The slippery six

There are six 7-limit linear temperaments which are on the list of 66
obtained from pairs of commas which did not turn up on the list of
505 obtained from pairs of ets. They seem to be ones which are so
closely tied to one particular et that they don't show up by studying
pairs. Also for some reason there are two 9/7-systems on the list.

(1) [6,10,10,-5,1,2] ets: 22

[0 2]
[3 1]
[5 1]
[5 2]

a = 7.98567775 / 22 (~9/7) ; b = 1/2
measure 3165

This is what I called a "unique facet" of the 22-et; that seems to be
why it does not turn up on the other list.

(2) [-4,-16,-9,24,3,-16] ets: 31

[ 0 1]
[ -4 3]
[-16 8]
[ -9 6]

a = 11.00394377 / 31 (~9/7) ; b = 1
measure 3197

A completely different supermajor third system, this one associated
to the 31-et.

(3) [-8,-13,-4,27,-16,-2] ets: 19

[ 0 1]
[-8 2]
[-13 3]
[-4 3]

a = .997597215 / 19; b = 1
measure 6149

The 19-et owns this one, obviously.

(4) [10,14,14,-7,6,-1] ets: 26

[0 2]
[5 2]
[7 3]
[7 4]

a = 3.026421762 / 26; b = 1/2
measure 8510

(5) [0,-12,-12,6,19,-19]

[ 0 12]
[ 0 19]
[-1 28]
[-1 34]

a = 23.40769169 cents; b = 100 cents
measure 9556

(6) [-2,4,-30,-81,42,11] ets: 46,80

[ 0 2]
[-1 4]
[ 2 3]
[-15 18]

a = 33.01588032 / 80 (~4/3); b = 1/2
measure 26079

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> There are six 7-limit linear temperaments which are on the list of
66
> obtained from pairs of commas which did not turn up on the list of
> 505 obtained from pairs of ets.

That's a good indication that Graham may have missed these too, since
he also started from pairs of ETs . . . Graham?

> They seem to be ones which are so
> closely tied to one particular et that they don't show up by
studying
> pairs. Also for some reason there are two 9/7-systems on the list.
>
> (1) [6,10,10,-5,1,2] ets: 22
>
> [0 2]
> [3 1]
> [5 1]
> [5 2]
>
> a = 7.98567775 / 22 (~9/7) ; b = 1/2

You know, I was just going to ask you what happened to this one, as I
remember it from the even earlier survey that you and Graham did,
coming from my list of commas.

> (5) [0,-12,-12,6,19,-19]
>
> [ 0 12]
> [ 0 19]
> [-1 28]
> [-1 34]
>
> a = 23.40769169 cents; b = 100 cents
> measure 9556

Oh yeah, this one again!

> (6) [-2,4,-30,-81,42,11] ets: 46,80
>
> [ 0 2]
> [-1 4]
> [ 2 3]
> [-15 18]
>
> a = 33.01588032 / 80 (~4/3); b = 1/2
> measure 26079

So this _isn't_ 46+34??

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > (6) [-2,4,-30,-81,42,11] ets: 46,80
> >
> > [ 0 2]
> > [-1 4]
> > [ 2 3]
> > [-15 18]
> >
> > a = 33.01588032 / 80 (~4/3); b = 1/2
> > measure 26079
>
> So this _isn't_ 46+34??

It is, but it won't show up as the *sum* of ets, only as the
difference. This is because the 34-et map in question is
h80 - h46, and that isn't h34 in the 7-limit, since h34(7)=95 and
(h80-h46)(7)=96. Maybe adding in a list of differences would be a
good idea.

The generator you get from the 46+34 method is 7/80, which
compliments 33/80 if that has you worried: 7/80 + 33/80 = 1/2.

The consequence of being 46+34 of course is that this system is a
hell of a lot better in the 5-limit than it is in the 7-limit; the
5-limit comma I get from the wedgie is 2048/2025--the diaschisma.
Graham devotes a web page to the diaschismic temperament as a 5-limit
temperament, where it makes a lot of sense.

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > (6) [-2,4,-30,-81,42,11] ets: 46,80
> > >
> > > [ 0 2]
> > > [-1 4]
> > > [ 2 3]
> > > [-15 18]
> > >
> > > a = 33.01588032 / 80 (~4/3); b = 1/2
> > > measure 26079
> >
> > So this _isn't_ 46+34??
>
> It is, but it won't show up as the *sum* of ets, only as the
> difference. This is because the 34-et map in question is
> h80 - h46, and that isn't h34 in the 7-limit, since h34(7)=95 and
> (h80-h46)(7)=96. Maybe adding in a list of differences would be a
> good idea.

Hmm . . . you keep avoiding my whining about consistency (most
recently with regard to 21), and this would seem to be a good place
to bring it up again. You told Graham that something like 46+34 to
you would be _defined_ so that the 80 would come out right, not
necessarily the individual ETs. Now you seem to be contradicting
yourself. What gives?

> The consequence of being 46+34 of course is that this system is a
> hell of a lot better in the 5-limit than it is in the 7-limit; the
> 5-limit comma I get from the wedgie is 2048/2025--the diaschisma.
> Graham devotes a web page to the diaschismic temperament as a 5-
limit
> temperament, where it makes a lot of sense.

And you brought up 80 when we were discussing ways of extending
diaschismic to 11-limit, if you recall . . . probably this same
mapping through the 7-limit.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Hmm . . . you keep avoiding my whining about consistency (most
> recently with regard to 21), and this would seem to be a good place
> to bring it up again. You told Graham that something like 46+34 to
> you would be _defined_ so that the 80 would come out right, not
> necessarily the individual ETs. Now you seem to be contradicting
> yourself. What gives?

By 46+34 I mean a particular system of generators in the 80-et, and
that is determined without reference to what the maps are. Graham
means by it the associated linear temperament, and that is *not*
determined without reference to the maps, and so is not strictly well-
defined. It is determined only mod 40 if you assume it should follow
the 46+34 of the 80-et.

> And you brought up 80 when we were discussing ways of extending
> diaschismic to 11-limit, if you recall . . . probably this same
> mapping through the 7-limit.

The point being, there was more than one sensible way to do it, which
were the same in the 80-et but not as linear temperaments.

Gene:
> > There are six 7-limit linear temperaments which are on the list of
> 66
> > obtained from pairs of commas which did not turn up on the list of
> > 505 obtained from pairs of ets.

Paul:
> That's a good indication that Graham may have missed these too, since
> he also started from pairs of ETs . . . Graham?

Yes, it looks like Gene's doing the same search as me, and so he's finding
the same weaknesses. I did point this one out before, so I'm not sure why
he isn't doing a different search, more in line with his thinking.

So again, if you take each consistent ET and choose each possible
generator, you can get a list of linear temperaments that way. It gets
messy, because there's more than one mapping for each generator. In fact,
that sounds much like the very problem we're trying to solve in the first
place.

If anybody wants to do some real work, this is something to look at.

Graham

> By 46+34 I mean a particular system of generators in the 80-et, and
> that is determined without reference to what the maps are. Graham
> means by it the associated linear temperament, and that is *not*
> determined without reference to the maps, and so is not strictly well-
> defined. It is determined only mod 40 if you assume it should follow
> the 46+34 of the 80-et.

Gene, when I called you on this before you were definitely talking about
temperaments. I wouldn't have mentioned it otherwise.

Graham

--- In tuning-math@y..., graham@m... wrote:

> Gene, when I called you on this before you were definitely talking
about
> temperaments. I wouldn't have mentioned it otherwise.

I was talking about 34&46, not 34+46; the first is not well-defined,
which was the point of my example.

--- In tuning-math@y..., graham@m... wrote:
> Gene:
> > > There are six 7-limit linear temperaments which are on the list
of
> > 66
> > > obtained from pairs of commas which did not turn up on the list
of
> > > 505 obtained from pairs of ets.
>
> Paul:
> > That's a good indication that Graham may have missed these too,
since
> > he also started from pairs of ETs . . . Graham?
>
> Yes, it looks like Gene's doing the same search as me, and so he's
finding
> the same weaknesses. I did point this one out before, so I'm not
sure why
> he isn't doing a different search, more in line with his thinking.

Well, he is -- he _also_ started from unison vectors and found the
slippery six (I bet there are simpler examples too). _My_ thinking
would be to _only_ start from unison vectors, _not_ ETs. I gave a
heuristic on _which_ unison vectors are most likely to help when they
are part of a reduced basis.

> So again, if you take each consistent ET and choose each possible
> generator, you can get a list of linear temperaments that way. It
gets
> messy, because there's more than one mapping for each generator.
In fact,
> that sounds much like the very problem we're trying to solve in the
first
> place.

Hmm . . . I'm not sure I see it that way.

> If anybody wants to do some real work, this is something to look at.
>
I guess so, but it's not a route I would probably travel.
>
> Graham