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.