Fokker refs

The references to Fokker's papers on periodicity blocks in harmonic
lattices are the following as they appeared in a periodical. Fokker
published a great number of papers in Dutch, English, German and French.
Many, if not all, are listed in the bibliography available at the Mills
anonymous ftp site (ella.mills.edu) in HTML format.

Fokker, A.D. "Selections from the harmonic lattice of perfect fifths and
major
thirds containing 12, 19, 22, 31, 41 or 53 notes", Proceedings of the
KNAW, Series B, vol. 71, 1968, pp. 251-266.

KNAW Koninklijke Nederlandse Academie van Wetenschappen

Fokker, A.D. "Unison vectors and periodicity blocks in the
three-dimensional
(3-5-7) harmonic lattice of notes,Proceedings of the KNAW,
Series B, vol. 72, 1969, pp. 153-168.

--John

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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?
>
>As Dan Wolf pointed out, these are not eikosanies, they are but dekanies. I
>am not used to seeing CPSs which don't contain 1 as a factor (BTW, octave
>equivalence is normally assumed in the CPS context). But let's see . . .
>
>2)5:
>
>3*5
>3*7
>3*9
>3*11
>5*7
>5*9
>5*11
>7*9
>7*11
>9*11
>
>3)5:
>
>3*5*7
>3*5*9
>3*5*11
>3*7*9
>3*7*11
>3*9*11
>5*7*9
>5*7*11
>5*9*11
>7*9*11
>
>How are they effectively the same? One is the inverse of the other, but are
>major and minor triads effectively the same?

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>'Scuse me for butting in, but it was an open letter. I certainly agree.
>I'm not sure what I can contribute to such an effort, but I'm on your
>side at least.

Well, I'm not sure what this effort will be or how best to focus it; the
misunderstanding that needs to be combatted here is diffuse and
widespread. I almost wanted to include you, but your concept of
higher-level consistency seems to support the other side. For example,
insisting that the approximations to 3-limit intervals be level-2
consistent means that you care about approximating certain 9-limit
intervals. But I claim that in order for any 9-limit intervals become
relevant (that is, for any just 9-limit ratios become relevant in
describing the effect/affect of any tempered intervals), all 7-limit
ratios would already be considered consonant (by an acoustical, not
cultural, evaluation), and therefore should be consistently expressed.
So insisting on full 9-limit consistency in this case would be more
appropriate.

Of course, you may simply want to exclude the number 7 from the
harmonies and include the number 9. I can understand if there is a
particular odd number that one wishes to exclude from the harmony and
therefore one does not need consistency for. Well, aside from my usual
consistency definition, which means all odd numbers up to the limit can
be included in ratios that are all consistent with one another, one can
concoct a more general definition based on any set of odd numbers. For
example, 22tET is consistent over the set of odd numbers
{3,5,7,9,11,15,17} -- 17-limit consistent if 13 is excluded.

On Mon, 4 May 1998, Paul H. Erlich wrote:
> I almost wanted to include you, but your concept of
> higher-level consistency seems to support the other side. For example,
> insisting that the approximations to 3-limit intervals be level-2
> consistent means that you care about approximating certain 9-limit
> intervals.

The implications of higher levels of consistency are varied, and I don't
want to get into another flamewar about it at this point, but just let
me say this:

(a) I wish to go beyond basic (level 1) consistency because it is
possible for an ET to be level 1 consistent and still err from a given
just ratio by nearly half the stepsize of the ET. An extreme example:
18TET is consistent to the 7-limit, but its 11-step "fifth" is only
barely better as an approximation to the 3/2 than its 10-step interval.
One is over 31 cents high, the other more than 35 cents low; the
difference between the two errors is less than four cents!

(b) There is no inconsistency ( 8-)> ) in supporting both higher-level
consistency and the use of odd limits over primes. In fact, it would be
rather difficult to adapt the idea of higher-level consistency to a
prime-limit paradigm.

(c) (semi-serious) I said I was on your side; isn't that good enough for
you? Don't look a gift horse in the mouth. 8-)>

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

Paul: There was a slight glitch in the format in my post.
The second scale is as you show and it has been posted before
by both Marion and me (I think).

--John