ExploreTuning1

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

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


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

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

This chart shows what's happening:

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

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

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

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

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

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

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

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

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

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

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

Which bring us to Dynamic Tonality.

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

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

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

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

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

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

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.  :-)

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.

Meantone temperament

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

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

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

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

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

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

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

Tonnetz

Since Euler, and especially since Hugo Riemann, the tonnetz has been thought of as being generated by a combination of major thirds and perfect fifths. Reading an excellent paper by Yale's Richard Cohn, I have suddenly realized how this traditional approach could be generalized using the Matrix/ThumMusic paradigm.

It is much more general to think of the tonnetz as being generated by octaves and (tempered) perfect fifths, just like everything else in the Matrix/ThumMusic paradigm.

Here's a portion of the Matrix's two-dimensional note-space expressed in the ThumMusic System's isomorphic note-layout:



Each note is of the form [alpha, beta] where alpha is the number of octaves (each of width P8) and beta is the number of perfect fifths (each of width P5) which, when added together, give the width of the indicated interval. For example:

• the origin note[0, 0] is zero octaves and zero perfect fifths away from itself, (0 * P8) + (0 * P5);
• note[1, 0] is one octave higher in pitch than the origin, (1 * P8) + (0 * P5);
• note[0, 1] is one P5 higher than in pitch the origin, (0 * P8) + (1 * P5);
• note[-2, 4] is two octaves lower, but four P5's higher, than the origin, (-2 * P8) + (4 * P5).

Assuming that the P8 is 1200 cents wide and the P5 is 700 cents wide, the notes of the note-matrix would have these widths:
Now, let's build a portion of the tonnetz on the note-matrix, following Cohn's paper:


The minor triad Q is surrounded by three major triads P, L, and R.

• P: Parallel;
• L: Leading-Tone Exchange;
• R: Relative.

The above construction of the Q, P, L, & R triads from octaves and tempered perfect fifths is much more general than the traditional construction, because these intervals are the generators of the syntonic temperament, so the tonnetz's properties are invariant across the syntonic tuning continuum, no matter what the specific width of the P5 (within the range 686-720). This continuum includes an infinite number of individual tunings, not just the small number of N-edo tunings (in which N mod 3 = 0) over which Cohn's paper generalizes the tonnetz' traditional construction.

Cohn's paper makes much of the toroidal topology of such equally-tempered tunings (as do many neo-Riemann theoreticians). This emphasis overlooks the syntonic temperament's general topology, which is cylindrical. The tonnetz' octave axis forms a closed loop around the cylinder; its axis of major thirds runs parallel to the cylinder's inifintely-long axis; and its axes of minor thirds and perfect fifths form spirals around the cylinder's inifintely-long axis. Many common chord progressions, such as the IV-vi-ii-V-I, require only the syntonic temperament's cylindrical topology (without which the ii below the vi would differ from the ii above the V by a syntonic comma).

At those points along the tuning continuum that correspond to an equal division of the octave, such as 12-edo, 17-edo, 19-edo, 31-edo, etc., the cylinder snaps into a torus. Each n-edo's toroidal tonnetz has (a) all of the properties of the cylindrical tonnetz, (b) all of the properties shared by all toroidal tonnetzs, and (b) the properties specific to that unique n-edo's tonnetz. These points of equal temperament are like beads on a string -- but what's really interesting is not the beads, but the string.

From Thumtronics' perspective, the potential of the neo-Riemannian PLR operations to provide an invariant basis for music theory across the whole syntonic tuning continuum is very exciting (I think). Or, to express the same thought from the neo-Riemannian perspective, the Matrix/ThumMusic paradigm may give neo-Riemannian theory the opportunity to expand its scope to embrace the entire syntonic tuning continuum, and perhaps also the tuning continua (and tonnetz') of other rank-2 temperaments (e.g., magic, hanson, schismatic, etc.). These other temperaments temper out different commas, so their tonnetz' will be different from the syntonic tonnetz, but the same general principles ought to apply (at some level of abstraction, anyway).

Cohn's paper (like Riemann himself) makes a number of statements regarding the relationship between the tonnetz and "acoustics" that are only true if one assumes that "acoustics" means "the Harmonic Series." Yet the Matrix/ThumMusic paradigm generalizes "acoustics" -- by dynamically aligning a timbre's partials with a tuning's notes, as specified by a temperament's defining intervals -- such that the relationship between the tonnetz and "acoustics" is 1:1. The Matrix/ThumMusic tonnetz is a direct embodiment of generalized musical reality.

I think I'd read something about the PLR approach to chord relationships, chord progressions, and the like before reading Cohn's paper, but it hadn't clicked. Now, it has definately clicked. I suspect that the PLR approach to chord relationships will prove to be a very powerful tool in the Matrix/ThumMusic System.

Cool bananas! :-)

[Update, Thur Sep 25th: A couple of prominent neo-Riemannians have (very) informally agreed (a) that the proposed application of neo-Riemannain theory to the syntonic tuning continuum appears to be both novel and interesting, and (b) that they would read the relevant Matrix/ThumMusic papers and get back to me.]