New section on 7-limit temperaments

I'm still working on the introduction to regular temperaments. I've added a section on 7-limit temperaments, which describes 7-limit wedgies and includes a brief mention of rank 3 (formerly known as "planar") temperaments, using starling as an example.

http://www.io.com/~hmiller/music/regular-temperaments.html

Whatever problems Yahoogroups was having seem to be fixed now, at least for the moment. Still, it wouldn't hurt to cc any comments to me in order to make sure that I get them.

I think I'm going to need to go back over this page once I've written the technical stuff and put in more examples of how and why this stuff is actually useful for music. I can just imagine someone seeing this for the first time and thinking it's a lot of technical nonsense without any practical use. But as far as the technical details, please take a look at the page and let me know if I've made any obvious mistakes, or if any part of the description isn't as clear as it should be.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> But as far as the technical details, please take a look
> at the page and let me know if I've made any obvious mistakes, or if
any
> part of the description isn't as clear as it should be.

Good stuff. However, I'd make a few changes.

(1) I don't think anything should be called a *Fokker* periodicity
block unless it is a paralleopiped block, which is what Fokker
discussed. And I don't think anything should be called a periodicity
block unless it is
convex.

(2) I'd say the same thing for starling as you said for meantone; the
generators of an octave, major third, and minor third "can be used".
For both meantone and starling, I think it is good to know that {2,3}
(for meantone) and {2,3,5} (for starling) can be used also. That is,
meantone is a tempered 3-limit, and starling a tempered 5-limit.

(3) I don't agree that calculating a wedge product of three vals is
more trouble than its worth. For one thing, this suggests its a lot of
trouble, which isn't true. You can write a triple wedge product
routine using the same method as your double wedge product routine:

wedgiesize = 0;
for (i = 0; i < valsize - 2; i++)
{
for (j = i + 1; j < valsize - 1; j++)
{
for (k = j + 1; j < valsize; k++)
{
wedgie[wedgiesize] = val1[i] * val2[j] *val3[k] +
val1[j] * val2[k] * val3[i] + val1[k] * val2[i] * val3[j] -
val1[j] * val2[i] * val3[k] - val1[k] * val2[j] * val3[i] -
val1[i] * val2[k] * val3[j]
wedgiesize++;
} } }

The other problem is that it suggests a triple wedge product doesn't
have much use. However, if for example you want to know what 7-limit
temperament you get from the standard vals for 12, 27 and 46, the
easist method is to take a triple wedge product. They also allow you
to uniquely identify and calculate a complexity for rank three
temperaments in any prime limit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> (2) I'd say the same thing for starling as you said for meantone; the
> generators of an octave, major third, and minor third "can be used".

Where does he say this about meantone?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > (2) I'd say the same thing for starling as you said for meantone; the
> > generators of an octave, major third, and minor third "can be used".
>
> Where does he say this about meantone?

On

http://www.io.com/~hmiller/music/regular-temperaments.html

we find "In the case of meantone, where the generator can be defined
as the meantone approximation of a perfect fourth (4:3)".

Gene Ward Smith wrote:
> Good stuff. However, I'd make a few changes.
> > (1) I don't think anything should be called a *Fokker* periodicity
> block unless it is a paralleopiped block, which is what Fokker
> discussed. And I don't think anything should be called a periodicity
> block unless it is > convex.

It's been a while since I read Paul's page, but I don't recall periodicity blocks needing to be convex. I'll have to check on that. I can change this to just "periodicity blocks".

> (2) I'd say the same thing for starling as you said for meantone; the
> generators of an octave, major third, and minor third "can be used".
> For both meantone and starling, I think it is good to know that {2,3}
> (for meantone) and {2,3,5} (for starling) can be used also. That is,
> meantone is a tempered 3-limit, and starling a tempered 5-limit.

How about something like:

"starling temperament", which uses generators of a minor third, a major third, and an octave (as one of many possible mappings)

I could also mention that TOP starling is <1199.01, 1900.39, 2788.61, 3366.05].

> (3) I don't agree that calculating a wedge product of three vals is
> more trouble than its worth. For one thing, this suggests its a lot of
> trouble, which isn't true. You can write a triple wedge product
> routine using the same method as your double wedge product routine:
> > wedgiesize = 0; > for (i = 0; i < valsize - 2; i++) > { > for (j = i + 1; j < valsize - 1; j++) > { > for (k = j + 1; j < valsize; k++) > { > wedgie[wedgiesize] = val1[i] * val2[j] *val3[k] +
> val1[j] * val2[k] * val3[i] + val1[k] * val2[i] * val3[j] -
> val1[j] * val2[i] * val3[k] - val1[k] * val2[j] * val3[i] -
> val1[i] * val2[k] * val3[j]
> wedgiesize++; > } } }
> > The other problem is that it suggests a triple wedge product doesn't
> have much use. However, if for example you want to know what 7-limit
> temperament you get from the standard vals for 12, 27 and 46, the
> easist method is to take a triple wedge product. They also allow you
> to uniquely identify and calculate a complexity for rank three
> temperaments in any prime limit.

I guess I just haven't been aware of the uses for triple wedge products; I've been mainly concerned with rank 2 temperaments. Thanks for the comments; I'll get the changes in the next update.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> It's been a while since I read Paul's page, but I don't recall
> periodicity blocks needing to be convex.

His examples are convex, and I don't think it makes much sense to use
nonconvex regions.

> How about something like:
>
> "starling temperament", which uses generators of a minor third, a major
> third, and an octave (as one of many possible mappings)

What about simply saying "can use generators of a minor third, a major
third, and an octave"?

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>>How about something like:
>>
>>"starling temperament", which uses generators of a minor third, a major >>third, and an octave (as one of many possible mappings)
> > > What about simply saying "can use generators of a minor third, a major
> third, and an octave"?

I think that sounds awkward in the context (introducing starling temperament to readers who probably are unfamiliar with it). How about:

"starling temperament", which can be defined using generators of a minor third, a major third, and an octave

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> "starling temperament", which can be defined using generators of a
minor
> third, a major third, and an octave

Sounds good, especially with a "for instance" tuning.

>> It's been a while since I read Paul's page, but I don't recall
>> periodicity blocks needing to be convex.
>
>His examples are convex, and I don't think it makes much sense to use
>nonconvex regions.

IIRC he mentions that any note in a block can be transposed by a
unison vector to get a non-convex block.

-Carl

At 08:49 PM 1/12/2006, you wrote:
>I'm still working on the introduction to regular temperaments. I've
>added a section on 7-limit temperaments, which describes 7-limit wedgies
>and includes a brief mention of rank 3 (formerly known as "planar")
>temperaments, using starling as an example.
>
>http://www.io.com/~hmiller/music/regular-temperaments.html

This is excellent, Herman. Thanks.

-Carl

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> "starling temperament", which uses generators of a minor third, a
major
> third, and an octave (as one of many possible mappings)

As one of many possible *bases*.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > (2) I'd say the same thing for starling as you said for
meantone; the
> > > generators of an octave, major third, and minor third "can be
used".
> >
> > Where does he say this about meantone?
>
> On
>
> http://www.io.com/~hmiller/music/regular-temperaments.html
>
> we find "In the case of meantone, where the generator can be defined
> as the meantone approximation of a perfect fourth (4:3)".

That's different from saying "the generators of an octave, major
third, and minor third can be used!" What am I missing?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >> It's been a while since I read Paul's page, but I don't recall
> >> periodicity blocks needing to be convex.
> >
> >His examples are convex, and I don't think it makes much sense to use
> >nonconvex regions.
>
> IIRC he mentions that any note in a block can be transposed by a
> unison vector to get a non-convex block.

Never said or implied the resulting block would be non-convex; in the
examples I give for this, it's still convex (in fact it's "even more
convex" in that it's closer to a circle) . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> > It's been a while since I read Paul's page, but I don't recall
> > periodicity blocks needing to be convex.
>
> His examples are convex, and I don't think it makes much sense to use
> nonconvex regions.

The melodic minor scale is nonconvex.