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.

Interval width changes across the syntonic tuning continuum

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

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

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:

1. 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
2. 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

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

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.

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

Isomorphism & diatonic set theory

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

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

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

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

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

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

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

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

...and 19:

...and 21:

...and so on, ad infinitum.

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

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

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

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

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

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

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

(b) They are symmetrical around Re.

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

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

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

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

Circle of Nths

Here's my latest Flex control:


The component shows the "Circle of Nths" for a given N and a given scale.

Hopefully, the component's controls are fairly self-explanatory.

Musical Issues
The component uses the interval-naming scheme discussed here.  In brief,
- There are two interval categories: perfect and imperfect.
- The only perfect interval is unison (and its octaves); all other intervals are imperfect.
- Each interval also has a quality: diminished, minor, perfect, major, or augmented.
- All imperfect intervals of diatonic Fa-mode (Lydian) have the quality "major" (including its fourth, the "major fourth").
- All imperfect intervals of diatonic Ti-mode (Locrian) have the quality "minor" (including its fifth, the "minor fifth").
- All perfect or major diatonic intervals, when chromatically widened, have the quality "augmented."
- All perfect of minor diatonic intervals, when chromatically narrowed, have the quality "diminished."
- The resulting interval-names are used to name the widths of the intervals of all scales, whether diatonic or not.  (Specifically, one does NOT generate interval names for non-diatonic scales by applying to them the name-generation algorithm described above; instead, one just names a non-diatonic scale's intervals using the corresponding interval-names generated for the diatonic scale.)  Actually, it may be that these names only make sense within the syntonic temperament; other temperaments, such as Magic, may require different interval-names. I haven't looked into these other temperaments enough yet to know for sure.

This use of color matches this animation of the relationships among the diatonic modes' intervals as one moves from mode to mode along the major-minor axis (which is also the Circle of Fifths).

Using this naming scheme makes it easy to see that, within the diatonic scale, all non-octave intervals occur in exactly two sizes (major and minor). This is Myhill's property, and it is the essential characteristic from which the other properties of the diatonic scale emerge (e.g., maximal evenness, cardinality equals variety, structure implies multiplicity, and being a well formed generated collection). It is also the property from which Dynamic Tonality arises. It is also easy to see that this property is not shared by any of the Prime Scales (i.e., those shown in the scale-selection combo box).
In his book Foundations of Diatonic Set Theory, Timothy Johnson uses a single-octave note-circle for all Circles of Nths. His Circle of Fifths, shown on Page 82, is one example. Using a single-octave circle shows the relationships among the notes clearly, whereas using (N-1)-octave circles shows the relationships among the intervals clearly.

Programming Issues
The sliders don't have tick marks or labels because I can't figure out how to make Spark sliders show these things. Halo sliders had a property, tickInterval, that I could set for this purpose, but Spark sliders don't. I spent a couple of hours searching the documentation and source code (always the best documentation), but couldn't find anything that looked right.

If you know how to decorate a Spark slider with tick marks and bounds labels, please let me know.

The component's interval-arrows are also drawn in a stupid manner -- by simply drawing a sequence of connected straight line segments. I'd rather use an elliptical Path, a la Degrafa/SVG, but Flex 4's FXG stuff -- despite being otherwise quite spiffy -- does not support elliptical paths (why not?).

This component is NOT a good example of how to use Flex 4's Spark architecture, because it doesn't. It is a very Halo-like component, making no use whatsoever of Spark's skinning or layout enhancements.

Now that I've got the basic control working to my satisfaction, I'll see if I can break it up according to the proper Spark-style architecture (components, skins, layouts).

The component is also a fairly egregious example of ravioli code (i.e., "encapsulated spagetti code"). The one component is doing way too much; its source file is nearly a thousand Lines Of Code long (1 K-LOC). That's nothing to compare to Flex 4's 12 K-LOC UIComponent, which is the Mother of All Ravilolis (of necessity) -- but it's still a signal that my component probably should be broken up.

I also need to learn how to bring MXML data into a library-based component. If anyone can tell me how to do that, I'd welcome the instruction.

Once I've re-architected the component to use Spark's new architecture, I should be able to change its superclass to Slider, and presto change-o, enable dragging a thumb around the "clock" to change its mode. Being able to interactively change the Circle on Nth's mode will make it easier for for a student to see the relationship between an interval's width and its degree in a given mode.  (The width of every 4th in the diatonic Circle of 4ths, for example, corresponds to the width of the 4th in the diatonic mode of the interval's starting-note.)

I'd also like to explore smooth animation of the component's state-changes. That way, when the control is changed from (say) being a Circle of 2nds to a Circle of 3rds, the note-labels can move around and change size slowly enough for the eye to follow, hopefully making the transition itself easier to understand.

Hence, this control is providing me with lots of opportunity to explore Flex 4's new architecture and "learn by doing."

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.  ;-)