Re: representation, fuzzy logic, optimal tuning, adaptive

>>>Hmm . . . so what is it about Pythagorean, 12-tET, and JI that put them >>>in
>>>the former category?

>>With them (and with the two others I described, overtone and acoustic),
>>it's easy to generate intervals with a few parameters without reference
>>to the construction of an entire scale (which comes a few pages later).

>I don't see how meantone is any different.

Neither are Quiggle temperaments, and I didn't include those either---
but I intend to someday. Since there's a class for representing intervals
in general, of course, meantone intervals can be represented; I just
hadn't worked out how to make a class to generate them by parameter.

>>My list of intervallic tuning paradigms was never meant to be exhaustive.
>>I represent intervals in a very general way, as fuzzy quantities of pitch
>>distance. The classes in the section you're looking at are just examples
>>of handy ways to produce interval objects by parameter.

>So perhaps meantone would fall under the Pythagorean umbrella?

Maybe, let me see, maybe the parameters would look like this:
n=number of "fifths" to transpose upward,
f=the interval we call a "fifth" (it could be anything),
t=the target interval we're trying to approximate.
Then the interval generated would be f+(t-f*n)/n?
Optionally it could return the temper itself, (t-f*n)/n?

>>a formula for generating?

>Sure -- there are a number of possibilities. For example, one can run
>a least-squares optimization on the three errors: the deviation of
>the fifth itself from 3:2, the deviation of the major sixth generated
>by three such fifths (minus an octave) from 5:3, and the deviation of
>the major third generated by four such fifths (minus two octaves)
>from 5:4. The result is 7/26-comma meantone temperament (Woolhouse
>1835), i.e. (3/2)*(80/81)^(7/26).

So, the parameters might be:
f=the "fifth"
c=the "comma"
x=the amount of comma to temper.
So the generated interval would be f*c^x?
That seems less natural and interesting than the above.

What would you suggest?

-m
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--- In tuning@y..., "Michael Saunders" <michaelsaunders7@h...> wrote:

> Maybe, let me see, maybe the parameters would look like this:
> n=number of "fifths" to transpose upward,
> f=the interval we call a "fifth" (it could be anything),
> t=the target interval we're trying to approximate.

t is redundant, isn't it?

As I see it, meantone is nothing more than a variation of
Pythagorean, where the fifths are uniformly flattened by a fraction
(typically 1/3 to 1/6) of a comma, in order to get much-improved
thirds and sixths. Since the thirds and sixths are automatically much-
improved, there isn't a need to additionally specify the target
interval we're trying to approximate, is there?

There is the issue of adaptive tuning, but I assume we're not talking
about that yet . . . we're talking about the basic paradigms . . .
right?

>>Maybe, let me see, maybe the parameters would look like this:
>>n=number of "fifths" to transpose upward,
>>f=the interval we call a "fifth" (it could be anything),
>>t=the target interval we're trying to approximate.

>t is redundant, isn't it?

Oh, no, without t it's just plain Pythagorean. You have to know the
interval you're trying to approximate in order to know how much
you have to temper, right? What I mean is, e.g.,
n=3
f=3/2=702c
t=5/3=884c
using, y=f+(t-(f*n))/n, we go up 3 fifths to 2106c=906c, so the
discrepency is 884c-906c=-22c. Divided amongst the fifths, it's
-22c/3=-7.33c. So, tempering the fifth by that much we generate
the interval y=-695c.

Another example, with a different "fifth":
n=6
f=9/8=204c
t=2/1=1200c
so, y=204+(1200-(204*6))/6=200c.

So that formula determines the tempered interval described
by the parameters.

>As I see it, meantone is nothing more than a variation of Pythagorean, >where the fifths are uniformly flattened by a fraction (typically 1/3 to >1/6) of a comma, in order to get much-improved thirds and sixths. Since the >thirds and sixths are automatically much-
>improved, there isn't a need to additionally specify the target interval >we're trying to approximate, is there?

Oh---if you want to start off by stating the tempered interval,
then there's no need for a special method to generate it. This is
why I neglected to define one. If you want to just name a
tempered interval and generate other intervals by stacking them,
then take another look at my Pythagorean class---you can define
any interval you like as the "fifth". I like the above procedure
though---of generating tempered intervals, though it's a lot like
my equal temperament object (since you can equally divide any
interval).

>There is the issue of adaptive tuning, but I assume we're not talking about >that yet . . . we're talking about the basic paradigms . . . right?

We're not even talking about tuning yet, just about how
to generate crisp intervals via parameters. Then you
take those (or others that you might define without bothering
with any paradigms or parameters) and build a network of them
to define a tuning.

-m

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--- In tuning@y..., "Michael Saunders" <michaelsaunders7@h...> wrote:

/tuning/topicId_23012.html#23092

>
> >> Maybe, let me see, maybe the parameters would look like this:
> >> n=number of "fifths" to transpose upward,
> >> f=the interval we call a "fifth" (it could be anything),
> >> t=the target interval we're trying to approximate.
>
> > t is redundant, isn't it?
>
> Oh, no, without t it's just plain Pythagorean. You have to know
> the interval you're trying to approximate in order to know how
> much you have to temper, right? What I mean is, e.g.,
> n=3
> f=3/2=702c
> t=5/3=884c
> using, y=f+(t-(f*n))/n, we go up 3 fifths to 2106c=906c, so the
> discrepency is 884c-906c=-22c. Divided amongst the fifths, it's
> -22c/3=-7.33c. So, tempering the fifth by that much we generate
> the interval y=-695c.
>
> Another example, with a different "fifth":
> n=6
> f=9/8=204c
> t=2/1=1200c
> so, y=204+(1200-(204*6))/6=200c.

What you're calling a "fifth" is what we generally call a
"generator", and later in this post you refer to it as a
"generating" interval.

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

--- In tuning@y..., "Michael Saunders" <michaelsaunders7@h...> wrote:
>
> >>Maybe, let me see, maybe the parameters would look like this:
> >>n=number of "fifths" to transpose upward,
> >>f=the interval we call a "fifth" (it could be anything),
> >>t=the target interval we're trying to approximate.
>
> >t is redundant, isn't it?
>
> Oh, no, without t it's just plain Pythagorean.

I thought it wouldn't be Pythagorean, since f would be, say, (3/2)*
(80/81)^(7/26).

> You have to know the
> interval you're trying to approximate in order to know how much
> you have to temper, right?

That's already built into the definition of meantone, and the
construction of the fifths.
>
> Another example, with a different "fifth":
> n=6
> f=9/8=204c
> t=2/1=1200c
> so, y=204+(1200-(204*6))/6=200c.

Well that's not meantone. Yes, there are many scales that are
analogous to meantone, such as my decatonic scales in 22-tET and the
MIRACLE scales we've been talking about. These could all be user-
defined paradigms, I assume. But standard musical notation is what
I'm focusing on in this particular discussion.
>
> So that formula determines the tempered interval described
> by the parameters.

Well, often there are more than two intervals involved in the
compromise . . .
>
>
> >There is the issue of adaptive tuning, but I assume we're not
talking about
> >that yet . . . we're talking about the basic paradigms . . . right?
>
> We're not even talking about tuning yet, just about how
> to generate crisp intervals via parameters. Then you
> take those (or others that you might define without bothering
> with any paradigms or parameters) and build a network of them
> to define a tuning.
>
Well, OK . . . perhaps it would be more interesting if you could
explain your generalized JI paradigm the way you've explained your
generalized Pythagorean paradigm.