Localization

An integral domain is a commutative ring without zero divisors, but
you don't need to know what that means; just language I might use. The
classic example is the integers, and that's what I want to focus on. A
multiplicative set S is a set such that the product of any two
elements in S is in S. If R is an integral domain, and S a
multiplicative set in R, then R_S, "R localized to S", is the set with
numerators elements of R, and denominators elements of S. For example,
if Z is the integers and S is the set of powers of two, then Z_S is
the dyadic rationals, the rational numbers n/2^i.

Now, an interesting factoid about localization is that localizing an
integral domain gives an integral domain. It occurs to me that this
kind of thing is potentially useful. For example, not only is the
product of two dyadic rational numbers dyadic, so is the sum. Scales
such as the dwarf scales, which includes for instance the
Ptolemy/Zarlino diatonic and the Ellis duodene, are comprised of
dyadic rationals so long as we don't transpose them. Hence,
it is tempting to take such a scale and transform it my adding a
dyadic rational, which makes another scale of dyadic rationals. This,
it seems to me, may moderate somewhat the slightly screwy results that
addition in general can lead to.

Some examples: 1-5/4-3/2, adding 1/2, gives 3/2-7/4-2, a 6:7:8 from a
4:5:6. 9/8-3/2-15/8 plus 1/2 goes to 13/8-2-19/8, a 13:16:19 chord. In
such manner we can have fun with Dwarf(<7 11 16|), a transposed
version of Zarlino, 9/8-5/4-45/32-3/2-27/16-15/8-2.

I hope this doesn't freak anyone out but things have been dead here
lately and it seemed like a good time to post such musings.

>Scales such as the dwarf scales, which includes for instance the
>Ptolemy/Zarlino diatonic and the Ellis duodene, are comprised of
>dyadic rationals so long as we don't transpose them.

How could transposing introduce irrational numbers into any
rational scale? (Example?)

>Hence, it is tempting to take such a scale and transform it my [sic]
>adding a dyadic rational, which makes another scale of dyadic
>rationals. This, it seems to me, may moderate somewhat the slightly
>screwy results that addition in general can lead to.
>
>Some examples: 1-5/4-3/2, adding 1/2, gives 3/2-7/4-2, a 6:7:8 from a
>4:5:6. 9/8-3/2-15/8 plus 1/2 goes to 13/8-2-19/8, a 13:16:19 chord.

The fact that it is guaranteed to produce rationals doesn't mean
a whole lot, does it? What else can be said of such transformations
to make them interesting?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >Scales such as the dwarf scales, which includes for instance the
> >Ptolemy/Zarlino diatonic and the Ellis duodene, are comprised of
> >dyadic rationals so long as we don't transpose them.
>
> How could transposing introduce irrational numbers into any
> rational scale? (Example?)

It can't, but it can make dyadic rationals not dyadic.

Here's Dwarf(<7 11 16|):

9/8-5/4-45/32-3/2-27/16-2

Everything in the scale, in any octave, is a dyadic rational interval
from the 1/1. Now multiply by 4/3 and reduce to the octave, and you have

9/8-5/4-4/3-3/2-5/3-15/8-2

A justly famous scale, but now everything is not a dyadic rational
interval from 1/1, but from 4/3.

> The fact that it is guaranteed to produce rationals doesn't mean
> a whole lot, does it? What else can be said of such transformations
> to make them interesting?

The fact that it's guaranteed to produce dyadic rationals only
probably isn't very interesting, but I thought I'd toss this out. You
can similarly limit the rationals you get by adding or substracting a
rational by sticking to other localizations also, but again, the
question is if that gets you anywhere. But for instance if you take
rational numbers with odd denominators, another localization, then
adding them together gets another rational number with an odd
denominator. It's a ring, closed under multiplication *and* addition.

Oh, well, it's been slow around here.

At 01:27 PM 5/7/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> >Scales such as the dwarf scales, which includes for instance the
>> >Ptolemy/Zarlino diatonic and the Ellis duodene, are comprised of
>> >dyadic rationals so long as we don't transpose them.
>>
>> How could transposing introduce irrational numbers into any
>> rational scale? (Example?)
>
>It can't, but it can make dyadic rationals not dyadic.
>
>Here's Dwarf(<7 11 16|):
>
>9/8-5/4-45/32-3/2-27/16-2
>
>Everything in the scale, in any octave, is a dyadic rational interval
>from the 1/1. Now multiply by 4/3 and reduce to the octave, and you have
>
>9/8-5/4-4/3-3/2-5/3-15/8-2
>
>A justly famous scale, but now everything is not a dyadic rational
>interval from 1/1, but from 4/3.

What's a "dyadic" rational?

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 01:27 PM 5/7/2006, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >>
> >> >Scales such as the dwarf scales, which includes for instance the
> >> >Ptolemy/Zarlino diatonic and the Ellis duodene, are comprised of
> >> >dyadic rationals so long as we don't transpose them.
> >>
> >> How could transposing introduce irrational numbers into any
> >> rational scale? (Example?)
> >
> >It can't, but it can make dyadic rationals not dyadic.
> >
> >Here's Dwarf(<7 11 16|):
> >
> >9/8-5/4-45/32-3/2-27/16-2

Gene, should this also include 15/8?

> >Everything in the scale, in any octave, is a dyadic rational
interval
> >from the 1/1. Now multiply by 4/3 and reduce to the octave, and
you have
> >
> >9/8-5/4-4/3-3/2-5/3-15/8-2
> >
> >A justly famous scale, but now everything is not a dyadic rational
> >interval from 1/1, but from 4/3.
>
> What's a "dyadic" rational?

I think this is Z_S or Integer/2^n right?

>
> -C.
>

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

> What's a "dyadic" rational?

There's a whole Wikipedia article on them:

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

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> > >Here's Dwarf(<7 11 16|):
> > >
> > >9/8-5/4-45/32-3/2-27/16-2
>
> Gene, should this also include 15/8?

Yes, sorry.

>>>> >Scales such as the dwarf scales, which includes for instance the
>>>> >Ptolemy/Zarlino diatonic and the Ellis duodene, are comprised of
>>>> >dyadic rationals so long as we don't transpose them.
>>>>
>>>> How could transposing introduce irrational numbers into any
>>>> rational scale? (Example?)
>>>
>>>It can't, but it can make dyadic rationals not dyadic.
>>>
>>>Here's Dwarf(<7 11 16|):
>>>
>>>9/8-5/4-45/32-3/2-27/16-2
>>>
>>>Everything in the scale, in any octave, is a dyadic rational interval
>>>from the 1/1. Now multiply by 4/3 and reduce to the octave, and you have
>>>
>>>9/8-5/4-4/3-3/2-5/3-15/8-2
>>>
>>>A justly famous scale, but now everything is not a dyadic rational
>>>interval from 1/1, but from 4/3.
>>
>>What's a "dyadic" rational?
>
>There's a whole Wikipedia article on them:
>
>http://en.wikipedia.org/wiki/Dyadic_rational

Now I understand.

-Carl