source file: m1399.txt
Date: Tue, 28 Apr 1998 12:08:13 -0500 (CDT)

Subject: <Gallagher>Totally new concept!</Gallagher>

From: Paul Hahn <Paul-Hahn@library.wustl.edu>

I've been working recently on verbalizing another concept I've developed
in my contemplations of various ETs, and I think perhaps it's time I
shared it with y'all to see what you think of it.  I have added the
following text to my personal webpage:

*** BEGIN QUOTED-WEBPAGE ***
   [The chart at the following URL] contains data on another concept which I call
   completeness. An ET is complete at a given harmonic limit if the basic
   intervals within that limit form a basis which spans that ET, if you
   think of the ET as a space or group. Example: 24TET is incomplete at
   the 5-limit because 5/4 is approximated by 8 steps and 3/2 by 14,
   which means no matter what combination of 5/4s and 3/2s (or 6/5s) you
   use, you can never generate those intervals which contain an odd
   number of steps.
   
   http://library.wustl.edu/~manynote/complete.txt
   
   So what do the numbers mean? Only those combinations of ET and (odd)
   limit have entries which are both consistent and complete. (I stopped,
   somwhat arbitrarily, at 300TET.) x/y means that the ET is x-level
   consistent at that limit (as in the above charts), and y is the
   diameter at which the ET is completed.
   
   So what, in turn, does diameter mean? If n-ET has diameter y at the
   m-limit, that means that there is at least one interval which requires
   combining y m-limit (primary) intervals to derive it, but no intervals
   which require more. Example: the primary intervals (consonances)
   within the 5-limit (the senario) are represented in 12TET by 3, 4, 5,
   7, 8, and 9 steps. 1 can be expressed as 4-3 (or 5-4, etc.), 2 by 5-3
   etc, and 6 by 3+3. (The derivations of 10 and 11 are analogous to
   those of their complements 2 and 1.) That completes the gamut of
   12TET, therefore the diameter of 12TET at the 5-limit is 2.
   
   A contrasting example: in 19TET, combinations of exactly 2 of the
   primary 5-limit consonances (5, 6, 8, 11, 13, 14) give you all the
   rest except for 4 (and its complement 15). 4 does, however, have a
   ternary derivation (4=5+5-6), so the diameter of 19TET at the 5-limit
   is 3.
*** END QUOTED-WEBPAGE ***

The point, in case it isn't clear, of introducing a parameter like
diameter is that it gives us another measure by which we can compare
ETs, i.e. an ET with a small diameter will be more "comprehensible" to
the ear/mind, whereas an ET with a large diameter will have intervals
which are interpreted as more complex and difficult to hear.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
    O
   /\        "Churchill? Can he run a hundred balls?"
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