Weighting schemes

[Charlie Jordan]
>> Accordion to Richard Morse <http://accordion.simplenet.com/wetdry.html>,
the
>> wet or musette tuning favored in France and Italy requires free reeds in
>> unison to be tuned 20 cents apart.

Certainly a slight mistuning (beats of a few hertz) sounds much more
interesting than phase-locked (consider also the 12-string guitar), but 50
cents (as mentioned by Paul Erlich) is pretty extreme. I know nothing about
accordions but I wonder whether the reeds are tuned separately but when
they are sounded together the slight acoustic coupling between them is
enough to pull them closer together in pitch. Perhaps the reeds have a very
low Q.

[Brett Barbaro]
>Thanks! That's interesting and helpful. I think the answer for "tolerance
of mistuning"
>is not some function that increases and then decreases for higher limit
ratios. I think
>the answer is a very sensitive function of musical context. Manuel's
perceptions may be
>right in some contexts, field of attraction may be more relevant in other
contexts, but
>typically different intervals will play different roles and not be
strictly comparable
>as if the only difference between them were a numerical one.
>
>So, I think I can safely say that for two-voice Renaissance and Baroque
music, where
>thirds and sixths greatly outnumber fifths (and fourths are considered
dissonant),
>something close to 2/7-comma meantone would be optimal. In other
situations, moving
>toward Pythagorean would be more appropriate.

Well of course. But I thought you were the one who started talking about
optimal diatonics as if such a thing had meaning outside a specific musical
context. And I think it does. There are things one cares about across many
musical styles. But in fact I assumed you were talking about using it for
new music and understood that your only consideration was that all the
conventionally consonant diatonic intervals *sounded* as close to Just as
possible.

Of course I agree with

[Manuel Op de Coul]
>One can speak about region of tolerance only in a broad manner because
>many factors play a role like the timbre, register, note duration, etc.

[Ara Sarkissian]
>... Correct me if i
>don't understand this right, but your scheme would place more importance in
>tuning the highest "limit" presented in a system, whereas the other
>emphasizes the lowest "limit" represented. That is, if the strict error is
>our measuring stick for the moment.
>
>For me, it makes sense to have the higher numbers presented more
>accurately, since i can either sense a fifth or i can't, the 3:2 being so
>fundamentally engraved/engrained/inbrained in my head. But then again, if
>the most "familiar" intervals (lowest ones) are off, then there's no hope
>in getting any sense out of a piece of music. So there has to be a balance
>somehow. It seems the attainment of this very balance has been the goal of
>theorists for a while now... So each scheme pulls towards the higher/lower
>"limits" to be approximated. To achieve a balance, perhaps you can somehow
>(and here i show my artist's badge, not that of a theorist!) merge the two
>? SOMEHOW ? Otherwise it seems we're just going in circles...
>but it's a fun circle nonetheless.

I'm very pleased to have an artist agree with me.

So Brett (or anyone), what's actually wrong with using "some [weighting]
function that increases and then decreases for higher limit ratios". If we
are to have a rule-of-thumb weighting scheme at all, it seems better than
the two conflicting alternatives.

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>So Brett (or anyone), what's actually wrong with using "some [weighting]
>function that increases and then decreases for higher limit ratios". If we
>are to have a rule-of-thumb weighting scheme at all, it seems better than
>the two conflicting alternatives.

Equal-weighting also seems better than the two conflicting alternatives, and
is simplest of all.

BUT:

I could see justifying what you're saying by considering two components of
dissonance, one which gives more tolerance to simpler ratios, and one which
gives more tolerance to more complex ones. The former would be due to
possible confusion with nearby ratios, while the latter would have to do
with the sensitivity on the intensity of beating or roughness on the amount
of mistuning. Then, taking the more stringent component for each interval
would lead to a weighting function like the one you describe. The first
component has to do with harmonic entropy, while the latter has to do with
the Plomp/Sethares models of dissonance. So can we begin to quantify your
idea?

[Me, Dave Keenan]
>>So Brett (or anyone), what's actually wrong with using "some [weighting]
>>function that increases and then decreases for higher limit ratios". If we
>>are to have a rule-of-thumb weighting scheme at all, it seems better than
>>the two conflicting alternatives.

[Paul Erlich]
>Equal-weighting also seems better than the two conflicting alternatives,
>and is simplest of all.

Yes indeed, and I've been using it for that reason since you first pointed
out these two conflicting tendencies to me many months ago.

>BUT:
>
>I could see justifying what you're saying by considering two components of
>dissonance, one which gives more tolerance to simpler ratios, and one which
>gives more tolerance to more complex ones.

I would have said "one which gives less tolerance to complex ratios, and
one which gives less tolerance to simpler ratios". But it means the same
thing.

>The former would be due to
>possible confusion with nearby ratios, while the latter would have to do
>with the sensitivity on the intensity of beating or roughness on the amount
>of mistuning. Then, taking the more stringent component for each interval
>would lead to a weighting function like the one you describe.

Yes! Absolutely.

>The first
>component has to do with harmonic entropy, while the latter has to do with
>the Plomp/Sethares models of dissonance. So can we begin to quantify your
>idea?

Maybe? Not sure if it's these two things, but it certainly is two different
things. But how? If it's to be a function of complexity we have to decide
what complexity measure to use. Or maybe that will just fall out.

Consider the relative rates of beating of the dyads in a chord. I assume
that, for a given dyad (that closely approximates a medium complexity
ratio), it is dominated by the difference in the frequencies with which
each note approximates the dyad's guide tone (lowest coinciding harmonic,
LCM). For example, for the otonal triad 6:8:11 the same number of cents
mistuning in any dyad (of course these are not possible simultaneously)
will lead to beat rates in the following proportions.
6:8 6:11 8:11
(2:3)
24 66 88 =
12 33 44

For the corresponding utonal triad 1/11:1/8:1/6 (= 24:33:44) they are
6:8 6:11 8:11
(2:3)
132 264 264 =
1 2 2

I suppose we could say that since this is the component that is important
for the more complex ratios, where utonal chords are highly dissonant
anyway, we should use the otonal version. But then which inversion should
be considered. If we go to 8:11:12 we get
8:12 11:12 8:11
(2:3) (6:11)
24 132 88 =
12 66 44

Twice the emphasis on errors in the 6:11. So I guess we only consider the
lowest numbered inversion. But notice that two intervals with the same
odd-limit 6:11 and 8:11 have different beat sensitivities in the 6:8:11
chord. In proportions of 6 to 8.

Incidentally this relates to the problem of optimally tempering the 6:8:11
into lumma's scale.

Regards,
-- Dave Keenan
http://dkeenan.com

> [Me, Dave Keenan]
> >>So Brett (or anyone), what's actually wrong with using "some [weighting]
> >>function that increases and then decreases for higher limit ratios". If we
> >>are to have a rule-of-thumb weighting scheme at all, it seems better than
> >>the two conflicting alternatives.
>
> [Paul Erlich]
> >Equal-weighting also seems better than the two conflicting alternatives,
> >and is simplest of all.
>
> Yes indeed, and I've been using it for that reason since you first pointed
> out these two conflicting tendencies to me many months ago.
>
> >BUT:
> >
> >I could see justifying what you're saying by considering two components of
> >dissonance, one which gives more tolerance to simpler ratios, and one which
> >gives more tolerance to more complex ones.
>
> I would have said "one which gives less tolerance to complex ratios, and
> one which gives less tolerance to simpler ratios". But it means the same
> thing.
>
> >The former would be due to
> >possible confusion with nearby ratios, while the latter would have to do
> >with the sensitivity on the intensity of beating or roughness on the amount
> >of mistuning. Then, taking the more stringent component for each interval
> >would lead to a weighting function like the one you describe.
>
> Yes! Absolutely.
>
> >The first
> >component has to do with harmonic entropy, while the latter has to do with
> >the Plomp/Sethares models of dissonance. So can we begin to quantify your
> >idea?
>
> Maybe? Not sure if it's these two things, but it certainly is two different
> things. But how? If it's to be a function of complexity we have to decide
> what complexity measure to use. Or maybe that will just fall out.
>
> Consider the relative rates of beating of the dyads in a chord. [...] I suppose we could say that since
> this is the component that is important
> for the more complex ratios

No -- I would say the opposite. If it weren't for beating/roughness, the tolerance for the simpler ratios
would be quite wide.

The error weighting function:

Error_weight(a:b) = Max( 6/Max(a,b), (a*b)/30) ), when a:b in lowest terms,

predicts approximately the same level of tolerance for each of the
following errors:

Intvl Error (cents)
1:1 3
1:2 6
2:3 9 (10c considered a wolf)
4:5 15 (14c in 12-tET)
5:6 18 (16c in 12-tET)
4:7 19
5:7 15 (17.5c in 12-tET and 22-tET
6:7 13
4:9 15
5:9 12
7:9 9
4:11 12
5:11 10
6:11 8
7:11 7
9:11 5

Anyone want to object to these or tweak the numbers a bit? Note that we are
only looking for a crude rule of thumb. Anything that we can even roughly
agree on will be an improvement over equal-weighting, or those two opposing
schemes.

Propose your own list of intervals and errors if you wish. e.g. If 10 cents
from a 2:3 is a borderline wolf fifth, how many cents error makes an
equally "wolfy" major third, minor third, etc. or an equally useful/useless
(because of indistinguishability from other nearby ratios) 7:9, 6:11, etc.

How much difference is there (in _relative_ weighting of errors) when we
consider the intervals standing alone as opposed to standing in chords?
I've been assuming that the intervals occur most often in subsets of the
hexad 4:5:6:7:9:11, since it's in the most consonant chords that we are
most concerned about the consonance of the intervals.

Regards,
-- Dave Keenan
http://dkeenan.com