A question about 5-limit temperaments

Dear intervalologists.

I was examining various 5-limit commas a while ago and I'd like to ask something. Suppose that A=81/80 and B=15625/15552. If we want to find a linear temperament with a period of 2/1 which tempers out A, we go for meantone, and to temper out B, we choose hanson. Then, let's do C=A/B = 78732/78125, which is tempered out in semisixth. And if we go on with D=C/B, we get a PSC (or "monzo", as you call it) of "-8 -14 13" which is only slightly over 5 cents. Is there any way to find a linear temperament where this small D could disappear? Perhaps I could find it out from the generator sizes in meantone, hanson and semisixth somehow? Or do these have no point here?

Thanks in advance.

Petr

Petr Pařízek wrote:
> Dear intervalologists.
> > I was examining various 5-limit commas a while ago and I'd like to ask > something. Suppose that A=81/80 and B=15625/15552. If we want to find a > linear temperament with a period of 2/1 which tempers out A, we go for > meantone, and to temper out B, we choose hanson. Then, let's do C=A/B = > 78732/78125, which is tempered out in semisixth. And if we go on with D=C/B, > we get a PSC (or "monzo", as you call it) of "-8 -14 13" which is only > slightly over 5 cents. Is there any way to find a linear temperament where > this small D could disappear? Perhaps I could find it out from the generator > sizes in meantone, hanson and semisixth somehow? Or do these have no point > here?
> > Thanks in advance.

Yes. Here it is with my old odd-limit library:

11/15, 884.7 cent generator

basis:
(1.0, 0.7372805782062406)

mapping by period and generator:
[(1, 0), (-8, 13), (-8, 14)]

mapping by steps:
[(11, 4), (16, 7), (24, 10)]

highest interval width: 14
complexity measure: 14 (15 for smallest MOS)
highest error: 0.000315 (0.378 cents)
unique

It's covered by 19-equal, as I suppose you could have predicted.

Graham

Graham wrote:

> Yes. Here it is with my old odd-limit library:

What library?

> [(1, 0), (-8, 13), (-8, 14)]

Wow! Whatever way you found it, thanks a lot. I had never thought that there
were even other ways of mapping 4+11 than the familiar hanson.

> It's covered by 19-equal, as I suppose you could have predicted.

Yes, and, sadly, that was the only thing that I was able to find out about
it.

Petr

PS: Do you know of any piece of software that could find linear temperaments
if I specified the period and a comma -- or possibly two commas?

Petr Pařízek wrote:
> Graham wrote:
> >> Yes. Here it is with my old odd-limit library:
> > What library?

The file temper.py in the zip file you get from

http://x31eq.com/temper.html

>> [(1, 0), (-8, 13), (-8, 14)]
> > Wow! Whatever way you found it, thanks a lot. I had never thought that there
> were even other ways of mapping 4+11 than the familiar hanson.
> >> It's covered by 19-equal, as I suppose you could have predicted.
> > Yes, and, sadly, that was the only thing that I was able to find out about
> it.
> > Petr
> > PS: Do you know of any piece of software that could find linear temperaments
> if I specified the period and a comma -- or possibly two commas?

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

Graham

Hi Petr,

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:

> PS: Do you know of any piece of software that could find
> linear temperaments if I specified the period and a comma
> -- or possibly two commas?

Tonescape will do it, and once you've created the tuning,
then you can use Tonescape to compose pieces in that tuning.

Tonescape can create tunings which have up to 7 generators,
i.e., are 7-dimensional.

If you use prime-factors as the generators, leaving it as
it is gives you JI with however many prime-factors you use,
up to 7.

Or you can temper out any or all of the commas, leaving
you with whatever type of temperament comes out of the process.

Or, you can use any given cent-sizes as your generators.
And you can mix prime-factors with non-prime generators
if you wish.

As a simple and easily understandable example:

* If you choose to have an identity-interval, and you
make it prime-factor 2, and use a 2-dimensional tonespace
with prime-factors 3 and 5, then you get 5-limit JI;

* If you temper out the 81/80, you get a theoretically
open-ended meantone chain, which you can then choose to
view as a helix in the Lattice window (or you may still
choose to view the rectangular or triangular JI lattice).

* If you temper out both the 81/80 and the comma you
chose (but your numbers seem to have the signs reversed),
[8 14, -13>, you get a temperament with 38 notes, which
apparently means that it is a 19-edo tempering of 3-5-space
which has torsion.

I've uploaded 4 screenshots of the Tonescape .space files
for this tuning:

* 3-5-space,uv=[8,14,-13]_rectangular.gif
* 3-5-space,uv=[8,14,-13]_closed-helix.gif
* 3-5-space,uvs=[8,14,-13],[-4,4,-1]_rectangular.gif
* 3-5-space,uvs=[8,14,-13],[-4,4,-1]_closed-3-loop.gif

This last one looks really interesting. Note that you
don't get the donut-shaped torus as in many other EDOs.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:

> I've uploaded 4 screenshots of the Tonescape .space files
> for this tuning:
>
> * 3-5-space,uv=[8,14,-13]_rectangular.gif
> * 3-5-space,uv=[8,14,-13]_closed-helix.gif
> * 3-5-space,uvs=[8,14,-13],[-4,4,-1]_rectangular.gif
> * 3-5-space,uvs=[8,14,-13],[-4,4,-1]_closed-3-loop.gif

Oops, my bad ... i forgot to post the URL:

/tuning/files/monz/3-5-
space,uv=[8,14,-13]

(delete the line-break in the URL, if there is one)

That takes you to the folder which contains all four graphics.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

Graham wrote:

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

Do I understand it right that it can only find 5-limit temperaments? It would help me a lot if I could also explore 7-limit temperaments this way. The other day, I was trying to manually find one that could temper out 2401/2400 and, for the time being, I'm still not sure how to do that.

petr

Why do you want to temper out this interval if you are not finding a
scale situation why you would want/need or be advantageous to do so?>
i am just curious how you got there

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
>
> Graham wrote:
>
> > http://x31eq.com/temper/vectors.html
>
> Do I understand it right that it can only find 5-limit temperaments?
It would help me a lot if I could also explore 7-limit temperaments
this way. The other day, I was trying to manually find one that could
temper out 2401/2400 and, for the time being, I'm still not sure how
to do that.
>
> petr
>

Petr Pařízek wrote:

> _> http://x31eq.com/temper/vectors.html_
> > Do I understand it right that it can only find 5-limit temperaments? It would > help me a lot if I could also explore 7-limit temperaments this way. The other > day, I was trying to manually find one that could temper out 2401/2400 and, for > the time being, I'm still not sure how to do that.

It can find rank 2 temperaments up to some reasonable limit (19?) provided you feed in the right ratios. It's quite forgiving about the format.

Graham

Graham wrote:

> It's quite forgiving about the format.

Then I must be doing something wrong. I tried 2401/2400 and it didn't show any results. Having thought it can only handle 2-dimmensional prime space, I tried 1029/1024 (which I think should work with a generator three times smaller than a fifth) and, again, no results were shown.

Petr

Kraig wrote:

> Why do you want to temper out this interval if you are not
> finding a scale situation why you would want/need or be
> advantageous to do so?

It's simply another of the many ways to find linear temperaments. If you choose to deliberately temper out a comma which is fairly small, then you can, of course, get a temperament that works really well. Supposes you want to find a linear temperament with a period of 2/1 where the 5-limit kleisma of 15625/15552 is tempered out. This gives you the regular hanson with the minor third as the generator. As the kleisma is only about 8 cents large, tempering it out makes the intervals only very slightly mistuned.

Petr

Petr Pařízek wrote:

> Then I must be doing something wrong. I tried 2401/2400 and it didn't show any > results. Having thought it can only handle 2-dimmensional prime space, I tried > 1029/1024 (which I think should work with a generator three times smaller than a > fifth) and, again, no results were shown.

Cut and paste that paragraph into the input box and it'll return miracle temperament for you.

Graham

Thanks.
This is quite in keeping with how many of the many of the Mt Meru
recurrent sequences are generated. --- In tuning@yahoogroups.com, Petr
Parízek <p.parizek@...> wrote:
>
> Kraig wrote:
>
> > Why do you want to temper out this interval if you are not
> > finding a scale situation why you would want/need or be
> > advantageous to do so?
>
> It's simply another of the many ways to find linear temperaments. If
you choose to deliberately temper out a comma which is fairly small,
then you can, of course, get a temperament that works really well.
Supposes you want to find a linear temperament with a period of 2/1
where the 5-limit kleisma of 15625/15552 is tempered out. This gives
you the regular hanson with the minor third as the generator. As the
kleisma is only about 8 cents large, tempering it out makes the
intervals only very slightly mistuned.
>
> Petr
>