Family commas

One way to sort out the family relationships is to use a comma scheme
which I have intended for some time to discuss, but which I may have
neglected to do. This is defining a linear temperament in terms of a
sequence of commas, each at a succesively higher prime limit, and each
with a minimal Tenney height given that all the previous commas are
fixed. This sort of whatzit reduction, for meantone, would say
meantone is the 81/80-temperament, dominant sevenths the [81/80,
36/35] temperament, septimal meantone the [81/80,
126/125]-temperament, flattone the [81/80, 525/512]-temperament. Then
11-limit meantone is the [81/80, 126/125, 385/384]-temperament and
huygens the [81/80, 126/125, 99/98]-temperament. And so forth.

It should be noted that while this keeps track of the familial
relationships, we don't necessarily get corresponding generators in
these family trees, nor do we necessarily get rid of contorsion.
7-limit ennealimmal is the [ennealimma, breedsma]-temperament, but the
wedge product of this has a common factor of 4. Hemiennealimmal is
then the [ennealimma, breedsma, lehmerisma]-temperament, again with a
common factor of 4, but now with non-corresponding generators.

> ... defining a linear temperament in terms of a
> sequence of commas, each at a succesively higher prime limit,
> and each with a minimal Tenney height given that all the
> previous commas are fixed. This sort of whatzit reduction,
> for meantone, would say meantone is the 81/80-temperament,
> dominant sevenths the [81/80, 36/35] temperament, septimal
> meantone the [81/80, 126/125]-temperament, flattone the
> [81/80, 525/512]-temperament. Then 11-limit meantone is
> the [81/80, 126/125, 385/384]-temperament and huygens the
> [81/80, 126/125, 99/98]-temperament. And so forth.

This immediately appeals to me more than generators.

I wonder how it relates to Paul's tratios. They involve
the LCM... I wonder what good that is.

Also, when you say "given that all the previous commas are
fixed", does this imply any relation to TM-reduction?

The words that are coming to mind are, temperament n should
be considered an extension of temperament m if m's TM-reduced
basis is a subset of n's. Does that make any sense?

> It should be noted that while this keeps track of the familial
> relationships, we don't necessarily get corresponding
> generators in these family trees, nor do we necessarily get
> rid of contorsion. 7-limit ennealimmal is the [ennealimma,
> breedsma]- temperament, but the wedge product of this has a
> common factor of 4.

Rats.

-Carl

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

> Also, when you say "given that all the previous commas are
> fixed", does this imply any relation to TM-reduction?

It's a different reduction--sequential reduction or something.

> The words that are coming to mind are, temperament n should
> be considered an extension of temperament m if m's TM-reduced
> basis is a subset of n's. Does that make any sense?

This won't work. It doesn't even work if you replace TM reduction with
sequential reduction, though that is better for this. The nexial
approach does it, however.

>> Also, when you say "given that all the previous commas are
>> fixed", does this imply any relation to TM-reduction?
>
>It's a different reduction--sequential reduction or something.
>
>> The words that are coming to mind are, temperament n should
>> be considered an extension of temperament m if m's TM-reduced
>> basis is a subset of n's. Does that make any sense?
>
>This won't work. It doesn't even work if you replace TM reduction with
>sequential reduction, though that is better for this. The nexial
>approach does it, however.

Noted.

>> >> >> This family stuff looks awesome. I wish I understood the
>> >> >> half of it. I'm surprised you're using generator sizes.
>> >> >> How do you standardize the generator representation? Forgive
>> >> >> me if this is old stuff, I haven't kept up.
>> >> >
>> >> >It's just the TOP tuning for the generators.
>> >>
>> >> How do you get a unique set of generators out of the TOP
>> >> tuning?
>> >
>> >One way is to apply the TOP tuning to a rational number generator
>> >which works as a reduced generator. For instance, with meantone that
>> >would be 4/3, with miracle 15/14 or 16/15,
>>
>> This is apparently not giving unique generators...
>
>Sure it does; miracle(15/14) = [0, 1] = miracle(16/15)

Right you are.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> One way to sort out the family relationships is to use a comma
scheme
> which I have intended for some time to discuss, but which I may have
> neglected to do. This is defining a linear temperament in terms of a
> sequence of commas, each at a succesively higher prime limit, and
each
> with a minimal Tenney height given that all the previous commas are
> fixed. This sort of whatzit reduction, for meantone, would say
> meantone is the 81/80-temperament, dominant sevenths the [81/80,
> 36/35] temperament, septimal meantone the [81/80,
> 126/125]-temperament, flattone the [81/80, 525/512]-temperament.
Then
> 11-limit meantone is the [81/80, 126/125, 385/384]-temperament and
> huygens the [81/80, 126/125, 99/98]-temperament. And so forth.
>
> It should be noted that while this keeps track of the familial
> relationships, we don't necessarily get corresponding generators in
> these family trees, nor do we necessarily get rid of contorsion.

How is it possible to get contorsion from commas? I don't think it is.

> 7-limit ennealimmal is the [ennealimma, breedsma]-temperament, but
the
> wedge product of this has a common factor of 4.

But this is torsion, not contorsion, right?

> Hemiennealimmal is
> then the [ennealimma, breedsma, lehmerisma]-temperament, again with
a
> common factor of 4, but now with non-corresponding generators.

Don't know what "non-corresponding" means.

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

> How is it possible to get contorsion from commas? I don't think it is.

Interpreting the GCD > 1 to mean contorsion rather than torsion seems
to make the most sense if you are considering it as representing a
temperament. You get torsion out of 648/625 and 2048/2025, but you
could also count it as contorsion by taking the <24 38 56| val
literally, as 24-equal, being contorted 12-equal, so that the mapping
of the 5-limit is not surjective. Otherwise you are looking at a rank
two group [Z, Z/2Z] as the image, which doesn't make a lot of sense
musically.

> > Hemiennealimmal is
> > then the [ennealimma, breedsma, lehmerisma]-temperament, again with
> a
> > common factor of 4, but now with non-corresponding generators.
>
> Don't know what "non-corresponding" means.

They are not approxmiately the same, nor are they mapped from the same
JI intervals.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > How is it possible to get contorsion from commas? I don't think
it is.
>
> Interpreting the GCD > 1 to mean contorsion rather than torsion
seems
> to make the most sense if you are considering it as representing a
> temperament. You get torsion out of 648/625 and 2048/2025, but you
> could also count it as contorsion by taking the <24 38 56| val
> literally, as 24-equal, being contorted 12-equal, so that the
mapping
> of the 5-limit is not surjective.

Yes, exactly! But this is not coming from commas, it's coming from a
val!! You seem to have mentally ignored that we were talking about
commas and somehow mentally substituted wedgies or something without
saying so. Or . . . (??)

> > > Hemiennealimmal is
> > > then the [ennealimma, breedsma, lehmerisma]-temperament, again
with
> > a
> > > common factor of 4, but now with non-corresponding generators.
> >
> > Don't know what "non-corresponding" means.
>
> They are not approxmiately the same, nor are they mapped from the
same
> JI intervals.

Hmm . . . can you elaborate on this with more detail, please?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > How is it possible to get contorsion from commas? I don't think
> it is.
> >
> > Interpreting the GCD > 1 to mean contorsion rather than torsion
> seems
> > to make the most sense if you are considering it as representing a
> > temperament. You get torsion out of 648/625 and 2048/2025, but you
> > could also count it as contorsion by taking the <24 38 56| val
> > literally, as 24-equal, being contorted 12-equal, so that the
> mapping
> > of the 5-limit is not surjective.
>
> Yes, exactly! But this is not coming from commas, it's coming from a
> val!!

And I got the val from the commas.

> > > > Hemiennealimmal is
> > > > then the [ennealimma, breedsma, lehmerisma]-temperament, again
> with
> > > a
> > > > common factor of 4, but now with non-corresponding generators.
> > >
> > > Don't know what "non-corresponding" means.
> >
> > They are not approxmiately the same, nor are they mapped from the
> same
> > JI intervals.
>
> Hmm . . . can you elaborate on this with more detail, please?

Ennealimmal has TOP generators [133.337, 49.024], which correspond to
27/25 and 36/35. Hemiennealimmal has TOP generators [66.669, 17.645],
which obviously differ, though 66.669-17.645 = 49.024, which we are
free to use for the second generator. The first generator, however, is
obviously not an approximate 27/25; we may now take the generators as
approximations of 80/77 and 99/98.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> > >
> > > > How is it possible to get contorsion from commas? I don't
think
> > it is.
> > >
> > > Interpreting the GCD > 1 to mean contorsion rather than torsion
> > seems
> > > to make the most sense if you are considering it as
representing a
> > > temperament. You get torsion out of 648/625 and 2048/2025, but
you
> > > could also count it as contorsion by taking the <24 38 56| val
> > > literally, as 24-equal, being contorted 12-equal, so that the
> > mapping
> > > of the 5-limit is not surjective.
> >
> > Yes, exactly! But this is not coming from commas, it's coming
from a
> > val!!
>
> And I got the val from the commas.

But if you go directly from the commas, it's unequivocally torsion
and not contorsion. Remember, soon after your joining, that Fokker
periodicity block, that you initially said represented 24-equal, but
then you retracted that and said it was really 12-equal with torsion?

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

> But if you go directly from the commas, it's unequivocally torsion
> and not contorsion. Remember, soon after your joining, that Fokker
> periodicity block, that you initially said represented 24-equal, but
> then you retracted that and said it was really 12-equal with torsion?

Which is why I picked this example. If we are thinking Fokker blocks,
it is torsion; however why do we have to be thinking Fokker blocks?
Admittedly, the contorsion interpretation may be a little articifical,
but there's nothing else for it to be, basically.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > But if you go directly from the commas, it's unequivocally
torsion
> > and not contorsion. Remember, soon after your joining, that
Fokker
> > periodicity block, that you initially said represented 24-equal,
but
> > then you retracted that and said it was really 12-equal with
torsion?
>
> Which is why I picked this example. If we are thinking Fokker
blocks,
> it is torsion; however why do we have to be thinking Fokker blocks?

Who said anything about Fokker blocks? If you start with JI (as true
temperaments do), and then temper out the two commas in question, you
end up with 12-equal, not 24-equal. It's that simple.

> Admittedly, the contorsion interpretation may be a little
articifical,
> but there's nothing else for it to be, basically.

??