CPS scales

Hi folks, no time just now to catch up on everything . . .

Anyway, here's a little something I wrote a while back:

http://tonalsoft.com/td/erlich/paul-cps.htm

Wherever you see
http://www.anaphoria.com/dal16.html (now defunct)
please substitute page 21 of
http://www.anaphoria.com/dal.PDF
-- a very beautiful diagram!!

The composer of my favorite piece of 2003 surely had to know *a
little* about the theory surrounding 19-equal.

Similarly, understanding the theory surrounding CPS scales will
probably not hurt his future explorations in that area ;)

When you get to the point in the article that says, "the dekanies are
interesting scales that you should work out for yourself as an
exercise", there should be a link over to Dave Keenan's wonderful
Excel "Tumbling Dekany" algorithmic composition / 3D movie:

http://www.uq.net.au/~zzdkeena/Music/StereoDekany.xls

There you can play with the 5 factors used to define the dekany, so
as to systematically explore the different dekany sounds available in
a given eikosany, for example. I suggest moving the "Distance at
which volume goes to zero" slider up from 150 to 180 or so, and to
really experience the justness of the intonation, you might want to
change the patch to the vibrato-less "021 Reed Organ" -- you might be
reminded of Harry Partch's Chromelodeon, but played in a more dreamy,
minimalist sort of way . . .

Strap on your headphones and 3D glasses and play!

Best wishes for the new year,
Paul

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
...
> When you get to the point in the article that says, "the dekanies are
> interesting scales that you should work out for yourself as an
> exercise", there should be a link over to Dave Keenan's wonderful
> Excel "Tumbling Dekany" algorithmic composition / 3D movie:
>
> http://www.uq.net.au/~zzdkeena/Music/StereoDekany.xls

Thnaks Paul, but the appropriate link now is

http://dkeenan.com/Music/StereoDekany.htm

That other one won't be around much longer.

All the best for the new year.

hi paul,

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> Hi folks, no time just now to catch up on everything . . .
>
> Anyway, here's a little something I wrote a while back:
>
> http://tonalsoft.com/td/erlich/paul-cps.htm
>
> Wherever you see
> http://www.anaphoria.com/dal16.html (now defunct)
> please substitute page 21 of
> http://www.anaphoria.com/dal.PDF
> -- a very beautiful diagram!!
>
> The composer of my favorite piece of 2003 surely had to know *a
> little* about the theory surrounding 19-equal.
>
> Similarly, understanding the theory surrounding CPS scales will
> probably not hurt his future explorations in that area ;)
>
> When you get to the point in the article that says, "the
> dekanies are interesting scales that you should work out
> for yourself as an exercise", there should be a link over
> to Dave Keenan's wonderful Excel "Tumbling Dekany"
> algorithmic composition / 3D movie:
>
> http://www.uq.net.au/~zzdkeena/Music/StereoDekany.xls
>
> There you can play with the 5 factors used to define the
> dekany, so as to systematically explore the different
> dekany sounds available in a given eikosany, for example.
> I suggest moving the "Distance at which volume goes to
> zero" slider up from 150 to 180 or so, and to really
> experience the justness of the intonation, you might want
> to change the patch to the vibrato-less "021 Reed Organ"
> -- you might be reminded of Harry Partch's Chromelodeon,
> but played in a more dreamy, minimalist sort of way . . .
>
> Strap on your headphones and 3D glasses and play!

thanks for that! it's been updated, with the new URLs for
Dave's tumbling dekany and _D'Alessandro_.

-monz

Paul and tuners,

Any comments about the scale I designed in Scala from 2 hexanies?

!
dual-hexany CPS
11
!
16/15
35/32
75/64
5/4
21/16
175/128
3/2
25/16
105/64
7/4
15/8

What I did was 2,4 [1,3,5,7] and 2,4[1,1/3,1/5,1/7]
I did 'delete 0' to make them both 'genuine CPSes'
I did a 'reduce 2/1' and 'sort'

Then, I rotated one of them to start with 16/15, and I rotated the other to
the only rotation that gave the union of the two 'subscales' 12 unique
pitches.....

Even if this is somehow not correct, I find that it sounds interesting, and
maybe you would like it too, if you played it on your keyboard.....

I'm interested to know anyone's thoughts or comments!

Best,
Aaron Krister Johnson

P.S. If I understand correctly, I think this scale is not 'proper'.

On Thursday 01 January 2004 04:17 pm, Paul Erlich wrote:
> Hi folks, no time just now to catch up on everything . . .
>
> Anyway, here's a little something I wrote a while back:
>
> http://tonalsoft.com/td/erlich/paul-cps.htm
>
> Wherever you see
> http://www.anaphoria.com/dal16.html (now defunct)
> please substitute page 21 of
> http://www.anaphoria.com/dal.PDF
> -- a very beautiful diagram!!
>
> The composer of my favorite piece of 2003 surely had to know *a
> little* about the theory surrounding 19-equal.
>
> Similarly, understanding the theory surrounding CPS scales will
> probably not hurt his future explorations in that area ;)
>
> When you get to the point in the article that says, "the dekanies are
> interesting scales that you should work out for yourself as an
> exercise", there should be a link over to Dave Keenan's wonderful
> Excel "Tumbling Dekany" algorithmic composition / 3D movie:
>
> http://www.uq.net.au/~zzdkeena/Music/StereoDekany.xls
>
> There you can play with the 5 factors used to define the dekany, so
> as to systematically explore the different dekany sounds available in
> a given eikosany, for example. I suggest moving the "Distance at
> which volume goes to zero" slider up from 150 to 180 or so, and to
> really experience the justness of the intonation, you might want to
> change the patch to the vibrato-less "021 Reed Organ" -- you might be
> reminded of Harry Partch's Chromelodeon, but played in a more dreamy,
> minimalist sort of way . . .
>
> Strap on your headphones and 3D glasses and play!
>
> Best wishes for the new year,
> Paul
>
>
> You do not need web access to participate. You may subscribe through
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--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

Hi folks

i am not sure if anyone is interested but pmask by
Maurizio Umberto Puxeddu, is an implementation of
cmask that enables score generation for csound.

I have hacked a few of the python scripts that come
with it to enable pythag,LY and Beardsley/Pagano 17
limit scales rather quickly. One can also set tet12
and create a class for extet12 if one sees fit. then
simply run the python script and voila, instant
parameter based indeterminate JI.

I would be interested to hear who else is using and
what they are using for JI/eXtets as it pertains to
computer software.

cheers

Pat

__________________________________
Do you Yahoo!?
Find out what made the Top Yahoo! Searches of 2003
http://search.yahoo.com/top2003

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

> Any comments about the scale I designed in Scala from 2 hexanies?
>
> !
> dual-hexany CPS
> 11
> !
> 16/15
> 35/32
> 75/64
> 5/4
> 21/16
> 175/128
> 3/2
> 25/16
> 105/64
> 7/4
> 15/8

The first comment is that what you've given is an 11-note scale with
interval of equivalence 15/8; what you wanted was this:

! dualhex.scl
Aaron Johnson's dual-hexany CPS January 2 2004
12
!
16/15
35/32
75/64
5/4
21/16
175/128
3/2
25/16
105/64
7/4
15/8
2

Scala knows not of it, but does tell us it is epimorphic but not
proper. It has two major tetrads (1--5/4--3/2--7/4) and two minor ones
(1--6/5--3/2--12/7) four each of major and minor triads, and is a
good candidate for tempering by 225/224.
>
> What I did was 2,4 [1,3,5,7] and 2,4[1,1/3,1/5,1/7]
> I did 'delete 0' to make them both 'genuine CPSes'
> I did a 'reduce 2/1' and 'sort'
>
> Then, I rotated one of them to start with 16/15, and I rotated the
other to
> the only rotation that gave the union of the two 'subscales' 12
unique
> pitches.....
>
> Even if this is somehow not correct, I find that it sounds
interesting, and
> maybe you would like it too, if you played it on your keyboard.....
>
> I'm interested to know anyone's thoughts or comments!
>
> Best,
> Aaron Krister Johnson
>
> P.S. If I understand correctly, I think this scale is not 'proper'.
>
> On Thursday 01 January 2004 04:17 pm, Paul Erlich wrote:
> > Hi folks, no time just now to catch up on everything . . .
> >
> > Anyway, here's a little something I wrote a while back:
> >
> > http://tonalsoft.com/td/erlich/paul-cps.htm
> >
> > Wherever you see
> > http://www.anaphoria.com/dal16.html (now defunct)
> > please substitute page 21 of
> > http://www.anaphoria.com/dal.PDF
> > -- a very beautiful diagram!!
> >
> > The composer of my favorite piece of 2003 surely had to know *a
> > little* about the theory surrounding 19-equal.
> >
> > Similarly, understanding the theory surrounding CPS scales will
> > probably not hurt his future explorations in that area ;)
> >
> > When you get to the point in the article that says, "the dekanies
are
> > interesting scales that you should work out for yourself as an
> > exercise", there should be a link over to Dave Keenan's wonderful
> > Excel "Tumbling Dekany" algorithmic composition / 3D movie:
> >
> > http://www.uq.net.au/~zzdkeena/Music/StereoDekany.xls
> >
> > There you can play with the 5 factors used to define the dekany,
so
> > as to systematically explore the different dekany sounds
available in
> > a given eikosany, for example. I suggest moving the "Distance at
> > which volume goes to zero" slider up from 150 to 180 or so, and to
> > really experience the justness of the intonation, you might want
to
> > change the patch to the vibrato-less "021 Reed Organ" -- you
might be
> > reminded of Harry Partch's Chromelodeon, but played in a more
dreamy,
> > minimalist sort of way . . .
> >
> > Strap on your headphones and 3D glasses and play!
> >
> > Best wishes for the new year,
> > Paul
> >
> >
> > You do not need web access to participate. You may subscribe
through
> > email. Send an empty email to one of these addresses:
> > tuning-subscribe@yahoogroups.com - join the tuning group.
> > tuning-unsubscribe@yahoogroups.com - unsubscribe from the
tuning group.
> > tuning-nomail@yahoogroups.com - put your email message delivery
on hold
> > for the tuning group. tuning-digest@yahoogroups.com - change your
> > subscription to daily digest mode. tuning-normal@yahoogroups.com -
change
> > your subscription to individual emails. tuning-
help@yahoogroups.com -
> > receive general help information.
> >
> >
> >
> > Yahoo! Groups Links
> >
> > To visit your group on the web, go to:
> > /tuning/
> >
> > To unsubscribe from this group, send an email to:
> > tuning-unsubscribe@yahoogroups.com
> >
> > Your use of Yahoo! Groups is subject to:
> > http://docs.yahoo.com/info/terms/
>
> --
> OCEAN, n. A body of water occupying about two-thirds of a world
made
> for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> Scala knows not of it, but does tell us it is epimorphic but not
> proper. It has two major tetrads (1--5/4--3/2--7/4) and two minor
ones
> (1--6/5--3/2--12/7) four each of major and minor triads, and is a
> good candidate for tempering by 225/224.

Saying it is a good candidate for 225/224-tempering isn't saying
much, because almost any 5 or 7 limit JI scale will be. However, dual-
hexany has five pure fifths plus one flat by 225/224, five pure 7/4s
plus one sharp by 225/224, three pure 7/6s and two flat by 225/224,
four pure 7/5s and one sharp by 7/5, and to top it all off two
different sizes of 9/7, neither of which is pure. Tempering out the
septimal kleisma 225/224 smooths all of this out.

The 225/224-planar temperament, for which the names Pauline and
Byzantine have been nominated but not seconded and for which I'd like
a name everyone can accept, is analogous in a way to meantone if we
assume the minor thirds are pure. In meantone, four (3/2)^4 ~ 5, and
if we choose exact fives, we get 1/4-comma meantone. In 225/224-
planar, (3/2)^2 * (5/4)^2 ~ 7/2, and choosing exact sevens gives us
1/4-kleismic.

Below is dual-hexany in 1/4-kleismic 225/224-planar:

! dualhexk.scl
Aaron Johnson's dual-hexany in 1/4-kleismic
12
!
115.587047
153.211740
268.798786
384.385833
468.853027
537.597573
700.027120
768.771666
853.238860
968.825906
1084.412953
1200.000000

>Any comments about the scale I designed in Scala from 2 hexanies?
>
>!
>dual-hexany CPS
> 11
>!
> 16/15
> 35/32
> 75/64
> 5/4
> 21/16
> 175/128
> 3/2
> 25/16
> 105/64
> 7/4
> 15/8
>
>What I did was 2,4 [1,3,5,7] and 2,4[1,1/3,1/5,1/7]
>I did 'delete 0' to make them both 'genuine CPSes'
>I did a 'reduce 2/1' and 'sort'

Then you will get two instances of this:

>0: 1/1 0.000 unison, perfect prime
>1: 7/6 266.871 septimal minor third
>2: 5/4 386.314 major third
>3: 35/24 653.185 septimal semi-diminished fifth
>4: 5/3 884.359 major sixth, BP sixth
>5: 7/4 968.826 harmonic seventh

Once again, the "otonal" and "utonal" versions are the
same. Why are you going to the trouble?

>Then, I rotated one of them to start with 16/15,

There's no 16/15... are you sure you did delete 0?

-Carl

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
> Paul and tuners,
>
> Any comments about the scale I designed in Scala from 2 hexanies?
>
> !
> dual-hexany CPS
> 11
> !
> 16/15
> 35/32
> 75/64
> 5/4
> 21/16
> 175/128
> 3/2
> 25/16
> 105/64
> 7/4
> 15/8
>
>
> What I did was 2,4 [1,3,5,7] and 2,4[1,1/3,1/5,1/7]
> I did 'delete 0' to make them both 'genuine CPSes'
> I did a 'reduce 2/1' and 'sort'
>
> Then, I rotated one of them to start with 16/15, and I rotated the
other to
> the only rotation that gave the union of the two 'subscales' 12
unique
> pitches.....
>
> Even if this is somehow not correct, I find that it sounds
interesting, and
> maybe you would like it too, if you played it on your keyboard.....
>
> I'm interested to know anyone's thoughts or comments!
>
> Best,
> Aaron Krister Johnson

Hi Aaron,

Let me start out by saying that yes, "it's somehow not correct", but
of course that shouldn't prevent you from making great music with it.
However, I think that if you bear with me and follow what I have to
say below, it will only help toward your eventual musical mastery of
CPS-based scales, by which I mean understanding of, hence control
over, the materials therein.

First of all, I'm confused, because you start off with

> dual-hexany CPS
> 11

indicating an 11-note per octave scale, but then you say

> 12 unique
> pitches.....

Anyway, I'm going to draw a lattice of your scale. As you will know
from my 'gentle introduction' I just posted about:

http://tonalsoft.com/td/erlich/paul-cps.htm

a hexany will look like an octahedron when latticed (and this is true
whether you write it as 2,4 [1,3,5,7] or as 2,4[1,1/3,1/5,1/7] since
those are two ways of deriving exactly the same structure). So a
scale consisting of two hexanies will look like two octahedra. If
they share 1 note, you'll have 11 notes all together. If they share 2
notes, you'll have 10 notes all together. Etc.

To see this lattice correctly on the web, click on 'Reply':

175/128
,'/ \`.
25/16/---\75/64
/|\/ \/|
/ |/\ /\|
/35/32----105/64
/,'/ \`.\ /,'/
5/4-/---\15/8 /
|\/ \/| /
|/\ /\|/
7/4------21/16
`.\ /,'
3/2

16/15

So what you have here, Aaron, are two hexanies (the two octahedra in
the lattice) with two notes in common (namely 35/32 and 15/8), making
10 notes all together, and one note totally disconnected from the two
hexanies, namely 16/15. You said that you 'rotated one of them to
start with 16/15', but clearly that did not work correctly. Anyway,
that accounts for 11 notes (as you said originally), not sure where
you see 12.

Since any of the consonant intervals of the hexany can be used as a
pair of notes to connect it with another hexany, and the hexany has
12 consonant intervals, there are 12/2 = 6 distinct (not counting
transpositions/modes) ways to make a 10-note scale out of two
hexanies (assuming [1,3,5,7] or any given set of 4 factors).

If you really want a 12-note scale, you could look at different ways
of adding 2 notes to each of the 6 possibilities above. There will
also be 3 distinct ways to make an 11-note scale out of two hexanies
(with one note in common), and you could look at different ways of
adding 1 note to each of those. Finally, there are quite a few ways
of combining two hexanies so that they are connected with consonance
but share no notes in common; this idea will immediately provide a
host of 12-note scales.

Probably too many possibilities to consider, though I'd be happy to
work some out for you if you wish. But it would be better if we could
add some further constraint to narrow down the search. Kraig is fond
of the CS property. For a 12-note scale, that would mean that on the
keyboard, you'd never be able to find two instances of the same
interval by playing intervals that would be different in 12-equal.
Sound reasonable? Then we just need to ask Gene (who's been doing
similar things on tuning-math lately) a question:

Gene:

Can you list all the distinct 12-note 7-limit periodicity blocks
which contain two hexanies?

Hopefully Gene can answer this without too much trouble, given his
mathematical virtuosity.

Enjoy your musical explorations, and I'm here for you if you need me,
Paul

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> wrote:
>
> > Any comments about the scale I designed in Scala from 2 hexanies?
> >
> > !
> > dual-hexany CPS
> > 11
> > !
> > 16/15
> > 35/32
> > 75/64
> > 5/4
> > 21/16
> > 175/128
> > 3/2
> > 25/16
> > 105/64
> > 7/4
> > 15/8
>
> The first comment is that what you've given is an 11-note scale
with
> interval of equivalence 15/8;

Looks like the 'delete 0' strategy is not a good one, then; not to
mention, what if one of the products does turn out to be a power of 2?

> what you wanted was this:
>
> ! dualhex.scl
> Aaron Johnson's dual-hexany CPS January 2 2004
> 12
> !
> 16/15
> 35/32
> 75/64
> 5/4
> 21/16
> 175/128
> 3/2
> 25/16
> 105/64
> 7/4
> 15/8
> 2

I disagree that that's what Aaron wanted. The note 2, or 1/1, or 2/1,
does not belong to either of the hexanies in this scale. (And neither
does 16/15.)

>Gene:
>
>Can you list all the distinct 12-note 7-limit periodicity blocks
>which contain two hexanies?

Paul Hahn did a similar search years ago, namely all rotations
of this structure...

35/32-----105/64
,'/ \`. ,' /
5/4-/---\-15/8/
/|\/ \/| /
/ |/\ /\|/
/ 7/4------21/16
/,'/ \`.\ /,'/
1/1-/---\-3/2 /
/|\/ \/| /
/ |/\ /\|/
/ 7/5------21/20
/,' `.\ /.'
8/5-------6/5

...which is a great way to get 2 hexanies in 12 tones. We didn't
know about Fokker blocks back then, but only in 3 versions are all
intervals 25/24 or larger...

a) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
b) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8
c) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/16, 7/4, 15/8

Version (c) is the only one covered by tetrads, and it is one put
forth by Wilson in 1967. Paul Hahn noted that (a) has a lower
diameter. I played around with them a bit, and preferred (c), and
I generally think chord coverage is an important thing.

-Carl

Correction...

a) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
b) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/64, 7/4, 15/8
c) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8

...(b) and (c) were switched below. The rest of the post was fine.

a) 35/32
,'/ \`.
5/4-/---\-15/8
/:\/ \/:
/ :/\ /\:
/ 7/4------21/16
/,'/ \`.\ /,'/
1/1-/---\-3/2 /
/:\/ \/: /
/ :/\ /\:/
/ 7/5------21/20
/,' \`.\ /,'/
8/5-----\-6/5 /
\ : /
\:/
42/25

c) 25/16
/:\
/ : \
/35/32\
/,'/ \`.\
5/4-/---\-15/8
/:\/ \/:
/ :/\ /\:
/ 7/4------21/16
/,'/ \`.\ /,'/
1/1-/---\-3/2 /
:\/ \/: /
:/\ /\:/
7/5------21/20
\`.\ /,'/
\ 6/5 /
\ : /
\:/
42/25

None of these scales are proper, and they all have the same largest
and smallest intervals. They each have six 5-limit ASS's, but (c) is
biased toward the 10:12:15:18.

History repeats itself! But none of you suckers can probably tell.
Even I, with my All Seeing Eye, can seldom tell. The list is a
black box which sucks history Away. It's baaad (worse than sine
waves). A Wiki doesn't have this problem. The list shines for
discussion, but not for publushin'.

-Carl

>>Gene:
>>
>>Can you list all the distinct 12-note 7-limit periodicity blocks
>>which contain two hexanies?
>
>Paul Hahn did a similar search years ago, namely all rotations
>of this structure...
>
> 35/32-----105/64
> ,'/ \`. ,' /
> 5/4-/---\-15/8/
> /|\/ \/| /
> / |/\ /\|/
> / 7/4------21/16
> /,'/ \`.\ /,'/
> 1/1-/---\-3/2 /
> /|\/ \/| /
> / |/\ /\|/
> / 7/5------21/20
> /,' `.\ /.'
> 8/5-------6/5
>
>...which is a great way to get 2 hexanies in 12 tones. We didn't
>know about Fokker blocks back then, but only in 3 versions are all
>intervals 25/24 or larger...
>
>a) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
>b) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8
>c) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/16, 7/4, 15/8
>
>Version (c) is the only one covered by tetrads, and it is one put
>forth by Wilson in 1967. Paul Hahn noted that (a) has a lower
>diameter. I played around with them a bit, and preferred (c), and
>I generally think chord coverage is an important thing.
>
>-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> I disagree that that's what Aaron wanted. The note 2, or 1/1, or 2/1,
> does not belong to either of the hexanies in this scale. (And neither
> does 16/15.)

The note 1/1 is *always* included if you give something as a Scala
.scl file, and it hardly seems likely another interval of equivalence
than 2
was in anyone's mind. Of course, you could be right and what I gave
wasn't what Aaron wanted, but then, what else would be?

>> I disagree that that's what Aaron wanted. The note 2, or 1/1, or 2/1,
>> does not belong to either of the hexanies in this scale. (And neither
>> does 16/15.)
>
>The note 1/1 is *always* included if you give something as a Scala
>.scl file, and it hardly seems likely another interval of equivalence
>than 2 was in anyone's mind. Of course, you could be right and what
>I gave wasn't what Aaron wanted, but then, what else would be?

Hello? Earth to Paul and Gene? I've already shown that Aaron forgot
a delete 0 somewhere, or other trivial error.

-Carl

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > I disagree that that's what Aaron wanted. The note 2, or 1/1, or
2/1,
> > does not belong to either of the hexanies in this scale. (And
neither
> > does 16/15.)
>
> The note 1/1 is *always* included if you give something as a Scala
> .scl file, and it hardly seems likely another interval of
equivalence
> than 2
> was in anyone's mind.

Just because the note 1/1 or 2/1 isn't included in someone's list of
pitches, there's no reason to assume they meant for the interval of
equivalence to be anything other than 2.

Scala clearly needs better ways of dealing with such things. Let's
assume the interval of equivalence is 2. If the convention is that
1/1 is always included, then scala needs to transpose the CPS so that
one of the notes becomes 1/1. As I mentioned before, 'delete 0' is
not a general solution, and clearly ran Aaron into trouble.

>

Hi!Aaron!
I latticed your scale out below and suggested a note besides the 16/15 which seem unconnected to the rest. It seems like i have seen this one before. is this one of Hahn's scales?
http://anaphoria.com/aarondualhex.gif

>
>
> Message: 12
> Date: Fri, 2 Jan 2004 10:54:56 -0600
> From: "Aaron K. Johnson" <akjmicro@comcast.net>
> Subject: Re: CPS scales--do you like this?
>
> Paul and tuners,
>
> Any comments about the scale I designed in Scala from 2 hexanies?
>
> !
> dual-hexany CPS
> 11
> !
> 16/15
> 35/32
> 75/64
> 5/4
> 21/16
> 175/128
> 3/2
> 25/16
> 105/64
> 7/4
> 15/8
>
> What I did was 2,4 [1,3,5,7] and 2,4[1,1/3,1/5,1/7]
> I did 'delete 0' to make them both 'genuine CPSes'
> I did a 'reduce 2/1' and 'sort'
>
> Then, I rotated one of them to start with 16/15, and I rotated the other to
> the only rotation that gave the union of the two 'subscales' 12 unique
> pitches.....
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Tuners,

I should have written that I did an 'expanded CPS'. I'm sorry I forgot this
crucial information.

What this means is, in a nutshell, I did
2,4[1,3,5,7],
3,4[1,3,5,7],
and
4,4[1,3,5,7]

for the otonals, and the fractional versions for the utonals:

2,4[1,1/3,1/5,1/7]
etc....

The rest of my description is, I think correct. So, I do believe I understand
the concept of CPS, although I'm new to it, and I left out a key concept in
my description which meant you all thought I was clueless, or that Scala
needs fixing....

Best,
Aaron Krister Johnson

P.S. I would have been impressed if any of you could have deduced what I had
done from the supplied data ;)

On Friday 02 January 2004 04:31 pm, Paul Erlich wrote:
> Date:
> Today 04:31:54 pm
>
>
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
>
> wrote:
> > Paul and tuners,
> >
> > Any comments about the scale I designed in Scala from 2 hexanies?
> >
> > !
> > dual-hexany CPS
> > 11
> > !
> > 16/15
> > 35/32
> > 75/64
> > 5/4
> > 21/16
> > 175/128
> > 3/2
> > 25/16
> > 105/64
> > 7/4
> > 15/8
> >
> >
> > What I did was 2,4 [1,3,5,7] and 2,4[1,1/3,1/5,1/7]
> > I did 'delete 0' to make them both 'genuine CPSes'
> > I did a 'reduce 2/1' and 'sort'
> >
> > Then, I rotated one of them to start with 16/15, and I rotated the
>
> other to
>
> > the only rotation that gave the union of the two 'subscales' 12
>
> unique
>
> > pitches.....
> >
> > Even if this is somehow not correct, I find that it sounds
>
> interesting, and
>
> > maybe you would like it too, if you played it on your keyboard.....
> >
> > I'm interested to know anyone's thoughts or comments!
> >
> > Best,
> > Aaron Krister Johnson
>
> Hi Aaron,
>
> Let me start out by saying that yes, "it's somehow not correct", but
> of course that shouldn't prevent you from making great music with it.
> However, I think that if you bear with me and follow what I have to
> say below, it will only help toward your eventual musical mastery of
> CPS-based scales, by which I mean understanding of, hence control
> over, the materials therein.
>
> First of all, I'm confused, because you start off with
>
> > dual-hexany CPS
> > 11
>
> indicating an 11-note per octave scale, but then you say
>
> > 12 unique
> > pitches.....
>
> Anyway, I'm going to draw a lattice of your scale. As you will know
> from my 'gentle introduction' I just posted about:
>
> http://tonalsoft.com/td/erlich/paul-cps.htm
>
> a hexany will look like an octahedron when latticed (and this is true
> whether you write it as 2,4 [1,3,5,7] or as 2,4[1,1/3,1/5,1/7] since
> those are two ways of deriving exactly the same structure). So a
> scale consisting of two hexanies will look like two octahedra. If
> they share 1 note, you'll have 11 notes all together. If they share 2
> notes, you'll have 10 notes all together. Etc.
>
> To see this lattice correctly on the web, click on 'Reply':
>
> 175/128
> ,'/ \`.
> 25/16/---\75/64
> /|\/ \/|
> / |/\ /\|
> /35/32----105/64
> /,'/ \`.\ /,'/
> 5/4-/---\15/8 /
> |\/ \/| /
> |/\ /\|/
> 7/4------21/16
> `.\ /,'
> 3/2
>
>
>
>
> 16/15
>
>
> So what you have here, Aaron, are two hexanies (the two octahedra in
> the lattice) with two notes in common (namely 35/32 and 15/8), making
> 10 notes all together, and one note totally disconnected from the two
> hexanies, namely 16/15. You said that you 'rotated one of them to
> start with 16/15', but clearly that did not work correctly. Anyway,
> that accounts for 11 notes (as you said originally), not sure where
> you see 12.
>
> Since any of the consonant intervals of the hexany can be used as a
> pair of notes to connect it with another hexany, and the hexany has
> 12 consonant intervals, there are 12/2 = 6 distinct (not counting
> transpositions/modes) ways to make a 10-note scale out of two
> hexanies (assuming [1,3,5,7] or any given set of 4 factors).
>
> If you really want a 12-note scale, you could look at different ways
> of adding 2 notes to each of the 6 possibilities above. There will
> also be 3 distinct ways to make an 11-note scale out of two hexanies
> (with one note in common), and you could look at different ways of
> adding 1 note to each of those. Finally, there are quite a few ways
> of combining two hexanies so that they are connected with consonance
> but share no notes in common; this idea will immediately provide a
> host of 12-note scales.
>
> Probably too many possibilities to consider, though I'd be happy to
> work some out for you if you wish. But it would be better if we could
> add some further constraint to narrow down the search. Kraig is fond
> of the CS property. For a 12-note scale, that would mean that on the
> keyboard, you'd never be able to find two instances of the same
> interval by playing intervals that would be different in 12-equal.
> Sound reasonable? Then we just need to ask Gene (who's been doing
> similar things on tuning-math lately) a question:
>
> Gene:
>
> Can you list all the distinct 12-note 7-limit periodicity blocks
> which contain two hexanies?
>
> Hopefully Gene can answer this without too much trouble, given his
> mathematical virtuosity.
>
> Enjoy your musical explorations, and I'm here for you if you need me,
> Paul
>
>
> You do not need web access to participate. You may subscribe through
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--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Correction...
>
> a) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
> b) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/64, 7/4,
15/8
> c) 1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4,
15/8

I tried Scala 2.2m on these, and as Carl indicated the third is due
to Wilson. The first two Scala didn't know about; they are inversely
equivalent, while the Wilson one is self-equivalent. All are
epimorphic and none are proper, and they follow the seemingly near-
universal rule of being good candidates for Marvel in either its 7 or
11 limit incarnation.

>I tried Scala 2.2m on these, and as Carl indicated the third is due
>to Wilson. The first two Scala didn't know about; they are inversely
>equivalent, while the Wilson one is self-equivalent. All are
>epimorphic and none are proper, and they follow the seemingly near-
>universal rule of being good candidates for Marvel in either its 7 or
>11 limit incarnation.

The 7-limit Marvel version of the Wilson "class" scale should be
submitted to the scale archive (and the list).

-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> Just because the note 1/1 or 2/1 isn't included in someone's list
of
> pitches, there's no reason to assume they meant for the interval of
> equivalence to be anything other than 2.

Which was my point.

> Scala clearly needs better ways of dealing with such things.

I think Manuel has been very logical in the way he treats scales; I
haven't messed with how it deals with CPS. For scales, the 1/1 can
always be assumed, so he assumes it. The last listed interval can be
taken to be the interval of equivalence, so he so takes it.

Let's
> assume the interval of equivalence is 2. If the convention is that
> 1/1 is always included, then scala needs to transpose the CPS so
that
> one of the notes becomes 1/1.

You are assuming the CPS is meant as a Scala scale; if so, it does
need a 1/1 reference point.

>> Let's assume the interval of equivalence is 2. If the convention
>> is that 1/1 is always included, then scala needs to transpose the
>> CPS so that one of the notes becomes 1/1.
>
>You are assuming the CPS is meant as a Scala scale; if so, it does
>need a 1/1 reference point.

What I think Paul means is that the 1/1 should be a note of the
CPS, not an external 1/1. Which is what you get if you "delete 0".
IOW, I think the Scala implementation is excellent.

-Carl

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
> Tuners,
>
> I should have written that I did an 'expanded CPS'. I'm sorry I
forgot this
> crucial information.
>
> What this means is, in a nutshell, I did
> 2,4[1,3,5,7],

That's a hexany.

> 3,4[1,3,5,7],

That's a utonal tetrad; it can also be expressed as 1,4
[1,1/3,1/5,1/7].

> 4,4[1,3,5,7]

That's a single note

> for the otonals, and the fractional versions for the utonals:
>
> 2,4[1,1/3,1/5,1/7]
> etc....

So you'd get a hexany, an otonal tetrad, and a single note.

> The rest of my description is, I think correct.

I'll take your word for it, but maybe you could hold my hand and take
me through it again, this time with everything above?

> So, I do believe I understand
> the concept of CPS, although I'm new to it, and I left out a key
concept in
> my description which meant you all thought I was clueless, or that
Scala
> needs fixing....
>
> Best,
> Aaron Krister Johnson

So Aaron, can you please explain what 16/15 is doing in the scale --
since as you can see from my lattice, it's totally unconnected to all
the other notes? What exactly is it you were trying to do?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Let's assume the interval of equivalence is 2. If the convention
> >> is that 1/1 is always included, then scala needs to transpose the
> >> CPS so that one of the notes becomes 1/1.
> >
> >You are assuming the CPS is meant as a Scala scale; if so, it does
> >need a 1/1 reference point.
>
> What I think Paul means is that the 1/1 should be a note of the
> CPS, not an external 1/1.

Correct.

> Which is what you get if you "delete 0".

Again -- not if one of the products is itself a power of 2.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Gene:
> >
> >Can you list all the distinct 12-note 7-limit periodicity blocks
> >which contain two hexanies?
>
> Paul Hahn did a similar search years ago, namely all rotations
> of this structure...
>
> 35/32-----105/64
> ,'/ \`. ,' /
> 5/4-/---\-15/8/
> /|\/ \/| /
> / |/\ /\|/
> / 7/4------21/16
> /,'/ \`.\ /,'/
> 1/1-/---\-3/2 /
> /|\/ \/| /
> / |/\ /\|/
> / 7/5------21/20
> /,' `.\ /.'
> 8/5-------6/5
>
> ...which is a great way to get 2 hexanies in 12 tones.

That is the tuning of Daniel Wolf's "TRIO: The Sands" from September,
1986, published in Xenharmonikon 10. The recording with flute,
fretless banjo and bassoon is great, like microtonal Jo Kondo.

Gabor

> What this means is, in a nutshell, I did
> 2,4[1,3,5,7],
> 3,4[1,3,5,7],
> and
> 4,4[1,3,5,7]
>
> for the otonals, and the fractional versions for the utonals:
>
> 2,4[1,1/3,1/5,1/7]
> etc....

As discussed, the 2,4 pairs result in an identical scale.
The 4,4 pairs will both result in unity. The 3,4[1,3,5,7]
will give a utonal tetrad, and the 3,4[1,1/3,1/5,1/7] an
otonal tetrad. Superposing these, I get an 8-tone scale...

!
7/6
5/4
7/5
35/24
3/2
5/3
7/4
2/1
!

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > What this means is, in a nutshell, I did
> > 2,4[1,3,5,7],
> > 3,4[1,3,5,7],
> > and
> > 4,4[1,3,5,7]
> >
> > for the otonals, and the fractional versions for the utonals:
> >
> > 2,4[1,1/3,1/5,1/7]
> > etc....
>
> As discussed, the 2,4 pairs result in an identical scale.
> The 4,4 pairs will both result in unity. The 3,4[1,3,5,7]
> will give a utonal tetrad, and the 3,4[1,1/3,1/5,1/7] an
> otonal tetrad. Superposing these, I get an 8-tone scale...
>
> !
> 7/6
> 5/4
> 7/5
> 35/24
> 3/2
> 5/3
> 7/4
> 2/1
> !
>
> -Carl

You're making some assumptions about how the structures are pitched
relative to one another . . . 8 is indeed the fewest number of tones
that will contain all these structures . . . but in any case it seems
like a stretch to end up with a nice connected chunk of lattice and
then a single disconnected tone, as Aaron J. did.

>> Which is what you get if you "delete 0".
>
>Again -- not if one of the products is itself a power of 2.

Again, you're wrong. If you have a failing case now would
be the time to produce it.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Which is what you get if you "delete 0".
> >
> >Again -- not if one of the products is itself a power of 2.
>
> Again, you're wrong. If you have a failing case now would
> be the time to produce it.
>
> -Carl

Oops -- I forgot that Scala will call such a pitch '2/1' rather
than '1/1'.

But wouldn't it be clearer if you didn't have to say 'delete 0'?

>> > What this means is, in a nutshell, I did
>> > 2,4[1,3,5,7],
>> > 3,4[1,3,5,7],
>> > and
>> > 4,4[1,3,5,7]
>> >
>> > for the otonals, and the fractional versions for the utonals:
>> >
>> > 2,4[1,1/3,1/5,1/7]
>> > etc....
>>
>> As discussed, the 2,4 pairs result in an identical scale.
>> The 4,4 pairs will both result in unity. The 3,4[1,3,5,7]
>> will give a utonal tetrad, and the 3,4[1,1/3,1/5,1/7] an
>> otonal tetrad. Superposing these, I get an 8-tone scale...
>>
>> !
>> 7/6
>> 5/4
>> 7/5
>> 35/24
>> 3/2
>> 5/3
>> 7/4
>> 2/1
>> !
>>
>> -Carl
>
>You're making some assumptions about how the structures are pitched
>relative to one another . . .

Indeed, I'm just slapping them together as Scala gives them to me,
which is my first guess as to what Aaron might have done.

>but in any case it seems
>like a stretch to end up with a nice connected chunk of lattice and
>then a single disconnected tone, as Aaron J. did.

Indeed; the simplest way to get the 16/15 would seem to be to forget
a delete 0 after doing one of the 2,4[]s.

-Carl

>Oops -- I forgot that Scala will call such a pitch '2/1' rather
>than '1/1'.
>
>But wouldn't it be clearer if you didn't have to say 'delete 0'?

I could go either way on it. This way, one can see the
factors from the 'outside' before picking a root... just a
thought. In any case, I suspect Manuel has more important
things on his plate at the moment.

-Carl

--- In tuning@yahoogroups.com, "alternativetuning"
<alternativetuning@y...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >Gene:
> > >
> > >Can you list all the distinct 12-note 7-limit periodicity blocks
> > >which contain two hexanies?
> >
> > Paul Hahn did a similar search years ago, namely all rotations
> > of this structure...
> >
> > 35/32-----105/64
> > ,'/ \`. ,' /
> > 5/4-/---\-15/8/
> > /|\/ \/| /
> > / |/\ /\|/
> > / 7/4------21/16
> > /,'/ \`.\ /,'/
> > 1/1-/---\-3/2 /
> > /|\/ \/| /
> > / |/\ /\|/
> > / 7/5------21/20
> > /,' `.\ /.'
> > 8/5-------6/5
> >
> > ...which is a great way to get 2 hexanies in 12 tones.
>
> That is the tuning of Daniel Wolf's "TRIO: The Sands" from
September,
> 1986, published in Xenharmonikon 10. The recording with flute,
> fretless banjo and bassoon is great, like microtonal Jo Kondo.
>
> Gabor

It's also known as a 3.5.5.7 Euler-Fokker Genus ('rooted' on 8/5),
though many of the equally-appealing rotations Carl was referring to
cannot be so easily expressed with the Genus concept.

Kraig-

That's neato!

I posted elsewhere a correction in the description of my process: I had in
fact done an 'expanded CPS' from 2-4 factors out of 4 using otonal and utonal
[1,3,5,7], but combined by modal rotation so that there were 12 unique tones.

Don't know if that is an unprecidented process or not, but I'm desperate to
explore 12-tones CPS's by any means!

Cheers,
Aaron.

On Friday 02 January 2004 06:02 pm, kraig grady wrote:
> Hi!Aaron!
> I latticed your scale out below and suggested a note besides the 16/15
> which seem unconnected to the rest. It seems like i have seen this one
> before. is this one of Hahn's scales? http://anaphoria.com/aarondualhex.gif
>
> > Message: 12
> > Date: Fri, 2 Jan 2004 10:54:56 -0600
> > From: "Aaron K. Johnson" <akjmicro@comcast.net>
> > Subject: Re: CPS scales--do you like this?
> >
> > Paul and tuners,
> >
> > Any comments about the scale I designed in Scala from 2 hexanies?
> >
> > !
> > dual-hexany CPS
> > 11
> > !
> > 16/15
> > 35/32
> > 75/64
> > 5/4
> > 21/16
> > 175/128
> > 3/2
> > 25/16
> > 105/64
> > 7/4
> > 15/8
> >
> > What I did was 2,4 [1,3,5,7] and 2,4[1,1/3,1/5,1/7]
> > I did 'delete 0' to make them both 'genuine CPSes'
> > I did a 'reduce 2/1' and 'sort'
> >
> > Then, I rotated one of them to start with 16/15, and I rotated the other
> > to the only rotation that gave the union of the two 'subscales' 12 unique
> > pitches.....
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island
> http://www.anaphoria.com
> The Wandering Medicine Show
> KXLU 88.9 FM WED 8-9PM PST
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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> your subscription to individual emails. tuning-help@yahoogroups.com -
> receive general help information.
>
>
> Yahoo! Groups Links
>
> To visit your group on the web, go to:
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>
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--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The 7-limit Marvel version of the Wilson "class" scale should be
> submitted to the scale archive (and the list).

If you want to try this stuff yourself, here are some Maple routines:

with(padic, ordp):

marvr := proc(q)
# map of 7-limit interval to 5-limit 225/224-planar
2^ordp(q,2)*(32768/10935)^ordp(q,3)*(32768/6561)^ordp(q,5)*
(36028797018963968/5147278302366225)^ordp(q,7)
end:

marvk := proc(q)
# map of 7-limit interval to 1/4-kleismic 225/224-planar
ce(2^ordp(q,2) * (2016/25)^(ordp(q,3)/4) * (5600/9)^(ordp(q,5)/4) *
7^ordp(q,7))
end:

marvel := proc(q)
# map of 11-limit interval to 225/224 & 385/384 planar (Marvel)
minimax
ce(2^ordp(q,2) * (26873856/1375)^(ordp(q,3)/9) * (2097152000/1089)^
(ordp(q,5)/9)
* (41472/121)^(ordp(q,7)/3) * (326940477095936/140625)^(ordp
(q,11)/9)) end:

The rational Marvel, "marvr", may be useful for keeping track of
which Marvel scales are the same, but Scala doesn't handle big
integers so I don't advice trying to use it to create Scala scl files.

Below I give 7 and 11 limit versions of Wilson Class; I use your
version because it has a tetrad over 1/1. By the way, what does your
best of 12-note JI scales look like now?

! wilcmarv7.scl
Wilson Class scale in 1/4-kleisma Marvel
12
!
84.467193
153.211740
315.641287
384.385833
468.853027
584.440073
700.027120
768.771666
900.081360
968.825906
1084.412953
1200.000000

! wilcmarv11.scl
Wilson Class scale in 11-limit minimax Marvel
12
!
85.393114
151.994179
316.998408
383.599473
468.992587
584.795233
700.597880
767.198946
901.793641
968.394706
1084.197353
1200.000000

--- In tuning@yahoogroups.com, "alternativetuning"
<alternativetuning@y...> wrote:

> That is the tuning of Daniel Wolf's "TRIO: The Sands" from
September,
> 1986, published in Xenharmonikon 10. The recording with flute,
> fretless banjo and bassoon is great, like microtonal Jo Kondo.

Did Wolf invent it?

>If you want to try this stuff yourself, here are some Maple routines:
//snip//
>! wilcmarv7.scl
>Wilson Class scale in 1/4-kleisma Marvel
>12
>!
>84.467193
>153.211740
>315.641287
>384.385833
>468.853027
>584.440073
>700.027120
>768.771666
>900.081360
>968.825906
>1084.412953
>1200.000000

Thanks Gene!! NB Kurt.

Hey, this is very similar to wafso12, a 225:224 temperament of
the prism scale.

>By the way, what does your best of 12-note JI scales look like now?

In the 7-limit? The same it's looked for 5 years. Prism, class and
Lester. See my folder in the files area of this list. I haven't
gone thru your recent classification effort, but I was under the
impression it was all 5-limit pre-temperament.

-Carl

Paul Erlich wrote:
> So Aaron, can you please explain what 16/15 is doing in the scale --
> since as you can see from my lattice, it's totally unconnected to all
> the other notes? What exactly is it you were trying to do?

Hi Paul,

Yes, it's a hexany, a tetrad, and a single note. It's called, in Scala, an
'extended CPS'. Read the docs before asking me more....!!!!

(in scala "File->New->Combination Product Set", then under 'kind', pick
'extended', and the 'Combination count' parameter is '2', and Last
Combination count is '4'....do this for [1,3,5,7], delete 0,reduce, sort,
and the same for [1,1/3,1/5,1/7]. Now you have some redundant notes, so rotate
each subset until you have unique notes...voila!)

I was at play, not trying to 'do' anything. Experimentation.....my goal was to
see what I could get by merging an otonal and utonal 'extended CPS', and if
necessary, rotate one or another of them ('key' in Scala-ese) until between
the two subsets, there were unique pitches--twelve of them. If there are
problems with symmetry, etc. (eg 16/15) then oh, well...!!! It is interesting
that Kraig grady wrote back with a lattice that took the 16/15 and
substituted it for a 525/512, which would make it a Euler-Fokker lattice!!!!

Again, I was just playing with blocks, like a child, innocent and pure ;)

Sometime when I have more time, I'll explain it step by step, with Scala
output.

Best,
Aaron.

On Friday 02 January 2004 06:40 pm, Paul Erlich wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
>
> wrote:
> > Tuners,
> >
> > I should have written that I did an 'expanded CPS'. I'm sorry I
>
> forgot this
>
> > crucial information.
> >
> > What this means is, in a nutshell, I did
> > 2,4[1,3,5,7],
>
> That's a hexany.
>
> > 3,4[1,3,5,7],
>
> That's a utonal tetrad; it can also be expressed as 1,4
> [1,1/3,1/5,1/7].
>
> > 4,4[1,3,5,7]
>
> That's a single note
>
> > for the otonals, and the fractional versions for the utonals:
> >
> > 2,4[1,1/3,1/5,1/7]
> > etc....
>
> So you'd get a hexany, an otonal tetrad, and a single note.
>
> > The rest of my description is, I think correct.
>
> I'll take your word for it, but maybe you could hold my hand and take
> me through it again, this time with everything above?
>
> > So, I do believe I understand
> > the concept of CPS, although I'm new to it, and I left out a key
>
> concept in
>
> > my description which meant you all thought I was clueless, or that
>
> Scala
>
> > needs fixing....
> >
> > Best,
> > Aaron Krister Johnson
>
> So Aaron, can you please explain what 16/15 is doing in the scale --
> since as you can see from my lattice, it's totally unconnected to all
> the other notes? What exactly is it you were trying to do?
>
>
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--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

>>! wilcmarv7.scl
>>Wilson Class scale in 1/4-kleisma Marvel
>>12
>>!
>>84.467193
>>153.211740
>>315.641287
>>384.385833
>>468.853027
>>584.440073
>>700.027120
>>768.771666
>>900.081360
>>968.825906
>>1084.412953
>>1200.000000
//
>Hey, this is very similar to wafso12, a 225:224 temperament of
>the prism scale.

In particular, they both contain the same tempered tetrads, but
wafso provides 6 of them while this scale provides 4 (no more
than Wilson's scale in JI). Which makes sense to me, since the
2-hexany structure was conceived by myself and Paul Hahn (and
proabably Wilson) as a pure JI structure, while the prism scale
was conceived by myself strictly for purposes of tempering by
225:224.

-Carl

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

> If there are
> problems with symmetry, etc. (eg 16/15) then oh, well...!!!

The problem wasn't with symmetry, it was that the note 16/15 was far
removed from the rest of the lattice. The point of CPS is to find
nice maximally compact, maximally connected structures in the lattice
that don't have a 'tonal center' the way Partch diamonds do. As you
can see in the lattices I drew, each hexany looks like an octahedron,
or viewed two-dimensionally, like a Star of David inscribed in a
hexagon. Each note of the hexany is connected by consonances to four
other notes in the hexany.

10 notes of your scale comprised 2 hexanies with 2 notes ( in common.
In addition to the consonances within each hexany, the juxtaposition
creates 8 new consonances connecting a notes in one hexany to a note
in the other hexany. As I mentioned, there would be 5 other ways of
doing this, but in any case it's very much in line with the idea of
maximally compact, maximally connected structures in the lattice.

Then there was, far away and all by its lonesome, the pitch 16/15,
forming no consonant intervals with the other 10 (or was it 11? you
never answered me) notes in your scale.

Hopefully you can understand my reaction a little better now.

> Sometime when I have more time, I'll explain it step by step, with
Scala
> output.

Looking forward to it,
Paul

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>! wilcmarv7.scl
> >>Wilson Class scale in 1/4-kleisma Marvel

> >Hey, this is very similar to wafso12, a 225:224 temperament of
> >the prism scale.

I've been calling your wafso12 "lumma", because that's the name
Manuel gave it in the scl archives. Neither name seems very logical,
given that they are Marvel temperings of a scale called "prism", but
Manuel is calling "prism" "lumma7". He's also got some of my prism
temperings in there.

Meanwhile, your "lester" he calls "grady7", but probably should have
called it "centaur", since it says it was published by Kraig under
that name in 1987 in Xenharmonikon 16. All of which leads to the
question--who is Lester?

You know, there used to be more of this scales on your canonical list.

> In particular, they both contain the same tempered tetrads, but
> wafso provides 6 of them while this scale provides 4 (no more
> than Wilson's scale in JI). Which makes sense to me, since the
> 2-hexany structure was conceived by myself and Paul Hahn (and
> proabably Wilson) as a pure JI structure, while the prism scale
> was conceived by myself strictly for purposes of tempering by
> 225:224.
>
> -Carl

>> >>! wilcmarv7.scl
>> >>Wilson Class scale in 1/4-kleisma Marvel
>
>> >Hey, this is very similar to wafso12, a 225:224 temperament of
>> >the prism scale.
>
>I've been calling your wafso12 "lumma", because that's the name
>Manuel gave it in the scl archives.

Oh, I see that. I don't like the archives, myself, because they're
very sloppy and contain a lot of crappy scales. I maintain my own
version.

>Neither name seems very logical,
>given that they are Marvel temperings of a scale called "prism",
>but Manuel is calling "prism" "lumma7". He's also got some of my
>prism temperings in there.

Wafso stands for within a fly's s*** of. Dave Keenan produced
the tempered version, so I gave him naming rights.

>Meanwhile, your "lester" he calls "grady7", but probably should
>have called it "centaur", since it says it was published by Kraig
>under that name in 1987 in Xenharmonikon 16. All of which leads
>to the question--who is Lester?

Lester is the make of my piano, I independently discovered this
scale and tuned my piano to it. Grady once said (I think) that
Wilson came up with the tuning.

>You know, there used to be more of this scales on your canonical
>list.

12-tone 7-limit JI? I don't think so. Maybe you're thinking of
stellhex or 12max7? They're very uneven.

-Carl

>
>

it is a full CPS of 4 elements one out of four through the four out of four set.
The differance between Euler and the CPS is that Euler did not think as 1 as a real element
except at the start and then he drops it. It might seem like nothing but because of it he missed
the hexany and the bigger Eikosany.

I highly recommend playing with blocks. i have come up with more this way than any other methods.
But i am possibly more visually oriented as compared to mathematically. In this case it was easy
to see what worked better than the 16/15 long befor i knowew what ratio it was.
No one mentioned that it has two hexanies a 5/4 apart?

>
> From: "Aaron K. Johnson" <akjmicro@comcast.net>
> Subject: Re: Re: CPS scales--correction
>
> It is interesting
> that Kraig grady wrote back with a lattice that took the 16/15 and
> substituted it for a 525/512, which would make it a Euler-Fokker lattice!!!!
>
> Again, I was just playing with blocks, like a child, innocent and pure ;)
>
> Sometime when I have more time, I'll explain it step by step, with Scala
> output.
>
> Best,
> Aaron.

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>

no this is one of the very few ones he overlooked somehow. The scales dates back to 1982 as i was
my first pump organ and wanted to have both harmonic and subharmonics series that i could hear
sustained. I used it in my first film Embryo Without Tears.

>
> From: "Carl Lumma" <ekin@lumma.org>
> Subject: Re: Wilson Class in Marvel
>
>
>
> Lester is the make of my piano, I independently discovered this
> scale and tuned my piano to it. Grady once said (I think) that
> Wilson came up with the tuning.
>
>

>
>
> -Carl
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>The rational Marvel, "marvr", may be useful for keeping track of
>which Marvel scales are the same, but Scala doesn't handle big
>integers so I don't advice trying to use it to create Scala scl files.
>
>Below I give 7 and 11 limit versions of Wilson Class; I use your
>version because it has a tetrad over 1/1. By the way, what does your
>best of 12-note JI scales look like now?

Perhaps Prism should be replaced by this in strict JI...

! pris.scl
!
Optimized (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 scale.
12
!
16/15
28/25
7/6
5/4
4/3
7/5
3/2
8/5
5/3
7/4
28/15
2/1
!
! 30 intervals 25 triads 5 tetrads
! Equivalent to lumma.scl under 225/224.

-Carl

[I wrote...]
>Perhaps Prism should be replaced by this in strict JI...
>
>! pris.scl
>!
> Optimized (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 scale.
> 12
>!
> 16/15
> 28/25
> 7/6
> 5/4
> 4/3
> 7/5
> 3/2
> 8/5
> 5/3
> 7/4
> 28/15
> 2/1
>!
>! 30 intervals 25 triads 5 tetrads
>! Equivalent to lumma.scl under 225/224.

This differs from lumma.scl by only one pitch; it swaps
112/75 for 3/2...

! 5/3-----------5/4
! /:\ /:\
! / : \ / : \
! / : \ / : \
! / 7/6-----------7/4 \
! / ,'/ \`. \ / ,'/ `. \
! /.' / \ `.\ /.' / `.\
! 4/3--/-----\--1/1--/--------3/2
! /:\ / \ /: /
! / : / \ : /
! / :/ \ / \:/
! / 28/15----------7/5
! / ,' \`. \ / ,'/
! /.' \ `.\ /.' /
! 16/15-------\--8/5 /
! \ : /
! \ : /
! \:/
! 28/25

-C.

>> Lester is the make of my piano, I independently discovered this
>> scale and tuned my piano to it. Grady once said (I think) that
>> Wilson came up with the tuning.
>
> no this is one of the very few ones he overlooked somehow. The
> scales dates back to 1982 as i was my first pump organ and wanted
> to have both harmonic and subharmonics series that i could hear
> sustained. I used it in my first film Embryo Without Tears.

Thanks for straightening that out.

-Carl

Paul wrote:
>But wouldn't it be clearer if you didn't have to say 'delete 0'?

It would be slightly easier if one doesn't want the
1/1, but make it more difficult in case one wants to keep it,
that's why I made the trade-off the way I did.

Manuel

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >! 30 intervals 25 triads 5 tetrads
> >! Equivalent to lumma.scl under 225/224.
>
> This differs from lumma.scl by only one pitch; it swaps
> 112/75 for 3/2...

Manuel already has it listed under "smithgw_pris.scl", by the way.

On Fri, 02 Jan 2004 20:25:24 -0000, "Gene Ward Smith" <gwsmith@svpal.org>
wrote:

>Saying it is a good candidate for 225/224-tempering isn't saying
>much, because almost any 5 or 7 limit JI scale will be. However, dual-
>hexany has five pure fifths plus one flat by 225/224, five pure 7/4s
>plus one sharp by 225/224, three pure 7/6s and two flat by 225/224,
>four pure 7/5s and one sharp by 7/5, and to top it all off two
>different sizes of 9/7, neither of which is pure. Tempering out the
>septimal kleisma 225/224 smooths all of this out.
>
>The 225/224-planar temperament, for which the names Pauline and
>Byzantine have been nominated but not seconded and for which I'd like
>a name everyone can accept, is analogous in a way to meantone if we
>assume the minor thirds are pure. In meantone, four (3/2)^4 ~ 5, and
>if we choose exact fives, we get 1/4-comma meantone. In 225/224-
>planar, (3/2)^2 * (5/4)^2 ~ 7/2, and choosing exact sevens gives us
>1/4-kleismic.

I admit I haven't been following the tuning-math list much, and I don't
know what names if any have been suggested for planar temperaments (other
than starling, 126/125), but the 225/224 does seem like one that we should
have a name for, especially with its connection with meantone. Many of the
more common meantone ET's, including 19- and 31-ET (but not 55-ET), are
compatible with this temperament. It also works with a number of common
schismic ET's, such as 41- and 53-ET (but not 65-ET), and a handful of
other useful ET's such as 22- and 72-ET. Other commas have their uses in
particular contexts (such as 64/63 in conjunction with porcupine
temperament), but this one seems to be one of the more productive ones for
7-limit harmony in general.

"Kleismic" would be a good name if it wasn't already in use; perhaps
"septimal kleismic"? But that might be confusing, since it's not
specifically related to kleismic temperament, and is incompatible with some
of the more common kleismic ET's like 15 and 34. "Septismic"? "Kleptimal"?

I wonder who was the first composer to write an augmented sixth in meantone
temperament? That doesn't seem like a question that's easily answered.

I suggest following the pattern of "starling" and naming 7-limit commas
after birds. So for instance, 225/224 might be "eagle", and 50/49 might be
"sparrow". But that might be confusing with "vulture" already referring to
a 5-limit comma.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_50781.html#50850

> >> Let's assume the interval of equivalence is 2. If the convention
> >> is that 1/1 is always included, then scala needs to transpose the
> >> CPS so that one of the notes becomes 1/1.
> >
> >You are assuming the CPS is meant as a Scala scale; if so, it does
> >need a 1/1 reference point.
>
> What I think Paul means is that the 1/1 should be a note of the
> CPS, not an external 1/1. Which is what you get if you "delete 0".
> IOW, I think the Scala implementation is excellent.
>
> -Carl

***Umm, am I getting confused again?? Isn't a crucial concept of the
CPS the fact that there really *is* no clear, definable 1/1??

I remember having some trouble with the "delete 0" concept when I was
messing with these. I'm assuming this was Aaron's quandary?

The whole idea of "deleting a 0" seemed a little unnecessary to me :)
but I learned it was crucial to the process...

J. Pehrson

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> I admit I haven't been following the tuning-math list much, and I
don't
> know what names if any have been suggested for planar temperaments
(other
> than starling, 126/125), but the 225/224 does seem like one that we
should
> have a name for, especially with its connection with meantone.

I've just named it "Marvel" and so far no one has objected. The name
is supposed to cover both that and {225/224,385/384}-planar, and
therefore has more to do with its connection to Miracle.

> I wonder who was the first composer to write an augmented sixth in
meantone
> temperament? That doesn't seem like a question that's easily
answered.

I would be interesting to find who from the meantone era made regular
use of it. I'd also like to know if anyone made regular use of the
supermajor triads, something which might be ignored by people with 12-
equal on the brain.

> I suggest following the pattern of "starling" and naming 7-limit
commas
> after birds. So for instance, 225/224 might be "eagle", and 50/49
might be
> "sparrow". But that might be confusing with "vulture" already
referring to
> a 5-limit comma.

I did propose "Thrush" for {126/125,176/175}-planar, but once again
if we were to follow the pattern of linear temperament naming, it
would make sense to call this (11-limit) Starling, as it seems to be
the best way of extending it.

Hello Aaron!

"Aaron K. Johnson" wrote:

> Don't you mean that Wilson, not Euler, drops the 1/1? I thought that was his
> innovation when coming up with CPS...

Erv uses a 1/1 or a one in the whole set although he labels it just the factor
1.
The purpose of the CPS was not to create uncentered structures , but to find
the tightest economical set of base pitches. In the process he discovered a
series of Subsets that are uncentered like the Hexany and the Eikosany that
stand on their own. The interesting features of these uncentered structures are
the new structural possibilities and relations they exhibit. It
relationship/interaction to the diamond is quite interesting. My subjective
observation is that the eikosany offers quite a bit more of things to play with
than the diamond.

>
>
> Anyway, it seems that an 'extended CPS', or 'full' as you call it, ends up
> being equivalent to a Euler-Fokker lattice anyway if you go from 1-out-of-n
> to 4-out-of-n, no? At least Manuel wrote it as such in the Scala help
> dialog...

Actually i think there are two of these full CPS's. (before i am corrected
again!)

>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
> >
>
> it is a full CPS of 4 elements one out of four through the four
out of four set.
> The differance between Euler and the CPS is that Euler did not
think as 1 as a real element
> except at the start and then he drops it.

Euler does not drop the 1 anywhere.

> It might seem like nothing but because of it he missed
> the hexany and the bigger Eikosany.

The hexany, and especially the Eikosany, is far removed from any
Euler structure. Wilson's own lattices make the distinction clearer --
CPS scales like the hexany and Eikosany are symmetrical in the
various triangular lattices of consonances, while Euler Genuses
(Geni?) are somewhat symmetrical only in the rectangular lattices of
primes. As long as you're assuming octave-equivalence, the former is
a more effective display, and thus CPSs have the more meaningful
symmetry. For octave-specific constructs that don't employ octave
transposition or reduction or repetition or equivalence, the
Euler/rectangular method is a bit more meaningful.

> No one mentioned that it has two hexanies a 5/4 apart?

I mentioned the two hexanies quite a few times and it was obvious
from the lattice I drew that they were a 5/4 apart.

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >But wouldn't it be clearer if you didn't have to say 'delete 0'?
>
> It would be slightly easier if one doesn't want the
> 1/1, but make it more difficult in case one wants to keep it,
> that's why I made the trade-off the way I did.
>
> Manuel

Why would one want to keep it? It seems to go against the whole
concept, symmetry and everything.

On Saturday 03 January 2004 02:23 am, kraig grady wrote:
> it is a full CPS of 4 elements one out of four through the four out of
> four set. The differance between Euler and the CPS is that Euler did not
> think as 1 as a real element except at the start and then he drops it. It
> might seem like nothing but because of it he missed the hexany and the
> bigger Eikosany.

Don't you mean that Wilson, not Euler, drops the 1/1? I thought that was his
innovation when coming up with CPS...

Anyway, it seems that an 'extended CPS', or 'full' as you call it, ends up
being equivalent to a Euler-Fokker lattice anyway if you go from 1-out-of-n
to 4-out-of-n, no? At least Manuel wrote it as such in the Scala help
dialog...

> I highly recommend playing with blocks. i have come up with more this way
> than any other methods. But i am possibly more visually oriented as
> compared to mathematically. In this case it was easy to see what worked
> better than the 16/15 long befor i knowew what ratio it was. No one
> mentioned that it has two hexanies a 5/4 apart?

Interesting!

Best,
Aaron.

> > From: "Aaron K. Johnson" <akjmicro@comcast.net>
> > Subject: Re: Re: CPS scales--correction
> >
> > It is interesting
> > that Kraig grady wrote back with a lattice that took the 16/15 and
> > substituted it for a 525/512, which would make it a Euler-Fokker
> > lattice!!!!
> >
> > Again, I was just playing with blocks, like a child, innocent and pure ;)
> >
> > Sometime when I have more time, I'll explain it step by step, with Scala
> > output.
> >
> > Best,
> > Aaron.
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island
> http://www.anaphoria.com
> The Wandering Medicine Show
> KXLU 88.9 FM WED 8-9PM PST
>
>
>
>
> You do not need web access to participate. You may subscribe through
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--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

>***Umm, am I getting confused again?? Isn't a crucial concept of the
>CPS the fact that there really *is* no clear, definable 1/1??

The chords in such scales are not centered around any single point in
the scale. To write the scale down one still has to pick a note to be
"1/1". The point is that any note is as good a choice here as any
other. The same is *not* true for the main CPS competition, Partch's
tonality diamonds.

-Carl

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Hello Aaron!
>
>
> "Aaron K. Johnson" wrote:
>
> > Don't you mean that Wilson, not Euler, drops the 1/1? I thought
that was his
> > innovation when coming up with CPS...

Aaron, did you delete the message in which you wrote this? I'd like
to help clarify . . . for now I would go over the 'gentle
introduction' I created along with figure 19 of
Wilson's 'D'Allessandro' article . . . 'Dropping the 1/1' is not a
helpful way of thinking about CPS scales, despite what Scala may lead
you to believe -- CPSs stand on their own symmetry, and they are
quite different, in general, from any Euler scales.

> > Anyway, it seems that an 'extended CPS', or 'full' as you call
it, ends up
> > being equivalent to a Euler-Fokker lattice anyway if you go from
1-out-of-n
> > to 4-out-of-n, no?

This is true when you use a particular method of pitching the 4 CPSs
relative to one another. However, there are other methods that are
equally appealing from the standpoint of the symmetry that the CPSs
embody in the first place. I would *love* to show you some examples,
if you aren't completely sick of me already . . .

Aaron J.,

I just came across your message. Somehow Kraig's reply showed up
earlier in the archives than your original message.

Aaron, where do you live again? It would be great to meet in person.
I had a far easier time explaining CPS scales (and other matters) to
Monz when we met in person than would have been possible otherwise.
Also, it would be great to jam with you! I too love immersing myself
in a ridiculous variety of musical situations -- from atonal acoustic
guitar bebop to folk-rock 3-part harmony singing . . . on New Year's
Eve I drummed with a DJ and played keyboard bass with an African
master tunesmith . . . life is good!

-Paul

>
> From: "Paul Erlich" <paul@stretch-music.com>
> Subject: Re: CPS scales--correction
>
>
> Euler does not drop the 1 anywhere.

He uses it at the beginning and then never again as a factor. you never see 1x3x5 in Euler for
example
of how he doesn't use 1 as a factor

>
>
> > It might seem like nothing but because of it he missed
> > the hexany and the bigger Eikosany.
>
> The hexany, and especially the Eikosany, is far removed from any
> Euler structure.

except they are often embedded in them but you can't see them because he doesn't use 1 as a factor

http://www.anaphoria.com/Euler.PDF page three shows how can happen on a larger scale.

> Wilson's own lattices make the distinction clearer --
> CPS scales like the hexany and Eikosany are symmetrical in the
> various triangular lattices of consonances, while Euler Genuses
> (Geni?) are somewhat symmetrical only in the rectangular lattices of
> primes.

Yes they are generated in a close but lightly different way that causes this disdinction

> As long as you're assuming octave-equivalence, the former is
> a more effective display, and thus CPSs have the more meaningful
> symmetry. For octave-specific constructs that don't employ octave
> transposition or reduction or repetition or equivalence, the
> Euler/rectangular method is a bit more meaningful.
>
> > No one mentioned that it has two hexanies a 5/4 apart?
>
> I mentioned the two hexanies quite a few times and it was obvious
> from the lattice I drew that they were a 5/4 apart.

correct, missed it as it is easy to do!
I still have a hard time reading lattices in e-mail form which is why i always go for gif.

>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >***Umm, am I getting confused again?? Isn't a crucial concept of
the
> >CPS the fact that there really *is* no clear, definable 1/1??
>
> The chords in such scales are not centered around any single point
in
> the scale. To write the scale down one still has to pick a note to
be
> "1/1". The point is that any note is as good a choice here as any
> other.

Well, that may be the theory, but I think most people will agree that
of two notes a fifth apart, the lower one is usually a better choice
to be "1/1" than the upper one . . . Partch, at least at the time he
wrote _Genesis_, might have stubbornly disputed this, so to him at
that time, your assertion that for CPSs, "any note is as good a
choice here as any other" would probably have rung true.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
> > From: "Paul Erlich" <paul@s...>
> > Subject: Re: CPS scales--correction
> >
> >
> > Euler does not drop the 1 anywhere.
>
> He uses it at the beginning and then never again as a factor. you
never see 1x3x5 in Euler for
> example
> of how he doesn't use 1 as a factor

Kraig, 1x3x5 = 3x5, which occurs in all such Genuses (Geni?). So you
could just as correctly say he uses 1 *everywhere* as a factor! Or
you could just as correctly say he never uses 1 explicitly at all,
and write a genus as

3^0*5^0--------3^1*5^0
...|..............|
...|..............|
...|..............|
...|..............|
...|..............|
...|..............|
...|..............|
3^0*5^1--------3^1*5^1

etc.

>Well, that may be the theory, but I think most people will agree that
>of two notes a fifth apart, the lower one is usually a better choice
>to be "1/1" than the upper one . . . Partch, at least at the time he
>wrote _Genesis_, might have stubbornly disputed this, so to him at
>that time, your assertion that for CPSs, "any note is as good a
>choice here as any other" would probably have rung true.

Of course, and in general when the entire CPS is considered a
chord there may be a preferred orientation. But my remarks were
strictly from the point of view of common-tone relations among
the chords of the scale.

-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> The hexany, and especially the Eikosany, is far removed from any
> Euler structure. Wilson's own lattices make the distinction
clearer --
> CPS scales like the hexany and Eikosany are symmetrical in the
> various triangular lattices of consonances, while Euler Genuses
> (Geni?) are somewhat symmetrical only in the rectangular lattices
of
> primes.

If you take any discrete set of points L in n-space, you can define
the Voroni cell associated to any point P in L as the region defined
by the set of points at least as close to P as to any other point in
L. This chops up the space into regions with disjoint interiors. A
vertex of a Voroni cell which is a local maximum in terms of distance
from points of L is called a hole. If it is an absolute maximum, it
is called a deep hole, otherwise it is a shallow hole. The holes of
the 5-limit triangular lattice (all deep) are the centers of the
triangles--and so correspond to triads. The 7-limit symmetrical
("A3") lattice has shallow holes corresponding to tetrads, and deep
holes which are octahedra, or hexanies.

In general, it makes sense to consider all points at a fixed distance
from a vertex or hole of a lattice L in terms of scale construction;
I'd call them holistic but that leaves out the verticies. It's such a
good name I may use it anyway. :)

Paul,

I;ve done a bit of reading on CPS's, and I think I understand them a quite a
bit more than you think I do.

I understand, for instance, that any pitch in a CPS can be the 'center' or
1/1, and it is rotatable, etc.

So, don't get too hyper when I say 'drop the 1/1'....

But, there is always more to learn, and clarify, so fire away if you must.

-Aaron.

On Saturday 03 January 2004 04:31 pm, Paul Erlich wrote:
> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> > Hello Aaron!
> >
> > "Aaron K. Johnson" wrote:
> > > Don't you mean that Wilson, not Euler, drops the 1/1? I thought
>
> that was his
>
> > > innovation when coming up with CPS...
>
> Aaron, did you delete the message in which you wrote this? I'd like
> to help clarify . . . for now I would go over the 'gentle
> introduction' I created along with figure 19 of
> Wilson's 'D'Allessandro' article . . . 'Dropping the 1/1' is not a
> helpful way of thinking about CPS scales, despite what Scala may lead
> you to believe -- CPSs stand on their own symmetry, and they are
> quite different, in general, from any Euler scales.
>
> > > Anyway, it seems that an 'extended CPS', or 'full' as you call
>
> it, ends up
>
> > > being equivalent to a Euler-Fokker lattice anyway if you go from
>
> 1-out-of-n
>
> > > to 4-out-of-n, no?
>
> This is true when you use a particular method of pitching the 4 CPSs
> relative to one another. However, there are other methods that are
> equally appealing from the standpoint of the symmetry that the CPSs
> embody in the first place. I would *love* to show you some examples,
> if you aren't completely sick of me already . . .
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold
> for the tuning group. tuning-digest@yahoogroups.com - change your
> subscription to daily digest mode. tuning-normal@yahoogroups.com - change
> your subscription to individual emails. tuning-help@yahoogroups.com -
> receive general help information.
>
>
>
> Yahoo! Groups Links
>
> To visit your group on the web, go to:
> /tuning/
>
> To unsubscribe from this group, send an email to:
> tuning-unsubscribe@yahoogroups.com
>
> Your use of Yahoo! Groups is subject to:
> http://docs.yahoo.com/info/terms/

--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

On Saturday 03 January 2004 04:40 pm, Paul Erlich wrote:
> Aaron J.,
>
> I just came across your message. Somehow Kraig's reply showed up
> earlier in the archives than your original message.
>
> Aaron, where do you live again? It would be great to meet in person.

Chicago...it would be great!!!!

and to talk over CPS's.

BTW, how do you explor CPS tunings--homebuilt instuments? Csound? 12-tone
keyboards?

Best,
Aaron.

> I had a far easier time explaining CPS scales (and other matters) to
> Monz when we met in person than would have been possible otherwise.
> Also, it would be great to jam with you! I too love immersing myself
> in a ridiculous variety of musical situations -- from atonal acoustic
> guitar bebop to folk-rock 3-part harmony singing . . . on New Year's
> Eve I drummed with a DJ and played keyboard bass with an African
> master tunesmith . . . life is good!
>
> -Paul
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold
> for the tuning group. tuning-digest@yahoogroups.com - change your
> subscription to daily digest mode. tuning-normal@yahoogroups.com - change
> your subscription to individual emails. tuning-help@yahoogroups.com -
> receive general help information.
>
>
>
> Yahoo! Groups Links
>
> To visit your group on the web, go to:
> /tuning/
>
> To unsubscribe from this group, send an email to:
> tuning-unsubscribe@yahoogroups.com
>
> Your use of Yahoo! Groups is subject to:
> http://docs.yahoo.com/info/terms/

--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
> On Saturday 03 January 2004 04:40 pm, Paul Erlich wrote:
> > Aaron J.,
> >
> > I just came across your message. Somehow Kraig's reply showed up
> > earlier in the archives than your original message.
> >
> > Aaron, where do you live again? It would be great to meet in
person.
>
> Chicago...it would be great!!!!

I played 9 gigs in Chicago on a blues tour I did in '98. Buddy Guy's,
Shuba's, etc. . . . Unfortunately I'm somewhat stuck in Boston at the
moment. You're welcome to visit, crash at my place, etc.

> and to talk over CPS's.
>
> BTW, how do you explor CPS tunings--homebuilt instuments? Csound?
12-tone
> keyboards?

My Ensoniq keyboard is fully retunable and I have 22-equal and 31-
equal guitars. I don't know if I've ever felt true musical "magic"
while playing with CPSs but still find them extraordinarily
fascinating theoretically -- as a scan through the archives will
attest.

Did you get a chance to spend some time with Dave Keenan's "Tumbling
Dekany"? My hours with that probably represent the bulk of my musical
CPS experience.

Cheers,
Paul

hi paul,

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> Kraig, 1x3x5 = 3x5, which occurs in all such Genuses (Geni?).

it's interesting that people keep using this same erroneous
plural for "genus". the correct plural is "genera".

http://tonalsoft.com/enc/genus.htm

(and again, at the moment my server seems to be down right now.)

-monz

Paul wrote:
>Why would one want to keep it? It seems to go against the whole
>concept, symmetry and everything.

A CPS plus 1/1 also has symmetry. But it's up to the user to answer
that question. I don't see a reason to assume that no user ever
wouldn't want to keep it.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Paul wrote:
> >Why would one want to keep it? It seems to go against the whole
> >concept, symmetry and everything.
>
> A CPS plus 1/1 also has symmetry.

Not nearly as much. A hexany has all the symmetries of the octahedron
and cube, while a hexany plus 1/1 (that is, a hexany with a tetrad on
one face) has only the symmetry of a triangular pyramid (not even
tetrahedron).

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>

/tuning/topicId_50781.html#50967

wrote:
> On Saturday 03 January 2004 04:40 pm, Paul Erlich wrote:
> > Aaron J.,
> >
> > I just came across your message. Somehow Kraig's reply showed up
> > earlier in the archives than your original message.
> >
> > Aaron, where do you live again? It would be great to meet in
person.
>
> Chicago...it would be great!!!!
>
> and to talk over CPS's.
>
> BTW, how do you explor CPS tunings--homebuilt instuments? Csound?
12-tone
> keyboards?
>
> Best,
> Aaron.
>

***Kraig Grady has done extensive work with original acoustic
instruments and, personally, I worked with Scala-retuned
synthesizers...

J. Pehrson