Talk:ABC, High Quality Commas, and Epimericity/WikispacesArchive

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All discussion below is archived from the Wikispaces export in its original unaltered form.
Please do not add any new discussion to this archive page.
All new discussion should go on Talk:ABC, High Quality Commas, and Epimericity.

Rocky the flying squirrel?

Is Noam Elkies "Rocky the flying squirrel"? Doesn't look like it, but I can't find any post by Elkies on this subject.

- clumma November 20, 2012, 01:00:16 PM UTC-0800

I've never seen or heard Noam Elkies call himself anything but "Noam Elkies".

- genewardsmith November 20, 2012, 03:39:13 PM UTC-0800

I guess, that the link automatically leads to the statement with the highest rating.

So, if you link to a new topic, you get the original statement.

If that becomes questionable, the links will immediately lead to the reason for this.

A good idea, I think.

- xenwolf November 21, 2012, 12:45:47 AM UTC-0800

I didn't get Elkies opinion in a post, by the way, but in email.

- genewardsmith November 21, 2012, 05:15:55 AM UTC-0800

What musical interpretation does the radical have?

It's not clear to me what the musical importance of log(n)/rad(n) is. The radical is the product of primes dividing n, d, and n-d; how did n-d get in there and why?

- mbattaglia1 September 07, 2012, 11:57:18 AM UTC-0700

Its not too clear to me either. It seems like the (n-d) term is just an arbitrary term thrown in there in order to ensure that there is a finite number of bounded "high-quality" commas.

I'm also still curious why it is (n-d) and not (n+d), which is what wikipedia is suggesting we should use. Does it have to do with d being in the denominator?

- Sarzadoce September 07, 2012, 12:30:24 PM UTC-0700

I think the radical business does tend to screw things up, and I also don't think that allowing n/d to be a power is a good idea from a music point of view. But the relationship to a ton of deep mathematics and the way "quality" gets around the necessity of introducing p-limits makes it very much worth mentioning, I think. Can we find a better conjecture for musical purposes? My epimericity conjecture has been around a while, but how about tweaking ABC?

- genewardsmith September 07, 2012, 01:00:18 PM UTC-0700

What else does ABC imply? Does it say anything about ratios with quality < 1? Does it establish any upper bound on quality?

- Sarzadoce September 07, 2012, 01:25:11 PM UTC-0700

It doesn't establish an upper bound, but it would be reasonable to conjecture there is one. The highest known quality for a comma is 1.63, for 6436343/6436341.

- genewardsmith September 07, 2012, 01:43:23 PM UTC-0700

That comma tells us (23/9)^5 = 109.0000339 is almost exactly 109. Don't know if there is much hope of a decent musical use for that.

- genewardsmith September 07, 2012, 01:54:10 PM UTC-0700

I have a suspicion that Stormer's theorem could help provide an upper bound.

- Sarzadoce September 07, 2012, 02:20:24 PM UTC-0700

In what capacity does quality remove the need for limits?

The ABC conjecture implies the Thue-Siegel-Roth theorem, which looks like it might have a few interesting consequences. So for any algebraic irrational number a and rational p/q, TSR proves that for any ɛ > 0, |a - p/q| < 1/q^(2+ɛ) has only finitely many solutions in coprime integers p and q.

If you want to interpret p/q as an interval and a as an irrational approximating it, then note that |a - p/q| is the -linear- rather than logarithmic error of p/q as approximated by a, so I'm not quite sure how that's going to turn out.

If you want to interpret p/q as some logarithmic division of the octave, e.g. what we usually write as p\q, and a as some irrational (perhaps the log of an interval), then this tells you that there are finitely many EDOs that approximate that interval better than a certain error bound which is dependent on q (the size of the EDO) and ɛ, which is one free parameter.

- mbattaglia1 September 07, 2012, 03:00:39 PM UTC-0700

I guess another thing is that Gene's formulation of it, where the triple (a, b, c) corresponds to d+(n-d)=n, has the quality as log(n)/log(rad(n*d*(n-d))). So this looks for commas where the difference tone between the numerator and denominator tends to be in the same subgroup of primes as the comma itself. I guess that's maybe good if you believe in recurrent sequence chords and stuff.

It so far to me looks like the rad(n*d*(n-d)) thing works because rad(n*d*(n-d)) correlates well with (n-d) when (n-d) is low. So for epimoric or low-degree epimeric ratios, you end up with a function that generally goes up as n increases and goes down when (n-d) increases, and since low-degree epimeric ratios are things we like for other reasons, it looks like the sequence of high-quality intervals magically spits out a sequence of musically useful ratios.

Where the quality measure diverges from this is that rad(n*d*(n-d)) will be low if (n-d) is really high, but (n-d) is in the same limit or subgroup as the ratio n/d itself. So for instance, a 5-limit ratio which is extremely far from epimoric, but which so happens to have the property that its difference tone is also 5-limit, will be really high-quality.

Is this musically correct?

- mbattaglia1 September 07, 2012, 03:55:57 PM UTC-0700