Question about a rank two complexity measure

If <0 a3 a5 ... ap| is the generator mapping of a rank two temperament with P periods to the octave, and <0 b3 b5 ... bp| is the same val in weighted coordinates, then P*(max(0,b3,b5,...bp)-min(0,b3,b5,...bp)) is a complexity measure which is less than or equal to L-infinity wedgie complexity. Can anyone recall if this complexity measure has been considered before? I may want to make use of it.

On Tue, Jan 24, 2012 at 1:50 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> If <0 a3 a5 ... ap| is the generator mapping of a rank two temperament with P periods to the octave, and <0 b3 b5 ... bp| is the same val in weighted coordinates, then P*(max(0,b3,b5,...bp)-min(0,b3,b5,...bp)) is a complexity measure which is less than or equal to L-infinity wedgie complexity. Can anyone recall if this complexity measure has been considered before? I may want to make use of it.

I just searched through my notes on something I've half-developed I
was calling "range complexity" a little while ago, but it doesn't seem
like quite the same thing. I went through a conversation with Graham
where we went over some different wedgie complexity types but I didn't
see it either.

May I ask what properties you like about this complexity measure?

-Mike

On Tue, Jan 24, 2012 at 1:50 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> If <0 a3 a5 ... ap| is the generator mapping of a rank two temperament with P periods to the octave, and <0 b3 b5 ... bp| is the same val in weighted coordinates, then P*(max(0,b3,b5,...bp)-min(0,b3,b5,...bp)) is a complexity measure which is less than or equal to L-infinity wedgie complexity. Can anyone recall if this complexity measure has been considered before? I may want to make use of it.

Wait a sec, isn't this just the Graham complexity?

-Mike

On Tue, Jan 24, 2012 at 2:03 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> On Tue, Jan 24, 2012 at 1:50 AM, genewardsmith
> <genewardsmith@sbcglobal.net> wrote:
>>
>> If <0 a3 a5 ... ap| is the generator mapping of a rank two temperament with P periods to the octave, and <0 b3 b5 ... bp| is the same val in weighted coordinates, then P*(max(0,b3,b5,...bp)-min(0,b3,b5,...bp)) is a complexity measure which is less than or equal to L-infinity wedgie complexity. Can anyone recall if this complexity measure has been considered before? I may want to make use of it.
>
> Wait a sec, isn't this just the Graham complexity?

Ack, sorry, I'm not reading thoroughly at all right now. I guess it's
like "weighted Graham complexity" or something, but the definition on
the wiki defines Graham complexity over a set of arbitrary pitch
classes, not just the primes or intervals within a certain odd-limit.
So it's kind of like weighted Graham complexity or weighted odd-limit
complexity or something.

I'm not sure if it's been used. A search for "weighted odd-limit
complexity" and "weighted Graham complexity" turns up nothing.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> May I ask what properties you like about this complexity measure?

It's a nice measure with which to do a search on wedgies below a certain complexity cutoff; TE is something of a mess in comparison. You can also bound generator steps to get to an interval in terms of Tenney height pretty easily. It's sort of like Graham complexity, but easier to compute and to work with on a search.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Ack, sorry, I'm not reading thoroughly at all right now. I guess it's
> like "weighted Graham complexity" or something, but the definition on
> the wiki defines Graham complexity over a set of arbitrary pitch
> classes, not just the primes or intervals within a certain odd-limit.
> So it's kind of like weighted Graham complexity or weighted odd-limit
> complexity or something.

You might think of it as *unweighted* Graham complexity in the limit as more and more intervals are included in your set of pitch classes. You remove all issues of odd limits. For instance, 7-limit Graham complexity weights 3, 5 and 7 the same, whereas 9-limit says that 3 is weighted at twice of what 5 and 7 get. This complexity, if you assign 3 to a weight of 100, gives 5 a weight of 68 and 7 a weight of 56, and 9 of course a weight of 50.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

I just wrote this:
http://xenharmonic.wikispaces.com/Tenney+complexity

I'm not that happy with the name; now would be the time to suggest a change.

On Tue, Jan 24, 2012 at 8:40 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Ack, sorry, I'm not reading thoroughly at all right now. I guess it's
> > like "weighted Graham complexity" or something, but the definition on
> > the wiki defines Graham complexity over a set of arbitrary pitch
> > classes, not just the primes or intervals within a certain odd-limit.
> > So it's kind of like weighted Graham complexity or weighted odd-limit
> > complexity or something.
>
> You might think of it as *unweighted* Graham complexity in the limit as more and more intervals are included in your set of pitch classes.

1) I thought Graham complexity was already unweighted
2) I thought this thing was weighted? You said to calculate it
relative to b1, b2, b3, .. bp in your post, which you said was
weighted coordinates.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> 1) I thought Graham complexity was already unweighted

It is.

> 2) I thought this thing was weighted? You said to calculate it
> relative to b1, b2, b3, .. bp in your post, which you said was
> weighted coordinates.

I did. But it all works out. Note that in the article, it's odd height which appears in the inequality at the end.