Ragismatic temperament (was Re: [tuning] Re: A tuning-math question -- finding 2D temperaments from 4D ratios)

I wrote:

> I'm only afraid it would take me whole days to get to them because finding
> the lowest multipliers by hand is terribly slow, or at least when I'm > doing
> it -- but I've done it many times already so I'll do it a few times more

I've just tried it with 4375/4374. This is what came out:
Generator = 9/5
4 generators = 21/2

Now this is interesting ... Wouldn't it be better then to eventually map the 3D version of the ragismatic temperament using generators of either 9/5 and 3/1 or 9/5 and 7/2 rather than having the generators themselves approximate 3/1 and 5/1? Wouldn't the prime approximations then need fewer generators?

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:

> Now this is interesting ... Wouldn't it be better then to eventually map the
> 3D version of the ragismatic temperament using generators of either 9/5 and
> 3/1 or 9/5 and 7/2 rather than having the generators themselves approximate
> 3/1 and 5/1? Wouldn't the prime approximations then need fewer generators?

The lowest-complexity basis for the lattice of pitch classes would be
[9/5, 5/3], or [10/9, 6/5] if you prefer. That tells us the most efficient way of representing intervals in ragismic temperament is in terms of 2, 9/5 and 5/3.

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

> The lowest-complexity basis for the lattice of pitch classes would be
> [9/5, 5/3], or [10/9, 6/5] if you prefer. That tells us the most efficient way of representing intervals in ragismic temperament is in terms of 2, 9/5 and 5/3.
>

If I remove the restriction of octave equivalence and make 2 take its chances with the rest of the primes, then the Minkowski minimal basis in order of increasing complexity is 9/5, 5/3 and 2.

Gene wrote:

> If I remove the restriction of octave equivalence and make 2 take its > chances with the rest of the primes, then
> the Minkowski minimal basis in order of increasing complexity is 9/5, 5/3 > and 2.

Okay, this suggests something.
If the period was 2/1 and the generators were a tempered 9/5 and 5/3, may I ask what the 3D prime mapping would be like then?

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> Gene wrote:
>
> > If I remove the restriction of octave equivalence and make 2 take its
> > chances with the rest of the primes, then
> > the Minkowski minimal basis in order of increasing complexity is 9/5, 5/3
> > and 2.
>
> Okay, this suggests something.
> If the period was 2/1 and the generators were a tempered 9/5 and 5/3, may I
> ask what the 3D prime mapping would be like then?

I don't understand the question, I'm afraid. The 7-limit minimax tuning says to make 9/5 2/7 ragisma sharp and 5/3 1/7 ragisma flat. The 9-limit minimax tuning is 2/5 ragisma sharp for 9/5, and 2/5 ragisma flat for 5/3.

Gene wrote:

> I don't understand the question, I'm afraid.

I meant what would be the period and generator counts representing the primes 2 to 7 if the generators were 9/5 and 5/3. So far I can only say that 2/1 would be (1, 0, 0) but I don't know about the rest.

Petr

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:
>
> Gene wrote:
>
> > I don't understand the question, I'm afraid.
>
> I meant what would be the period and generator counts representing the primes 2 to 7 if the generators were 9/5 and 5/3. So far I can only say that 2/1 would be (1, 0, 0) but I don't know about the rest.

If I take the matrix for [2, 9/5, 5/3, 4375/4374] I get

[|1 0 0 0>, |0 2 -1 0>, |0 -1 1 0>, |-1 -7 4 1>]

Inverting gives

[<1 0 0 1|, <0 1 1 0|, <0 1 2 0|, <1 3 -1 1|]

Hence 3 = (9/5)*(5/3), 5 = (9/5)*(5/3)^2, 7 = 2*(9/5)^3*(5/3)^(-1)*(4375/4374).

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

> If I take the matrix for [2, 9/5, 5/3, 4375/4374] I get
>
> [|1 0 0 0>, |0 2 -1 0>, |0 -1 1 0>, |-1 -7 4 1>]
>
> Inverting gives
>
> [<1 0 0 1|, <0 1 1 0|, <0 1 2 0|, <1 3 -1 1|]

Sorry, [<1 0 0 0|, <0 1 1 0|, <0 1 2 0|, <1 3 -1 1|]

Gene wrote:

> Sorry, [<1 0 0 0|, <0 1 1 0|, <0 1 2 0|, <1 3 -1 1|]

Gene, thanks, this is just awesome. What do you think about the idea of possibly having a list of 3D temperament mappings using the generators of the lowest complexity as in the example you've just given? I think that a lot of these tunings would not lookd so "discouraging" to me then as they do now ... I mean, the usual mapping for the ragismic temperament requires 7 steps of 3/1 and 4 steps of 1/5 to arrive at the target 7/2. This means that in order to get a 7/10, you need to do at least 12 steps, which can be quite a lot for such a "highly favored" interval. However, with this alternative mapping, a full pentad like 1:3:5:7:9 could be achieved in a much more efficient way.
Petr

Petr wrote

> Gene, thanks, this is just awesome. What do you think about the
> idea of possibly having a list of 3D temperament mappings using
> the generators of the lowest complexity as in the example you've
> just given? I think that a lot of these tunings would not looked
> so "discouraging" to me then as they do now ... I mean, the usual
> mapping for the ragismic temperament requires 7 steps of 3/1 and
> 4 steps of 1/5 to arrive at the target 7/2. This means that in
> order to get a 7/10, you need to do at least 12 steps, which can
> be quite a lot for such a "highly favored" interval. However,
> with this alternative mapping, a full pentad like 1:3:5:7:9 could
> be achieved in a much more efficient way.
> Petr

A very promising development indeed! -Carl

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:
>
> Gene wrote:
>
> > Sorry, [<1 0 0 0|, <0 1 1 0|, <0 1 2 0|, <1 3 -1 1|]
>
> Gene, thanks, this is just awesome. What do you think about the idea of possibly having a list of 3D temperament mappings using the generators of the lowest complexity as in the example you've just given?

In some cases that already exists, in the form of "Lattice basis" for some of the planar temperaments. But a separate page with more examples might be nice. In order to make it easier to compute, I'd want to use LLL and not bother with the Minkowski basis; also, I need to know if you want to assume octave equivalence, not assume it, or want both.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@> wrote:
> >
> > Gene wrote:
> >
> > > Sorry, [<1 0 0 0|, <0 1 1 0|, <0 1 2 0|, <1 3 -1 1|]
> >
> > Gene, thanks, this is just awesome. What do you think about the idea of possibly having a list of 3D temperament mappings using the generators of the lowest complexity as in the example you've just given?
>
> In some cases that already exists, in the form of "Lattice basis"
> for some of the planar temperaments.

I see. So am I right in assuming I could read it out of some information on the Xenwiki pages, for example?

> But a separate page with more examples might be nice.

I would definitely appreciate such a thing. And I believe I wouldn't be alone.

> In order to make it easier to compute, I'd want to use LLL and not
> bother with the Minkowski basis;

Are you saying that the results might be different depending on whether you do it one way or the other?

> also, I need to know if you want to assume octave equivalence,
> not assume it, or want both.

Well, I personally would prefer not to do that but I can understand that a lot of people would hate me for this.

You know, it's the thing which often happens to me with 2D temperaments as well -- I say that the meantone generator is a fifth because then you need the lowest number of periods to map the target ratios -- and someone will always insist that it should be a fourth.

Petr

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:

> > In some cases that already exists, in the form of "Lattice basis"
> > for some of the planar temperaments.

> I see. So am I right in assuming I could read it out of some information on the Xenwiki pages, for example?

Sure. Look at the pages for the various families of temperaments, and spot anything which says "Lattice basis" and gives lengths and angles.

> > In order to make it easier to compute, I'd want to use LLL and not
> > bother with the Minkowski basis;
>
> Are you saying that the results might be different depending on whether you do it one way or the other?

Sure; but I just wrote code for LLL and am not happy with the result as a final result, though it would be fine as preparation for finding a Minkowski temperamental complexity basis.

> > also, I need to know if you want to assume octave equivalence,
> > not assume it, or want both.
>
> Well, I personally would prefer not to do that but I can understand that a lot of people would hate me for this.

Not to do what??

> You know, it's the thing which often happens to me with 2D temperaments as well -- I say that the meantone generator is a fifth because then you need the lowest number of periods to map the target ratios -- and someone will always insist that it should be a fourth.

That's not what I'm talking about--I'm asking if you should force 2 or some fraction thereof to be one of the generators, and look at a basis for the pitch classes, not the pitches, which is what I've been doing. You get nice 2D lattices which tell you about the planar lattice in question.

"petrparizek2000" <petrparizek2000@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith"
> <genewardsmith@...> wrote:

> > In order to make it easier to compute, I'd want to use
> > LLL and not bother with the Minkowski basis;
>
> Are you saying that the results might be different
> depending on whether you do it one way or the other?

TLLL -- LLL on a Tenney-Euclidean lattice -- looks good to
me. It means the bases have to be orthogonal, which works
fine for octave-specific just intonation, but not
temperamental complexity. I think JI is fine in this case
because the generators constitute a subgroup and you want
to find the simplest generators.

Carl's method for finding unison vectors to use to generate
periodicity blocks was like this. I forget if he invoked
LLL but the results were consistent.

Sometimes LLL will disagree with Minkowski. LLL is
simpler, more widely available, and tries to get orthogonal
as well as short vectors.

Staying with octave-specific vectors makes sense. If the
2:1 works as a generator, it should always be the shortest
possible generator, and so come out of the LLL reduction.
You can then throw it away and reduce the other generators
if you so choose. If the 2:1 isn't a generator, don't you
want a ratio for the period?

What I'm not clear about is where the set of generators
you're reducing comes from. It's something other people
have asked about.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What I'm not clear about is where the set of generators
> you're reducing comes from. It's something other people
> have asked about.

Using the pseudoinverse to get fractional monzo generators, clearing denominators, adding in the commas to the resulting monzos, saturating the result and massaging this further in ways I would need to examine the code to recall.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@> wrote:
>
> > > In some cases that already exists, in the form of "Lattice basis"
> > > for some of the planar temperaments.
>
> > I see. So am I right in assuming I could read it out of some information on the Xenwiki pages, for example?
>
> Sure. Look at the pages for the various families of temperaments,
> and spot anything which says "Lattice basis" and gives lengths and angles.

Good to know this, thx.

> > > In order to make it easier to compute, I'd want to use LLL and not
> > > bother with the Minkowski basis;
> >
> > Are you saying that the results might be different depending on whether you do it one way or the other?
>
> Sure; but I just wrote code for LLL and am not happy with the result
> as a final result, though it would be fine as preparation for
> finding a Minkowski temperamental complexity basis.

If the TM basis approach can suggest a less complex mapping, then I would probably prefer if you could use that one; but I'll leave the final decision upon you. If I knew how to do that, I would try to find such a pair of generators which makes the complexity as low as possible in terms of prime mapping. What I'm not sure about, however, is whether a substantial number of temperaments would come out with different generators if I wanted the lowest possible complexity for a complete 9-limit chord like 2:3:5:7:9. If that should be the case, then I don't know how to decide whether I should go for one mapping or the other.

> That's not what I'm talking about--I'm asking if you should force 2
> or some fraction thereof to be one of the generators, and look at a
> basis for the pitch classes, not the pitches, which is what I've
> been doing. You get nice 2D lattices which tell you about the
> planar lattice in question.

Well, let's first see if I'm on the right track. In the 2D case, the generator is 9/5 and the period is then 7/2, if I'm correct. In the 3D case where the period is 2/1, the two generators in question would be 9/5 and 5/3, if I've understood what's been said so far. And this 3D temperament is what I'm interested in. But I'm not sure if that answers your question.

Petr

Hi all- This is a fascinating thread but I'm moving house
so I can't reply more fully. But I thought I'd suggest
that Petr work out five examples his way, and then have
Gene and/or Graham can try their methods to the same examples
and results can be compared.

-Carl

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:

> Well, let's first see if I'm on the right track. In the 2D case,
> the generator is 9/5 and the period is then 7/2, if I'm correct.
> In the 3D case where the period is 2/1, the two generators in
> question would be 9/5 and 5/3, if I've understood what's been
> said so far. And this 3D temperament is what I'm interested in.
> But I'm not sure if that answers your question.
>
> Petr
>

Carl wrote:

> Hi all- This is a fascinating thread but I'm moving house
> so I can't reply more fully. But I thought I'd suggest
>that Petr work out five examples his way, and then have
> Gene and/or Graham can try their methods to the same examples
> and results can be compared.

Carl, I would be happy to do that but I'm not sure which of the two topics you're now referring to.
My earlier idea was to temper out 7-limit factors using 2D temperaments which approximate non-prime intervals -- like, for example, the one which uses 1 generator for 6/5 and 3 generators for 12/7. This is what I can surely do, and I'm looking forward to the results.
The other idea was to find such generators for 3D temperaments which give them the lowest possible complexity in terms of approximating 1:3:5:7 (and possiby 9) if the equivalence interval is 2/1. Unfortunately, I'm not mathematically skilled enough to be able to do that. That's why I've asked Gene about it.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
> > Hi all- This is a fascinating thread but I'm moving house
> > so I can't reply more fully. But I thought I'd suggest
> >that Petr work out five examples his way, and then have
> > Gene and/or Graham can try their methods to the same examples
> > and results can be compared.
>
> Carl, I would be happy to do that but I'm not sure which of
> the two topics you're now referring to
> My earlier idea was to temper out 7-limit factors using
> 2D temperaments which approximate non-prime intervals

Hi Petr - it's this one. -Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What I'm not clear about is where the set of generators
> you're reducing comes from. It's something other people
> have asked about.

I've put up an answer for this in the form of a Xenwiki web page:

http://xenharmonic.wikispaces.com/Transversal+generators

If there are questions about this likely to become technical, they should probably be asked on tuning-math.