More on decatonic transposability -- five views

I wrote,

>Anyway, my decatonic system has a form of transposability, only slightly
>more complex than "transpoability" as you've defined it.

Let me explain about decatonic transposability.

There are two forms of the decatonic scale:

Symmetrical decatonic: L s s s s L s s s s
Pentachordal " " : L s s s L s s s s s

Note that all rotations are considered equally interesting/valid here.

First, first note that the symmetrical decatonic is a "mode of limited
transposition" -- which means you can transpose it a half-octave with _no_
changes in the pitch set.

Second, note that the pentachordal decatonic, though not an MOS, still has a
"characterstic interval": s s s s s (or L s s s L) -- which only occurs in
one out of ten positions in the scale. Now, as you've pointed out, in the
MOS case altering the characteristic interval (by moving either endpoint) so
that it becomes the "perfect" interval (namely the generator of the MOS)
results in a rotation of the scale. In this case, however, doing the
alteration (so that it becomes the much more common "perfect decsixth")
results in a symmetrical decatonic scale! So rather than rotating, you've
"jumped" from one scale to the other.

Third, you can "jump back". In the symmetrical decatonic scale, the closest
thing to a "characteristic interval" is s s s s (or L s s s s L), which only
occurs in two out of ten positions in the scale. Altering one endpoint of
one occurence of this, so that you get the much more common interval
("perfect decfifth" or "perfect decseventh"), results in a pentachordal
decatonic scale!

Fourth, Dan Stearns has stated that he views the symmetrical decatonic as an
MOS, with generator L+3*s or s, and with periodicity at the half-octave
instead of the octave. On this view the scale is simply

L s s s s,

the characteristic interval is s s s s, and the MOS rule applies -- the
scale just rotates when you alter the characteristic interval. In terms of
the usual octave periodicity, though, this means that _two_, rather than
one, pitches have changed as a result of this rotation.

Fifth, either of the decatonic scales can be considered to be two
intertwined pentatonic scales, both of the form

L+s s+s L+s s+s s+s.

In the pentachordal decatonic scale, the two pentatonics are an s apart from
one another, while in the symmetrical decatonic scale, they are a
half-octave (L+4*s) apart. Either way, each pentatonic scale is an MOS with
generator (L+s)+(s+s) or (L+s)+2*(s+s), and if you apply "transposability"
to both pentatonics simultaneously, you're applying it to the full decatonic
-- you'll be rotating it, and _two_ pitches will change. (In the symmetrical
decatonic case, this fifth view results in the same transpositions as the
fourth view.)

Thus we see that there are a plethora of simple "modulation" operations
within the decatonic framework.

The first leaves a sym. dec. scale with all pitches unchanged.

The second changes a pent. dec. scale to one of two possible rotations of
the sym. dec. scale; one pitch changes.

The third changes a sym. dec. scale to one of _four_ possible rotations of
the sym. dec. scale; one pitch changes.

The fourth changes a sym. dec. scale to one of two possible rotations of
itself; two pitches changes.

The fifth changes either kind of decatonic scale to one of two possible
rotations of itself; two pitches change.

Hope this was enjoyable to you and that you will include these scales in
what you're working on.

P.S. Stephen Soderberg may note that I did not make use of any ratios or
overtones in the above -- it was purely "set theory" (right?) But, I note
that these properties are merely the _result_ of analyzing the decatonic
scales after they have been constructed -- none of these properties were
criteria used to derive the scales in the first place.

You'd find a similar pattern of transpositions among, for example, the class
of eight-tone scales in 18-tone equal temperament that can be constructed
from steps of 3/18 oct. and 2/18 oct. . . .however you won't find me
claiming that these scales contain any semblance of "consonant harmony" or
"tonality".

On the other hand, the class of twelve-tone scales in 26-tone equal
temperament follows a similar pattern too, and _does_ have many of the
properties I consider necessary for modal and perhaps even tonal harmony.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_20868.html#20868

>
> P.S. Stephen Soderberg may note that I did not make use of any
ratios or overtones in the above -- it was purely "set theory"
(right?) But, I note that these properties are merely the _result_ of
analyzing the decatonic scales after they have been constructed --
none of these properties were criteria used to derive the scales in
the first place.
>
> You'd find a similar pattern of transpositions among, for example,
the class of eight-tone scales in 18-tone equal temperament that can
be constructed from steps of 3/18 oct. and 2/18 oct. . . .however you
won't find me claiming that these scales contain any semblance of
"consonant harmony" or "tonality".
>
> On the other hand, the class of twelve-tone scales in 26-tone equal
> temperament follows a similar pattern too, and _does_ have many of
the properties I consider necessary for modal and perhaps even tonal
harmony.

Actually, this is one of the "problems" in "traditional" set theory
as I see it, particularly when it is applied to works of the so
called "free atonal" school of chromaticists early in the 20th
Century, before the 12-tone system had been codified.

There is LOTS of analysis that describes localized patterns in such
analysis... say a 0,4,7 set for a a major triad, etc., that doesn't
take into account, I believe, the UNDERLYING tonal or structural
basis of 12-tET and the harmonic possibilities...

Such analysis misses the AUDIBLE overtone series relationships of
such music. Imagine analyzing, for example, Berg's Violin Concerto
as simply a series of sets (well, in that case, entire tone rows) with
no reference whatever to the overtone series or a harmonic basis...

_______ _____ _ ____
Joseph Pehrson