Complexity terminology wars

There are a few different meanings of the word "complexity." Here's a
roundup as best as I can tell of the ones in common usage:

1) (JI) "Complexity": applied to untempered monzos, and gives you
their weighted distance as measured by some norm along the JI lattice.
e.g. "3/2 is lower in complexity than 10/9."
2) (Temperamental) "Complexity": applies to untempered monzos and a
regular temperament; tells you how many generators it takes to get to
a representation of said monzo. e.g. "In porcupine, 10/9 is lower in
complexity than 3/2."
3) (TE) "Complexity": measures the complexity of an entire temperament
all at once. e.g. "meantone temperament is lower in complexity than
amity temperament."

Recently I was really tripped up by this one:
4) "Octave-Equivalent TE Seminorm": This seems to be a type of
"temperamental complexity." I initially thought, from context, it was
a form of JI complexity in that it applied to the JI lattice and just
ignored powers of 2/1. I'd also probably think it was a type of TE
complexity if I didn't know any better and couldn't realize from
context that we weren't talking about temperaments.

I sometimes see these terms get mixed together, forming statements
like "in amity, all of the lower-complexity intervals are more
complex." I guess this usage started because people didn't want to say
"in amity, all of the more consonant intervals are more complex,"
since "consonant" can be a loaded term.

Is there a consistent terminological convention that I'm unaware of?
If not, I find the current terminology somewhat confusing, especially
as I'm now trying to develop something called "Spectral complexity,"
which is a type of (#3) complexity. I suggest, if context makes it
unclear which type of complexity we're talking about, that the first
type be specifically called "just complexity," the second type be
called "temperamental complexity," and I have no idea what to call the
third one, because "temperamental complexity" is already taken. Maybe
"homomorphism complexity" or "icon complexity" would be a good way to
go.

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> Is there a consistent terminological convention that I'm
> unaware of? If not, I find the current terminology
> somewhat confusing, especially as I'm now trying to
> develop something called "Spectral complexity," which is
> a type of (#3) complexity. I suggest, if context makes it
> unclear which type of complexity we're talking about,
> that the first type be specifically called "just
> complexity," the second type be called "temperamental
> complexity," and I have no idea what to call the third
> one, because "temperamental complexity" is already taken.
> Maybe "homomorphism complexity" or "icon complexity"
> would be a good way to go.

Just complexity is temperamental complexity applied to just
intonation. That is, you take the mapping of
just intonation to itself (an automorphism) and plug it
into the temperamental complexity formula as if just
intonation were a temperament.

Graham

On Tue, Sep 27, 2011 at 11:18 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> Just complexity is temperamental complexity applied to just
> intonation. That is, you take the mapping of
> just intonation to itself (an automorphism) and plug it
> into the temperamental complexity formula as if just
> intonation were a temperament.

Yes, I understand that, but people sometimes use the term "complexity"
by itself to mean the "just complexity" of a tempered interval. This
implicitly assumes we've taken some kind of transversal of the
tempered space, picking the lowest-complexity JI element from each
quotient group of JI mod the kernel that maps onto that point in the
space, and we're measuring the complexity of that. For example, the
statement "all of the low-complexity intervals in Amity are more
complex" assumes this meaning for the phrase "low-complexity," but
assumes no transversal when stating "more complex" at the end.

Also, the fact that both of these typically involve the TE norm, but
that "TE complexity" refers to the complexity of an entire temperament
all at once, I think is really confusing. What do I call it if I want
to specifically refer to the use of the Tenney-weighted L2 norm in
measuring just or temperamental complexity? Temperamental-TE
Complexity vs just TE Complexity, with the latter actually being the
one referring to the complexity of temperaments?

-Mike