Back to buzz

On the last episode, Carl suggested a number of listening examples to
test my model, which is that periodicity buzz originates in the
cochlea. He also proposed an alternative model in which periodicity
buzz actually originates in the brain, and stems from how "spiky" the
time-domain waveform looks, which also proposes that the brain sums
together the outputs from each auditory filter and to see the original
time-domain waveform again. Here are the last two messages,
synthesized into one reply:

On Tue, Apr 26, 2011 at 8:12 PM, Carl Lumma <carl@...> wrote:
>
> > Now, if you're trying to make the argument that the brain adds
> > up the time domain waveform and sees the original again,
> > and then gets buzz from it
>
> Yep, that's what I proposed (you clipped it).
> [snip]
> > Anyway, assuming you understand that,
>
> It looked interesting, but I'm afraid to say you lost me.

My thoughts are that this whole thing makes a few assumptions that I
don't think are worth making:

1) Being able to filter a signal into different parts and then sum
them to get the original signal implies that the filters are linear.
Or, if they're not linear, they'd have to be nonlinear in such a
fashion that all of the nonlinear components of the signal cancel upon
the summing to get the original signal. That would be quite a feat.

2) Hysteresis (http://en.wikipedia.org/wiki/Hysteresis) would make
this even more of a feat. Your model would assume that the phase
response of each auditory filter is magically tweaked so that perfect
reconstruction of the waveform is possible just by summing. Or, if
not, that some kind of allpass filter with the inverse of the phase
response of the ear exists in the brain. This would be even more of a
feat.

As for what I lost you on, the point is that when you say that spikier
waveforms = more buzz, what you're saying is phase coherence = more
buzz. Phase coherence and spiky waveforms are the same thing - one
doesn't "cause" the other. Words like "time-domain" and
"frequency-domain" apply to the perspectives that we take to analyze a
system's behavior, not the behavior of the system itself.

> > This still doesn't propose a mechanism for why spikier
> > waveforms would produce more buzz than non-spikier waveforms.
> > Filterbanks propose such a mechanism.
> (from another msg)
> > That still wouldn't explain why there's buzz, no.
>
> Why not? And if the output of the filterbank is
> resynthesized, how does the filterbank supply such a
> mechanism?

Because all you've noticed are that spikier waveforms tend to exhibit
buzz, but you haven't explained why. Why should spikier waveforms
exhibit a perceptual "buzz"-like quality at all? What process in the
brain should cause some kind of perceptual buzzing over that? And
while we're at it, you don't seem to think that impulse trains exhibit
buzz, but they're the spikiest waveforms of them all. Why is this?

The filterbank supplies such a mechanism because it is a
time-frequency representation of the signal, and as such is subject to
the Fourier uncertainty principle. Because it's subject to the Fourier
uncertainty principle, it can only have perfect frequency resolution
if it has zero time resolution. When you combine this with a "spiky
waveform," the result is
1) the waveform will get caught weakly in the filters whose frequency
response is intermediate to that of the constituent tones,
2) the output from those filter will have an amplitude envelope,
3) and the AM envelopes generated from multiple tones will be in sync
with one another if the waveform is spikier, which is another way of
saying they'll be in sync with one another if the phase response of
the signal is linear, because this is what spikiness is.

It also explains why impulse trains should exhibit a different kind of
buzz than normal chords, but I've already explained that.

From the other thread:
> Right, so can you confirm those are sine tones? And, the
> example I mentioned was 1/1s up to 750 Hz.

Yes, those are sines. I can't generate any more examples now as I'm
about to drive cross country tomorrow. I will say that I hear, at 750
Hz, no buzzing at all for 9/5, lots of buzzing at 16/15, and a whole
spectrum of intermediate buzz stages in between. At 7/5 I don't really
hear any buzzing at all, and if I do it's on the minimum bare
threshold of detectability. If the volume is high, I guess maybe I
hear something, but it seems to have more to do with the appearance of
combination tones than anything else.

-Mike

Mike,

I haven't really followed closely the discussion about periodicity
buzz, sorry about that. I always thought that periodicity buzz means
that sound of well-tuned otonalities that you hear when the virtual
pitch starts to approach infrapitch region. I also think that that
characteristic sound is most evident when the constituent tones of
the otonality are rather high while there is a strong but low virtual
pitch. So higher limit otonalities would do the trick. Low frequency
sawtooths or impulse trains don't quite sound the same.

Kalle

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On the last episode, Carl suggested a number of listening examples to
> test my model, which is that periodicity buzz originates in the
> cochlea. He also proposed an alternative model in which periodicity
> buzz actually originates in the brain, and stems from how "spiky" the
> time-domain waveform looks, which also proposes that the brain sums
> together the outputs from each auditory filter and to see the original
> time-domain waveform again. Here are the last two messages,
> synthesized into one reply:
>
> On Tue, Apr 26, 2011 at 8:12 PM, Carl Lumma <carl@...> wrote:
> >
> > > Now, if you're trying to make the argument that the brain adds
> > > up the time domain waveform and sees the original again,
> > > and then gets buzz from it
> >
> > Yep, that's what I proposed (you clipped it).
> > [snip]
> > > Anyway, assuming you understand that,
> >
> > It looked interesting, but I'm afraid to say you lost me.
>
> My thoughts are that this whole thing makes a few assumptions that I
> don't think are worth making:
>
> 1) Being able to filter a signal into different parts and then sum
> them to get the original signal implies that the filters are linear.
> Or, if they're not linear, they'd have to be nonlinear in such a
> fashion that all of the nonlinear components of the signal cancel upon
> the summing to get the original signal. That would be quite a feat.
>
> 2) Hysteresis (http://en.wikipedia.org/wiki/Hysteresis) would make
> this even more of a feat. Your model would assume that the phase
> response of each auditory filter is magically tweaked so that perfect
> reconstruction of the waveform is possible just by summing. Or, if
> not, that some kind of allpass filter with the inverse of the phase
> response of the ear exists in the brain. This would be even more of a
> feat.
>
> As for what I lost you on, the point is that when you say that spikier
> waveforms = more buzz, what you're saying is phase coherence = more
> buzz. Phase coherence and spiky waveforms are the same thing - one
> doesn't "cause" the other. Words like "time-domain" and
> "frequency-domain" apply to the perspectives that we take to analyze a
> system's behavior, not the behavior of the system itself.
>
> > > This still doesn't propose a mechanism for why spikier
> > > waveforms would produce more buzz than non-spikier waveforms.
> > > Filterbanks propose such a mechanism.
> > (from another msg)
> > > That still wouldn't explain why there's buzz, no.
> >
> > Why not? And if the output of the filterbank is
> > resynthesized, how does the filterbank supply such a
> > mechanism?
>
> Because all you've noticed are that spikier waveforms tend to exhibit
> buzz, but you haven't explained why. Why should spikier waveforms
> exhibit a perceptual "buzz"-like quality at all? What process in the
> brain should cause some kind of perceptual buzzing over that? And
> while we're at it, you don't seem to think that impulse trains exhibit
> buzz, but they're the spikiest waveforms of them all. Why is this?
>
> The filterbank supplies such a mechanism because it is a
> time-frequency representation of the signal, and as such is subject to
> the Fourier uncertainty principle. Because it's subject to the Fourier
> uncertainty principle, it can only have perfect frequency resolution
> if it has zero time resolution. When you combine this with a "spiky
> waveform," the result is
> 1) the waveform will get caught weakly in the filters whose frequency
> response is intermediate to that of the constituent tones,
> 2) the output from those filter will have an amplitude envelope,
> 3) and the AM envelopes generated from multiple tones will be in sync
> with one another if the waveform is spikier, which is another way of
> saying they'll be in sync with one another if the phase response of
> the signal is linear, because this is what spikiness is.
>
> It also explains why impulse trains should exhibit a different kind of
> buzz than normal chords, but I've already explained that.
>
> From the other thread:
> > Right, so can you confirm those are sine tones? And, the
> > example I mentioned was 1/1s up to 750 Hz.
>
> Yes, those are sines. I can't generate any more examples now as I'm
> about to drive cross country tomorrow. I will say that I hear, at 750
> Hz, no buzzing at all for 9/5, lots of buzzing at 16/15, and a whole
> spectrum of intermediate buzz stages in between. At 7/5 I don't really
> hear any buzzing at all, and if I do it's on the minimum bare
> threshold of detectability. If the volume is high, I guess maybe I
> hear something, but it seems to have more to do with the appearance of
> combination tones than anything else.
>
> -Mike
>

What about all of the linear evenness stuff? You seemed to agree with it here

/tuning/topicId_95699.html#95708

It has been shown that periodicity buzz generally correlates more with
this linear spacing property than with actual periodicity. See here

http://www.mikebattagliamusic.com/music/buzztempering.wav

The first two to me sound like they have even buzz, and the last one
sounds like it has chorusy, warbly buzz. The first one is 5:7:9, the
second one is a 5:7:9 stretched so that the frequency difference
between the two is even, and the last is 1/1 17/12 20/11, which is
actually periodic but produces buzz that's less clear. Also check the
examples here

/tuning/topicId_95699.html#95699

All of the "mystery" examples, as well as the "alt" examples, are
linearly stretched and not actually periodic. If you alter the linear
spacing as Michael tried to do by actually rationalizing it, but in
such a way that for a:b:c b-a != k*(c-b), you'll get this warbly buzz.
The buzz is also phase-sensitive and for any chord, the buzz can be
attenuated by screwing with the phase of the notes. Examples are
provided of this as well. Gammatone filter plots are also provided and
predict this behavior, which means there's a good bet that it's
happening in the cochlea.

Assuming the hypothesis of cochlear origin is correct, then the reason
impulse trains sound different is that they contain partials of equal
volume up till infinity. So while periodicity buzz in this model is
the sound of AM between the constituent tones, for an impulse train
you reach the point where the higher partials get so close together
that they just sound completely unpitched, and you get a periodic,
recurring, high-passed "thwack" sound on top of the buzz. So if the
perceptual quality of buzz is that the notes have some AM on them (or
the timbral quality of the fused tone fluctuates a bit, which would be
the same thing), then an impulse train will contain this plus an
unpitched recurring thwack sound on top of it, which is not what you'd
normally call buzz.

-Mike

On Thu, May 5, 2011 at 9:27 AM, Kalle Aho <kalleaho@...> wrote:
>
> Mike,
>
> I haven't really followed closely the discussion about periodicity
> buzz, sorry about that. I always thought that periodicity buzz means
> that sound of well-tuned otonalities that you hear when the virtual
> pitch starts to approach infrapitch region. I also think that that
> characteristic sound is most evident when the constituent tones of
> the otonality are rather high while there is a strong but low virtual
> pitch. So higher limit otonalities would do the trick. Low frequency
> sawtooths or impulse trains don't quite sound the same.
>
> Kalle

--- Mike Battaglia <battaglia01@...> wrote:

> My thoughts are that this whole thing makes a few assumptions
> that I don't think are worth making:
> 1) Being able to filter a signal into different parts and then
> sum them to get the original signal implies that the filters are
> linear. Or, if they're not linear, they'd have to be nonlinear
> in such a fashion that all of the nonlinear components of the
> signal cancel upon the summing to get the original signal. That
> would be quite a feat.

The cochlea is essentially a compressor, and compression
is nonlinear. But obviously this nonlinearity is highly
desirable. The entire apparatus is nanoengineered to do
exactly what you say: slice and dice sound and put it back
together so that qualities of interest are retained. It is,
in fact, one of the more impressive feats in nature.

> 2) Hysteresis would make this even more of a feat. Your model
> would assume that the phase response of each auditory filter
> is magically tweaked so that perfect reconstruction of the
> waveform is possible just by summing. Or, if not, that some
> kind of allpass filter with the inverse of the phase response
> of the ear exists in the brain. This would be even more of a
> feat.

The relevant test is whether setting phases for maximum or
minimum spikiness in the time-domain is audible. I believe
you did a few examples that suggest it is.

> What process in the brain should cause some kind of perceptual
> buzzing over that?

I'm not sure.

> And while we're at it, you don't seem to think that impulse
> trains exhibit buzz, but they're the spikiest waveforms of
> them all. Why is this?

The phenomenon of periodicity buzz seems to involve a pitched
sound that buzzes. Impulse trains buzz, but they are not
pitched sounds that buzz.

> The filterbank supplies such a mechanism because it is ...
> subject to the Fourier uncertainty principle, it can only have
> perfect frequency resolution if it has zero time resolution.
> When you combine this with a "spiky waveform," the result is
>
> 1) the waveform will get caught weakly in the filters whose
> frequency response is intermediate to that of the constituent
> tones,
> 2) the output from those filter will have an amplitude envelope,
> 3) and the AM envelopes generated from multiple tones will be
> in sync with one another if the waveform is spikier, which is
> another way of saying they'll be in sync with one another if
> the phase response of the signal is linear, because this is what
> spikiness is.

I don't understand 1). Regarding 2), all the filters have
amplitude envelopes. Point 3) seems to be what I'm saying,
and seems independent of points 1) & 2).

> > Right, so can you confirm those are sine tones? And, the
> > example I mentioned was 1/1s up to 750 Hz.
>
> Yes, those are sines. I can't generate any more examples now
> as I'm about to drive cross country tomorrow.

Ah! where to? Headed to Microfest?

> I will say that I hear, at 750 Hz, no buzzing at all for 9/5,
> lots of buzzing at 16/15, and a whole spectrum of intermediate
> buzz stages in between. At 7/5 I don't really hear any buzzing
> at all, and if I do it's on the minimum bare threshold of
> detectability. If the volume is high, I guess maybe I hear
> something, but it seems to have more to do with the appearance
> of combination tones than anything else.

With sines, mono into both ears through headphones with good
DAC and preamp at moderate volume levels, 9/5 and 9/4 hardly
buzz no matter where I root them. 7/5 does buzz -- maybe I
would call it a "characteristic graininess" -- and seems to
do so equally well whether I root it on 200, 300, 400, 500,
600, or 750 Hz. Specifically, its rate increases with
increasing 5= frequency but its loudness seems to stay about
the same. It's heard as buzz (AM) in the perceived pitch.

16/15 beats, but does not have a pitch.

Trying more complex ratios near in size to 6/5, 9/7 etc.,
the effect does not seem to be terribly dependent on tuning.
However some ratios sound 'wrong', apparently because of
combination tones clashing with stuff. Sigh.

-Carl

On May 5, 2011, at 4:48 PM, Carl Lumma <carl@...> wrote:

--- Mike Battaglia <battaglia01@...> wrote:

> My thoughts are that this whole thing makes a few assumptions
> that I don't think are worth making:
> 1) Being able to filter a signal into different parts and then
> sum them to get the original signal implies that the filters are
> linear. Or, if they're not linear, they'd have to be nonlinear
> in such a fashion that all of the nonlinear components of the
> signal cancel upon the summing to get the original signal. That
> would be quite a feat.

The cochlea is essentially a compressor, and compression
is nonlinear. But obviously this nonlinearity is highly
desirable. The entire apparatus is nanoengineered to do
exactly what you say: slice and dice sound and put it back
together so that qualities of interest are retained. It is,
in fact, one of the more impressive feats in nature.

The compressive aspect of the cochlea is only one source of nonlinear
behavior. Time-varying amplification will not cause a problem, neither in
terms of combination tones nor in terms of hysteresis. But what about the
transduction between each channel in the basilar membrane to the auditory
nerve? What about the brainstem and the cochlear nucleus and so on? From
what I understand no part of this is linear at all, with the whole thing
playing a role more akin to a feature extractor than a transparent
transducer.

> 2) Hysteresis would make this even more of a feat. Your model
> would assume that the phase response of each auditory filter
> is magically tweaked so that perfect reconstruction of the
> waveform is possible just by summing. Or, if not, that some
> kind of allpass filter with the inverse of the phase response
> of the ear exists in the brain. This would be even more of a
> feat.

The relevant test is whether setting phases for maximum or
minimum spikiness in the time-domain is audible. I believe
you did a few examples that suggest it is.

That tests whether phase-coherent notes will buzz more strongly than
non-coherent notes. It doesn't give us any information about the hysteresis
of the auditory system, nor does it suggest a neural origin over a cochlear
one.

> And while we're at it, you don't seem to think that impulse
> trains exhibit buzz, but they're the spikiest waveforms of
> them all. Why is this?

The phenomenon of periodicity buzz seems to involve a pitched
sound that buzzes. Impulse trains buzz, but they are not
pitched sounds that buzz.

Impulse trains are pitched. They're differentiated sawtooth waves.

> The filterbank supplies such a mechanism because it is ...
> subject to the Fourier uncertainty principle, it can only have
> perfect frequency resolution if it has zero time resolution.
> When you combine this with a "spiky waveform," the result is
>
> 1) the waveform will get caught weakly in the filters whose
> frequency response is intermediate to that of the constituent
> tones,
> 2) the output from those filter will have an amplitude envelope,
> 3) and the AM envelopes generated from multiple tones will be
> in sync with one another if the waveform is spikier, which is
> another way of saying they'll be in sync with one another if
> the phase response of the signal is linear, because this is what
> spikiness is.

I don't understand 1).

For xxx Hz and yyy Hz, the beating will occur more strongly at (xxx+yyy)/2
than at xxx or yyy Hz.

Regarding 2), all the filters have
amplitude envelopes.

I said the filtered outputs will have amplitude envelopes, not the filters
themselves. In this case the outputs will have sinusoidal envelopes.

Point 3) seems to be what I'm saying,
and seems independent of points 1) & 2).

Right, and I'm saying I agree. Spikier waveforms do produce more buzz, and
cochlear models demonstrate how this percept could result from the original
physical property. If you have a better modelthen I'm all ears. But my
original statement, that phase coherence strengthens buzz, is equivalent to
what you said. You've just restated the hypothesis in the time domain
instead of the frequency domain.

> > Right, so can you confirm those are sine tones? And, the
> > example I mentioned was 1/1s up to 750 Hz.
>
> Yes, those are sines. I can't generate any more examples now
> as I'm about to drive cross country tomorrow.

Ah! where to? Headed to Microfest?

New York! Let's see if I can get 22-equal at Smalls.

With sines, mono into both ears through headphones with good
DAC and preamp at moderate volume levels, 9/5 and 9/4 hardly
buzz no matter where I root them. 7/5 does buzz -- maybe I
would call it a "characteristic graininess" -- and seems to
do so equally well whether I root it on 200, 300, 400, 500,
600, or 750 Hz. Specifically, its rate increases with
increasing 5= frequency but its loudness seems to stay about
the same. It's heard as buzz (AM) in the perceived pitch.

The timbre of the perceived pitch should buzz if its constituent harmonics
are changing in volume.

16/15 beats, but does not have a pitch.

In what way does it not have a pitch? You mean a virtual pitch?

Trying more complex ratios near in size to 6/5, 9/7 etc.,
the effect does not seem to be terribly dependent on tuning.
However some ratios sound 'wrong', apparently because of
combination tones clashing with stuff. Sigh.

Yeah. Sigh indeed. This won't ever be fully be put to rest unless we can do
a study in which the listening equipment and A-weighted volume is completely
controlled for. Until then we'll just have to keep winging it

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> The compressive aspect of the cochlea is only one source of
> nonlinear behavior. [snip]
> From what I understand no part of this is linear at all, with
> the whole thing playing a role more akin to a feature extractor
> than a transparent transducer.

Yes, I think that's right. And wherever music is concerned,
it tends to be the 'features' that are important. That's why
looking at the auditory system from a traditional signal-
processing perspective is often misguided in my opinion.

>>> 1) the waveform will get caught weakly in the filters whose
>>> frequency response is intermediate to that of the constituent
>>> tones,
>>> 2) the output from those filter will have an amplitude envelope,
>>> 3) and the AM envelopes generated from multiple tones will be
>>> in sync with one another if the waveform is spikier, which is
>>> another way of saying they'll be in sync with one another if
>>> the phase response of the signal is linear, because this is
>>> what spikiness is.
[snip]
>> Point 3) seems to be what I'm saying,
>> and seems independent of points 1) & 2).
>
> Right, and I'm saying I agree. Spikier waveforms do produce
> more buzz, and cochlear models demonstrate how this percept
> could result from the original physical property. If you have
> a better model then I'm all ears. But my original statement,
> that phase coherence strengthens buzz, is equivalent to what
> you said. You've just restated the hypothesis in the time
> domain instead of the frequency domain.

I think we're just arguing about where the interference
occurs. In a time-domain plot, it occurs in the plotting
algorithm. In human hearing, you're saying it occurs in
the cochlear filters, and I think it occurs higher up.
I'm not sure how to differentiate these, except that the
cochlear mechanism must predict rapidly decreasing buzz
loudness with increasing interval width.

> > Yes, those are sines. I can't generate any more examples now
> > as I'm about to drive cross country tomorrow.
>
> Ah! where to? Headed to Microfest?
>
> New York! Let's see if I can get 22-equal at Smalls.

Excellent. Here is what New York sounded like to me in '97:
http://lumma.org/music/score/NewYork.pdf
http://lumma.org/music/score/midi/NewYork_human.mid

> 16/15 beats, but does not have a pitch.
>
> In what way does it not have a pitch? You mean a
> virtual pitch?

All pitch is virtual pitch. In the case of 16/15, the pitch
is indeterminate to me, so I say it has none.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> What about all of the linear evenness stuff? You seemed to agree with it here
>
> /tuning/topicId_95699.html#95708

That was a conditional sentence.

> It has been shown that periodicity buzz generally correlates more with
> this linear spacing property than with actual periodicity.

But the buzz you are describing seems to correlate with the *periodically*
occurring spikes in the waveform.

> See here
>
> http://www.mikebattagliamusic.com/music/buzztempering.wav
>
> The first two to me sound like they have even buzz, and the last one
> sounds like it has chorusy, warbly buzz. The first one is 5:7:9, the
> second one is a 5:7:9 stretched so that the frequency difference
> between the two is even, and the last is 1/1 17/12 20/11, which is
> actually periodic but produces buzz that's less clear.

These don't sound at all like periodicity buzz to me. Listen to this:

http://sonic-arts.org/monzo/haircut/haircutlattices.htm

The high-limit otonalities have that sound.

> Also check the examples here
>
> /tuning/topicId_95699.html#95699

These have that sound, even the 5:6:7 has some of it because it is
lower now. I also hear a strong infrapitch sensation with the
mystery phi example. Periodicity seems to be a good description
what it phenomenally sounds like, as if something is repeating.
I don't care if the waveform is really periodic.

Kalle

> On Thu, May 5, 2011 at 9:27 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > Mike,
> >
> > I haven't really followed closely the discussion about periodicity
> > buzz, sorry about that. I always thought that periodicity buzz means
> > that sound of well-tuned otonalities that you hear when the virtual
> > pitch starts to approach infrapitch region. I also think that that
> > characteristic sound is most evident when the constituent tones of
> > the otonality are rather high while there is a strong but low virtual
> > pitch. So higher limit otonalities would do the trick. Low frequency
> > sawtooths or impulse trains don't quite sound the same.
> >
> > Kalle
>

On Sat, May 7, 2011 at 11:19 AM, Kalle Aho <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > What about all of the linear evenness stuff? You seemed to agree with it here
> >
> > /tuning/topicId_95699.html#95708
>
> That was a conditional sentence.

You're right, it was conditional on the notion that no other chords
could also produce buzz. If this were the case, then linearly spaced
chords would form only a subset of all of the potentially buzzing
chords that there are. But I haven't found any other chords that can
create buzz like that, and I've tried lots of stuff. If you have any
suggestions for chords that we could test that would violate the
linear spacing stuff but you think will also cause buzz, I can
synthesize examples to test.

> > It has been shown that periodicity buzz generally correlates more with
> > this linear spacing property than with actual periodicity.
>
> But the buzz you are describing seems to correlate with the *periodically*
> occurring spikes in the waveform.

I'm pretty sure that this is not coincidental and that these perfectly
periodic spikes occur when chords are linearly spaced. Think about the
time-domain response of the hairs in the ear to the incoming
waveforms, and how the spikes might cause synchronized AM vs
asynchronous AM if the spikes weren't periodic.

> > The first two to me sound like they have even buzz, and the last one
> > sounds like it has chorusy, warbly buzz. The first one is 5:7:9, the
> > second one is a 5:7:9 stretched so that the frequency difference
> > between the two is even, and the last is 1/1 17/12 20/11, which is
> > actually periodic but produces buzz that's less clear.
>
> These don't sound at all like periodicity buzz to me. Listen to this:
>
> http://sonic-arts.org/monzo/haircut/haircutlattices.htm
>
> The high-limit otonalities have that sound.

Well, so far, everyone has said they've heard buzz on the example I
provided. Carl made the additional note that although there was some
kind of chorus in the last example, it didn't destroy the buzz. So
perhaps what you're calling periodicity buzz is a different phenomenon
than we've been hearing.

I'm not sure exactly what phenomenon you're describing as buzz, but
these chords are ridiculously awesome! Joe Monz wrote this? Is he a
jazz musician? How the hell did I miss this guy being around?

But yeah, over the otonalities I hear some kind of rhythmic amplitude
fluctuation, same as I do over the examples I posted, just more
intense, since there are more notes and harmonic timbres involved.

> > Also check the examples here
> >
> > /tuning/topicId_95699.html#95699
>
> These have that sound, even the 5:6:7 has some of it because it is
> lower now. I also hear a strong infrapitch sensation with the
> mystery phi example. Periodicity seems to be a good description
> what it phenomenally sounds like, as if something is repeating.
> I don't care if the waveform is really periodic.

What do you mean by "infrapitch?" But maybe it just has to do with the
varying individual response of the auditory filter that you didn't
hear it with the 5:7:9 above. Either way, you didn't seem to even hear
it with the truly periodic 5:7:9, but note that we all hear it more as
the signal gets closer.

-Mike

PS: damn, this Monzo composition is awesome

On Fri, May 6, 2011 at 4:11 AM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > The compressive aspect of the cochlea is only one source of
> > nonlinear behavior. [snip]
> > From what I understand no part of this is linear at all, with
> > the whole thing playing a role more akin to a feature extractor
> > than a transparent transducer.
>
> Yes, I think that's right. And wherever music is concerned,
> it tends to be the 'features' that are important. That's why
> looking at the auditory system from a traditional signal-
> processing perspective is often misguided in my opinion.

First off, signal processing is a branch of pure mathematics, and can
theoretically handle any transfer function that takes place inside any
signal chain. It doesn't matter whether it's the ridiculous sort of
nonlinear stuff occurring in the brain, or the more simple linear
stuff that a capacitor might do in a circuit. If in the past I've ever
simplified the true nature of what's going on in the brain, as with my
"filterbank of comb filters" idea, it's only because I'm trying to see
if the simplified model might work well enough to be usable in
practice, because the kind of nonlinear dynamics that are involved in
the real brain use math that's extraordinarily complicated. This isn't
a claim that there really is a true, perfectly linear signal chain all
along the length of the auditory system, but just that the results
from simplifications like that might be good enough for government
work anyway. I'm not sure when I've ever proposed such a
simplification outside of the filterbank idea so you'll have to remind
me if I have.

But what you have said above is the argument that I'm using against
your suggestion that periodicity buzz is caused by an algorithm in the
brain, and one that would require the perfect reconstruction of the
time-domain waveform. I see no evidence for that, and lots of evidence
that the opposite is instead true. The fact that the original time
domain waveform is spiky could very well have profound impacts all
across the length of the signal chain, but to state that the brain
puts it all back together and then has a second algorithm to
re-process it, and that this second algorithm leads to buzz, seems
rather unlikely. While I won't say it's impossible, as biological
feats of nature are quite common, I will say it's not worth assuming
unless there's some reason to assume it.

Actually, this discussion might not be necessary, as the example below
demonstrates very clearly an example of where more spikiness != more
buzz.

> > Right, and I'm saying I agree. Spikier waveforms do produce
> > more buzz, and cochlear models demonstrate how this percept
> > could result from the original physical property. If you have
> > a better model then I'm all ears. But my original statement,
> > that phase coherence strengthens buzz, is equivalent to what
> > you said. You've just restated the hypothesis in the time
> > domain instead of the frequency domain.
>
> I think we're just arguing about where the interference
> occurs. In a time-domain plot, it occurs in the plotting
> algorithm. In human hearing, you're saying it occurs in
> the cochlear filters, and I think it occurs higher up.
> I'm not sure how to differentiate these, except that the
> cochlear mechanism must predict rapidly decreasing buzz
> loudness with increasing interval width.

I'm not sure that it means "rapidly" decreasing buzz loudness with
increasing interval width. It means generally decreasing buzz loudness
with interval width. As you've stated, the cochlea is a compressor,
the full details of which are continually emerging, so we have no idea
how rapid the effective perceptual decline should be. The cochlear
amplifier is certainly going to be interacting with things in ways
that I have no idea how to predict. However, the gammatone toolbox has
predicted that even the 7/5 you first requested at 750 Hz will have
weak buzz between the two tones, so it's decent enough for now.

Either way, the important thing is that this model does predict that
there will be a decline with increasing distance, which the
"spikiness" model doesn't predict. For example:

http://www.mikebattagliamusic.com/music/buzzrolloffcomparison.png
http://www.mikebattagliamusic.com/music/buzzrollofftest.wav

These are all with cosines, which should lead to optimal spikiness.
Note that 2/1 is the spikiest, but it is in no way buzzier than 7/6.
This is inconsistent with the hypothesis that more spikiness = more
buzzing, and is consistent with the hypothesis that buzzing has to do
with critical band roughness. For an even more blatant example, here
it is with triads:

http://www.mikebattagliamusic.com/music/buzzrolloffcomparisonchords.png
http://www.mikebattagliamusic.com/music/buzzrollofftestchords.wav

1:2:3 is way spikier than 6:7:8, but does not generate more buzz.

> Excellent. Here is what New York sounded like to me in '97:
> http://lumma.org/music/score/NewYork.pdf
> http://lumma.org/music/score/midi/NewYork_human.mid

Nice! This should be turned into a choral arrangement. You could
probably get EWQL symphonic choirs on it.

> > In what way does it not have a pitch? You mean a
> > virtual pitch?
>
> All pitch is virtual pitch. In the case of 16/15, the pitch
> is indeterminate to me, so I say it has none.

I hear it as having two pitches a semitone apart, but no clear
fundamental pitch.

-Mike

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

> First off, signal processing is a branch of pure mathematics, and can
> theoretically handle any transfer function that takes place inside any
> signal chain.

Signal processing is a branch of engineering mathematics which makes it applied math by definition. The theory of abelian groups or of the Riemann zeta function is pure math, even if you are so misguided as to apply it to music theory. Linear algebra is pure math, but with an applied flavor because there are so many applications. These distinctions, of course, are not intrinsic to the subjects themselves and may seem a little arbitrary, but they are important to mathematicians in sorting things out in practice, like who gets what office and where.

Hi Mike,

Wow, thanks for the compliments! Glad you like "Invisible Haircut"
so much ... i think it's one of my best little riffs.

I can't categorize myself as a "jazz musician", because i have
composed in a wide variety of different styles, and while i
am a huge fan of many jazz musicians, my musical idols are
Mahler, Beethoven, and Schoenberg (in that order), and i
also really love and have been heavily influenced by
the Beatles and Bruce Springsteen.

I joined this list around June 1998 and was very active
until about 2007. Since then i've become "married with
children" and have just been too busy too keep checking in.

While i am still closely associated with Sonic Arts, i have
moved most of my own personal stuff to my own website, and
my List of Works is here:

http://tonalsoft.com/monzo/worklist/worklist.aspx

If you like my jazz style, you probably would like
"3 Plus 4", which is also exists in a JI version.
(links are about 1/4 of the way down that page)

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 7, 2011 at 11:19 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > These don't sound at all like periodicity buzz to me.
> > Listen to this:
> >
> > http://sonic-arts.org/monzo/haircut/haircutlattices.htm
> >
> > The high-limit otonalities have that sound.
>
> <snip>
>
> I'm not sure exactly what phenomenon you're describing as buzz,
> but these chords are ridiculously awesome! Joe Monz wrote this?
> Is he jazz musician? How the hell did I miss this guy being
> around?
>
> <snip>
>
> PS: damn, this Monzo composition is awesome

--- Mike Battaglia <battaglia01@...> wrote:

> > Yes, I think that's right. And wherever music is concerned,
> > it tends to be the 'features' that are important. That's why
> > looking at the auditory system from a traditional signal-
> > processing perspective is often misguided in my opinion.
>
> First off, signal processing is a branch of pure mathematics,
> and can theoretically handle any transfer function that takes
> place inside any signal chain. It doesn't matter whether it's
> the ridiculous sort of nonlinear stuff occurring in the brain,
> or the more simple linear stuff that a capacitor might do in a
> circuit.

I used the qualifier "traditional" to try to head off this
objection. Of course there is a filter that corresponds to
human hearing, the question is whether the usual frequency-
domain methods will help you find it.

> to state that the brain puts it all back together and then
> has a second algorithm to re-process it ... seems rather
> unlikely.

It's not unlikely because it's a known fact. It's how pitch
works of course, and almost all the types of scene analysis
that make up hearing.

> lots of evidence that the opposite is instead true.

Such as?

> Either way, the important thing is that this model does predict
> that there will be a decline with increasing distance, which
> the "spikiness" model doesn't predict. For example:
> http://www.mikebattagliamusic.com/music/buzzrolloffcomparison.png
> http://www.mikebattagliamusic.com/music/buzzrollofftest.wav
> These are all with cosines, which should lead to optimal
> spikiness. Note that 2/1 is the spikiest, but it is in no way
> buzzier than 7/6.

Why is 2/1 spikiest? Of course I haven't supplied a definition
of spiky, so that's on me. I hear buzz only on 7/6 and 7/5,
and on 7/5 only weakly (or maybe rapidly).

> For an even more blatant example, here it is with triads:
> http://www.mikebattagliamusic.com/music
> /buzzrolloffcomparisonchords.png
> http://www.mikebattagliamusic.com/music/buzzrollofftestchords.wav
> 1:2:3 is way spikier than 6:7:8, but does not generate more buzz.

Or is it that the buzz is too rapid for 1:2:3?

> > Excellent. Here is what New York sounded like to me in '97:
> > http://lumma.org/music/score/NewYork.pdf
> > http://lumma.org/music/score/midi/NewYork_human.mid
>
> Nice! This should be turned into a choral arrangement.
> You could probably get EWQL symphonic choirs on it.

I've always heard it as pretty percussive and piano
oriented... the left hand in bars 8 & 9 for instance are
blues piano inspired (but cut down to 2 beats, because
there's no time for 4 in New York). And the cluster chords
later on were inspired by Henry Cowell. But you know, it
might be interesting... thanks for the suggestion.

-Carl

On Sun, May 8, 2011 at 10:54 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > First off, signal processing is a branch of pure mathematics, and can
> > theoretically handle any transfer function that takes place inside any
> > signal chain.
>
> Signal processing is a branch of engineering mathematics which makes it applied math by definition. The theory of abelian groups or of the Riemann zeta function is pure math, even if you are so misguided as to apply it to music theory. Linear algebra is pure math, but with an applied flavor because there are so many applications. These distinctions, of course, are not intrinsic to the subjects themselves and may seem a little arbitrary, but they are important to mathematicians in sorting things out in practice, like who gets what office and where.

Would the Fourier transform, or the Laplace transform, or the concept
of a transfer function, or the concept of a filter not be pure
mathematics? If so, what gets the "pure" designator?

-Mike

On Sun, May 8, 2011 at 1:34 PM, monz <joemonz@...> wrote:
>
> Hi Mike,
>
> Wow, thanks for the compliments! Glad you like "Invisible Haircut"
> so much ... i think it's one of my best little riffs.
>
> I can't categorize myself as a "jazz musician", because i have
> composed in a wide variety of different styles, and while i
> am a huge fan of many jazz musicians, my musical idols are
> Mahler, Beethoven, and Schoenberg (in that order), and i
> also really love and have been heavily influenced by
> the Beatles and Bruce Springsteen.

I wouldn't say I'm a strict "jazz musician" either, but my question
was more along the lines of whether or not you have formal jazz
training? I see you studied at MSM (right on, man) and to be honest I
never expected to see 13-limit renditions of altered chords and
diminished dominant chords on this list. Are you familiar with all of
the usual jazz theory, or did you just use your ears to find those
chords? Just curious, because I thought I was the only guy around here
who was hip to those sounds.

> While i am still closely associated with Sonic Arts, i have
> moved most of my own personal stuff to my own website, and
> my List of Works is here:
>
> http://tonalsoft.com/monzo/worklist/worklist.aspx
>
> If you like my jazz style, you probably would like
> "3 Plus 4", which is also exists in a JI version.
> (links are about 1/4 of the way down that page)

This is sick! When did you write this? More importantly, where did you
write this? Sounds like Gamble and Huff mixed with the Rascals mixed
with the Rocky theme or something. This is a major trip for me right
now.

-Mike

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

> Would the Fourier transform, or the Laplace transform, or the concept
> of a transfer function, or the concept of a filter not be pure
> mathematics? If so, what gets the "pure" designator?

Fourier or Laplace transform is classic pure math. Transfer function is engineering math again. Don't ask me to supply a formula to make sense of it all. You could try reading this:

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

Hi Mike,

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, May 8, 2011 at 1:34 PM, monz <joemonz@...> wrote:
> >
> > I can't categorize myself as a "jazz musician", because i have
> > composed in a wide variety of different styles, and while i
> > am a huge fan of many jazz musicians, my musical idols are
> > Mahler, Beethoven, and Schoenberg (in that order), and i
> > also really love and have been heavily influenced by
> > the Beatles and Bruce Springsteen.
>
> I wouldn't say I'm a strict "jazz musician" either, but my
> question was more along the lines of whether or not you have
> formal jazz training? I see you studied at MSM (right on, man)
> and to be honest I never expected to see 13-limit renditions
> of altered chords and diminished dominant chords on this list.
> Are you familiar with all of the usual jazz theory, or did you
> just use your ears to find those chords? Just curious, because
> I thought I was the only guy around here who was hip to those
> sounds.

If you by "formal" that i learned jazz theory in a school,
then no -- but i did learn a lot of it from books. At the time
i wrote "Invisible Haircut" i was playing in an R&B/top-40 band,
and also had gotten very much into Antonio Carlos Jobim. So
in much of what i was composing at that time i was deliberately
trying to make the chords as weird as possible but still
sound cool. Also, note once again that i really love Mahler
and Schoenberg, so there are influences from them too.

> > http://tonalsoft.com/monzo/worklist/worklist.aspx
> >
> > If you like my jazz style, you probably would like
> > "3 Plus 4", which is also exists in a JI version.
> > (links are about 1/4 of the way down that page)
>
> This is sick! When did you write this? More importantly,
> where did you write this? Sounds like Gamble and Huff mixed
> with the Rascals mixed with the Rocky theme or something.
> This is a major trip for me right now.

I composed the main theme for "Invisible Haircut" on
12 February 1990, in 12-edo, but had the desire from
the beginning to "justify" it and so immediately set out
figuring out common-tone chord progressions on paper.
This was way before i had any good computer software
to do this stuff, so it was all done with pencil and
hand calculator, and lots of hand-drawn diagrams.

Having to go thru that laborious procedure is exactly why
i had thought up the idea for Tonescape about 6 years before,
but it didn't become a reality until 2005.

And BTW, Invisible Haircut is 13-limit plus 19,
that is, 19-limit but without using 17.

Until 1993 this tune did not have a name. At that time
a good friend of mine produced and directed a play he
had written called "Invisible Haircut" in New York
and asked me to write some "warped" incidental music
for it, so i used that tune as the main theme, and
expanded and varied it for the rest of the music.

I composed 3 Plus 4 on 5 January 1992, all in my head
as i was driving around, then when i got home that night
i fired up Cakewalk and recorded the intro, main theme
(at the beginning and end), and backing tracks in the
middle for two solos. A good friend of mine came over
a few days later and recorded about 8 or 9 solos and
we picked our two favorites and put them in. This was
all in 12-edo.

3 Plus 4 remained like that until 1998, when i used
pitch-bend in Cakewalk to transform it into 11-limit JI.
Again i did it mainly on "paper" (actually on the computer
desktop this time), except for one particular chord about
which i've written quite often on this list: the D#-minor
chord which comes at the climax of each verse, right
when the main drum parts kick in and just before the
horns play the hook. I wanted a "darker" (i.e., smaller)
minor-3rd than 6:5, so i tried both 7:6 and 19:16, but
settled by ear on 75:64 as being exactly what i wanted.

This many years down the road, when i listen to 3 Plus 4
there are a few notes here and there that i would tune
differently, but overall i really like this piece a lot.
Glad you do too!

Oh, and you are pretty close with your location guesses:
i lived in the Germantown neighborhood of Philadelphia
when i wrote both of them.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

On Sun, May 8, 2011 at 8:46 PM, Carl Lumma <carl@...> wrote:
>
> > First off, signal processing is a branch of pure mathematics,
> > and can theoretically handle any transfer function that takes
> > place inside any signal chain. It doesn't matter whether it's
> > the ridiculous sort of nonlinear stuff occurring in the brain,
> > or the more simple linear stuff that a capacitor might do in a
> > circuit.
>
> I used the qualifier "traditional" to try to head off this
> objection. Of course there is a filter that corresponds to
> human hearing, the question is whether the usual frequency-
> domain methods will help you find it.

I'm not sure I understand. What exactly do you mean by "traditional,"
and what are the usual frequency-domain methods that you think are
falling short? I also don't know if I'd treat human hearing as a
filter, I'd treat it as a feature space with some feature extraction
stuff going on, most of which does not involve the exact preservation
of the original time domain waveform in any sense. I guess the two are
equivalent if you really want to nitpick.

> > to state that the brain puts it all back together and then
> > has a second algorithm to re-process it ... seems rather
> > unlikely.
>
> It's not unlikely because it's a known fact. It's how pitch
> works of course, and almost all the types of scene analysis
> that make up hearing.

I don't think it is a known fact that the brain takes the ridiculously
processed input that's coming out of the cochlea and brain stem and so
on, and then reconstructs the time-domain signal exactly. I've never
heard anyone put this out there as a serious theory, and it's
definitely not necessary for F0 analysis to take place or for anything
else. If you have evidence that the brain actually reconstructs the
waveform, then I'll gladly change my tune.

> > Either way, the important thing is that this model does predict
> > that there will be a decline with increasing distance, which
> > the "spikiness" model doesn't predict. For example:
> > http://www.mikebattagliamusic.com/music/buzzrolloffcomparison.png
> > http://www.mikebattagliamusic.com/music/buzzrollofftest.wav
> > These are all with cosines, which should lead to optimal
> > spikiness. Note that 2/1 is the spikiest, but it is in no way
> > buzzier than 7/6.
>
> Why is 2/1 spikiest? Of course I haven't supplied a definition
> of spiky, so that's on me. I hear buzz only on 7/6 and 7/5,
> and on 7/5 only weakly (or maybe rapidly).
//snip
>
> Or is it that the buzz is too rapid for 1:2:3?

I was going by, I guess, some intuitive combination of the standard
deviation and mean of amplitude with respect to time. But this is
probably a good stopping point for this discussion - I can't test for
the effects of an unknown spikiness property if I don't know what it
is.

> > Nice! This should be turned into a choral arrangement.
> > You could probably get EWQL symphonic choirs on it.
>
> I've always heard it as pretty percussive and piano
> oriented... the left hand in bars 8 & 9 for instance are
> blues piano inspired (but cut down to 2 beats, because
> there's no time for 4 in New York). And the cluster chords
> later on were inspired by Henry Cowell. But you know, it
> might be interesting... thanks for the suggestion.

It's also got some pretty hip harmonies in it. You have a gift, my
friend! A hip and somewhat modal gift.

-Mike

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> Until 1993 this tune did not have a name. At that time
> a good friend of mine produced and directed a play he
> had written called "Invisible Haircut" in New York
> and asked me to write some "warped" incidental music
> for it, so i used that tune as the main theme, and
> expanded and varied it for the rest of the music.

I always thought that the name was a pun on periodicity buzz and a buzz
cut. :D

Kalle

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 7, 2011 at 11:19 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > What about all of the linear evenness stuff? You seemed to agree with it here
> > >
> > > /tuning/topicId_95699.html#95708
> >
> > That was a conditional sentence.
>
> You're right, it was conditional on the notion that no other chords
> could also produce buzz. If this were the case, then linearly spaced
> chords would form only a subset of all of the potentially buzzing
> chords that there are. But I haven't found any other chords that can
> create buzz like that, and I've tried lots of stuff. If you have any
> suggestions for chords that we could test that would violate the
> linear spacing stuff but you think will also cause buzz, I can
> synthesize examples to test.

It's hard to come up with high-limit chords that have no linear
spacing anywhere and at the same time have a strong virtual pitch.
Anyway, I think your phi example convinces me that that the
periodicity buzz is not a virtual pitch.

> > > It has been shown that periodicity buzz generally correlates more with
> > > this linear spacing property than with actual periodicity.
> >
> > But the buzz you are describing seems to correlate with the *periodically*
> > occurring spikes in the waveform.
>
> I'm pretty sure that this is not coincidental and that these perfectly
> periodic spikes occur when chords are linearly spaced. Think about the
> time-domain response of the hairs in the ear to the incoming
> waveforms, and how the spikes might cause synchronized AM vs
> asynchronous AM if the spikes weren't periodic.

Right.

> > > The first two to me sound like they have even buzz, and the last one
> > > sounds like it has chorusy, warbly buzz. The first one is 5:7:9, the
> > > second one is a 5:7:9 stretched so that the frequency difference
> > > between the two is even, and the last is 1/1 17/12 20/11, which is
> > > actually periodic but produces buzz that's less clear.
> >
> > These don't sound at all like periodicity buzz to me. Listen to this:
> >
> > http://sonic-arts.org/monzo/haircut/haircutlattices.htm
> >
> > The high-limit otonalities have that sound.
>
> Well, so far, everyone has said they've heard buzz on the example I
> provided. Carl made the additional note that although there was some
> kind of chorus in the last example, it didn't destroy the buzz. So
> perhaps what you're calling periodicity buzz is a different phenomenon
> than we've been hearing.

Or it could just be that I personally hear it only at lower
frequencies (meaning the frequency of the buzz/envelope spikes).

> I'm not sure exactly what phenomenon you're describing as buzz, but
> these chords are ridiculously awesome! Joe Monz wrote this? Is he a
> jazz musician? How the hell did I miss this guy being around?
>
> But yeah, over the otonalities I hear some kind of rhythmic amplitude
> fluctuation, same as I do over the examples I posted, just more
> intense, since there are more notes and harmonic timbres involved.

Paul described it as periodicity buzz at the time:

/tuning/topicId_1727.html#1788

> > > Also check the examples here
> > >
> > > /tuning/topicId_95699.html#95699
> >
> > These have that sound, even the 5:6:7 has some of it because it is
> > lower now. I also hear a strong infrapitch sensation with the
> > mystery phi example. Periodicity seems to be a good description
> > what it phenomenally sounds like, as if something is repeating.
> > I don't care if the waveform is really periodic.
>
> What do you mean by "infrapitch?" But maybe it just has to do with the
> varying individual response of the auditory filter that you didn't
> hear it with the 5:7:9 above. Either way, you didn't seem to even hear
> it with the truly periodic 5:7:9, but note that we all hear it more as
> the signal gets closer.

Infrapitch is "the perception of acoustic iterance", I read that
somewhere. :)
Not quite rhythm and not quite pitch but something between. I think
that the frequency ranges of infrapitch and pitch somewhat overlap.

Kalle

Hi Joe,

Would you mind if I render Invisible Hair Cut with Garritan Jazz and Big
Band sample set listing you as composer and posting it on my site or making
it available for you to post it to your website?

I host the work of a few composers at micro.soonlabel.com

Thanks,

Chris

On Sun, May 8, 2011 at 10:58 PM, monz <joemonz@...> wrote:

>
>
> Hi Mike,
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Sun, May 8, 2011 at 1:34 PM, monz <joemonz@...> wrote:
> > >
> > > I can't categorize myself as a "jazz musician", because i have
> > > composed in a wide variety of different styles, and while i
> > > am a huge fan of many jazz musicians, my musical idols are
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>
>
>
>

Hi Chris ... sure, no problem. I'd be honored, and would
love to hear your rendition.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Hi Joe,
>
> Would you mind if I render Invisible Hair Cut with Garritan
> Jazz and Big Band sample set listing you as composer and posting
> it on my site or making it available for you to post it to your
> website?
>
> I host the work of a few composers at micro.soonlabel.com

On Sun, May 8, 2011 at 10:58 PM, monz <joemonz@...> wrote:
>
> If you by "formal" that i learned jazz theory in a school,
> then no -- but i did learn a lot of it from books. At the time
> i wrote "Invisible Haircut" i was playing in an R&B/top-40 band,
> and also had gotten very much into Antonio Carlos Jobim. So
> in much of what i was composing at that time i was deliberately
> trying to make the chords as weird as possible but still
> sound cool. Also, note once again that i really love Mahler
> and Schoenberg, so there are influences from them too.

Yeah, I guess formal's not what I mean, really, but the fact that
you're into music like this at all is pretty sweet.

> 3 Plus 4 remained like that until 1998, when i used
> pitch-bend in Cakewalk to transform it into 11-limit JI.
> Again i did it mainly on "paper" (actually on the computer
> desktop this time), except for one particular chord about
> which i've written quite often on this list: the D#-minor
> chord which comes at the climax of each verse, right
> when the main drum parts kick in and just before the
> horns play the hook. I wanted a "darker" (i.e., smaller)
> minor-3rd than 6:5, so i tried both 7:6 and 19:16, but
> settled by ear on 75:64 as being exactly what i wanted.

So you can hear the difference between 7/6 and 75/64? I can't really
hear it, personally. But is your goal generally to compose in JI then?
I haven't seen anyone on here really focus on JI for a long time,
being as it seems to be all about comma pumps these days.

> Oh, and you are pretty close with your location guesses:
> i lived in the Germantown neighborhood of Philadelphia
> when i wrote both of them.

Haha, alright, close enough. Never lived up there myself, but I kept
going up there for a while for this jam session at "Larose Jazz Club."
It was alright.

-Mike

On Mon, May 9, 2011 at 10:21 AM, Kalle Aho <kalleaho@...> wrote:
>
> It's hard to come up with high-limit chords that have no linear
> spacing anywhere and at the same time have a strong virtual pitch.
> Anyway, I think your phi example convinces me that that the
> periodicity buzz is not a virtual pitch.

It doesn't really have to be linear spacing, it can be a 2:1 spacing
or something like that too (like 16:17:19 or something). The gammatone
plots give a better visual descriptor of what I think is going on.

> > I'm pretty sure that this is not coincidental and that these perfectly
> > periodic spikes occur when chords are linearly spaced. Think about the
> > time-domain response of the hairs in the ear to the incoming
> > waveforms, and how the spikes might cause synchronized AM vs
> > asynchronous AM if the spikes weren't periodic.
>
> Right.

You should also check my recent response to Carl - 2/1 is much spikier
than 7/6, but 7/6 buzzes way more. But since you seem to be more about
larger chords, perhaps you won't be satisfied until I demonstrate an
example with pentads or something. 1:2:3:4:5 is I believe spikier than
any other 5-note pentad out there (it'll look like an impulse train
convolved with a sinc function), but I guarantee you that it won't
buzz more than the 5-note phi pentad I posted. Spikiness doesn't
decline with dyad width, but buzz does decline with dyad width, which
is why I currently think it has to do with roughness.

> > Well, so far, everyone has said they've heard buzz on the example I
> > provided. Carl made the additional note that although there was some
> > kind of chorus in the last example, it didn't destroy the buzz. So
> > perhaps what you're calling periodicity buzz is a different phenomenon
> > than we've been hearing.
>
> Or it could just be that I personally hear it only at lower
> frequencies (meaning the frequency of the buzz/envelope spikes).

It certainly could be that.

> > But yeah, over the otonalities I hear some kind of rhythmic amplitude
> > fluctuation, same as I do over the examples I posted, just more
> > intense, since there are more notes and harmonic timbres involved.
>
> Paul described it as periodicity buzz at the time:
>
> /tuning/topicId_1727.html#1788

I think we could certainly call it a periodic buzzing, but
"periodicity buzz" is just misleading. It only happens to occur over
periodic waveforms by coincidence. If we were using phi-based timbres,
in which the phi-harmonic series was 1:phi:2phi-1:3phi-2:4phi-3:etc,
the points of maximum buzzing would be much different. I'd be happy to
demonstrate this if you'd like, but it might take a minute. My main
computer isn't set up yet, so I only have my older MATLAB setup to
work with in which I don't have all of my tools set up.

> Infrapitch is "the perception of acoustic iterance", I read that
> somewhere. :)
> Not quite rhythm and not quite pitch but something between. I think
> that the frequency ranges of infrapitch and pitch somewhat overlap.

OK, I see. Yes, I think so as well. I suppose it would to do with the
fact that the time-frequency response of the ear changes across the
spectrum, right?

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> You should also check my recent response to Carl - 2/1 is much
> spikier than 7/6, but 7/6 buzzes way more.

I guess you haven't checked my reply. I don't see how 2/1
is spikier - in fact, its envelope is apparently flat.
Or at least, has a period of little more than a millisecond
(59 units). 7/6 has an envelope period of about 350 units.

> I think we could certainly call it a periodic buzzing, but
> "periodicity buzz" is just misleading.

Oh, drop it. Both Kalle and I assumed the weaker meaning.

-Carl

On Tue, May 10, 2011 at 12:37 AM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > You should also check my recent response to Carl - 2/1 is much
> > spikier than 7/6, but 7/6 buzzes way more.
>
> I guess you haven't checked my reply. I don't see how 2/1
> is spikier - in fact, its envelope is apparently flat.
> Or at least, has a period of little more than a millisecond
> (59 units). 7/6 has an envelope period of about 350 units.

I did check it, and I responded. I didn't get a response from you.

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

Sorry I missed this,

> > > to state that the brain puts it all back together and then
> > > has a second algorithm to re-process it ... seems rather
> > > unlikely.
> >
> > It's not unlikely because it's a known fact. It's how pitch
> > works of course, and almost all the types of scene analysis
> > that make up hearing.
>
> I don't think it is a known fact that the brain takes the
> ridiculously processed input that's coming out of the cochlea
> and brain stem and so on, and then reconstructs the
> time-domain signal exactly.

It doesn't reconstruct it exactly, and it needn't to see buzz.
Cochlear hair bundles output sparse spike trains that are
phase-locked to the maxima of the excitation on the basilar
membrane at that point, with the amplitude of the excitation
being encoded by the spiking rate. The overall envelope of
spikes across hair bundles, as well as the interspike timing,
are known to be used in pitch perception. See for example
http://dx.doi.org/10.1371/journal.pone.0000369
See here for a diagram http://min.us/lnkytq

It's known these auditory nerve signals are integrated in the
inferior colliculus, where there are cells that are specially
tuned to enhance amplitude modulation. You might try googling
"inferior colliculus MTF" for some of the literature on this.

The time-domain signal is even detectable in brainstem
responses through electrodes on the scalp. See here
http://min.us/lkTy4a
Apparently if these signals are amplified and played back,
speech can even be understood in them!

> > Or is it that the buzz is too rapid for 1:2:3?
>
> I was going by, I guess, some intuitive combination of the
> standard deviation and mean of amplitude with respect to time.
> But this is probably a good stopping point for this
> discussion - I can't test for the effects of an unknown
> spikiness property if I don't know what it is.

Good point. I'm thinking if the buzz is 'audible envelope',
it must require something like

* a pitched sound
* expressible as carrier + AM with AM frequency in the
realm of 10 - 100 Hz.
* perhaps the AM frequency should != f0 (the pitch)

Ok, that still has a long way to go, but that's what's been
floating around in my head.

> It's also got some pretty hip harmonies in it. You have
> a gift, my friend! A hip and somewhat modal gift.

Thanks! I don't understand modes or extended chords
(beyond 7th chords) at all - at least not in a way I can
access when composing or jamming.

-Carl

Thanks, I'll see what I can do!

Chris

On Mon, May 9, 2011 at 3:06 PM, monz <joemonz@...> wrote:

>
>
> Hi Chris ... sure, no problem. I'd be honored, and would
> love to hear your rendition.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > Hi Joe,
> >
> > Would you mind if I render Invisible Hair Cut with Garritan
> > Jazz and Big Band sample set listing you as composer and posting
> > it on my site or making it available for you to post it to your
> > website?
> >
> > I host the work of a few composers at micro.soonlabel.com
>
>
>

On Tue, May 10, 2011 at 1:44 AM, Carl Lumma <carl@...> wrote:
>
> > I don't think it is a known fact that the brain takes the
> > ridiculously processed input that's coming out of the cochlea
> > and brain stem and so on, and then reconstructs the
> > time-domain signal exactly.
>
> It doesn't reconstruct it exactly, and it needn't to see buzz.
> Cochlear hair bundles output sparse spike trains that are
> phase-locked to the maxima of the excitation on the basilar
> membrane at that point, with the amplitude of the excitation
> being encoded by the spiking rate. The overall envelope of
> spikes across hair bundles, as well as the interspike timing,
> are known to be used in pitch perception.

Alright, but if the overall envelope is related to pitch perception,
then that means that "difference tones" are related to pitch
perception as well.

> See for example
> http://dx.doi.org/10.1371/journal.pone.0000369
> See here for a diagram http://min.us/lnkytq

I'm going through this now, this is a good read. It looks like my
point about difference tones is represented in this study as well,
with them modeling pitch as a combination of the envelope and
fine-structure pitch percepts. This is a really simple model and could
easily be coded up in MATLAB, if I understand it correctly - looks
like they're just averaging the envelope and fine structure responses
whenever they fall within the half-octave. But if your tones are 300,
500, and 700 Hz, then the envelope will be 200 Hz, and the average
will give you 350 Hz. So I'm not sure if I'm misunderstanding some
part of it.

I'll have to read through this again, as I'm pretty beat now and this
was just my first readthrough of it. It doesn't look like rules out
cochlear generation for buzz, just suggests perhaps another spot that
it could be generated. Keep in mind again that we didn't hear buzz
when the signal was split between two ears, so it has to occur between
binaural integration.

> It's known these auditory nerve signals are integrated in the
> inferior colliculus, where there are cells that are specially
> tuned to enhance amplitude modulation. You might try googling
> "inferior colliculus MTF" for some of the literature on this.

Will check it out, last paper has left me brain fried.

> > > Or is it that the buzz is too rapid for 1:2:3?
> >
> > I was going by, I guess, some intuitive combination of the
> > standard deviation and mean of amplitude with respect to time.
> > But this is probably a good stopping point for this
> > discussion - I can't test for the effects of an unknown
> > spikiness property if I don't know what it is.
>
> Good point. I'm thinking if the buzz is 'audible envelope',
> it must require something like
>
> * a pitched sound
> * expressible as carrier + AM with AM frequency in the
> realm of 10 - 100 Hz.
> * perhaps the AM frequency should != f0 (the pitch)
>
> Ok, that still has a long way to go, but that's what's been
> floating around in my head.

Why the 100 Hz limit?

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> Alright, but if the overall envelope is related to pitch
> perception, then that means that "difference tones" are
> related to pitch perception as well.

"Difference tones" usually means tones are heard,
not envelope.

> > http://dx.doi.org/10.1371/journal.pone.0000369

> if I understand it correctly - looks
> like they're just averaging the envelope and fine structure
> responses whenever they fall within the half-octave. But
> if your tones are 300, 500, and 700 Hz, then the envelope
> will be 200 Hz, and the average will give you 350 Hz.
> So I'm not sure if I'm misunderstanding some part of it.

It's an average of the two, I don't know about the half-
octave part. I haven't read the paper completely, it just
came up in a google search. Needless to say, I'm not
endorsing their algorithm.

> Keep in mind again that we didn't hear buzz when the signal
> was split between two ears, so it has to occur between
> binaural integration.

Yep.

> Why the 100 Hz limit?

10ms is really short to hear any kind of envelope changes, no?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@> wrote:
>
> > Alright, but if the overall envelope is related to pitch
> > perception, then that means that "difference tones" are
> > related to pitch perception as well.
>
> "Difference tones" usually means tones are heard,
> not envelope.

An envelope with periodic amplitude fluctuation is created by combination.

On Wed, May 11, 2011 at 6:45 PM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > Alright, but if the overall envelope is related to pitch
> > perception, then that means that "difference tones" are
> > related to pitch perception as well.
>
> "Difference tones" usually means tones are heard,
> not envelope.

If you're saying that the envelope is in some sense "heard" as a
pitch, then that means that combination frequencies are in some sense
"heard" as a pitch, because that's what's going to result from that
sort of operation on the waveform.

> > > http://dx.doi.org/10.1371/journal.pone.0000369
>
> > if I understand it correctly - looks
> > like they're just averaging the envelope and fine structure
> > responses whenever they fall within the half-octave. But
> > if your tones are 300, 500, and 700 Hz, then the envelope
> > will be 200 Hz, and the average will give you 350 Hz.
> > So I'm not sure if I'm misunderstanding some part of it.
>
> It's an average of the two, I don't know about the half-
> octave part. I haven't read the paper completely, it just
> came up in a google search. Needless to say, I'm not
> endorsing their algorithm.

It worked for this particular case, e.g. where they gradually increase
the inharmonicity of the waveform and average the two waveforms
together. If I understand it correctly, the model would break down in
the case I presented above. The reason is that they did to the
waveform what you accused me of doing, which is performing an
oversimplified frequency domain analysis on the filtered signal. It
looks like the cochlea generates a transient spike at either every
zero-crossing or every instance in which the vibrating hair turns
around (same difference), and that this spike's amplitude is in some
sense proportional to the amplitude of the oscillation. Thus, if the
hair's oscillation is undergoing some kind of amplitude modulation,
the spikes will have AM in the same shape as the envelope itself. For
them to just take the entire signal and put it in the frequency domain
will definitely stick the envelope frequency in there as a component
tone, where normally it doesn't for just a sine with AM on it, because
this whole "spikes" concept is a ridiculously severe nonlinearity
that's going to make stuff like that happen. So that's expected.

But what they did is just take the Fourier transform of the whole
thing and then do some sketchy math to estimate the resultant virtual
pitch. Their model wasn't an attempt to be an accurate neurological
model, but rather a simple metric for pitch estimation that,
unfortunately, looks like it'll break down for cases like the one I
mentioned (unless I am really misunderstanding something). Their
oversimplified pure-frequency domain analysis doesn't comment on the
time-frequency spread of the resulting waveform, hence it doesn't say
anything at all about how or where buzz could occur, which requires a
frequency to fluctuate in volume over time.

If they instead use a short-time Fourier transform with a windowing
function, then that gives us a second filterbank-like structure where
buzz could occur, and it would also suggest a second place for another
critical band to exist. I've seen a double critical band hypothesis
proposed in the literature, but I honestly don't know much about it. I
am willing to accept that buzz could be generated at a second spectral
filtering stage like that, but I didn't want to buy into this double
critical band theory without seeing some proof of it, so I said it was
cochlear in origin. Maybe an exploration of the cochlear model of buzz
will lead to a prediction that experiment violates, and then ends up
providing more proof of the double critical band hypothesis that way.
Or maybe people already believe it, I dunno, like I said, I'm not up
to date on this concept. All I know is that it would entail that all
of our roughness experiments have been measuring the effects of two
critical bands and attributing their combined interaction solely to
the cochlea, so I'm not sure what kind of radical reinterpretation
that would require of the existing literature.

> > Why the 100 Hz limit?
>
> 10ms is really short to hear any kind of envelope changes, no?

I guess so, but the reason that I believe is the case is because once
the spikes get close enough together, they also start to signal tones
that are far enough apart to be well-resolved by the cochlea. If we're
taking the cochlea out of this process, where does the limitation come
from?

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> > "Difference tones" usually means tones are heard,
> > not envelope.
>
> If you're saying that the envelope is in some sense "heard"
> as a pitch,

I'm saying it's heard as an envelope. They're saying they
found evidence that the envelope period influences pitch
perception. That may be true, but it needn't be true as
far as buzz is concerned.

> It looks like the cochlea generates a transient spike at either
> every zero-crossing or every instance in which the vibrating
> hair turns around (same difference), and that this spike's
> amplitude is in some sense proportional to the amplitude of the
> oscillation.

No, the spiking rate is proportional to the amplitude of the
stimulus, and the spike timing is phase-locked to the frequency
of the stimulus. (as I wrote already)

-Carl

On Thu, May 12, 2011 at 6:06 PM, Carl Lumma <carl@...> wrote:
>
> > It looks like the cochlea generates a transient spike at either
> > every zero-crossing or every instance in which the vibrating
> > hair turns around (same difference), and that this spike's
> > amplitude is in some sense proportional to the amplitude of the
> > oscillation.
>
> No, the spiking rate is proportional to the amplitude of the
> stimulus, and the spike timing is phase-locked to the frequency
> of the stimulus. (as I wrote already)

You did write it, but being as spiking rate and spike timing are the
same thing, I didn't understand what you meant. What I wrote was what
I thought the study was saying.

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> > No, the spiking rate is proportional to the amplitude of the
> > stimulus, and the spike timing is phase-locked to the frequency
> > of the stimulus. (as I wrote already)
>
> You did write it, but being as spiking rate and spike timing
> are the same thing, I didn't understand what you meant.

Not the same. I think I wrote "overall" or "aggregate"
firing rate, vs. timing being phase-locked to the stimulus
maxima. You can imagine a spike train at every maximum,
with spikes deleted at random. The quieter the stimulus
the more you delete.

-Carl

I did some renderings of Carl's piece New York

playable
http://micro.soonlabel.com/carl_lumma/new_york/new_york.html
down-loadable
http://micro.soonlabel.com/carl_lumma/new_york

These are in 12 equal because I didn't find instructions to the contrary.
If Carl has an alternative tuning in mind I can give it a go.

Chris

>
> > New York! Let's see if I can get 22-equal at Smalls.
>
> Excellent. Here is what New York sounded like to me in '97:
> http://lumma.org/music/score/NewYork.pdf
> http://lumma.org/music/score/midi/NewYork_human.mid
>
>
> > 16/15 beats, but does not have a pitch.
> >
> > In what way does it not have a pitch? You mean a
> > virtual pitch?
>
> All pitch is virtual pitch. In the case of 16/15, the pitch
> is indeterminate to me, so I say it has none.
>
> -Carl
>
>
>

I want to thank Chris for being inspired to do these -- they
came out great! He's added the drum track and added a touch
of funk to the bass. My fav is the brass version but the EP
version is good too. Tuning is still 12-ET.

-Carl

--- Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I did some renderings of Carl's piece New York
>
> playable
> http://micro.soonlabel.com/carl_lumma/new_york/new_york.html
> down-loadable
> http://micro.soonlabel.com/carl_lumma/new_york
>
> These are in 12 equal because I didn't find instructions to
> the contrary.
> If Carl has an alternative tuning in mind I can give it a go.
>
> Chris
>

Ok, here it is.

http://micro.soonlabel.com/joe_monzo/invisible_hair_cut_sax.mp3

I had to transpose everything up a minor 3rd to get all of the bass line in
range of the samples.
The lead is now flute & flute plus alto sax

If I messed up something please tell me.

Chris

On Mon, May 9, 2011 at 3:06 PM, monz <joemonz@...> wrote:

>
>
> Hi Chris ... sure, no problem. I'd be honored, and would
> love to hear your rendition.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > Hi Joe,
> >
> > Would you mind if I render Invisible Hair Cut with Garritan
> > Jazz and Big Band sample set listing you as composer and posting
> > it on my site or making it available for you to post it to your
> > website?
> >
> > I host the work of a few composers at micro.soonlabel.com
>
>
>

I added a Hollywood strings only version

http://micro.soonlabel.com/carl_lumma/new_york/Lumma-New-York-HS.mp3

or play online

http://micro.soonlabel.com/carl_lumma/new_york/new_york.html

On Sat, May 14, 2011 at 3:15 PM, Carl Lumma <carl@...> wrote:

>
>
> I want to thank Chris for being inspired to do these -- they
> came out great! He's added the drum track and added a touch
> of funk to the bass. My fav is the brass version but the EP
> version is good too. Tuning is still 12-ET.
>
> -Carl
>
>
> --- Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > I did some renderings of Carl's piece New York
> >
> > playable
> > http://micro.soonlabel.com/carl_lumma/new_york/new_york.html
> > down-loadable
> > http://micro.soonlabel.com/carl_lumma/new_york
> >
> > These are in 12 equal because I didn't find instructions to
> > the contrary.
> > If Carl has an alternative tuning in mind I can give it a go.
> >
> > Chris
> >
>
>
>

Thanks Chris! It's amazing what Kontakt can do. -Carl

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I added a Hollywood strings only version
>
> http://micro.soonlabel.com/carl_lumma/new_york/Lumma-New-York-HS.mp3
>
> or play online
>
> http://micro.soonlabel.com/carl_lumma/new_york/new_york.html
>
>

Hollywood strings is different from Kontakt. Though kontakt does come with a
good string section.

On Tue, May 17, 2011 at 3:40 PM, Carl Lumma <carl@lumma.org> wrote:

>
>
> Thanks Chris! It's amazing what Kontakt can do. -Carl
>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > I added a Hollywood strings only version
> >
> > http://micro.soonlabel.com/carl_lumma/new_york/Lumma-New-York-HS.mp3
> >
> > or play online
> >
> > http://micro.soonlabel.com/carl_lumma/new_york/new_york.html
> >
> >
>
>
>

Ah, thanks. -C.

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Hollywood strings is different from Kontakt. Though kontakt does
> come with a good string section.
>

Back to this

On Fri, May 13, 2011 at 3:38 AM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > > No, the spiking rate is proportional to the amplitude of the
> > > stimulus, and the spike timing is phase-locked to the frequency
> > > of the stimulus. (as I wrote already)
> >
> > You did write it, but being as spiking rate and spike timing
> > are the same thing, I didn't understand what you meant.
>
> Not the same. I think I wrote "overall" or "aggregate"
> firing rate, vs. timing being phase-locked to the stimulus
> maxima. You can imagine a spike train at every maximum,
> with spikes deleted at random. The quieter the stimulus
> the more you delete.

In the study referenced, it looks like there are spike trains with
envelopes, no deletions. If there were deletions, one I think should
expect to see subharmonics in the signal. I don't see anything like
that. If there's something that I've missed in this article, feel free
to point it out.

-Mike