The Scale Tree vs. Regular Temperaments

This is based of a recent off-list discussion with Mike B.:

Trying to draw relationships between temperaments and the scale tree, Mike made a point that I interpreted as saying at infinity-EDO, nothing is tempered out, and you have an infinity of pitches based on infinite number of prime-dimensions, each of which is infinitely large, but at 0-EDO, you essentially have a single note where all of an infinite number of commas are tempered out. So it seems like all temperaments really arise from a *de-tempering* of commas from 0-EDO. But I don't know if this idea makes any sense, because I can't see how de-tempering a single comma will result in another EDO. I can conceive of the boundaries, but I can't conceive of a temperament in which all possible commas except *one* are tempered out. Any thoughts?

-Igs

On Wed, Feb 2, 2011 at 2:47 PM, cityoftheasleep <igliashon@...> wrote:
>
> This is based of a recent off-list discussion with Mike B.:
>
> Trying to draw relationships between temperaments and the scale tree, Mike made a point that I interpreted as saying at infinity-EDO, nothing is tempered out, and you have an infinity of pitches based on infinite number of prime-dimensions, each of which is infinitely large, but at 0-EDO, you essentially have a single note where all of an infinite number of commas are tempered out. So it seems like all temperaments really arise from a *de-tempering* of commas from 0-EDO. But I don't know if this idea makes any sense, because I can't see how de-tempering a single comma will result in another EDO. I can conceive of the boundaries, but I can't conceive of a temperament in which all possible commas except *one* are tempered out. Any thoughts?

That would be 1-EDx/x, where x/x is the comma you're not tempering out.

-Mike

On Wed, Feb 2, 2011 at 3:04 PM, Mike Battaglia <battaglia01@...> wrote:
> On Wed, Feb 2, 2011 at 2:47 PM, cityoftheasleep <igliashon@sbcglobal.net> wrote:
>> Trying to draw relationships between temperaments and the scale tree, Mike made a point that I interpreted as saying at infinity-EDO, nothing is tempered out, and you have an infinity of pitches based on infinite number of prime-dimensions, each of which is infinitely large, but at 0-EDO, you essentially have a single note where all of an infinite number of commas are tempered out. So it seems like all temperaments really arise from a *de-tempering* of commas from 0-EDO. But I don't know if this idea makes any sense, because I can't see how de-tempering a single comma will result in another EDO. I can conceive of the boundaries, but I can't conceive of a temperament in which all possible commas except *one* are tempered out. Any thoughts?
>
> That would be 1-EDx/x, where x/x is the comma you're not tempering out.

To clarify further, these are my initial half-formed thoughts, so they
probably don't make complete sense. I wasn't going to post them until
I had things worked out more, so tread with extreme caution. but since
Igs brings it up, I might as well explain what train of thought I'm
on.

The "detempering" in this sense is not the same thing as detempering
in regular mapping in that it doesn't elevate the dimensionality of
the space you're working within. 1-EDO puts us at rank 1, and the only
interval there is is 2/1. Then, if we're trying to -ACTUALLY- detemper
one step further to go to rank-2, you could end up with an unspecified
1L1s scale in which all possible choices of generator (except for 2/1)
are equated. I used the word that they're "tempered together" in my
email, which was probably a poor choice. I am denoting, in a sense,
"paradigms" rather than temperaments.

What I'm saying is that in this 1L1s scale, the generator could be
anything from 0 to 600 cents, which we could represent as being -any-
generator except for 2/1 and its multiples. So Igs brought up 3L1s
scales - assuming we're in the 5-limit, the generator could be either
6/5 or 5/4, so I thought it would make sense to say that the generator
is both, and we're in some "dicot"-based primitive, as per the recent
definition of primitive.

The children of 3L1s are going to be 4L3s and 3L4s; the former maps
best to things like hanson and amity, which have generators of 6/5,
and the latter maps best to things like dicot and magic, which have
generators of 5/4. So as you go further down the scale tree, the two
split off. So in light of that, rather than saying that 2L3s maps to
the 3-limit pythagorean val, it makes more sense to say that mavila
and meantone are "tempered together" there, and then split off at 5L2s
and 2L5s.

This is obviously a different sense of "tempering" than with usual
regular temperament, which is why I say that we're talking about
paradigms. A paradigm is going to be something like a fuzzy set of
vals with a membership function for each one, and different
temperaments fit into different paradigms. In this case, we were
exploring how to apply paradigms to nodes on the scale tree. For 1L1s
your generator can range from 0 to 600 cents, which equates 25/24,
16/15, 10/9, 9/8, 6/5, 5/4, and 4/3, etc. This puts us at what, rank
negative 3? But 1L1s "equates" them in a different sense.

I think that looking at projective Tenney tuning space, but taking the
"inverse" of it in such a way that you look at space(1/x, 1/y) rather
than space(x,y), might be interesting here.

-Mike