RE: Re: non-centricity of CPS/Pythagorean Hexany piece

Paul H. Erlich wrote:

> I guess I was questioning that because some pairs of intervals (like
> (81*1)/(9*3) and (81*3)/(81*1)) would sound exactly the same -- so why would
> they "look" so different? Only if you preconceived them to, I would argue.

Perhaps. So valid hexanies are composed of twenty distinctly different sized intervals?

The above don't sound the same unless you sound them in isolation from each other, outside the structure. As you know, they produce
the same interval but at different pitches. Their relationship to each other is what I'm seeing as lying on the face of the
octahedron. If the same intervals at different pitches can't communicate a shift to a different side of a geometrical figure, then how
can Erv's same-pitch-for-two-products thing work? Seems to me I'm not pushing the abstraction as far as he is.

The shape changes because of viewing it from a different angle. If the A face looks like an equilateral triangle, the B face will not
look equilateral from that same angle. I'm seeing the octahedron as made of lines, not planes, so I can see the B from its "back side"
while looking at the A face on. This is all completely relative. Since a geometrical figure can be viewed from any angle, I don't see
how _any_ direct, absolute relationship could ever be established between chord shapes and the sides of a figure in a CPS structure.
If they could, since we're using "fair dice" as our figures, all chords would have to have the same shape. That's obviously not going
to happen with a CPS structure. I don't think you're saying it should work that way, but I'm stating that I don't think so either.

It's the _kinds of relationships_ between the faces of a shape as viewed from any one particular angle that should be able to be
reflected musically. Now, no one is going to hear a piece of music in a hexany and immediately be reminded of 3D geometry. Well,
geometricians might but not because the musical structure gives them such clues. :-) The point, it seems to me, is to reflect the
beauty and symmetry of the shape by musical analogy, not to make the shape itself appear in the listener's imagination.

So what do other composers seek to accomplish with CPS structures? Kraig? Carl?

> Beautiful piece! I really don't see what it has to do with any of the
> properties of the hexany, but it's a lovely piece in the Pythagorean
> hexatonic scale.

Thank you. I stuck religiously to the chords dictated by the faces of the octahedron, and followed logical paths around it, making use
of the common tones to pivot from face to face. What other properties of a hexany can be revealed musically?

If I had been composing simply in a hexatonic scale, I would have had many more triads and a bunch of tetrads and pentads at my
disposal, not just the 8 triads of the hexany. Also, I would have had far more melodic freedom. But I don't see the point of a merely
hexatonic Pythagorean scale. The circle "closes" at five and seven tones, thus providing complete systems. Six Pythagorean tones
don't provide a complete system when viewed as a string of 3:2s. It's an open circle. They should form a cohesive set when placed on
the 6 points of an octahedron, as far as I can see. What am I missing?

Oh, I see I never answered one of your questions about "complete triads." As you may have guessed, I was making the mistake of looking
at it from a diatonic perspective, counting only triads with a 1-3-5 (diatonic relative scale number) relationship as "complete." That
appears to have nothing to do with CPS.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

http://www.egroups.com/message/tuning/15125

> So what do other composers seek to accomplish with CPS structures?
Kraig? Carl?

Hi David...

Well, I did one, which you may or may not find interesting:

http://artists.mp3s.com/artist_song/749/749501.html

_____________ ____ _
Joseph Pehrson

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:
> Paul H. Erlich wrote:
>
> > I guess I was questioning that because some pairs of intervals (like
> > (81*1)/(9*3) and (81*3)/(81*1)) would sound exactly the same -- so why would
> > they "look" so different? Only if you preconceived them to, I would argue.
>
> Perhaps. So valid hexanies are composed of twenty distinctly different sized intervals?

No. Much less that that. However, the pair of intervals I mentioned above look completely
different on your hexany diagram -- one is a direct connection between two adjacent
notes, while the other relates two notes on opposite sides of the hexany.

> The above don't sound the same unless you sound them in isolation from each other, outside the structure. As you know, they produce
> the same interval but at different pitches. Their relationship to each other is what I'm seeing as lying on the face of the
> octahedron. If the same intervals at different pitches can't communicate a shift to a different side of a geometrical figure, then how
> can Erv's same-pitch-for-two-products thing work? Seems to me I'm not pushing the abstraction as far as he is.

In Erv's diagrams a given interval will normally appear in different places in the diagram
with the same length and the same direction. That's what lattices are all about. In your
hexany that's not happening.
>
> The shape changes because of viewing it from a different angle. If the A face looks like an equilateral triangle, the B face will not
> look equilateral from that same angle. I'm seeing the octahedron as made of lines, not planes, so I can see the B from its "back side"
> while looking at the A face on. This is all completely relative. Since a geometrical figure can be viewed from any angle, I don't see
> how _any_ direct, absolute relationship could ever be established between chord shapes and the sides of a figure in a CPS structure.

It can, and is. In the lattice (even yours), a triangle in a given orientation will always
represent the same kind of chord, with the same intervals (octave-equivalence
assumed). The eight triangles in the hexany are all in different orientations (you can't
make one coincide with another just by sliding), thus they should all be different kinds of
chords. In the 1-3-5-7 hexany, they are: 1:3:5, 1:3:7, 1:5:7, 3:5:7, 1/(1:3:5), 1/(1:3:7), 1/(1:5:7),
1/(3:5:7).
>
> If I had been composing simply in a hexatonic scale, I would have had many more triads and a bunch of tetrads and pentads at my
> disposal, not just the 8 triads of the hexany. Also, I would have had far more melodic freedom. But I don't see the point of a merely
> hexatonic Pythagorean scale. The circle "closes" at five and seven tones, thus providing complete systems. Six Pythagorean tones
> don't provide a complete system when viewed as a string of 3:2s. It's an open circle. They should form a cohesive set when placed on
> the 6 points of an octahedron, as far as I can see. What am I missing?

David, you could make a 2)4 1-9-27-81 hexany using the exact same Pythagorean
hexatonic scale -- the same six pitches. This would give you a different set of 8 triads and
different melodic choices. However, if you "modulated" from one hexany to the other at
various points in your piece, would they be perceived as modulations?

In your original hexany, consider the following chords:

a) (81*1):(9*3):(9*1)
b) (1*9):(1*3):(3*9)
c) (81*1):(1*3):(81*3)

Of these, (b) and (c) are triads you're allowing yourself to use, while (a) is not. However,
(a) is more consonant than (c), and just as consonant as (b). Normally, the "disallowed"
triads in the hexany should be at least as dissonant as the "allowed" ones. That's kind of
the point of the hexany.

I would argue that you could map the six notes of the hexatonic scale to the octahedron
any old way, disregarding products of factors, etc., and still end up with just as nice and
logical a piece of music, because you're a talented composer.

> Oh, I see I never answered one of your questions about "complete triads." As you may have guessed, I was making the mistake of looking
> at it from a diatonic perspective, counting only triads with a 1-3-5 (diatonic relative scale number) relationship as "complete." That
> appears to have nothing to do with CPS.

I don't even remember when that came up . . . Anyway, did I mention I really liked your
piece?

Paul,

Thank you for your patient explanation and your encouraging words.
"Lattice" was the key - it's like a section of a lattice, not merely
an abstraction. I think I get it now. Of course, I said that last
time. :-) We'll see.

David Finnamore

Hi David,

>So what do other composers seek to accomplish with CPS structures? Kraig?
>Carl?

Actually, I've never composed strictly within a CPS, at least not
intentionally (other than a few demos which I'll post shortly).

One thing I'm very interested in as a composer is leaving diatonicity and
exploring a style where (often simple) melodies are derrived from non-
diatonic common-tone modulations, and subject to heavy variation (with the
interesting catch that the borders between variations are intentionally
blurred -- but that's another topic for another list). Inspired by late
Beethovan, YES, and Jazz, and much 20th-century music, this is the
direction I've taken with 12-tET so far. Anywho, I think CPSs are a great
way to familiarize oneself with the common-tone progressions of a given
limit, and I like the way their incomplete chords help thin out higher-
limit utonal sonorities. For these reasons, I plan on spending some
quality time with the pentadekany and eikosany at some point. Eventually,
however, I'd imagine I'd work in free-JI (and perhaps even extend
common-tone possibilities somewhat with temperament), but armed with my
CPS experience.

Of course, I'm also interested in new types of diatonicity (it's not
diatonicity I'm bored of... just 5-limit 7-tone diatonicity!) like Paul
Erlich's decatonic scales in 22-tET, or Easley Blackwood's decatonic in
15-tET (his 15-tone etude is one of my favorite microtonal tunes). To that
end, the hexany and dekany might prove to be good theoretical tools for
locating "generalized diatonic" scales.

BTW, I like your hexany piece too! And your contribution to the tape swap
was way cool, as I've said here before!

-Carl