Re: Replication vectors and arrays -- for Paul Erlich

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Replication vectors and vector arrays:
Another approach to "dimensional" tunings
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[Hello, everyone -- those familiar with my Xeno-Gothic tuning may find
this article not so surprising, and I hope in any case that it may
have some musical interest apart from the mathematical intricacies
which may be involved.]

One pleasure of artful theory is that it can set many lines of
variation in motion. Paul Erlich's recent post on "unison vectors"
leads me to propose the concept of a "replication vector" as a rather
technical term for an approach to generating tunings which I find
congenial.

Also, while the concept of "dimensionality" is often used in reference
to the number of primes (or primes greater than 2) used as the basis
of a just tuning, I would like to present a different type of
dimensionality which might or might not have already been defined many
times before.

Taking Paul Erlich's post as a starting point, I will also begin with
Pythagorean tuning, or 3-limit just intonation (3-limit JI). Here is a
typical Pythagorean chromatic scale of Western European theory around
the 14th century, with lots of familiar and very useful intervals,
shown here in rounded cents:

114 294 612 792 996
0 204 408 498 702 906 1110 1200

Now we come to a possible "bug" of this JI system which can actually
be an engaging feature: the fact that 12 fifths do not quite close the
system, but rather exceed seven octaves by an amount called the
Pythagorean comma (531441:524288, ~23.46 cents).

This "gap" in the tuning may be viewed rather as an open door for
generating new intervals by what has been called a "spiral of fifths."

While "spiral" is an evocative image, another way of expressing the
potential that an interval such as the Pythagorean comma gives for
larger tunings is a two-dimensional table such as this:

138 294 636 816 1020
24 228 432 522 726 930 1134 1224
------------------------------------------------------------------------
114 294 612 792 996
0 204 408 498 702 906 1110 1200
------------------------------------------------------------------------
90 270 588 768 972
-24 180 384 474 678 882 1086 1176

Here we have a "12 x 3" array: 12 notes in each row, and 3 rows (which
might, for example, represent three 12-note keyboards each tuned a
Pythagorean comma apart. For convenience, I have trated this comma as
an even 24 cents.

In this scheme, I term the comma a "replication vector," a distance at
which identical 12-note sets may be repeated in order to generate a
larger tuning. The resulting tuning -- here 36 (12 x 3) notes -- I
refer to as a "vector array."

Applying this rather abstruse language to musical practice, we might
have a set of three 12-note keyboards tuned in pure 3:2 fifths a
Pythagorean comma apart. Or, more parsimoniously, we might simply have
two such keyboards, agreeing to call either one the "standard"
keyboard as might be convenient in order to obtain intervals either
wider or narrower than the usual ones by our "replication vector,"
here ~23.46 cents. Such an arrangement opens three possibilities:

(1) Each keyboard can form a musical "universe" in itself including
all of the usual concords, dual-purpose intervals, and discords for
14th-century Western European music. Without actually using any new
vertical intervals generated by simultaneously combining notes on
different keyboards, we might sometimes "shift" from one keyboard or
universe to another, making a "comma jump."

(2) Combining notes on different keyboards gives us new intervals,
such as an extra-wide major sixth at around 930 cents or a narrow
minor seventh at around 972 cents. These two intervals, for example,
invite especially efficient cadential resolutions to an octave and a
fifth respectively by stepwise contrary motion. A narrow major third
of 384 cents, in contrast, may be less efficient than the usual
interval of 408 cents at cadences, but an interesting "diversion" or
element of vertical color at noncadential points.

(3) We may also use the "replication vector" as a direct melodic
interval in a progression such as the following (C4 is middle C, and
higher numbers show higher octaves):

E4 F#4 G4
(702) (996) (1200)
Db4 C#4 D4
(384) (408) (702)
A3 G3

Here the first sonority is a "coloristic" sonority of A3-Db4-E4, a
_quinta fissa_ or "split fifth" with a narrow major third below and a
wide minor third (~318 cents) above. Then the middle voice has a comma
inflection (Db4-C#4) moving it to a usual major third of ~408 cents,
more tense and more actively "striving" for expansion to the fifth
G3-D4, at the same time as the upper voice moves from the fifth to the
usual major sixth A3-F#4 with its directed goal of the octave G3-G4.

While I know of no medieval example calling for direct melodic motion
by a Pythagorean comma, this example is inspired by a progression
opening an early 15th-century keyboard piece. If we wished to bring
out a contrast between a "diversionary" quality for the opening
sonority and a more directed "cadential" quality for the sixth and
third immediately before the fifth and octave, then the direct
inflection has a curious musical logic.

Please note that I do not necessarily recommend such direct
inflections in performing early 15th-century music -- although using
both A3-C#4 and A3-Db4 in the course of piece might well have been
done on a suitable keyboard with 15 or more notes per octave -- but
suggest the availability of such a nuance in modern compositions or
improvisations in a "Xeno-Gothic" style.

In short, I would like to suggest a kind of "two-dimensional"
structure based on an array of keyboards (actual or conceptual)
located at an interesting distance provided by a "replication vector"
such as the 23.46-cent comma of 3-limit JI.

Such vectors might be of interest either as factors of variation to
generate new vertical intervals, or as possible direct melodic
intervals. Sometimes there can be a factor of serendipity: a
replication vector which might to arise quite incidentally to the main
aesthetic goals of a tuning may turn out to have dramatic vertical and
melodic potentials for musical expressiveness.

While my example here happens to be Pythagorean tuning, a JI system,
the same principle can apply to other kinds of tunings, including
tunings mixing just intervals (other than the unison and octave) with
tempered ones. Replication vectors in such "mixed and tempered music"
will be a topic of a post I'm now preparing.

Most respectfully,

Margo Schulter
mschulter@value.net