[sdiy] The ideal quantizer, was: Music

[David G. Dixon]

> >> Just run all the white keys from E to E, and voila!  The E
> >> phrygian mode.
> >
> > ...
> >
> > Sorry, I didn't mean to bore the entire list with that little bit
> > of music
> > theory.  I thought I was sending a private message to Scott.  My
> > bad.  :)
> 
> I find this subject very interesting.
> I've been wondering for quite some time how a good universal
> quantizer would look like. This would be a module which takes
> continuous (pitch cv) input and outputs pitch control voltages
> according to a scale. It would take as few parameters as possible and
> give as many musically useful results as possible while not omitting
> musically useful results.

Tthe Intellijel Designs "uScale" does pretty much exactly what you're after
and costs less than $200.  It has a little keyboard on the front with all 12
chromatic tones represented by little black and white buttons, and you can
select whatever chord or scale you desire.

> Ok, the other project I'm working on is a perpetuum mobile ;-)
> 
> One approach is, have 12 switches. Switch on means note of the
> chromatic scale belongs to target scale. This gives you (2^12)-1
> possibilities (-1 because all switches off doesn't make sense). It's
> obvious that there must be a lot of scales that are less useful.
> OTOH, you're restricted to scales that are subsets of the chromatic
> scale, and repeat (at least) every octave. Also, unless you already
> know how a "musically useful" scale looks like, you'll have a hard
> time finding good results by randomly toggling the switches.

It's not that hard: musically useful scales have lots of 1- and 2-semitone
intervals.  Very rarely will a scale have a 3-semitone interval (the
Hungarian scale and the "harmonic minor" come to mind, and they have one
each), and very rarely (almost never) more than 1-semitone interval in a
row.  Exceptions to that last rule are the so-called "blue notes" or
appoggiature, essentially chromatic leading tones above or below the notes
of the prevailing tonic.  Thinking about the mixolydian mode (G-G on all
white keys, or C-C with Bb instead of B), these leading tones would be the
b3, the b5, and the (natural) 7.  That would make ten notes in the scale,
leaving only two "unmusical" tones.  Hence, it really depends on how you use
the "unmusical" notes.  This probably can't be done well with a quantizer
under CV control.  Hence, I would stick to the rules.

Most scales contain 7 notes (heptatonic) but there are two important 8-note
(octatonic) scales and a few important 5-note (pentatonic) and 6-note
(hexatonic) scales, many (but certainly not all) can be viewed as limited
subsets of the heptatonic scales.

I have classified heptatonic scales as falling into two main categories:
"normal" ones (i.e., those without 3-semitone intervals or consecutive
1-semitone intervals), and "weird" ones.  Of the normal ones, there are
really only two scales: major and (ascending) melodic minor, each with seven
unique modes (including the home scale).  In fact, these are the only two
possible "normal" heptatonic scales.  The major scale is what you find on
the white keys of the piano, and has the repeating pattern of 2 and 3 black
keys representing the "adiatonic" (or extra-scalar) tones.  The ascending
melodic minor scale (often just called "melodic minor" by jazzers) is what
you would get if the black keys on the piano were arranged in repeating
patterns of 1 and 4.  There are no other possibilities in the "normal"
category.

The only important "octatonic" scale is the diminished scale.  This has two
modes: "whole-half" and "half-whole", referring to the first two intervals
in each.  These are very prevalent in late 19th century orchestral music
(Debussy loved them) and in jazz.  The only important hexatonic scale is the
"whole tone" scale, made up of all whole-tone intervals.  These modes
consist of repeating patterns semitones.  In fact, a whole sub-family of
such modes can be constructed, and this system of modes were systematized
and extensively used by Olivier Messiaen (he called them the "modes of
limited transposition" -- I refer you to his excellent book, "Technique de
mon langage musicale" for details.  He invented a whole compendium of other
scales and harmonies which are chronicled in his exhaustive 7-volume
treatise, "Traite de rythme, de couleur, et d'ornithologie").

Pentatonic scales are the bread and butter of rock guitar solos, as they
sound good in simple harmonic contexts and fall nicely under the fingers on
a guitar fretboard.  Most of them are simple reductions of regular major
scales and modes, but some of them are not.  I refer here to the Japanese
system of pentatonic modes, an interesting topic in an of itself, and also
discussed at length in Messiaen's "Traite".

Probably the best books on modes and related harmony that I've found are the
two volume set, "Modal Jazz Composition and Harmony, Volumes 1 and 2" by Ron
Miller from the University of Miami (Advance Music).  Highly recommended.

I could go on and on about this, but I think I'll stop there.

> What I'm thinking of is:
> Parameter 1: "big" interval. This is the interval at which the scale
> repeats. Usually set to one octave.
> Parameter 2: "little" interval. This is the interval of the equal-
> tempered "super-scale" which the target scale is a subset of. For
> western scales this is set to a semitone.
> Parameter 3: number of notes per "big" interval (usually octave). Set
> to 7 for diatonic scales, to 5 for pentatonic, to 6 for whole-tone
> scale and so on.

This is an interesting approach.  To really make good on it, it would be
useful to have two octaves over which to make the pattern.

> Further parameters could be "transpose" and "mode". "Transpose" would
> be just an offset at the output. It's questionable if "mode" is
> required at all. In David's example, C ionian, D dorian and so on all
> share the same notes. A "mode" parameter could perhaps sweep from C
> ionian to C dorian to C phrygian and so on.

Transposing the scale to a different puts a mode on the same note (as long
as that note belongs to the mode).  Hence, you are right that "mode"
probably isn't required, but that means that the "legal" transposition
shifts are limited to seven.  Otherwise, you'd be "modulating" to a
different key altogether.

> Still another parameter (or parameters) is needed, perhaps called
> "unevenness". In a heptatonic scale there are five whole tone
> intervals and two semitone intervals between neighbouring notes. This
> parameter could determine how far apart the two semitone intervals
> are placed. Increasing this parameter further, minor thirds would be
> introduced, along with more semitone intervals. In the extreme
> position, the heptatonic scale would look like C-C#-D-D#-E-F-F#-C.
> 
> The goal is, finding nice scales without having to know a lot about
> music theory. Any ideas?

Yeah: learn the theory!  As I said, there aren't really that many truly
useful scales (in Western music, anyway), and really understanding how to
use them is an extremely rewarding thing.