A hunch for Hahn

I suspect there is a strong relationship between my quest for well-formed,
so to speak, periodicity blocks, and Paul Hahn's notion of an ET having a
consistency level greater than or equal to the diameter at which it is
completed. See Paul H's page http://library.wustl.edu/~manynote/music.html
for an explanation of his terms.

I should make a "philosophical" point here. I have often stated that Paul
Hahn's higher-level consistency measures are irrelevant for evaluating the
quality of an ET. I still believe this is true, in the sense I meant it. But
periodicity blocks represent a way of approximating ETs with JI. So if one
insists on an "organic" derivation of ETs, starting with JI, then finding
unison vectors, and finally tempering the resulting periodicity block, then
Paul Hahn's concept may be very important in measuring the plausibility of
this process. But if one is allowed to invent ETs from the air, as is so
easy to do these days, then one can end up with some "inorganic" solutions
that nonetheless contain many consonant sonorities, and higher-level
consistency is irrelevant. I think both philosophies have merit, depending
on the historical context.

On Wed, 29 Sep 1999, Paul H. Erlich wrote:
> I suspect there is a strong relationship between my quest for well-formed,
> so to speak, periodicity blocks, and Paul Hahn's notion of an ET having a
> consistency level greater than or equal to the diameter at which it is
> completed. See Paul H's page http://library.wustl.edu/~manynote/music.html
> for an explanation of his terms.

My intuition (and it shouldn't be too hard to prove) is that consistency
>= diameter is a sufficient but not necessary condition for what you're
looking for--as I've noted in past postings, ETs that fulfil that
condition are rather few and far between.

[ . . . ]
> periodicity blocks represent a way of approximating ETs with JI. So if one
> insists on an "organic" derivation of ETs, starting with JI, then finding
> unison vectors, and finally tempering the resulting periodicity block, then
> Paul Hahn's concept may be very important in measuring the plausibility of
> this process. But if one is allowed to invent ETs from the air, as is so
> easy to do these days, then one can end up with some "inorganic" solutions
> that nonetheless contain many consonant sonorities, and higher-level
> consistency is irrelevant.

I'd meant to say something long ago about how higher level consistency
seemed more important when you're going in the JI -> ET direction than
the reverse, but never got around to it. Someday I'll get the time to
organize all these thoughts and post them. If only I were Margo--she
seems to knock out lengthy theoretical-historical-philosophical
treatises at the drop of a hat . . .

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "'Jever take'n try to give an ironclad leave to
-\-\-- o yourself from a three-rail billiard shot?"

I wrote,

>I suspect there is a strong relationship between my quest for well-formed,
so to speak, periodicity >blocks, and Paul Hahn's notion of an ET having a
consistency level greater than or equal to the >diameter at which it is
completed. See Paul H's page http://library.wustl.edu/~manynote/music.html
for >an explanation of his terms.

I note that Paul Hahn's table of such ETs, considering odd limits up to 11
and ETs up to 300, runs as follows:

2, 3, 4, 5, 6, 7, 12, 15, 17, 19, 31, 34, 41, 53, 65, 72, 118, 171

Meanwhile, I posted a 22-tone periodicity block in which all the unison
vectors (and their combinations) are smaller than the smallest step size.
Perhaps that block should not be considered well-formed, though, since it
had holes in it! That means that certain notes in the block would not be
accessible through consonant intervals, contrary to what Hahn's construction
requires.

More in a moment . . .

>My intuition (and it shouldn't be too hard to prove) is that consistency
>>= diameter is a sufficient but not necessary condition for what you're
>looking for--as I've noted in past postings, ETs that fulfil that
>condition are rather few and far between.

Right. Consistency at a given level implies _all_ possible paths through the
lattice with length up to that level will agree with the ET representation.
If we remember Patrick Ozzard Low's notion of fractional consistency, where
some intervals agree and some don't, then perhaps what I'm looking for
(along with a provision for no holes) is equivalent to some suitably chosen
subset of paths, whose length is equal to the diameter, being consistent
with one another. Maybe Mr. Hahn can help make this more precise.

More to come . . .

Fokker detailed three 22-tone periodicity blocks. Of the two 5-limit ones,
one has some steps smaller than a unison vector, and the other one has
holes. Let's look at the 7-limit one (ooh, I wish I had Canright's program
now!):

The matrix is

-1 3 2
-7 -1 3
2 2 -1

and the lattice coordinates and cents values are:

-4 -1 2 1144
-3 -2 1 490
-3 -1 1 877
-3 0 2 1032
-3 1 2 218
-2 -1 1 379
-2 0 1 765
-1 -2 0 925
-1 -1 0 112
-1 0 1 267
-1 1 1 653
0 -1 0 814
0 0 0 0
1 -2 -1 161
1 -1 -1 547
1 0 0 702
1 1 0 1088
2 -1 -1 49
2 0 -1 435
2 1 0 590
3 -1 -2 982
3 1 -1 323

The steps range from 49 cents to 63 cents, clearly larger than any
combination (without repetition) of the unison vectors, which are 5.4, 6.5,
and 7.7 cents. Let's use the rotated representation of the lattice:

1
/|\
/ | \
/ 7 \
/,' `.\
3---------5

490
\
\
\ 1144
\ ,'
877
/
925 /
\ /
\ /
\ 379
\ ,' \
112 \
/ \ 1032
161 / \ ,' \
\ / 765 \
\ / / \
\ 814 / \
\ ,' \ / 218
547 \ /
/ \ 267
/ \ ,' \
/ 0 \
/ / \
49 / \
,' \ / 653
982 \ /
\ 702
\ ,' \
435 \
\
\
1088
/
/
/
/
590
,'
323

Clearly this is fully connected (that's what I meant by not having "holes").
So it represents a way 22-tET could be "organically" derived from JI via a
"well-formed" periodicity block. Interestingly, each note in the scale is
consonant with at most three other notes: either a 5:4 down, a 4:3 down,
and/or a 7:6 up, or the reverse.

It is no coincidence that these three (or six, if you count inversions)
consonances are better approximated in 22-equal than the other three, 6:5,
8:7, and 7:5 (six if you count their inversions). Unfortunately, no
consonant triads or tetrads can be created from this configuraion of those
three intervals, as the graph above shows.

The notes falling into two distinct categories like this is a very
interesting phenomenon (any comments from Paul H. would be much appreciated)
but on second glance doesn't appear very organic (unless you mean organic
chemistry in which case you might see benzene rings in the graph above!).
Even if, for whatever reason, you chose 5:4, 4:3, and 7:6 as your basic
consonances, why would you construct them only in one direction on half the
notes and only in the opposite direction on the other half? Very strange.

Is there a more "organic" 22-tone periodicity block to be found, which has
all steps larger than the unison vectors (and hopefully even larger than
their sum), and is fully connected?

Paul! and the other heads of block here!
I still can't seem to break the code. I understand matrixes a bit And
Grahams explanation doesn't help because it lacks enough singular examples.
Although once he explained it and I thought i got it but somehow looking at the
chart below tells me something is wrong. How do you get the latter from the
former.? why is there no -7 for instance.

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Fokker detailed three 22-tone periodicity blocks. Of the two 5-limit ones,
> one has some steps smaller than a unison vector, and the other one has
> holes. Let's look at the 7-limit one (ooh, I wish I had Canright's program
> now!):
>
> The matrix is
>
> -1 3 2
> -7 -1 3
> 2 2 -1
>
> and the lattice coordinates and cents values are:
>
> -4 -1 2 1144
> -3 -2 1 490
> -3 -1 1 877
> -3 0 2 1032
> -3 1 2 218
> -2 -1 1 379
> -2 0 1 765
> -1 -2 0 925
> -1 -1 0 112
> -1 0 1 267
> -1 1 1 653
> 0 -1 0 814
> 0 0 0 0
> 1 -2 -1 161
> 1 -1 -1 547
> 1 0 0 702
> 1 1 0 1088
> 2 -1 -1 49
> 2 0 -1 435
> 2 1 0 590
> 3 -1 -2 982
> 3 1 -1 323
>
> The steps range from 49 cents to 63 cents, clearly larger than any
> combination (without repetition) of the unison vectors, which are 5.4, 6.5,
> and 7.7 cents. Let's use the rotated representation of the lattice:
>
> 1
> /|\
> / | \
> / 7 \
> /,' `.\
> 3---------5
>
> 490
> \
> \
> \ 1144
> \ ,'
> 877
> /
> 925 /
> \ /
> \ /
> \ 379
> \ ,' \
> 112 \
> / \ 1032
> 161 / \ ,' \
> \ / 765 \
> \ / / \
> \ 814 / \
> \ ,' \ / 218
> 547 \ /
> / \ 267
> / \ ,' \
> / 0 \
> / / \
> 49 / \
> ,' \ / 653
> 982 \ /
> \ 702
> \ ,' \
> 435 \
> \
> \
> 1088
> /
> /
> /
> /
> 590
> ,'
> 323
>
> Clearly this is fully connected (that's what I meant by not having "holes").
> So it represents a way 22-tET could be "organically" derived from JI via a
> "well-formed" periodicity block. Interestingly, each note in the scale is
> consonant with at most three other notes: either a 5:4 down, a 4:3 down,
> and/or a 7:6 up, or the reverse.
>
> It is no coincidence that these three (or six, if you count inversions)
> consonances are better approximated in 22-equal than the other three, 6:5,
> 8:7, and 7:5 (six if you count their inversions). Unfortunately, no
> consonant triads or tetrads can be created from this configuraion of those
> three intervals, as the graph above shows.
>
> The notes falling into two distinct categories like this is a very
> interesting phenomenon (any comments from Paul H. would be much appreciated)
> but on second glance doesn't appear very organic (unless you mean organic
> chemistry in which case you might see benzene rings in the graph above!).
> Even if, for whatever reason, you chose 5:4, 4:3, and 7:6 as your basic
> consonances, why would you construct them only in one direction on half the
> notes and only in the opposite direction on the other half? Very strange.
>
> Is there a more "organic" 22-tone periodicity block to be found, which has
> all steps larger than the unison vectors (and hopefully even larger than
> their sum), and is fully connected?
>
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

In-Reply-To: <938684587.18792@onelist.com>
Kraig Grady, digest 335.22 wrote

> Paul! and the other heads of block here!
> I still can't seem to break the code. I understand matrixes a bit
> And
> Grahams explanation doesn't help because it lacks enough singular
> examples.

I need to redo that page. Does anybody know of a simple matrix tutorial
on the web, purely from the maths perspective? The best I could find
running through Google is:

http://www.aha.ru/~pervago/articles/matrix.txt

Kraig,

The table with four columns shows the power of 3 in the first column, the
power of 5 in the second column, the power of 7 in the third column, and the
cents value in the fourth column. The first three columns are taken directly
from Fokker's paper. These 22 notes are those contained within the
parallelopiped enclosed by the vectors (-1 3 2), (-7 -1 3), and (2 2 -1).
Those vectors again correspond to musical intervals defined as 3 to the
first element, 5 to the second element, and 7 to the third element. They are
all very small intervals (5.4, 6.5, and 7.7 cents) and are called unison
vectors. One property of a periodicity block so defined: Any note in the
infinite (in this case, 3D) lattice is equivalent, via transpositions by
unison vectors, to one and only one note in the periodicity block (in this
case, of 22 notes). In the lattice diagram, I showed all 7-limit consonant
intervals in the scale. Let me know if anything else needs elaboration.

-Paul