Periodicity and buzz again

Hi folks.

Hey, I haven't been here for almost half a year (not counting my post about Christmas variations) and after a quick skim through the past messages in my mailbox, I'm realizing that essentially nothing substantial (except of Gene's nice scale ideas) has happened here until January when Mike started giving examples of periodicity buzz.

Now, let me say something to all who have contributed to this thread since its start. I can't speak for too many people here as I couldn't get through all the "buzz" messages carefully one by one. But at least my general feeling from what I've got is this:

People here seem to be divided into two groups. One group of people has some particular experience with periodicity and trying to share it with others, they explain some phenomena in terms of changes to the actual sound (what happens when you add this, remove this, amplify this, phase-shift this). Then there's the other group of people who are so much focused on the question of "why we hear it the way we do" that they occasionally seem to either dismiss a part of the former group's explanation or claim it can't have much to do with it even before reading it carefully or studying the given examples in appropriate detail. Moreover, from time to time, they seem to go right onto the subjective part of the topic without first asking themselves if they could possibly prove or disprove their view with a help in the "objective part". If not, there's obviously not much to say then; if they could, it's up to them whether they want to. To this, I can say maybe one thing: People like Mike B have dedicated a large amount of their time to make examples of the matters they try to document both in sound and in words. If someone isn't as involved in Hilbert transforms or other phase shifts as Mike might be, I hope there are other ways they might respond than things like "I was asking about *this* and you're answering with *that* which has absolutely nothing to do with it" when it actually *has* something to do with it. As some of you may know, I've examined acoustical periodicity for some time already and I was always on the other "extreme"; i.e. not so much interested in to why we hear it the way we do but rather what happens to the "shapes" of the waveforms, regardless of the question if someone is actually listening to them at that moment or not. Through this method, I was able to figure a large part of the explanation why I heard many sounds the way I did, which convinced me that a large part of what I was experiencing could possibly be explained to other people if their hearing is "sensitive enough" (I mean, don't know about you but I could possibly recognize, after hearing a 7-tone chord, what tones it was made of -- unless they're played in the very low registers; if you played something like A0-C1-E1-G#1-B1-D2-F#2 to me, I would probably recognize only some of the upper tones).

Some time ago, we had a similar discussion here and I quickly run into conflicts just because I wanted to explain a part of the purely "acoustical" stuff before trying to hassle with the question of human hearing in particular (and I say "trying to" as I would probably never be interested into that part on my own). Although "virtual fundamental" is something else than the "fundamental frequency" of a period, someone eventually claimed that I had said it was the same. Since VF is not objectively measurable while the fundamental frequency is, a statement like that can't lead the discussion forward a great deal. Later, when I was, together with other people, trying to explain how phase-shifted echoes influence the frequency ratios of the resulting period, all of us were told that this has nothing to do with the topic. Eventually, I swapped a couple of offlist messages with Mike B and we did some experiments with these phenomena but it seemed noone else cared then -- except Rick who was eventually excluded or he had left or I'm not sure what's happened to him. This time, Mike has done the great thing that he took a short sine sweep and repeated it periodically, and why is this causing so much controversy if it really is an example of a periodic spectrum (i.e. a proper harmonic series) with different phases for each harmonic?

Anyway, before we start discussing sharp timbres, if you listen to sines of 2300Hz and 2000Hz together, don't you hear 1700Hz softly in a similar sine-like timbre as well and also 300Hz in a sharper timbre? And don't you hear 1400Hz if you raise the volume a bit? And don't you hear 1100Hz if you raise it further? And if you do, does it tell you something very clear about a part of this whole topic? To me it certainly does. Try running this simple dyad through a gentle overdriving effect and then scanning the spectrum through a long sine sweep convolution. You'll hear 200Hz, 500, 800, 1100, 1400, 1700, ... But not 300. 300 is the result of the amplitude modulation (that may be the reason for the narrowband-like timbre) but the spectrum itself doesn't contain this frequency even after being overdriven because the phase relations which are 1/300-of-a-second apart actually differ by 120 degrees. The best way to demonstrate what I mean would be to make impulses of phase-shifting echoes. But I'm not sure who else apart from Mike has some tools for convolution and which format these impulses should be in. But if you like, I'll be happy to do that.

Petr

2011/1/21 Petr Pařízek <petrparizek2000@...>
>
> Hey, I haven't been here for almost half a year (not counting my post about
> Christmas variations) and after a quick skim through the past messages in my
> mailbox, I'm realizing that essentially nothing substantial (except of
> Gene's nice scale ideas) has happened here until January when Mike started
> giving examples of periodicity buzz.

There were a few interesting things. I'll message you offlist to get
you up to speed.

> Now, let me say something to all who have contributed to this thread since
> its start. I can't speak for too many people here as I couldn't get through
> all the "buzz" messages carefully one by one.

I'm sure everyone feels the same way. It's a lot to sort through and
also, frankly, involves some pretty boring stuff. There's still a lot
more to do, with my ultimate goal here to delve a little further into
the concept of "tuning error," what it means, and hopefully come up
with a new metric for measuring the error of a tuning. I'll post some
kind of recap when I'm all done.

> If someone isn't as involved in Hilbert transforms or other phase
> shifts as Mike might be, I hope there are other ways they might respond than
> things like "I was asking about *this* and you're answering with *that*
> which has absolutely nothing to do with it" when it actually *has* something
> to do with it.

I think you might be talking about the discussion about whether the
buzzing present in impulse trains and sawtooth waves is related to
periodicity buzz. I haven't really expounded on this yet, but I came
across an explanation about why people might differentiate the two:
For normal periodicity buzz, you end up with AM for the notes in the
chord. For a buzz or sawtooth waveform, when you end up at like the
50th harmonic or the like, the harmonics are so close together that I
don't think that the auditory filters in the cochlea can separate the
notes out -at all-. When you get to that stage, periodicity buzz takes
on a different perceptual gestalt: in the lower register, you still
get the AM for each note, but in the upper register, there are simply
so many notes packed closely together that the entire thing starts to
simply sound like recurring, periodic, unpitched impulses. Meaning
that the AM of individual pitches does NOT occur in the upper
register, because the notes are so close together you don't even hear
pitch at all. The AM occurs only in the lower register.

So the fact that you still, technically, only have ratios in a
harmonic series up there contributes to the periodic nature of this
nonpitched sound, but the frequencies are simply too close together
for the auditory filters in the cochlea to differentiate them. So when
people first hear this sound, they're like "WTF is this?! This isn't
periodicity buzz." Well, it still is, or rather it's generated by the
same mechanism, but it generates a different percept than we're used
to, since we don't routinely listen to chords like 92:93:94:95:96:97
and such.

> As some of you may know, I've examined acoustical periodicity
> for some time already and I was always on the other "extreme"; i.e. not so
> much interested in to why we hear it the way we do but rather what happens
> to the "shapes" of the waveforms, regardless of the question if someone is
> actually listening to them at that moment or not.

You should start playing around with the Gammatone toolkit in MATLAB.
That's where I think this approach really takes off. You don't just
examine the waveform of the signal, but you also run the signal
through each filter and examine the output waveform of each one of
THOSE signals. This can yield some really interesting insights.

> Eventually, I swapped a couple of offlist messages with
> Mike B and we did some experiments with these phenomena but it seemed noone
> else cared then -- except Rick who was eventually excluded or he had left or
> I'm not sure what's happened to him.

LOL well, Rick Ballan, dear to my heart though he may be, was kind of
a special case. :)

> This time, Mike has done the great
> thing that he took a short sine sweep and repeated it periodically, and why
> is this causing so much controversy if it really is an example of a periodic
> spectrum (i.e. a proper harmonic series) with different phases for each
> harmonic?

I partially blame myself - this stuff is really complicated, and I
find myself struggling to explain the nuances of all of it while still
being brief. There is no short explanation for a lot of it, and so I'm
aiming for comprehensiveness rather than brevity. The end result is
that people don't have any idea what I'm saying sometimes, because I
just drop a lot of DSP jargon without going into detail about what it
all means. This is a catch-22, because if I go into detail about what
it all means, I end up being less brief, and people tune it out. The
solution is for me to write some kind of page on the xenharmonic wiki
about signal processing as applied to music theory, but that'll have
to be later...

> Anyway, before we start discussing sharp timbres, if you listen to sines of
> 2300Hz and 2000Hz together, don't you hear 1700Hz softly in a similar
> sine-like timbre as well and also 300Hz in a sharper timbre? And don't you
> hear 1400Hz if you raise the volume a bit? And don't you hear 1100Hz if you
> raise it further? And if you do, does it tell you something very clear about
> a part of this whole topic?

I think that the very clear thing that it tells me is this:

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

Read the thing about DPOAE's. Your ear, effectively, is distorting the
signal. We had a pretty intense discussion about all of this over the
summer with regards to the new model of cochlear amplification that is
coming out.

> To me it certainly does. Try running this simple
> dyad through a gentle overdriving effect and then scanning the spectrum
> through a long sine sweep convolution. You'll hear 200Hz, 500, 800, 1100,
> 1400, 1700, ... But not 300. 300 is the result of the amplitude modulation
> (that may be the reason for the narrowband-like timbre)

I'm not sure that this is exactly what's going on though. That's
actually an interesting idea, that this buzzing we're delving into
actually does cause the activation of the frequency differential. I'm
not sure I know enough about the way the signal is passed from the
cochlea through to the auditory nerve to comment on this one way or
another. But without modeling any such nonlinear effects, it looks
like the 300 Hz is basically represented not as an actual pitch in the
signal, but as a literal 300 Hz amplitude modulation for the notes.
The reason we hear 300 Hz, as per your example above, is probably
related to DPOAE's.

> but the spectrum itself doesn't contain this frequency even after being overdriven because
> the phase relations which are 1/300-of-a-second apart actually differ by 120
> degrees.

Or put more simply, when you run the dyad through an overdrive effect,
you're only going to get even-order distortion, and 300 Hz won't be
present there anyway. Instead of running it through overdrive like
that, which can be modeled as arctan(k*y), where 0<k<Inf is the
distortion scaling factor and y is the current sample, do
log(1+1/k+y), where 0<k<Inf is the distortion scaling factor, and then
convolve with a chirp, and you'll hear the 300 Hz just fine. This
produces asymmetrical distortion, so you get odd-order distortion
products too. So if we're hearing 300 Hz, and it really is due to
otoacoustic emissions, then the nonlinear effects taking place in the
cochlea clearly have to asymmetrically distort the waveform.

-Mike