Yahoo Tuning Groups Ultimate Backup tuning One eikosany donut please

Hey folks, I just understood the structure of the eikosany (and CPS in
general) for the first time (pathetic I know). What an awesome thing. Major
kudos to Erv Wilson.

I have to say that neither Monz's dictionary entries nor Kraig's web pages
did it for me, although they helped. It was Paul Erlich's simple
explanation that made it click. My how easily something so simple in
concept (but not in conception) can be obscured.

The hard part for me was that I was looking for how one generates _ratios_
for the pitches. But in fact one doesn't. One simply generates _integers_.
And (in general) no note corresponds to 1. Of course these integers can
become ratios by scaling them all by the same amount or by scaling them to
the same octave.

I couldn't help thinking, hey this eikosany thing is at least 4
dimensional, assuming products of 3 at a time from {1,3,5,7,9,11}) so it
sure suffers when mapped to 2 dimensions, whether in Wilson's
10-star-inside-a-10-gon diagram or his bosanquet-style keyboard mapping.

I want it mapped to the surface of a donut. One just large enough to get my
fingers inside, with 20 buttons distributed over the surface. To see how to
map it, just take the standard 10-star-inside-a-10-gon diagram and raise
every second note of the star above the page while lowering the others. the
outer decagon can get a bit zig-zaggy too.

You can build this approximately in zometool but it's too big (with 10 long
reds in the middle, 10 medium yellows around the outside and 10 short reds
joining them).

I would limit it to 20 notes, but not to one octave (about 2.5 octaves) as
follows. Simply take products of 3 at a time from {4,5,6,7,9,11} and don't
reduce them to the same octave. The "scale" would have no octave but it
would be a brilliant toy to let people hear the maximum number of different
types of 11-limit consonance with the fewest notes (?), even if it was just
some cheesy battery operated thing that only did sawtooth waves with no
dynamics.

I've also looked at microtempering it (distributing the 224:225 and
384:385). By my rough count this would add about 17 additional dyads to the
already 96 (?). This may complete some pentads. Eikosanies are normally
limited to tetrads (for complete pairwise consonance).

This microtempering would avoid phase-locking and would only introduce
1.1 c errors in the 2:3's,
max 3.2 c errors in ratios of 5,
max 2.4 c errors in ratios of 7,
max 2.1 c errors in ratios of 9,
max 1.6 c errors in ratios of 11.

Regards,

-- Dave Keenan
http://dkeenan.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

David C Keenan wrote:

> I couldn't help thinking, hey this eikosany thing is at least 4
> dimensional, assuming products of 3 at a time from {1,3,5,7,9,11}) so it
> sure suffers when mapped to 2 dimensions, whether in Wilson's
> 10-star-inside-a-10-gon diagram or his bosanquet-style keyboard mapping.

Erv stated that it was 12 dimensional In that it should look like a donut from 12 different
directions

> I would limit it to 20 notes, but not to one octave (about 2.5 octaves) as
> follows. Simply take products of 3 at a time from {4,5,6,7,9,11} and don't
> reduce them to the same octave. The "scale" would have no octave but it
> would be a brilliant toy to let people hear the maximum number of different
> types of 11-limit consonance with the fewest notes (?), even if it was just
> some cheesy battery operated thing that only did sawtooth waves with no
> dynamics.

I used to have an instrument called the tree that used the 10 tone cycle as the pattern to
hang bars that I made. I explored quite a few lattices of this type. the article in Xen XI
(which ! should put up) shows another application of this type of spacing.

http://www.anaphoria.com/images/xenXI-15.gif

Here the idea was to place the entire 1-3-5-7-9-11 CPS (36 notes) on a 31 tone keyboard. On
examination you have one note in one octave with the other possibility in the other. Such a
technique can be used in other ways to great advantage. here the tuning spans 2 octaves and a
"large whole tone"

> I've also looked at microtempering it (distributing the 224:225 and
> 384:385). By my rough count this would add about 17 additional dyads to the
> already 96 (?). This may complete some pentads. Eikosanies are normally
> limited to tetrads (for complete pairwise consonance).

Such things are possible and might seem advantageous, but as some one who has used this tuning
extensively, you would be undermining what the tuning does. Every tetrad has a unique
relationship to the whole and each pair of hexanies, in fact every scale has a strongly
defined relationship to the whole that becomes musically meaningful. Inversions are always
found at the most dissonant points, now if you get rid of these dissonance you have undermined
the meaning.

> This microtempering would avoid phase-locking and would only introduce
> 1.1 c errors in the 2:3's,
> max 3.2 c errors in ratios of 5,
> max 2.4 c errors in ratios of 7,
> max 2.1 c errors in ratios of 9,
> max 1.6 c errors in ratios of 11.
>
> Regards,
>
> -- Dave Keenan
> http://dkeenan.com

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Kraig Grady wrote:
>Erv stated that it was 12 dimensional In that it should look like a donut
>from 12 different directions

Hmm. I can't see that that makes it 12 dimensional. In the sense that a
hexany is three dimensional (an octahedron), an eikosany is 5 dimensional.
But it can be reduced to four (with some loss of symmetry) when one of the
odd factors is a product of the others. e.g. 9 or 15.

I wrote:
>>I've also looked at microtempering it (distributing the 224:225 and
>>384:385). By my rough count this would add about 17 additional dyads to
>>the already 96 (?). This may complete
>>some pentads. Eikosanies are normally limited to tetrads (for complete
>>pairwise consonance).

Kraig replied
>Such things are possible and might seem advantageous, but as some one who
>has used this tuning extensively, you would
>be undermining what the tuning does. Every tetrad has a unique relationship
>to the whole and each pair of hexanies, in
>fact every scale has a strongly defined relationship to the whole that
>becomes musically meaningful. Inversions are always
>found at the most dissonant points, now if you get rid of these dissonance
>you have undermined the meaning.

Maybe I don't understand what you are saying here but I can't see how
microtempering could get rid of the most dissonant intervals. The only
intervals that it brings into use as consonances, are those which are
already only 7.7 c (224:225) or 4.5 c (384:385) away from Just. And just
because they become available doesn't mean you have to use them.

But maybe you're right. Maybe even this would undermine the "logic" of it.
But isn't it already somewhat undermined by the fact that 1:3 = 3:9?

Certainly the Eikosany doesn't have anywhere near as high a proportion of
these almost-consonant intervals as does Dean Drummond's 31 note
zoomoozaphone tuning (11-limit diamond plus 16/15 and 15/8), which I once
described as "crying out for it".

Regards,
-- Dave Keenan
http://dkeenan.com

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Kraig wrote,

>Erv stated that it was 12 dimensional In that it should look like a donut
from 12 different
>directions

hmm . . . why is it 12 dimensional? As I understood Erv's diagrams:

(1,3) -> 1 dimensional
(1,3,5) -> 2 dimensional
(1,3,5,7) -> 3 dimensional
(1,3,5,7,9) -> 4 dimensional
(1,3,5,7,9,11) -> 5 dimensional

>Here the idea was to place the entire 1-3-5-7-9-11 CPS (36 notes)

You mean genus or "grand slam", not CPS, correct?

>Such things are possible and might seem advantageous, but as some one who
has used this tuning
>extensively, you would be undermining what the tuning does. Every tetrad
has a unique
>relationship to the whole and each pair of hexanies, in fact every scale
has a strongly
>defined relationship to the whole that becomes musically meaningful.
Inversions are always
>found at the most dissonant points, now if you get rid of these dissonance
you have undermined
>the meaning.

Sounds a lot like my admonition to John deLaubenfels not to make the
characteristic dissonance of the diatonic scale into a 7-limit smoothness.

🔗Kraig Grady <kraiggrady@anaphoria.com>

David C Keenan wrote:

> Certainly the Eikosany doesn't have anywhere near as high a proportion of
> these almost-consonant intervals as does Dean Drummond's 31 note
> zoomoozaphone tuning (11-limit diamond plus 16/15 and 15/8), which I once
> described as "crying out for it".

It is a shame he doesn't carry it out to 43 tones and have a real scale:-). The diamond is
much more condensed around a single tone and Partch would use these near relationships. The
advantage with the Eikosany is that you can more to more remote tonal areas, all with less
tones. Each tone will function in 6 different ways-3 harmonically and 3 subharmonically each
tone in its own unique fashion

>
>
> Regards,
> -- Dave Keenan
> http://dkeenan.com
>
>
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-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

🔗Joseph Pehrson <pehrson@pubmedia.com>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14663

> Kraig wrote,
>
> >Erv stated that it was 12 dimensional In that it should look like
a
donut from 12 different
> >directions
>
> hmm . . . why is it 12 dimensional? As I understood Erv's diagrams:
>
> (1,3) -> 1 dimensional
> (1,3,5) -> 2 dimensional
> (1,3,5,7) -> 3 dimensional
> (1,3,5,7,9) -> 4 dimensional
> (1,3,5,7,9,11) -> 5 dimensional
>

Ummm. Could somebody please explain to me in LAYMAN'S terms this bit
about the dimensions?? I thought there were three visible
dimensions... like we use for the lattices, and isn't the 4th
dimension TIME....??

Or are these different definitions of "dimensions."

And then, if you *DO* have more than three dimensions, how do you get
a DONUT??

I DONOT understand it...

Thanks!

________ ____ __ __ _
Joseph Pehrson

🔗Kraig Grady <kraiggrady@anaphoria.com>

"Paul H. Erlich" wrote:

> Kraig wrote,
>
> >Erv stated that it was 12 dimensional In that it should look like a donut
> from 12 different
> >directions
>
> hmm . . . why is it 12 dimensional? As I understood Erv's diagrams:

maybe it was six- because to see the symmetries in the structure it should look like a donut
fronm 12 different directions something you can't do in 3-d to say the least.

>
>
> (1,3) -> 1 dimensional
> (1,3,5) -> 2 dimensional
> (1,3,5,7) -> 3 dimensional
> (1,3,5,7,9) -> 4 dimensional
> (1,3,5,7,9,11) -> 5 dimensional
>
> >Here the idea was to place the entire 1-3-5-7-9-11 CPS (36 notes)
>
> You mean genus or "grand slam", not CPS, correct?

Dallesandro runs through the full combinations of 0f 1 out of 6 set to the 6 out of 6 set.
these are all CPS's

>
> Sounds a lot like my admonition to John deLaubenfels not to make the
> characteristic dissonance of the diatonic scale into a 7-limit smoothness.

As Carl Jung said, "There is no life without tension." ( I bet Dan Sterns likes that)

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Kraig Grady <kraiggrady@anaphoria.com>

Joseph !

Joseph Pehrson wrote:

> Ummm. Could somebody please explain to me in LAYMAN'S terms this bit
> about the dimensions?? I thought there were three visible
> dimensions... like we use for the lattices, and isn't the 4th
> dimension TIME....??

yes there are only 4 dimensions but if we could see all the repeated symmetries within the
Eikosany we would need 6 , of course we can't do this. The donut happens to be a way of
seeing the Eikosany. Imagine you looked down on something and it looked like a donut ,then you
moved 60 degrees down and you look at it again and it still looked like a donut and so on .

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

🔗Joseph Pehrson <pehrson@pubmedia.com>

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>

http://www.egroups.com/message/tuning/14680

>
> yes there are only 4 dimensions but if we could see all the
repeated
symmetries within the Eikosany we would need 6 , of course we can't
do this. The donut happens to be a way of seeing the Eikosany.
Imagine
you looked down on something and it looked like a donut ,then you
> moved 60 degrees down and you look at it again and it still looked
like a donut and so on .
>
> >
>
Hi Kraig!

You mean I'm actually inside the "donut hole" when I'm looking at the
donut again??

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> http://www.egroups.com/message/tuning/14641
>
> Hey folks, I just understood the structure of the eikosany (and
> CPS in general) for the first time (pathetic I know). What an
> awesome thing. Major kudos to Erv Wilson.
>
> I have to say that neither Monz's dictionary entries nor Kraig's
> web pages did it for me, although they helped. It was Paul
> Erlich's simple explanation that made it click.

Paul or Dave, can you please send this explanation to me so that
I can include it in my definition?

> I want it mapped to the surface of a donut. One just large
> enough to get my fingers inside, with 20 buttons distributed
> over the surface.

The latest developments in neo-Riemannian music-theory
(see the entire issue of _Journal of Music Theory_ devoted to
it, volume 42.2, fall 1998) feature essays with diagrams which
frequently map the 12-tET pitches to a torus (donut).

It's the only way 12-tET theorists can get the lattice concept
to work in a closed system.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14711
> For an attempt to help you visualize an object in four spacial
dimensions,
> go slowly through
http://www.geom.umn.edu/docs/outreach/4-cube/top.html.

This is very interesting, but the "movies" go VERY fast... like about
3 seconds each, in both Netscape and Internet Explorer...

Is that what they're SUPPOSED to do??

Joseph

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Carl Lumma <CLUMMA@NNI.COM>

>>For an attempt to help you visualize an object in four spacial
>>dimensions, go slowly through
>>
>>http://www.geom.umn.edu/docs/outreach/4-cube/top.html.
>
>This is very interesting, but the "movies" go VERY fast... like about
>3 seconds each, in both Netscape and Internet Explorer...

Joseph et all,

You may want to check out the following sites...

http://dogfeathers.com/java/hyprcube.html
http://darkwing.uoregon.edu/~koch/java/FourD.html
http://www.graphics.cornell.edu/~gordon/peek/

-Carl

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Carl Lumma <CLUMMA@NNI.COM>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Why don't these applets work for me? Do I need to install Java or something?

-----Original Message-----
From: Carl Lumma [mailto:CLUMMA@NNI.COM]
Sent: Friday, October 20, 2000 12:42 PM
To: tuning@egroups.com
Subject: [tuning] Re: donot understand donut

Joseph et all,

You may want to check out the following sites...

http://dogfeathers.com/java/hyprcube.html
http://darkwing.uoregon.edu/~koch/java/FourD.html
http://www.graphics.cornell.edu/~gordon/peek/

-Carl

You do not need web access to participate. You may subscribe through
email. Send an empty email to one of these addresses:
tuning-subscribe@egroups.com - join the tuning group.
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the tuning group.
tuning-digest@egroups.com - change your subscription to daily digest mode.
tuning-normal@egroups.com - change your subscription to individual emails.

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/tuning/14763

>
> Joseph et all,
>
> You may want to check out the following sites...
>
> http://dogfeathers.com/java/hyprcube.html
> http://darkwing.uoregon.edu/~koch/java/FourD.html
> http://www.graphics.cornell.edu/~gordon/peek/
>
> -Carl

Here are Carl Lumma's links again for the "multidimensional" sites...
they just keep "rotating and rotating" which is nice. The
"dogfeathers" one is particularly splendid...

🔗Carl Lumma <CLUMMA@NNI.COM>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗M. Edward Borasky <znmeb@teleport.com>

Awesome!! So, how can we transform something with thousands of edges, faces
and vertices into a piece of music?
--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

Cold leftover pizza: it's not just for breakfast any more!

> -----Original Message-----
> From: Paul H. Erlich [mailto:PERLICH@ACADIAN-ASSET.COM]
> Sent: Friday, October 20, 2000 9:42 PM
> To: 'tuning@egroups.com'
> Subject: RE: [tuning] RE: Re: donot understand donut
>
>
> OK guys, this is funny . . .
>
> http://members.aol.com/Polycell/swirlprm.gif
>
> I got to it from
>
> http://members.aol.com/Polycell/uniform.html
>
> These people are insane! We need to import some of them!
>
>
> You do not need web access to participate. You may subscribe through
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> digest mode.
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>
>
>