Yahoo Tuning Groups Ultimate Backup tuning Questions about Eikosany, etc. (was Re: MMM and non12comp)

On Monday 29 November 2004 12:30 am, Kraig Grady wrote:

> Whether it is having a consonance environment without a strong tonal
> center as with the eikosany.

Could some one explain why the eikosany is 'without a strong tonal center'?
By my way of thinking, this would mean that it is short on 3/2's, but that
doesnt appear to be the case....plus, tonal center is as much a function of
repetition, etc. as of other factors, wouldn't you agree?

> scales where the tones are equally consonant/dissonant as in the Scales
> of Mt. Meru ( recurrent sequences),

Kraig, I looked at Wilson's notes on these, and was confused. I got that these
were Fibonacci-related, or quasi-Fibonacci-related, but I don't understand
how one goes about building a set of JI ratios from this stuff. Can you or
someone else who understands this give an example?

Best,
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Carl Lumma <ekin@lumma.org>

>> Whether it is having a consonance environment without a strong
>> tonal center as with the eikosany.
>
>Could some one explain why the eikosany is 'without a strong tonal
>center'? By my way of thinking, this would mean that it is short
>on 3/2's, but that doesnt appear to be the case....plus, tonal
>center is as much a function of repetition, etc. as of other
>factors, wouldn't you agree?

Compositional factors such as repetition, etc. will override
scale-based theory, as always. This, however does not invalidate
theory.

3/2s are strong establishers of tonality, but if a scale has more
than one 3/2, what can be said for the tonal center of the overall
scale? Counting 3/2s only gives a very rough answer to this
question. How they are distributed gives another.

Note that the term "eikosany" has been applied to more than one
scale. It is always a 3|6 CPS, but the factors, though usually
1.3.5.7.9.11, can vary.

Depending on the factors, the presence of 3/2s and near-3/2s
will change. So we can make an assumption -- that all intervals
in the space established by the factors are equally good
establishers of tonality. This isn't true, but whatever.

In this case, we notice that the Eikosany is symmetrical around
a point at which there is no note. This means that any note
of the structure is as good a root as another (under the
assumption). In particular, this stands in contrast to the
other high-density (lots of consonances per notes) structures
in extended JI -- tonality diamonds -- all the complete chords
of which share a single common note. (A composer once said he
could always recognize Partch's music because it was always in G.)

This is just one way of explaining the sort of thinking that
goes into a claim like 'without a strong tonal center'.

-Carl

🔗Aaron K. Johnson <akjmicro@comcast.net>

On Monday 29 November 2004 04:04 pm, Carl Lumma wrote:
> >> Whether it is having a consonance environment without a strong
> >> tonal center as with the eikosany.
> >
> >Could some one explain why the eikosany is 'without a strong tonal
> >center'? By my way of thinking, this would mean that it is short
> >on 3/2's, but that doesnt appear to be the case....plus, tonal
> >center is as much a function of repetition, etc. as of other
> >factors, wouldn't you agree?
>
> Compositional factors such as repetition, etc. will override
> scale-based theory, as always. This, however does not invalidate
> theory.

Right. Could we agree that theory sets up certain 'assumptions' or 'axioms' or
tendencies which could be then violated?

> 3/2s are strong establishers of tonality, but if a scale has more
> than one 3/2, what can be said for the tonal center of the overall
> scale? Counting 3/2s only gives a very rough answer to this
> question. How they are distributed gives another.

Yes. I would say that one might have tonality 'without traditional structures'
if the distribution didn't allow, for instance, 'ii-V-I' type progressions.
We already have this in strict 5-limit JI anyhow, because of the 'bad'
supertonic fifth on 'ii'...

> Note that the term "eikosany" has been applied to more than one
> scale. It is always a 3|6 CPS, but the factors, though usually
> 1.3.5.7.9.11, can vary.
>
> Depending on the factors, the presence of 3/2s and near-3/2s
> will change. So we can make an assumption -- that all intervals
> in the space established by the factors are equally good
> establishers of tonality. This isn't true, but whatever.
>
> In this case, we notice that the Eikosany is symmetrical around
> a point at which there is no note. This means that any note
> of the structure is as good a root as another (under the
> assumption). In particular, this stands in contrast to the
> other high-density (lots of consonances per notes) structures
> in extended JI -- tonality diamonds -- all the complete chords
> of which share a single common note. (A composer once said he
> could always recognize Partch's music because it was always in G.)

I see. One could modulate more freely away from an established 'I' then?

> This is just one way of explaining the sort of thinking that
> goes into a claim like 'without a strong tonal center'.

Thanks, Carl! As always, I think writing music or experimenting for oneself
gives the best understanding. And I asked a while back about 2-note CPS'es,
and there aren't many. This I think limits my contact with the full resources
of a set like Eikosany. And my desire and ability to implement CPS's of
larger cardinality is limited right now by time and predilection. I would
have to write custom software to do it, similar in function to the ET
software I've already written. It can be done when I have free time, and with
a Handels' "Messiah" to practice coming up, I won't hold my breath. I gotta
nail those pedal parts...

Right now, I'm also more drawn to non-harmonic pitch sets (irrational
relations, if you will). But I work in phases. Sometimes JI seems the way to
go.

-A.

Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Carl Lumma <ekin@lumma.org>

>> >> Whether it is having a consonance environment without a strong
>> >> tonal center as with the eikosany.
>> >
>> >Could some one explain why the eikosany is 'without a strong tonal
>> >center'? By my way of thinking, this would mean that it is short
>> >on 3/2's, but that doesnt appear to be the case....plus, tonal
>> >center is as much a function of repetition, etc. as of other
>> >factors, wouldn't you agree?
>>
>> Compositional factors such as repetition, etc. will override
>> scale-based theory, as always. This, however does not invalidate
>> theory.
>
>Right. Could we agree that theory sets up certain 'assumptions' or
>'axioms' or tendencies which could be then violated?

Yes, this is perhaps necessary when working out a grossly
incomplete theory.

>> 3/2s are strong establishers of tonality, but if a scale has more
>> than one 3/2, what can be said for the tonal center of the overall
>> scale? Counting 3/2s only gives a very rough answer to this
>> question. How they are distributed gives another.
>
>Yes. I would say that one might have tonality 'without traditional
>structures' if the distribution didn't allow, for instance, 'ii-V-I'
>type progressions. We already have this in strict 5-limit JI anyhow,
>because of the 'bad' supertonic fifth on 'ii'...

Ja. Well, in certain subsets of 5-limit JI. In real 5-limit JI
all chords are complete.

>> Note that the term "eikosany" has been applied to more than one
>> scale. It is always a 3|6 CPS, but the factors, though usually
>> 1.3.5.7.9.11, can vary.
>>
>> Depending on the factors, the presence of 3/2s and near-3/2s
>> will change. So we can make an assumption -- that all intervals
>> in the space established by the factors are equally good
>> establishers of tonality. This isn't true, but whatever.
>>
>> In this case, we notice that the Eikosany is symmetrical around
>> a point at which there is no note. This means that any note
>> of the structure is as good a root as another (under the
>> assumption). In particular, this stands in contrast to the
>> other high-density (lots of consonances per notes) structures
>> in extended JI -- tonality diamonds -- all the complete chords
>> of which share a single common note. (A composer once said he
>> could always recognize Partch's music because it was always in G.)
>
>I see. One could modulate more freely away from an established
>'I' then?

If you believe that the root of any 11-limit tetrad is its '1'
(whether or not it's in the tetrad), then yes. A diagram is
helpful (thanks to Erv Wilson)...

http://lumma.org/tuning/13,14.gif

Do you grok this?

>> This is just one way of explaining the sort of thinking that
>> goes into a claim like 'without a strong tonal center'.
>
>Thanks, Carl! As always, I think writing music or experimenting for
>oneself gives the best understanding. And I asked a while back about
>2-note CPS'es, and there aren't many.

2-note? I don't recall this investigation of yours... Oh, 12-note
ones. There may not be many 12-note CPSs, but there are many ways
to stick CPSs together to get 12 notes. But if you're into the
11-limit, why not just take a 12-note subset of the eikosany?
You can, for example, get chord progressions like this...

http://lumma.org/tuning/eikosany_modulation_2.mp3

It's just a 12-note mapping. Actually, it sounds good in 12...

http://lumma.org/tuning/eikosany_modulation_2.mid

-Carl

🔗Aaron K. Johnson <akjmicro@comcast.net>

On Wednesday 01 December 2004 11:40 am, Carl Lumma wrote:
> >> >> Whether it is having a consonance environment without a strong
> >> >> tonal center as with the eikosany.
> >> >
> >> >Could some one explain why the eikosany is 'without a strong tonal
> >> >center'? By my way of thinking, this would mean that it is short
> >> >on 3/2's, but that doesnt appear to be the case....plus, tonal
> >> >center is as much a function of repetition, etc. as of other
> >> >factors, wouldn't you agree?
> >>
> >> Compositional factors such as repetition, etc. will override
> >> scale-based theory, as always. This, however does not invalidate
> >> theory.
> >
> >Right. Could we agree that theory sets up certain 'assumptions' or
> >'axioms' or tendencies which could be then violated?
>
> Yes, this is perhaps necessary when working out a grossly
> incomplete theory.

Right-O.

> >> 3/2s are strong establishers of tonality, but if a scale has more
> >> than one 3/2, what can be said for the tonal center of the overall
> >> scale? Counting 3/2s only gives a very rough answer to this
> >> question. How they are distributed gives another.
> >
> >Yes. I would say that one might have tonality 'without traditional
> >structures' if the distribution didn't allow, for instance, 'ii-V-I'
> >type progressions. We already have this in strict 5-limit JI anyhow,
> >because of the 'bad' supertonic fifth on 'ii'...
>
> Ja. Well, in certain subsets of 5-limit JI. In real 5-limit JI
> all chords are complete.

Yes, I was thinking of the traditional 4x3 5-limit lattice as an example, so
designed for the traditional keyboard.

We still have a problem with commas in traditional chord progressions however.
Some people don't like the effect of common tones slipping around by
microtonal intervals. I do in certain cases, but I can see how one wouldn't
always want this, and seek a mean-tone solution to a traditional
voice-leading progression.

Personally, I think JI music should be different from traditional music, and
thus use different voice-leading rules (for instance, one might avoid
ii-V-I). Otherwise, what's the point?

> >> Note that the term "eikosany" has been applied to more than one
> >> scale. It is always a 3|6 CPS, but the factors, though usually
> >> 1.3.5.7.9.11, can vary.
> >>
> >> Depending on the factors, the presence of 3/2s and near-3/2s
> >> will change. So we can make an assumption -- that all intervals
> >> in the space established by the factors are equally good
> >> establishers of tonality. This isn't true, but whatever.
> >>
> >> In this case, we notice that the Eikosany is symmetrical around
> >> a point at which there is no note. This means that any note
> >> of the structure is as good a root as another (under the
> >> assumption). In particular, this stands in contrast to the
> >> other high-density (lots of consonances per notes) structures
> >> in extended JI -- tonality diamonds -- all the complete chords
> >> of which share a single common note. (A composer once said he
> >> could always recognize Partch's music because it was always in G.)
> >
> >I see. One could modulate more freely away from an established
> >'I' then?
>
> If you believe that the root of any 11-limit tetrad is its '1'
> (whether or not it's in the tetrad), then yes. A diagram is
> helpful (thanks to Erv Wilson)...
>
> http://lumma.org/tuning/13,14.gif
>
> Do you grok this?

It would help if this were a .tiff or .jpg or .png.... this particular .gif is
really pixellated when I try to zoom in to read the fine print. I think I get
the idea. Boy those diagrams are pretty to look at. It's easy to understand
that the theory alone has an aesthetic beauty.

> >> This is just one way of explaining the sort of thinking that
> >> goes into a claim like 'without a strong tonal center'.
> >
> >Thanks, Carl! As always, I think writing music or experimenting for
> >oneself gives the best understanding. And I asked a while back about
> >2-note CPS'es, and there aren't many.
>
> 2-note? I don't recall this investigation of yours... Oh, 12-note
> ones.

Ha-ha!

> There may not be many 12-note CPSs, but there are many ways
> to stick CPSs together to get 12 notes. But if you're into the
> 11-limit, why not just take a 12-note subset of the eikosany?
> You can, for example, get chord progressions like this...
>
> http://lumma.org/tuning/eikosany_modulation_2.mp3

Very cool. Very Prent Rodgers-esque timbrally, too....

> It's just a 12-note mapping. Actually, it sounds good in 12...
>
> http://lumma.org/tuning/eikosany_modulation_2.mid

It does sound good in 12-equal!

Best,
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Carl Lumma <ekin@lumma.org>

>> >> Note that the term "eikosany" has been applied to more than one
>> >> scale. It is always a 3|6 CPS, but the factors, though usually
>> >> 1.3.5.7.9.11, can vary.
>> >>
>> >> Depending on the factors, the presence of 3/2s and near-3/2s
>> >> will change. So we can make an assumption -- that all intervals
>> >> in the space established by the factors are equally good
>> >> establishers of tonality. This isn't true, but whatever.
>> >>
>> >> In this case, we notice that the Eikosany is symmetrical around
>> >> a point at which there is no note. This means that any note
>> >> of the structure is as good a root as another (under the
>> >> assumption). In particular, this stands in contrast to the
>> >> other high-density (lots of consonances per notes) structures
>> >> in extended JI -- tonality diamonds -- all the complete chords
>> >> of which share a single common note. (A composer once said he
>> >> could always recognize Partch's music because it was always in G.)
>> >
>> >I see. One could modulate more freely away from an established
>> >'I' then?
>>
>> If you believe that the root of any 11-limit tetrad is its '1'
>> (whether or not it's in the tetrad), then yes. A diagram is
>> helpful (thanks to Erv Wilson)...
>>
>> http://lumma.org/tuning/13,14.gif
>>
>> Do you grok this?
>
>It would help if this were a .tiff or .jpg or .png.... this particular
>.gif is really pixellated when I try to zoom in to read the fine print.

jpeg is a worse choice than gif or png for print. tiff is completely
obsolete. png would be better for peripheral reasons but not quality
reasons since both gif and png are lossless. The problem is the
quality of the scan.

>I think I get the idea. Boy those diagrams are pretty to look at.
>It's easy to understandthat the theory alone has an aesthetic beauty.

What you might be missing due to the unreadability of the text is
that the pentagon/star thing (four of them across the bottom)
represents an 11-limit hexad (otonal if upward-pointing, utonal
if downward-pointing). The point in the middle is 1, and the other
identities are the vertices of the pentagon. You can see that in
a tonality diamond (at right), all the chords involve a single point
in the dead center. Whereas in the eikosany (left), the center
is an empty hole. In fact there is no note common to all its
chords.

>> There may not be many 12-note CPSs, but there are many ways
>> to stick CPSs together to get 12 notes. But if you're into the
>> 11-limit, why not just take a 12-note subset of the eikosany?
>> You can, for example, get chord progressions like this...
>>
>> http://lumma.org/tuning/eikosany_modulation_2.mp3
>
>Very cool. Very Prent Rodgers-esque timbrally, too....

IIRC that's the GM reed organ patch that comes stock on a
Sound Blaster Live.

>> It's just a 12-note mapping. Actually, it sounds good in 12...
>>
>> http://lumma.org/tuning/eikosany_modulation_2.mid
>
>It does sound good in 12-equal!

Rock.

-Carl

🔗Aaron K. Johnson <akjmicro@comcast.net>

On Wednesday 01 December 2004 01:45 pm, Carl Lumma wrote:

> jpeg is a worse choice than gif or png for print.

Depends on the compression level.

> tiff is completely
> obsolete.

Try telling that to a graphics artist! It's still a standard lossless format
for professional CYMK print houses. My CD cover (Divide by Pi) was a TIFF
master.

Tiff look beautiful. JPL uses them for master copies of astro-photos, too.

> png would be better for peripheral reasons but not quality
> reasons since both gif and png are lossless. The problem is the
> quality of the scan.

There's another problem with GIF--it's an 8-bit color format. Loseless or not,
gif just plain sucks for serious work. Just about it's only good point is
being a low-bandwidth format, and it's standard color map scheme for the web.

> What you might be missing due to the unreadability of the text is
> that the pentagon/star thing (four of them across the bottom)
> represents an 11-limit hexad (otonal if upward-pointing, utonal
> if downward-pointing). The point in the middle is 1, and the other
> identities are the vertices of the pentagon. You can see that in
> a tonality diamond (at right), all the chords involve a single point
> in the dead center. Whereas in the eikosany (left), the center
> is an empty hole. In fact there is no note common to all its
> chords.

Thanks for that explanation....

Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Carl Lumma <ekin@lumma.org>

>> jpeg is a worse choice than gif or png for print.
>
>Depends on the compression level.

There are rare cases when jpeg is better for text.
jpeg2000 and kin are far better.

>> tiff is completely
>> obsolete.
>
>Try telling that to a graphics artist! It's still a standard lossless
>format for professional CYMK print houses. My CD cover (Divide by Pi)
>was a TIFF master.

I know, we still use tiff at the magazine. But that doesn't change
the fact that the format blowz. IIRC, it doesn't even specify
compression. I suppose png lacks a CMYK mode... forgot about that.

>> png would be better for peripheral reasons but not quality
>> reasons since both gif and png are lossless. The problem is the
>> quality of the scan.
>
>There's another problem with GIF--it's an 8-bit color format.

Indexed color seldom makes sense beyond 8 bits. So the real
problem with gif is that it doesn't have an RGB mode; a situation
png rights. But a sample like this should be indexed with
very few colors indeed. I didn't make this file, so I don't
really know what's going on under the hood.

-Carl

🔗David Beardsley <db@biink.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Aaron K. Johnson <akjmicro@comcast.net>

On Wednesday 01 December 2004 05:57 pm, Carl Lumma wrote:
> >> jpeg is a worse choice than gif or png for print.
> >
> >Depends on the compression level.
>
> There are rare cases when jpeg is better for text.
> jpeg2000 and kin are far better.
>
> >> tiff is completely
> >> obsolete.
> >
> >Try telling that to a graphics artist! It's still a standard lossless
> >format for professional CYMK print houses. My CD cover (Divide by Pi)
> >was a TIFF master.
>
> I know, we still use tiff at the magazine. But that doesn't change
> the fact that the format blowz. IIRC, it doesn't even specify
> compression. I suppose png lacks a CMYK mode... forgot about that.

I say below we should take this to metatuning...but first, a good background:

http://home.earthlink.net/~ritter/tiff/#whatis

Why does tiff blow? It's simply a rather simple professional standard format,
designed to meet print-shop particular needs for rasterization and publishing
of images. Yes, it has its limitations, I suppose.

Why should we forget about CYMK mode?

As for obsolescence, I think that's an empirical question whose answer in this
case is decidedly 'no'. Something is obsolete only when a small minority of
people still use it (I would say LISP and FORTRAN are obsolete, save
academia, and parenthesis-lovers). If Keyboard magazine still uses .tiff, it
ain't obsolete.

Even were it true, let's also not forget that obsolete things still hold sway,
and are enlightening--like LISP and Latin are (both are incredibly good to
know)--and influence future designs. But I digress. I think Carl, what you
meant by saying obsolete was "I don't like it or condone it's use myself".
Right? ;)

>
> Indexed color seldom makes sense beyond 8 bits.

Stop right there---explain why you think that...

Let's take this to metatuning.

Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Aaron K. Johnson <akjmicro@comcast.net>

On Wednesday 01 December 2004 11:40 am, Carl Lumma wrote:

> http://lumma.org/tuning/13,14.gif
>
> Do you grok this?

I *do* have another question after doing my own research on Kraig Grady's site
in the Wilson archives, at Wilson's text, which didn't answer it for me.

I wasn't able to figure out how the star-like 'Eikosany' figure was derived
from the bottom pentagonal figure. Wilson just says it was without fleshing
out just how he did it. The factors on the outside of the star figure don't
relate proportionally by factors that the pentagonal model do, as is claimed.
So I'm a bit confused. Or missing some information.

OTOH, I do notice a certain pattern in the rotational symmetry of the 'star'
figure, where for instance, the '1' factor occurs as every other vertex on
the outside, etc. But that seems arbitrary. And wouldn't it be better to do
this diagram with the actual resultant ratios listed, and not the generating
factors, so that we might intuit the pentachords which are formed as a result
of the process?

Kraig? Carl? Monz? Gene? Paul, if you are lurking? Othere experts on Wilson
out there lurking, who I have neglected to mention?

-A.

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Carl Lumma <ekin@lumma.org>

>> http://lumma.org/tuning/13,14.gif
>> Do you grok this?
>
>I *do* have another question after doing my own research on
>Kraig Grady's site in the Wilson archives, at Wilson's text,
>which didn't answer it for me.
>
>I wasn't able to figure out how the star-like 'Eikosany' figure
>was derived from the bottom pentagonal figure. Wilson just says
>it was without fleshing out just how he did it.

From what I know of Erv, I think he's really trying to be as
clear as possible. And in a sense, I think he is. I wouldn't
want him to hand-hold any more than he does. Play and
exploration are an inseparable part of his approach. The fun
of it is figuring it out for yourself.

>The factors on the outside of the star figure don't relate
>proportionally by factors that the pentagonal model do,
>as is claimed.

The figures are to scale. Remember the eikosany contains
4-tone subsets of the hexads. Can you see them?

>So I'm a bit confused. Or missing some information.
>
>OTOH, I do notice a certain pattern in the rotational symmetry of
>the 'star' figure, where for instance, the '1' factor occurs as
>every other vertex on the outside, etc. But that seems arbitrary.

The eikosany itself is not consonant. You're barking up the wrong
tree viewing it as a whole.

But start with the diamond. The component chords are easier to
spot there.

>And wouldn't it be better to do this diagram with the actual
>resultant ratios listed, and not the generating factors, so that
>we might intuit the pentachords which are formed as a result
>of the process?

No.

-Carl

🔗Aaron K. Johnson <akjmicro@comcast.net>

On Friday 03 December 2004 03:20 pm, Carl Lumma wrote:
> >> http://lumma.org/tuning/13,14.gif
> >> Do you grok this?
> >
> >I *do* have another question after doing my own research on
> >Kraig Grady's site in the Wilson archives, at Wilson's text,
> >which didn't answer it for me.
> >
> >I wasn't able to figure out how the star-like 'Eikosany' figure
> >was derived from the bottom pentagonal figure. Wilson just says
> >it was without fleshing out just how he did it.
>
> From what I know of Erv, I think he's really trying to be as
> clear as possible. And in a sense, I think he is. I wouldn't
> want him to hand-hold any more than he does. Play and
> exploration are an inseparable part of his approach. The fun
> of it is figuring it out for yourself.

That's true, but I don't have a lot of leisure time to play around with this
right now, and I suppose if no one wants to 'hand-hold' I'll have to wait.

Sigh.

> >The factors on the outside of the star figure don't relate
> >proportionally by factors that the pentagonal model do,
> >as is claimed.
>
> The figures are to scale. Remember the eikosany contains
> 4-tone subsets of the hexads. Can you see them?

Not yet. I'm assuming by a hexad you mean 1,3,5,7,9,11

> >So I'm a bit confused. Or missing some information.
> >
> >OTOH, I do notice a certain pattern in the rotational symmetry of
> >the 'star' figure, where for instance, the '1' factor occurs as
> >every other vertex on the outside, etc. But that seems arbitrary.
>
> The eikosany itself is not consonant. You're barking up the wrong
> tree viewing it as a whole.
>
> But start with the diamond. The component chords are easier to
> spot there.

The diamond makes sense to me.

> >And wouldn't it be better to do this diagram with the actual
> >resultant ratios listed, and not the generating factors, so that
> >we might intuit the pentachords which are formed as a result
> >of the process?
>
> No.

I'm glad it's so black and white for you. I always believe that more
information about interelationships of things is better than less, so I still
say yes it would be, or at the very least, it couldn't hurt. If there's no
relationship, I'll find out soon enough. What I know about math and the
universe tells me there is a good chance there is. But, I'll do it myself
when I have the time, perhaps I'll be better in my ability to answer
questions for the next person having done it, who knows.

Paul Erlich has offered to explore this together on the phone sometime,
thanks, we can drop this topic if you are sick of it....

Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Carl Lumma <ekin@lumma.org>

>> >> http://lumma.org/tuning/13,14.gif
>> >> Do you grok this?
>> >
>> >I *do* have another question after doing my own research on
>> >Kraig Grady's site in the Wilson archives, at Wilson's text,
>> >which didn't answer it for me.
>> >
>> >I wasn't able to figure out how the star-like 'Eikosany' figure
>> >was derived from the bottom pentagonal figure. Wilson just says
>> >it was without fleshing out just how he did it.
>>
>> From what I know of Erv, I think he's really trying to be as
>> clear as possible. And in a sense, I think he is. I wouldn't
>> want him to hand-hold any more than he does. Play and
>> exploration are an inseparable part of his approach. The fun
>> of it is figuring it out for yourself.
>
>That's true, but I don't have a lot of leisure time to play around
>with this right now, and I suppose if no one wants to 'hand-hold'
>I'll have to wait.
>
>Sigh.

What do you think I'm trying to do?

>> >The factors on the outside of the star figure don't relate
>> >proportionally by factors that the pentagonal model do,
>> >as is claimed.
>>
>> The figures are to scale. Remember the eikosany contains
>> 4-tone subsets of the hexads. Can you see them?
>
>Not yet. I'm assuming by a hexad you mean 1,3,5,7,9,11

That's a otonal or harmonic hexad (as opposed to a utonal or
subharmonic hexad). The four stars across the bottom are hexads.

>> >So I'm a bit confused. Or missing some information.
>> >
>> >OTOH, I do notice a certain pattern in the rotational symmetry of
>> >the 'star' figure, where for instance, the '1' factor occurs as
>> >every other vertex on the outside, etc. But that seems arbitrary.
>>
>> The eikosany itself is not consonant. You're barking up the wrong
>> tree viewing it as a whole.
>>
>> But start with the diamond. The component chords are easier to
>> spot there.
>
>The diamond makes sense to me.

How many hexads does it contain?

>> >And wouldn't it be better to do this diagram with the actual
>> >resultant ratios listed, and not the generating factors, so that
>> >we might intuit the pentachords which are formed as a result
>> >of the process?
>>
>> No.
>
>I'm glad it's so black and white for you. I always believe that
>more information about interelationships of things is better than
>less,

Maybe I misunderstood your question. What's a resultant ratio?

Information about the interrelationships is precisely what Erv's
approach gives. The edges are consonant intervals, and their
identities are shown by their orientation on the page. The
vertices are pitches (or in an octave-equivalent interpretation,
"notes"). Their names as absolute ratios would change wildly
depending on which point was chosen as 1/1.

>Paul Erlich has offered to explore this together on the phone
>sometime, thanks, we can drop this topic if you are sick of it....

I'm not at all sick of it. It sounds like you are! I think
you're trying too hard (it's really simple).

-Carl

🔗Pete McRae <ambassadorbob@yahoo.com>

I have a "hand-holding" question, if I may:

What's the best way to get Scala to reduce a CPS to a 1-octave span? I have a set of ratios I expect to see when it comes up right. I got close, but time ran out on me... (I was working on a .tun file for the Cronox, to match my other instruments.)

Carl Lumma <ekin@lumma.org> wrote:

What do you think I'm trying to do?

That's a otonal or harmonic hexad (as opposed to a utonal or
subharmonic hexad). The four stars across the bottom are hexads.

How many hexads does it contain?

Maybe I misunderstood your question. What's a resultant ratio?

>Paul Erlich has offered to explore this together on the phone
>sometime, thanks, we can drop this topic if you are sick of it....

I'm not at all sick of it. It sounds like you are! I think
you're trying too hard (it's really simple).

-Carl

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🔗Carl Lumma <ekin@lumma.org>

>I have a "hand-holding" question, if I may:
>
>What's the best way to get Scala to reduce a CPS to a 1-octave span?
>I have a set of ratios I expect to see when it comes up right. I
>got close, but time ran out on me... (I was working on a .tun file
>for the Cronox, to match my other instruments.)

The procedure to get a CPS in Scala is not completely straightforward.
Have you checked the docs? I bet Manuel explains it there.

There was a good discussion of this on one of the lists when Aaron
started his CPS investigations...

...Well, now I can't find it. Lemmee see, I think it's...

() "cps"; tell it the "combination count" and what factors you
want, ie for a 7-limit hexany: 4, 2 and then 1 3 5 7.

() "del 0" to get rid of the 1/1.

() "normalize" to refactor to a single octave and sort.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Aaron K. Johnson <akjmicro@comcast.net>

🔗Carl Lumma <ekin@lumma.org>

🔗Aaron K. Johnson <akjmicro@comcast.net>

On Friday 03 December 2004 04:06 pm, Carl Lumma wrote:

> Information about the interrelationships is precisely what Erv's
> approach gives. The edges are consonant intervals, and their
> identities are shown by their orientation on the page. The
> vertices are pitches (or in an octave-equivalent interpretation,
> "notes"). Their names as absolute ratios would change wildly
> depending on which point was chosen as 1/1.

I knew the last bit, so I suppose you're right that the abstraction is more
helpful to see the overall logic. For me, sometimes I need to go in the
direction of specific concrete examples to general abstract principles.
That's what I was getting at....

> >Paul Erlich has offered to explore this together on the phone
> >sometime, thanks, we can drop this topic if you are sick of it....
>
> I'm not at all sick of it. It sounds like you are! I think
> you're trying too hard (it's really simple).

Actually, I wasn't trying hard enough. Right now, I'm whacking myself with a
cat o' nine tails.

Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

🔗Pete McRae <ambassadorbob@yahoo.com>

Yeah, I messed around with it. Not quite what I expected, but maybe I just need to keep trying slightly different tweaks, like which degree to delete besides 0, or something.

I liked your article, too. Thanks!

Carl Lumma <ekin@lumma.org> wrote:

The procedure to get a CPS in Scala is not completely straightforward.
Have you checked the docs? I bet Manuel explains it there.

There was a good discussion of this on one of the lists when Aaron
started his CPS investigations...

...Well, now I can't find it. Lemmee see, I think it's...

() "cps"; tell it the "combination count" and what factors you
want, ie for a 7-limit hexany: 4, 2 and then 1 3 5 7.

() "del 0" to get rid of the 1/1.

() "normalize" to refactor to a single octave and sort.

-Carl

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🔗Carl Lumma <ekin@lumma.org>

>>>I have a "hand-holding" question, if I may:
>>>
>>>What's the best way to get Scala to reduce a CPS to a 1-octave span?
>>>I have a set of ratios I expect to see when it comes up right. I
>>>got close, but time ran out on me... (I was working on a .tun file
>>>for the Cronox, to match my other instruments.)
>>
>>
>>The procedure to get a CPS in Scala is not completely straightforward.
>>Have you checked the docs? I bet Manuel explains it there.
>>
>>There was a good discussion of this on one of the lists when Aaron
>>started his CPS investigations...
>>
>>...Well, now I can't find it. Lemmee see, I think it's...
>>
>>() "cps"; tell it the "combination count" and what factors you
>>want, ie for a 7-limit hexany: 4, 2 and then 1 3 5 7.
>>
>>() "del 0" to get rid of the 1/1.
>>
>>() "normalize" to refactor to a single octave and sort.
>
>
>Yeah, I messed around with it. Not quite what I expected, but
>maybe I just need to keep trying slightly different tweaks,
>like which degree to delete besides 0, or something.
>
>I liked your article, too. Thanks!

If you want a CPS, I believe you in fact need to delete
degree 0. You might get some cool results deleting something
else, though!

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Aaron K. Johnson <akjmicro@comcast.net>

🔗Carl Lumma <ekin@lumma.org>

>> >> In particular, this stands in contrast to the
>> >> other high-density (lots of consonances per notes) structures
>> >> in extended JI -- tonality diamonds -- all the complete chords
>> >> of which share a single common note. (A composer once said he
>> >> could always recognize Partch's music because it was always in G.)
>> >
>> >(Gene wrote)There are more than just these two high-density structures,
>> >surely.
>>
>> Indeed. I meant, 'the other famous one'. Though I suppose
>> Euler-Fokker genuses are fairly well-known...
>
>Shall we enumerate all the known high-density structures for fun?
>
>I'll start:
>
>1) CPSes
>2) Tonality Diamonds
>3) Euler-Fokker genera

2.5) stellated CPSs

...I can't think of any more existing named structures at the
moment, but one can pretty much take any convex swath of tone
space.

Gene has done lots of nice work on tuning-math... finding balls
containing all pitches within a certain radius of 1/l, for a
variety of different lattice-distance measures.

-Carl