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Graef article on rationalization of scales

🔗Gene Ward Smith <gwsmith@svpal.org>

1/27/2004 11:15:47 AM

Does anyone know why Barlow changed Euler's (p-1) weighting to
2(p-1+1/(2p))?

Graef gives examples of "rationaizing" 12-equal and 1/4-comma
meantone in the 5-limit. Rationalizing means moving to a nearby value
so that a certain badness figure is minimized, where we look at the
entire matrix of intervals and work locally (interval pair by
interval pair) not globally. Adding the last condition makes the
problem much more complicated, and I don't see the point in it. He
got the duodene by rationalizing equal temperament, and
syndie2=fogliano1 by rationalizing 1/4-comma.

🔗Carl Lumma <ekin@lumma.org>

1/27/2004 1:13:25 PM

>Does anyone know why Barlow changed Euler's (p-1) weighting to
>2(p-1+1/(2p))?
>
>Graef gives examples of "rationaizing" 12-equal and 1/4-comma
>meantone in the 5-limit. Rationalizing means moving to a nearby value
>so that a certain badness figure is minimized, where we look at the
>entire matrix of intervals and work locally (interval pair by
>interval pair) not globally. Adding the last condition makes the
>problem much more complicated, and I don't see the point in it. He
>got the duodene by rationalizing equal temperament, and
>syndie2=fogliano1 by rationalizing 1/4-comma.

This raises an interesting question. What is our approved method
for finding Fokker blocks for an arbitrary irrational scale?
Such a method would surely make Graf's look silly.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

1/27/2004 3:09:00 PM

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Does anyone know why Barlow changed Euler's (p-1) weighting to
> >2(p-1+1/(2p))?
> >
> >Graef gives examples of "rationaizing" 12-equal and 1/4-comma
> >meantone in the 5-limit. Rationalizing means moving to a nearby
value
> >so that a certain badness figure is minimized, where we look at
the
> >entire matrix of intervals and work locally (interval pair by
> >interval pair) not globally. Adding the last condition makes the
> >problem much more complicated, and I don't see the point in it. He
> >got the duodene by rationalizing equal temperament, and
> >syndie2=fogliano1 by rationalizing 1/4-comma.
>
> This raises an interesting question. What is our approved method
> for finding Fokker blocks for an arbitrary irrational scale?
> Such a method would surely make Graf's look silly.

All such methods are silly, but I prefer the hexagonal (rhombic
dodecahedral, etc.) or Kees blocks that result from the min-"odd-
limit" criterion. But the whole idea of rationalizing a tempered
scale is completely backwards and misses the point in a big way.

🔗Carl Lumma <ekin@lumma.org>

1/27/2004 6:29:50 PM

>> This raises an interesting question. What is our approved method
>> for finding Fokker blocks for an arbitrary irrational scale?
>> Such a method would surely make Graf's look silly.
>
>All such methods are silly,

Perhaps you mean Graf's idea of people wanting "just" versions of
arbitrary scales is silly. That's for sure.

A method which could show when there is (and isn't) a reasonable
Fokker-block interp. of, say scales taken from field measurements
silly? I think not.

>but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees
>blocks that result from the min-"odd-limit" criterion. But the
>whole idea of rationalizing a tempered scale is completely
>backwards and misses the point in a big way.

I think you missed my point.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

1/28/2004 2:57:42 PM

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> This raises an interesting question. What is our approved method
> >> for finding Fokker blocks for an arbitrary irrational scale?
> >> Such a method would surely make Graf's look silly.
> >
> >All such methods are silly,
>
> Perhaps you mean Graf's idea of people wanting "just" versions of
> arbitrary scales is silly. That's for sure.

Yup.

> >but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees
> >blocks that result from the min-"odd-limit" criterion. But the
> >whole idea of rationalizing a tempered scale is completely
> >backwards and misses the point in a big way.
>
> I think you missed my point.

Would you, then, clarify your point, perhaps with examples?

🔗Carl Lumma <ekin@lumma.org>

1/28/2004 5:11:34 PM

>> >> This raises an interesting question. What is our approved method
>> >> for finding Fokker blocks for an arbitrary irrational scale?
>> >> Such a method would surely make Graf's look silly.
>> >
>> >All such methods are silly,
>>
>> Perhaps you mean Graf's idea of people wanting "just" versions of
>> arbitrary scales is silly. That's for sure.
>
>Yup.
>
>> >but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees
>> >blocks that result from the min-"odd-limit" criterion. But the
>> >whole idea of rationalizing a tempered scale is completely
>> >backwards and misses the point in a big way.
>>
>> I think you missed my point.
>
>Would you, then, clarify your point, perhaps with examples?

That the diatonic scale is a 'good' PB seems like the best example.
Since you practically single-handedly launched the 'popular scales
are good PBs' program, I find it highly unusual that you are now
asking me what it is.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

1/28/2004 5:29:27 PM

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> This raises an interesting question. What is our approved
method
> >> >> for finding Fokker blocks for an arbitrary irrational scale?
> >> >> Such a method would surely make Graf's look silly.
> >> >
> >> >All such methods are silly,
> >>
> >> Perhaps you mean Graf's idea of people wanting "just" versions of
> >> arbitrary scales is silly. That's for sure.
> >
> >Yup.
> >
> >> >but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees
> >> >blocks that result from the min-"odd-limit" criterion. But the
> >> >whole idea of rationalizing a tempered scale is completely
> >> >backwards and misses the point in a big way.
> >>
> >> I think you missed my point.
> >
> >Would you, then, clarify your point, perhaps with examples?
>
> That the diatonic scale is a 'good' PB seems like the best example.

Sure, but when 81:80 is tempered out, it's a good "periodicity strip"
(figure 5 in TFoT), which is even better, and doesn't require you to
arbitrarily rationalize the pitches.

> Since you practically single-handedly launched the 'popular scales
> are good PBs' program, I find it highly unusual that you are now
> asking me what it is.

Now I understand you better. Yet, 'popular scales' will often have a
large number of plausible derivations from a PB, in terms of its
shape, its position in the lattice, and the unison vectors involved,
so going from the scale to the PB still seems like a step backwards,
a step from greater generality to lesser generality.

🔗Carl Lumma <ekin@lumma.org>

1/28/2004 6:51:30 PM

>> Since you practically single-handedly launched the 'popular scales
>> are good PBs' program, I find it highly unusual that you are now
>> asking me what it is.
>
>Now I understand you better. Yet, 'popular scales' will often have a
>large number of plausible derivations from a PB,

You mean to a PB, I think?

>in terms of its shape, its position in the lattice, and the unison
>vectors involved, so going from the scale to the PB still seems like
>a step backwards, a step from greater generality to lesser generality.

True, but we're not necessarily interested in a particular one.
If we are, it seems Gene has or is close to having tools to transform
between them, define a canonical one, etc. But if your 'the basis
doesn't matter' reasoning applied to blocks, we might say that all
alternate PB versions ought to have the same goodness. If we could
then further show that historical scales correspond to gooder blocks
than random scales...

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

1/28/2004 7:19:32 PM

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Since you practically single-handedly launched the 'popular
scales
> >> are good PBs' program, I find it highly unusual that you are now
> >> asking me what it is.
> >
> >Now I understand you better. Yet, 'popular scales' will often have
a
> >large number of plausible derivations from a PB,
>
> You mean to a PB, I think?

A gets derived *from* B, it doesn't get derived *to* B, right?

> But if your 'the basis
> doesn't matter' reasoning applied to blocks,

It only applies to the *temperament kernel*, or to the commatic
unison vectors -- not the chromatic ones.

> If we could
> then further show that historical scales correspond to gooder blocks
> than random scales...

Since Gene is done enumerating 12-note Fokker blocks, I hope he will
treat us to a collection of 5-note ones soon . . .

🔗Carl Lumma <ekin@lumma.org>

1/28/2004 7:28:24 PM

>> >Now I understand you better. Yet, 'popular scales' will often have
>> >a large number of plausible derivations from a PB,
>>
>> You mean to a PB, I think?
>
>A gets derived *from* B, it doesn't get derived *to* B, right?

Sure. I guess what threw me is that you expect multiple PBs,
and "from a PB" sounds very singular.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

1/28/2004 11:40:30 PM

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> That the diatonic scale is a 'good' PB seems like the best example.

Is PB a synonym for Fokker block, or is it more general, and if so,
how precisely is it defined?

🔗Carl Lumma <ekin@lumma.org>

1/29/2004 2:07:00 AM

>> That the diatonic scale is a 'good' PB seems like the best example.
>
>Is PB a synonym for Fokker block, or is it more general, and if so,
>how precisely is it defined?

My impression was that PB is weaker than Fokker, the later requiring
epimorphism and monotonicity (neither of which I have a solid
understanding of) and that the former requires, well, nothing more
than the correct number of commas that, when all of them are tempered
out, gives an equal temperament.

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

1/29/2004 1:29:42 PM

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> That the diatonic scale is a 'good' PB seems like the best
example.
> >
> >Is PB a synonym for Fokker block, or is it more general, and if
so,
> >how precisely is it defined?
>
> My impression was that PB is weaker than Fokker, the later requiring
> epimorphism and monotonicity (neither of which I have a solid
> understanding of) and that the former requires, well, nothing more
> than the correct number of commas that, when all of them are
tempered
> out, gives an equal temperament.
>
> -Carl

Pretty much. We've been through this before, whereupon Gene
defined "block" for convex PB and "semiblock" for something "not too
concave" or something, if my vague recollection is reliable.

🔗Paul Erlich <perlich@aya.yale.edu>

1/29/2004 1:31:49 PM

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> That the diatonic scale is a 'good' PB seems like the best
example.
> >
> >Is PB a synonym for Fokker block, or is it more general, and if
so,
> >how precisely is it defined?
>
> My impression was that PB is weaker than Fokker, the later requiring
> epimorphism and monotonicity (neither of which I have a solid
> understanding of) and that the former requires, well, nothing more
> than the correct number of commas that, when all of them are
tempered
> out, gives an equal temperament.
>
> -Carl

Actually, all PBs always have epimorphism, but I don't know what
monotonicity is. Fokker blocks are parallelograms/parallelepipeds --
bounded exactly by the linearly independent set of unison vectors.

🔗Gene Ward Smith <gwsmith@svpal.org>

1/29/2004 6:46:28 PM

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Pretty much. We've been through this before, whereupon Gene
> defined "block" for convex PB and "semiblock" for something "not
too
> concave" or something, if my vague recollection is reliable.

But we never settled anything, did we?

🔗Paul Erlich <perlich@aya.yale.edu>

1/30/2004 1:44:17 PM

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Pretty much. We've been through this before, whereupon Gene
> > defined "block" for convex PB and "semiblock" for something "not
> too
> > concave" or something, if my vague recollection is reliable.
>
> But we never settled anything, did we?

I thought you were solid with your definitions of these. What remains
unsettled?