Re: non-centricity of CPS

I think it is an exageration to say that any CPS has no preferred tonal
center. All consonant dyads are not equally consonant and otonal chords are
not equal to utonal. It seems as though the note corresponding to the
product of the smallest factors would be preferred. But still it's an
extraordinary acheivement to get as close to atonal as this, in JI.

I think that complete atonality can only be obtained by using all the notes
from some equal temperament (equal division of the octave). But that is
boring. So is the complete tonality of a diamond. The partial tonality of a
CPS is much more interesting.

Here's a dekany drawn as a standard 7-prime-limit lattice (3:9 same as 1:3,
no edges drawn for ratios of nine). This is the CPS formed by taking the
product of two at a time from {1,3,5,7,9}

5*7
,'/ \`.
1*5-/---\-3*5-------5*9
|\/ \/| /|\
|/\ /\| / | \
1*7-------3*7-------7*9 \
`.\ /,' /,' `.\
1*3-------1*9-------3*9

Here it is with meantone-ish names.

Fx
,'/ \`.
A -/---\- E ------- B
|\/ \/| /|\
|/\ /\| / | \
D#------- A#------- E# \
`.\ /,' /,' `.\
C ------- G ------- D

It seems clear to me that some notes will have a much stronger pull than
others because they are involved in more consonances that are simple and
otonal.

-- Dave Keenan
http://dkeenan.com

Kraig Grady wrote:

> the suspension of tonality by a cycle/chain/shpere of consonance.

Nice! I just noticed that the simplest 3- and 5-limit hexanies (1,3,9,81 and 1,3,5,9) possibly answer a riddle I've been puzzling over
for a while. A lot of old European folk tunes - traditional tunes, I mean - sound pentatonic at first listen, but on analysis I find
they are actually hexatonic. For "major" melodies, either the 4th or 7th (diatonically speaking), but not both, will seemingly be
"added." At first I thought they were just passing tones "borrowed" from the diatonic scale but that didn't hold up to further
scrutiny. Where, I wondered, did the sixth note come from? Was this an intermediate stage between pentatonicism and diatonicism? That
makes no sense at all.

However, both of the above mentioned hexanies provide two modes each that seem to fit the rustic hexatonic scales nicely. From what I
hear, there is evidence of similar scales in Africa, though that may result from an imposed Western viewpoint.

Perhaps the tendency to "suspend" tonality explains why some of these melodies feel modal rather than tonal, even though they are in
neither pentatonic nor diatonic modes. A classic example is the early American Appalachian hymn tune "What Wondrous Love Is This?":

D D C E G A | G E D | D C E | A C B A G A G E D ...

Ahh, a closed system to explain a common phenomenon! :-) Wouldn't that be nice.

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

David Finnamore wrote,

>I just noticed that the simplest 3- and 5-limit hexanies (1,3,9,81 and
1,3,5,9)

I think the 81 is way to high a number for the hexany to work as such.

The normal hexatonic scale is just a chain of 5 (possibly meantone) fifths.

> From: David J. Finnamore [mailto:daeron@bellsouth.net]
> Sent: Tuesday, October 24, 2000 9:38 AM
> To: tuning@egroups.com
> Subject: [tuning] Re: non-centricity of CPS
> Perhaps the tendency to "suspend" tonality explains why some of
> these melodies feel modal rather than tonal, even though they are in
> neither pentatonic nor diatonic modes. A classic example is the
> early American Appalachian hymn tune "What Wondrous Love Is This?":
>
> D D C E G A | G E D | D C E | A C B A G A G E D ...
>
> Ahh, a closed system to explain a common phenomenon! :-)
> Wouldn't that be nice.

Hmmm ... the version of "Wondrous Love" that I know has two As in the first
group where you have "G A". I used to sing this (a capella) and play it on
an Appalachian dulcimer (that's the little lap-sized one without the
hammers), which is tuned for modal music. I don't have the list of modes
that dulcimer players use handy; I'm sure it's on the web, though. It starts
on D and ends on D, and the only note missing is F. I think it counts as
modal even with a missing note. Now I suppose I have to check my dulcimer to
see if the frets are spaced for just intonation or something else :-).
--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

Cold leftover pizza: it's not just for breakfast any more!

For a dulcimer in the key of D (3-string variety) see

http://www.sksmithmusic.com/virtual_classroom/modes.html

--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

Cold leftover pizza: it's not just for breakfast any more!

And a potentially microtonal dulcimer!!

http://www.servtech.com/~dwilder/bear-meadow/flexifrets.html

"What wondrous love is this, oh my soul, oh my soul?"
--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

Cold leftover pizza: it's not just for breakfast any more!

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> David Finnamore wrote,
>
> >I just noticed that the simplest 3- and 5-limit hexanies (1,3,9,81
and
> 1,3,5,9)
>
> I think the 81 is way to high a number for the hexany to work as
such.
>
> The normal hexatonic scale is just a chain of 5 (possibly meantone)
fifths.

That's exactly what this hexany produces - a chain of 5 fifths; just
ones in this case. As it turns out, it only produces two complete
triads. But since every tone has one or two others in the system
that are only 3:2 away, I don't see why any of them should be
regarded as "too high." It's all relative. Of course, I haven't
tried it yet, either. Maybe I should shut up and go do my homework.

David Finnamore

Hi, Ed,

I don't think all dulcimers have to be tuned (or fretted) the same. I bought

one years ago from the guy who made it. The frets were exactly under the

nodal points where you touch the strings to get the harmonics, so it can

play just intonation, but with only 3 strings the chords are limited,

and also it isn't very loud because I changed the strings to nylon strings.

(The body shape doesn't seem to amplify it as much as the shape of,

for example, a classical guitar.)

Best,

Lydia Ayers

I wrote,

>> I think the 81 is way to high a number for the hexany to work as
such.
>>
>> The normal hexatonic scale is just a chain of 5 (possibly meantone)
fifths.

David Finnamore wrote,

>That's exactly what this hexany produces - a chain of 5 fifths; just
>ones in this case.

yesss...

>As it turns out, it only produces two complete
>triads.

What do you mean?

>But since every tone has one or two others in the system
>that are only 3:2 away, I don't see why any of them should be
>regarded as "too high." It's all relative. Of course, I haven't
>tried it yet, either. Maybe I should shut up and go do my homework.

What I meant was, the hexany formalism would regard the interval between
81*9 and 3*1, i.e., the 243:128, as a consonant interval. So I don't think
the hexany formalism is particularly useful when applied to this scale.

I wrote to David Finnamore,

>What I meant was, the hexany formalism would regard the interval between
>81*9 and 3*1, i.e., the 243:128, as a consonant interval. So I don't think
>the hexany formalism is particularly useful when applied to this scale.

That's not right. What I meant was, the hexany formalism would regard the
interval between 3*81 and 3*1, i.e., an 81:64, as a consonant interval,
while it would regard the interval between 1*81 and 3*9, i.e., a 3:2, as a
dissonant interval. Hence, I don't think the hexany formalism is
particularly useful when applied to this scale.

> -----Original Message-----
> From: Lydia Ayers [mailto:LAYERS@CS.UST.HK]
> The frets were exactly under the
> nodal points where you touch the strings to get the harmonics, so it can
> play just intonation, but with only 3 strings the chords are limited,
> and also it isn't very loud because I changed the strings to
> nylon strings.
> (The body shape doesn't seem to amplify it as much as the shape of,
> for example, a classical guitar.)

Typical "traditional" Appalachian dulcimer music uses the two lower strings
as drones, with the melody on the highest (pitched) string. This string is
usually closest to the player and is worked with a stick rather than the
fingers, while the strings are strummed with a turkey feather. Anyhow,
that's the way *I* learned to play it. Nowadays anything that can be done
will be done, including (gasp) putting a capo on to change the pitch :-(. My
dulcimer has four strings; I usually tune the two highest strings together,
which gives the melody more strength, and I use metal strings. I'm pretty
sure mine is 12-TET fretted, because dulcimer / folk guitar duos were quite
common in the 1960s. If they're still in print, you can hear some
first-class music of this type from Richard and Mimi Farina.
--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

Cold leftover pizza: it's not just for breakfast any more!

Paul H. Erlich wrote:

> the hexany formalism would regard the
> interval between 3*81 and 3*1, i.e., an 81:64, as a consonant interval,
> while it would regard the interval between 1*81 and 3*9, i.e., a 3:2, as a
> dissonant interval. Hence, I don't think the hexany formalism is
> particularly useful when applied to this scale.

That makes sense. As I play with it, though, it still seems to reveal sonically the shape of the octagon. Perhaps what I'm doing with
it is somewhat different that "usual" thing - another "diverse (and sometimes bizarre)" version of the idea.

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

David Finnamore wrote,

>As I play with it, though, it still seems to reveal sonically the shape of
the octagon.

The octahedron? How so?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> David Finnamore wrote,
>
> >As I play with it, though, it still seems to reveal sonically the
shape of
> the octagon.
>
> The octahedron? How so?

Octahedron, sorry. I'll try to get to where I can post a meaningful
audio example. Of course, I could be fooling myself - imposing on
the sound images I was already seeing in my head. It seems like if I
picture one of the triads as a triangle, I can "hear" it shifting to
each of the other triangles at different angles as I play them as
progressions. It's a little bit like in 12 EDO how a full diminished
7th chord sounds like a square, and an augmented chord sounds like a
large equilateral triangle. With the 1,3,9,81 hexany, the chords
have different shapes of that kind, of course. It appears that they
should all have the same shape - when you look at each "face on."
But you can also see them as if viewing the whole octahedron from a
single viewpoint, so that the changing chord shapes are analogous to
the 2-D projection of each face from that angle. Then you can "hear"
the chord progression shifting around from face to face. I think.

David Finnamore

David Finnamore wrote,

> It seems like if I picture one of the triads as a triangle, I can
"hear" it shifting to each of the other triangles at different angles
as I play them as progressions. It's a little bit like in 12 EDO how
a full diminished 7th chord sounds like a square, and an augmented
chord sounds like a large equilateral triangle.

Hmm, then I think you'd also sort of have to imagine a 4:5:6 as
sounding like a (LOG(N)-LOG(D))*(360/LOG(2)) triangle as opposed to a
360/D*N triangle (etc., etc.)...

Personally, I really like these sorts of collisions between
logarithmic and linear designs, and while I do think that most folks
think like you are here -- ETs as equidistant audio visual objects --
it's tough to imagine a simple little spinning geometric interface
were both could be clearly discernable at the same time in the context
of the analogy you make... it'd get pretty "messy" in hurry.

Though to my mind this seems more "natural" anyway. It's definitely
much closer to what I see/hear/feel when I look into any given (macro)
clump of the natural world...

--d.stearns

David Finnamore wrote,

>With the 1,3,9,81 hexany, the chords
>have different shapes of that kind, of course. It appears that they
>should all have the same shape - when you look at each "face on."
>But you can also see them as if viewing the whole octahedron from a
>single viewpoint, so that the changing chord shapes are analogous to
>the 2-D projection of each face from that angle. Then you can "hear"
>the chord progression shifting around from face to face. I think.

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.