Meantone, Partch

Re Meantone: "Linear" is an historical term (Ellis?) contrasting with
"cyclic" or closed. I agree that in general Meantone tone systems are 2-D,
but under the
common assumption of octave equivalence, the dimensionality may be reduced
to 1. In which case, both ET's and meantone tunings are generated by one
interval, the tempered fifth. Tempered systems are closed and have a finite
number N of
points on the line of fifths, while meantone are open and infinite. In this
sense they are "linear" and open as opposed to closed and cyclic.

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.

Paul Rapoport has also done much work in this area.

Wu"rschmidt, I believe, was the first to identify "defining intervals" for
tunings and to distinguish them from "constructing intervals," the
successive intervals of the JI scales corresponding to each of the ET's as
defined by this method (there are many possible sets of defining intervals
for any given ET and there are many possible sets of 5 limit intervals
interpretable as 12-tet.)

Re Partch: HP's dual use of ratios to label scale degrees and to define
functions can be somewhat confusing at first as Marion has pointed out.
Partch assumed octave equivalence in deriving his theories, but did
recognize that inversions have different sounds and inn his music used a
variety of voicings.
Hence, sometimes 1/1 appears in the middle of a chord where the voicing
would indicate that the ratio should be 1/2, 2/1, 4/1 etc.

As for odd numbers (or factors) in the denominators, they determine the
roots of his otonal or harmonic series chords, though his term "numerary
nexus," has not caught on. They also indicate the harmonic distances from
the common tonic, the 1/1 (tempered G). When HP writes 8/5 1/1 6/5, the
notation is a shorthand for 8/5 10/5 12/5 a major triad on the root 8/5,
a minor sixth above his 1/1 G. This notation makes the harmonic relations
clear, while somewhat obscuring the melodic movement and actual voicing.
Usually, he's clear enough as to the actual inversion and voicing.
Otherwise, he'd have to make the register explicit and increase the number
of symbols, e.g., 1/3, 2/3, 4/3, 8/3 etc. for
4/3 (roughly C) in various registers.. Such schematic spellings are not
unknown in conventional music theory.

For the subharmonic (utonal) chords, the odd factors in numerator serve the
same function. The Diamond diagram may clarify his meaning.

One really has to read Partch's theoretical sections as an introduction and
outline, not as a detailed composition manual.


--John

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Paul Erlich wrote:

''22tET can be considered 5-dimensional because
it consistently represents all ratios of 3, 5, 7, 9, and 11.''

Interestingly, Erv Wilson discovered that the 3,5,7,9,11 Eikosany cannot be
mapped without conflict onto 22 tones while the 3,7,9,11,15 can - and makes
a quite convenient keyboarding.

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Daniel Wolf wrote,

>Interestingly, Erv Wilson discovered that the 3,5,7,9,11 Eikosany cannot >be
>mapped without conflict onto 22 tones while the 3,7,9,11,15 can - and mak>es
>a quite convenient keyboarding.

Is "the" 3,5,7,9,11 Eikosany 2)5 or 3)5? Same question for "the"
3,7,9,11,15.

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Paul Erlich wrote:

''Is "the" 3,5,7,9,11 Eikosany 2)5 or 3)5? Same question for "the"
3,7,9,11,15.''

Mea culpa! I dropped the ''1'' from both eikosanies. I meant to compare the
3)6 (1,3,5,7,9,11) with the 3)6 (1,3,7,9,11,15)

2)5 or 3)5 will give dekanies (incidentally, I find dekanies to be the most
useful subsets of Eikosanies, and often modulate a dekany by a hexany).

For those not up on their CPS (combination-product set) terminology, for a
given set of 5 generating intervals (A,B,C,D,E), a dekany will have all
combinations of two elements (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE),
while an eikosany will have all combinations of three elements from a setof six,

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On Wed, 9 Jul 1997, Paul H. Erlich wrote:
> Is "the" 3,5,7,9,11 Eikosany 2)5 or 3)5?

Aren't they effectively the same?

--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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On Wed, 9 Jul 1997, Daniel Wolf wrote:
> (incidentally, I find dekanies to be the most
>useful subsets of Eikosanies, and often modulate a dekany by a hexany).

This sounds interesting but I'm not sure what it means. How do you
modulate a dekany by a hexany? Does this mean to transpose a dekany so
that the transposed and untransposed dekanies have the hexany in
common?

--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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Paul Hahn wrote:

How do you
modulate a dekany by a hexany? Taken any dekany, treat one degree as 1/1. Take any hexany, treat one
degree as 1/1. Now transpose (multiply) the entire dekany by each remaining
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Paul Erlich wrote:

>Topologically, meantone tunings wrap the 5-limit plane onto an infinite
>cylinder. I like to see ETs as having as many dimensions as odd factors
>with which they are consistent. ETs consistent with the 5-limit wrap
>said plane onto a torus.

A 2-D scale is linear with octave invariance, because each point
will lie on a line drawn on Paul's cylinder, at the intersections
with the original lattice. I generally don't like this idea of
"linear temperaments" because it places too strong an emphasis on
octave invariance. However, if you define a temperament as "a
formula that produces a scale" that would make the difference.

Applying octave invariance would wrap 3-D pitch space into a
hypercylinder. Note that there would be two straight lines
connecting any two points in this hypercylinder, so this space
would also imply inversional invariance. A meantone would then
wrap this into a toroidal hyperprism, and an ET a hypertorus.
Quite something, given that an integer CET or TET would be a
finite set of points in this hypertorus, having a Hausdorff
dimension of zero. Anyway, back to Earth:

>22tET can be considered 5-dimensional because
>it consistently represents all ratios of 3, 5, 7, 9, and 11.

This is a 1-D scale approximating 5-D harmony in my terminology,
including 2 and implying 9. Call it an ET approximating the
11-limit if you prefer. However, I would certainly not say that
the scale is 5 dimensional, rather that you are mapping it to a
5-dimensional space. The number of points in the full 5-D real
pitch space depends linearly upon the number of octaves you look
at, hence the scale has a Hausdorff dimension of 1. I wouldn't
say that adding a direction for 9 alters the dimension. A
definition, then:

The dimension of a scale is the smallest number of fundamental
intervals (FIs) such that every interval in the scale is a linear
combination of integer multiples (phew!) of those FIs. This
implies that none of the FIs can be a linear combination of
integer multiples of the others. The FIs would be real numbers
denoting the interval size in octaves, c*nts, or some such unit.
Distinct prime numbers always constitute a set of FIs. Note the
analogy reals->vectors, integers->scalars.

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