Re:Perception of n-limit intervals

Since there've been a number of postings on this recently, thought some
maths I've proved relevant to this could be of interest.

Idea is to show that you can approximate any n-limit ratio to any desired
accuracy using n-1 limit ratios.

Example, try a 5-limit approx to 7/6.

Want

(1+e)7/6 = 5^m/3^n with abs(e) as small as you like.

So taking logs
log(1+e) + log(7/6) = m log 5 - n log 3. with abs(log(1+e)) as small as you
like.

One can apply Kronecker's theorem in two dimensions: states that
m*a + n*b
is dense in the reals for m, n integer provided a/b is irrational.

In this case, need to show that log 5 / log 3 is irrational.

Prove this by usual method of supposing it is rational, and derive a
contradiction.

Suppose log 5 / log 3 = r/s. for some r,s integer.

Then s log 5 = r log 3.
So 5^s = 3^r
Contradiction (by unique factorisation theorem for numbers).

That was an example, but clearly method is completely general.

To state result as a theorem:

Theorem
Let a, b be such that a has a prime factor that doesn't divide into b, and
vice versa.
Then the set (a^m/b^n: m, n integer} is dense in the positive reals.
Proof
By assumptions, log(a)/log(b) is irrational.
Result follows by Kronecker's theorem in two dimensions.
http://mathworld.wolfram.com/KroneckersApproximationTheorem.html
(see the explanation of how restriction on alpha to [0, 1] can be removed)
See also Hardy and Wright, chapter on Kronecker's theorem.
Q.E.D.

So 5-ness merges into 7-ness at some point.

However this leaves it open whether there is a quality of 5-ness or 7-ness
etc of ratios.

Some words such as colour words are what philosphers refer to as vague
predicates. For instance, blue can shade to white (as the sky does on some
days, blue at zenith, shading to white at horizon). You can make
subdivisions, e.g. pale blue, but question still arises, where does pale
blue end. You can define a cut off point at which blue ends and white
begins. However what interests philosophers about this is that if the colour
shades sufficiently gradually, it will be impossible to distinguish by eye
between a shade, and another one that is minutely paler in shade.

The eye will see shades to either side of the cut off point as identical.
Then you can use an argument called the sorites, or heap, and show that if a
shade is blue, the next one is as well, because it looks identical to the
eye, then the next, until eventually you prove to yourself that white is
blue, and find yourself in a quandry.

(The actual example most often used in papers is red shading to orange, but
I rather prefer to use blue shading to white because one can actually see
this phenomenon when the weather is right, and have a go at it
experimentally, seeing if one can decide on a definite cut of point for the
colour blue, or not. One can use, say, the twigs of a tree as a marker, and
ask which is the last one to be in front of a blue part of the sky, and
which the first before a white part, and actually carry out the argument,
and baffle oneself a little by it).

This raises interesting questions about how words have meanings, and vague
predicates have been the subject of numerous philosophical papers.

However, no-one denies that colour words are meaningful and useful, the
debate is only about why they are, and how it works.

In the same way, 5-ness could shade gradually into 7-ness, and both words
could have useful meanings.

Interesting to see what those who feel they can hear a quality of 5-ness or
7-ness in ratios think.

I've done a program that generates successive n-limit (and non n-1 limit)
approximations to any interval.

Compiled as Win 95/98 32 bit console app.
http://www.robertwalker.f9.co.uk/nlimit.exe [46 K]
c source code:
http://www.robertwalker.f9.co.uk/nlimit.c

Examples of use:

>nlimits 5 7 6
Successive 5-limit and non 3-limit approx. to 7/6
1/1 5/4 6/5 75/64 729/625

>nlimits 7 11 10
Successive 7-limit and non 5-limit approx. to 11/10
1/1 7/6 8/7 15/14 28/25 35/32 54/49 19683/17920 36015/32768 65536/59535

Robert Walker

http://www.robertwalker.f9.co.uk/fts_beta/fts_beta_download.htm

Hi Robert -- you've touched upon one of the reasons I don't really believe
in prime-affect, or prime-limit as an indication of anything acoustical.
Also, I use harmonic entropy to determine which intervals can really said to
be perceived as ratios (when presented as dyads), and which dissolve into a
confused set of approximations to simpler ratios.

Hi Paul,

I've had a look at your Harmonic entropy ideas, and they look interesting.

Posting was meant to leave open whether there is a quality of 5-ness or
7-ness etc, and just present the theorem as a result. Bearing in mind the
posting on Indian srutis (message 13329), it seems to me there could well be
something in it, but prob. limited to small-ish ratios.

Update on nlimit prog:

Prog nlimit found successive approx by looking at n-limit ratios with
denominator and denumerator less than 100,000. It could find any n-limit
ratios, not just the ones of the type described in the theorem.

It did it by testing all possible denumerators up to the max value of
100,000. This only got to within one or two cents most of the time, because
of the limit on the denumerator. Code could be re-compiled to extend the
limit somewhat, but not much, because of practical time and memory limits.

I've done another one that finds ratios only of the form described by the
theorem, but can do up to any size of denominator one wants (within reason).

http://www.robertwalker.f9.co.uk/nlimits.exe [44 Kb]

Source code:
http://www.robertwalker.f9.co.uk/nlimits.c

Ex. of use
>nlimits 5 7 11 9
Successive approx. to 11/9 in form 5^m/7^n
1/1 5^5/7^4 5^11/7^9 5^63/7^52 5^144/7^119 5^277/7^229

Values in cents
0 456.26 330.02 358.82 338.89 347.77

Ditto with word cents after each entry (can use to paste into FTS scale
window)
0 cents 456.26 cents 330.02 cents 358.82 cents 338.89 cents 347.77 cents

11/9 = 347.41 cents

Robert

http://www.robertwalker.f9.co.uk/fts_beta/fts_beta_download.htm

(FTS min req. Win 95/98 + soundcard)

Robert Walker wrote,

>Posting was meant to leave open whether there is a quality of 5-ness or
>7-ness etc, and just present the theorem as a result. Bearing in mind the
>posting on Indian srutis (message 13329), it seems to me there could well
be
>something in it, but prob. limited to small-ish ratios.

For tuning _systems_ such as the sruti system, prime limit is definitely
relevant, as the srutis in a 5-prime-limit systems can all be tuned from one
another using 5-odd-limit ratios. However, for individual dyads out of
context, it is my contention that prime limit says nothing.