Involution maps and Rothenberg, comparing rank 2 temperaments, comma pumps

This is in relation to Petr's latest thread about comma pumps.

As you are all aware, these comma pumps sound awesome. But this is
only because we aren't fluent in negri and semisixths, so to speak. If
we were, we'd just hear these pumps as normal chord progressions, and
be like "OK, so what?" But we're not - instead we all hear it as some
kind of auditory illusion that blows our minds. So what's going on?

What's going on is that we're subconsciously applying some form of
internal (most likely 12-tet) mapping to everything we hear, and comma
pumps violate this template as per Rothenberg. Meaning - if you're
playing a comma pump, then I as a listener am mentally following it in
my head. Finally, at a certain point, something in my perception
"shifts," and voila, the comma pump illusion takes place.

This mysterious process can be broken down by seeing how different
intervals in things like porcupine "line up" if you remap them in
12-tet, and finding the areas where the internal logic becomes
inconsistent. There is a pattern to these inconsistencies - if you
apply a 12-tet mapping to porcupine, for example, there will be some
kind of pattern in where things break down. Furthermore, this pattern
will be different from what you get if you apply a 12-tet mapping to
negri instead, or to Hanson, or to 5-limit JI. This is what I think
leads to "the characteristic sound" for each comma pump.

For example, let's apply a 12-tet mapping to porcupine[7]. In
porcupine, if you start stacking minor thirds, C-Eb-Gb-Bbb - that Bbb
is going to be equal to 16/9, which maps to 10\12. On the other hand,
each 6/5 will map to 3\12, but three of those on top of each other
maps to only 9\12, not 10\12. So there's a discrepancy here, and that
discrepancy happens to be 250/243 itself: in 12-equal, 250/243 maps to
1\12, but in a porcupine tuning, it maps to a unison. So any time
250/243 is involved, you'll end up getting a shift of 1\12 somewhere.
The same applies to 16875/16384 as well, where you'll also get a shift
of 1\12 in 12-equal. However, the pattern of where exactly this shift
will occur differs from that of porcupine, and this is what leads to
the characteristic negri "sound," vs the characteristic porcupine
"sound." This should all sound tautological.

So, what I'm interested in is figuring out a way to flesh out exactly
what the porcupine pattern is vs the negri pattern. That is, if you
take a negri temperament and re-map everything to 12-tet, where will
things map "inconsistently?" Where does 250/243 manifest, and how?
It's the difference between 81/80 and 25/24, and the difference
between 10/9 and 27/25, and the difference between a bunch of other
stuff as well, but how can we apply this realization to the above
paradigm?

Gene seems to have done some work on something similar here:
http://www.archive.org/details/MusicForYourEars

Except in this case I'm trying to send a rank-2 temperament
(porcupine) to a rank-1 temperament (12-equal), so I'm not sure how
that'll work out.

-Mike

On Fri, Apr 22, 2011 at 4:08 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Gene seems to have done some work on something similar here:
> http://www.archive.org/details/MusicForYourEars
>
> Except in this case I'm trying to send a rank-2 temperament
> (porcupine) to a rank-1 temperament (12-equal), so I'm not sure how
> that'll work out.

To add to this, this whole thing will still work and be much simpler
if you keep it within rank-2, e.g. cross-apply porcupine to meantone.
The same thing ends up happening as if you try to apply meantone to
5-limit JI, or 5-limit JI to an inconsistent temperament - there are
times when the internal logic of it breaks down, and you have
ambiguity which can be exploited in comma pumps.

So with the above example, 3 6/5's on top of each other in porcupine
produces 16/9, which is a minor 7th in meantone, whereas 3 stacked
6/5's in meantone produces a diminished 7th. Thus, in meantone you end
up a chromatic semitone off from where you'd end up in porcupine. This
is because in JI, the two actually differ by 250/243, which is equated
with a chromatic semitone in meantone, and is equated with a unison in
porcupine. I'm having trouble seeing the big picture here, but there
has to be some way to utilize this tautology to algorithmically create
a map of how porcupine, in general, "lays out" with respect to
meantone.

Or, if we're assuming you're growing up in Negri-ville, you can use
this map to see how meantone lays out with Negri and hence where your
perception will break that way.

This would be tremendously useful to me as a type of Rosetta stone,
enabling one to easily figure out how another temperament lays out in
relation to chromatic hearing. Since these other temperaments will
warp the chromatic template in a certain, characteristic type of way,
that would be useful in giving me a stepping stone to then grasp
porcupine or negri logic directly. Furthermore, being easily able to
cross-reference temperaments would lead to a more generalized approach
for cognizing music, and one in which you don't have to try and "shut
your diatonic hearing off" to appreciate other tunings, but rather see
the big picture of how all of these things co-relate.

If someone can point out the larger pattern here, it would be muchly
appreciated.

-Mike