dinner is served . . .

hey george, take a look at this curve:

/harmonic_entropy/files/dyadic/secortst.g
if

i left it unlabeled for your fun and amusement . . .

(if you don't like seeing all those tiny local minima, not taking the
exponential should help . . . let me know).

anyhow, the global maximum here is at 67 cents.

the next most discordant intervals are kind of in that vicinity, but
don't form a contiguous region the way they did with the original,
smooth harmonic entropy formulation.

then, the next most discordant interval (within the octave) is at
1139 cents.

then more in the vicinities of 67 and 1139, with the caveat above.

then, 758, followed by 757, followed by *750* . . .

let me know if these values are ok for you . . . if not, there's
plenty more that can be tried on tomorrow's menu . . .

--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote
[#603]:
> --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > [pe:]
> > > there's also the s (resolution)
> > > parameter which can always be tweaked to give more or less
importance
> > > to more complex ratios (the better your hearing resolution, the
more
> > > easily you can identify the complex ratios "as such", because
the
> > > complex ratios are in more "crowded" areas amongst all the
ratios).
> >
> > For that we can refer to my graph:
> >
> > /tuning-math/files/secor/consonce.gif
> >
> > It appears that the point at which I no longer heard n:d as a
local
> > consonance is when n*d reached a value around 150 (with 16:9 and
> > 17:10). So margo2.gif would have a little more sensitivity than
what
> > I observed.
>
> i think you're misunderstanding how s works here in the vos-based
> curve. also, my question above would seem to be relevant again --
> could it be that you're simply not noticing tinier and tinier bumps
> in the road, corresponding to more and more complex ratios?

I think we're talking about the same thing, but I just didn't say it
the right way. By sensitivity, I didn't mean y-axis resolution, but
rather sensitivity to hearing higher harmonics, which would translate
to seeing bumps for more complex ratios. I think that the bumps
should disappear when n*d gets up to around 150, which would not make
16:9 and 17:10 appear as local maximum points of consonance. So I
didn't want to see as many bumps as in margo2.gif. Is that making
any sense?

> now, notice that in margo2.gif, the most discordant interval not
near
> 80 cents is between 1100 and 1200 cents, and the most discordant
> interval not near either of these is between 700 and 800 cents.
this
> seems very promising for being able to attain the specifications
you
> requested. however, please note that "near" in this context doesn't
> mean quite what it meant in the original, "rounder" harmonic
entropy
> formulation -- even very near to the extreme peaks of discordance,
> there can be significant interruptions in the "plateau", 14:9 being
a
> perfect example for margo2.gif.

I have no problem with the larger one at 14:9. The dips at 17:11 and
19:12 are the ones that I would want to minimize.

> anyway, i'm now computing a version of margo2.gif where an s of 1%
> will be assumed -- should be hot off the grill within the hour . . .

And here it is:

--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote
[#605]:
> hey george, take a look at this curve:
>
>
/harmonic_entropy/files/dyadic/secortst.g
if
>
> i left it unlabeled for your fun and amusement . . .
>
> (if you don't like seeing all those tiny local minima, not taking
the
> exponential should help . . . let me know).
>
> anyhow, the global maximum here is at 67 cents.

I imported the file into Paint and saved it as a bitmap so I could
read the x-coordinate for each point in the graph, which I can
convert to and from cents using a spreadsheet. The maximum in the
graph looks like a flat line ranging from 61 to 75 cents, so this
looks pretty good.

> the next most discordant intervals are kind of in that vicinity,
but
> don't form a contiguous region the way they did with the original,
> smooth harmonic entropy formulation.
>
> then, the next most discordant interval (within the octave) is at
> 1139 cents.

This is another flat line from 1133 to 1142 cents. This is also
pretty close to what I had, which was 1145.

> then more in the vicinities of 67 and 1139, with the caveat above.
>
> then, 758, followed by 757, followed by *750* . . .

I had 740 as a local maximum dissonance in this region, which was
somewhat different. That was the result of only a single experiment
in a single register, however. I need to do more testing on that, as
well as with a flattened fifth.

> let me know if these values are ok for you . . . if not, there's
> plenty more that can be tried on tomorrow's menu . . .

So smoothing out the curve a little (via the s resolution) so that
16:9, 17:10, and more complex ratios no longer appear as local
maximum points of consonance (or only as tiny dips) would be the only
change I would recommend at this point.

--George

--- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:

> > > It appears that the point at which I no longer heard n:d as a
> local
> > > consonance is when n*d reached a value around 150 (with 16:9
and
> > > 17:10). So margo2.gif would have a little more sensitivity
than
> what
> > > I observed.
> >
> > i think you're misunderstanding how s works here in the vos-based
> > curve. also, my question above would seem to be relevant again --
> > could it be that you're simply not noticing tinier and tinier
bumps
> > in the road, corresponding to more and more complex ratios?
>
> I think we're talking about the same thing, but I just didn't say
it
> the right way. By sensitivity, I didn't mean y-axis resolution,
but
> rather sensitivity to hearing higher harmonics, which would
translate
> to seeing bumps for more complex ratios.

actually, it doesn't work that way in the harmonic entropy model.
firstly, hearing harmonics is not even an issue, rather it's
*recognizing* notes *as* harmonics of an implied fundamental.
secondly, once you've reached a certain, fairly low point in the
harmonic series, ascribing equal importance to higher and higher
harmonics does *not* translate to seeing more bumps for more complex
ratios. rather, it remains the (relatively) simple ratios which
correspond to the visible bumps, because (and only because) these are
the regions in which the concentration of ratios (in whatever,
usually ridiculously high, limit is assumed) is lower; near the
simplest ratios one finds the *lowest* concentration of ratios (in
whatever limit).

harmonic entropy measures how crowded the ratios are.

> I think that the bumps
> should disappear when n*d gets up to around 150, which would not
make
> 16:9 and 17:10 appear as local maximum points of consonance. So I
> didn't want to see as many bumps as in margo2.gif. Is that making
> any sense?

sure, but could it be that these are merely such tiny bumps in the
road that your tires just plow over them as if they weren't even
there?

>
> > anyway, i'm now computing a version of margo2.gif where an s of
1%
> > will be assumed -- should be hot off the grill within the
hour . . .
>
> And here it is:
>
> --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
wrote
> [#605]:
> > hey george, take a look at this curve:
> >
> >
>
/harmonic_entropy/files/dyadic/secortst.g
> if
> >
> > i left it unlabeled for your fun and amusement . . .
> >
> > (if you don't like seeing all those tiny local minima, not taking
> the
> > exponential should help . . . let me know).
> >
> > anyhow, the global maximum here is at 67 cents.
>
> I imported the file into Paint and saved it as a bitmap so I could
> read the x-coordinate for each point in the graph, which I can
> convert to and from cents using a spreadsheet. The maximum in the
> graph looks like a flat line ranging from 61 to 75 cents, so this
> looks pretty good.
>
> > the next most discordant intervals are kind of in that vicinity,
> but
> > don't form a contiguous region the way they did with the
original,
> > smooth harmonic entropy formulation.
> >
> > then, the next most discordant interval (within the octave) is at
> > 1139 cents.
>
> This is another flat line from 1133 to 1142 cents. This is also
> pretty close to what I had, which was 1145.
>
> > then more in the vicinities of 67 and 1139, with the caveat above.
> >
> > then, 758, followed by 757, followed by *750* . . .
>
> I had 740 as a local maximum dissonance in this region, which was
> somewhat different. That was the result of only a single
experiment
> in a single register, however. I need to do more testing on that,
as
> well as with a flattened fifth.
>
> > let me know if these values are ok for you . . . if not, there's
> > plenty more that can be tried on tomorrow's menu . . .
>
> So smoothing out the curve a little (via the s resolution) so that
> 16:9, 17:10, and more complex ratios no longer appear as local
> maximum points of consonance (or only as tiny dips) would be the
only
> change I would recommend at this point.

as i mentioned, not taking the exponential, but looking at the actual
entropy itself, remedies this to some extent. here is the resulting
graph:

/harmonic_entropy/files/dyadic/secorts2.g
if

please be as picky as you like -- there are still a lot of things one
can tweak.

--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > I think that the bumps
> > should disappear when n*d gets up to around 150, which would not
make
> > 16:9 and 17:10 appear as local maximum points of consonance. So
I
> > didn't want to see as many bumps as in margo2.gif. Is that
making
> > any sense?
>
> sure, but could it be that these are merely such tiny bumps in the
> road that your tires just plow over them as if they weren't even
> there?

I guess you could put it that way. But I would like the curve to
correspond to what is actually heard with musical tones (i.e., tones
in which the harmonics are not as rich as in a sawtooth waveform).

> > ... So smoothing out the curve a little (via the s resolution) so
that
> > 16:9, 17:10, and more complex ratios no longer appear as local
> > maximum points of consonance (or only as tiny dips) would be the
only
> > change I would recommend at this point.
>
> as i mentioned, not taking the exponential, but looking at the
actual
> entropy itself, remedies this to some extent. here is the resulting
graph:
>
>
/harmonic_entropy/files/dyadic/secorts2.g
if

That's a bit closer to what I had in mind.

However, one unexpected difference occurs at the local maximum of
dissonance at ~750 cents. In this latest curve it's a little bit
less than the local maximum at ~652 cents, whereas in secortst.gif it
was somewhat greater. The earlier curve agreed with my observation
(I think Helmholtz made it too) that the worst (badly out-of-tune)
wide fifths are more dissonant than the worst narrow fifths.

However, I think that the locations of those maximums (above and
below both the fifth and fourth) may be a more difficult thing to
resolve. I expect that a local maximum dissonance closer to 740
cents would also result in a global maximum dissonance at an location
somewhat different from 67 cents (but smaller or larger?).

> please be as picky as you like -- there are still a lot of things
one
> can tweak.

It's nice to hear that. What are some of those things, and what
would the anticipated effect(s) be?

--George

--- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
wrote:
> > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > > I think that the bumps
> > > should disappear when n*d gets up to around 150, which would
not
> make
> > > 16:9 and 17:10 appear as local maximum points of consonance.
So
> I
> > > didn't want to see as many bumps as in margo2.gif. Is that
> making
> > > any sense?
> >
> > sure, but could it be that these are merely such tiny bumps in
the
> > road that your tires just plow over them as if they weren't even
> > there?
>
> I guess you could put it that way. But I would like the curve to
> correspond to what is actually heard with musical tones (i.e.,
tones
> in which the harmonics are not as rich as in a sawtooth waveform).

human voices, bowed strings, brasses, all of these can have audible
harmonics into the 40s. but again, this doesn't concern harmonics
themselves; it concerns the capacity for the interval to be *heard
as* a pair of harmonics over a missing fundamental. i might infer
from your own remarks that even sine waves possess this capacity,
yes? so isn't it possible that for a "generic" timbre, there are
these tiny features in the discordance curve, so tiny that even
incrementing by 2 cents at a time, and listening with a relatively
good sensitivity to changes in discordance, is going to be unlikely
to notice?

> > > ... So smoothing out the curve a little (via the s resolution)
so
> that
> > > 16:9, 17:10, and more complex ratios no longer appear as local
> > > maximum points of consonance (or only as tiny dips) would be
the
> only
> > > change I would recommend at this point.
> >
> > as i mentioned, not taking the exponential, but looking at the
> actual
> > entropy itself, remedies this to some extent. here is the
resulting
> graph:
> >
> >
>
/harmonic_entropy/files/dyadic/secorts2.g
> if
>
> That's a bit closer to what I had in mind.
>
> However, one unexpected difference occurs at the local maximum of
> dissonance at ~750 cents. In this latest curve it's a little bit
> less than the local maximum at ~652 cents, whereas in secortst.gif
it
> was somewhat greater.

whoops! this shouldn't have been possible -- as the first graph was
merely the exponential of the second, the two should have shared the
same rank-order. but i inadvertantly changed the tenney seed limit
from 65536 to 10000. sorry! here's the (unexponetiated) harmonic
entropy curve with tenney seed limit 65536:

/harmonic_entropy/files/dyadic/secorts3.g
if

should agree perfectly in rank order with secortst.gif, which is the
exponential of this exact harmonic entropy function.

> However, I think that the locations of those maximums (above and
> below both the fifth and fourth) may be a more difficult thing to
> resolve. I expect that a local maximum dissonance closer to 740
> cents would also result in a global maximum dissonance at an
location
> somewhat different from 67 cents (but smaller or larger?).

if we simply reduce s lower than its current value of .01, then i
expect we could get the 740, but the 67 would probably drop to a
lower value (holding the tenney seed limit constant).

> > please be as picky as you like -- there are still a lot of things
> one
> > can tweak.
>
> It's nice to hear that. What are some of those things, and what
> would the anticipated effect(s) be?

take a look at my discussion of the algorithm that i posted here
yesterday. we could try much lower values for the tenney seed limit.
we could use the actual mediant-to-mediant widths instead of the
1/sqrt(n*d) approximation for them. either (or, especially, both
together) of these moves would result in a bit more "wackiness" or
idiosyncratic behavior in the curves, which will be highly dependent
on the particular value of the tenney seed limit that we choose --
other than that, i can't make any firm predictions (though i bet a
good mathematician such as gene could figure out these behaviors),
but we may well find a curve that suits you among the results. maybe
even *raising* the tenney seed limit while increasing s could help.
we could also try including unreduced ratios. we could even try a
different functional form for the uncertainty -- so far the only two
tried have been the bell curve (resulting in the earlier round-
looking curves, which produced too few local minima of discordance
for your taste) and the absolute exponential decay curve (all the
recent ones, which have the "pointy" appearance at the local minima
of discordance).

sharpening my knives,
chef paul

--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
> wrote:
> > > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > >
> > > > I think that the bumps
> > > > should disappear when n*d gets up to around 150, which would
> not
> > make
> > > > 16:9 and 17:10 appear as local maximum points of consonance.
> So
> > I
> > > > didn't want to see as many bumps as in margo2.gif. Is that
> > making
> > > > any sense?
> > >
> > > sure, but could it be that these are merely such tiny bumps in
> the
> > > road that your tires just plow over them as if they weren't
even
> > > there?
> >
> > I guess you could put it that way. But I would like the curve to
> > correspond to what is actually heard with musical tones (i.e.,
> tones
> > in which the harmonics are not as rich as in a sawtooth waveform).
>
> human voices, bowed strings, brasses, all of these can have audible
> harmonics into the 40s. but again, this doesn't concern harmonics
> themselves; it concerns the capacity for the interval to be *heard
> as* a pair of harmonics over a missing fundamental.

We seem to be looking at this in two different (but related) ways:
you're considering the tones in the intervals as partials over a
missing fundamental, while I'm thinking of them as fundamentals with
barely audible coincident partials.

> i might infer
> from your own remarks that even sine waves possess this capacity,
> yes?

From your viewpoint, yes.

> so isn't it possible that for a "generic" timbre, there are
> these tiny features in the discordance curve, so tiny that even
> incrementing by 2 cents at a time, and listening with a relatively
> good sensitivity to changes in discordance, is going to be unlikely
> to notice?

Yes, that's possible.

> > >
> > > ... as i mentioned, not taking the exponential, but looking at
the
> > actual
> > > entropy itself, remedies this to some extent. here is the
> resulting
> > graph:
>
/harmonic_entropy/files/dyadic/secorts2.g
if
> >
> > That's a bit closer to what I had in mind.
> >
> > However, one unexpected difference occurs at the local maximum of
> > dissonance at ~750 cents. In this latest curve it's a little bit
> > less than the local maximum at ~652 cents, whereas in
secortst.gif
> it
> > was somewhat greater.
>
> whoops! this shouldn't have been possible -- as the first graph was
> merely the exponential of the second, the two should have shared
the
> same rank-order. but i inadvertantly changed the tenney seed limit
> from 65536 to 10000. sorry! here's the (unexponetiated) harmonic
> entropy curve with tenney seed limit 65536:
>
>
/harmonic_entropy/files/dyadic/secorts3.g
if
>
> should agree perfectly in rank order with secortst.gif, which is
the
> exponential of this exact harmonic entropy function.

Okay, that's just what the customer ordered!

> > However, I think that the locations of those maximums (above and
> > below both the fifth and fourth) may be a more difficult thing to
> > resolve. I expect that a local maximum dissonance closer to 740
> > cents would also result in a global maximum dissonance at an
> location
> > somewhat different from 67 cents (but smaller or larger?).
>
> if we simply reduce s lower than its current value of .01, then i
> expect we could get the 740, but the 67 would probably drop to a
> lower value (holding the tenney seed limit constant).

I still have to listen to intervals approaching 3:2 and 4:3 from both
directions (with some variation in register) to see if my previous
observations regarding local maximums of dissonance can be
confirmed. So for the time being, secorts3.gif will be for me the
entree of choice. (My compliments to the chef!)

> > > please be as picky as you like -- there are still a lot of
things
> > one
> > > can tweak.
> >
> > It's nice to hear that. What are some of those things, and what
> > would the anticipated effect(s) be?
>
> take a look at my discussion of the algorithm that i posted here
> yesterday.

I'll do that.

> we could try much lower values for the tenney seed limit.
> we could use the actual mediant-to-mediant widths instead of the
> 1/sqrt(n*d) approximation for them. either (or, especially, both
> together) of these moves would result in a bit more "wackiness" or
> idiosyncratic behavior in the curves, which will be highly
dependent
> on the particular value of the tenney seed limit that we choose --
> other than that, i can't make any firm predictions (though i bet a
> good mathematician such as gene could figure out these behaviors),
> but we may well find a curve that suits you among the results.
maybe
> even *raising* the tenney seed limit while increasing s could help.
> we could also try including unreduced ratios. we could even try a
> different functional form for the uncertainty -- so far the only
two
> tried have been the bell curve (resulting in the earlier round-
> looking curves, which produced too few local minima of discordance
> for your taste) and the absolute exponential decay curve (all the
> recent ones, which have the "pointy" appearance at the local minima
> of discordance).
>
> sharpening my knives,
> chef paul

Chef Paul is to be commended for his patience in catering to a
(possibly) difficult customer. Please give ample warning if the
patience starts wearing thin -- I would not want the chef suddenly to
begin throwing those knives.

--George

--- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
wrote:
> > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > > --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
> > wrote:
> > > > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > > >
> > > > > I think that the bumps
> > > > > should disappear when n*d gets up to around 150, which
would
> > not
> > > make
> > > > > 16:9 and 17:10 appear as local maximum points of
consonance.
> > So
> > > I
> > > > > didn't want to see as many bumps as in margo2.gif. Is that
> > > making
> > > > > any sense?
> > > >
> > > > sure, but could it be that these are merely such tiny bumps
in
> > the
> > > > road that your tires just plow over them as if they weren't
> even
> > > > there?
> > >
> > > I guess you could put it that way. But I would like the curve
to
> > > correspond to what is actually heard with musical tones (i.e.,
> > tones
> > > in which the harmonics are not as rich as in a sawtooth
waveform).
> >
> > human voices, bowed strings, brasses, all of these can have
audible
> > harmonics into the 40s. but again, this doesn't concern harmonics
> > themselves; it concerns the capacity for the interval to be
*heard
> > as* a pair of harmonics over a missing fundamental.
>
> We seem to be looking at this in two different (but related) ways:
> you're considering the tones in the intervals as partials over a
> missing fundamental, while I'm thinking of them as fundamentals
with
> barely audible coincident partials.

if consonance were simply a matter of coincident partials, then
utonal chords would be at least as consonant as their otonal
counterparts -- q.v. fokker, vogel, etc. yet you've described even a
simple 7-limit utonal chords as sounding like a "parody". there would
seem to be some inconsistency in your view, then . . . ?

> So for the time being, secorts3.gif will be for me the
> entree of choice. (My compliments to the chef!)

why thank you! this function could be written as a single-line
mathematical expression, given fancy enough symbology for relative
primality of indices.

> Chef Paul is to be commended for his patience in catering to a
> (possibly) difficult customer. Please give ample warning if the
> patience starts wearing thin -- I would not want the chef suddenly
to
> begin throwing those knives.

hee hee!

--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > We seem to be looking at this in two different (but related)
ways:
> > you're considering the tones in the intervals as partials over a
> > missing fundamental, while I'm thinking of them as fundamentals
> with
> > barely audible coincident partials.
>
> if consonance were simply a matter of coincident partials, then
> utonal chords would be at least as consonant as their otonal
> counterparts -- q.v. fokker, vogel, etc. yet you've described even
a
> simple 7-limit utonal chords as sounding like a "parody". there
would
> seem to be some inconsistency in your view, then . . . ?

If we're only considering intervals, then otonal/utonal has no
bearing.

Do you also have a calculation of harmonic entropy for chords?

--George

--- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
wrote:
> > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > >
> > > We seem to be looking at this in two different (but related)
> ways:
> > > you're considering the tones in the intervals as partials over
a
> > > missing fundamental, while I'm thinking of them as fundamentals
> > with
> > > barely audible coincident partials.
> >
> > if consonance were simply a matter of coincident partials, then
> > utonal chords would be at least as consonant as their otonal
> > counterparts -- q.v. fokker, vogel, etc. yet you've described
even
> a
> > simple 7-limit utonal chords as sounding like a "parody". there
> would
> > seem to be some inconsistency in your view, then . . . ?
>
> If we're only considering intervals, then otonal/utonal has no
> bearing.

exactly. you seem to be missing my point. read it over again.

are you saying that completely different factors are at work in
determining the consonance of dyads than those at work in determining
the consonance of triads or tetrads? wouldn't it be simpler to assume
the same factors were at work?

> Do you also have a calculation of harmonic entropy for chords?

well, it's pretty much been mapped out on this list, but the
calculation has never actually been done. i've been waiting for
someone to offer a computational shortcut. but my new computer is
pretty fast; maybe i'll fire it up . . .

--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...
wrote:
> > > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > > >
> > > > We seem to be looking at this in two different (but related)
ways:
> > > > you're considering the tones in the intervals as partials
over a
> > > > missing fundamental, while I'm thinking of them as
fundamentals with
> > > > barely audible coincident partials.
> > >
> > > if consonance were simply a matter of coincident partials, then
> > > utonal chords would be at least as consonant as their otonal
> > > counterparts -- q.v. fokker, vogel, etc. yet you've described
even
> > a
> > > simple 7-limit utonal chords as sounding like a "parody". there
would
> > > seem to be some inconsistency in your view, then . . . ?
> >
> > If we're only considering intervals, then otonal/utonal has no
> > bearing.
>
> exactly. you seem to be missing my point. read it over again.
>
> are you saying that completely different factors are at work in
> determining the consonance of dyads than those at work in
determining
> the consonance of triads or tetrads? wouldn't it be simpler to
assume
> the same factors were at work?

Of course the same factors are at work. We perceive dissonance as
disturbances (i.e., beating or roughness) due to two things: partials
and combinational tones. (Do you know of any others?)

I believe that combinational tones are relatively unimportant in
evaluating the sonance of intervals, but they play a more significant
role in how we perceive chords. In chords lacking close
approximations to single-digit ratios, combinational tones probably
play the primary role. (And in intervals approaching a unison they
would have virtually no effect.)

These two things are of relative importance, depending on the
circumstances. So I guess I've made your point -- they both enter
into the equation.

> > Do you also have a calculation of harmonic entropy for chords?
>
> well, it's pretty much been mapped out on this list, but the
> calculation has never actually been done. i've been waiting for
> someone to offer a computational shortcut. but my new computer is
> pretty fast; maybe i'll fire it up . . .

For starters, I would be interested in comparing the values you get
for 4:5:6, 10:12:15, 6:7:9, and 14:18:21.

--George

--- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
wrote:
> > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > > --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...
> wrote:
> > > > --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> > > > >
> > > > > We seem to be looking at this in two different (but
related)
> ways:
> > > > > you're considering the tones in the intervals as partials
> over a
> > > > > missing fundamental, while I'm thinking of them as
> fundamentals with
> > > > > barely audible coincident partials.
> > > >
> > > > if consonance were simply a matter of coincident partials,
then
> > > > utonal chords would be at least as consonant as their otonal
> > > > counterparts -- q.v. fokker, vogel, etc. yet you've described
> even
> > > a
> > > > simple 7-limit utonal chords as sounding like a "parody".
there
> would
> > > > seem to be some inconsistency in your view, then . . . ?
> > >
> > > If we're only considering intervals, then otonal/utonal has no
> > > bearing.
> >
> > exactly. you seem to be missing my point. read it over again.
> >
> > are you saying that completely different factors are at work in
> > determining the consonance of dyads than those at work in
> determining
> > the consonance of triads or tetrads? wouldn't it be simpler to
> assume
> > the same factors were at work?
>
> Of course the same factors are at work. We perceive dissonance as
> disturbances (i.e., beating or roughness) due to two things:
partials
> and combinational tones. (Do you know of any others?)

yes, i believe that the virtual pitch phenomenon (measurably distinct
from combinational tone effects), which most recently seems to have
acquired a neurological explanation in terms of a revived version of
the old "periodicity pitch" paradigms, is one of the most important
factors. take a look at the book _harmony: a psychoacoustical
approach_ by richard parncutt for but one (not very good, but it
makes the point) attempt to work out the implications of such a view,
back when pattern-matching rather than neural timing was the
predominant explanation for virtual pitch effects. for a more recent
understanding, i'd suggest striking up a conversation with hans
straub over on the SpecMus list.
>
> > > Do you also have a calculation of harmonic entropy for chords?
> >
> > well, it's pretty much been mapped out on this list, but the
> > calculation has never actually been done. i've been waiting for
> > someone to offer a computational shortcut. but my new computer is
> > pretty fast; maybe i'll fire it up . . .
>
> For starters, I would be interested in comparing the values you get
> for 4:5:6, 10:12:15, 6:7:9, and 14:18:21.

if you look over this list, you'll see that i've always claimed that
harmonic entropy is only one component of discordance, while
roughness is another. so, while i'm pretty sure the triadic harmonic
entropy calculation would rank these triads simply in the order of
the size of their numbers (specifically their product or geometric
mean), a roughness calculation would show 4:5:6 and 10:12:15 as
approximately tied for most consonant. how to combine the harmonic
entropy piece and the roughness piece into an overall discordance
measure is not a problem i can forsee a full or elegant or even
universal solution to. we are fortunate that for dyads, harmonic
entropy considerations and roughness considerations predict pretty
much the same kinds of discordance curves, so in the dyadic case we
can "pretend" that harmonic entropy *is* discordance.

for the tetrads we listened to, it seems harmonic entropy was a
*stronger* determinant of the consonance/dissonance rankings than
roughness. however, 9:11:13:15 was perceived as considerably more
discordant than other tetrads with similar geometric mean, so it
seemed roughness did have *some* importance to most listeners.
however, you seem to feel that 9:11:13:15 is extremely concordant,
which is why i made the remarks i did that are quoted way up at the
top of this message. maybe you could look them over one more time so
that we might each approach a better understanding of what the other
is saying/thinking.

best,
paul

--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote:
>
> for the tetrads we listened to, it seems harmonic entropy was a
> *stronger* determinant of the consonance/dissonance rankings than
> roughness. however, 9:11:13:15 was perceived as considerably more
> discordant than other tetrads with similar geometric mean, so it
> seemed roughness did have *some* importance to most listeners.
> however, you seem to feel that 9:11:13:15 is extremely concordant,
> which is why i made the remarks i did that are quoted way up at the
> top of this message. maybe you could look them over one more time
so
> that we might each approach a better understanding of what the
other
> is saying/thinking.
>
> best,
> paul

Could you tell me what some of those "other tetrads" were?

But whatever they were, I suspect that we are not in disagreement on
this, because, after reflecting on this a bit, I now realize that I
was thinking of 9:11:13:15 as more consonant than *other* (non-
isoharmonic) chords containing lower primes, but which would have a
higher geometric mean.

Here's an example in which I would consider the isoharmonic chord
more consonant than another chord containing a lower prime --
14:18:21 vs. 7:9:11 -- in which only the top tone differs. In
essence, replacing 2:3 and 6:7 with more dissonant intervals -- 7:11
and 9:11 -- makes the chord more consonant.

Would you agree?

--George

--- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...>
wrote:
> >
> > for the tetrads we listened to, it seems harmonic entropy was a
> > *stronger* determinant of the consonance/dissonance rankings than
> > roughness. however, 9:11:13:15 was perceived as considerably more
> > discordant than other tetrads with similar geometric mean, so it
> > seemed roughness did have *some* importance to most listeners.
> > however, you seem to feel that 9:11:13:15 is extremely
concordant,
> > which is why i made the remarks i did that are quoted way up at
the
> > top of this message. maybe you could look them over one more time
> so
> > that we might each approach a better understanding of what the
> other
> > is saying/thinking.
> >
> > best,
> > paul
>
> Could you tell me what some of those "other tetrads" were?

look at joseph pehrson's "tuning lab" page on mp3.com.

> But whatever they were, I suspect that we are not in disagreement
on
> this, because, after reflecting on this a bit, I now realize that I
> was thinking of 9:11:13:15 as more consonant than *other* (non-
> isoharmonic) chords containing lower primes, but which would have a
> higher geometric mean.

the question of primality (absolute primality, that is) doesn't
directly come into any of the components of discordance, in my
opinion.

> Here's an example in which I would consider the isoharmonic chord
> more consonant than another chord containing a lower prime --
> 14:18:21 vs. 7:9:11 -- in which only the top tone differs. In
> essence, replacing 2:3 and 6:7 with more dissonant intervals --
7:11
> and 9:11 -- makes the chord more consonant.
>
> Would you agree?

that's a tough call, but certainly the geometric mean is lower for
7:9:11, and since these are fairly low numbers, i can predict with
some confidence that the triadic harmonic entropy model will find
7:9:11 less discordant than 14:18:21. the roughness is probably lower
for 14:18:21, though, and the relative importance of harmonic entropy
and roughness appears to be very sensitive to timbre, register,
loudness, and individual. you seem to be an individual for whom
harmonic entropy is more important than roughness . . . i'm going to
produce some strikingly beautiful hexagonal charts of triadic
discordance for you (look around the files section for a preview),
but it may be a while . . . still hoping for some help from the math
gurus like gene . . .