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.

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.

Dynamic Tonality Demo Video

You can find a video demonstration of Dynamic Tonality here:
http://www.youtube.com/watch?v=Nd4h8vmEsQM

The sound quality is terrible, because I had to record it from my laptop, the microphone jack on which is busted, and using demo-creation software which couldn't tap into the speakers directly -- so the sound you're hearing is coming out of its speakers and into the laptop's built-in mic, which is a recipe for feedback. Noisy fan, too. Please accept my apologies for this.

But, that being said, the demo still makes the point -- clearly, I hope -- that the Thummer keyboard’s note-layout makes microtonal music brain-dead simple, by exposing tonal intervals consistently in every tuning of the syntonic tuning continuum.

Next month, a paper is being published in the peer-reviewed Journal of Mathematics and Music which rigorously proves that the Thummer's Wicki/Hayden note-layout is optimal for controlling Dynamic Tonality. No other note-layout -- not the Janko, nor Fokker, nor Bosanquet, etc. -- packs so many octaves of tonally-relevant intervals into such a small area over such a wide tuning range.

It's easy to dismiss microtonality as an irrelevant fringe interest that has no appeal whatsoever to mass-market consumers. But this ignores both history and current practice, in which tuning matters.

Currently, Western musicians bend their notes constantly – intoning them towards Just Intonation, Pythagorean tuning, expressive exaggerations thereof, or blue notes. Monophonic instruments have dominated Western orchestras in part because they allowed such note-by-note intonation. Tuning matters. The Thummer allows musicians to intone notes polyphonically -- bending many notes at once towards their Pythagorean tuning, for example (with the sharps getting sharper and the flats flatter).

Also, there is a big wide world out there beyond the West, and many non-Western cultures use non-Western tunings. The Thummer's keyboard has the same fingering in 7-edo (related to Thai & Mandinka music) and 5-edo (related to Indonesian music) as it does in Western 12-edo. Even the Turkish 53-edo schismatic temperament fits the Thummer's note-layout, too (albeit with different note-choices than are used in the syntonic temperament, e.g. d4’s in place of M3’s). The Thummer supports all of these different cultures' tunings. To musicians from non-Western cultures, or to Western musicians who wish to learn about or to mix and match the music of non-Western cultures, tuning matters.

Historically, 12-edo is recent, only having been widely adopted between 1850 and 1900, give or take. Before that, Pythagorean tuning, 1/4-comma meantone, and various well temperaments dominated Western tuning for thousands of years. All of these pre-modern Western tunings have the same fingering on the Thummer's keyboard, too. You can see a piece of the soft-synth's controller for Just and irregular tunings in the above-mentioned Dynamic Tuning video, to the left of the tuning slider, towards the top of the screen (look for the phrase "Minor JI"). If you want to play music in its historically-accurate tuning (albeit perhaps on a modern instrument), then tuning matters.

In addition to past and current practice, one should also consider the future. The new musical effects enabled by Dynamic Tonality -- polyphonic tuning bends, new chord progressions (!), temperament modulations, and the like -- enable entirely new styles and forms of music. Consider the expansion of form enabled by the chromaticism of the Romantic period, or the staggering popularity of the non-equally-tempered blues scale over the last hundred years. Tuning matters.

These ideas may seem complicated, because Dynamic Tonality is brand new. However, as you can see/hear from the demo video, Dynamic Tonality is brain-dead simple to USE. You just change the tuning -- by wiggling one of the Thummer’s joysticks, perhaps -- and cool new musical effects happen. You don't have to understand prime numbers, ratios, logarithms, or any of the other arcana of tuning theory. You just wiggle a friggin' joystick. The Thummer knows music theory, so you don't have to.

Hostorically speaking, every change in tuning -- Pythagorean to 1/4-comma, 1/4-comma to well-tempered, well-tempered to 12-ET -- has expanded music's possibilities. Some of these initially seemed complicated and perhaps even diabolical, largely because these tunings moved notes away from their alignment with harmonic partials. But Dynamic Tonality generalizes the relationship between the Harmonic Series and Just Intonation by adjusting a timbre's partials (in real time) to align with the notes of the current tuning, then one gets pure consonance all across the syntonic temperament's tuning range -- as you can hear in the demo (through the noise of the lousy recording -- sorry). So again: you don't need to know music theory to use this stuff; the Thummer knows music theory, so you don't have to.

In short: one of the main reasons to prefer the Wicki/Hayden note-layout over all other isomorphic layouts is that it enables unique support for Dynamic Tonality.

The ThumMusic System was also designed with Dynamic Tonality in mind. It emphasizes those aspects of music -- intervals, and the relationships among intervals -- which are invariant in the music of the past, the present, and the future, across many different cultures, while deprecating those aspects of music – most notably tying each note to a fixed pitch -- which assume a single, static tuning, unique to one time, place, and culture.

Or that’s the idea, anyway. ;-)