Top 20

I started from 990 pairs of ets, from which I got 505 linear 7-limit
temperaments. The top 20 in terms of step^3 cents turned out to be:

(1) [2,3,1,-6,4,0] <21/20,27/25>

(2) [1,-1,0,3,3,-4] <8/7,15/14>

(3) [0,2,2,-1,-3,3] <9/8,15/14>

(4) [4,2,2,-1,8,6] <25/24,49/48>

(5) [2,1,3,4,1,-3] <15/14,25/24>

(6) [2,1,-1,-5,7,-3] <21/20,25/24>

(7) [2,-1,1,5,4,-6] <15/14,35/32>

(8) [1,-1,1,5,1,-4] <7/6,16/15>

(9) [1,-1,-2,-2,6,-4] <16/15,21/20>

(10) [4,4,4,-2,5,-3] <36/35,50/49>

(11) [18,27,18,-34,22,1] <2401/2400,4375/4374> Ennealimmal

(12) [2,-2,1,8,4,-8] <16/15,49/48>

(13) [0,0,3,7,-5,0] <10/9,16/15>

(14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not having a
name--"septimal kleismic" maybe?

(15) [0,5,0,-14,0,8] <28/27,49/48>

(16) [6,-7,-2,15,20,-25] <225/224,1029/1024> Miracle

(17) [2,-4,-4,2,12,-11] <50/49,64/63> Paultone

(18) [2,-2,-2,1,9,-8] <16/15,50/49>

(19) [10,9,7,-9,17,-9] <126/125,1728/1715> This one should have a
name if it doesn't already. If I call it "nonkleismic" will that
force someone to come up with a good one?

(20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-et
meantone.

Gene, this is shaping up to be an immense contribution you're making
to tuning theory.

> I started from 990 pairs of ets, from which I got 505 linear 7-
limit
> temperaments.

You'll also try starting from an expanded list of UVs, correct?

The top 20 in terms of step^3 cents

How did you decide on this criterion? Would you please try

Z^(step^(1/3)) cents

where you're free to pick Z to be 2 or e or whatever.

turned out to be:

wedgie univectors

> (1) [2,3,1,-6,4,0] <21/20,27/25>

JI block (what simple UVs complete a TMR (TM-reduced) basis for this)?

> (2) [1,-1,0,3,3,-4] <8/7,15/14>

JI block (ditto)

> (3) [0,2,2,-1,-3,3] <9/8,15/14>

JI (ditto)

> (4) [4,2,2,-1,8,6] <25/24,49/48>

JI or Planar (ditto)

> (5) [2,1,3,4,1,-3] <15/14,25/24>

JI (ditto)

> (6) [2,1,-1,-5,7,-3] <21/20,25/24>

JI "

> (7) [2,-1,1,5,4,-6] <15/14,35/32>

"

> (8) [1,-1,1,5,1,-4] <7/6,16/15>
"
> (9) [1,-1,-2,-2,6,-4] <16/15,21/20>
"
> (10) [4,4,4,-2,5,-3] <36/35,50/49>

JI or Planar "

> (11) [18,27,18,-34,22,1] <2401/2400,4375/4374> Ennealimmal

You win! But somewhere out there, I wonder . . .
What are some manageable MOSs of this?

> (12) [2,-2,1,8,4,-8] <16/15,49/48>

JI or Planar (ditto)
>
> (13) [0,0,3,7,-5,0] <10/9,16/15>

JI "
>
> (14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not having a
> name--"septimal kleismic" maybe?

Please post details. Is this Dave Keenan's chain-of-minor-thirds
thingy? It loses on tetrachordality.

> (15) [0,5,0,-14,0,8] <28/27,49/48>

JI or Planar
>
> (16) [6,-7,-2,15,20,-25] <225/224,1029/1024> Miracle
>
> (17) [2,-4,-4,2,12,-11] <50/49,64/63> Paultone
>
> (18) [2,-2,-2,1,9,-8] <16/15,50/49>

JI or Planar
>
> (19) [10,9,7,-9,17,-9] <126/125,1728/1715> This one should have a
> name if it doesn't already. If I call it "nonkleismic" will that
> force someone to come up with a good one?

Is this Graham's #1 7-limit? And he missed ennealimmal because . . . ?

> (20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-et
> meantone.

What's the generator?

Where is Huygens meantone in all this?

Would you do the 5-limit too, that would be so cool!

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Gene, this is shaping up to be an immense contribution you're
making
> to tuning theory.

Thanks.

> > I started from 990 pairs of ets, from which I got 505 linear 7-
> limit
> > temperaments.
>
> You'll also try starting from an expanded list of UVs, correct?

I'm going to merge lists, and then expand by taking sums of wedge
invariants, but I need a decision on cut-offs. I am thinking the end
product would be additively closed--a list where any sum or
difference of two wedge invariants on the list was beyond the cut-
off; but I have 173 in this list below 10000 already, so there's also
a question of how many of these we can handle.

> The top 20 in terms of step^3 cents
>
> How did you decide on this criterion? Would you please try
>
> Z^(step^(1/3)) cents

Well, I could but what's the rationale? Cubic growth is already
enough to give us a finite list; we don't need expondential growth.
> wedgie univectors
>
> > (1) [2,3,1,-6,4,0] <21/20,27/25>
>
> JI block (what simple UVs complete a TMR (TM-reduced) basis for
this)?

There are far too many answers to this question.
<25/24,28/27,21/20,27/25> makes a nice basis for a notation, but
there are far too many of those also.
Would a list of ets help?
> > (11) [18,27,18,-34,22,1] <2401/2400,4375/4374> Ennealimmal

> You win! But somewhere out there, I wonder . . .
> What are some manageable MOSs of this?

27 or 45 notes would be good--or even 72. 45 notes is just two more
than the Partch 43, and gives a large supply of essentially just
7-limit harmonies.

> > (14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not having
a
> > name--"septimal kleismic" maybe?
>
> Please post details. Is this Dave Keenan's chain-of-minor-thirds
> thingy? It loses on tetrachordality.

From Graham's page I got the idea this was supposed to be 5-limit,
but in fact Keenan views it as 7-limit, so "kleismic" is the official
name.
> > (20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-et
> > meantone.
>
> What's the generator?

A sharp fifth, but otherwise it's like 12-et. The sharp fifth reduces
the errors of the 7-limit at the expense of the 3 and 5 limits, and
this one could be promoted as a temperament for the bold and daring,
or those who engage in Setheresization.

> Where is Huygens meantone in all this?

Coming up soon, I'd guess. Should I keep on going?

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:

> I'm going to merge lists, and then expand by taking sums of wedge
> invariants, but I need a decision on cut-offs. I am thinking the
end
> product would be additively closed--a list where any sum or
> difference of two wedge invariants on the list was beyond the cut-
> off; but I have 173 in this list below 10000 already, so there's
also
> a question of how many of these we can handle.

Maybe Matlab would help. Do you have it? Can you write programs for
it in matrix notation?

> > How did you decide on this criterion? Would you please try
> >
> > Z^(step^(1/3)) cents
>
> Well, I could but what's the rationale?

You said it sounded plausible that the amount of tempering associated
with a unison vector was

(n-d)/(d*log(d))

which is

(n-d)/(2^length*length)

in the Tenney lattice. Now if a 3-d (my way) orthogonal block
typically has "step" notes, then the tempering along each unison
vector will typically involve a length of step^(1/3) . . . so this
becomes

(n-d)/(2^(step^(1/3))*length)

Now if we say our 'badness measure' is proportional to amount of
tempering times length, we have

badness = (n-d)/(2^(step^(1/3)))

Now in general, it seems that any worthwhile 7-limit temperament can
be described with roughly orthogonal superparticular unison vectors
(I kinda asked you about this sorta) . . . so it seems that we can say

n-d = 1

and make our goodness measure

2^(step^(1/3))

Is that some sloppy thinking or what (but shouldn't the exponential
part be right)?

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

> Now in general, it seems that any worthwhile 7-limit temperament
can
> be described with roughly orthogonal superparticular unison vectors
> (I kinda asked you about this sorta) . . . so it seems that we can
say

This is how you sneak in exponential growth, but is it plausible? The
TM reduced basis I get for a lot of good temperaments (eg. Miracle)
are not all superparticular.

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Now in general, it seems that any worthwhile 7-limit temperament
> can
> > be described with roughly orthogonal superparticular unison
vectors
> > (I kinda asked you about this sorta) . . . so it seems that we
can
> say
>
> This is how you sneak in exponential growth, but is it plausible?
The
> TM reduced basis I get for a lot of good temperaments (eg. Miracle)
> are not all superparticular.

2401:2400 and 225:224 are roughly orthogonal. So it _can_ . . . how
about others?

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

> 2401:2400 and 225:224 are roughly orthogonal. So it _can_ . . . how
> about others?

I don't think you can make <2401/2400, 65625/65536> superparticular.
What about <2401/2400, 3136/3125>?

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > 2401:2400 and 225:224 are roughly orthogonal. So it _can_ . . .
how
> > about others?
>
> I don't think you can make <2401/2400, 65625/65536> superparticular.
> What about <2401/2400, 3136/3125>?

If you can't, just think of (n-d) as an additional penalty for
complexity. Length alone isn't much of a penalty -- it's sorta like
step^(1/3)!

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

> If you can't, just think of (n-d) as an additional penalty for
> complexity. Length alone isn't much of a penalty -- it's sorta like
> step^(1/3)!

Hey Gene -- something's wrong with my thinking here . . . note that
the cents error _is_ the amount of tempering! So my criterion would
be applied _without_ multiplying by the cents error . . . it would be
a decent criterion with which to _constrain a search_, but definitely
not for a final ranking . . .

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

> Would you do the 5-limit too, that would be so cool!

I'll do that; it isn't very hard, since it is defined by a single
comma. In fact, it's connected to the wedge invariant, since the
comma defining the corresponding 5-limit linear temperament can be
read off the wedge invariant. If I do that for Ennealimmal, I do in
fact get the ennealimma, so it's in there. For Miracle, it is
(3/2)/(16/15)^6 = 34171875/33554432 (Ampersand's comma.)

Paul wrote:

> Graham's ... missed ennealimmal because . . . ?

It's too complex. I get a complexity of 27, but 7-limit temperaments are
capped at 18. Also, I only consider the first 20 consistent ETs in that
list, which goes up to 42 for the 7-limit, and you need 27 and 45 for
ennealimmal.

Anyway, I have it now

3/8, 49.0 cent generator

basis:
(0.111111111111, 0.0408387831857)

mapping by period and generator:
[(9, 0), (15, -2), (22, -3), (26, -2)]

mapping by steps:
[(45, 27), (71, 43), (104, 63), (126, 76)]

unison vectors:
[[-5, -1, -2, 4], [-1, -7, 4, 1]]

highest interval width: 3
complexity measure: 27 (45 for smallest MOS)
highest error: 0.000170 (0.204 cents)
unique

I'll add it to the catalog sometime. It should be at the top of the
7-limit microtemperaments at <http://x31eq.com/limit7.micro>.
It isn't in my local copy, but I think that's out of date. I'll have a
look when I connect to send this.

Graham

Gene,

As these are linear temperaments, could you also include the generator
and the period in your lists?

thanks,

--Dan Stearns

--- In tuning-math@y..., genewardsmith@j... wrote:
> I started from 990 pairs of ets, from which I got 505 linear 7-limit
> temperaments. The top 20 in terms of step^3 cents turned out to be:
>
> (1) [2,3,1,-6,4,0] <21/20,27/25>
>
> (2) [1,-1,0,3,3,-4] <8/7,15/14>
>
> (3) [0,2,2,-1,-3,3] <9/8,15/14>
>
> (4) [4,2,2,-1,8,6] <25/24,49/48>
>
> (5) [2,1,3,4,1,-3] <15/14,25/24>
>
> (6) [2,1,-1,-5,7,-3] <21/20,25/24>
>
> (7) [2,-1,1,5,4,-6] <15/14,35/32>
>
> (8) [1,-1,1,5,1,-4] <7/6,16/15>
>
> (9) [1,-1,-2,-2,6,-4] <16/15,21/20>
>
> (10) [4,4,4,-2,5,-3] <36/35,50/49>
>
> (11) [18,27,18,-34,22,1] <2401/2400,4375/4374> Ennealimmal
>
> (12) [2,-2,1,8,4,-8] <16/15,49/48>
>
> (13) [0,0,3,7,-5,0] <10/9,16/15>
>
> (14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not having a
> name--"septimal kleismic" maybe?
>
> (15) [0,5,0,-14,0,8] <28/27,49/48>
>
> (16) [6,-7,-2,15,20,-25] <225/224,1029/1024> Miracle
>
> (17) [2,-4,-4,2,12,-11] <50/49,64/63> Paultone
>
> (18) [2,-2,-2,1,9,-8] <16/15,50/49>
>
> (19) [10,9,7,-9,17,-9] <126/125,1728/1715> This one should have a
> name if it doesn't already. If I call it "nonkleismic" will that
> force someone to come up with a good one?
>
> (20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-et
> meantone.

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Gene,
>
> As these are linear temperaments, could you also include the
generator
> and the period in your lists?
>
> thanks,
>
> --Dan Stearns

Yes -- this would answer much of what went unanswered in my questions.

Also, where's double-diatonic (14+12)? I wouldn't think that should
be too much worse than paultone, but . . . can you show exactly
how "step" is computed, with an example (no wedgies please)?

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Gene, this is shaping up to be an immense contribution you're
> making
> > to tuning theory.
>
> Thanks.
>
> > > I started from 990 pairs of ets, from which I got 505 linear 7-
> > limit
> > > temperaments.
> >
> > You'll also try starting from an expanded list of UVs, correct?
>
> I'm going to merge lists, and then expand by taking sums of wedge
> invariants, but I need a decision on cut-offs. I am thinking the
end
> product would be additively closed--a list where any sum or
> difference of two wedge invariants on the list was beyond the cut-
> off; but I have 173 in this list below 10000 already, so there's
also
> a question of how many of these we can handle.
>
> > The top 20 in terms of step^3 cents
> >
> > How did you decide on this criterion? Would you please try
> >
> > Z^(step^(1/3)) cents
>
> Well, I could but what's the rationale? Cubic growth is already
> enough to give us a finite list; we don't need expondential growth.

So what's the rationale for cubic growth as opposed to any other
function that gives you

> > wedgie univectors
> >
> > > (1) [2,3,1,-6,4,0] <21/20,27/25>
> >
> > JI block (what simple UVs complete a TMR (TM-reduced) basis for
> this)?
>
> There are far too many answers to this question.
> <25/24,28/27,21/20,27/25> makes a nice basis for a notation, but
> there are far too many of those also.
> Would a list of ets help?

How about just the usual details -- generator, mapping.

> > > (11) [18,27,18,-34,22,1] <2401/2400,4375/4374> Ennealimmal
>
> > You win! But somewhere out there, I wonder . . .
> > What are some manageable MOSs of this?
>
> 27 or 45 notes would be good--or even 72. 45 notes is just two more
> than the Partch 43, and gives a large supply of essentially just
> 7-limit harmonies.

Not more than MIRACLE-41, though, does it?
>
> > > (14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not
having
> a
> > > name--"septimal kleismic" maybe?
> >
> > Please post details. Is this Dave Keenan's chain-of-minor-thirds
> > thingy? It loses on tetrachordality.
>
> From Graham's page I got the idea this was supposed to be 5-limit,
> but in fact Keenan views it as 7-limit, so "kleismic" is the
official
> name.

Is this Dave Keenan's chain-of-minor-thirds thingy?

> > > (20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-
et
> > > meantone.
> >
> > What's the generator?
>
> A sharp fifth, but otherwise it's like 12-et.

I derived this several years ago, so I forget the cents value. 704?
>
> > Where is Huygens meantone in all this?
>
> Coming up soon, I'd guess. Should I keep on going?

Please do, unless you think you may be missing some due to the
limitations of your search.

Gene, for some reason the message that contains the questions I was
referring to just got posted to the website now. Some sort of
internet bottleneck, I suppose. So you can't be blamed for not having
answered them!

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

> So what's the rationale for cubic growth as opposed to any other
> function that gives you

Just that it is a simple function with faster than quadratic growth,
but not a great deal faster. When in a polynomial growth situation,
one normally uses x^n for some expondent n which need not be an
integer.

> How about just the usual details -- generator, mapping.

I really had put the list out for a preliminary review, to get
feedback on whether the ordering seemed to make sense. Why don't I
work on it some more and see what I get?

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

> I really had put the list out for a preliminary review, to get
> feedback on whether the ordering seemed to make sense.

I really can't complain!

> Why don't I
> work on it some more and see what I get?

Awesome!

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:

> Just that it is a simple function with faster than quadratic
growth,
> but not a great deal faster. When in a polynomial growth situation,
> one normally uses x^n for some expondent n which need not be an
> integer.

step^3 measures the number of possible triads in the typical
scale . . . so I guess it makes some sense . . .

graham@microtonal.co.uk () wrote:

> I'll add it to the catalog sometime. It should be at the top of the
> 7-limit microtemperaments at
> <http://x31eq.com/limit7.micro>. It isn't in my local copy,
> but I think that's out of date. I'll have a look when I connect to
> send this.

It was there.

I've added files with a .cubed suffix to show my version of the new figure
of demerit (I don't do all this RMS stuff). Doesn't look like an
improvement to me, but I've still got the safety harness on.

If you want to play with the parameters, get the source code. See, as
usual, <http://x31eq.com/temper.html>.

Graham

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

> I've added files with a .cubed suffix to show my version of the new
figure
> of demerit (I don't do all this RMS stuff). Doesn't look like an
> improvement to me, but I've still got the safety harness on.

It's an alternative to the safety harness, not an improvement.