Mike S's "Dorian"-ish scale [discussion continued from MMM] - well temperaments having fractional rank

I'm continuing the discussion from MMM here.

On Fri, Nov 26, 2010 at 12:05 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> Calling it rank six wasn't my idea. The point is, the smallest number of generators which can express every note of the scale is six; you can regard it as a scale of the {2, 3, 5, 7, 31, 223} JI subgroup of the 223-limit. But that's on paper; the 223-limit doesn't have much connection to what people hear; in fact, the 23-limit is pushing it. The basic features of the scale are in place if you retune it using three generators, so if you are going to try to classify scales by ranks Mike Battaglia's idea of calling it rank three seems to have some justice.

I actually initially threw the rank 3 idea out there without thinking
too much about it, but upon further analysis, it seems to occupy a
rank somewhere "between" two and three, in an abstract sense. If this
were a distorted meantone, then 81/80 would be tempered out, and if it
were a distorted 5-limit JI, then 81/80 wouldn't be tempered out. It's
kind of "half" tempered out.

Observe the interval sizes for Michael's scale, organized by step size:

1: 117.925, 119.443, 188.452, 190.004, 190.115, 196.198, 197.861
2: 307.895, 307.930, 308.041, 315.641, 386.314, 387.866
3: 498.045, 504.094, 504.239, 505.757, 505.791, 576.318
4: 623.682, 694.209, 694.243, 695.761, 695.906, 701.955
5: 812.134, 813.686, 884.359, 891.959, 892.070, 892.105
6: 1002.139, 1003.802, 1009.885, 1009.996, 1011.548, 1080.557, 1082.075

Now I'll categorize them again, informally, into groups based on some
common error. I will define the error in this case to be two cents.
This is an attempt to "squeeze" the rank-6 tuning we have here into a
smaller rank-3 system. L means Large, m means medium, and s means
small:

1: [s: 117.925, 119.443], [m: 188.452, 190.004, 190.115], [L: 196.198, 197.861]
2: [s: 307.895, 307.930, 308.041] [m: 315.641], [L: 386.314, 387.866]
3: [s: 498.045], [m: 504.094, 504.239, 505.757, 505.791], [L: 576.318]

Every entry in each category are within two cents of one another. The
last three rows are redundant since they're just octave-inverted
versions of the first three, so I didn't put them.

Compare this to 5-limit JI's interval sizes:
1: [s: 111.731], [m: 182.404], [L: 203.910] -- L-m = 21.506
2: [s: 294.135], [m: 315.641], [L: 386.314] -- m-s = 21.506
3: [s: 498.045], [m: 519.551], [L: 590.224] -- m-s = 21.506

Here are 1/4-comma meantone's, in which there are only L's and s's,
but I'm going to write it as Lms anyway for reasons that will become
apparent shortly:
1: [s: 117.108], [m: 193.157], [L: 193.157] -- L-m = 0
2: [s: 310.265], [m: 310.265], [L: 386.314] -- m-s = 0
3: [s: 503.422], [m: 503.422], [L: 579.471] -- m-s = 0

Tempering out 81/80 collapses two of the classes in each category to
just one class. Note that in a sense, the splitting of each of these
classes into the two diverging ones is what defines a temperament as
being rank-3, rather than rank-2. In 5-limit JI, there are two classes
in each row that diverge by 81/80, which is 21.5 cents, and it's rank
3. In 1/4-comma meantone, they diverge by 0 cents, and it's rank 2.

So let's revisit Michael's temperament, and just "collapse" each class
to its mean value to simplify things:
1: [s: 118.684], [m: 189.524], [L: 197.030] -- L-m = 7.506
2: [s: 307.955] [m: 315.641], [L: 387.090] -- m-s = 7.686
3: [s: 498.045], [m: 504.970], [L: 576.318] -- m-s = 6.925

So the syntonic comma has been partially tempered out to be about 1/3
of the length that it was originally. The mean divergence between the
relevant classes here is 7.372, and 7.372/21.506 is about 0.343, so
perhaps Michael's scale could be said to have a rank of 2.343.

There is a bit of mathematical glue missing here that perhaps Gene or
Graham would have some better insights on - perhaps it's possible to
describe any temperament as any other temperament in which something
is "almost tempered out." Or perhaps an inconsistent temperament could
be viewed as a HIGHER rank version of a consistent one.

-Mike

On Fri, Nov 26, 2010 at 2:33 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> There is a bit of mathematical glue missing here that perhaps Gene or
> Graham would have some better insights on - perhaps it's possible to
> describe any temperament as any other temperament in which something
> is "almost tempered out." Or perhaps an inconsistent temperament could
> be viewed as a HIGHER rank version of a consistent one.

Also, to simplify all of this, the algorithm I presented has five parts to it:
1) Grouping of similarly-sized intervals into "interval classes" that
fall under a certain error
2) Mapping those interval classes to suitably picked JI intervals
3) Identifying the interval in question that we're trying to determine
if it's tempered out
4) Figuring out "how much" that interval is actually tempered out
5) Determining the "fractional rank" of the temperament

It seems to me that #1 could be obtained by retro-recursively applying
2, 3, 4, and 5 back to the original set, but I'm not sure how exactly
to do that. It would be kind of stupid if you had a version of
quarter-comma meantone that had 10000 tiny variations for each
generator, and you ended up figuring out what fractional rank the
resultant temperament occupied, and it turned out to be something like
10.532, which is much closer to 1 than 10000, but still doesn't make
much sense.

I'm also not sure how to apply the algorithm in a broader sense to
scales that aren't MOS.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> So the syntonic comma has been partially tempered out to be about 1/3
> of the length that it was originally. The mean divergence between the
> relevant classes here is 7.372, and 7.372/21.506 is about 0.343, so
> perhaps Michael's scale could be said to have a rank of 2.343.
>
> There is a bit of mathematical glue missing here that perhaps Gene or
> Graham would have some better insights on - perhaps it's possible to
> describe any temperament as any other temperament in which something
> is "almost tempered out."

We've looked before at temperaments which shrink some particular comma without tempering it out. This can definitely relate one temperament to another: for example, if you temper out the comma (81/80)^5/(128/125) then you'll find 81/80 is shrunk without being eliminated.

On Fri, Nov 26, 2010 at 3:20 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> We've looked before at temperaments which shrink some particular comma without tempering it out. This can definitely relate one temperament to another: for example, if you temper out the comma (81/80)^5/(128/125) then you'll find 81/80 is shrunk without being eliminated.

I'm confused - do you mean for the TOP equivalent of that tuning? If
not, then wouldn't 81/80's size depend on what generator you pick -
couldn't you tune it to give a pure 81/80 if you want?

Maybe this is something I don't understand about regular mapping - how
do you relate the generator size to the temperament you're working in?
If you're working in 5-limit JI, and you temper out 81/80, you go down
to rank-2. There's an infinitely large range of sizes that you could
assign to the generator, but only a small subset of these will be
useful or recognizable.

One of these useful generator sizes is 700 cents. If you pick it,
although you aren't actually tempering out 128/125 in the mathematical
formalism, it's gone. So what happens - is that rank-1 now? or does it
depend on the mapping you use?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > We've looked before at temperaments which shrink some particular comma without tempering it out. This can definitely relate one temperament to another: for example, if you temper out the comma (81/80)^5/(128/125) then you'll find 81/80 is shrunk without being eliminated.
>
> I'm confused - do you mean for the TOP equivalent of that tuning? If
> not, then wouldn't 81/80's size depend on what generator you pick -
> couldn't you tune it to give a pure 81/80 if you want?

Any reasonable tuning choice will shrink it. For example, TE tuning shrinks it to 43.7%, minimax tuning to 38.1%, and least squares to 42.3%.

> Maybe this is something I don't understand about regular mapping - how
> do you relate the generator size to the temperament you're working in?

Use some method to select a tuning. This could be simply a matter of finding a suitable et, or some method such as TOP, TE, minimax or least squares.

> One of these useful generator sizes is 700 cents. If you pick it,
> although you aren't actually tempering out 128/125 in the mathematical
> formalism, it's gone. So what happens - is that rank-1 now?

A period of 700 cents and an octave of 1200 cents is rank 1, yes.

On Fri, Nov 26, 2010 at 4:23 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> > One of these useful generator sizes is 700 cents. If you pick it,
> > although you aren't actually tempering out 128/125 in the mathematical
> > formalism, it's gone. So what happens - is that rank-1 now?
>
> A period of 700 cents and an octave of 1200 cents is rank 1, yes.

I think I'm being unclear - I'm trying to, for my own understanding,
separate the mathematical formalism from the psychoacoustic guidelines
we use to make useful tunings, and from the ad hoc decisions we make
about when to jump from mapping to mapping.

Let's say you're in 5-limit JI, and you temper out only 81/80. Then
let's say you pick a generator of exactly 697.5 cents. As opposed to a
700 cent generator, where the chain obviously closes after 12
iterations of the generator, this seems more like it would be a rank 2
meantone.

But, if I really pick a generator width of 697.5 cents and a period of
1200 cents, then both of these are part of 160-et. So every pitch in
the system is really a member of 160-tet, which means some stupidly
small additional comma is being tempered out as well with this
generator.

So my question is, which of the following is the "correct" view, from
a purely mathematical standpoint, of this pathological example?

1) This is defined as a rank 2 temperament, because in the very
beginning I said "you temper out only 81/80," meaning that we're going
to be looking at this tuning from a perspective in which only 81/80
and no additional comma is being tempered out, even though it may not
be useful
2) This is defined as a rank 1 temperament, because 697.5 cents and
1200 cents are both multiples of 7.5 cents, which means that 7.5 cents
is a "simpler" generator that generates both of them
3) The concept of having a generator that's 697.5 cents and 1200 cents
doesn't lead to any particular rank-1 or rank-2 mapping - you could
analyze it as a subset of 160-et, or as a rank-2 system in which it so
happens that your choice of generator leads to overlapping intervals

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> So my question is, which of the following is the "correct" view, from
> a purely mathematical standpoint, of this pathological example?

Either, both, all three, whatever. I like to look at it in terms of a doodad called a commutative triangle. Rather than try to draw it, I'll see if this link does the trick:

http://tinyurl.com/285vb82

The idea is that we can separate the abstract temperament from the particular tuning, so that while a temperament might be rank r, the tuning might be of lesser rank. The group of the abstract temperament can be expressed purely abstractly. If, for instance, you name it by means of the wedgie W, then if m is a monzo of the correct type (ie, limit) then the tempered note corresponding to it can be represented by the interior product Wvm ("v" being what I'm using to notate the interior product.) This is an abelian group of the right rank and other properties, which can then be mapped to any chosen tuning.

For example, the wedgie for septimal meantone is <<1 4 10 4 13 12||. Call this W. Then Wv|1 0 0 0> = <0 -1 -4 -10| and Wv|-1 1 0 0> = <1 1 0 -3| are the interior products from the period 2 and generator 3/2. Every interior product with W is a linear combination of these two, so we can map the abstract group to any particular tuning we choose.

Interior products: http://mathworld.wolfram.com/InteriorProduct.html

On Fri, Nov 26, 2010 at 6:00 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> The idea is that we can separate the abstract temperament from the particular tuning, so that while a temperament might be rank r, the tuning might be of lesser rank. The group of the abstract temperament can be expressed purely abstractly. If, for instance, you name it by means of the wedgie W, then if m is a monzo of the correct type (ie, limit) then the tempered note corresponding to it can be represented by the interior product Wvm ("v" being what I'm using to notate the interior product.) This is an abelian group of the right rank and other properties, which can then be mapped to any chosen tuning.

OK, I think I understand, although I'll need to do a bit more reading
about the interior product. So what I'm trying to do then is extend
the free abelian structure to include well temperaments, which would
be some kind of "free abelian groupoid" (not sure what the proper term
would be), which would have a set of linearly independent basis
vectors that somehow has a "fractional" cardinality.

As an abstract concept, this wouldn't apply for temperaments in which
an interval shrinks but is not willfully tempered out. It would
specifically apply to a temperament in which the goal is to take an
interval and "half-temper" or unequally temper it out; i.e., it would
apply to well-temperaments.

It's a bit over my head how to do this precisely, and the algorithm in
my first post in this thread was just my initial attempt to do
something in that vein. It would be nice to define the structure in
such a way that

- If we want to map 5-limit JI as a well-temperament of 5-limit JI, it
ends up with rank 3
- If we want to map meantone as a well-temperament of 5-limit JI, it
ends up with rank 2
- If we want to map 12-equal as a well-temperament of 5-limit JI, we
get a rank of 1
- If we want to map Werckmeister III as a well-temperament of 5-limit
JI, it ends up being between rank 1 and 2
- If we want to map Michael S's scale as a well-temperament of 5-limit
JI, it ends up being between rank 2 and 3
- If we want to map Michael S's scale as a well-temperament of 7-limit
JI, it ends up being between rank 3 and 4

I'm not quite sure how to set things up like this... Maybe this could
get involved:

http://en.wikipedia.org/wiki/Minkowski_dimension

-Mike