete crap. Given all of the patents that have been filed, it is surely possible to find the average cost (and standard deviation) of patent filing by number of claims, number of descriptive pages, number of drawings, number of countries, number of languages, etc. This is basic data-mining.

However, if starry-eyed inventors knew in advance what the process might cost, they would be less likely to invest in it. Hence, it is in the patent attorney's interest to keep the costs of the process as opaque as possible, while leaving its early stages inexpensive. It's a scam, made worse by the high barriers to entry erected to limit entry to patent attorney's profession, thereby limiting competition and increasing the margins that patent attorneys can charge.

As with so many other aspects of modern global capitalism, the patent system has been skewed away from "defending the rights of the little guy" towards being a tool for their oppression. The big guys can afford to work the legislative/industrial system, in which legislators sell -- for "campaign contributions" and other graft -- the right to expropriate money from the little guys. The little guys just get screwed.

So, why bother to invent at all? Well, because -- as Canada Bill Jones is quoted as saying -- "Of course the game is rigged! But it's the only game in town, and if you don't bet, you can't win."

Rather than spending a fortune on patent attorneys, I submit that one should file one's own patents -- keeping the cost as low as possible. Then, once you've produced a product based on the patent that is generating a profit stream, buy patent insurance to defend those profits. A million-dollar policy currently costs around US$25K/year.

The most carefully-crafted patent is worthless if you can't afford to defend it, whereas even a poorly-crafted patent, backed up by a million-dollar defense fund, will deter infringers.

As always, while justice whispers, money talks.

States in Flex 4 (entry 2)

2009-12-12T18:19:00.000-06:00

This app looks the same as the last one I posted, but it implements the transitions between states using Flex's 'transitions" mechanism, rather than just executing onEnterState event handling code (source code here):

Other differences include:
1) on entry to State1, the label for the current state is spun at the same time as the narration is played, and
2) on entry to State2, a second sound is played after "test narration two" completes.

It shows the same clipping problem in IE on Windows. I'm going to have to look into that.

States in Flex 4

2009-12-11T16:15:00.001-06:00

One way to architect the code for a "lesson" is as a series of states. Flex 4 (now in beta) has improved its support for states, so this seems like a reasonable approach to explore.

Here's a program that demonstrates a trivial use of states in Flex 4 Beta 2 (source code here):

A ButtonBar at the bottom is initialized with the names of each of three states, and a label reflects the name of the currently-selected state. When the ButtonBar is clicked, the state is changed. Whenever a new state is entered, a short narration is played.

To use this approach in my courseware, I'll need to break each lesson up into a series of states, and define the actions to be taken whenever a new state is entered. Flex 4's new-and-more-general animation model should make it easier to define these actions than was previously the case.

I have no idea whether this is the "best practice" way to architect interactive courseware in Flex. I'm kinda flying blind here. Suggestions welcome.

[Postscript: All three ButtonBar-buttons display properly in Safari on my Mac, but the rightmost part of the StateTest1 widget, including most of the third ButtonBar button, is clipped off in Internet Explorer on my Windows machine. I don't know why. Welcome to the joys of cross-browser incompatibility, I guess.]

Sound synthesis solution...for now

2009-12-10T17:55:00.001-06:00

Here's a new version of SoundTest that sounds much better (source code here):

The difference is that this new version calls sound-generating code from a beta version of the SonoFlash library. Assuming that this library will be offered at a reasonable price once it is released in final form, I expect to use it for most of my courseware's lessons.

Adobe has a list of requested features that one can vote for, and MIDI is on the list, but apparently it has never made the cut. Perhaps Adobe should acquire SonoFlash, and use its team (and cross-platform sound engine) to implement MIDI in Flash.

Sound synthesis problem

2009-12-08T14:24:00.000-06:00

I spent the last month packing and moving, but I'm now back online (at last!).

Why doesn't this code work?

Here's what works:

Click either "Play" button; a note will sound, and the button's name will change to "Stop".
Click the same button again to silence the note; the button's name will change back to "Play."
Click the other "Play" button; a different note will sound (an octave higher than the first note); click it again to silence it.

So far, so good. :-)

Here's what doesn't work:

Click on either "Play" button, sounding a first note.
Click on the other "Play" button, sounding a second note while the first button is still sounding.

Bad things now happen. The sounds do not blend (as one would expect octaves to do), but instead grate against each other in a nasty alternating, overlapping grind. Also, the application's performance slows to a crawl. Click on either button, to silence its note, and you'll see that it takes forever for the button to respond to the click.

I'm not sure how to fix this. The code seems to be working as intended. The flaw seems to be in how I'm using Flex/Flash's sound architecture -- that is, my tactics are correct, but my strategy's wrong.

Comments welcome. :-)

Superfreakonomics, global warming, and music

2009-11-13T11:33:00.000-06:00

A contrarian chapter on global warming in Superfreakonomics has sparked intense controvery. I am partcularly intriqued by the "shape" of the debate -- the kinds of arguments that are being used in both directions.

I found the following paragraph, from this Bloomberg blog post, to be particularly interesting:

Dubner wonders why everyone is so angry. In part, it’s because the book’s blithe remedies – “We could end this debate and be done with it, and move on to problems that are harder to solve,” Levitt told the U.K. Guardian newspaper – are an insult to the thousands of scientists who have devoted their careers to this crisis.

Hmmm. Let's paraphrase that paragraph as follows.

Mr. Inventor wonders why everyone is so angry. In part, it’s because his invention's blithe remedy is an insult to the thousands of people who have devoted their careers to this crisis.

People who have devoted their careers to Paradigm A can become very angry when you make a case for Paradigm B. Their anger spills over from attacks on Paradigm B (which are an intrinsic part of the scientific process) to personal attacks on those who dare to back it (which are an intrinsic part of human nature).

If JIMS Isomorphic Music System starts to gain momentum, I can reasonably expect to have such anger directed at me, for daring to challenge established music theoretical and music education orthodoxy. In a way, this would be a sign of progress. Such anger would prove that the Establishment was on the defensive, which is a big step forward for something that is currently beneath the Establishment's notice.