Progressive Complexity

In response to this comment on my post describing Lesson 005.1...


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All of your criticisms above make perfectly valid points. I have what I *think* are good reasons for structuring JiMS the way I am, but in the absence of hard scientific evidence as to the efficacy of one structure vs. another, there is very ample room for differences of opinion. I value, very highly, such critical feedback, because it forces me to revisit my assumptions and see if they have any rational basis. That said...Many of the concepts that I'm teaching in JiMS with specifics, could be taught with greater generality. Here's why I'm starting with narrow, concrete specifics, such as the assumption of the syntonic temperament in 12-TET tuning. I could design JiMS such that it did not assume the syntonic temperament (i.e., alpha = P8, beta = tempered "perfect" fifth, with first comma tempered to unison) or 12-TET tuning (i.e., P8=1200 cents and P5=700 cents). Instead, I could teach that alpha and beta could be any interval. This would be a temperament-neutral approach. So, why am I basing JiMS firmly on the syntonic temperament...and with a fixed octave, to boot? Firstly, because JiMS is based on the philosophy of "progressive complexity," which states that "the simplest approximations of the truth should be presented first, with increasingly complex approximations added later, only as necessary." (I'm quoting myself here, but some prominent educator somewhere must have said something similar.) So, JiMS starts with the diatonic scale in the syntonic scale, in 12-TET tuning, and will progress through the chromatic scale also in 12-TET. It's only when JiMS needs to introduce the enharmonic scale -- which I've decided to do WAY later -- that the concept of "tuning" needs to be introduced. After that, JiMS can introduce Dynamic Tonality. Somewhere right before the introduction of Dynamic Tonality, I'll introduce the notion that the octave's width can be tempered in the syntonic temperament, too. After Dynamic Tonality, I can introduce non-syntonic temperaments. Remember the immediate goal of JiMS: to dramatically increase the efficiency with which non-musicians gain sufficient musical knowledge to write their own pop/rock music. To achieve this goal, students do not need to learn about non-syntonic temperaments. That being said, JiMS' *ultimate* goal is to establish a new universal paradigm of music which expands the frontiers of tonality. I believe that this ultimate goal cannot be achieved without first achieving JiMS immediate goal. Therefore, the immediate goal must take precedence in all of JiMS' design decisions. Specifically, I can't allow JiMS early lessons to be made more complex in order to facilitate the later introduction of non-syntonic temperaments. My recent re-writing of Lessons 4 and 5 is a manifestation of this design trade-off. I have a tendency to want to introduce non-12-TET and non-syntonic ideas sooner than necessary, because I think that JiMS' ability to support those concepts distinguishes it from traditional approaches, and because I think that these concepts are WAY COOL. However, it's simply not efficient to introduce those concepts too early. Secondly, the musical invariances on which JiMS is based -- transpositional invariance, tuning invariance, and cardinality invariance -- are all invariant only within a single given temperament. None of them are invariant across temperaments. I can't teach these invariances in the abstract; I can only teach them in the concrete context of a given temperament. Combined with the doctrine of progressive complexity, the temperament-dependency of musical invariances requires me to teach them within a given temperament (and tuning) first, and only then to generalize those teachings across (tunings and) temperaments. For example, consider the syntonic and schismatic temperaments. Both have the same generators -- P8 and P5 -- so they can share the same note-names, note-layouts, and staff notation. However, their comma sequence differs, and hence so does their mapping of partials to notes. For example, from a fundamental on Do, the syntonic temperament maps the fifth partial to Mi, while the schismatic maps it to Fe. As a result, the "shape" of the major triad (for example) is different, on JiMS keyboard and the JiMS staff, even across temperaments as closely-related as the syntonic and the schismatic. Combined with the doctrine of progressive complexity, this variance-across-temperaments requires JiMS to start by assuming the use of a single (tuning and) temperament, introducing other (tunings and) temperaments only when necessary (and it isn't necessary until after Dynamic Tonality has been introduced, using the syntonic temperament). Thirdly, the music of human cultures in the real world seems to be strongly biased towards temperaments generated by the P8 and P5, including the syntonic and schismatic. The syntonic temperament's tuning continuum includes nearly all of the tunings ever used by human cultures; the exceptions are arguably schismatic, such as Turkish and arguably some Indian music. This bias may arise from the human ear/brain/mind's apparent use, for the detection and tracking of tonal relationships, of a "map of the regions" generated by P8's and P5's (as exposed by Petr Janata's brain-scans). Such a map of the regions is the dual graph of the Wicki/JiMS note-layout, and hence is topologically identical to it. Combined with the doctrine of progressive complexity, this human cultural bias towards the syntonic temperament leads me to choose it as the basis of JiMS, with the added complexity of other temperaments being added to JiMS only much later in the lesson sequence. Please note that all of this is "just talk," however. I don't have any scientific evidence that proves that these design choices lead to the most efficient and accurate acquisition of musical knowledge by musical novices, or that graduates of such an educational program are able to advance the state of the art faster and/or more creatively. But then, no contrary evidence exists to refute these claims, either. The efficacy of alternative paradigms in achieving such objectives is an under-studied area. I hope that, as JiMS becomes available, such rigorous studies can be carried out. In the meantime, you're not wrong, and I'm not right. We just disagree as to when, in the sequence of ideas, non-12-TET tunings and non-syntonic temperaments should be introduced. On the other hand, we agree that these concepts SHOULD be introduced as soon as possible -- an agreement that differentiates us from the vast majority of music educators and music theorists, who neither understand nor care about these concepts. I hope that you will find this response to be in the cooperative, exploratory spirit of give-and-take in which it is intended.

Lesson 005.1

My latest lesson is Lesson 005.1 (source code here), which replaces Lesson 005:


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

This lesson defines "scale" and "diatonic scale," and introduces JiMS keyboard -- i.e., the mapping of the Wicki/Thummer note-layout to the computer keyboard.

My lessons are starting to look a lot like PowerPoint presentations, except that their "graphics" are often interactive (e.g., JiMS keyboard).  I've always liked PowerPoint, so the similarity is fine with me.

Although I was very strongly tempted to introduce other scales and even tunings at this point in the lessons. However, there is absolutely no advantage to the student in introducing those concepts now; they would just be a confusing distraction -- and the student's advantage must win all such design trade-offs. Hence, my decision to re-write Lessons 4 and 5, to provide a leaner, cleaner sequence of concepts.

I expect the next lesson (6) to introduce the term "mode," and discuss the modes of the diatonic scale. I think that I've got the components I need for that, but some of them haven't been used in a lesson before, and so will probably need to be tweaked...so don;t hold your breath for the next lesson.  ;-)

Lesson 004.1

My latest lesson is Lesson 004.1 (source code here):


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

This lesson introduces a number of new terms. Each is clearly defined, and each definition is followed by a question to help cement understanding of the definition.

I now expect to revise Lesson 5 to focus on defining the term "scale." This may require introducing the notion of "tuning," but I don't want to go there yet, so I'll avoid it if I can.

Once the notion of "scale" is defined, then we can dive right into the Diatonic Scale, including its modes, intervals, chords, etc.

Way behind

I'm way behind on developing new lessons, for three reasons:
(1) I had a short consulting contract that occupied a couple of weeks of my time
(2) I need to correct some pedagogical deficiencies in earlier lessons (e.g., introducing the term "note" without defining it in Lesson 4) rather than focus on new lessons, and
(3) I invested a couple of days learning more about Adobe's Text Layout Framework (TLF).

I need to understand TLF better because JiMS makes heavy use of superscripts -- in note-names such as Re⁰ -- which, in Flex, require the use of TLF.

But, I'm back in the saddle now. A corrected version of Lesson 4 should be posted tomorrow (Lesson 004.01), with a modified version of Lesson 5 to follow a few days thereafter.

ExploreTuning1

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.  ;-)

Marek Zabka: Let's Talk

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

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

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.

Lesson 004.0

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

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.

Lesson 2

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.

Lesson 001.0

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.

The Importance of a Good Notation

In his 1911 book An introduction to mathematics, Alfred North Whitehead wrote (p. 59):

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the [human] race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties.

Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. The consequential extension of the notation to decimal fractions was not accomplished till the seventeenth century. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.

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

Great quote, isn't it? 

It clarifies the two essential benefits of a good notation, to wit, that it:
1. Enables domain specialists to advance the state of the art; and it
2. Enables a higher percentage of non-specialists to master the domain's fundamentals.

That's a pretty powerful combination, which explains why notational improvements have been the key to so many of humanity's great leaps forward.

Likewise, JIMS Isomorphic Music System (JIMS)
1. Enables creative artists to advance the state of the art (through such novel effects as Dynamic Tonality), and
2. Enables a higher percentage of non-musicians to master the musical domain's fundamentals.

Or, at least, that's my claim.  Time will tell.  ;-)

In the meantime, today's music education establishment will continue to argue -- as Greek mathematicians did in their day -- that their domain's high failure rate is due to the inherent difficulty of their domain, not due to the imperfection of their notation (and instrumentation). Perhaps JIMS, too, will astonish domain experts by doing the impossible.

JMTP paper rejected

Today I received notification that my recent submission to the Journal of Music Theory Pedagogy (JMTP) was rejected.  The rejection letter is appended below.

The reasons for the rejection were many, but all boil down to this one, from Reviewer #2:

We have precious little time to teach students as it is now; if in their theory and/or aural skills classes they are dealing with a new notational system, I believe that will not support their progress in reading and performing traditional music.

One might paraphrase this as "Our patients are dying left and right, despite our bleeding them with the best leeches available. Your proposed 'germ theory' does nothing to improve the efficiency of our leeches, and hence has no place in the practice of modern medicine."

This submission/review/rejection process proved to me that academia, per se, will reject JIMS reflexively. This proof validates my decision to target my JIMS-based online courseware at "musically-inclined but not-formally-trained individual consumers" (using rock music rather than the Common Practice classics), thereby following a path very similar to that of Rosetta Stone's breakthrough language-learning courseware.

Once the first version of JIMS courseware is on a self-sustaining trajectory, I can produce separate version that is dumbed down (by using traditional note-names and staff notation), thereby meeting the needs of modern academia.

In the meantime, illigitimi non carborundum, and back to developing the courseware!

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

From: Steve Laitz [mailto:***]
Sent: Saturday, October 24, 2009 9:00 AM
To: Jim Plamondon; Andrew Milne; William Sethares
Subject: Re: JMTP & Paths to Musicianship

Dear Professors Plamondon, Milne, and Sethares,

I write to inform you that your article “Sightreading Music Theory” will not be accepted for publication by the Journal of Music Theory Pedagogy. Unfortunately, the reviewers found the paper to have little to do with its title: "Sight-Reading Music Theory." They were somewhat confused in that the paper appears to be a rewrite of The ThumMusic System.

The reviewers felt that the article is not relevant to pedagogy in general and it does not demonstrate direct application to the teaching of theory. One issue each reviewer voiced was a frustration concerning constant references that were “beyond the scope of this paper” and that these references were often directed to your own website for explanation. They felt that anything that is not common knowledge in the theory community needs to be explained within the paper.

The reviewers felt that the non-theorist who is teaching theory at the collegiate level—and there are plenty of these folks who read JMTP--will not find the paper particularly helpful in pedagogical matters. Further, even the professional music theorist will reap little benefit from the paper given that issues are explored but not clearly explained. For example, the acronym JIMS is used throughout the paper, but there is never an explanation for what the letters represent, except in the abstract, and even there, the letter “J” is not defined. Other examples include the “pitch buttons” and references to “playing pitches,” but without clear definition at that point in the article. Regarding the Thummer instrument and the related ThumMusic system, both of which you mention late in the article, this all requires far more explanation and, more importantly, direct focus on the applicability of the JIMS System and Thummer instrument to theory teaching.

Below are detailed comments from each of the three readers that I hope will help you, should you decide to revise the paper and submit it to another journal.

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

Reviewer #1:

Page 2, “scenario”: how can a HS band student have no background whatsoever and still play an instrument? How much does the author’s “etc.” include? Students would have to at least read music in a single clef in order to play, and presumably would know a piano keyboard. And that is one of my issues throughout this article—any pedagogy should help student musicians with the music they will encounter in any setting outside a theory classroom: in performance (printed scores); in analyzing scores in other classes, etc. Music is not going to be rewritten to accommodate a new technology.

Reviewer #2

1. Page 2, #3, Musical Isomorphisms, term “chromatic staff”: is he referring to a standard staff (capable of showing any pitch) or a specialized staff. I think this needs to be made clear.

2. Page 3: last sentence before section 3.2: re music-control interfaces: if he means piano layout, fretboard layout, etc, I think he should say that. I’m not sure most of us refer to the layout of notes on different instruments as music-control interfaces.

3. Page 4, the sentence that says the “extra” notes will become clear later – I’m not sure they do. Or at least, not without a lot of work on the reader’s part – see my note #21

4. Page 4, 2nd paragraph under Figure 3: “To play in C major . . . beyond the scope . . . ” – I think he needs to provide an explanation. If I understand correctly, this is somewhat equivalent to a transposing instrument – regardless of the key, the fingering always remains the same. Rather than changing horns, for example, one sets G as tonic, or A as tonic, etc.

5. Page 5, section 3.3: “Chromatic staff”: this 2nd reference suggests the chromatic staff is not our standard staff, yet I’m not sure how many people would know this reference. I am not familiar with it as far as I recall. So again, I think some explanation is in order.

6. Page 5, re Fig. 4: I think some examples would be in order to show what he is talking about.

7. Page 5, next paragraph, beginning “At the far left”: end of paragraph -- again, I think an example is in order.

8. Page 5, next paragraph, beginning: “All other symbols.” He cites his website for a detailed description of JIMS staff notation. Well, once again, if this article is supposed to explain “Sight-Reading Music Theory,” and the author is wishing to encourage support for this alternative system of notation, then I think fundamental information concerning these things is not at all beyond the scope of this paper, but belongs squarely in it.

9. Page 6: Are the terms “double harmonic major” and “double harmonic minor” common terms? They aren’t as far as I am aware, so again, should be explained, at least in terms of where these scales are found or employed.

10. Page 6: small point: Fig. 5: Make the outline of the white honeycomb cells darker – they are hard to see. But, if the scale dots should be white, how are they to be seen against the while honeycomb?

11. Page 6, Fig. 7 and 8 confuse me. I ultimately can see them the way the author intends, but it took some time and effort. Perhaps it would be better to use 2 diagrams to show first major, then minor triads. It took me some time to figure out what he was doing in this figure, which is duplicating each note (Do, do), to show how a single pitch fits into either a major or minor triad. If his system is supposed to make things simpler, this absolutely does not do that.

12. Page 7, Section 3.4.1: he says that we should minimize memorization load: Great! I’m all for using as few terms as possible, and not creating new ones, especially if they conflict with terms already in use for the same concept (one of my problems with Edwin Gordon’s writings). But he then introduces yet more terms to be memorized. Thought experiment or not, one cannot ignore the fact that students will be learning terms like Plagal and Authentic (and should!!) in music history classes, and or encountering them in various other classes and musical environments. In my view, music notation is not going to change. We have precious little time to teach students as it is now; if in their theory and/or aural skills classes they are dealing with a new notational system, I believe that will not support their progress in reading and performing traditional music. If a student has a gig, he or she needs to be completely conversant with standard music notation. Using the proposed set of terms would make it difficult for students trained that way to converse and play with other musicians, as I don’t think this will ever gain world-wide or even country-wide adoption.

13. Page 7: Similarly, bulleted items no. 3 and 4 strike me as confusing, and would add to what a student would have to memorize. (I admit that I am only vaguely aware of Nashville numbers, so I don’t know if this corresponds to them in some way, and I can see how this corresponds somewhat to how one would read a jazz chart. But it is still not really the same, so creates the necessity of learning yet another system on top of what students will need to learn in order to perform any standard notated music, or music from a lead sheet.)

14. Page 8, secondary dominants: maybe I’m slow, but I don’t get how his system makes this “entirely clear,” since in my experience, secondary dominants are never entirely clear to any but a very few students. So once again, I think he is asking too much of the reader to have to go to his website to understand how this would work (“beyond the scope of this paper”), since this explanation would seem to be at the core of what his title suggests. (Oh, wait – is he trying to drum up business on his website???)

15. Page 8, last paragraph, first sentence: “Likewise every occurrence . . .”— I’d like an example to see how exactly every Do-Fa-Sol-Do chord progression will look like every other.

16. Page 9, last bulleted item before section 4.2: I suppose this is correct, but how would it correspond to what students are seeing when they read actual music?

17. Page 9, section 4.2.1, Creative Power:
a. Again, one must go to his website to understand – very problematic

b. Syntonic temperament – again, one is forced to go to his website to try to learn what he is talking about. I used to know quite a bit about tuning and temperament, but I’m not sure from the article what he is talking about. Did he mean this was a Just system? That didn’t seem to be right. And how, btw, can a syntonic system be equal tempered? That is not my understanding of syntonic at all. Perhaps that is my own ignorance, but I’m guessing other readers would have a problem with this concept as well. So I did go to his cited publication to see what he meant. It cleared it up for me, but I think he needs to explain what he means in the article. I suspect that would not be clear to very many readers at all.

c. I think this could be pared down by saying something like “this system is capable of producing any number of tuning systems by simply setting the system via the controller.”

d. Next paragraph: very small point, but in the first line, the word “retaining” should be “retain.”

18. Page 10, fig. 10 – this seems to be getting beyond the scope of this paper, plus I’m not sure I completely understand his figure. I understand cents, I understand commas, I understand equal and non-equal tunings, but I don’t quite get what his figure is showing. And in fact, this whole section on tuning seems slightly out of place with what the first part of the article seems to be about. If the article is about pedagogical efficiency, and “sight-reading” theory (I don’t really think that is the appropriate title for this article, either), then this digression into tuning seems to me a bit out of place I believe this would require more explanation, but that would truly be beyond the scope of this paper.

19. Page 11, 2nd paragraph: “three full 8vas, of 19 buttons” – I think this needs more explanation: coming from a section on tuning, and tunings with many divisions of the 8va, this becomes confusing. I think he needs to specify that his 19 notes include enharmonic equivalents, including “De” and “My.” He says earlier on p. 4, below Table 1, that the need for these “extra” notes will become clear later in the paper, but he never again addresses that. This would appear to be the place to do so.

20. Page 11: Under Fig. 11, I have no idea what he means by 10 degrees of freedom.

21. Page 12, section 5, Metrics: umm, sorry, but I am once again confused. What does he mean that before Guido invented sight-singing his singers could sing but didn’t know any songs?? Of course they did. They learned them by rote and memorized them, just as any child learns the ABC song, Happy Birthday, etc, without ever learning to read music.

22. Page 12, section 5, second paragraph, 1st sentence: “ . . . quality of a music theorist” – I thought this was about teaching students, not music theorists?? Following sentence: “On the one hand”: this seems to be getting at a separate agenda.

23. Page 12, section 5: Small item towards end of 2nd paragraph: “To identify of key centers” – obviously, delete the work “of.”

24. Page 12, section 5, end of 2nd paragraph: “to recognize modulation . . . ” – he never demonstrated this earlier on, when he should have.

25. Page 12, Section 6, Previous Work, end of 1st paragraph, “most viewers absorbed the basics quickly.” As a reader, I would like to know what those basics included.

26. Page 13, top: Huh?? “this paper cannot and does not propose that JIMS be used today in music theory pedagogy.” I thought that this was what the article was supposed to be about?

Reviewer #3:

Page column line comment

1 2 15 On which page in Einstein’s article does this quote appear?

2 1 13 You should use a gender-neutral reference for the students. I would suggest you use “his/her” rather than “her.

4 Table 1 The fifth scale step in Tonic solfa is “Sol” and not “so.”

4 2 5 You write, “to play in C Major, one must indicate . . . that Do should sound the pitch C.” How is a student to determine what the tonic note is?

What are the criteria and where in this paper have you established this?

How does the student know if the piece is in major or minor?

5 2 1 You need a musical example to illustrate Ri as an “upward-pointing note-head” and Me as “a downward-pointing note-head.”

6 2 7 You need to have a page reference for Euler (is page 6 in Cohn’s article?).

7 2 5 I don’t agree with your assertion: “the ‘major scale’ and ‘minor scales’ are not scales at all.” A scale is a collection of notes that span the octave; the mode is the specific pattern of steps and half steps that encompass the octave. In his harmony book (Harmony, rev. ed. New York: W.W. Norton, 1948), Piston opines, “tonality is synonymous with key, modality with scale” (29).

7 2 20 Here I take exception to your premise that you are not “defining new terms for specialized uses.” Moreover, if a student is to be conversant and literate, s/he must know what “authentic” and “plagal” mean. Finally, for those of us who use do-based minor, reading la-re-fa-ti-mi-la does not allow me to audiate i-iv-VI-ii-V-I (aside from the infrequent progression of iv-VI!). Am I to assume mi means (in a minor) E-G#-B?

8 1 5 “3Do” reminds me of Percy Goetschius’s nomenclature in The Material Used in Musical Composition (New York: G. Schirmer, 1889). The difference resides in how it is written: in Goetschius IV2 = IV@. I doubt that the literate musician will know that “5So7” means V$.

8 2 14 How do I know that Re7 “is the dominant of the dominant” in a diatonic D-mode rather than the diatonic ii‡?

9 1 26 I do not believe that the ability “to transpose notation among clefs and keys; to identify key centers and key relationships; to recognize modulation to closely related keys; and so on” is irrelevant. A French horn player in band must know how to perform at sight an Eß part on his/her F horn.

13 1 1 This says it all: “Clearly, this paper cannot and does not propose that JIMS be used today in music theory pedagogy.” Moreover, nothing in this paper convinces me that JIMS will “improve the efficiency of theorist-training.”

References. By this I assume you mean a bibliography. However, some of the documentation is inaccurate or missing. For example, the ISBN for d’Arezzo is 1-896926-186 (not 978-1896926186). For Cohen, where was the book published?

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

Steve Laitz
Editor, Journal of Music Theory Pedagogy

Myhill's Property and Interval Names

The characteristic of having two versions of each simple interval is known in diatonic set theory as Myhill's property, and it is the source of many other musically-significant characteristics.

It seems to me that this property is more-easily exposed and explained if the two versions of each simple interval are named consistently, e.g., major and minor, rather than calling some perfect, some augmented, and some diminished.

The Major-Minor Axis

I've written a little Flash application to show how the modes of the diatonic scale relate to one another, using JIMS' Keyboard.  It's not intended to be courseware, but rather to answer some questions that have come up in a different forum (which is why I didn't clean up its bugs).

To run the app, click here.

There are two button on the lower-right corner of the screen:
- Minor-ward: Moves the mode one degree towards the minor end of the major-minor axis (Ti).
- Major-ward: Moves the mode one degree towards the major end of the major-minor axis (Fa).

Initially, the app just shows JIMS Keyboard.

1. Click once on the "Minor-ward" button. The simple diatonic intervals (henceforth, "intervals") of Fa-mode (Lydian) will be revealed. All of its intervals are major, because Fa-mode is the major endpoint of the major-minor axis.  Note that this discussion uses the renaming of "perfect" intervals discussed here.
2. Click again on Minor-Ward button. Fa-mode's pattern of (major) intervals will be slid up-and-right to Do, the next note minor-ward along the diatonic major-minor axis. Fa-mode's pattern of intervals fits Do-mode just fine, except for one interval: the major fourth. In the previous mode, it ended on Ti, but in this mode, it ends off the diatonic scale, on Fi. Therefore, we must replace the previous mode's major 4th with a minor 4th. By shifting the interval's endpoint to Fa, we get Do-mode's minor 4th.
3. Click on Minor-ward again. Do-mode's pattern of intervals, including the m4, will be slid along the major-minor axis to So. Do-mode's pattern of intervals fits So-mode just fine, except for one interval: the major 7th. In the previous mode, it ended on Ti, but in this mode, it ends off the diatonic scale, on Fi. Therefore, we must replace the previous mode's major 7th with a minor 7th. By shifting the interval's endpoint to Fa, we get So-mode's minor 7th.
4. Click on Minor-ward again, shifting the previous mode's pattern of intervals to Re-mode. Again, the previous mode's intervals all fit Re-mode just fine, except for the major third, which ends off the diatonic scale, on Fi. Replacing the major 3rd (ending on Fi) with a major third (ending on Fa), we get the intervals of Re-mode (half major, half minor).

• By now, the pattern should be clear: at each step minor-ward along the major-minor axis, the only interval changed in width is the (major) interval ending on Ti, which is replaced by a (minor) interval ending on Fa.

1. Click on Minor-ward again to see the Ti-ending major 6th change to a Fa-ending minor 6th.
2. Click on Minor-ward again to see the Ti-ending major 2nd change to a Fa-ending minor 2nd. 
3. Click on Minor-ward again to see the Ti-ending major 5th change to a Fa-ending minor 5th.

Now, we've arrived at Ti-mode, at the minor end of the major-minor axis.  All of its intervals are minor.

1. To go back down the axis in the other direction, click on Major-ward. All of Ti-mode's intervals will fit Mi-mode just fine, except for Ti-mode's minor 5th. In Ti-mode, this 5th ended on Fa, but now it falls off the diatonic scale onto Te -- so it must be switched to end on Ti, instead, giving Mi-mode its major 5th.
2. Before clicking on Major-ward again, identify the interval that ends on Fa. It's Mi-mode's minor 2nd. That's the interval that will be changed when moving down the major-minor axis to La-mode.
3. Click on Major-ward, and watch Mi-mode's Fa-ending minor 2nd be replaced by a Ti-ending major 2nd in La-mode. Which interval will be replaced next? The one that ends on Fa. Which one is that? La-mode's minor 6th.
4. Click on Major-ward again to see La-mode's Fa-ending minor 6th be replaced with Re-mode's Ti-ending major 6th.
5. Click on Major-ward again to se Re-mode's Fa-ending minor 3rd be replaced by So-mode's Ti-ending major 3rd.
6. Again, and So-mode's Fa-ending minor 7th is replaced by Do-mode's Ti-ending major 7th.
7. Again, and Do-mode's Fa-ending minor 4th is replaced by Fa-mode's Ti-ending major 4th.

Unfortunately, code bugs prevent you from going back up the axis, or from reversing course mid-way along the axis.

Nonetheless, this simple app usefully exposes some of music's patterns:

1. Fa-mode (Lydian) is "the most major" mode, and Ti-mode (Locrian) the "most minor," each being at extreme ends of the major-minor axis, which runs along an axis of major fifths.
2. Moving up the axis towards minor, the (major) interval ending on Ti will be swapped for the (minor) interval ending on Fa.
3. Moving down the axis towards major, the (minor) interval ending on Fa will be swapped for the (major) interval ending on Ti.
4. Re-mode (Dorian) is half-major and half-minor, giving it a uniquely-ambiguous position along the axis.
5. Stepping from Do-mode to So-mode changes an odd-numbered degree (7th), and so does the adjacent step from So-mode to Re-mode (3rd).  These are the ONLY two adjacent steps along the major-minor axis which both change odd-numbered intervals. This is significant, because tonal harmony is based on stacking odd-numbered degrees (that is, 3rds) in the mode of a given chord's root. (Similarly, the steps Re-to-La-to-Mi change the 6th and 2nd degrees, which might matter more to stack-of-4ths [quartian] harmony, as found in some jazz, than to stack-of-thirds [tertian] harmony).
6. The traditional names for the 4ths and 5ths obscure the consistency of these patterns. These intervals' names should follow the same pattern as the other two-value diatonic intervals, that is, the larger size (traditionally "augmented 4th" and "perfect 5th") should both be called "major," and the smaller size (traditionally "perfect 4th" and "diminished 5th") should be called "minor." The only intervals that should be called "perfect" are unison and its octaves, because they alone are distinguished by having only one size in the diatonic scale.
7. The The traditional names for the 4ths and 5ths also obscure the potential consistency of the "diminished" and "augmented" names. Once the names of the 4ths and 5ths are regularized, then "augmented" and "diminished" intervals can be recognized as referring consistently to chromatic alterations of diatonic intervals.

Much more betterish.  ;-)

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.

JIMS paper rejected (MTO)

On Monday, I received an email notifying me that my paper on JIMS Isomorphic Music System (JIMS) was rejected by the peer-reviewed journal Music Theory Online.

The rejection letter read as follows:

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From: Matthew Shaftel
Sent: Monday, July 06, 2009 1:01 PM
To: Jim Plamondon
Subject: Re: MTO Submissions

Dear Jim,

I have just spoken informally with both reviewers and we are in agreement that your submission is not appropriate for MTO. It's underlying scenario is not appropriate to our audience (who mostly teach students with musical backgrounds), and the technology you describe seems far too cumbersome for entry-level theory at either the High-School or College level. In addition, we agreed that the articles advocacy of the Thummer (and subsequent open-source iterations thereof), is simply not appropriate for an academic journal like MTO.

I am sorry that we cannot give you any better news, but I wish you luck in your continued endeavors.

Sincerely,

Matthew

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Today I responded as follows:

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From: Jim Plamondon
Sent: Wednesday, July 08, 2009 12:24 PM
To: 'Matthew Shaftel'
Subject: RE: MTO Submissions

Matthew –

To be more specific, it appears to me that your rejection is based on three claims:

1. JIMS is not compatible with traditional instruments and notation, and therefore not appropriate for students and teachers who have already mastered both. One could argue identically that Guido’s sight-reading technology was of no value to those who had already memorized the Church’s canon, as was the norm before his technology took root. Backward compatibility with established technologies is relevant only up to a point. Historically, if a new technology offers an efficiency-gain of 100% to 200% (i.e., twice or three times the previous efficiency), then compatibility with previous technologies becomes irrelevant.
2. JIMS is cumbersome. “Cumbersome” is defined as “unwieldy because of heaviness and bulk,” or “troublesome or onerous.” I can’t see how any aspect of JIMS can be fairly described as cumbersome, compared to traditional instruments & notation. I suspect that what you really mean is that JIMS “is not what I already know,” which is just a re-statement of Claim #1 above.
3. The paper’s advocacy [of the Thummer] is unacceptable. If an alternative technology is worthy of contending with the status quo, then it must provide sufficient benefits to overcome the inertia of the status quo. However, when an alternative technology is new, there has not yet been sufficient opportunity (by definition) to develop rigorous evidence to support its claimed benefits (else it would no longer be “new”). The factually-based description of *potential* advantages is, for a truly new technology, the most scientific discourse possible. Consider, for example, the metric system; before its adoption, there was considerable scientific discourse on its potential benefits – that is, “advocacy,” by your definition – which could not be rigorously proven until it had been adopted by at least one country. If JIMS is incompatible with traditional instruments, then it must be shown to be compatible with at least one novel instrument that delivers at least the expressive power of traditional instruments. This is not advocacy; it is counter-argument to the argument of incompatibility…which brings us back to Claim #1.

Hence, your only substantial criticism of the JIMS paper is that JIMS is incompatible with traditional instruments & notations. Yet this incompatibility will be shared by *any* paradigm-shifting improvement to the status quo in music theory and/or music theory education. To use this criticism as the basis of rejecting the JMS paper is to enshrine, as an editorial goal, the MTO’s reactionary defense of the status quo.

I sincerely doubt that this is your intent. You strike me as a good, reasonable, and serious person. I respect your good will, and hope that you respect mine, too. :-)

Consider your criticism of the JIMS paper’s “advocacy.” As you have noted, JIMS is incompatible with traditional notation and instruments. This incompatibility is likely to be noted by the paper’s readers, too, if only because I state it explicitly. A reader might then wonder if learning music using JIMS were a dead end, leaving no opportunity for expressive performance. Therefore, it is necessary, in any paper that introduces JIMS, to counter this inevitable argument with a counter-argument, to wit, that (a) Thummer-like instruments can be made; that (b) they can offer up to 10 degrees of freedom; that (c) this is more expressive potential than that offered by any other polyphonic instrument; and that (d) they offer the unique ability to control the novel effects of dynamic tonality. If, as your rejection suggests, the cost of incompatibility is high, then the benefits offered by any proposed alternative must be high, too. To argue that the description of such potential benefits is unacceptable “advocacy” is to require only the negative consequences of incompatibility to be discussed. This is clearly a reactionary, pro-status-quo bias.

Likewise, if the mere description of the potential advantages of an as-yet unimplemented technology is deemed to constitute unacceptable advocacy thereof, then no scientific cooperation in the implementation of such a technology is possible, because such cooperation requires exactly the shared awareness that’s blocked by this criterion. It’s a Catch-22.

Consider, for example, Mendeleev’s initial paper describing his Periodic Table, in which he described (a) a number of proposed corrections to previously-accepted atomic weights and (b) predictions that previously-unknown elements existed with the weights and properties described by “holes” in his Table. He had no scientific evidence whatsoever to support these claims; indeed, his claimed “corrections” to established atomic weights flew in the face of all previous evidence. Instead, his paper invited its readers to collaborate in conducting the experiments necessary to prove or disprove his radical claims. Those experiments subsequently bore out his claims, and enshrined the Periodic Table as one of the greatest discoveries of science.

Yet according to the standards of compatibility and advocacy you describe above, Mendeleev’s paper should not have been published, because it was incompatible with the status quo and “advocated” – that is, described – an alternative to it. Had his paper not been published, then no one would have known of its predictions, and hence none of the research to prove or disprove those predictions would have taken place (for decades, at least).

Likewise, when Wegner proposed his theory of continental drift – which has since become the basis of modern geology’s plate tectonics – his suggestions were derided as being incompatible with the prevailing theory of geosynclines, and his papers were often rejected on grounds similar to your claim of “advocacy,” because they described ways in which his theory resolved previously unresolved issues in geology, paleontology, and paleoclimatology – exactly as my JIMS paper described ways in which JIMS enables greater ease of learning, expressive potential, and freedom of tuning.

These are not isolated incidents. The history of science is rife with such examples; it is a well-recognized problem at the intersection of the peer-review system and paradigm-shifting ideas (see here, here, and here).

Your rejection of the JIMS paper on the grounds of incompatibility and advocacy is, I submit, an example of exactly this kind of implicit, reactionary, pro-status-quo bias. This is not because you’re a bad person, but rather because you are merely human, and have humanity’s inherent weaknesses…as do I. ;-)

I do not purport to have a solution to this systematic problem. I would, however, encourage you to look critically at the MTO’s use of “incompatibility and advocacy” as publication criteria; to consider these criteria’s roles in suppressing potentially paradigm-shifting innovation; and to reconsider the use of these criteria in the interest of truly advancing the state of the art.

Which is what such journals are all about, right?

Thanks! :-)

--- Jim

From: Jim Plamondon [mailto:jim@thumtronics.com]
Sent: Wednesday, July 08, 2009 9:18 AM
To: 'Matthew Shaftel'
Subject: RE: MTO Submissions

Bummer. Semmelweis reflex in action.

Irony, or Abstraction?

The irony of my previous post is not lost on me. On the one hand, I am advocating that others learn music using a non-standard system (JiMS) due to the benefits of its higher level of abstraction, while at the same time choosing myself to learn a standard (programming) system despite the costs of its lower level of abstraction.

This analogy is misplaced, however, because a single level of abstraction removes the inconsistency.

Adobe's ActionScript/Flex/Flash toolset isn't the "standard programming system" today; today's standard is Java. Adobe's toolset is an alternative to today's standard that offers efficiency benefits in the development of Rich Internet Applications (RIAs). Therefore, if one views what I'm doing as learning to develop RIAs, then I'm choosing the non-standard, high-efficiency approach -- just as I am suggesting others do by using JiMS to do when learning to develop music (so to speak).

Re: The Center of All Things

In JiMS iGetIt! Music System, the notes of the diatonic scale (in any key) are named "Do Re Mi Fa So La Ti." What are the names of the notes of other scales?

I am not aware of any standard mapping of these other scales' notes to tonic solfa names, nor of any standard rules for defining such a mapping, nor of any standard criteria for comparing one mapping-rule to another. One possible mapping-rule might be to "maximize the correspondence between one scale and another."

For example, when mapping the notes of the a given scale to tonic solfa, one might reasonably choose to map them to that set of tonic solfa names that maximzed the given scale's correspondence with the Diatonic scale. Using this mapping-rule, the Harmonic Major and Harmonic Minor scales would be mapped to a set of note-names that differed from the Diatonic by only one note. The Harmonic Minor would have Si instead of So, while the Harmonic Major would have Le instead of La. Those are perfectly cromulent mappings, which usefully expose the similarity of these scales to the Diatonic.

However, that's note the mapping-rule that I used in the just-posted online version of JiMS iGetIt! keyboard, however.

Instead, I chose to use a different mapping-rule: maximize the extent to which the scale is centered on Re. Using this rule, it is obvious that all of the Prime Scales

• include the notes So, Re, and La
• are either
• symmetrical around Re (Diatonic, Melodic, Neapolitan, Double Harmonic), or
• are not symmetrical (Harmonic Major and Harmonic Minor), but are reflections of each other around the Re axis.

I don't know that this mapping-rule is any better, and it may be much worse. By what criteria should different mapping-rules be judged? In the development of JiMS, I had only one criterion throughout: maximizing the efficiency of learning. However, I'm not sure that this criterion delivers a clear answer on this mapping-rule question. Exposing the consistency of the Prime Scales' So-Re-La core is a good thing, but is it better (i.e., more efficiency-increasing) than exposing the near-identicality of the Diatonic and Harmonic scales? I don't know.

Certainly the "Diatonic maximization" rule hews most closely to music-ed tradition, which sees all scales in terms of their difference from the "major scale" (including the other modes of the diatonic scale, which is just bizarre). I happily admit to having a knee-jerk reaction against traditional musical thinking. This reaction forces me to ask "why?" about absolutely everything, which has proven to be a very useful habit. However, if I can't find an efficiency-increasing alternative to a traditional practice, then I'll go with the traditional practice, to meet the secondary criterion of "maximizing compatibility."

For example, I would *love* to change the tonic solfa names to something more meaningful, so that the vowels of the names in the diatonic scale meant something, instead of actively conflicting with meaning as the diatonic vowels currently do. ('i' means 'sharp,' except for Mi and Ti; 'e' means 'flat,' except for Re. This is exactly the kind of inconsistency that makes music so hard.) One alternative set of names would be "Do Ra Ma Fo So La Te", with 'u' meaning flat and 'i' meaning sharp, 'o' meaning the root of a major triad, 'a' meaning the root of a minor triad, and 'e' meaning the root of a diminished triad. There are two problems with this alternative naming: it is meaningful only for the diatonic scale, and it would irritate everyone who had already learned the traditional solfa names. The first concern eliminates any efficiency benefit; the second imposes an efficiency cost. Hence, JiMS uses the traditional solfa names, despite their irritating lack of meaning.

What note-name mapping-rule should JiMS use for non-diatonic scales?

Your comments welcome. :-)