Re: [tuning] Digest Number 3438-the diamond

I might add that Erv Wilson pointed out to me that Augusto Novaro in his
1925 booklet presented a set of pitches identical to the 9-limit
diamond, but he derived it by process other than the interlocking
harmonic and subharmonic chords that Partch used to generate the
Tonality Diamond.
See Novaro's text on the Anaphoria site.

I was collaborating with Erv at the time he came up with the CPS and my
recollection is that
he was thinking of ways to relate chords as well as trying to find the
most useful set of
pitch bases (or 'keys') in which to print out the set of just intonation
tables I was computing for him at the USCD computer center.

--John

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--- In tuning@yahoogroups.com, John Chalmers <JHCHALMERS@U...> wrote:

> I was collaborating with Erv at the time he came up with the CPS and my
> recollection is that
> he was thinking of ways to relate chords as well as trying to find the
> most useful set of
> pitch bases (or 'keys') in which to print out the set of just intonation
> tables I was computing for him at the USCD computer center.

By the way, I just showed over on tuning-math that the eikosany is a
scale of ball type, another example of which is the diamond. One way
to describe what that means is that you can take the centroid, or
center point, of the eikosany, which is |0 3/2 1/2 1/2 1/2>, and draw
an ellipsoidal region around it which contains all of the notes of the
eikosany and no other notes. This was my project for the day and it
would be interesting to find a less laborious way of determining what
is, and what is not, a scale of ball type.

Message: 12 Date: Thu, 10 Mar 2005 17:42:33 -0800
From: John Chalmers <JHCHALMERS@UCSD.EDU>
Subject: Re: Digest Number 3438-the diamond

I might add that Erv Wilson pointed out to me that Augusto Novaro in his
1925 booklet presented a set of pitches identical to the 9-limit
diamond, but he derived it by process other than the interlocking
harmonic and subharmonic chords that Partch used to generate the
Tonality Diamond.

(Actually it is 1927) and hopefully by the fall we will have the bilingual edition for sale. translation is complete
It is hard to guess how Partch came up with the diamond, except that Forester attributes it to Meyer triadic diamond diagram ( see his site) I had assumed he examined the common tone modulations of the hexad. He did fill in the gaps in the tuning to get as close to an MOS as his base structured allowed.
I am not quite sure when he met with Schlesinger or had been exposed to her work before the meeting. on Page 15 of novaro we can see a diagram that resembles the method of Schlesinger mapping the subharmonic scales, in this case it is the harmonic,
interestingly he adds notes to the diamond to fill in the gaps.
It is uncanny that the two never met

I was collaborating with Erv at the time he came up with the CPS and my
recollection is that
he was thinking of ways to relate chords as well as trying to find the
most useful set of
pitch bases (or 'keys') in which to print out the set of just intonation
tables I was computing for him at the USCD computer center

this is well documented in http://www.anaphoria.com/dal.PDF
pages 1, 6 and 18

--
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

I am not sure that the eikosany can be given a single center point, but possibly i assume you are doing this just to generate the structure here.
In the letter to chalmers he tries to show how it can be generated from 12 different points.
for instance what tone does this center point coincide with.

But in order to take this whole discussion into a positive direction, i would be curious to know ...

what a ball type scale is
if there are more variations than the diamond and the eikosany
and if so how they overlap as it would interesting to compare with the what one finds with what we already have.
if this could be done with ratios and or lattices we can look at, i imagine it will be better understood.

By the way, I just showed over on tuning-math that the eikosany is a
scale of ball type, another example of which is the diamond. One way
to describe what that means is that you can take the centroid, or
center point, of the eikosany, which is |0 3/2 1/2 1/2 1/2>, and draw
an ellipsoidal region around it which contains all of the notes of the
eikosany and no other notes. This was my project for the day and it
would be interesting to find a less laborious way of determining what
is, and what is not, a scale of ball type.

> >

--
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, Kraig Grady <kraiggrady@a...> wrote:

> I am not sure that the eikosany can be given a single center point, but
> possibly i assume you are doing this just to generate the structure
here.
> In the letter to chalmers he tries to show how it can be generated
from
> 12 different points.
> for instance what tone does this center point coincide with.

The center point is 3^(3/2) 5^(1/2) 7^(1/2) 11^(1/2).

> But in order to take this whole discussion into a positive
direction, i
> would be curious to know ...
>
> what a ball type scale is
> if there are more variations than the diamond and the eikosany
> and if so how they overlap as it would interesting to compare with the
> what one finds with what we already have.

Suppose we take a four dimensional lattice of pitch classes for the
11-limit, in which {1,5,7,9,11} is a regular 4-simplex, or
pentachoron--the 4-dimensional analog of a regular tetrahedron. In
this lattice therefore 9 has the same distance from the unison as 5,7,
and 11, and so of course 3 has half of that distance. If we use this
metric, and draw hyperspheres around the center point I gave above, we
find we first touch a "shell" of six intervals equidistant from the
center, {45, 63, 99, 105, 165, 231}. Three of these are of the form
9*p, where p is 5,7, or 11, and the other three are of the form
3*p*q, where p and q are distinct primes from 5,7, and 11. Then we hit
a second shell, consisting of twelve more points, which gives us
{15, 21, 33, 35, 55, 77, 135, 189, 297, 315, 495, 693}. This consists
of 3*p, 27*p, 9*p*q, and p*q tones. Thirdly we hit a shell consisting
of just two tones, 27 and 385. We can keep on adding as many shells as
we like, but stopping here gives the eikosany.

There are many more ball scales than these, of course, and
noneuclidean metrics, such as the Hahn metric which has been discussed
on the tuning-math list, add to the possibilities. It is also possible
to do balls of chords, rather than of notes directly. By seeing that
the hexany, p-limit diamond, and eikosany are balls we put them in
this broader context.

hi Kraig,

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

> <snip>
>
> It is hard to guess how Partch came up with the diamond,
> except that Forester attributes it to Meyer triadic diamond
> diagram ( see his site)

this question has come up here before, and i've addressed it:

/tuning/topicId_43101.html#43120

there's is no doubt in my mind at all that Partch adapted
Meyer's diagram, to create the tonality diamond. Meyer's
looks exactly the same except that he uses integer proportions
instead of ratios.

i thought i made a scan of Meyer's diagram and had it
up on the web somewhere, but apparently not ... and now
i no longer have a working scanner ... oh well. i'll
see if i can do it sometime (but since i just moved, now
i'm not even sure where i have my copy of Meyer's book ...)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> hi Kraig,
>
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> > <snip>
> >
> > It is hard to guess how Partch came up with the diamond,
> > except that Forester attributes it to Meyer triadic diamond
> > diagram ( see his site)
>
>
>
> this question has come up here before, and i've addressed it:
>
> /tuning/topicId_43101.html#43120
>
>
> there's is no doubt in my mind at all that Partch adapted
> Meyer's diagram, to create the tonality diamond. Meyer's
> looks exactly the same except that he uses integer proportions
> instead of ratios.
>
> i thought i made a scan of Meyer's diagram and had it
> up on the web somewhere, but apparently not ... and now
> i no longer have a working scanner ... oh well. i'll
> see if i can do it sometime (but since i just moved, now
> i'm not even sure where i have my copy of Meyer's book ...)

aha! thanks to Yahoo's lousy search engine, i found
what i was remembering incorrectly ... i had made an ASCII
version of Meyer's diagram on this list a long time ago:

/tuning/topicId_14360.html#14360

(you have to click "reply" on the Yahoo interface to see
it formatted properly)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > there's is no doubt in my mind at all that Partch adapted
> > Meyer's diagram, to create the tonality diamond. Meyer's
> > looks exactly the same except that he uses integer proportions
> > instead of ratios.

i've finally made a graphic of this, and put it into the
files section of this list:

/tuning/files/
meyer_tonality_diamond_precursor.gif

(delete the line break, or use this instead:)

http://tinyurl.com/58ztf

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > > there's is no doubt in my mind at all that Partch adapted
> > > Meyer's diagram, to create the tonality diamond. Meyer's
> > > looks exactly the same except that he uses integer proportions
> > > instead of ratios.
>
>
> i've finally made a graphic of this, and put it into the
> files section of this list:
>
> /tuning/files/
> meyer_tonality_diamond_precursor.gif
>
> (delete the line break, or use this instead:)
>
> http://tinyurl.com/58ztf

hmmm ... actually, it's obvious from this diagram that
Meyer's values are in cents. he may have had another version
of the diagram which used integer proportions ... at any rate,
integer proportions is the format he uses throughout his book.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> /tuning/files/
> meyer_tonality_diamond_precursor.gif

Why is this a "precursor"? Isn't it just the diamond?

>> I am not sure that the eikosany can be given a single center point,
>> but possibly i assume you are doing this just to generate the
>> structure here.
>> In the letter to chalmers he tries to show how it can be generated
>> from 12 different points. for instance what tone does this center
>> point coincide with.
>
>The center point is 3^(3/2) 5^(1/2) 7^(1/2) 11^(1/2).

You probably realize this Kraig, but there's no pitch at
this point. The center of the diamond is one of its own
pitches.

-Carl

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@c...>
wrote:
If we use this
> metric, and draw hyperspheres around the center point I gave above,
we
> find we first touch a "shell" of six intervals equidistant from the
> center, {45, 63, 99, 105, 165, 231}. Three of these are of the form
> 9*p, where p is 5,7, or 11, and the other three are of the form
> 3*p*q, where p and q are distinct primes from 5,7, and 11.

This hexany-like structure, corresponding to a hole of the lattice,
strikes me as something which should be interesting to explore for
fans of the {1,3,5,7} hexany. It's not quite the same kind of thing,
but has many similarities. Scala knows not of it; has anyone seen it
before?

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

> This hexany-like structure, corresponding to a hole of the lattice,
> strikes me as something which should be interesting to explore for
> fans of the {1,3,5,7} hexany. It's not quite the same kind of
thing,
> but has many similarities. Scala knows not of it; has anyone seen
it
> before?

Well, duh. If I divide out a common factor of 3, I get
{15,21,33,35,55,77}, and that is just the {3,5,7,11} combinations of
two CPS. So, these are lattice holes, and the eikosany appears
surrounding this kind of hole.

hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > /tuning/files/
> > meyer_tonality_diamond_precursor.gif
>
> Why is this a "precursor"? Isn't it just the diamond?

yes, i understand that from a mathematical point of view
there's nothing special which differentiates Meyer's diamond
from Partch's.

my point is adressing the historical question of where
Partch got the idea for the diamond ... it didn't just
spring from his mind spontaneously -- it was inspired by
Partch's knowledge of Meyer's book, which is abundantly
cited in Partch's own book.

i pointed this out because there is speculation that
Partch may have gotten the idea from Novaro, but i haven't
read Novaro's works in detail and don't know how well
acquainted Partch was with them. but i know for sure that
he had read Meyer.

-monz

<>
From what we know at the present , the only known copy of the 1927 book of Novaro is in the library of congress and was 'found' accidently by mark Rankin thinking he was going to get his 1953 book of the same name. Unless there is a copy somewhere else , it is pretty hard to imagine how Partch could have gotten it from Novaro. I know of no one claiming otherwise
On the question of Meyer and Partch though.
Could we not say that Meyer just took the Lambdoma of Nicomachis
and applied the 3 limit consonant structure to a 5 limit one.
Possibly Partch saw Meyer work in the same light. he was after all a greek revivalist
and it is to this that he regresses in order to start anew
<> Partch did more than just taking the diamond out to 11 limit, he filled in the blanks that made up a 41 tone scale with two alternates.
Partch in other words made a musically viable tuning that included the dianmond, but that was not all of it.

In a sense it is an example of an MOS created by the mans ear and musical instinct.
Meyer exibited no understanding of his triadic diamond in the context of a linear scale or what it would take to make it musically useful.

yes, i understand that from a mathematical point of view
there's nothing special which differentiates Meyer's diamond
from Partch's.

my point is adressing the historical question of where
Partch got the idea for the diamond ... it didn't just
spring from his mind spontaneously -- it was inspired by
Partch's knowledge of Meyer's book, which is abundantly
cited in Partch's own book.

i pointed this out because there is speculation that
Partch may have gotten the idea from Novaro, but i haven't
read Novaro's works in detail and don't know how well
acquainted Partch was with them. but i know for sure that
he had read Meyer.

-monz

>
> >

--
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

hi Kraig,

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

> On the question of Meyer and Partch though.
> Could we not say that Meyer just took the Lambdoma of
> Nicomachis and applied the 3 limit consonant structure
> to a 5 limit one. Possibly Partch saw Meyer work in the
> same light. he was after all a greek revivalist and it
> is to this that he regresses in order to start anew

sure, the lambdoma is the origin of all the lattice stuff.
but i pretty sure that Meyer was not aware of the lambdoma.
and Meyer's "tonality diamond" is not 5-limit, it's 7-limit.

> Partch did more than just taking the diamond out to 11 limit,
> he filled in the blanks that made up a 41 tone scale with
> two alternates.

that's one way of view how he settled on 43 tones.

but keep in mind that he used many varying numbers of
tones in his tuning system for years before he finally
settled on 43.

> Partch in other words made a musically viable tuning
> that included the dianmond, but that was not all of it.

and i've pointed out before that, for all his railing
against pythagoreanism, he himself extended the diamond
by including pitches that were 3/2's up and down from
those in the diamond.

http://sonic-arts.org:80/monzo/partch/et/partch-on-et.htm

(almost halfway down the page)

> In a sense it is an example of an MOS created by the mans
> ear and musical instinct.

it's also a 41-tone periodicity-block, keeping in mind
the two pairs of alternates.

http://sonic-arts.org/td/erlich/partchpblock.htm

> Meyer exibited no understanding of his triadic diamond
> in the context of a linear scale or what it would take
> to make it musically useful.

have you read Meyer's book? are you certain that you
can make that claim?

admittedly, Meyer's "Musician's Arithmetic" is based on
the idea of small-integer-ratio JI harmony ... but i would
hesitate before saying the he "exhibited no understanding
of his triadic diamond in the context of a linear scale".
in fact, he starts the book out by examining the usual
diatonic major scale.

i'd have to read it again myself to say for sure whether
you're right or not ... and don't have time for that now.

-monz

K,

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> <> Partch did more than just taking the diamond out to 11 limit

In point of fact, Partch had diagrams of diamonds going out to the 17
limit.

When people speak about what Partch did, did not, or might have done,
it is probably a good idea that they not only have read "Genesis of a
Music", but also the Gilmore biography and most importantly "Enclosure
III" which contains much of Partch's thoughts and correspondence.
There is a lot of light shed on his early, theory-developing days, and
naturally the same kind of lighting after he put down the theorizing
and began in earnest to create works of music and theatre/drama.

Cheers,
Jon