virtual pitch and rootedness

Hi Kurt and All;

I believe the notion of "virtual pitch", "periodicity pitch",
or "implied fundamental" as it is variously called, first
came about due to an observation regarding additive synthesis:

1. A whole bunch of sine waves arranged in a harmonic
series sounds like a single tone with a pitch corresponding
to the fundamental of that series.

2. Failing to include the fundamental in the synthesis
does not change #1.

I've even heard that the frequency response of early telephone
systems did not cover the range of typical male voices, yet
this did not cause a problem. That may be an urban legend,
though.

As described in The Just Intonation Primer, this observation
cannot be explained by difference tones, as it happens even
if you:

() Do things at very low volumes, where difference tones
are insignificant.

() Present the stimuli binaurally.

() Put a bunch of noise in the signal.

In music, there is a notion of "root". I'll define this as
what letter is used to identify the chord above that music
in a chart (think jazz). There's no general consensus on
how to do this. Usually major triads like 4:5:6 are marked
C where the 4 is a C... 5:6:8 would most likely be marked C
as well, and 3:4:5. The notes C,Eb,Gb,Bb might be called a
C half-dim chord. BUT, the notes Eb,Gb,Bb,C would likely be
called an Ebmin6.

This just demonstrates that "inversions" sortof only hold
for triads. But in geneeral, the context of the music is
used to determine the spelling of a chord, and indeed
context is fundamental to perception. Because of this,
perhaps, musicians haven't worried about a rigorous formula
for stand-alone chords.

Ok, my claim is that these two concepts -- the virtual
fundamental and rootedness* -- are related. Maybe not
to the point of sharing an exact neurophysiological origin,
but related.

That's all I want to say for now.

-Carl

* disclaimer: the term "rootedness" has been used on these
lists before, and NO similarity between that usage and the
usage above should be assumed]

on 8/18/04 8:12 PM, Carl Lumma <ekin@lumma.org> wrote:

> I believe the notion of "virtual pitch", "periodicity pitch",
> or "implied fundamental" as it is variously called, first
> came about due to an observation regarding additive synthesis:
>
> 1. A whole bunch of sine waves arranged in a harmonic
> series sounds like a single tone with a pitch corresponding
> to the fundamental of that series.
>
> 2. Failing to include the fundamental in the synthesis
> does not change #1.
>
> I've even heard that the frequency response of early telephone
> systems did not cover the range of typical male voices, yet
> this did not cause a problem. That may be an urban legend,
> though.

Yes, its all coming back to me now. This is very clear and for me has no
relation in my experience to the general class of phenomena that I think can
be called "resultant tones" which certainly includes difference tones and
sum tones (and presumably *not* virtual pitch--or am I wrong?).

> As described in The Just Intonation Primer, this observation
> cannot be explained by difference tones, as it happens even
> if you:
>
> () Do things at very low volumes, where difference tones
> are insignificant.
>
> () Present the stimuli binaurally.
>
> () Put a bunch of noise in the signal.
>
> In music, there is a notion of "root". I'll define this as
> what letter is used to identify the chord above that music
> in a chart (think jazz). There's no general consensus on
> how to do this. Usually major triads like 4:5:6 are marked
> C where the 4 is a C... 5:6:8 would most likely be marked C
> as well, and 3:4:5. The notes C,Eb,Gb,Bb might be called a
> C half-dim chord. BUT, the notes Eb,Gb,Bb,C would likely be
> called an Ebmin6.
>
> This just demonstrates that "inversions" sortof only hold
> for triads. But in geneeral, the context of the music is
> used to determine the spelling of a chord, and indeed
> context is fundamental to perception. Because of this,
> perhaps, musicians haven't worried about a rigorous formula
> for stand-alone chords.
>
> Ok, my claim is that these two concepts -- the virtual
> fundamental and rootedness* -- are related. Maybe not
> to the point of sharing an exact neurophysiological origin,
> but related.

Yes, that's interesting, I'm not sure yet. A reinterpretion of what I
previously said is that rootedness sense does not appear to depend on
pitches being so exact that difference tones are heard (by me). That is I
can sense rooteness well enough in 12et where I hear no difference tones.
For the moment, difference tones and virtual pitch are the only 2
explanations I would have for hearing lower pitches that aren't present, so
*my* experience is consistent with your hypothesis. More could be found out
perhaps by using pure tones and seeing how far pitches can be altered from
just before the sense of rootedness is lost. However contextual cues
(including memory of a previously played chord) have to be dealt with since
they could interfere with the result.

-Kurt

>> Ok, my claim is that these two concepts -- the virtual
>> fundamental and rootedness* -- are related. Maybe not
>> to the point of sharing an exact neurophysiological origin,
>> but related.
>
>Yes, that's interesting, I'm not sure yet. A reinterpretion of
>what I previously said is that rootedness sense does not appear
>to depend on pitches being so exact that difference tones are
>heard (by me). That is I can sense rooteness well enough in
>12et where I hear no difference tones.

Yes, mildly inharmonic spectra do not destroy a sensation of
pitch, and temperament should not destroy a sense of root.

But temperament should not remove difference tones either --
they just go out of tune. You must be comparing two different
timbres, two different listening levels, or...?

>For the moment, difference tones and virtual pitch are the only
>2 explanations I would have for hearing lower pitches that
>aren't present, so *my* experience is consistent with your
>hypothesis.

My guess is that you've been describing difference tones.
To me they sound like a hollow, ethereal tone which I usually
have to focus on to hear, and which depends on the position
of my head, or at least phases in and out as my head moves.

I often find myself more aware of sum tones than of difference
tones, but this may be somewhat atypical.

-Carl

on 8/21/04 1:53 AM, Carl Lumma <ekin@lumma.org> wrote:

>>> Ok, my claim is that these two concepts -- the virtual
>>> fundamental and rootedness* -- are related. Maybe not
>>> to the point of sharing an exact neurophysiological origin,
>>> but related.
>>
>> Yes, that's interesting, I'm not sure yet. A reinterpretion of
>> what I previously said is that rootedness sense does not appear
>> to depend on pitches being so exact that difference tones are
>> heard (by me). That is I can sense rooteness well enough in
>> 12et where I hear no difference tones.
>
> Yes, mildly inharmonic spectra do not destroy a sensation of
> pitch, and temperament should not destroy a sense of root.
>
> But temperament should not remove difference tones either --
> they just go out of tune.

They go out of tune and so they don't reinforce each other. I never heard
difference tones until they were reinforced. That's my best guess. I
should try listening to dyads later (mostly I hear what I am talking about
in larger chords), in and out of tune, though even in that case a difference
tone might reinforce one of the two notes. Keep in mind I'm not talking
about pure tones. So there is a whole spectrum of harmonics in a single
note to participate and the actual array of difference tones can therefore
be quite large.

> You must be comparing two different
> timbres, two different listening levels, or...?

Regardless of listening levels, though I've never tested the "extremely low
level" case.

>> For the moment, difference tones and virtual pitch are the only
>> 2 explanations I would have for hearing lower pitches that
>> aren't present, so *my* experience is consistent with your
>> hypothesis.
>
> My guess is that you've been describing difference tones.
> To me they sound like a hollow, ethereal tone which I usually
> have to focus on to hear, and which depends on the position
> of my head, or at least phases in and out as my head moves.

I hear it as a buzzy quality (and so do most others that I've asked), which
I've attributed to the resultant timbre (is that a known concept?) of the
resultant lower-pitched note. In fact the timbre is much buzzier when
mapped onto a lower fundamental, because the harmonic spectrum goes much
higher, in "harmonic space". Buzzy timbres have a log of higher harmonics.

> I often find myself more aware of sum tones than of difference
> tones, but this may be somewhat atypical.

Strange.

-Kurt

>> My guess is that you've been describing difference tones.
>> To me they sound like a hollow, ethereal tone which I usually
>> have to focus on to hear, and which depends on the position
>> of my head, or at least phases in and out as my head moves.
>
>I hear it as a buzzy quality (and so do most others that I've
>asked), which I've attributed to the resultant timbre (is that
>a known concept?) of the resultant lower-pitched note. In
>fact the timbre is much buzzier when mapped onto a lower
>fundamental, because the harmonic spectrum goes much higher,
>in "harmonic space". Buzzy timbres have a log of higher
>harmonics.

Huh. Every description of difference tones I've ever read
is similar to the one I wrote there. They sound almost like
sine tones. If Paul were around, I'm sure he could suggest
some listening scenarios for sorting this out.

I had heard sum tones when tuning pianos, but hadn't noticed
difference tones until Kraig Grady and Erin Barnes played
an Anaphorian metalophone for me. It was something of a
ecstatic experience for me.

-Carl

on 8/21/04 1:02 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> My guess is that you've been describing difference tones.
>>> To me they sound like a hollow, ethereal tone which I usually
>>> have to focus on to hear, and which depends on the position
>>> of my head, or at least phases in and out as my head moves.
>>
>> I hear it as a buzzy quality (and so do most others that I've
>> asked), which I've attributed to the resultant timbre (is that
>> a known concept?) of the resultant lower-pitched note. In
>> fact the timbre is much buzzier when mapped onto a lower
>> fundamental, because the harmonic spectrum goes much higher,
>> in "harmonic space". Buzzy timbres have a log of higher
>> harmonics.
>
> Huh. Every description of difference tones I've ever read
> is similar to the one I wrote there. They sound almost like
> sine tones.

The may sound like sine tones if you put sine tones in (say two). If you
put a pair of "harmonic sets" in (two non-sine tones), the resultant
*includes* for example the differences between the corresponding harmonics
of each set, which if you do the math, turns out to be harmonics of the
fundamental difference tone. Thus the difference tone gets a timbre because
the original tones had a timbre. So it seems implausible to hear a sine
tone in that scenario. Yet that is only part of the picture because there
are more difference tones than that set: there are differences between all
pairs of harmonics of all tones present.

When you were here a couple weeks back you also talked about some
buzzing-like thing. You had your own unique name for it, maybe not
"buzzing". What was that?

> If Paul were around, I'm sure he could suggest
> some listening scenarios for sorting this out.
>
> I had heard sum tones when tuning pianos,

I heard sum tones I *think* when listening to a soprano with a really loud
voice a couple weeks ago. But this was clearly a case of driving my ears to
the point of distortion.

-Kurt

> but hadn't noticed
> difference tones until Kraig Grady and Erin Barnes played
> an Anaphorian metalophone for me. It was something of a
> ecstatic experience for me.

>>>> My guess is that you've been describing difference tones.
>>>> To me they sound like a hollow, ethereal tone which I usually
>>>> have to focus on to hear, and which depends on the position
>>>> of my head, or at least phases in and out as my head moves.
>>>
>>> I hear it as a buzzy quality (and so do most others that I've
>>> asked), which I've attributed to the resultant timbre (is that
>>> a known concept?) of the resultant lower-pitched note. In
>>> fact the timbre is much buzzier when mapped onto a lower
>>> fundamental, because the harmonic spectrum goes much higher,
>>> in "harmonic space". Buzzy timbres have a log of higher
>>> harmonics.
>>
>> Huh. Every description of difference tones I've ever read
>> is similar to the one I wrote there. They sound almost like
>> sine tones.
>
>The may sound like sine tones if you put sine tones in (say two).

I'd love to hear what you're hearing. Maybe tomorrow at 4?

>If you put a pair of "harmonic sets" in (two non-sine tones),
>the resultant *includes* for example the differences between the
>corresponding harmonics of each set,

Seems reasonable, though I've never read about this. There
are 2nd- and higher order combination tones, but I thought their
origin was in the 'modes' of the nonlinearities responsible for
the combination tone effect. Paul's the resident expert here.

>When you were here a couple weeks back you also talked about some
>buzzing-like thing. You had your own unique name for it, maybe
>not "buzzing". What was that?

Periodicity buzz! You might try googling yahoo groups for that
one. I've not experienced this as a pitch, but rather as a
texture. I hear it all over the Twining album.

-Carl

on 8/21/04 6:24 PM, Carl Lumma <ekin@lumma.org> wrote:

>>>> I hear it as a buzzy quality (and so do most others that I've
>>>> asked), which I've attributed to the resultant timbre (is that
>>>> a known concept?) of the resultant lower-pitched note. In
>>>> fact the timbre is much buzzier when mapped onto a lower
>>>> fundamental, because the harmonic spectrum goes much higher,
>>>> in "harmonic space". Buzzy timbres have a log of higher
>>>> harmonics.
>>>
>>> Huh. Every description of difference tones I've ever read
>>> is similar to the one I wrote there. They sound almost like
>>> sine tones.
>>
>> The may sound like sine tones if you put sine tones in (say two).
>
> I'd love to hear what you're hearing. Maybe tomorrow at 4?

Sure.

>> If you put a pair of "harmonic sets" in (two non-sine tones),
>> the resultant *includes* for example the differences between the
>> corresponding harmonics of each set,
>
> Seems reasonable, though I've never read about this. There
> are 2nd- and higher order combination tones, but I thought their
> origin was in the 'modes' of the nonlinearities responsible for
> the combination tone effect. Paul's the resident expert here.

Its pretty straightforward. There's no difference between mixing 4 notes
that each have 4 harmonics and mixing 16 pure sine tones (of the appropriate
pitches). Its identical. If nonlinearity is involved, cross-modulation
between all possible pairs of the 16 will occur.

Then there could be higher-order modulations (involving 3 or more together)
but I haven't heard people talk about this much. Maybe the amplitude of the
higher-order terms is small enough that they can usually be ignored in
ordinary situations.

>> When you were here a couple weeks back you also talked about some
>> buzzing-like thing. You had your own unique name for it, maybe
>> not "buzzing". What was that?
>
> Periodicity buzz! You might try googling yahoo groups for that
> one. I've not experienced this as a pitch, but rather as a
> texture.

> I hear it all over the Twining album.

So do I, but I can hear the pitch. To me a buzz has a pitch, or it wouldn't
be a buzz, it would be a hiss or some other name.

>
> -Carl