another complexity measure....

...from our discussion on EDO efficiency might be an integer which
expresses in what generation on the Stern-Brocot tree a given EDO
appears, e.g. we assume the necessity of the 3-limit, and look at
approximations to 3/2 starting with i/j of an octave, perhaps using
two well-established limits, like 4/7 oct. and 3/5 oct:

generation
1 [4/7 3/5] "parents"
2 [7/12]
3 [11/19 10/17]
4 [15/26 18/31 17/29 13/22]
5 [19/33 26/45 29/50 25/43 24/41 27/46 23/39 16/27]

To repeat, 'complexity', as Carl calls it, would then be the integer
number of the generation in which that EDO appears.

....but then we'd have to find a way to algorithmically account for
34-ED0, 15-EDO, etc. which don't appear on the tree.

Another way might be an analogue to Tenney's complexity: to add
numerator and denominator? This would tend to make complexity rise in
a non-linear way, which for this purpose, might be a good thing...

-Aaron.

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

> Another way might be an analogue to Tenney's complexity: to add
> numerator and denominator?

The difference would be of little significance, and the significance
would drop with increasing n. In line with what is sometimes done with
wedgies, you could divide by log2(p) for each prime, and then take the
maximum or average. For 12, you would get in the 7-limit
<12 19/log2(3) 28/log2(5) 34/log2(7)| = <12 11.99 12.06 12.11|,
leading to a slightly higher complexity.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "akjmicro" <aaron@> wrote:
>
> > Another way might be an analogue to Tenney's complexity: to add
> > numerator and denominator?
>
> The difference would be of little significance, and the significance
> would drop with increasing n. In line with what is sometimes done with
> wedgies, you could divide by log2(p) for each prime, and then take the
> maximum or average. For 12, you would get in the 7-limit
> <12 19/log2(3) 28/log2(5) 34/log2(7)| = <12 11.99 12.06 12.11|,
> leading to a slightly higher complexity.
>

Interesting....

I'm curious what you thought about the Stern-Brocot generational
complexity proposal?

-Aaron.

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

> I'm curious what you thought about the Stern-Brocot generational
> complexity proposal?

I didn't like it; you are working hard to get a number which isn't a
very good complexity measure, whereas you already have n, the number
of steps in an octave, which *is* a good complexity measure.

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

> Another way might be an analogue to Tenney's complexity: to add
> numerator and denominator?

How is that an analogue of Tenney's complexity?

> This would tend to make complexity rise in
> a non-linear way, which for this purpose, might be a good thing...
>
> -Aaron.

The complexity measure I've suggested for all temperaments, which is
the sum of the weighted absolute wedgie elements, agrees exactly with
the Tenney complexity of the vanishing comma (or "unison vector") in
cases where there's really only one of those (like meantone, where it's
81:80), and it agrees very, very well with the number of notes in the
ET case.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "akjmicro" <aaron@> wrote:
>
> > Another way might be an analogue to Tenney's complexity: to add
> > numerator and denominator?
>
> How is that an analogue of Tenney's complexity?

ok...is Tenney complexity the same as n*d. And forget what I said
about adding. Doesn't make sense.

> > This would tend to make complexity rise in
> > a non-linear way, which for this purpose, might be a good thing...
> >
> > -Aaron.
>
> The complexity measure I've suggested for all temperaments, which is
> the sum of the weighted absolute wedgie elements, agrees exactly with
> the Tenney complexity of the vanishing comma (or "unison vector") in
> cases where there's really only one of those (like meantone, where it's
> 81:80), and it agrees very, very well with the number of notes in the
> ET case.

I have a 145 IQ. But I still would like to hear that in English if you
get the chance. I see "wedgie" and I get turned off *immediatly*. I
think (I know) I understand "weighted" and "unison vector". I should
brush up on Tenney complexity, and I hate the word "wedgie".

BTW, I know what a Monzo is, but I don't think its good manners to
start naming shit after yourself (all due respect, Joe!). Couldn't we
call it something else. Shouldn't someone die before they receive that
kind of honor? ;)

Probably the best way for me to understand this would be to see some
code, preferably not in a LISP-like language. Python would be perfect,
obviously (Graham?).

Parenthesis in a computing language in this modern day and age suck.

-Aaron.

akjmicro wrote:

> ok...is Tenney complexity the same as n*d. And forget what I said
> about adding. Doesn't make sense.

Tenney distance is the log of n*d. If that's complexity, it's of an interval rather than a temperament.

>>The complexity measure I've suggested for all temperaments, which is >>the sum of the weighted absolute wedgie elements, agrees exactly with >>the Tenney complexity of the vanishing comma (or "unison vector") in >>cases where there's really only one of those (like meantone, where it's >>81:80), and it agrees very, very well with the number of notes in the >>ET case.
> > I have a 145 IQ. But I still would like to hear that in English if you
> get the chance. I see "wedgie" and I get turned off *immediatly*. I
> think (I know) I understand "weighted" and "unison vector". I should
> brush up on Tenney complexity, and I hate the word "wedgie". You need wedgies for this.

> Probably the best way for me to understand this would be to see some
> code, preferably not in a LISP-like language. Python would be perfect,
> obviously (Graham?).

Everything's in http://x31eq.com/temper/regular.zip

The file regular_wedgie.py uses the complexity Paul described above, and that file also has my simplest generic code for wedge products. There's a simpler way of doing the R2 case which I could pull from the revision control. Here's Herman's original C:

wedgiesize = 0;
for (i = 0; i < valsize - 1; i++)
{
for (j = i + 1; j < valsize; j++)
{
wedgie[wedgiesize] = val1[i] * val2[j] - val2[i] * val1[j];
wedgiesize++;
}
}

The file regular.py has two different Tenney-weighted prime complexities which don't need wedge products. And temper.py has the usual unweighted, odd-limit complexity.

Graham

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

> BTW, I know what a Monzo is, but I don't think its good manners to
> start naming shit after yourself (all due respect, Joe!). Couldn't we
> call it something else. Shouldn't someone die before they receive that
> kind of honor? ;)

That was my fault. I didn't set about naming it after him, I stumbled
into it. I wanted a word to stick in the comment lines of my Maple
programs, to distinguish an interval written as a ratio from an
interval written as a list of integers, both of which my programs
used. Since Monz was an enthusiast for the latter, I used his name.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "akjmicro" <aaron@> wrote:
>
> > BTW, I know what a Monzo is, but I don't think its good manners to
> > start naming shit after yourself (all due respect, Joe!). Couldn't we
> > call it something else. Shouldn't someone die before they receive that
> > kind of honor? ;)
>
> That was my fault. I didn't set about naming it after him, I stumbled
> into it. I wanted a word to stick in the comment lines of my Maple
> programs, to distinguish an interval written as a ratio from an
> interval written as a list of integers, both of which my programs
> used. Since Monz was an enthusiast for the latter, I used his name.
>

Ah...I see, my apologies, Joe!

-Aaron.

Hi Aaron,

--- In tuning-math@yahoogroups.com, "akjmicro" <aaron@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "akjmicro" <aaron@> wrote:
> >
> > > BTW, I know what a Monzo is, but I don't think its
> > > good manners to start naming shit after yourself
> > > (all due respect, Joe!). Couldn't we call it something
> > > else. Shouldn't someone die before they receive that
> > > kind of honor? ;)
> >
> > That was my fault. I didn't set about naming it after
> > him, I stumbled into it. I wanted a word to stick in
> > the comment lines of my Maple programs, to distinguish
> > an interval written as a ratio from an interval written
> > as a list of integers, both of which my programs used.
> > Since Monz was an enthusiast for the latter, I used his
> > name.
> >
>
> Ah...I see, my apologies, Joe!

Apology accepted. :)

Moral of the story: do more research before jumping to
unfounded assumptions. My Encyclopedia entry for the term
(http://tonalsoft.com/enc/m/monzo.aspx) clearly states that
Gene was its inventor.

The section of the paper i wrote in 1997 describing its use
is here:

http://sonic-arts.org/monzo/article/article.htm#pitchclass

I found out much later (just a couple of years ago) that
in passing Fokker also mentioned this method of describing
ratios, but altho it is implicitly a central part of his
theory, he never placed much emphasis on its use as a
notation (AFAIK).

I advocated the use of the "monzo" (or prime-factor
exponent vector, if you prefer to be long-winded) as
an accidental to accompany notes in the staff-notation
of just-intonation music.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "akjmicro" <aaron@> wrote:
>
> > BTW, I know what a Monzo is, but I don't think its good manners to
> > start naming shit after yourself (all due respect, Joe!). Couldn't
we
> > call it something else.

Yes. You could call it a prime exponent vector, which is what it has
been called for far longer than it has been called a monzo.

-- Dave Keenan