this T[n] business

1. Paul, could you brief us on how the T[n] method compares
to what you and Kalle were using before?

2. The last version of this thread thread (see msg. 6017)
left off with how to identify important commas. Gene's
mentioned square and triangular numbers as being better,
though I'm not sure why... though I imagine a high level of
non-primeness in general would be good, since it increases
chances of turning a simple interval into another simple
interval. For the same reason, complexity (Tenney or
heuristic) might be good.

I suggested some measures which included size, but Paul's
probably right that size doesn't have anything to do with it.
Tangentially though, I asserted that Tenney complexity is
tainted wrt size, since smaller ratios tend to have bigger
numbers.

I got fed up with Excel and tested this assertion in Scheme.
Using all ratios with Tenney height less than 3000, I counted
how many approximated to each degree of 23-, 50-, and 100-et.
I didn't bother to plot the results, but a roughly equal
number of ratios seem to fall in all bins in each case,
whether I enforced octave equivalence or not. So it seems my
assertion is wrong; simple ratios don't tend to be bigger.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> 1. Paul, could you brief us on how the T[n] method compares
> to what you and Kalle were using before?

eh . . . i think it's all the same, except that we identified
three particular chromatic unison vectors that were likely to produce
the major tetrads and minor tetrads using the same pattern of scale
steps -- gene is not focused on that property, but rather
investigating one chromatic unison vector at a time amongst quite a
few possibilities.

> 2. The last version of this thread thread (see msg. 6017)
> left off with how to identify important commas. Gene's
> mentioned square and triangular numbers as being better,
> though I'm not sure why...

don't know

> though I imagine a high level of
> non-primeness in general would be good, since it increases
> chances of turning a simple interval into another simple
> interval. For the same reason, complexity (Tenney or
> heuristic) might be good.

complexity would be bad (inverse complexity would be good), since
simple intervals would tend to turn to complex intervals, and since
you'd tend to get more notes in the scale.

> I suggested some measures which included size, but Paul's
> probably right that size doesn't have anything to do with it.
> Tangentially though, I asserted that Tenney complexity is
> tainted wrt size, since smaller ratios tend to have bigger
> numbers.

bigger numbers than what? than those in the ratios for larger
intervals? nope, not unless you're holding something constant such as
insisting on superparticularity.

> I got fed up with Excel and tested this assertion in Scheme.
> Using all ratios with Tenney height less than 3000, I counted
> how many approximated to each degree of 23-, 50-, and 100-et.
> I didn't bother to plot the results, but a roughly equal
> number of ratios seem to fall in all bins in each case,
> whether I enforced octave equivalence or not. So it seems my
> assertion is wrong; simple ratios don't tend to be bigger.

that's the great thing about tenney complexity (as opposed to farey,
mann, etc.)!

>> 2. The last version of this thread thread (see msg. 6017)
>> left off with how to identify important commas. Gene's
>> mentioned square and triangular numbers as being better,
>> though I'm not sure why...
>
>don't know

I suppose squareness and triangularity are types of
compositeness.

>> though I imagine a high level of
>> non-primeness in general would be good, since it increases
>> chances of turning a simple interval into another simple
>> interval. For the same reason, complexity (Tenney or
>> heuristic) might be good.
>
>complexity would be bad (inverse complexity would be good),

Yep, that's what I meant. Which is why n*d was in the den.
of my suggested measures.

>> I got fed up with Excel and tested this assertion in Scheme.
>> Using all ratios with Tenney height less than 3000, I counted
>> how many approximated to each degree of 23-, 50-, and 100-et.
>> I didn't bother to plot the results, but a roughly equal
>> number of ratios seem to fall in all bins in each case,
>> whether I enforced octave equivalence or not. So it seems my
>> assertion is wrong; simple ratios don't tend to be bigger.
>
>that's the great thing about tenney complexity (as opposed to
>farey, mann, etc.)!

Ah, you've said that before, I think!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> 2. The last version of this thread thread (see msg. 6017)
> >> left off with how to identify important commas. Gene's
> >> mentioned square and triangular numbers as being better,
> >> though I'm not sure why...
> >
> >don't know
>
> I suppose squareness and triangularity are types of
> compositeness.

It's nothing to worry about; n^2/(n^2-1) are commas between

9/8,8/7,7/6,6/5,5/4 etc.

whereas Triangle(n)/(Triangle(n-1)) are commas between

3/2,5/3,7/4,9/5 etc. These are the ones which are going to turn up in
this business, but it doesn't particularly help to know that. If we
know that 15/14, 16/15, 21/20, 25/24, 28/27, 36/35, 45/44, 49/48 are
worth checking, it doesn't add much to say that all of the numerators
are eithers squares or triangles.

>>>So it seems my assertion is wrong; simple ratios don't tend
>>>to be bigger.
>>
>>that's the great thing about tenney complexity (as opposed to
>>farey, mann, etc.)!
>
>Ah, you've said that before, I think!

I've just verified this for the Farey series.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>>So it seems my assertion is wrong; simple ratios don't tend
> >>>to be bigger.
> >>
> >>that's the great thing about tenney complexity (as opposed to
> >>farey, mann, etc.)!
> >
> >Ah, you've said that before, I think!
>
> I've just verified this for the Farey series.
>
> -Carl

something that *does* work like tenney in this way is the odd limit.

>that's the great thing about tenney complexity (as opposed to
>farey, mann, etc.)!
//
>
>something that *does* work like tenney in this way is the odd limit.

Hmm, this doesn't seem right; the diamond pitches aren't very
regular at all. IIRC they get thicker and thicker as they approach
the identity, but then there's a gap before hitting it.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >that's the great thing about tenney complexity (as opposed to
> >farey, mann, etc.)!
> //
> >
> >something that *does* work like tenney in this way is the odd
limit.
>
> Hmm, this doesn't seem right; the diamond pitches aren't very
> regular at all. IIRC they get thicker and thicker as they approach
> the identity, but then there's a gap before hitting it.
>
> -Carl

this is no more true than it is for tenney.