156edo, articulating septimal kleisma

hmmm ... this is interesting:

i was searching for an EDO which would articulate the
"septimal kleisma" [2 2 -1] = 225:224 (~7.711522991 cents).

156edo looked very good, since the "augmented-6th" 225:128 and
the "harmonic-minor-7th" 7:4 are approximated very closely
by is 127 and 126 degrees, respectively. thus, the
"septimal kleisma" is almost exactly 1 degree.

but then, when i made a bingo-card-lattice of 156edo
("perfect-5th" ~3:2 = 91 degrees, "major-3rd" ~5:4 = 50),
the representation of 225:128 turned out to be 126.

so at least in this mapping (which to me is the one
which makes the most sense ... i think ...), the
"septimal kleisma" is not articulated after all!

help.

-monz

Use DIVIDE/CONSISTENT 225/224
You see the size is zero in 156-ET, 152 is better.

Manuel

thanks, Manuel.

> From: <manuel.op.de.coul@eon-benelux.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, November 05, 2002 1:56 AM
> Subject: Re: [tuning-math] 156edo, articulating septimal kleisma
>

> Use DIVIDE/CONSISTENT 225/224
> You see the size is zero in 156-ET, 152 is better.
>
> Manuel

-monz

> i was searching for an EDO which would articulate the
> "septimal kleisma" [2 2 -1] = 225:224 (~7.711522991 cents).

You could really articulate it with ets like 27 and 37,
which both represent it as one step, or 26, which shrinks
it to -1 steps!

If you really want to articulate it accurately, why not
use JI?

If Gene/Paul would follow through and make the tree zoom
duals, you might be able to see this kind of thing at a
glance!

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> If Gene/Paul would follow through and make the tree zoom
> duals, you might be able to see this kind of thing at a
> glance!

Here you go:

We can consider 5-limit ets to be vals [a,b,c] with elements positive integers; they define a point in the projective plane. Since "a" is not zero, we can divide by "a" and get the point [b/a,c/a] in the affine plane as a coordinate patch. We now note that the points are all clustered about [log2(3),log2(5)], so we may move the origin to this point. We now have the "tree zoom" picture.

Similarly, if q is a comma, then we may consider it to be [a,b,c] where 2^a 3^b 5^c = q. We no longer can always divide by "a", so we cannot take "a=0" to be the line at infinity as we did with ets. Instead, we can take a+log2(3)*b + log2(5)*c=0 to be the line at infinity, since the comma is not 1. This means we can divide through by log2(q) (or cents, etc--which log map we use is not important) and
get [a/log2(q),b/log2(q),c/log2(q)]; taking the last two gives us an affine coordinate patch: [b/log2(q), c/log2(q)] which can be used to plot commas as points.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

>taking the last two gives us an affine coordinate patch: [b/log2(q),
>c/log2(q)] which can be used to plot commas as points.

interesting . . . thanks so much gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> hmmm ... this is interesting:
>
> i was searching for an EDO which would articulate the
> "septimal kleisma" [2 2 -1] = 225:224 (~7.711522991 cents).
>
> 156edo looked very good, since the "augmented-6th" 225:128 and
> the "harmonic-minor-7th" 7:4 are approximated very closely
> by is 127 and 126 degrees, respectively. thus, the
> "septimal kleisma" is almost exactly 1 degree.
>
> but then, when i made a bingo-card-lattice of 156edo
> ("perfect-5th" ~3:2 = 91 degrees, "major-3rd" ~5:4 = 50),
> the representation of 225:128 turned out to be 126.
>
> so at least in this mapping (which to me is the one
> which makes the most sense ... i think ...), the
> "septimal kleisma" is not articulated after all!
>
> help.

monz, this is really no different from 55-equal -- although the
*just* 80:81 (syntonic comma) is very nearly one degree of 55-equal,
the *native* 80:81 (syntonic comma) of 55-equal vanishes.

in that case you just need a much better 5-limit approximation -- 53-
equal of course does the trick.

in the current case you just need a much better 7-limit
approximation -- 152-equal, the universal tuning, will work. :)

similarly, one of your webpages claims that the schisma is well-
represented by one degree of 614-equal. of course, 612-equal would,
functionally, work much better, since it offers a far better 5-limit
approximation.

maybe we should start using the symbol 81;80 for the native syntonic
comma of any temperament?

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> If Gene/Paul would follow through and make the tree zoom
> duals, you might be able to see this kind of thing at a
> glance!
>

unfortunately, 225:224 has components in all three directions (3, 5,
and 7), so it might be hard without a nice VRML implementation of
this . . .

>>If Gene/Paul would follow through and make the tree zoom
>>duals, you might be able to see this kind of thing at a
>>glance!
>>
>
>unfortunately, 225:224 has components in all three directions (3,
>5, and 7), so it might be hard without a nice VRML implementation
>of this . . .

Maybe Robert Walker can help us there... For now, the 5-limit
would still be cool. I suspect even cooler than the et-centric
versions...

-Carl

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
>
> >taking the last two gives us an affine coordinate patch: [b/log2(q),
> >c/log2(q)] which can be used to plot commas as points.
>
> interesting . . . thanks so much gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Yer welcome--of course now you need to zoom out rather than in if you want to fool with this.

What would be really cool is a 3D applet, which let you look at a 7-limit picture from various directions. We could have either ets as points, linear temperaments as lines, and commas as planes, or a dual picture with commas as points, linear temperaments again as lines, and ets as planes.

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> Maybe Robert Walker can help us there... For now, the 5-limit
> would still be cool. I suspect even cooler than the et-centric
> versions...

Plus you can always use inversive geometry, turn either version inside-out, and replace the lines with circles.

> From: "Gene Ward Smith" <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, November 05, 2002 6:58 PM
> Subject: [tuning-math] Re: Tree zoom duals
>
>
> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> > --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> >
> > >taking the last two gives us an affine coordinate patch: [b/log2(q),
> > >c/log2(q)] which can be used to plot commas as points.
> >
> > interesting . . . thanks so much gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
>
> Yer welcome--of course now you need to zoom out rather than
> in if you want to fool with this.
>
> What would be really cool is a 3D applet, which let you look at
> a 7-limit picture from various directions. We could have either
> ets as points, linear temperaments as lines, and commas as planes,
> or a dual picture with commas as points, linear temperaments
> again as lines, and ets as planes.

this is all stuff that i've always intended from the beginning
to have available in my JustMusic software.

http://sonic-arts.org/monzo/justmusic/introtojm.htm

unfortunately, both the project and the Yahoo group have
been slumbering for quite some time. any chance we can
wake them up?

-monz
(still looking for a Microsoft Visual C++ programmer to help out)

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
> >
> > >taking the last two gives us an affine coordinate patch: [b/log2
(q),
> > >c/log2(q)] which can be used to plot commas as points.
> >
> > interesting . . . thanks so much
gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
>
> Yer welcome--of course now you need to zoom out rather than in if
>you want to fool with this.

what do you mean?

> What would be really cool is a 3D applet, which let you look at a 7-
>limit picture from various directions. We could have either ets as
>points, linear temperaments as lines, and commas as planes, or a
>dual picture with commas as points, linear temperaments again as
>lines, and ets as planes.

yup yup yup yup
yup!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> >
> > What would be really cool is a 3D applet, which let you look at
> > a 7-limit picture from various directions. We could have either
> > ets as points, linear temperaments as lines, and commas as planes,
> > or a dual picture with commas as points, linear temperaments
> > again as lines, and ets as planes.
>
>
>
> this is all stuff that i've always intended from the beginning
> to have available in my JustMusic software.
>
> http://sonic-arts.org/monzo/justmusic/introtojm.htm

is that really true? do you have any of the ideas above referenced
anywhere?

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning-math@yahoogroups.com>
Sent: Tuesday, November 05, 2002 11:55 PM
Subject: [tuning-math] Re: Tree zoom duals

> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > >
> > > What would be really cool is a 3D applet, which let you look at
> > > a 7-limit picture from various directions. We could have either
> > > ets as points, linear temperaments as lines, and commas as planes,
> > > or a dual picture with commas as points, linear temperaments
> > > again as lines, and ets as planes.
> >
> >
> >
> > this is all stuff that i've always intended from the beginning
> > to have available in my JustMusic software.
> >
> > http://sonic-arts.org/monzo/justmusic/introtojm.htm
>
> is that really true? do you have any of the ideas above referenced
> anywhere?

what's on the webpage and in the archives of the Yahoo JustMusic group
is all i've ever made public about it. the rest of my ideas are in
folders and folders of notes and old BASIC code ... stuff that i
haven't even looked at in over 5 years.

-monz
(*really* wishing that i could find that Visual C++ programmer!!)

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> > Yer welcome--of course now you need to zoom out rather than in if
> >you want to fool with this.
>
> what do you mean?

The high-octane ets crowd in towards the origin, so you zoom in to find them. The high-octane commas expand out to infinity, so you would need to zoom out to include them.

>The high-octane ets crowd in towards the origin, so you zoom in
>to find them. The high-octane commas expand out to infinity, so
>you would need to zoom out to include them.

Rad!

In both cases, the unzoomed versions show me what I want to see!

-Carl

--- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning-math@y...>
> Sent: Tuesday, November 05, 2002 11:55 PM
> Subject: [tuning-math] Re: Tree zoom duals
>
>
> > --- In tuning-math@y..., "monz" <monz@a...> wrote:
> >
> > > >
> > > > What would be really cool is a 3D applet, which let you look
at
> > > > a 7-limit picture from various directions. We could have
either
> > > > ets as points, linear temperaments as lines, and commas as
planes,
> > > > or a dual picture with commas as points, linear temperaments
> > > > again as lines, and ets as planes.
> > >
> > >
> > >
> > > this is all stuff that i've always intended from the beginning
> > > to have available in my JustMusic software.
> > >
> > > http://sonic-arts.org/monzo/justmusic/introtojm.htm
> >
> > is that really true? do you have any of the ideas above
referenced
> > anywhere?
>
>
>
> what's on the webpage and in the archives of the Yahoo JustMusic
group
> is all i've ever made public about it. the rest of my ideas are in
> folders and folders of notes and old BASIC code ... stuff that i
> haven't even looked at in over 5 years.

monz, if you dig up notes in your folders that mention even one of
these ideas, let alone all of them, i'll be absolutely stunned.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > > Yer welcome--of course now you need to zoom out rather than in
if
> > >you want to fool with this.
> >
> > what do you mean?
>
> The high-octane ets crowd in towards the origin, so you zoom in to
>find them. The high-octane commas expand out to infinity, so you
>would need to zoom out to include them.

right -- i realized this as soon as i signed off last night.

i also realized that each et line will intersect each axis at a
distance from the origin inversely proportional to that consonance's
error in that et. this will spare me the effort of having to
construct a list of commas.

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> i also realized that each et line will intersect each axis at a
> distance from the origin inversely proportional to that consonance's
> error in that et. this will spare me the effort of having to
> construct a list of commas.

Slick!