Distributing the complexity of a tempered dyad through dyad space with a Gaussian

Hey all,

I wanted to finally test this model whereby we come up with some JI
dyad v complexity plot and convolve the whole thing with a Gaussian.
This quick message will demonstrate the results of that approach.

First off, an explanation of what this means: I'm making almost the
opposite set of assumptions from HE. In this model, rather than
assuming the incoming dyad has a logarithmic (in cents) Gaussian
probability mistuning curve around it and that the basis dyads have
rectangular-shaped "domains" around them, I assume that the incoming
dyad has no mistuning curve, but that every dyad has a Gaussian-shaped
"field of attraction" around it. So by

- assigning each dyad a complexity (let's say n*d for now), and hence
a "simplicity" of 1/n*d
- and then plotting a Gaussian centered around that dyad
- and giving it a height equal to 1 over the aforementioned complexity
- and a standard deviation equivalent to Paul's "s" parameter
- and then adding all of the Gaussians together to get some resultant
curve (or creating a bunch of impulses and convolving it with a
Gaussian to compute everything in nlogn time)

the following "consonance curve" is produced (I used s=1% and a Tenney
series where n*d<=10000 for this example). All of these are on loglog
plots, where the y axis is log as well to make the nuances in the
curve easier to see:

/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_consonance_s_1.png

Taking 1/(this curve) gives the corresponding "dissonance curve,"
which will probably look really familiar:

/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1.png

The similarities between this and HE were really surprising at first.
This is probably mathematically really close or the same as the actual
HE calculation "under the hood," but I don't care enough to figure it
out anymore. I haven't labeled the minima or maxima because it's 7 AM
here and they're all exactly what we're expecting.

For lack of a better term, I'm calling this a "distributed complexity"
model for now. It can also be extended to triads or tetrads pretty
easily, so once Steve has his triadic HE calculation finished, I'll
see how this compares to that. As I'll address in a later post, this
expands elegantly to some more complex models for consonance - namely
adapting the Gaussian widths so that more complex intervals have a
"narrower" field of attraction than less complex ones. This will
"estimate" what my HLT entropy calculation will look like once I get
it done.

For a quick teaser for the adaptive widths version, check out the
following pictures, which I think might be more accurate to real-life
concordance than the fixed-widths version:

/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1_adaptive_smooth_overlay.png
/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_consonance_s_1_adaptive_smooth_overlay.png

Don't worry about what the two curves mean yet, and I've exaggerated
it a bit to demonstrate the "shape" of the curve and how it differs,
so there's no point in picking out specific cents values for it just
yet. Stay tuned...

-Mike

Look at this pattern (note = means approximately equals within a few cents)

((7/5)/(4/3))^(1/2) * 7/5 = 15/11 (input fractions = 7/5 and 4/3)
((5/4) / (6/5))^(1/2) * 6/5 = 11/9 (input fractions = 5/4 and 6/5)
((2/1) / (7/4))^(1/2) * 7/4 = 15/8 (input fractions = 2/1 and 7/4)
((5/3) / (8/5)) ^(1/2) * 8/5 = 18/11 (input fractions = 5/3 and 8/5)

This seems to be a general pattern in virtually all my favorite
intervals...that they are the logarithmic mid-point between the nearest two
major dips in the Harmonic Entropy curve.
It also (finally) may explain scientifically why such values sound better to me
than, say, 17/11 and other similar limit fractions.
-------------------
The only fraction I've found NOT in the HE curve that's needed to include all
the fractions I've found is 10/7 (which, of course, is still a fairly powerful
low-limit fraction)
((3/2) / (10/7))^(1/2) * 10/7 = 22/15 (input fractions = 3/2 and 10/7)
-----------------------

My question to you all is there any way to modify the HE curve that could
include these points?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> For lack of a better term, I'm calling this a "distributed complexity"
> model for now. It can also be extended to triads or tetrads pretty
> easily, so once Steve has his triadic HE calculation finished, I'll
> see how this compares to that.

Very nice work! I await the triads and tetrads with interest.

You could do the same thing with temperamental complexity instead, and I wonder if that has some sensible interpretation.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> For a quick teaser for the adaptive widths version, check out the
> following pictures, which I think might be more accurate to real-life
> concordance than the fixed-widths version:
>
> /tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1_adaptive_smooth_overlay.png
> /tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_consonance_s_1_adaptive_smooth_overlay.png

Dude, this is fascinating! Excellent work. Would it be much trouble to spit out a larger/higher-resolution graph?

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Dude, this is fascinating! Excellent work. Would it be much trouble to spit out a larger/higher-resolution graph?

As usual, these seem to be easier to view after conversion to an svg file. One thing you can do is create a very high resolution image file and convert it to a much smaller svg file, by the way.

MikeB>"/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1_adaptive_smooth_overlay.png"

The blue-line-indicated (adaptive width?!) version really catches my eye, it
seems to denote some new low-entropy areas NOT indicated by harmonic entropy
while keeping the strong low-entropy areas that HE has.
Although reading the graph with only markers every 500 cents is a bit
tough...any way to make a "high res" version of the graph with vertical lines
indicating every 10 cents or so?

In your first chart, it looks like it is a 7/2 that is standing out. Is that so or am i misreading it. If so it would imply some problem with Octave equivalence.
Not that i mind:).

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hey all,
>
> I wanted to finally test this model whereby we come up with some JI
> dyad v complexity plot and convolve the whole thing with a Gaussian.
> This quick message will demonstrate the results of that approach.
>
> First off, an explanation of what this means: I'm making almost the
> opposite set of assumptions from HE. In this model, rather than
> assuming the incoming dyad has a logarithmic (in cents) Gaussian
> probability mistuning curve around it and that the basis dyads have
> rectangular-shaped "domains" around them, I assume that the incoming
> dyad has no mistuning curve, but that every dyad has a Gaussian-shaped
> "field of attraction" around it. So by
>
> - assigning each dyad a complexity (let's say n*d for now), and hence
> a "simplicity" of 1/n*d
> - and then plotting a Gaussian centered around that dyad
> - and giving it a height equal to 1 over the aforementioned complexity
> - and a standard deviation equivalent to Paul's "s" parameter
> - and then adding all of the Gaussians together to get some resultant
> curve (or creating a bunch of impulses and convolving it with a
> Gaussian to compute everything in nlogn time)
>
> the following "consonance curve" is produced (I used s=1% and a Tenney
> series where n*d<=10000 for this example). All of these are on loglog
> plots, where the y axis is log as well to make the nuances in the
> curve easier to see:
>
> /tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_consonance_s_1.png
>
> Taking 1/(this curve) gives the corresponding "dissonance curve,"
> which will probably look really familiar:
>
> /tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1.png
>
> The similarities between this and HE were really surprising at first.
> This is probably mathematically really close or the same as the actual
> HE calculation "under the hood," but I don't care enough to figure it
> out anymore. I haven't labeled the minima or maxima because it's 7 AM
> here and they're all exactly what we're expecting.
>
> For lack of a better term, I'm calling this a "distributed complexity"
> model for now. It can also be extended to triads or tetrads pretty
> easily, so once Steve has his triadic HE calculation finished, I'll
> see how this compares to that. As I'll address in a later post, this
> expands elegantly to some more complex models for consonance - namely
> adapting the Gaussian widths so that more complex intervals have a
> "narrower" field of attraction than less complex ones. This will
> "estimate" what my HLT entropy calculation will look like once I get
> it done.
>
> For a quick teaser for the adaptive widths version, check out the
> following pictures, which I think might be more accurate to real-life
> concordance than the fixed-widths version:
>
> /tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1_adaptive_smooth_overlay.png
> /tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_consonance_s_1_adaptive_smooth_overlay.png
>
> Don't worry about what the two curves mean yet, and I've exaggerated
> it a bit to demonstrate the "shape" of the curve and how it differs,
> so there's no point in picking out specific cents values for it just
> yet. Stay tuned...
>
> -Mike
>

Large meta-reply

On Thu, Nov 4, 2010 at 2:15 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Very nice work! I await the triads and tetrads with interest.
>
> You could do the same thing with temperamental complexity instead, and I wonder if that has some sensible interpretation.

That's a good question. I really don't know. I'll try spitting some
tempered curves out for you once I get the triads sorted out. Maybe
some interpretation could be worked for the perception of melodies or
chord progressions, where movement by 4 just fifths really does lead
to a different noticeable end result than movement by 5/4.

On Thu, Nov 4, 2010 at 2:46 PM, cityoftheasleep <igliashon@...> wrote:
>
> Dude, this is fascinating! Excellent work. Would it be much trouble to spit out a larger/higher-resolution graph?
>
> -Igs

No problem, but keep in mind these two are nothing more than just very
basic expositional work for the moment to demonstrate the "shape" of
the graph - I don't have them tuned to human hearing constants yet. I
set 1/1's field of attraction equal to 17 cents and then made every
other field of attraction equal to 17/n+17/d for reasons I'd rather
not get into just yet. I think I screwed up on the calculation as well
and made every field of attraction twice as wide as it should be - the
field of attraction for 2/1 is way too large, for example. So take
specific values this curve is spitting out at different dyads with a
huge grain of salt.

/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1_adaptive_smooth_overlay_large.png
/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_consonance_s_1_adaptive_smooth_overlay_large.png

On Thu, Nov 4, 2010 at 3:15 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> > Dude, this is fascinating! Excellent work. Would it be much trouble to spit out a larger/higher-resolution graph?
>
> As usual, these seem to be easier to view after conversion to an svg file. One thing you can do is create a very high resolution image file and convert it to a much smaller svg file, by the way.

Sorry, I'm not sure how to export to SVG in MATLAB. Maybe newer
versions allow it...?

On Thu, Nov 4, 2010 at 3:19 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1_adaptive_smooth_overlay.png"
>
> The blue-line-indicated (adaptive width?!) version really catches my eye, it seems to denote some new low-entropy areas NOT indicated by harmonic entropy while keeping the strong low-entropy areas that HE has.
> Although reading the graph with only markers every 500 cents is a bit tough...any way to make a "high res" version of the graph with vertical lines indicating every 10 cents or so?

Check out my reply to Igs. The blue-line and the green-line ones are
both adaptive. The difference is that the green one has an additional
"smoothing" Gaussian overlaid on top of it, which might make sense if
you take it as representing something like the JND. I overshot in the
example images and smoothed it too much.

On Thu, Nov 4, 2010 at 4:04 PM, Brofessor <kraiggrady@...> wrote:
>
> In your first chart, it looks like it is a 7/2 that is standing out. Is that so or am i misreading it. If so it would imply some problem with Octave equivalence.
> Not that i mind:).

I think it's a 3/1 that you're seeing.

-Mike

I think the current thinking around here is that maxima of harmonic entropy might have an identifiable, er, identity, as well as minima.
This absolutely makes sense to me. I think it's the reason the "shadow interval" of 464 +/- 2 cents (or phi^2 modulo 2) works so well as a kind of "crossroads" or hinging point. You can't really put your finger on it.

I disagree with HE on the fuzziness of 11:9 and 16:13, as I hear them as sweet spots. I'm open to the idea that a little space inbetween them, right about the square root of 3/2, might also be a point of "shadow", though.

However, right in there you've got a spot where there's an acoustic phenomenon I haven't seen mentioned before: the beating is symmetrical about the fifth partial, and for typical musical ranges it's a harmonic buzz in the transition zone from being percieved as beats and being percieved as a low tone. You can reckon it out and listen for it on headphones, it's pretty distinct. That, it seems to me, might actually be the signpost of a truly middle third. Whereas in my experience a half dozen cents up or down from there and you're in the postal zones, so to speak, of the coincidences of harmonic partial at either 11 and 9 or 16 and 13.

Of course everything I just said is about reasonable and sharable names for things, not "numbers" as such. We're not going to be able to have much of a discussion, or share tooling methods, if I were to refuse to translate "muddy green, shaped like this..." into the perfectly reasonable translation "11:9".

-Cameron Bobro

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Look at this pattern (note = means approximately equals within a few cents)
>
> ((7/5)/(4/3))^(1/2) * 7/5 = 15/11 (input fractions = 7/5 and 4/3)
> ((5/4) / (6/5))^(1/2) * 6/5 = 11/9 (input fractions = 5/4 and 6/5)
> ((2/1) / (7/4))^(1/2) * 7/4 = 15/8 (input fractions = 2/1 and 7/4)
> ((5/3) / (8/5)) ^(1/2) * 8/5 = 18/11 (input fractions = 5/3 and 8/5)
>
> This seems to be a general pattern in virtually all my favorite
> intervals...that they are the logarithmic mid-point between the nearest two
> major dips in the Harmonic Entropy curve.
> It also (finally) may explain scientifically why such values sound better to me
> than, say, 17/11 and other similar limit fractions.
> -------------------
> The only fraction I've found NOT in the HE curve that's needed to include all
> the fractions I've found is 10/7 (which, of course, is still a fairly powerful
> low-limit fraction)
> ((3/2) / (10/7))^(1/2) * 10/7 = 22/15 (input fractions = 3/2 and 10/7)
> -----------------------
>
>
> My question to you all is there any way to modify the HE curve that could
> include these points?
>

Gene wrote:

> > so once Steve has his triadic HE calculation finished, I'll
> > see how this compares to that.
>
> Very nice work! I await the triads and tetrads with interest.

There's no need to wait, because there's plenty left to do
comparing to dyadic HE before even considering triads.

-Carl

Gene wrote:

> > Dude, this is fascinating! Excellent work. Would it be
> > much trouble to spit out a larger/higher-resolution graph?
>
> As usual, these seem to be easier to view after conversion to
> an svg file. One thing you can do is create a very high
> resolution image file and convert it to a much smaller
> svg file, by the way.

Ideally, Mike could produce SVG directly from Matlab (or
whatever package he's using). Don't know if it supports
that. But some tabular data would also be good.

-Carl

Mike wrote:

> This will "estimate" what my HLT entropy calculation will
> look like once I get it done.

What's HLT stand for? -C.

Highly ludicrous tantrums?

Oz.

--

✩ ✩ ✩
www.ozanyarman.com

Carl Lumma wrote:
> Mike wrote:
>
>> This will "estimate" what my HLT entropy calculation will
>> look like once I get it done.
>
> What's HLT stand for? -C.
>
>

Cameron>"I disagree with HE on the fuzziness of 11:9 and 16:13, as I hear them
as sweet spots."
Agreed on 11/9, not so much on 16/13...but I do note that the area a bit
above 11/9 (nearer to 16/13) sounds virtually as "sweet" to me as 11/9. The
area between 15/11 and 11/8 seems to act much the same way to me.

Cameron>"However, right in there you've got a spot where there's an acoustic
phenomenon I haven't seen mentioned before: the beating is symmetrical about the
fifth partial, and for typical musical ranges it's a harmonic buzz in the
transition zone from being percieved as beats and being percieved as a low
tone."

Fascinating...so how, mathematically, are you doing this calculation of
"symmetry"?

On Thu, Nov 4, 2010 at 8:15 PM, Carl Lumma <carl@...> wrote:
>
> There's no need to wait, because there's plenty left to do
> comparing to dyadic HE before even considering triads.

Feel free to compare. I have no interest in finding out the
mathematical relationship between them anymore. This is just another
model for consonance, deriving from a second set of perfectly
reasonable assumptions, that happens to look exactly the same as the
other one. Maybe there are subtle features differentiating the two
that I don't know about.

-Mike

On Thu, Nov 4, 2010 at 8:37 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > This will "estimate" what my HLT entropy calculation will
> > look like once I get it done.
>
> What's HLT stand for? -C.

Harmonic Laplace Transform. It's the idea I was talking about a while
ago that represents a signal as the sum of exponentially damped sines
+ damped harmonics. The Laplace transform is just the sum of
exponentially damped (complex) sines, so I'm calling this for lack of
a better name the "Harmonic Laplace Transform." The discrete
equivalent will be the Harmonic Z-Transform.

-Mike

Mike wrote:

> Feel free to compare. I have no interest in finding out the
> mathematical relationship between them anymore. This is just
> another model for consonance, deriving from a second set of
> perfectly reasonable assumptions, that happens to look exactly
> the same as the other one. Maybe there are subtle features
> differentiating the two that I don't know about.

I'm afraid I don't share your enthusiasm, and neither, it
seems, does Paul. Harmonic entropy was pretty extensively
tested over a period of years. If you aren't interested in
doing any comparisons why should others be?

-Carl

On Fri, Nov 5, 2010 at 2:09 AM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > Feel free to compare. I have no interest in finding out the
> > mathematical relationship between them anymore. This is just
> > another model for consonance, deriving from a second set of
> > perfectly reasonable assumptions, that happens to look exactly
> > the same as the other one. Maybe there are subtle features
> > differentiating the two that I don't know about.
>
> I'm afraid I don't share your enthusiasm, and neither, it
> seems, does Paul. Harmonic entropy was pretty extensively
> tested over a period of years. If you aren't interested in
> doing any comparisons why should others be?

Hi Carl,

I'm sorry you don't share my enthusiasm. This is a very simple model
for consonance that stems from some pretty basic assumptions. It
happens to generate something pretty similar to the HE curve. I don't
really care too much to find out the two are mathematically related.
It's a pretty big job and I don't really have the time for it.

I also should probably make clear that I don't think that HE is a
completely infallable model either, and my goal is not to "regenerate"
the HE curve.

-Mike

> On Fri, Nov 5, 2010 at 2:09 AM, Carl Lumma <carl@...> wrote:
>> I'm afraid I don't share your enthusiasm, and neither, it
>> seems, does Paul. Harmonic entropy was pretty extensively
>> tested over a period of years. If you aren't interested in
>> doing any comparisons why should others be?

I just realized I might be misinterpreting this. Are you asking me to
post the minima and maxima? If so, just be patient for a bit.

But if you're asking me to work out how mathematically the convolution
integral is related to the entropy summation - no thanks.

-Mike

Hi Mike,

> I'm sorry you don't share my enthusiasm. This is a very simple
> model for consonance that stems from some pretty basic
> assumptions.

It's not quite clear what those assumptions are.

> It happens to generate something pretty similar to the HE curve.

I can draw something similar to an HE curve freehand.
So what?

> I don't really care too much to find out the two are
> mathematically related.

I haven't asked about that, I've asked if the results are
related. Until you publish tabular data we can't check
that even if we wanted to.

> my goal is not to "regenerate" the HE curve.

What's your goal then?

-Carl

Alright, for your viewing pleasure, extrema have been calculated. I've
posted the "dissonance" version of my equal-widths plot, so it's
easier for everyone to compare to HE.

I've noticed that my curve seems to "sync up" better with the HE curve
if a slightly lower value of s is used, something like 4/5 of what
we're comparing it to. I'm using a modified Gaussian function so it
might be that mine is scaled slightly differently than Paul's, or it
might be a difference in the way mine is calculated.

Here it is for n*d<10000, s = 1%:

/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1_extrema.png

Here's the classic HE curve with s=1.2%. Note all the maxima are
within a few cents of one another vs the my curve@1.0 version:

/tuning/files/dyadic/default.gif

Now to compare, here's my curve with s=1.2%. Note that s=1.2% is the
equivalent of something like s=1.5% with classic HE:

/tuning/files/MikeBattaglia/DistributedComplexityModeling/dc_log_dissonance_s_1.2_extrema.png

Here's classic HE @ s=1.5%:

/tuning/files/dyadic/t2_015_13p2877.jpg

-Mike

On Fri, Nov 5, 2010 at 4:08 AM, Carl Lumma <carl@...> wrote:
>
> Hi Mike,
>
> > I'm sorry you don't share my enthusiasm. This is a very simple
> > model for consonance that stems from some pretty basic
> > assumptions.
>
> It's not quite clear what those assumptions are.

I've said this in the first post and offlist, but again

- I'm assuming that complexity is somehow related to dissonance
- This implies that 1/complexity is related to consonance
- I'm assuming that every JI interval doesn't just stop being
perceived as that interval when it's mistuned, but that mistunings of
that interval will still be perceived as that interval
- To formalize this assumption I have given every interval a Gaussian
shaped (in log space) "field of attraction" with width s that
surrounds that interval
- I then plot the consonance of each interval as 1/complexity, and
then use the Gaussian to distribute the consonance of each interval
sideways in dyad space. When Gaussians overlap, they add
- I decided n*d was a good start for complexity, but a more
conceptually rigorous version might be this 1/n^2+1/d^2 measure I
described to you offlist

> > It happens to generate something pretty similar to the HE curve.
>
> I can draw something similar to an HE curve freehand.
> So what?

So your freehand curve isn't going to be generated automatically from
an algorithm that stems from the simple assumptions above.

> > my goal is not to "regenerate" the HE curve.
>
> What's your goal then?

My goal was just to see what kind of curve would emerge if I did the
above. It happens to look pretty similar to HE. I spent a lot of time
trying to figure out how they're mathematically related, and now I'm
giving up, since I'd rather move onto the adaptive-widths version. It
makes an equal number of assumptions as does HE and you should feel
free to compare them. There will probably be subtle differences.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > It's not quite clear what those assumptions are.
>
> I've said this in the first post and offlist, but again

What I mean is, I don't see how these translate into robust
assumptions. But since we're discussing it offlist let's take
it all there.

-Carl

What is you thoughts on proportional triads as also being consonant?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Nov 5, 2010 at 4:08 AM, Carl Lumma <carl@...> wrote:
> >
> > Hi Mike,
> >
> > > I'm sorry you don't share my enthusiasm. This is a very simple
> > > model for consonance that stems from some pretty basic
> > > assumptions.
> >
> > It's not quite clear what those assumptions are.
>
> I've said this in the first post and offlist, but again
>
> - I'm assuming that complexity is somehow related to dissonance
> - This implies that 1/complexity is related to consonance
> - I'm assuming that every JI interval doesn't just stop being
> perceived as that interval when it's mistuned, but that mistunings of
> that interval will still be perceived as that interval
> - To formalize this assumption I have given every interval a Gaussian
> shaped (in log space) "field of attraction" with width s that
> surrounds that interval
> - I then plot the consonance of each interval as 1/complexity, and
> then use the Gaussian to distribute the consonance of each interval
> sideways in dyad space. When Gaussians overlap, they add
> - I decided n*d was a good start for complexity, but a more
> conceptually rigorous version might be this 1/n^2+1/d^2 measure I
> described to you offlist
>
> > > It happens to generate something pretty similar to the HE curve.
> >
> > I can draw something similar to an HE curve freehand.
> > So what?
>
> So your freehand curve isn't going to be generated automatically from
> an algorithm that stems from the simple assumptions above.
>
> > > my goal is not to "regenerate" the HE curve.
> >
> > What's your goal then?
>
> My goal was just to see what kind of curve would emerge if I did the
> above. It happens to look pretty similar to HE. I spent a lot of time
> trying to figure out how they're mathematically related, and now I'm
> giving up, since I'd rather move onto the adaptive-widths version. It
> makes an equal number of assumptions as does HE and you should feel
> free to compare them. There will probably be subtle differences.
>
> -Mike
>

On Fri, Nov 5, 2010 at 7:19 AM, Brofessor <kraiggrady@...> wrote:
>
> What is you thoughts on proportional triads as also being consonant?

What do you mean by proportional triads?

-Mike

these are found in recurrent sequences such as the fib series. If one has the series 8 -13-21 and one moves the top number down to 8-10.5-13 one can see that this triad will have the same difference tones in octaves and if one is using this series as a scale will also be a scale member hence reinforcing other notes one is going to hear.
This is not my favorite here . but things like the 32-37-42 is one i use allot and even the 43-50-57 works in a '6-7-8' sense within my meta-slendro.
Actually my biggest problem with this scale is that the most dissonant thing i can do, playing all 12 tones of my series does not sound that dissonant at all. This makes it hard for me to explain it as simply an appox. to JI vicinity, although i think you are on the right track.
If i were to pursuit the problem ( which i am not ) i would assume two ways of having consonance both with share the property of producing acoustical coincidence. So for me HE would have to lie in between these.
Although i do think if you composed music in such a way that it was only based on one of these It would work, so i am not trying to undermine where you are. just another option.
I was thinking of this earlier today with a concept of a "rhythmic entropy" and Stephen Taylor had played me horograms where the durations of long to short where 1.618 and it sounded perfectly correct and even.
It was a bit uncanny.
So my thought was it would have to exist in duration somewhere between additive durations and phi ones.
The ideal entropic state would be if one could not tell which the proportion was or say it might not sound the same way twice.
This it seems would be what it would be like with pitch also.

But we have complete continuums of all of these. JI intervals, PHI intervals and HE intervals and at a certain point they are all going to be the same. Noble mediants and EDOs intervals
we are left with context to define and when we hear music. I think these types of "codes" are presented to us from which we define everything else. So we will follow yours more, but how do you turn this into a scale?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Nov 5, 2010 at 7:19 AM, Brofessor <kraiggrady@...> wrote:
> >
> > What is you thoughts on proportional triads as also being consonant?
>
> What do you mean by proportional triads?
>
> -Mike
>

Carl to Mike B>"I'm afraid I don't share your enthusiasm, and neither, it seems,
does Paul. Harmonic entropy was pretty extensively tested over a period of
years. If you aren't interested in doing any comparisons why should others be?"

Carl, if that really is your issue, why not make an effort to help test
Mike's theory yourself and encourage others to do so? The way the "Mike B's
theory lacks Harmonic Entropy's extended testing" gap would be closed. I
figure, you can obviously test "SHE"'s validity without testing it directly
against HE (even though they are two completely different models).
I just don't see any point in limiting a list of "valid consonance models" to
only "models that can be directly interpreted as versions of older ones (if I'm
hearing you correctly)...on the surface it only seems to put an artificial limit
on innovation (including "even" rigorously mathematical and scientifically based
innovation). What exactly is the point here?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> - I decided n*d was a good start for complexity, but a more
> conceptually rigorous version might be this 1/n^2+1/d^2 measure I
> described to you offlist

Why is that more conceptually rigorous? It's certainly different, at any rate: 5/3 and 15/1 are the same in terms of n*d, but 1/5^2+1/3^2 is 34/225, whereas 1/15^2 + 1/1^2 is 226/225. That's 226 versus 34!

A few side question (from someone who knows what Harmonic Entropy represents and
implies, but has little clue how it is calculated):

A) Is Tenney Height the basis from which where the "low entropy points are
calculated?
B) Does "s" in the formula represent the simple standard deviation AKA square
root of variance from each trough/low-entropy-point? How is the area around
each peak calculated (I'm guessing if two distributions collided they are simply
summed)? I've never done statistics with multiple bell curves at once...

My suspicion is that Harmonic Entropy looks to me, on the surface, to be a
process of building bell curves around Tenney Height (which seems to assume
Tenney Height = complexity) and the curves represent the chance something will
be interpreted as being of that complexity. And in such a case, perhaps,
plugging something other than simple Tenney Height (such of Tenney Height
weighted by critical band or a list of ratios shown as "lease dissonant" in a
comprehensive listening test played by, say, a guitar) may give better results
and likely find a few new low-entropy point options while preserving the current
ones pretty well.

On Fri, Nov 5, 2010 at 2:43 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > - I decided n*d was a good start for complexity, but a more
> > conceptually rigorous version might be this 1/n^2+1/d^2 measure I
> > described to you offlist
>
> Why is that more conceptually rigorous? It's certainly different, at any rate: 5/3 and 15/1 are the same in terms of n*d, but 1/5^2+1/3^2 is 34/225, whereas 1/15^2 + 1/1^2 is 226/225. That's 226 versus 34!

It comes out of an idea that I had a long time ago: that the array of
combination-sensitive neurons in the brain might effectively form some
kind of filterbank, or set of "harmonic sieves," or however you'd like
to think of it, for all of the frequencies across the hearing
spectrum.

Carl suggested that the ideal spectral rolloff for the most
"concordant" sound is 1/N^2, since that's the approximate spectrum of
the human voice. So it makes sense that the filters in the brain would
be tuned to something similar (or in actuality, tuned to the inverse
of this, so that the crosscorrelation of the filter and the signal
would be highest if the signal has a rolloff of 1/N^2).

So if that were the case, then any dyad n/d would contribute 1/n^2 +
1/d^2 energy to that filter, which is n^2d^2/n^2+d^2. This is close to
0.5*(n*d) if n-d is small (superparticular), and as n-d gets larger it
diverges.

But I screwed up in my initial message to you: 1/n^2 + 1/d^2 is more
like an analogue of "simplicity" in that 1/1 gives the highest result,
and 2/1 gives the next-highest result, and so on. So this would mean
that 15/1 ends up being less "complex" than 5/3. Why? Because what
this is really trying to model, my perhaps ill-advised choice of the
word "complexity" aside, is how strong the volume of the VF produced
will be, or how "clear" it will be. For 15/1, the 1 is being played
right there, so there's no estimation that needs to take place. For
5/3, the volume of the VF will be less clear than 15/1.

The question is, then, whether or not 15/1 will really be heard as a
complex 1 with only a 15th harmonic, or whether the 15 will be heard
as a flattened 2, or perhaps whether it'll be heard as inharmonic.
This is where the Gaussian spreading comes into play, which will
reveal whether or not 15 gets its own consonance maximum or not.

It's a new toy to play around with, and I'm still exploring the
implications of it. Perhaps complexity is a bad term for it, since
that already refers to something else.

-Mike

On Fri, Nov 5, 2010 at 6:44 PM, Mike Battaglia <battaglia01@...> wrote:
>
> So if that were the case, then any dyad n/d would contribute 1/n^2 +
> 1/d^2 energy to that filter, which is n^2d^2/n^2+d^2. This is close to
> 0.5*(n*d) if n-d is small (superparticular), and as n-d gets larger it
> diverges.

Sorry, screwed this up here. 1/n^2 + 1/d^2 is (n^2+d^2)/(n^2*d^2). And
this is close to 2/(n*d) if n-d is small.

Keep in mind that the choice of the exponent being 2 is just an
initial guess based off of Carl's speech idea, and that it all might
turn out to make more sense if the rolloff is 1, or somewhere between
1 and 2.

-Mike

On Fri, Nov 5, 2010 at 4:57 PM, Michael <djtrancendance@...> wrote:
>
> A few side question (from someone who knows what Harmonic Entropy represents and implies, but has little clue how it is calculated):
>
> A) Is Tenney Height the basis from which where the "low entropy points are calculated?

Indirectly.

> B) Does "s" in the formula represent the simple standard deviation AKA square root of variance from each trough/low-entropy-point?  How is the area around each peak calculated (I'm guessing if two distributions collided they are simply summed)?  I've never done statistics with multiple bell curves at once...

Indirectly. But what you're actually describing is the model that I've
just come up with - these characteristics emerge indirectly out of the
HE model. The fact that the minima are proportional to Tenney Height
is because of the assumption in HE that every interval has a
rectangular width that's equal to (pick your favorite average) between
each interval and its nearest neighbor, which ends up being roughly
proportional to sqrt(n*d). If every interval were given an equal
domain, for example, the same thing wouldn't be happening. I think
that the sqrt(n*d) reflects something about the distribution of the
rationals in a log-lin space.

The "s" ends up representing the standard deviation around each
minimum, but it directly ends up representing the standard deviation
for the bell curve that represents the fictional "incoming dyad" that
you're trying to measure. If you instead give the incoming dyad a Vos
curve, the minima end up being Vos curve-shaped.

> My suspicion is that Harmonic Entropy looks to me, on the surface, to be a process of building bell curves around Tenney Height (which seems to assume Tenney Height = complexity) and the curves represent the chance something will be interpreted as being of that complexity.  And in such a case, perhaps, plugging something other than simple Tenney Height (such of Tenney Height weighted by critical band or a list of ratios shown as "lease dissonant" in a comprehensive listening test played by, say, a guitar) may give better results and likely find a few new low-entropy point options while preserving the current ones pretty well.

On the other hand, what you just described above is exactly what I'm
doing with my model - I assume every interval has a complexity of n*d
(Tenney Height squared) and then build a bell curve around each one,
and sum them where they overlap. Except I instead plot the inverse of
complexity (which is 1/(n*d)) and then do the above, and do 1/(the
resultant curve) to get the answer.

For some reason, putting in sqrt(n*d) makes the curve not work,
although putting in n*d and then taking the sqrt of the summed curve
at the end makes it work. I'm not sure why this is yet. I'm also
exploring replacing the 1/n*d with 1/n^2+1/d^2. It doesn't seem to
change the minima but it gives the whole curve an overall sloping
trend.

I'm also trying to mess around with giving the bell curves a standard
deviation that's a function of the interval's complexity, which means
that 11/8 will have a "narrower" field of attraction than 5/3. I think
this offers a number of advantages over the other curve and I'm
interested in seeing how it'll hold up in a listening test.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> On the other hand, what you just described above is exactly what I'm
> doing with my model - I assume every interval has a complexity of n*d
> (Tenney Height squared) and then build a bell curve around each one,
> and sum them where they overlap. Except I instead plot the inverse of
> complexity (which is 1/(n*d)) and then do the above, and do 1/(the
> resultant curve) to get the answer.

I thought this was how HE was calculated.

> For some reason, putting in sqrt(n*d) makes the curve not work,
> although putting in n*d and then taking the sqrt of the summed curve
> at the end makes it work. I'm not sure why this is yet.

1/(n*d) is 1/t^2 where t is the geometric mean, whereas 1/sqrt(n*d) is 1/t. It's not surprising to find 1/t^2 has better convergence properties than 1/t if convergence is the problem.

On Fri, Nov 5, 2010 at 8:25 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > On the other hand, what you just described above is exactly what I'm
> > doing with my model - I assume every interval has a complexity of n*d
> > (Tenney Height squared) and then build a bell curve around each one,
> > and sum them where they overlap. Except I instead plot the inverse of
> > complexity (which is 1/(n*d)) and then do the above, and do 1/(the
> > resultant curve) to get the answer.
>
> I thought this was how HE was calculated.

Not quite - just indirectly. Here's a paper on how it's calculated:

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/HarmonicEntropy.pdf

If you replace the Gaussian around the incoming dyad with |e(x)|, the
resultant curve has |e(x)| shaped valleys. And if you look at the
mediant-to-mediant width plot in that paper - you will notice that
placing that on a semilogx plot, and then convolve it with a Gaussian
that looks "straight" in log space, you get something that looks
almost identical to an upside down HE curve. If you set s to 0, so
that you're convolving the plot with a Dirac delta, you get the same
thing. Yep.

So my original goal with this model, once upon a time, was to
"regenerate" the HE curve and work out how the convolution integral is
related to the entropy summation, and treat this as a computational
speedup.

At this point I'm giving up because I've spent a lot of time on that
and it's led nowhere, and I'd rather just compute the model by itself,
since the assumptions producing it are reasonable enough to just see
where it leads without worrying about how the math relates to HE's
math.

-Mike

On Fri, Nov 5, 2010 at 7:48 AM, Brofessor <kraiggrady@...> wrote:
>
> This is not my favorite here . but things like the 32-37-42 is one i use allot and even the 43-50-57 works in a '6-7-8' sense within my meta-slendro.
> Actually my biggest problem with this scale is that the most dissonant thing i can do, playing all 12 tones of my series does not sound that dissonant at all. This makes it hard for me to explain it as simply an appox. to JI vicinity, although i think you are on the right track.

Well, one thing is that I don't think that it has anything to do with
actual acoustic difference tones. I've seen some interesting research
on first-order difference tones maybe having something to do with the
brain, but I haven't delved into it enough.

What I've noticed is that 5:7:9:11 sounds great - everything's
separated by 2. But 5:8:11:14 doesn't sound as good. That is, 2 seems
to be the limit - when things are separated by 3, it doesn't work out
as well anymore.

> If i were to pursuit the problem ( which i am not ) i would assume two ways of having consonance both with share the property of producing acoustical coincidence. So for me HE would have to lie in between these.

I could probably come up with a model that takes (n-d) into account,
or if we're dealing with a:b:c, that takes (c-b)+(b-a) into account,
just for fun.

> I was thinking of this earlier today with a concept of a "rhythmic entropy" and Stephen Taylor had played me horograms where the durations of long to short where 1.618 and it sounded perfectly correct and even.
> It was a bit uncanny.

That's pretty cool, do you have a link to anything like this? I'm
having trouble picturing it.

> But we have complete continuums of all of these. JI intervals, PHI intervals and HE intervals and at a certain point they are all going to be the same. Noble mediants and EDOs intervals
> we are left with context to define and when we hear music. I think these types of "codes" are presented to us from which we define everything else. So we will follow yours more, but how do you turn this into a scale?

I'm not sure how to turn this into a scale - I'm just kind of doing
this for fun. I'm interested in seeing what happens if tetrads get
involved.

-Mike

Mike,

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Carl suggested that the ideal spectral rolloff for the most
> "concordant" sound is 1/N^2, since that's the approximate spectrum of
> the human voice. So it makes sense that the filters in the brain would
> be tuned to something similar (or in actuality, tuned to the inverse
> of this, so that the crosscorrelation of the filter and the signal
> would be highest if the signal has a rolloff of 1/N^2).

What if a spectrum has a rolloff of 1/N^3? Would the sound be less
concordant than with 1/N^2?

Kalle

On Sat, Nov 6, 2010 at 6:14 AM, Kalle Aho <kalleaho@...> wrote:
>
> Mike,
>
> What if a spectrum has a rolloff of 1/N^3? Would the sound be less
> concordant than with 1/N^2?

This is purely speculation, but let's assume that 1/N^2 is the "ideal
rolloff" for the sake of this discussion. This means that if the
rolloff is brighter than 1/N^2, like 1/N or just 1, that the harmonics
will themselves be audible as sinusoids in their own right, and so the
sound will flip flop between hearing the harmonics completely "fused"
and hearing them as isolated. This is higher entropy than a sound
which has no such ambiguity, because in the model I'm trying to
construct, it's going to cause a lot more of the harmonic filters to
fire. Keep in mind we're used to and like this phenomenon: when you
play a just major chord, you aren't only hearing the fundamental of
the 4:5:6, you're hearing each of the notes in the chord as its own
fundamental as well. That is, C-E-G will seem to flipflop between a
virtual C 2 octaves down, and between the C, E, and G being individual
notes. Maybe you'll get the C one octave down popping out as well,
from the 2:3 on the end.

Anyway, let's say the rolloff is steeper, so something like 1/N^3.
This means that while you won't have the ambiguity, it just isn't
going to cause the filter to resonate as strongly as with 1/N^2. If
I'm estimating my model correctly, it also means that the ambiguity in
that fundamental will sort of spread to the surrounding pitches more,
whereas adding harmonics will "focus" it. So getting closest to the
"sweet spot" will bring a sort of "pitch clarity" to the sound.

-Mike

Mike B,
thanks for posting the link to Sethares' description of H.E. - I hadn't seen that one before. I have a few quibbles with your explanations to Michael S.; although I'm quite sure you have understood it well yourself, I mention them in the hope that it will help him.

MikeB> The fact that the minima are proportional to Tenney Height ...

Did you mean that? I'm not sure what it means.

MikeB> ... is because of the assumption in HE that every interval has a rectangular width that's equal to (pick your favorite average) between each interval and its nearest neighbor, ...

Every member of the Farey set (or Tenney set, or whatever) *does* have a region around it, e.g. all points on the real line that are nearer to that member than any other; that's not the assumption (!). Note also that the region is one-dimensional (not rectangular). One assumption is that the width of that region is significant in calculating the probability that the "test" interval ("i" in Sethares' doc) will be heard as that member of the set. Another assumption is that it would be better to use the mediant-mediant distance rather than the mean-mean distance I described above. Note that the accurate calculation calls for the area under the bell-curve between the mediants; it is an approximation to assume the area is rectangular (actually, I have been wondering if there is any benefit in making it at least trapezoidal).

MikeB> ... which ends up being roughly proportional to sqrt(n*d). If every interval were given an equal domain, for example, the same thing wouldn't be happening.

I agree with all of that.

> The "s" ends up representing the standard deviation around each minimum ...

Again, not sure that's true (or I don't understand it).

> ... but it directly ends up representing the standard deviation for the bell curve that represents the fictional "incoming dyad" that you're trying to measure.

I think you mean "started out" rather than "ends up" here?

> ... If you instead give the incoming dyad a Vos curve, the minima end up being Vos curve-shaped.

Do you mean for Erlich's H.E. or for your model, or both? I haven't tried this; I intend to.

> On the other hand, what you just described above is exactly what I'm
> doing with my model ...

It sounds interesting; the shape of your curve around the minima looks wider than for H.E. ...

Actually, I have a quibble with Sethares' paper too, but perhaps I'll save that for another time.

Steve M.

On Sat, Nov 6, 2010 at 8:32 AM, martinsj013 <martinsj@...> wrote:
>
> Mike B,
> thanks for posting the link to Sethares' description of H.E. - I hadn't seen that one before. I have a few quibbles with your explanations to Michael S.; although I'm quite sure you have understood it well yourself, I mention them in the hope that it will help him.
>
> MikeB> The fact that the minima are proportional to Tenney Height ...
>
> Did you mean that? I'm not sure what it means.

The minima of the curve generally have a height that ends up being
proportional to sqrt(n*d), yes?

> MikeB> ... is because of the assumption in HE that every interval has a rectangular width that's equal to (pick your favorite average) between each interval and its nearest neighbor, ...
>
> Every member of the Farey set (or Tenney set, or whatever) *does* have a region around it, e.g. all points on the real line that are nearer to that member than any other; that's not the assumption (!). Note also that the region is one-dimensional (not rectangular). One assumption is that the width of that region is significant in calculating the probability that the "test" interval ("i" in Sethares' doc) will be heard as that member of the set. Another assumption is that it would be better to use the mediant-mediant distance rather than the mean-mean distance I described above. Note that the accurate calculation calls for the area under the bell-curve between the mediants; it is an approximation to assume the area is rectangular (actually, I have been wondering if there is any benefit in making it at least trapezoidal).

Whether you use the mediant-to-mediant or mean-to-mean distance, if
you use something like a Farey series or a Tenney series, you end up
getting the pattern where the intervals have a width proportional to
complexity. What would happen if you used a different series to
enumerate the rationals...? Let's say you came up with a Tenney
series, and then at the very end of the calculation you took every
interval i in the series and then added i - 0.1 and i + 0.1 to the
series, and then still did mediant-to-mediant widths. That's another
perfectly acceptable way to enumerate the rationals, but I doubt the
curve would be too useful.

Paul spoke of the Tenney, Farey, and Mann series as having some kind
of "unideterminant" property that caused the curve to come out right.
I'm not sure what that means though. Either way, here's another
thought experiment - what if you set things up so that every interval
had a width of 5 cents, and you just allowed the domains to overlap?
Or every interval had a width of (total cents range)/(number of
intervals), so that the probability curve still summed to 1?

The fact that the HE curve ends up looking the way it does comes
directly out of the way the widths are set up.

>> The "s" ends up representing the standard deviation around each minimum ...
> Again, not sure that's true (or I don't understand it).

Check out some of the HE graphs. When s is smaller, the width of the
valleys around the minima shrinks; when it's larger, it expands. A
larger s = larger valleys.

> > ... but it directly ends up representing the standard deviation for the bell curve that represents the fictional "incoming dyad" that you're trying to measure.
>
> I think you mean "started out" rather than "ends up" here?

Right, sorry.

> > ... If you instead give the incoming dyad a Vos curve, the minima end up being Vos curve-shaped.
>
> Do you mean for Erlich's H.E. or for your model, or both? I haven't tried this; I intend to.

Both. Paul has some graphs of this on the harmonic_entropy list.

> > On the other hand, what you just described above is exactly what I'm
> > doing with my model ...
>
> It sounds interesting; the shape of your curve around the minima looks wider than for H.E. ...

For some reason an s in the DC model syncs up with an slightly larger
s in HE. The rule of thumb I've been using is DC s = 4/5 * HE s. So if
you put s in the DC computation as being 1%, it ends up being most
similar to HE when HE's s is set to about 1.2%.

If you're interested in working out how the convolution relates to HE,
which is something I had started doing and given up, perhaps a good
place to start is working out the "differential entropy" equivalent of
the HE calculation:

http://en.wikipedia.org/wiki/Differential_entropy

If we could perhaps get that worked out, and come up with the exact
equation that results for some s and assuming sqrt(n*d) widths, then I
could probably rewrite the whole thing in terms of a convolution
integral, assuming that this doesn't just end up being the largest
coincidence in the world.

-Mike

http://x31eq.com/temper/net.html

Well, Mr. Breed's temperament finder is providing me with dozens of scales and hours of amusement. I feel like some tourist pulling the lever on a one-armed bandit, over and over.

I'm looking for all possible reasonably good 46-pitch tunings.

If you provide it with strange parameters, like three large temperaments and a 3-limit, it will sometimes spit out bizarre scales with notes not in ascending order.

Another thing that puzzles me, is that that I've come across different versions of scales such as Rodan with 46 pitches, but with a much lower fifth.

Many of the scales have a fifth (3/2-ish) way above 715 cents.

The 4/3 is often higher than 500 cents, although I don't know why.

Anyway, thanks to G. Breed, I'm happily wasting time and building up my scale library with a range of tunings with low to high fifths.

Wish I knew what relation the parameters I type in have to the output--it's still mysterious to me, at this point.

Caleb

>
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> The minima of the curve generally have a height that ends up being
> proportional to sqrt(n*d), yes?

I haven't noticed this or read about it - I'll look into it - is it in "the literature"?

> ...if you use something like a Farey series or a Tenney series, you end up getting the pattern where the intervals have a width proportional to complexity ... another perfectly acceptable way to enumerate the rationals, but I doubt the curve would be too useful. ... The fact that the HE curve ends up looking the way it does comes directly out of the way the widths are set up.

I completely agree that the choice of the Tenney series is crucial, and is an assumption - it was just that your wording did not capture that for me at least (and perhaps mine didn't either!).

> Check out some of the HE graphs. When s is smaller, the width of the
> valleys around the minima shrinks; when it's larger, it expands. A
> larger s = larger valleys.

OK, I see what you mean.

SM> Do you mean for Erlich's H.E. or for your model, or both? I haven't tried this; I intend to.
> Both. Paul has some graphs of this on the harmonic_entropy list.

Thank you.

> If you're interested in working out how the convolution relates to HE, which is something I had started doing and given up, perhaps a good place to start is working out the "differential entropy" equivalent of the HE calculation.

Not sure I'm up to it, but will take a look.

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> > The minima of the curve generally have a height that ends up being
> > proportional to sqrt(n*d), yes?
>
> I haven't noticed this or read about it - I'll look into it -
> is it in "the literature"?
>

Hi Steve,

I thought we'd discussed this -- remember the graphs showing
the relationship for dyads, and the one showing the areas of
the voronoi cells for triads? It's the basis for using Tenney
height instead of finding a true area on the plot.

-Carl

MikeB>"I'm also trying to mess around with giving the bell curves a standard
deviation that's a function of the interval's complexity, which means
that 11/8 will have a "narrower" field of attraction than 5/3. I think
this offers a number of advantages over the other curve and I'm
interested in seeing how it'll hold up in a listening test."

I have two main issues with Tenney Height (not to say that it's wrong, but that
I believe it could do better at handling the following situations):

1) That intervals like 3/2 are considered exponentially better ever than
ones like 5/3 (while they don't sound that much better in reality to me). Why
do simple intervals have such large dips in the dissonance curve even over only
slightly less simple one? It all seems so skewed... I'm wondering if you have
found using 1/n^2+1/d^2 (your idea) helps eliminate this extreme skewing toward
"very simple intervals"?

2) That it seems to completely throw out intervals like 16/11 as being
"unbearably complex" when I've found they are tolerable and represent fairly
deep "dips" in the dissonance curve, but have a far less wide field of
attraction (this may well be revolved by using your "curved standard deviation"
model based on interval complexity)

3) (and this is a tough one)...why do things like 22/15 sound significantly
more consonant than 16/11 even though 16/11has a far lower Tenney Height (same
goes for 9/8 vs. 18/11)? One thing that obviously appears missing is critical
band dissonance (quite important, it seems, for any fraction closer than 5/4 or
so, such as 9/8). But, to be honest, I don't have any clue how to explain the
22/15 vs. 16/11 example (the only guess I can think of numerically is that the
fact 22/15 is right smack between 3/2 and 10/7 on the exponential scale may have
something to do with it and 10/7, odd limit wise, is still "only" 7-limit).

Mike wrote:

> If you're interested in working out how the convolution relates
> to HE, which is something I had started doing and given up,
> perhaps a good place to start is working out the "differential
> entropy" equivalent of the HE calculation:
>
> http://en.wikipedia.org/wiki/Differential_entropy
>
> If we could perhaps get that worked out, and come up with the
> exact equation that results for some s and assuming sqrt(n*d)
> widths, then I could probably rewrite the whole thing in terms
> of a convolution integral,

Sounds fancy. But as I keep trying to say offlist, I think
all that's needed is discrete entropy, of the partition of the
convolved total at each point.

-Carl

Mike wrote:

> Carl suggested that the ideal spectral rolloff for the most
> "concordant" sound is 1/N^2, since that's the approximate
> spectrum of the human voice. So it makes sense that the filters
> in the brain would be tuned to something similar (or in
> actuality, tuned to the inverse of this, so that the
> crosscorrelation of the filter and the signal would be highest
> if the signal has a rolloff of 1/N^2).
>
> So if that were the case, then any dyad n/d would contribute
> 1/n^2 + 1/d^2 energy to that filter

We should talk about the filterbank as a whole, not individual
neurons. As a gross oversimplification, it consists of neurons
that respond to a specific pair of absolute frequencies. It's
thought the neurons learn these frequencies during neonatal
exposure to speech (whether they remain plastic into adulthood
and by how much is not known).

We would expect that the filterbank has better coverage for
the most salient parts of human speech, and that means the
low-numbered harmonics. Now, 1/N^2 is the rolloff at the
vocal folds, but the resonances of the vocal tract modify
this considerably. Purves' lab has determined this average
modification by analyzing a large corpus of recorded speech
in different languages. But just taking 1/N^2 for now, we
expect the response of the filterbank to a dyad of two sine
waves n/d to be proportional to the percentage of the total
energy of a vocal sound found in its two partials n & d.
If x is the energy of its fundamental then

inf
Sum x/(n^2) converges to 1.64493x
n=1

is its energy overall. So the percentage is

(x/n^2 + x/d^2) / 1.64493x

which does not depend on x and is a constant factor of

(d^2+n^2)/(d^2*n^2)

That's a long winded way of saying that, since the series
converges, Mike's 1/n^2 + 1/d^2 is right. I haven't followed
all the other variants he's given but perhaps I'll try to
track those down now.

-Carl

Hi Gene,

> Why is that more conceptually rigorous? It's certainly different,
> at any rate: 5/3 and 15/1 are the same in terms of n*d, but
> 1/5^2+1/3^2 is 34/225, whereas 1/15^2 + 1/1^2 is 226/225.
> That's 226 versus 34!

These rules of thumb are always subject to SPAN, the caps
indicating that it's a special term for the falloff of both
concordance *and* discordance at large interval sizes.
Let's try

Tenney Mike
2:1 0.50 1.25
4:3 0.08 0.17
5:3 0.06 0.15
7:4 0.04 0.08
11:4 0.02 0.07

They both seem reasonable, though I might prefer Tenney on
11:4 vs 7:4. And of course, the addition isn't desirable if
we're doing prime-based stuff for regular temperaments and
so forth, so Tenney has the edge there also.

One thing about the quasi-plausible story I just told about
the combination-sensitive neurons is that it's based on the
incoming dyad being made of sines. In real musical situations
where a dyad is played with two complex tones at roughly
equal volume... we get pairs, (n d), (2n 2d) etc that rolloff
with the square of the coefficients. But the brain may not
know or care they're paired, it just sees a bunch of partials
and neurons can respond to things like (2n d). Who knows what
rule of thumb that leads to... seems like there's still a
chance the 1/n^2 + 1/d^2 rule would hold...

-Carl

MikeB> The minima of the curve generally have a height that ends up being proportional to sqrt(n*d), yes?

SteveM> I haven't noticed this or read about it - I'll look into it - is it in "the literature"?

Carl> I thought we'd discussed this -- remember the graphs showing
> the relationship for dyads, and the one showing the areas of
> the voronoi cells for triads? It's the basis for using Tenney
> height instead of finding a true area on the plot.

OK, I know about using Tenney height instead of area (I doubt I would have done the Triadic calculation without it); I think Mike is saying that the heights of the resulting minima (dyadic) have a relationship to Tenney height. Or am I getting confused?

Steve M.

Steve wrote:

> > I thought we'd discussed this -- remember the graphs showing
> > the relationship for dyads, and the one showing the areas of
> > the voronoi cells for triads? It's the basis for using Tenney
> > height instead of finding a true area on the plot.
>
> OK, I know about using Tenney height instead of area (I doubt
> I would have done the Triadic calculation without it); I think
> Mike is saying that the heights of the resulting minima (dyadic)
> have a relationship to Tenney height. Or am I getting confused?

No, you got it. The latter is taken as evidence that the
former is correct. Actually, the dyadic minima come out
proportional to the Tenney height of their nearby rationals
even if n+d and a Mann series is used on the front end, and
even if the original mediant-mediant widths (in a Farey series)
are used. So Paul concluded all signs pointed to Tenney.
Make sense?

On a slightly related note, are you removing all JI chords
under the Tenney limit that aren't in lowest terms (e.g.
4:6:8 would not be included when seeding the space)?

-Carl

On Sat, Nov 6, 2010 at 3:30 PM, Carl Lumma <carl@...> wrote:
>
> We should talk about the filterbank as a whole, not individual
> neurons. As a gross oversimplification, it consists of neurons
> that respond to a specific pair of absolute frequencies. It's
> thought the neurons learn these frequencies during neonatal
> exposure to speech (whether they remain plastic into adulthood
> and by how much is not known).

I thought that 1/N^2 might be a good basic starting point, but if you
know any literature that gives more insight into the specific filter
characteristics, that would be a better one. I'm trying to use the
"harmonic sieve" approach, which is a term I've sometimes seen
reflected in the medical literature for when they use FFR's to look at
the frequency output.

> That's a long winded way of saying that, since the series
> converges, Mike's 1/n^2 + 1/d^2 is right. I haven't followed
> all the other variants he's given but perhaps I'll try to
> track those down now.

I'm not sure that series convergence is important though, here, since
these filters are ideally going to be damped, so they won't resonate
forever. This is what I haven't worked out yet.

There's also the notion that these filters might be impulse trains,
that have no rolloff at all (all harmonics, amplitude of 1), but that
they might be damped (damping in the time domain causes frequency
spreading in the frequency domain), and that the incoming signal being
spread out logarithmically to reflect some kind of error (a la HE)
might just cause the simple ones to be emphasized entirely because of
span. For now it seems sensible to start with 1/n^2.

What I really need is more reading on Goldstein's stuff - hopefully I
can find some reading that doesn't cost $25 a paper...

-Mike

On Sat, Nov 6, 2010 at 2:42 PM, Michael <djtrancendance@...> wrote:
>
> I have two main issues with Tenney Height (not to say that it's wrong, but that I believe it could do better at handling the following situations):
>
>      1) That intervals like 3/2 are considered exponentially better ever than ones like 5/3 (while they don't sound that much better in reality to me).  Why do simple intervals have such large dips in the dissonance curve even over only slightly less simple one?  It all seems so skewed...  I'm wondering if you have found using 1/n^2+1/d^2 (your idea) helps eliminate this extreme skewing toward "very simple intervals"?

It really depends, I guess, on what you want to model. If you can come
up with a better metric for consonance, I can model it. Try playing
with 1/n^2 + 1/d^2 and seeing how you like it.

>    3) (and this is a tough one)...why do things like 22/15 sound significantly more consonant than 16/11 even though 16/11has a far lower Tenney Height (same goes for 9/8 vs. 18/11)?  One thing that obviously appears  missing is critical band dissonance (quite important, it seems, for any fraction closer than 5/4 or so, such as 9/8).  But, to be honest, I don't have any clue how to explain the 22/15 vs. 16/11 example (the only guess I can think of numerically is that the fact 22/15 is right smack between 3/2 and 10/7 on the exponential scale may have something to do with it and 10/7, odd limit wise, is still "only" 7-limit).

I don't know. I'm not sure that I hear it the same way.

-Mike

On Sat, Nov 6, 2010 at 2:45 PM, Carl Lumma <carl@...> wrote:
>
>
> Sounds fancy. But as I keep trying to say offlist, I think
> all that's needed is discrete entropy, of the partition of the
> convolved total at each point.

I'm confused by this - what do you mean "the partition of the
convolved total?" If you're asking for a measure of how much each JI
interval contributes to a particular point in dyad space, the formula
for in the DC model is, taken from our conversation offlist

Let's define G_s(x) as a Gaussian of standard deviation s and with a
mean of 0. So the energy that any interval n/d would contribute to
some target dyad t is going to be (n*d)*G_s(t-cents(n/d)).

Are you asking me to figure out how much energy each JI interval
contributes to a dyad in HE?

-Mike

Mike B>"Try playing with 1/n^2 + 1/d^2 and seeing how you like it."
Now if I understand your formula correctly this is how it plays out...
1/4 + 1/1 = 1.25 (2/1)
1/9 + 1/4 =.3611 (3/2)
1/16 + 1/9 = .1736 (4/3)
1/25 + 1/16 = .1025 (5/4)

Still seems to say 4/3 is around half as consonant as 3/2...still very steep
for my tastes. And the 2/1 being over 3 times as consonant as 3/2 and around 6
times as consonant as 4/3 seems borderline insane vs. what I hear from those
dyads.

>"If you can come up with a better metric for consonance, I can model it. "
Well...better for my ears is all I can guarantee, but here's a shot....
dissonance = cubed root(n) + sqrt(d)
<the idea is the denominator's being higher counts more than the
numerator's being higher>

2/1 = 1.25992 + 1 = 2.25992
3/2 = 1.4422 + 1.4142 = 2.85641
4/3 = 1.5874 + 1.732 = 3.3194
5/4 = 1.7099 + 2 = 3.7099
11/9 = 2.224 + 3 = 5.224
11/6 = 2.224 + 2.449 = 4.673 (a bit high, should ideally be only a bit over
5/4's dissonance since there's so little critical band dissonance for such a
wide interval)
18/11 = 2.6207 + 3.3166 = 5.937 (a bit high, should ideally be a bit lower than
11/9's dissonance for the same critical band dissonance reason)

Note that, ultimately, I believe there should be some factor of critical band
dissonance bonus added for anything over 8/7 up to 5/4 (IE lower the 18/11's
dissonance rating by about 1.0)...but this formula at least seems "in the
ballpark" to me.

On Sat, Nov 6, 2010 at 8:29 PM, Michael <djtrancendance@...> wrote:
>
> Mike B>"Try playing with 1/n^2 + 1/d^2 and seeing how you like it."
>   Now if I understand your formula correctly this is how it plays out...
> 1/4 + 1/1 = 1.25 (2/1)
> 1/9 + 1/4 =.3611 (3/2)
> 1/16 + 1/9 = .1736 (4/3)
> 1/25 + 1/16 =  .1025 (5/4)
>
>    Still seems to say 4/3 is around half as consonant as 3/2...still very steep for my tastes.  And the 2/1 being over 3 times as consonant as 3/2 and around 6 times as consonant as 4/3 seems borderline insane vs. what I hear from those dyads.

Try 1/n + 1/d. Or something in between. The point of that metric is to
measure the strength of the virtual pitch that's produced. Perhaps an
exponent of 2 is too steep. The consonance of anything/1 is going to
be way higher than everything else, since the 1 is right there in the
chord.

Also try log(1/n^2 + 1/d^2). The idea is that we generally hear things
logarithmically, so if a dyad produces 1/4 the energy of another dyad,
what that means is that it's 6 dB lower.

I think sometimes what you're listening for when you say a dyad is
"consonant" is the clarity of the periodicity beating that it
produces, rather than the strength of the virtual pitch.

>    Note that, ultimately, I believe there should be some factor of critical band dissonance bonus added for anything over 8/7 up to 5/4 (IE lower the 18/11's dissonance rating by about 1.0)...but this formula at least seems "in the ballpark" to me.

OK, I'll spit out a DC model with this in mind. We'll see how it goes.

The model isn't really refined enough yet, though, to place much
weight on the relative amplitude of each one.

-Mike

On Sat, Nov 6, 2010 at 8:08 PM, Mike Battaglia <battaglia01@...> wrote:
>
> Let's define G_s(x) as a Gaussian of standard deviation s and with a
> mean of 0. So the energy that any interval n/d would contribute to
> some target dyad t is going to be (n*d)*G_s(t-cents(n/d)).

Sorry, that first term should be 1/(n*d). In general, it's
1/complexity * G_s(t-cents(n/d)).

-Mike

Mike wrote:

> I thought that 1/N^2 might be a good basic starting point, but
> if you know any literature that gives more insight into the
> specific filter characteristics, that would be a better one.

We discussed the Purves papers on several occasions...

> > That's a long winded way of saying that, since the series
> > converges, Mike's 1/n^2 + 1/d^2 is right. I haven't followed
> > all the other variants he's given but perhaps I'll try to
> > track those down now.
>
> I'm not sure that series convergence is important though,
> here, since these filters are ideally going to be damped,
> so they won't resonate forever. This is what I haven't
> worked out yet.

It's important because it means your expression represents
the right thing (the percentage).

> There's also the notion that these filters

You say filters plural... not sure what you mean.
What are these filters and how do they work?

> What I really need is more reading on Goldstein's stuff -
> hopefully I can find some reading that doesn't cost
> $25 a paper...

Oh, lame. I'll put them in my folder:

/tuning/files/Goldstein,Houtsma-CentralOriginPitch.pdf

/tuning/files/Goldstein-OptimumProcessor.pdf

I'm not sure they'll be of any use in your current efforts
but I did think you might have some particularly good insights
on them. The latter is one of the most-cited papers in
psychoacoustics. One interesting criticism is that the data
were gathered mostly on one person (Houtsma), and briefly on
a few others (including Goldstein himself). Another is that
he seems to have a 'just so' answer for everything... That
said I haven't studied either paper deeply.

-Carl

-- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Are you asking me to figure out how much energy each JI interval
> contributes to a dyad in HE?

No, in DC. -Carl

I wrote:

> I'll put them in my folder:

Sorry, darn URLs...

/tuning/files/CarlLumma/Goldstein,Houtsma-CentralOriginPitch.pdf

/tuning/files/CarlLumma/Goldstein-OptimumProcessor.pdf

-Carl

MikeB>"Try 1/n + 1/d. Also try log(1/n^2 + 1/d^2)."

All off these appear to have similar problems to my ears...they seem to say
the dyads that are very simple going to fairly simple have a much higher level
of dissonance going down (IE 2/1 is many times as consonant as 3/2 is many times
as consonant as 4/3....)

>"I think sometimes what you're listening for when you say a dyad is "consonant"
>is the clarity of the periodicity beating that it produces, rather than the
>strength of the virtual pitch."

I am simply listening for tension and admittedly don't care much how it is
produced (IE critical band, periodicity, degree of reference toward a virtual
pitch, or otherwise) any more than consideration of such ideas helps me align
the values. My method of looking at it is similar to Plomp and Llevelt's (where
the whole theory was based indirectly on listening tests): to align the math to
near-exactly match listening tests rather than start with the math and see if
the listening tests confirm it to any significant degree.

Me> "Note that, ultimately, I believe there should be some factor of critical
band dissonance bonus added for anything over 8/7 up to 5/4 (IE lower the
18/11's dissonance rating by about 1.0)...but this formula at least seems "in
the ballpark" to me."

MikeB>"OK, I'll spit out a DC model with this in mind. We'll see how it goes."

Thank you. I'm hoping it goes a good ways toward solving the issue with
fractions with higher numerator but that are well spread out (like 10/7, 11/6,
15/8, 9/7...) often getting low consonance ratings despite being relatively wide
(and IMVHO not very tense) intervals.

On Sat, Nov 6, 2010 at 9:08 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"Try 1/n + 1/d.  Also try log(1/n^2 + 1/d^2)."
>
>   All off these appear to have similar problems to my ears...they seem to say the dyads that are very simple going to fairly simple have a much higher level of dissonance going down (IE 2/1 is many times as consonant as 3/2 is many times as consonant as 4/3....)

Not the log one...

> >"I think sometimes what you're listening for when you say a dyad is "consonant" is the clarity of the periodicity beating that it produces, rather than the strength of the virtual pitch."
>
>    I am simply listening for tension and admittedly don't care much how it is produced (IE critical band, periodicity, degree of reference toward a virtual pitch, or otherwise) any more than consideration of such ideas helps me align the values.  My method of looking at it is similar to Plomp and Llevelt's (where the whole theory was based indirectly on listening tests): to align the math to near-exactly match listening tests rather than start with the math and see if the listening tests confirm it to any significant degree.

OK, but I'm only looking at virtual pitch for the purposes of this
model. This is only supposed to measure virtual pitch. I've just seen
a medical study on how harmonic periodicity is generally the most
universal factor playing into "consonance" across cultures and stuff.
But if you want to come up with some kind of weighted average, go for
it.

> Me>   "Note that, ultimately, I believe there should be some factor of critical band dissonance bonus added for anything over 8/7 up to 5/4 (IE lower the 18/11's dissonance rating by about 1.0)...but this formula at least seems "in the ballpark" to me."
>
> MikeB>"OK, I'll spit out a DC model with this in mind. We'll see how it goes."
>
>   Thank you.  I'm hoping it goes a good ways toward solving the issue with fractions with higher numerator but that are well spread out (like 10/7, 11/6, 15/8, 9/7...) often getting low consonance ratings despite being relatively wide (and IMVHO not very tense) intervals.

But you're talking about something totally different now. A tritone is
pretty wide, but also very tense. Do you want me to model critical
band dissonance?

-Mike

On Sat, Nov 6, 2010 at 8:59 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > I thought that 1/N^2 might be a good basic starting point, but
> > if you know any literature that gives more insight into the
> > specific filter characteristics, that would be a better one.
>
> We discussed the Purves papers on several occasions...

Were those the ones that discussed combination-sensitive neurons, but
that contained only multiples of 2 and 3? I wasn't sure how those
would apply, or how they'd be aggregated into larger filters that deal
with things like 5.

> > > That's a long winded way of saying that, since the series
> > > converges, Mike's 1/n^2 + 1/d^2 is right. I haven't followed
> > > all the other variants he's given but perhaps I'll try to
> > > track those down now.
> >
> > I'm not sure that series convergence is important though,
> > here, since these filters are ideally going to be damped,
> > so they won't resonate forever. This is what I haven't
> > worked out yet.
>
> It's important because it means your expression represents
> the right thing (the percentage).

Yes, but we're not going to actually filter the signal from -Inf to
Inf. Put another way, a 440 Hz sine wave, if you take the forward
Fourier Transform of it, does not converge to a finite amount over
infinity - you end up getting an infinite spike at 440 and 0
everywhere else (a delta function).

Wait, I think I understand what you're saying now. But that would mean
it's impossible to create a filterbank that has a rolloff of 1/N?
Maybe I don't get it...

But one thing I thought about recently is that these can't actually BE
filters, but I guess waveforms that are constantly being
crosscorrelated with the input signal. After all, if you send in a
waveform with a 1/N^2 rolloff, and then filter it with a filter that
also has a 1/N^2 filter, then the resultant signal is going to have a
1/N^4 rolloff, as the already-soft higher harmonics will become even
softer as they're filtered by the soft filter harmonics. We get the
opposite effect. Crosscorrelation is going to give us what we want, I
think.

So perhaps we should think of it that way from now on. This is a model
that assumes that rather than the brain doing some ingenious
autocorrelation on the signal, that it's constantly crosscorrelating
it with this "filterbank" of neurons. Assuming that perspective,
everything you said applies, although I still don't understand the
convergence thing 100%...

> > There's also the notion that these filters
>
> You say filters plural... not sure what you mean.
> What are these filters and how do they work?

I just mean the individual filters in the filterbank.

> I'm not sure they'll be of any use in your current efforts
> but I did think you might have some particularly good insights
> on them. The latter is one of the most-cited papers in
> psychoacoustics. One interesting criticism is that the data
> were gathered mostly on one person (Houtsma), and briefly on
> a few others (including Goldstein himself). Another is that
> he seems to have a 'just so' answer for everything... That
> said I haven't studied either paper deeply.

Thanks man, I'll check these out.

-Mike

Mike wrote:

> > We discussed the Purves papers on several occasions...
>
> Were those the ones that discussed combination-sensitive
> neurons,

Nope, Purves' group doesn't do cell biology

http://www.purveslab.net

> but that contained only multiples of 2 and 3? I wasn't sure
> how those would apply, or how they'd be aggregated into larger
> filters that deal with things like 5.

The only way I know to measure these responses is with in vivo
single-cell recording. Not a technique you can normally use on
humans, and even in animals you can only record from a few dozen
cells before the animal perishes. So the studies done to date
have together sampled only hundreds of neurons, compared to many
thousands in each animal. So we may just not have seen the
higher harmonics if relatively few neurons respond to them.

Kadia & Wang is the best study for our purposes (I've mentioned
it here before) and the first one that establishes without a
doubt the existence of simple frequency ratio detectors in the
primate (marmoset) brain. Unfortunately, like most papers,
their data are hard to read off the tiny and poorly-labeled
figures, but they did find 5:1 and 5:2. Also relevant to the
present discussion, they found that when the higher tone was
quieter than the lower, its presence increased the neuron's
firing rate, but when it was louder it inhibited it!

> > It's important because it means your expression represents
> > the right thing (the percentage).
>
> Wait, I think I understand what you're saying now. But that
> would mean it's impossible to create a filterbank that has a
> rolloff of 1/N? Maybe I don't get it...

You have to look at what I originally wrote:

"we expect the response of the filterbank to a dyad of two
sine waves n/d to be proportional to the **percentage** of
the total energy of a vocal sound found in its two
partials n & d."

Obviously there is finite energy in any human vocalization,
because N does not go to infinity. But in our abstract
representation, it does (or you have to pick an arbitrary
cutoff). So yes, to get a percentage out of 1/N I think you
have to choose a cutoff for N. No?

> But one thing I thought about recently is that these can't
> actually BE filters, but I guess waveforms that are constantly
> being crosscorrelated with the input signal. After all, if
> you send in a waveform with a 1/N^2 rolloff, and then filter
> it with a filter that also has a 1/N^2 filter, then the
> resultant signal is going to have a 1/N^4 rolloff,

You know way more about filters than me, but I thought you
said earlier the filter would have the inverse response.
That's seemingly backed up by one neuron's plot n Kadia & Wang.
It had CFs an octave apart. With the lower tone at 70dB, the
maximum facilitation occurred when the upper tone was at 30dB.
It declined sharply as the lower tone was made louder, and the
neuron essentially shut off when both tones were 70dB.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
[using Tenney height instead of area]
[heights of the resulting minima have a relationship to Tenney height]
> ... The latter is taken as evidence that the
> former is correct.

Ah, I had not made that connection before - and I'll need to think about it to convince myself.

> Actually, the dyadic minima come out
> proportional to the Tenney height of their nearby rationals
> even if n+d and a Mann series is used on the front end, and
> even if the original mediant-mediant widths (in a Farey series)
> are used. So Paul concluded all signs pointed to Tenney.
> Make sense?

Do you (and Paul and Mike B) mean for example that the HE height/result for 2:3 is sqrt(2*3)/sqrt(3*4) times the height for 3:4? I have not done any measurements but am surprised if so. One of my graphs seems to show that, with increasing N (=Tenney limit) not only does the "background level" of HE increase, but the depth of the minimum in relation to the "background level" seems to DEcrease - so we couldn't maintain a constant ratio with increasing N?

>
> On a slightly related note, are you removing all JI chords
> under the Tenney limit that aren't in lowest terms (e.g.
> 4:6:8 would not be included when seeding the space)?

Yes I am removing them; I hope that is what you expected. Although I seem to remember your mentioning that leaving them in (at least in the dyadic case) could give us something (was it greater discrimination between values, that Igs was wanting?)

Steve M.

Hi Steve,

> Do you (and Paul and Mike B) mean for example that the HE
> height/result for 2:3 is sqrt(2*3)/sqrt(3*4) times the height
> for 3:4? I have not done any measurements but am surprised
> if so. One of my graphs seems to show that, with increasing
> N (=Tenney limit) not only does the "background level" of HE
> increase, but the depth of the minimum in relation to the
> "background level" seems to DEcrease - so we couldn't maintain
> a constant ratio with increasing N?

Sorry, I always gloss over this. The mediant-to-mediant widths
are proportional to 1/sqrt(n*d) for dyads, and the voronoi cell
areas to 1/geomean(a*b*c) for triads. Exp(entropy) is supposed
to be proportional to n*d for the dyadic minima, but it doesn't
look like that's true now that I try it...

> > On a slightly related note, are you removing all JI chords
> > under the Tenney limit that aren't in lowest terms (e.g.
> > 4:6:8 would not be included when seeding the space)?
>
> Yes I am removing them; I hope that is what you expected.
> Although I seem to remember your mentioning that leaving them
> in (at least in the dyadic case) could give us something (was
> it greater discrimination between values, that Igs was wanting?)

Paul tried dyadic entropy with unreduced ratios included and
got something that looked like exp(entropy), which Igs liked.
See here:
/harmonic_entropy/topicId_267.html#271

-Carl

I wrote:

> You know way more about filters than me, but I thought you
> said earlier the filter would have the inverse response.
> That's seemingly backed up by one neuron's plot n Kadia & Wang.
> It had CFs an octave apart. With the lower tone at 70dB, the
> maximum facilitation occurred when the upper tone was at 30dB.
> It declined sharply as the lower tone was made louder, and the
> neuron essentially shut off when both tones were 70dB.

D'oh! that should be "declined sharply as the _upper_ tone was
made louder..." -C.

It would certainly go a ways in explaining why sawtooth waves
are not really concordant. -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I wrote:
>
> > You know way more about filters than me, but I thought you
> > said earlier the filter would have the inverse response.
> > That's seemingly backed up by one neuron's plot n Kadia & Wang.
> > It had CFs an octave apart. With the lower tone at 70dB, the
> > maximum facilitation occurred when the upper tone was at 30dB.
> > It declined sharply as the lower tone was made louder, and the
> > neuron essentially shut off when both tones were 70dB.
>
> D'oh! that should be "declined sharply as the _upper_ tone was
> made louder..." -C.
>

On Thu, Nov 18, 2010 at 2:56 PM, Carl Lumma <carl@...> wrote:
>
> I wrote:
>
> > You know way more about filters than me, but I thought you
> > said earlier the filter would have the inverse response.
> > That's seemingly backed up by one neuron's plot n Kadia & Wang.
> > It had CFs an octave apart. With the lower tone at 70dB, the
> > maximum facilitation occurred when the upper tone was at 30dB.
> > It declined sharply as the lower tone was made louder, and the
> > neuron essentially shut off when both tones were 70dB.
>
> D'oh! that should be "declined sharply as the _upper_ tone was
> made louder..." -C.

Awesome. So that lends some credence to this approach then. So what
this model is conceptually equivalent to saying, then, is:

There has been a lot of speculation that some process in the brain is
performing something similar to an autocorrelation on the incoming
signal to find periodicity information. This approach diverges from
that by postulating that the brain is performing a huge set of
cross-correlations for every possible fundamental frequency, and
whichever ones match the signal end up "lighting up" that part of the
brain.

If the brain is, instead, performing some kind of autocorrelation on
the signal, and this model is actually TOO robust for what the brain
is doing, then there will be several "auditory illusions" that exist
that this model doesn't predict.

This is I think related to the question of whether or not these
neuronal "filters" reflect individual parallel pitch processors
(PPPs), or if there still exists a Goldstein-esque central pitch
processor (CPP), and these get "activated" based on what the CPP
decides. So either these are individual filters, or they reflect some
kind of "memory" in the brain where the CPP would store the results of
its analysis.

-Mike

On Thu, Nov 18, 2010 at 3:06 PM, Carl Lumma <carl@...> wrote:
>
> It would certainly go a ways in explaining why sawtooth waves
> are not really concordant. -Carl

In what sense do you not hear them as concordant? Do you hear them as
discordant beyond the obnoxious "buzziness" of most aliased digital
sawtooths these days?

Another thought, for future listening tests: perhaps the most
perfectly concordant sound is one in which the fundamental is most
focused, but the overtones themselves generate no independent pitch of
their own.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Nov 18, 2010 at 3:06 PM, Carl Lumma <carl@...> wrote:
> >
> > It would certainly go a ways in explaining why sawtooth waves
> > are not really concordant. -Carl
>
> In what sense do you not hear them as concordant?

I think it's generally agreed they are harsher than other
common waveforms used in subtractive synthesis. With chords,
I find wider voicings must be used.

> Do you hear them as discordant beyond the obnoxious
> "buzziness" of most aliased digital sawtooths these days?

I can't say I've noticed this problem. Any instrument-
culprits come to mind?

> Another thought, for future listening tests: perhaps the most
> perfectly concordant sound is one in which the fundamental
> is most focused, but the overtones themselves generate no
> independent pitch of their own.

Not sure what you mean... -Carl

On Tue, Nov 23, 2010 at 3:26 PM, Carl Lumma <carl@...> wrote:
>
> > Do you hear them as discordant beyond the obnoxious
> > "buzziness" of most aliased digital sawtooths these days?
>
> I can't say I've noticed this problem. Any instrument-
> culprits come to mind?

I think my Nord Lead 2 had this problem, but basically: in the digital
realm, if you just "draw" a sawtooth wave in the time domain, all of
the harmonics above the Nyquist frequency will "reflect" about it
(alias) and come back down into audible range. You can think of it
also as jitter, or as an artifact of the fact that the sawtooth can
never "jump" back down between samples, so to make it jump back at the
next nearest sample introduces a rounding error into the mix. The
spectrum ends up looking like this:

http://www.music.mcgill.ca/~gary/307/week5/img12.gif

This is a more extreme example than we're used to, but the best I
could find. Here's an example of aliased vs anti-aliased sawtooths for
a sample rate of 22050 Hz:
http://en.wikipedia.org/wiki/File:Sawtooth-aliasingdemo.ogg

Since most DAC's these days are operating at 44100 Hz instead, the
aliasing will be half as severe as that (you'll never hear 1760 Hz
aliasing as bad at a 44100 Hz sample rate).

> > Another thought, for future listening tests: perhaps the most
> > perfectly concordant sound is one in which the fundamental
> > is most focused, but the overtones themselves generate no
> > independent pitch of their own.
>
> Not sure what you mean... -Carl

I mean that if you listen to the sound of an impulse train, which is a
differentiated sawtooth and hence has no rolloff at all, you can pick
out individual harmonics in the sound. In general, for most sounds,
you can pick out the individual harmonics. So perhaps a good way to
define the "most concordant" sound is to define it as one in which the
sound is so fused that you can't pick out those individual harmonics
anymore (i.e. the rolloff is less than 1/N), but the fundamental is
still "clearer" than that of a sine wave, in which you also can't pick
out individual harmonics because there are none.

-Mike

Mike wrote:
>
> I think my Nord Lead 2 had this problem, but basically: in
> the digital realm, if you just "draw" a sawtooth wave in the
> time domain, all of the harmonics above the Nyquist frequency

Yes, I understand aliasing. I didn't think anybody was
making instruments these days without proper filters.
Nord in particular would be the last I'd expect. I've only
ever played the first Nord Lead and I don't think I ever
tried a plain sawtooth.

> Here's an example of aliased vs anti-aliased sawtooths for
> a sample rate of 22050 Hz:
> http://en.wikipedia.org/wiki/File:Sawtooth-aliasingdemo.ogg
>
> Since most DAC's these days are operating at 44100 Hz instead,

I bet the DAC in your Nord is 96K. Hm, if you have the 2X it is

http://www.clavia.se/main.asp?tm=Products&clpm=Nord_Lead_2X

> sound is so fused that you can't pick out those individual
> harmonics anymore (i.e. the rolloff is less than 1/N), but the
> fundamental is still "clearer" than that of a sine wave, in
> which you also can't pick out individual harmonics because
> there are none.

I wonder what roughness-based models give for a bare sawtooth
vs. a bare triangle...

http://eceserv0.ece.wisc.edu/~sethares/comprog.html
http://musicalgorithms.ewu.edu/algorithms/Roughness.html

-Carl