Tripod notation and asymmetricized versions of symmetrical scales

On Thu, Feb 10, 2011 at 5:57 AM, Graham Breed <gbreed@...> wrote:
>
> There are other complexities involved. There are new, or
> obscure, technical terms. I say "Didymic" instead of
> "5-limit" which avoids a number but doesn't simplify
> anything. I introduce terms for the basic Magic
> intervals. There are two new variants of staff notation
> which are unavoidable given that they're what I was writing
> about.

I think that all of this is really good - I haven't seen this before.
Or I think I did once, a few years ago, but didn't understand any of
it back then.

So here's a question then: if we're talking about magic temperament,
then you're splitting the octave up into three unequal feet. But if
we're talking about augmented temperament, then you're splitting the
octave into three equal feet. And I've long noticed that magic is like
an asymmetrical or expanded version of augmented, and magic[7] and
augmented[6] share a lot of the same chord types. You start off with a
major chord, then if you move it "diatonic"ally over you get something
sounding like a major 7 chord in inversion with the third missing,
etc.

In the same way, kleismic[7] is sort of an asymmetrical version of
diminished[8], although the analogy doesn't work quite as well, and
blackwood[10] is like a symmetricized diatonic scale. Is there some
kind of relationship here I'm missing?

-Mike

Mike Battaglia <battaglia01@...> wrote:

> So here's a question then: if we're talking about magic
> temperament, then you're splitting the octave up into
> three unequal feet. But if we're talking about augmented
> temperament, then you're splitting the octave into three
> equal feet. And I've long noticed that magic is like an
> asymmetrical or expanded version of augmented, and
> magic[7] and augmented[6] share a lot of the same chord
> types. You start off with a major chord, then if you move
> it "diatonic"ally over you get something sounding like a
> major 7 chord in inversion with the third missing, etc.

You say that's a question, but it looks like an explanation
to me.

In Schoenberg's related notation, for 12-equal, the
augmented triads are equally tempered. So you could
interpret that as a notation for Augmented. You can also
say this is what Tripod Notation gives when you allow 5:4
and 9:7 to be tempered the same (36:35 tempered out) along
with the marvel temperament. August is what it seems to be
called.

> In the same way, kleismic[7] is sort of an asymmetrical
> version of diminished[8], although the analogy doesn't
> work quite as well, and blackwood[10] is like a
> symmetricized diatonic scale. Is there some kind of
> relationship here I'm missing?

The defining asymmetrical diminished tetrad would be

6:5 * 6:5 * 6:5 * 7:6 =~ 2:1

In vector form,

3<1,1,-1,0] + <-1,-1,0,1] =~ <1,0,0,0]
<2,2,-3,1] =~ <1,0,0,0]
<1,2,-3,1] =~ <0]

That means 126:125 would be tempered out. This defines a
starling temperament, so Tetrapod Notation would be for
Starling. The kleismic temperament (extension of Hanson)
seems to be Keemum:

http://x31eq.com/cgi-bin/rt.cgi?ets=19_15&limit=7

Graham

On Thu, Feb 10, 2011 at 2:42 PM, Graham Breed <gbreed@...> wrote:
>
> That means 126:125 would be tempered out. This defines a
> starling temperament, so Tetrapod Notation would be for
> Starling. The kleismic temperament (extension of Hanson)
> seems to be Keemum:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=19_15&limit=7

That's it! That's the precise temperament I was after - I was messing
with different kleismic generators today and couldn't get it.

So do you have any idea how to extend this to other tunings? Like, say
you want to symmetricize porcupine. How might one go about doing this?
I was thinking that 8&16 might be a good idea, but I'm not sure if it
actually is anymore.

-Mike

Mike Battaglia <battaglia01@...> wrote:

> So do you have any idea how to extend this to other
> tunings? Like, say you want to symmetricize porcupine.
> How might one go about doing this? I was thinking that
> 8&16 might be a good idea, but I'm not sure if it
> actually is anymore.

Normally, if you're looking for a notation, you take a
reasonably sized proper MOS. That doesn't work for Magic
because there isn't a proper MOS between 3 and 19 notes.
The tripod scale is a hack to get a staff notation out of
a 3 note MOS. I don't see why we should be looking for
Tripod-like solutions where there is an MOS of a reasonable
size.

Porcupine comes out as 7&15 for me, so it has a juicy 7
note generalized diatonic. I guess there's no proper MOS
between 1 and 7 so it won't divide into feet.

Graham