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.  :-)

Separation of concerns in music education

Computer science has an important concept called separation of concerns, first described by Dijkstra in his 1974 paper On the role of scientific thought as follows:

Let me try to explain to you, what to my taste is characteristic for all intelligent thinking. It is, that one is willing to study in depth an aspect of one's subject matter in isolation for the sake of its own consistency, all the time knowing that one is occupying oneself only with one of the aspects. [...]

It is what I sometimes have called "the separation of concerns", which, even if not perfectly possible, is yet the only available technique for effective ordering of one's thoughts, that I know of. This is what I mean by "focusing one's attention upon some aspect": it does not mean ignoring the other aspects, it is just doing justice to the fact that from this aspect's point of view, the other is irrelevant.

JIMS is "concerned" with one thing, and one thing only: the efficiency with which the concepts of tonal music can be learned.

This does not mean that I consider other musical "concerns" (such as performance skills, music history, ehtnomusicology, etc.) to be unimportant.  Quite the contrary! Rather, it is only by studying each concern in isolation that its unique characteristics can be understood with sufficient clarity to enable the concern to be re-integrated with other related concerns.

In this video, starting about 8:50m in, Dijkstra compares writing software to writing music, and discusses the impact of higher-level programming languages—that is, languages that express programming concepts at a higher level of abstraction. This discussion is apt because one could consider JIMS to be a "higher level language" for music, compared to traditional notational and gestural languages.

In the above video at 17:25m, there's a quote about elegance from EWD 1284 which reads,

In some ways programs are among the most complicated artefacts mankind ever trued to design, and personally I find it fascinating to see that reasoning about them is so much aided by simple, elegant devices such as predicate calculus and lattice theory. After more than 45 years in the field, I am still convinced that in computing, elegance is not a dispensable luxury but a quality that decides between success and failure; in this connection I gratefully quote from The Concise Oxford Dictionary a definition of "elegant", viz. "ingeniously simple and effective". Amen.

The use of "simple and effective devices" (such as isomorphic note-layouts, solfege, and the tonnetz) is precisely how JIMS intends to help students "reason about music."

Diatonic Set Theory

I've been reading up on diatonic set theory, using Timothy Johnson's Foundations of Diatonic Set Theory and various scholarly papers (thank God for Google!).  It all seems to be based firmly on the syntonic temperament (that is, on stacks of tempered perfect fifths, in which the syntonic comma is tempered to unison).

This is absolutely the right simplifying assumption to make initially. Now, however, it seems reasonable to explore the application of its findings to other temperaments (such as Magic). Presumably, it will be discovered that some the "global" rules apply across a well-defined subset of all possible temperaments, and that each temperament has its own "local" rules.

Knowing which rules are global, and which local rules exist in any given temperament (such as Magic), could go a long way towards defining the intrinsic music theories of these alternative temperaments -- temperaments that now have, for the first time ever, the possibility of local consonance and dynamic tonality.

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.  ;-)