Hey Carl

Hey Carl, just wanted to let you know that my "Hypothesis" is
dedicated to you. You asked what a 2D and higher-dimensional
generalization of an MOS might be. There was a lot of talk on the
Tuning list about trivalent scales (scales where each generic
interval has exactly three specific step sizes) for a while but those
seem too rarefied to be the "right" answer. I believe we're on the
right track with the Hypothesis. Namely:

periodicity block with 1 unison vector not tempered out --> MOS
periodicity block with 2 unison vectors not tempered out --> 2DGMOS
periodicity block with 3 unison vectors not tempered out --> 3DGMOS

etc.

Examples in the second category would include the JI major scale,
where neither of the 2 unison vectors are tempered out; and Dave
Keenan's 31-tone 11-limit planar microtemperament, where 2 of the 4
unison vectors are tempered out.

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> If you look at the old BP posts I linked yesterday and the recent
> comma-chroma posts, I'm using a generalization that would find the
> simplest 1 or 2D "generators" as identities that fall within a given
> range (i.e., 2:3 for the 1D 7-tone Pythagorean or 4:5:6 for the 2D
> syntonic diatonic).
>
> This is of course one way to generate periodicity blocks that'll
> accomplish what you've outlined here to two dimensions -- only it
does
> it by starting with some arbitrary MOS index (i.e., x small steps
and
> y large steps) and not a set of UVs.
>
Interesting . . . so what would be the 2D "generator" for Dave
Keenan's 31-tone planar microtemperament?

> Hey Carl, just wanted to let you know that my "Hypothesis" is
> dedicated to you.

Neato! It's quite an honor to have such a cool hypothesis
dedicated to you.

> You asked what a 2D and higher-dimensional generalization of an
> MOS might be. There was a lot of talk on the Tuning list about
> trivalent scales (scales where each generic interval has exactly
> three specific step sizes) for a while but those seem too rarefied
> to be the "right" answer.

Yeah, "trihill" never seemed right.

> Keenan's 31-tone 11-limit planar microtemperament, where 2 of the 4
> unison vectors are tempered out.

Is that the pre-Canasta one (Canasta having three unison vectors
tempered out... 'zthat right? And 31-tet all four?)

-Carl

--- In tuning-math@y..., carl@l... wrote:
>
> > Keenan's 31-tone 11-limit planar microtemperament, where 2 of the
4
> > unison vectors are tempered out.
>
> Is that the pre-Canasta one (Canasta having three unison vectors
> tempered out... 'zthat right? And 31-tet all four?)
>
> -Carl

You got it!

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
>
> What is Dave Keenan's 31-tone planar microtemperament exactly... I
> either missed it or have forgotten it -- any links?
>
> --Dan Stearns

It's keenan5.scl in the Scala archive.

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Hi Paul,
> >
> > What is Dave Keenan's 31-tone planar microtemperament exactly... I
> > either missed it or have forgotten it -- any links?
> >
> > --Dan Stearns
>
> It's keenan5.scl in the Scala archive.

Dave, where's the tuning list post where you describe it, lattice it,
and tell us which unison vectors are tempered out and which aren't? I
was just looking at it a few days ago, but now I can't find it.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Dave, where's the tuning list post where you describe it, lattice
it,
> and tell us which unison vectors are tempered out and which aren't?
I
> was just looking at it a few days ago, but now I can't find it.

/tuning/topicId_7202.html#7202
/tuning/topicId_7202.html#7279
/tuning/topicId_7341.html#7341

I don't say anything about commas _not_ tempered out. Those tempered
out are 224:225 and 384:385.

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Dave, where's the tuning list post where you describe it, lattice
> it,
> > and tell us which unison vectors are tempered out and which
aren't?
> I
> > was just looking at it a few days ago, but now I can't find it.
>
> /tuning/topicId_7202.html#7202
> /tuning/topicId_7202.html#7279
> /tuning/topicId_7341.html#7341
>
> I don't say anything about commas _not_ tempered out. Those
tempered
> out are 224:225 and 384:385.

I'm almost positive it was a different message I was looking at.
Dang . . . should have bookmarked it. Anyway, the first two above may
just confuse people.

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave,
>
> I don't use Scala, could you either give a link to an old post or
just
> repost the scale in cents?
>
> thanks,
>
> --Dan Stearns

Hi Dan,

As it turns out, the scale that appears as keenan5.scl doesn't have
only 3 step sizes. It's the second scale I give in the same post, (as
a "near miss") that has only 3 step-sizes. Although I give a lattice
for it in that post, I didn't give the cents.
/tuning/topicId_7341.html#7341
So here they are:

0.0 s
30.40953308 L
79.61111475 m
115.8026469 m
151.994179 L
201.1957607 s
231.6052938 m
267.7968259 L
316.9984076 s
347.4079406 m
383.5994728 L
432.8010544 m
468.9925866 s
499.4021197 L
548.6037013 m
584.7952335 s
615.2047665 m
651.3962987 L
700.5978803 s
731.0074134 m
767.1989456 L
816.4005272 m
852.5920594 s
883.0015924 L
932.2031741 m
968.3947062 s
998.8042393 L
1048.005821 m
1084.197353 m
1120.388885 L
1169.590467 s
1200.0

What we want to know is:

What are the generators for this particular 31 note hyper-MOS of a
planar temperament?
How you do you find them?
Are they unique?
What is the mapping from generators to primes?
How do you construct this tuning from the generators?
How do you construct other examples of the same planar temperament
from them?
How do you find linear temperaments that cover them?
How do you make only those examples having exactly 3 step sizes
(hyper-MOS)?
Of those, how do you make only strictly proper ones?
Is there a smaller strictly-proper 3-step-size scale in this planar
temperament?
Are there different 3-step-size scales in this planar temperament,
having 31 notes?

We can answer all the corresponding questions for linear temperaments.

I think I know the answer to some of these questions for this
particular planar temperament, but not for planar temperaments in
general.

Any light you can shed will be much appreciated.
-- Dave Keenan

Here's a set of unison vectors for the tuning in the previous post.

3 5 7 11
--------------------------
3 7 0 0
4 -1 0 0
2 2 -1 0
-1 1 1 1

It has a determinant of 31, the first two vectors are chromatic (not
tempered out), the last two are commatic (tempered out).

3 7
4 -1

has a determinant of -31 by itself.

Is it possible that although the scale has 3 step sizes and is
symmetrical, it is not a hyper-MOS?
-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Here's a set of unison vectors for the tuning in the previous post.
>
> 3 5 7 11
> --------------------------
> 3 7 0 0
> 4 -1 0 0
> 2 2 -1 0
> -1 1 1 1

FWIW, the Fokker parallelopiped PB corresponding to these four unison
vectors is entirely within the 3-5 plane:

cents numerator denominator
0 1 1
41.059 128 125
70.672 25 24
111.73 16 15
162.85 1125 1024
203.91 9 8
223.46 256 225
274.58 75 64
315.64 6 5
335.19 4096 3375
386.31 5 4
427.37 32 25
478.49 675 512
498.04 4 3
539.1 512 375
590.22 45 32
609.78 64 45
660.9 375 256
701.96 3 2
721.51 1024 675
772.63 25 16
813.69 8 5
864.81 3375 2048
884.36 5 3
925.42 128 75
976.54 225 128
996.09 16 9
1037.1 2048 1125
1088.3 15 8
1129.3 48 25
1158.9 125 64

and contains the following step sizes:

41.059
29.614
41.059
51.12
41.059
19.553
51.12
41.059
19.553
51.12
41.059
51.12
19.553
41.059
51.12
19.553
51.12
41.059
19.553
51.12
41.059
51.12
19.553
41.059
51.12
19.553
41.059
51.12
41.059
29.614
41.059

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave,
>
> <<Is it possible that although the scale has 3 step sizes and is
> symmetrical, it is not a hyper-MOS?>>
>
> Right, that's sort of what I just posted, but then again I'm not
sure
> of exactly what definition of "hyper-MOS" we're going by!

Perhaps Dave is trying to proceed by analogy from, say, the situation
where a scale like 2 2 1 2 1 2 2 in 12-tET has 2 step sizes and is
symmetrical, but is not an MOS?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Hi Dave,
> >
> > <<Is it possible that although the scale has 3 step sizes and is
> > symmetrical, it is not a hyper-MOS?>>
> >
> > Right, that's sort of what I just posted, but then again I'm not
> sure
> > of exactly what definition of "hyper-MOS" we're going by!
>
> Perhaps Dave is trying to proceed by analogy from, say, the
situation
> where a scale like 2 2 1 2 1 2 2 in 12-tET has 2 step sizes and is
> symmetrical, but is not an MOS?

Yes. "hyper-MOS" as something that can be "generated" (in this case
presumably by a pair of generators).

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Dave Keenan wrote,
>
> <<"hyper-MOS" as something that can be "generated" (in this case
> presumably by a pair of generators).>>
>
> Good, that's what I thought.

I have no idea what this means.

Reaching in the dark, I'd say my best guess of what hyper-MOS might turn out to mean might
be something like something Clampitt brought up on the tuning list:

For any generic interval (at least the ones with cardinality relatively prime with the cardinality of
the scale), look at the cycle of that interval, and note the pattern of sizes. For some sensible (or
perhaps all) mappings from the full range of sizes to two size "classes", the cycle expressed in
terms of these "classes" is an MOS pattern. . . . ?
>
> I'd say the answer is no then... certainly not 2D, or "hyper-MOS", as
> it relates to the possible 1D "miracle" generators anyway (though I
> don't have any expedient way to check that, and it's mostly just a
> guess that's based on looking at it and having worked with these types
> of things a bit).

Still can't see what a 2D generator could mean.
>
> If I'm correct, doesn't that sink Paul's hypothesis, or at least
> permanently unhinge it from the possibility of a simple single
> generator to double generator, MOS to hyper-MOS?

The hypothesis itself doesn't mention hyper-MOS, let along 2D generators.

Here's why the hypothesis should work.

Take an n-dimensional lattice, and pick n independent unison vectors. Use these to divide the
lattice into parallelograms or parallelepipeds or hyperparallelepipeds, Fokker style. Each one
contains an identical copy of a single scale (the PB) with N notes. Any vector in the lattice now
corresponds to a single generic interval in this scale no matter where the vector is placed (if the
PB is CS, which it normally should be). Now suppose all but one of the unison vectors are
tempered out. The "wolves" now divide the lattice into parallel strips, or layers, or hyper-layers.
The "width" of each of these, along the direction of the chromatic unison vector (the one that
remains untempered), is equal to the length of exactly one of this chromatic unison vector.

Now let's go back to "any vector in the lattice". This vector, added to itself over and over, will
land one back at a pitch in the same equivalence class as the pitch one started with, after N
iterations (and more often if the vector represents a generic interval whose cardinality is not
relatively prime with N). In general, the vector will have a length that is some fraction M/N of the
width of one strip/layer/hyperlayer, measured in the direction of this vector (NOT in the direction
of the chromatic unison vector). M must be an integer, since after N iterations, you're guaranteed
to be in a point in the same equivalence class as where you started, hence you must be an
exact integer M strips/layers/hyperlayers away. As a special example, the generator has length
1/N of the width of one strip/layer/hyperlayer, measured in the direction of the generator.
Anyhow, each occurence of the vector will cross either floor(M/N) or ceiling(M/N) boundaries
between strips/layers/hyperlayers. Now, each time one crosses one of these boundaries in a
given direction, one shifts by a chromatic unison vector. Hence each specific occurence of the
generic interval in question will be shifted by either floor(M/N) or ceiling(M/N) chromatic unison
vectors. Thus there will be only two specific sizes of the interval in question, and their difference
will be exactly 1 of the chromatic unison vector. And since the vectors in the chain are equally
spaced and the boundaries are equally spaced, the pattern of these two sizes will be an MOS
pattern.

QED -- right?

I'm quite confused as to why various 11-limit PB interpretations of the blackjack scale, put forth
by various people, turned out _not_ to be equivalent to the blackjack MOS, when I calculated
the hyperparallelepiped corresponding to these suggestions. This should be looked into.

Anyway, a hyper-MOS should have the property that turning all but one of its chromatic unison
vectors into commatic unison vectors (i.e., tempering them out) results in an MOS, _no matter
which UV is chosen to remain chromatic_. Now the Clampitt-derived intuition I had in the last
message is making a lot more sense to me . . . hopefully to you all too . . . see, any
"reasonable" classificiation of specific sizes of a given generic interval into two classes should be
able to be formulated as simply tempering out certain differences between the specific sizes.
And since (as we have seen above, but extending from the 1D MOS case), any difference
between the specific sizes of a generic interval must be a combination of one each of some
subset of the set of chromatic unison vectors, tempering out any of these unison vectors will
reduce the number of specific sizes that occur . . . another thought is that trivalency is a very
special case . . . in general a D-dimensional hyper-MOS will have up to 2^D specific sizes for
each generic interval, since there are D chromatic unison vectors, each of which defines a set of
parallel boundaries in the lattice, and the number of these boundaries crossed by a specific
instance of a given vector in the lattice has 2 possible values . . . may certain PB geometries
allow one to derive a tighter upper bound than 2^D???

----- Original Message -----
From: Paul Erlich <paul@stretch-music.com>
To: <tuning-math@yahoogroups.com>
Sent: Saturday, July 28, 2001 11:40 AM
Subject: [tuning-math] Re: Hey Carl

> Here's why the hypothesis should work.
>
> Take an n-dimensional lattice, and pick n independent
> unison vectors. Use these to divide the lattice into
> parallelograms or parallelepipeds or hyperparallelepipeds,
> Fokker style. Each one contains an identical copy of
> a single scale (the PB) with N notes. Any vector in the
> lattice now corresponds to a single generic interval
> in this scale no matter where the vector is placed
> (if the PB is CS, which it normally should be). Now
> suppose all but one of the unison vectors are tempered
> out. The "wolves" now divide the lattice into parallel
> strips, or layers, or hyper-layers. The "width" of each
> of these, along the direction of the chromatic unison
> vector (the one that remains untempered), is equal to
> the length of exactly one of this chromatic unison vector.
> <etc. ... snip>

Paul, could you draw this process on some lattices.
I'll put it in a webpage if you put the whole hypothesis
together with nice graphics.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
>
> Lots of stuff to chew on there, but here's some quick thoughts...
>
> <<Still can't see what a 2D generator could mean.>>
>
> When I say it I mean it in a very simple way -- as I see it it's
> exactly the generalized difference between the 7-tone Pythagorean
> single 1D plane built from 2:3s, and the classic JI, 5-limit
diatonic
> 2D plane built from 2:3s and 4:5s.

So wouldn't any connected, convex chunk of an N-dimensional JI
lattice be "generatable" by this definition?
>
>
> <<Here's why the hypothesis should work [SNIP] QED -- right?>>
>
> Well it might make sense right off to some folks but it's a lot for
me
> to follow right now without some examples... do you think you could
> flesh out the steps with some?

I need to draw some diagrams for you . . .
>
>
> <<another thought is that trivalency is a very special case>>
>
> Certainly when compared to "bivalency" it's not as omnipresent (as
> I've said before all M-out-of-N sets are bivalent within a period),
> but I could come up with several trivalent examples pretty easily,
so
> I don't think it's exactly "very special".

By "very special" here I didn't mean to imply that it would be hard
for you to find examples . . . just that in general, a 2D hyper-MOS
would have up to four specific sizes for each generic interval.