Fokker

John Chalmers wrote:

>BTW, Fokker used the idea of defining intervals in 2 and 3-D tones spaces
>in a somewhat different way. He chose sets of "unison vectors," which he
>set to 0, in 3x5 and 3x5x7 tone spaces to define temperaments as
>"periodicity blocks" in the space. For example, the diesis and the syntonic
>comma define a repeating block of 12 pitches as the area defined by the
>absolute value of the cross product of the vectors describing these
>intervals. (The box product is used for 3 intervals in 3-D space.) I
>believe the concept can be extended to higher dimensions by computing the
>absolute values of the determinants of the square matrices whose row
>vectors are the defining intervals.

12tet is the temperament where octaves are perfect, and both the
syntonic comma (-4 4 -1)H and the diesis (0 -1 2)H are zero. In
my matrix representation, these conditions can be written:

( 1 0 0) (1 0 0)
( 7 0 -3)H' (0 0 0)H
(-4 4 -1) (0 0 0)

Where H is the metric of pitch space, and H' is its approximation
in 12tet. To get H', you multiply the right hand side of the
equation by the inverse of the matrix on the left. This general
method works for all ETs that approximate some kind of JI. For
the perfect octave case, you only need the left hand column of
the inverted matrix and the number of steps in an octave is +/-
the determinant of the original matrix. This will also be one
dimension of the cross product of the two zeroed intervals. It is
also the determinant of the square matrix formed by the zeroed
intervals without the octave column, and this will work for any
dimension. So, it looks like I was thinking along the same lines
as Fokker. Did he come up with any other stuff like this? Is
there a book I can read?

Graham

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Paul Hahn wrote:

''Hmm, sounds more like modulation in the signal-processing sense than inthe musical sense of the word. Also, the result has, potentially, up to
sixty pitches! Isn't that rather overwhelming, cognitively speaking?''

In fact, modulation of this sort is a primary resource in 20th century
composition. Stravinsky's tonal resources included playing the octotonic
collection against each diatonic transposition (and vice versa); Partch and
Schoenberg multiplied a set of tone by their inversion (the row box and
diamond are remarkably strange bedfellows); Boulez's ''multiplication''
technique is another species of the same.

As to the number of pitches, all of the tones of a hexany multiplied by adexany fit onto a stellate eikosany (which adds the pitches needed to fill
out all of the hexads of an eikosany). Wilson has done some impressive work
with the 70 tone, 4(8 ''Hebdomekontany'', including construction of a
metallophone with all pitches; using similar kinds of transposition, I find
it easy the tonal relationships easy hear, if difficult to play. Of course,
any of these might be mapped onto closed temperaments... As to the cognitive digestibility of such transpositions, it is entirely a
compositional problem - to compose in such a way that the tonal
relationships are projected audibly. I would venture that it is easier with
a dekany and a hexany based on simple harmonic ratios than with any
non-trivial 12 tone row.

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On Wed, 9 Jul 1997, Daniel Wolf wrote:
> In fact, modulation of this sort is a primary resource in 20th century
> composition. Stravinsky's tonal resources included playing the octotonic
> collection against each diatonic transposition (and vice versa); Partch a> nd
> Schoenberg multiplied a set of tone by their inversion (the row box and
> diamond are remarkably strange bedfellows); Boulez's ''multiplication''
> technique is another species of the same.

Okay, I misunderstood you before. I had thought that one might be using
all ~60 tones within a short space of time, when in fact you mean
(paraphrasing) transposing a ten-note scale through six different tonal
centers. That sounds eminently comprehensible.

--pH http://library.wustl.edu/~manynote <*>
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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