RE: [tuning] Re: OK, Pierre, now we're talking!

I wrote,

<< The problem is that it breaks down for high enough a and b (let alone
what happens for irrational ratios). For example, 30001:20000 is
clearly more consonant than 17:13, though S is much larger for the
first than for the second. >>

Pierre wrote,

>It will be hard to find better example to illustrate that we have to pay
>serious attention at theoric context. I don't propose sonance definition as
>a measure in practical context. If 30001:20000 "is" very consonant, it's
>clearly because 30001:20000 "is perceived" as 3:2 ratio. Rationality of
>sonance, or rationality of width, are a matter of brain. 30001:20000 exists
>in (abstract) space of acoustical measured values, but don't exist in brain
>as category for sonance apprehension.

The problem with this model is that it would require an arbitrary range
around every simple ratio, as well as a definition of which ratios are
simple and which are not, and would allow the brain to hear any interval
within the range as the central ratio, but those outside the range would be
more dissonant, unless we crossed over into the range of a simpler ratio.
Too many arbitrary assumptions for me, and more importantly, it disagrees
with experience, which suggests a smooth sonance curve (such as the ones
harmonic entropy or Helmholtz/Plomp/Sethares/Kameoka&Kuriagawa methods
produce).

And I still have no idea how temperament figures into your ratio-diagrams.

>For example, I can translate and develop what I have soon written privately
>in french. We can construct three total distinct orders on all pairs of
>rationals in using criteria (ab) or (a+b) or max(a,b) for reduced ratio a/b
>in comparaison relation. With criteria

> (ab) 7/2 < 3/5 for (7x2) < (3X5)
> (a+b) 7/2 > 3/5 for (7+2) > (3+5)
> max(a,b) 7/2 > 3/5 for (7) > (5)

>All rationals are completly ordered by one or other of these criteria. Two
>ratios can be equal but all ratios are comparable. These total orders are
>very different. Comparing 7/2, 3/5 et 7/1, we have (14)(15)(7) for (ab),
>(9)(8)(8) for (a+b) and (7)(5)(7) for max(a,b).

I have tried these very three criteria within the harmonic entropy
calculation, and the resulting curves are quite similar locally, differing
mainly in global features like overall slope (i.e., they agree when
comparing intervals of similar sizes, but not intervals of very different
sizes.

>But partial order of adjacent ratios in Stern-Brocot tree is compatible
>with the three total orders defined.

OK, interesting!

>(As you know, adjacent ratios a/b and
>c/d, on Stern-Brocot tree, are binded by relation ad - bc = 1).

As is true for the adjacent ratios in the set defined by an upper bound for
any of the three criteria above.

Now, what does the partial ordering of the Stern-Brocot tree allow you to
do?

>What is important for music is what is perceived. But for mathematical
>standpoint, congruence is important. The three criteria have interesting
>properties. But the following examples showing octave translation pointed
>to (ab) as best candidate for logarithmic transformation.

> 7/8 7/4 7/2 7/1 14/1 1/4 1/2 1 2/1 4/1

>(ab) 56 28 14 7 14 4 2 1 2 4

>(a+b) 15 11 9 8 15 5 3 1 3 5

>max(a,b) 8 7 7 7 14 4 2 1 2 4

Please explain why (ab) is the best candidate. It's certainly a very
attractive one, in that it allows Tenney's geometrical interpretation of
harmonic distance using a rectangular lattice with prime axes (including 2)
and units along each axis proportional to the log of the corresponding
prime. But on the basis of perception, I find it hard to verify that, say,
35:1 and 7:5 are identically sonant.

What do you mean by "congruence"?

I wrote,
>
>> Please explain why (ab) is the best candidate. It's certainly a
>> very attractive one, in that it allows Tenney's geometrical
>> interpretation of harmonic distance using a rectangular lattice
>> with prime axes (including 2) and units along each axis
>> proportional to the log of the corresponding prime.

Joe Monzo wrote,

>Paul, isn't that explanatory last clause in your statement
>precisely a description of my lattice formula? If it is (and
>it seems to be to me), then doesn't my formula also 'allow
>Tenney's geometrical interpretation of harmonic distance'?

There are three important differences between Tenney's lattice distance and
yours.

1. (most important) The version of Tenney's approach I'm referring to here
includes an axis for the prime 2. The correspondence between sonance and
distance wouldn't work right for octave-invariant pitch classes mapped
without a "2" axis -- the pitches have to be octave-specific.

2. Units along Tenney's axes are proportional to the log of the
corresponding prime, while units along your axes are proportional to the
prime itself.

3. Tenney's distance is a city-block distance -- one travels only parallel
the axes, making sharp turns where necessary, never taking diagonal
shortcuts across them. Though Tenney's lattice is usually considered to be
rectangular, the angles between the axes are irrelevant, since all distances
are measured along the axes in discrete steps. Meanwhile, I've seen you
measure distance in your lattice using Euclidean distance -- the length of a
straight line connecting the points -- and hence your distance depends very
strongly on the idiosyncratic angles that you use between your axes.
Actually, in the two-dimensional representations you use, it is quite
possible for two points very distant from one another in Tenney's lattice
(even ignoring the "2" axis) to be virtually on top of one another in your
lattice.