Yahoo Tuning Groups Ultimate Backup tuning Hey, what am I working on...

Referring to a Dan and Paul thread, I saw something
related to something I've been playing with. I have
stated here before my interest in transposable
structures. I have stated in the past a 'rule' which
turned out to insufficient. Currently I'm using a
progressive subdivision algorithm, I'll just show
in 12 (working in interval space).

12
7 5
5 2 5
3 2 2 3 2
2 1 2 2 2 1 2

This is a tree of transposable structures. What
makes them transposable is that finding the
characteristic interval and altering it at either
endpoint such that it becomes the 'normal'
interval causes the scale to be rotated (i.e.
turn into one of its modes).

Then David J. Finnamore stated :

> Subject: Term for horagramic division
>
> In each ring of a horagram, the previous ring's large
> interval (L) is divided into a new L and s (small
> interval). As it does so, it also adopts the "s" from
> the previous ring as either it's "s" or it's new "L."
> (See http://www.anaphoria.com/wilson.html ) It seems to
> me that there's something "proper" about a ring's "s"
> becoming the next ring's "L," as opposed to adjacent
> rings sharing an "s" size. The term "proper" already
> has an important meaning in the world of scales so it
> probably should not be called something like "proper
> subdivision." For now, I'm calling it "sequential
> subdivision." But that feels a little clumsy and
> vague. Any suggestions?
>

...so obviously what I'm doing is well understood. I
have since stopped using an L and s nomenclature since
whether
LsLss

turned into

LssLsss or sLLsLLL

was not important to whether or not it was transposable.
Guess I'll have to go to the Wilson archive but first...

So David, what are you looking for? (I noticed that
the branches of Ls -> Lss -> Lsss -> Lssss didin't
seem to go anywhere interesting, but there seemsed to
be interesting things where SOME steps do this)...

Paul and Dan, I've been following
your thread on decatonics. I found a 10-note
transposable set which I would label...

baa baa baaa

The characteristic interval is aaa (or baabaab).

In a subsequent post I'll probably go into all kinds
of interesting points about this (and other) such
systems but first...

1) Paul, why didn't you think that transposability
was of any relevence in your definition of
tonality? As far as I know, this is the only
transposable decatonic system, and it didn't
appear in your paper (I don't think) nor is
it representable in 22edo.

2) Is this orderring above, "maximally even"?

3) One tuning for this scale is

b a a a b a a b a a
0 57 204 351 498 555 702 849 906 1053 1200

which was derived by making baabaa = 3:2. A pretty
accurate renderring in 24tet would be 1333133133.

Is this tuning "proper"? I would say no becuase the
largest third (333) is larger than the smallest
fourth (1331). Is there a special word fo improper
but unique?

4) Is this MOS? If I start on the 849 term and
continuously add 351 I fill out exactly this
scale at the time a third required stepsize
appears...

5) any questions I forgot?

Bob Valentine

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Robert wrote,

>1) Paul, why didn't you think that transposability
> was of any relevence in your definition of
> tonality? As far as I know, this is the only
> transposable decatonic system, and it didn't
> appear in your paper (I don't think) nor is
> it representable in 22edo.

Transposability (in a general sense) is important in music that's going to
modulate. But I don't thing modulation is a prerequisite for tonality.

Anyway, my decatonic system has a form of transposability, only slightly
more complex than "transpoability" as you've defined it. This is evident
from a table of decatonic key signatures. To see a correct version of that,
you may order my paper _The Forms of Tonality_ from Bill Alves.

>2) Is this orderring above, "maximally even"?

baa baa baaa

might be maximally even depending on what b and a are. If b and a are 1 and
2 or 2 and 1, yes. If b and a are 1 and 3 or 3 and 1, no. Maximal evenness
is only defined with respect to the cardinality of the "chromatic set".
Another reason I don't like maximal evenness -- why should you have to
postulate a chromatic set in a musical style which doesn't make use of one?

>Is this tuning "proper"? I would say no becuase the
>largest third (333) is larger than the smallest
>fourth (1331). Is there a special word fo improper
>but unique?

You mean there's only one impropriety? Like the Pythagorean diatonic?

>4) Is this MOS? If I start on the 849 term and
> continuously add 351 I fill out exactly this
> scale at the time a third required stepsize
> appears...

Yes, your "tree of transposable structures" is exactly the same concept as
MOS. The horagram rings are all the MOSs for a given generator; though
Wilson's golden horagrams are constructed from noble generators, Kraig Grady
has used rhythmic horagram structures which show that the concept is
applicable to equal divisions as well.

🔗David J. Finnamore <daeron@bellsouth.net>

Bob Valentine wrote:

> Currently I'm using a
> progressive subdivision algorithm, I'll just show
> in 12 (working in interval space).
>
> 12
> 7 5
> 5 2 5
> 3 2 2 3 2
> 2 1 2 2 2 1 2
>
> This is a tree of transposable structures.

Well, it looks like it's related to the Lambdoma, Wilson's Scale Tree, and the Stern-Brocott (sp?)
Tree. Although I can't say I understand any one of those well enough to say for sure. But it looks
reminiscent of them. Paul? Dan?

> have since stopped using an L and s nomenclature since
> whether
> LsLss
>
> turned into
>
> LssLsss or sLLsLLL

> was not important to whether or not it was transposable.

Why would that make you abandon Ls nomenclature?

> Guess I'll have to go to the Wilson archive but first...

I'd highly recommend it. Plan to stay there awhile; it's not the sort of stuff that jumps right out
at you. A bit like interpreting Egyptian hieroglyphs at first. But once it starts sinking in, the
light bulbs will come on like crazy.

> So David, what are you looking for?

Well, I found it. Or rather, Kraig suggested it to me. Since the pattern of L -> Ls is known as a
"rabbit sequence" by analogy to rabbit procreation, I'm calling rings that receive their _L_ from the
previous ring's _s_ "mature rings." It might not be totally clear as to why just from that brief
description but it will become clear with a little meditation on the horagrams.

> (I noticed that
> the branches of Ls -> Lss -> Lsss -> Lssss didin't
> seem to go anywhere interesting, but there seemsed to
> be interesting things where SOME steps do this)...

Yes. Once an _s_ "grows up," or "matures," as I prefer to say, it passes its size to the next ring
as a new _L_. Thus it doesn't continue to be Lssssss... forever. Depending on the size of the
generator, a great variety of Ls strings can result. Again, the horagrams crystalize this concept.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Robert C Valentine <BVAL@IIL.INTEL.COM>

Thanks for answering David.

>
> > I have since stopped using an L and s nomenclature since
> > whether
> > LsLss
> >
> > turned into
> >
> > LssLsss or sLLsLLL
>
> > was not important to whether or not it was transposable.
>
> Why would that make you abandon Ls nomenclature?
>

As I said, from a standpoint of transposability, LssLsss
and sLLsLLL are equivalent to 'baabaaa'. Removing the
'loaded letters' 'L' and 's' allows me to thing of
the system in a manner allowing an automatic search through
all the possibilitys.

What I will be playing with this weekend is given one
of these transposable systems (which I understand better
in the past few days partly fom Paul, partly from a
brief visit to the Anaphoria archives),
how do I get something that might be an interesting
tonal/modal system. The type of things I would look for are
some sort of system of small-number-ratio approximations
(what we can call consonances), which are approachable from
the characteristic interval by contrary motion (I want my
musical investigations to start with basic counterpoint
that I am familiar with). Given that, I plan to automate
the search, for instance in the 8-tone transposable
system

baabaaba (equivalent to LssLssLs or sLLsLLsL)

1 < a < 2^(1/5)

( 2 )
b = ( --- )^(1/3)
( a^5 )

...which by moving to cents we can turn into a really
simple program (and making our minimal chroma some
hearable interval, say 8c)

from a = 8, a < 240, step 1
b = ( 1200 - ( 5 * a ) ) / 3

...and now take the generated tuning of the transposable
structure and apply tests on it to see if it fills any
useful criteria. Assuming we assign these criteria
in a meaningful manner, out will pop a few candidates
that may either be musically useful or (more likely)
reveal problems in the underlying assumptions.

The basic criteria I'll be trying are

minimize sum of 'complexity' of all intervals

or a variation

minimize sum of 'complexity' of all intervals
except for the characteristic interval(s)

>
> > Guess I'll have to go to the Wilson archive but first...
>
> I'd highly recommend it. Plan to stay there awhile; it's
> not the sort of stuff that jumps right out
> at you. A bit like interpreting Egyptian hieroglyphs at
> first. But once it starts sinking in, the
> light bulbs will come on like crazy.
>

Yeah, I have to admit being scared off by it on my visits
but I did see (in 35 hand-written pages) the same trees
written to many more places than I'd taken mine.

>
> > (I noticed that
> > the branches of Ls -> Lss -> Lsss -> Lssss didin't
> > seem to go anywhere interesting, but there seemsed to
> > be interesting things where SOME steps do this)...
>
> Yes. Once an _s_ "grows up," or "matures," as I prefer
> to say, it passes its size to the next ring
> as a new _L_. Thus it doesn't continue to be Lssssss...
> forever. Depending on the size of the generator, a great
> variety of Ls strings can result. Again,
> the horagrams crystalize this concept.
>

When generating the trees, it was important for me to
use Ls notation to figure out what the next fork would turn
into. However, once I decided to look at the "7 tone
structures" I could collapse the "mirror" patterns into one.

Bob Valentine