12-equal/12-wt is the only possible temperament eliminating the syntonic, pythagorean, and septimal commas?

What the title says. Just making sure I have this right.

So is it that the only possible temperament eliminating 81/80,
531441/524288, and 64/63 is 12-equal, or just some 12-note subset in
general (possibly a well-temperament)?

Furthermore, from what I'm gathering, eliminating any one of the above
intervals leads to a few possible rank-2 temperaments:

Dropping the vanishing 64/63 yields meantone temperament
Dropping the vanishing 81/80 yields pajara temperament
Dropping the vanishing 531441/524288 yields dominant temperament

Do I have this right here? This is after like a 4 hour long marathon
reading and number crunching session here, and I want to make sure I'm
on the right track. It'd be awfully nice to finally have figured out
what's going on.

Thanks,
Mike

Bah, the title got cut off. The full title was supposed to be:

"12-equal/12-wt is the only possible temperament eliminating the syntonic, pythagorean, and septimal commas?"

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> What the title says. Just making sure I have this right.
>
> So is it that the only possible temperament eliminating 81/80,
> 531441/524288, and 64/63 is 12-equal, or just some 12-note subset in
> general (possibly a well-temperament)?
>
> Furthermore, from what I'm gathering, eliminating any one of the above
> intervals leads to a few possible rank-2 temperaments:
>
> Dropping the vanishing 64/63 yields meantone temperament
> Dropping the vanishing 81/80 yields pajara temperament
> Dropping the vanishing 531441/524288 yields dominant temperament
>
> Do I have this right here? This is after like a 4 hour long marathon
> reading and number crunching session here, and I want to make sure I'm
> on the right track. It'd be awfully nice to finally have figured out
> what's going on.
>
> Thanks,
> Mike
>

Hi Mike.

Well, you've just said it all. And you're right. If you mix meantone and dominant, you get 12-equal. And tempering out the Pyth. comma results in 12-equal just by itself. I actually don't think I could add much to that.

Petr

>"Furthermore, from what I'm gathering, eliminating any one of the above
intervals leads to a few possible rank-2 temperaments:

Dropping the vanishing 64/63 yields meantone temperament
Dropping the vanishing 81/80 yields pajara temperament
Dropping the vanishing 531441/524288 yields dominant temperament"

Good grief...IMVHO, this seems to say despite the plethora of discussion in the past about the vast superiority (or inferiority) between these few and several theories how to generate them...and they turn out to be very very similar. ...And, if I have it right, ultimately related to comma-tempered versions of the older-than-dirt Pythagorean tuning.
Any reason(s) why they wouldn't be...or why we should go on for ages debating which of the above is better?

Well, wouldn't tempering out the Pyth. comma lead to a whole series of
12-equals?

Like, in 3-limit space (a line), tempering out the pyth. comma would lead to
12-equal.

Then, in 5-limit space (a plane), tempering out the pyth. comma would lead
to a bunch of 12-equals, all separated by powers of 5... And then in that
same 5-limit space, tempering out the pyth. comma AND 81/80 would lead to a
single ring of 12-equals again.

And then if you extend meantone to 7-limit space, you get another set of
12-equals, separated by powers of 4... then making 7/4 and 16/9 a unison
would collapse the whole thing again back to 12-equal.

I think... :)

-Mike

On Tue, Feb 9, 2010 at 6:25 AM, Petr Parízek <p.parizek@...> wrote:

>
>
> Hi Mike.
>
> Well, you've just said it all. And you're right. If you mix meantone and
> dominant, you get 12-equal. And tempering out the Pyth. comma results in
> 12-equal just by itself. I actually don't think I could add much to that.
>
> Petr
>
>
>
>

how much is the license to use Battaglia tuning?

On Tue, Feb 9, 2010 at 3:21 PM, Mike Battaglia <battaglia01@...>wrote:

>
>
> Well, wouldn't tempering out the Pyth. comma lead to a whole series of
> 12-equals?
>
> Like, in 3-limit space (a line), tempering out the pyth. comma would lead
> to 12-equal.
>
> Then, in 5-limit space (a plane), tempering out the pyth. comma would lead
> to a bunch of 12-equals, all separated by powers of 5... And then in that
> same 5-limit space, tempering out the pyth. comma AND 81/80 would lead to a
> single ring of 12-equals again.
>
> And then if you extend meantone to 7-limit space, you get another set of
> 12-equals, separated by powers of 4... then making 7/4 and 16/9 a unison
> would collapse the whole thing again back to 12-equal.
>
> I think... :)
>
> -Mike
>
>
> On Tue, Feb 9, 2010 at 6:25 AM, Petr Parízek <p.parizek@...> wrote:
>
>>
>>
>> Hi Mike.
>>
>> Well, you've just said it all. And you're right. If you mix meantone and
>> dominant, you get 12-equal. And tempering out the Pyth. comma results in
>> 12-equal just by itself. I actually don't think I could add much to that.
>>
>> Petr
>>
>>
>>
>
>
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> What the title says. Just making sure I have this right.
[...]
> Dropping the vanishing 64/63 yields meantone temperament
> Dropping the vanishing 81/80 yields pajara temperament
> Dropping the vanishing 531441/524288 yields dominant temperament

You're combining commas that live in different spaces.
64/64 is 7-limit, 81/80 is 5-limit, and the Pythag. comma is
3-limit. As you probably know, in the 3-limit we only need a
single comma to get an rank 1 temperament, and the Pythagorean
comma gives us 12 (and its multiples up to 300, though these
are torsional systems since they contain many independent
chains of fifths).

In the 5-limit, we need two commas to get an ET. There are
an infinite number of comma pairs that will give the same ET.
This is where some of the finer points of regular temperament
theory, contributed mostly by Gene and not fully understood
by me, come into play. Basically, he showed how we can find
the pair where the commas are simplest, using something called
Tenney-Minkowski reduction.

http://tonalsoft.com/enc/t/tm-basis.aspx

As shown here, the 5-limit TM basis for 12-ET is
{81/80, 128/125}. (And again, multiples of 12 up to 36.)

In the 7-limit we need three commas, and you can get 12 from
{36/35, 50/49, 64/63}, {81/80, 126/125, 36/35}, etc. I'm
not sure which is the TM-reduced basis.

-Carl

You're right. Although I was talking to Paul on my facebook and asked him
the same question -- whether tempering out the commas mentioned in the OP
also would yield 24-tet, 36-tet, etc -- and he said no, since picking those
as unison vectors yields a 12-note PB. I think what it is is that in 24-tet
64/63 would no longer count as vanishing, since 7/4 would map to the 950
cent interval, and 16/9 would map to the 1000 cent interval. I could be
wrong though.

What I'm interested in is finding the temperament that eliminates the
difference between 64/63, 81/80, and 531441/524288 (and hence the other
intervals you mentioned), and turns them all into a single "comma." If the
math works, that might prove to be very useful for educational (and
conceptual) purposes.

-Mike

On Tue, Feb 9, 2010 at 4:45 PM, Carl Lumma <carl@lumma.org> wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> <battaglia01@...> wrote:
> >
> > What the title says. Just making sure I have this right.
> [...]
>
> > Dropping the vanishing 64/63 yields meantone temperament
> > Dropping the vanishing 81/80 yields pajara temperament
> > Dropping the vanishing 531441/524288 yields dominant temperament
>
> You're combining commas that live in different spaces.
> 64/64 is 7-limit, 81/80 is 5-limit, and the Pythag. comma is
> 3-limit. As you probably know, in the 3-limit we only need a
> single comma to get an rank 1 temperament, and the Pythagorean
> comma gives us 12 (and its multiples up to 300, though these
> are torsional systems since they contain many independent
> chains of fifths).
>
> In the 5-limit, we need two commas to get an ET. There are
> an infinite number of comma pairs that will give the same ET.
> This is where some of the finer points of regular temperament
> theory, contributed mostly by Gene and not fully understood
> by me, come into play. Basically, he showed how we can find
> the pair where the commas are simplest, using something called
> Tenney-Minkowski reduction.
>
> http://tonalsoft.com/enc/t/tm-basis.aspx
>
> As shown here, the 5-limit TM basis for 12-ET is
> {81/80, 128/125}. (And again, multiples of 12 up to 36.)
>
> In the 7-limit we need three commas, and you can get 12 from
> {36/35, 50/49, 64/63}, {81/80, 126/125, 36/35}, etc. I'm
> not sure which is the TM-reduced basis.
>
> -Carl
>
>
>

Mike Battaglia wrote:
> What the title says. Just making sure I have this right.
> > So is it that the only possible temperament eliminating 81/80,
> 531441/524288, and 64/63 is 12-equal, or just some 12-note subset in
> general (possibly a well-temperament)?

Up to the 7-limit, that sounds right.

> Furthermore, from what I'm gathering, eliminating any one of the above
> intervals leads to a few possible rank-2 temperaments:
> > Dropping the vanishing 64/63 yields meantone temperament
> Dropping the vanishing 81/80 yields pajara temperament
> Dropping the vanishing 531441/524288 yields dominant temperament
> > Do I have this right here? This is after like a 4 hour long marathon
> reading and number crunching session here, and I want to make sure I'm
> on the right track. It'd be awfully nice to finally have figured out
> what's going on.

Meantone only tempers out 81/80; the 7-limit version of meantone tempers out 126/125, 225/224, etc. but not 531441/524288. From the tuning-math list we have the very useful tool of the "wedge product", which lets us identify a temperament from any combination of commas. In the case of 81/80 and 531441/524288, we end up with 12-ET, not meantone. More precisely, the temperament that tempers out 81/80 and 531441/524288 is <<12, 19, 28]], which is the usual 5-limit mapping of 12-ET. Meantone is <<1, 4, 4]] (5-limit) or <<1, 4, 10, 4, 13, 12]] (7-limit).

But if you do a 7-limit version of the wedge product you end up with <<0, 0, 12, 0, 19, 28]], which is a rank 2 temperament, identified in Paul Erlich's "Middle Path" paper as catler (presumably named after Jon Catler). You can think of it as two (or more) copies of 12-ET offset by a small amount (around 25 cents).

Similarly, wedging 64/63 with 531441/524288 gives you another rank-2 variation of 12-ET, <<0, 12, 0, 19, 0 -34]] (I don't know a name for this one).

But you're correct about dominant temperament from 81/80 and 64/63.

On 10 February 2010 03:52, Mike Battaglia <battaglia01@...> wrote:
>
>
> You're right. Although I was talking to Paul on my facebook
> and asked him the same question -- whether tempering out
> the commas mentioned in the OP also would yield 24-tet,
> 36-tet, etc -- and he said no, since picking those as unison
> vectors yields a 12-note PB. I think what it is is that in 24-tet
> 64/63 would no longer count as vanishing, since 7/4 would
> map to the 950 cent interval, and 16/9 would map to the
> 1000 cent interval. I could be wrong though.

It depends on your perspective. 24-tet tempers out those
commas. The algorithms are usually defined not to give
these "contorted" temperaments. They're often defined to
remove "torsion" on the input as well. If you gave unison
vectors for a 24 note periodicity block, you might find they
still yield 12-tet, because that's probably what you wanted.

> What I'm interested in is finding the temperament that
> eliminates the difference between 64/63, 81/80, and
> 531441/524288 (and hence the other intervals you
> mentioned), and turns them all into a single "comma."
> If the math works, that might prove to be very useful for
> educational (and conceptual) purposes.

So calculate the ratios corresponding to the "difference"
and put them into my temperament finder.

Graham

> Meantone only tempers out 81/80; the 7-limit version of meantone tempers
> out 126/125, 225/224, etc. but not 531441/524288.

Whoa, you're entirely right. Not sure how I missed that.

> From the tuning-math
> list we have the very useful tool of the "wedge product", which lets us
> identify a temperament from any combination of commas. In the case of
> 81/80 and 531441/524288, we end up with 12-ET, not meantone. More
> precisely, the temperament that tempers out 81/80 and 531441/524288 is
> <<12, 19, 28]], which is the usual 5-limit mapping of 12-ET. Meantone is
> <<1, 4, 4]] (5-limit) or <<1, 4, 10, 4, 13, 12]] (7-limit).
>
> But if you do a 7-limit version of the wedge product you end up with
> <<0, 0, 12, 0, 19, 28]], which is a rank 2 temperament, identified in
> Paul Erlich's "Middle Path" paper as catler (presumably named after Jon
> Catler). You can think of it as two (or more) copies of 12-ET offset by
> a small amount (around 25 cents).
>
> Similarly, wedging 64/63 with 531441/524288 gives you another rank-2
> variation of 12-ET, <<0, 12, 0, 19, 0 -34]] (I don't know a name for
> this one).

Thanks for the info - I'll have to do some reading on this. Never
understood the concept of the wedge product before.

-Mike

> It depends on your perspective. 24-tet tempers out those
> commas. The algorithms are usually defined not to give
> these "contorted" temperaments. They're often defined to
> remove "torsion" on the input as well. If you gave unison
> vectors for a 24 note periodicity block, you might find they
> still yield 12-tet, because that's probably what you wanted.

Ah, so that's what this mysterious torsion is I've been hearing about lately.

> So calculate the ratios corresponding to the "difference"
> and put them into my temperament finder.

I wasn't aware that there was a temperament finder. Sounds useful.

-Mike

> Graham
>

On 13 February 2010 14:27, Mike Battaglia <battaglia01@...> wrote:

> I wasn't aware that there was a temperament finder. Sounds useful.

It got mentioned on a recent thread (as recently as my e-mail backlog,
anyway). Here it is:

http://x31eq.com/temper/vectors.html

Put in your ratios and away you go!

Graham