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