What is the name of this?

Harry Partch's diamond is iteresting and I think many people who have seen it have thought of the same generalization as occured to me.

In this generalization, the Partch diamond scale can be constructed by doing something like
[1,3,5,7,9,11] x [1/1,1/3,1/5,1/7,1/9,1/11]

where x means something similar to cartesian product (I'm sure you can figure out what I mean). If we also introduce a unary operator ~ which reprociates every element in a set, we can type it even more concisely:

[1,3,5,7,9,11] x ~[1,3,5,7,9,11]

Now an obvious thing one might want to try out is a scale like:

[1,3,5,7,9,11] x [1,3,5,7,9,11]

or more concisely

[1,3,5,7,9,11]^2

As another example, the first 8 harmonics of the tonic, dominant and subdominant could be expressed as:

[1,2,3,4,5,6,7,8] x [1,3/2,2/3]

I don't think this concept is new, but also I haven't found any mention of it anywhere. The closest thing I've seen is Combination Product Set,
but that concept seems more complicated. So if anyone knows the name...

-Magnus Jonsson

--- In tuning@yahoogroups.com, Magnus Jonsson <magnus@s...> wrote:
>
> Harry Partch's diamond is iteresting and I think many people who
have seen
> it have thought of the same generalization as occured to me.
>
> In this generalization, the Partch diamond scale can be
constructed by
> doing something like
> [1,3,5,7,9,11] x [1/1,1/3,1/5,1/7,1/9,1/11]
>
> where x means something similar to cartesian product (I'm sure you
can
> figure out what I mean). If we also introduce a unary operator ~
which
> reprociates every element in a set, we can type it even more
concisely:
>
> [1,3,5,7,9,11] x ~[1,3,5,7,9,11]
>
> Now an obvious thing one might want to try out is a scale like:
>
> [1,3,5,7,9,11] x [1,3,5,7,9,11]
>
> or more concisely
>
> [1,3,5,7,9,11]^2
>
> As another example, the first 8 harmonics of the tonic, dominant
and
> subdominant could be expressed as:
>
> [1,2,3,4,5,6,7,8] x [1,3/2,2/3]
>
> I don't think this concept is new, but also I haven't found any
mention of
> it anywhere. The closest thing I've seen is Combination Product
Set,
> but that concept seems more complicated. So if anyone knows the
name...
>
> -Magnus Jonsson

It sounds similar to Boulez' chord multiplication, but in JI,
although the technique can be adapted to any set of tones, sequence
or inclusion of various musical parameters/dimentions (the ppppp to
fffff dynamic scale, 128 MIDI values, woodwind articulations etc).
So in essence you have some set of tones, or harmonics, and you are
transposing the same harmonic pattern over each (or particular)
member. Why multiply your special series over the fourth and fifth.
Back in the 20th century I wrote some interpretations of 10
Messiaen modes in JI, based on simple multiplications, similar to
how we analyze the modes to identify melodic resources inherent in
each scale.

Hi Magnus,

> or more concisely
>
> [1,3,5,7,9,11]^2

This is called (by Erv Wilson and in the Scala documentation)
a cross set or Carthesian cross set.

> As another example, the first 8 harmonics of the tonic,
> dominant and subdominant could be expressed as:
>
> [1,2,3,4,5,6,7,8] x [1,3/2,2/3]
>
> I don't think this concept is new, but also I haven't found
> any mention of it anywhere.

This latter idea is the basis of many scales. David Canright,
Kurt Bigler, and myself have used used 1,3,5,7,9,11 x [1,3] -type
12-note scales. Jules Siegel released at least 2 albums of
music in [1,3,5,7,9,11,13,15]^2 (see the JI Network Store).

> The closest thing I've seen is Combination Product Set,
> but that concept seems more complicated.

Yes, that's a bit different.

-Carl

Hi Magnus!
I know that some people at oberlin in the 80's were playing around with this scale.
i remember cause one ofthem was another Kraig with a K. THe second example, the only thing close i can think of are the helixsongs which are harmonic series a 3/2 apart.
the other example a while back someone had a tetradic or pentatic diamond on the tonic dominant and subdominant.
Paul might remember.

The nice thing about the diamond though is that it creats it own opposites so that each tone hase at least two uses. you get less variety with straight harmonics of harmonics

From: Magnus Jonsson <magnus@smartelectronix.com>
Subject: What is the name of this?

Harry Partch's diamond is iteresting and I think many people who have seen it have thought of the same generalization as occured to me.

In this generalization, the Partch diamond scale can be constructed by doing something like
[1,3,5,7,9,11] x [1/1,1/3,1/5,1/7,1/9,1/11]

where x means something similar to cartesian product (I'm sure you can figure out what I mean). If we also introduce a unary operator ~ which reprociates every element in a set, we can type it even more concisely:

[1,3,5,7,9,11] x ~[1,3,5,7,9,11]

Now an obvious thing one might want to try out is a scale like:

[1,3,5,7,9,11] x [1,3,5,7,9,11]

or more concisely

[1,3,5,7,9,11]^2

As another example, the first 8 harmonics of the tonic, dominant and subdominant could be expressed as:

[1,2,3,4,5,6,7,8] x [1,3/2,2/3]

I don't think this concept is new, but also I haven't found any mention of it anywhere. The closest thing I've seen is Combination Product Set,
but that concept seems more complicated. So if anyone knows the name...

-Magnus Jonsson

>
>
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Magnus Jonsson <magnus@s...> wrote:
>
> Harry Partch's diamond is iteresting and I think many people who
have seen
> it have thought of the same generalization as occured to me.
>
> In this generalization, the Partch diamond scale can be constructed
by
> doing something like
> [1,3,5,7,9,11] x [1/1,1/3,1/5,1/7,1/9,1/11]
>
> where x means something similar to cartesian product (I'm sure you
can
> figure out what I mean).

Hi Magnus,

If I may comment . . .

Yes. This is how many people have interpreted the Diamond. However,
reading Partch's Genesis, it seems that a specification closer to his
own concerns would consider the Diamond to consist of 1/1 and all the
notes at least somewhat "consonant" (by Partch's definition) with
1/1.

> If we also introduce a unary operator ~ which
> reprociates every element in a set, we can type it even more
concisely:
>
> [1,3,5,7,9,11] x ~[1,3,5,7,9,11]
>
> Now an obvious thing one might want to try out is a scale like:
>
> [1,3,5,7,9,11] x [1,3,5,7,9,11]
>
> or more concisely
>
> [1,3,5,7,9,11]^2

A third distinct possibility here is ~[1,3,5,7,9,11] x ~
[1,3,5,7,9,11], or "~[1,3,5,7,9,11])^2"

The diamond has more symmetries in the lattice of consonances (where
each consonant interval gets a "rung" of unit length) than the
("otonal") cross set or the inverse (or "utonal") cross set. The 11-
limit is a lot for ASCII graphics to handle (not to mention the 'odd'
question of 9:3 vs. 3:1), so let's come down to the 7-limit.

You'll need to click on "reply" to view these correctly if you're
reading off the website.

The 7-limit diamond:

5/3-------5/4
/|\`. ,'/|\
/ | \10/7 / | \
/ 7/6-\/|\/-7/4 \
/,' \`/\|/\'/ `.\
4/3-----/-1/1-\-----3/2
\`. /,'/|\`.\ ,'/
\ 8/7------12/7 /
\ | / 7/5 \ | /
\|/,' `.\|/
8/5-------6/5

The 7-limit otonal cross-set:

25/16
/|\
/ | \
/35/32\
/,'/|\`.\
5/4-/-|-\15/8
/|\/49/32\/ \
/ |/,' `.\ \
/ 7/4------21/16\
/,' `.\ /,' `.\
1/1-------3/2-------9/8

The 7-limit utonal cross-set:

16/9-------4/3-------1/1
\`. ,'/ \`. ,'/
\32/21/---\-8/7 /
\ |\/. ,\/| /
\|/\64/49/\|/
16/15------8/5
\`.\|/,'/
\64/35/
\ | /
\|/
32/25

Meanwhile, the 2)4 [1,3,5,7] CPS (Combination Product Set) scale, the
hexany, the six-note octahedron of notes found in the center of each
of the above cross sets, does have all the symmetries of the diamond.
(The note actually appearing in the 2D center of the diagrams --
49/32 or 64/49 -- is supposed to be the closest or farthest
(respectively) from the viewer in 3D, so doesn't count as part of the
3D center, which is the hexany.) So does the stellated hexany. If you
transpose the otonal and/or utonal cross sets so that their central
hexany is the same set of six notes, the resulting union of the two
happens to be the stellated hexany, with 14 notes in all. I have more
3D-looking diagrams of the 7-limit Diamond, Hexany, and Stellated
Hexany here:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

In the 11-limit, the 3)6 [1,3,5,7,9,11] CPS is called the Eikosany,
and has 20 notes. See if this document by Erv Wilson makes sense to
you:

http://www.anaphoria.com/dal.PDF

The Stellated Eikosany has a lot more notes, the exact number having
been a subject of debate here in July 2000. :)

This very structure is the opening discussion in
http://anaphoria.com/CPStoC-pt1.PDF

>
>Message: 21 > Date: Tue, 13 Sep 2005 20:15:04 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>Subject: Re: What is the name of this?
>
>
> >
>reply" to view these correctly if you're >reading off the website.
>
>The 7-limit diamond:
> > 5/3-------5/4
> /|\`. ,'/|\
> / | \10/7 / | \
> / 7/6-\/|\/-7/4 \
> /,' \`/\|/\'/ `.\
>4/3-----/-1/1-\-----3/2
> \`. /,'/|\`.\ ,'/
> \ 8/7------12/7 /
> \ | / 7/5 \ | /
> \|/,' `.\|/
> 8/5-------6/5
>
>The 7-limit otonal cross-set:
>
> 25/16
> /|\
> / | \
> /35/32\
> /,'/|\`.\
> 5/4-/-|-\15/8
> /|\/49/32\/ \
> / |/,' `.\ \
> / 7/4------21/16\ > /,' `.\ /,' `.\
>1/1-------3/2-------9/8
>
>The 7-limit utonal cross-set:
>
>16/9-------4/3-------1/1
> \`. ,'/ \`. ,'/
> \32/21/---\-8/7 /
> \ |\/. ,\/| /
> \|/\64/49/\|/
> 16/15------8/5
> \`.\|/,'/
> \64/35/
> \ | /
> \|/
> 32/25
>
>Meanwhile, the 2)4 [1,3,5,7] CPS (Combination Product Set) scale, the >hexany, the six-note octahedron of notes found in the center of each >of the above cross sets, does have all the symmetries of the diamond. >(The note actually appearing in the 2D center of the diagrams -- >49/32 or 64/49 -- is supposed to be the closest or farthest >(respectively) from the viewer in 3D, so doesn't count as part of the >3D center, which is the hexany.) So does the stellated hexany. If you >transpose the otonal and/or utonal cross sets so that their central >hexany is the same set of six notes, the resulting union of the two >happens to be the stellated hexany, with 14 notes in all. I have more >3D-looking diagrams of the 7-limit Diamond, Hexany, and Stellated >Hexany here:
>
>http://lumma.org/tuning/erlich/erlich-tFoT.pdf
>
>In the 11-limit, the 3)6 [1,3,5,7,9,11] CPS is called the Eikosany, >and has 20 notes. See if this document by Erv Wilson makes sense to >you:
>
>http://www.anaphoria.com/dal.PDF
>
>The Stellated Eikosany has a lot more notes, the exact number having >been a subject of debate here in July 2000. :)
> >

>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

You might think I plagiarized the beginning of this document below
but actually I never saw it before. Thanks for sharing it, looking
forward to reading the rest when I get a moment!

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> This very structure is the opening discussion in
> http://anaphoria.com/CPStoC-pt1.PDF
>
> >
> >Message: 21
> > Date: Tue, 13 Sep 2005 20:15:04 -0000
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> >Subject: Re: What is the name of this?
> >
> >
> >
> >
> >reply" to view these correctly if you're
> >reading off the website.
> >
> >The 7-limit diamond:
> >
> > 5/3-------5/4
> > /|\`. ,'/|\
> > / | \10/7 / | \
> > / 7/6-\/|\/-7/4 \
> > /,' \`/\|/\'/ `.\
> >4/3-----/-1/1-\-----3/2
> > \`. /,'/|\`.\ ,'/
> > \ 8/7------12/7 /
> > \ | / 7/5 \ | /
> > \|/,' `.\|/
> > 8/5-------6/5
> >
> >The 7-limit otonal cross-set:
> >
> > 25/16
> > /|\
> > / | \
> > /35/32\
> > /,'/|\`.\
> > 5/4-/-|-\15/8
> > /|\/49/32\/ \
> > / |/,' `.\ \
> > / 7/4------21/16\
> > /,' `.\ /,' `.\
> >1/1-------3/2-------9/8
> >
> >The 7-limit utonal cross-set:
> >
> >16/9-------4/3-------1/1
> > \`. ,'/ \`. ,'/
> > \32/21/---\-8/7 /
> > \ |\/. ,\/| /
> > \|/\64/49/\|/
> > 16/15------8/5
> > \`.\|/,'/
> > \64/35/
> > \ | /
> > \|/
> > 32/25
> >
> >Meanwhile, the 2)4 [1,3,5,7] CPS (Combination Product Set) scale,
the
> >hexany, the six-note octahedron of notes found in the center of
each
> >of the above cross sets, does have all the symmetries of the
diamond.
> >(The note actually appearing in the 2D center of the diagrams --
> >49/32 or 64/49 -- is supposed to be the closest or farthest
> >(respectively) from the viewer in 3D, so doesn't count as part of
the
> >3D center, which is the hexany.) So does the stellated hexany. If
you
> >transpose the otonal and/or utonal cross sets so that their
central
> >hexany is the same set of six notes, the resulting union of the
two
> >happens to be the stellated hexany, with 14 notes in all. I have
more
> >3D-looking diagrams of the 7-limit Diamond, Hexany, and Stellated
> >Hexany here:
> >
> >http://lumma.org/tuning/erlich/erlich-tFoT.pdf
> >
> >In the 11-limit, the 3)6 [1,3,5,7,9,11] CPS is called the
Eikosany,
> >and has 20 notes. See if this document by Erv Wilson makes sense
to
> >you:
> >
> >http://www.anaphoria.com/dal.PDF
> >
> >The Stellated Eikosany has a lot more notes, the exact number
having
> >been a subject of debate here in July 2000. :)
> >
> >
>
> >
> >
> >
> >
> >
> >
> >
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

>> http://anaphoria.com/dal.PDF

> You might think I plagiarized the beginning of this document below
> but actually I never saw it before. Thanks for sharing it, looking
> forward to reading the rest when I get a moment!

You've never seen this article before?!

-C.

No . but i am surprised you didn't know it was up (as is part 2 also) but you might have been off line.
This assumed i did mention it here which i had hope i did!

I tried rearranged Erv's work in recent months from chronologically to what i consider from easiest to more difficult.
someone else might have a different idea of this and sometime other factors played into it.
it still could change again
http://www.anaphoria.com/wilson.html
I mention this is case there are other things there you might not have seen

From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

You might think I plagiarized the beginning of this document below but actually I never saw it before. Thanks for sharing it, looking forward to reading the rest when I get a moment!
\

>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

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

> >> http://anaphoria.com/dal.PDF
>
> > You might think I plagiarized the beginning of this document below
> > but actually I never saw it before. Thanks for sharing it, looking
> > forward to reading the rest when I get a moment!
>
> You've never seen this article before?!

I've seen http://anaphoria.com/dal.PDF plenty of times before, but not
the document I was referring to in the quoted snip above.

wallyesterpaulrus wrote:

>--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> >
>>>>http://anaphoria.com/dal.PDF
>>>> >>>>
>>>You might think I plagiarized the beginning of this document below >>>but actually I never saw it before. Thanks for sharing it, looking >>>forward to reading the rest when I get a moment!
>>> >>>
>>You've never seen this article before?!
>> >>
>
>I've seen http://anaphoria.com/dal.PDF plenty of times before, but not >the document I was referring to in the quoted snip above.
> >
Maybe there's a problem with your monitor.

--
* David Beardsley
* microtonal guitar
* http://biink.com/db

Thanks Kraig, I'll have to review the site and read whatever "new"
articles I can find when I get a chance. Your work in sharing these
writings with us is greatly appreciated!

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> No . but i am surprised you didn't know it was up (as is part 2
also) but you might have been off line.
> This assumed i did mention it here which i had hope i did!
>
> I tried rearranged Erv's work in recent months from chronologically
to what i consider from easiest to more difficult.
> someone else might have a different idea of this and sometime other
factors played into it.
> it still could change again
> http://www.anaphoria.com/wilson.html
> I mention this is case there are other things there you might not
have seen
>
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> You might think I plagiarized the beginning of this document below
> but actually I never saw it before. Thanks for sharing it, looking
> forward to reading the rest when I get a moment!
> \
>
> >
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Sorry -- in case there's any confusion, the document that I *hadn't*
seen before is http://anaphoria.com/CPStoC-pt1.PDF . . .

--- In tuning@yahoogroups.com, David Beardsley <db@b...> wrote:
> wallyesterpaulrus wrote:
>
> >--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> >
> >
> >>>>http://anaphoria.com/dal.PDF
> >>>>
> >>>>
> >>>You might think I plagiarized the beginning of this document
below
> >>>but actually I never saw it before. Thanks for sharing it,
looking
> >>>forward to reading the rest when I get a moment!
> >>>
> >>>
> >>You've never seen this article before?!
> >>
> >>
> >
> >I've seen http://anaphoria.com/dal.PDF plenty of times before, but
not
> >the document I was referring to in the quoted snip above.
> >
> >
> Maybe there's a problem with your monitor.
>
> --
> * David Beardsley
> * microtonal guitar
> * http://biink.com/db

> > >> http://anaphoria.com/dal.PDF
> >
> > > You might think I plagiarized the beginning of this document
> > > below but actually I never saw it before. Thanks for sharing
> > > it, looking forward to reading the rest when I get a moment!
> >
> > You've never seen this article before?!
>
> I've seen http://anaphoria.com/dal.PDF plenty of times before,
> but not the document I was referring to in the quoted snip above.

I musn't have seen it either. What document is that?

-Carl

> > > > You might think I plagiarized the beginning of this document
> > > > below but actually I never saw it before. Thanks for sharing
> > > > it, looking forward to reading the rest when I get a moment!
> > >
> > > You've never seen this article before?!
> >
> > I've seen http://anaphoria.com/dal.PDF plenty of times before,
> > but not the document I was referring to in the quoted snip above.
>
> I musn't have seen it either. What document is that?

I see it was:

http://anaphoria.com/CPStoC-pt1.PDF

I *hadn't* seen that.

-Carl

Carl Lumma wrote:

>>>>>You might think I plagiarized the beginning of this document
>>>>>below but actually I never saw it before. Thanks for sharing
>>>>>it, looking forward to reading the rest when I get a moment!
>>>>> >>>>>
>>>>You've never seen this article before?!
>>>> >>>>
>>>I've seen http://anaphoria.com/dal.PDF plenty of times before,
>>>but not the document I was referring to in the quoted snip above.
>>> >>>
>>I musn't have seen it either. What document is that?
>> >>
>
>I see it was:
>
>http://anaphoria.com/CPStoC-pt1.PDF
>
>I *hadn't* seen that.
>
Maybe there's something wrong with your monitor. Check with Paul.

* David Beardsley
* microtonal guitar
* http://biink.com/db