virtual pitch and rootedness

>From: Carl Lumma <ekin@l...>
>Date: Wed Aug 18, 2004 11:12 pm
>Subject: 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.

This observation, and its specific application to questions of
musical harmony, goes back to Rameau -- centuries before additive
synthesis was possible.

>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.

It's true not only of early telephones, but modern telephones as
well. The typical fundamental of male voices is still below the lower
end of the frequency response range of most telephones.

>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.

More importantly, when the sine waves are shifted by a constant Hz
addition, the difference tones remain exactly the same, but the
virtual pitch will be heard to shift. So clearly the two are
different phenomena. Perhaps the subject of inharmonic spectra was a
little too "far from home" for the JI Primer, but it demonstrates the
distinction in the most unequivocal way.

>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.

Some triads, like C D G, tend to be called different names (Csus2,
D7sus4, Gsus4) depending on the inversion. Meanwhile, most tetrads
use the same name regardless of which inversion is used.

> >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.
>
> This observation, and its specific application to questions of
> musical harmony, goes back to Rameau -- centuries before
> additive synthesis was possible.

Precisely what was his observation? Did he call it "virtual
pitch"? Of course, additive synthesis predates Rameau.

> >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.
>
> It's true not only of early telephones, but modern telephones as
> well. The typical fundamental of male voices is still below the
> lower end of the frequency response range of most telephones.

Really?

> >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.
>
> More importantly, when the sine waves are shifted by a constant
> Hz addition, the difference tones remain exactly the same, but
> the virtual pitch will be heard to shift. So clearly the two are
> different phenomena. Perhaps the subject of inharmonic spectra
> was a little too "far from home" for the JI Primer, but it
> demonstrates the distinction in the most unequivocal way.

That *is* a better demonstration.

> >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.
>
> Some triads, like C D G, tend to be called different names
> (Csus2, D7sus4, Gsus4) depending on the inversion. Meanwhile,
> most tetrads use the same name regardless of which inversion
> is used.

I was using the stronger meaning of "triad" there, but
mixing it with the weaker meaning of "tetrad". It's true
that chord size isn't the issue, though in the case of
utonal chord inversions it may be.

-Carl

--- In harmonic_entropy@yahoogroups.com, "Carl Lumma" <ekin@l...>
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.
> >
> > This observation, and its specific application to questions of
> > musical harmony, goes back to Rameau -- centuries before
> > additive synthesis was possible.
>
> Precisely what was his observation?

The harmonic series had been just recently discovered. Rameau was the
first to combine what are known today as different inversions of a
chord into a single category, which is still used in most chord
analyses today. Before him, the modern concept of "major
triad", "minor triad", etc., invariant as to inversion, did not
exist. Anyway, Rameau observed that the roots of chords corresponded
to the fundamental of harmonic series that included many or all of
the chord tones as partials. He had to 'massage' things a bit, of
course, for the case of minor chords. He then developed an entire
theory of tonal harmony from this observation. It's the same
observation you state above. If you want it a little more precisely,
I suggest obtaining Rameau's book (I saw it in Boston's Tower Records
before it became Virgin).

> Did he call it "virtual
> pitch"?

No.

> Of course, additive synthesis predates Rameau.

Hmm . . . is it that obvious? I can't think of an example. Can you
fill me in? It doesn't seem possible, since even producing a sine
wave was far beyond the technology of the time, let alone combining
different sine wave sources together.

> > >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.
> >
> > It's true not only of early telephones, but modern telephones as
> > well. The typical fundamental of male voices is still below the
> > lower end of the frequency response range of most telephones.
>
> Really?

Yes -- dig up some specs and you'll readily see this is the case.

>> Precisely what was his observation?
>
>The harmonic series had been just recently discovered. Rameau was
>the first to combine what are known today as different inversions
>of a chord into a single category, which is still used in most
>chord analyses today. Before him, the modern concept of "major
>triad", "minor triad", etc., invariant as to inversion, did not
>exist. Anyway, Rameau observed that the roots of chords corresponded
>to the fundamental of harmonic series that included many or all of
>the chord tones as partials. He had to 'massage' things a bit, of
>course, for the case of minor chords. He then developed an entire
>theory of tonal harmony from this observation. It's the same
>observation you state above. If you want it a little more precisely,
>I suggest obtaining Rameau's book (I saw it in Boston's Tower
>Records before it became Virgin).

Cool. Maybe I'll check it out at the library.

>> Of course, additive synthesis predates Rameau.
>
>Hmm . . . is it that obvious? I can't think of an example. Can you
>fill me in? It doesn't seem possible, since even producing a sine
>wave was far beyond the technology of the time, let alone combining
>different sine wave sources together.

Pipe organs, chiefly. "Additive synthesis" isn't restricted to
techniques that allow control over individual partials. Analog
synthesis with a few rich waveforms is sometimes called additive.

>> > >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.
>> >
>> > It's true not only of early telephones, but modern telephones
>> > as well. The typical fundamental of male voices is still below
>> > the lower end of the frequency response range of most telephones.
>>
>> Really?
>
>Yes -- dig up some specs and you'll readily see this is the case.

I wonder why.... the lines support supersonic frequencies for
DSL, and even the smallest/cheapest condenser mic ought to go
down to 80Hz.

-Carl

At 11:43 AM 20/08/2004, Carl wrote:
>I wonder why.... the lines support supersonic frequencies for
>DSL, and even the smallest/cheapest condenser mic ought to go
>down to 80Hz.

The lines from a subscriber to the first exchange support those frequencies, but between exchanges, whether via copper, optic fibre, microwave, satellite, undersea cable, all rely on being able to use any extra bandwidth to carry multiple calls simultaneously. The less bandwidth used for each call, the more can be carried at once. So each call is limited to the range of 300 Hz to 3.4 kHz (or sometimes as little as 400 Hz to 3.0 kHz) because experiments showed that limiting it like this makes very little difference to the intelligibility of speech (and even music).

>>I wonder why.... the lines support supersonic frequencies for
>>DSL, and even the smallest/cheapest condenser mic ought to go
>>down to 80Hz.
>
>The lines from a subscriber to the first exchange support those
>frequencies, but between exchanges, whether via copper, optic fibre,
>microwave, satellite, undersea cable, all rely on being able to use
>any extra bandwidth to carry multiple calls simultaneously. The less
>bandwidth used for each call, the more can be carried at once. So
>each call is limited to the range of 300 Hz to 3.4 kHz (or sometimes
>as little as 400 Hz to 3.0 kHz) because experiments showed that
>limiting it like this makes very little difference to the
>intelligibility of speech (and even music).

This checks out. You're a knowledgeable fellow, Dave K.!

-Carl

At 12:40 PM 20/08/2004, Carl wrote:
>This checks out. You're a knowledgeable fellow, Dave K.!

Nah. Just happen to have been trained as a Telecom technician in my younger days (and used Google to check that my memory wasn't failing me). "300 Hz to 3.4 kHz" was like a mantra in Telecom training school.

I wrote...

> This checks out. You're a knowledgeable fellow, Dave K.!

And Paul too! (duh)

-C.

on 8/19/04 2:02 PM, Carl Lumma <ekin@...> wrote:

>> More importantly, when the sine waves are shifted by a constant
>> Hz addition, the difference tones remain exactly the same, but
>> the virtual pitch will be heard to shift. So clearly the two are
>> different phenomena. Perhaps the subject of inharmonic spectra
>> was a little too "far from home" for the JI Primer, but it
>> demonstrates the distinction in the most unequivocal way.
>
> That *is* a better demonstration.

We can use this when we get together to listen. My digital organ software
supports a constant Hz offset via the "celeste" feature. This should be
interesting.

-Kurt

>>> More importantly, when the sine waves are shifted by a constant
>>> Hz addition, the difference tones remain exactly the same, but
>>> the virtual pitch will be heard to shift. So clearly the two are
>>> different phenomena. Perhaps the subject of inharmonic spectra
>>> was a little too "far from home" for the JI Primer, but it
>>> demonstrates the distinction in the most unequivocal way.
>>
>> That *is* a better demonstration.
>
>We can use this when we get together to listen. My digital organ
>software supports a constant Hz offset via the "celeste" feature.
>This should be interesting.

But it doesn't make sine waves... or does it?

-Carl

on 8/22/04 12:43 AM, Carl Lumma <ekin@...> wrote:

>>>> More importantly, when the sine waves are shifted by a constant
>>>> Hz addition, the difference tones remain exactly the same, but
>>>> the virtual pitch will be heard to shift. So clearly the two are
>>>> different phenomena. Perhaps the subject of inharmonic spectra
>>>> was a little too "far from home" for the JI Primer, but it
>>>> demonstrates the distinction in the most unequivocal way.
>>>
>>> That *is* a better demonstration.
>>
>> We can use this when we get together to listen. My digital organ
>> software supports a constant Hz offset via the "celeste" feature.
>> This should be interesting.
>
> But it doesn't make sine waves... or does it?

Right, I was thinking it would offset all the partials, but that was a
nonsense thought. However it does make sine waves as an option and can be
enhanced fairly easily to apply the offset to each partial. But maybe not
today.

-Kurt

on 8/22/04 3:36 PM, Kurt Bigler <kkb@...> wrote:

> on 8/22/04 12:43 AM, Carl Lumma <ekin@...> wrote:
>
>>>>> More importantly, when the sine waves are shifted by a constant
>>>>> Hz addition, the difference tones remain exactly the same, but
>>>>> the virtual pitch will be heard to shift. So clearly the two are
>>>>> different phenomena. Perhaps the subject of inharmonic spectra
>>>>> was a little too "far from home" for the JI Primer, but it
>>>>> demonstrates the distinction in the most unequivocal way.
>>>>
>>>> That *is* a better demonstration.
>>>
>>> We can use this when we get together to listen. My digital organ
>>> software supports a constant Hz offset via the "celeste" feature.
>>> This should be interesting.
>>
>> But it doesn't make sine waves... or does it?
>
> Right, I was thinking it would offset all the partials, but that was a
> nonsense thought.

It turned out my original thought was correct and my rethinking wrong.

> However it does make sine waves as an option and can be
> enhanced fairly easily to apply the offset to each partial. But maybe not
> today.

Just to report the results...

Well as it turned out I didn't need to make that enhancement. The code
already worked that way, but I tweaked it a little to increase the range of
the frequency offsets, etc. and Carl was here with me and heard the results.

I set up a sequence of partials 1 through 9, with slightly decreasing
amplitudes, a "rich" timbre as Carl said. (Well less rich than many being
limited to 9 harmonics.) What we heard when I played a 8:9:10:11 chord with
that "timbre" when I offset the partials upwards was that the "resultant"
pitch went up too.

So we were hearing *mainly* the implied fundamental (virtual pitch), *not*
the difference tones as I had always suspected.

And in fact this is the same phenomenon I am very familiar with and recently
called "implied fundamental" which I then thought was incorrect because I
had *assumed* I was hearing difference tones. So that was a surprising
result for me.

We heard some other things too, some of which depended more on head
position, but we didn't have enough time to investigate it all thoroughly.
The primary result was that the main additional pitch hear was the virtual
pitch based on Pauls proposed experiment. A secondary result was that Carl
was hearing at least a 2nd virtual pitches which was approximately the 3rd
harmonic of what I will call the main one. This is no surprise since the
3rd partial was present in all 4 pitches in the 8:9:10:11 chord and in the
frequency-offset case, the partial are not exactly harmonic. And therefore
hearing an additional virtual pitch seems to me to make sense. The ear can
do a lot of different things--though perhaps not all at once. I don't think
Carl reported hearing both virtual pitches at the same time.

-Kurt

--- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Just to report the results...
>
> Well as it turned out I didn't need to make that enhancement. The
code
> already worked that way, but I tweaked it a little to increase the
range of
> the frequency offsets, etc. and Carl was here with me and heard the
results.
>
> I set up a sequence of partials 1 through 9, with slightly
decreasing
> amplitudes, a "rich" timbre as Carl said. (Well less rich than
many being
> limited to 9 harmonics.) What we heard when I played a 8:9:10:11
chord with
> that "timbre" when I offset the partials upwards was that
the "resultant"
> pitch went up too.

I'm confused. Can you explain what you did a little more fully?
Wouldn't the main thing you hear when 8:9:10:11 is played with
inharmonic timbres be that it starts to beat?

I would have recommended a different experiment -- construct a
*timbre* where the only partials are, say, 5, 6, 7, 8, and 9; offset
them by a constant Hz offset; and observe the effect on the heard
pitch.

> So we were hearing *mainly* the implied fundamental (virtual
pitch), *not*
> the difference tones as I had always suspected.

Hopefully the experiment I'm suggesting would lead you to this
conclusion too. I'm kind of missing why your experiment led to it.

> And in fact this is the same phenomenon I am very familiar with and
recently
> called "implied fundamental" which I then thought was incorrect
because I
> had *assumed* I was hearing difference tones. So that was a
surprising
> result for me.

Well, I'm glad you learned something, and welcome to (in a sense) the
main idea behind this list! :)

Paul,

Sorry so slow getting back on this. Somehow I just missed your message
until now...

on 8/24/04 9:13 PM, wallyesterpaulrus <wallyesterpaulrus@...> wrote:

> --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Just to report the results...
>>
>> Well as it turned out I didn't need to make that enhancement. The
> code
>> already worked that way, but I tweaked it a little to increase the
> range of
>> the frequency offsets, etc. and Carl was here with me and heard the
> results.
>>
>> I set up a sequence of partials 1 through 9, with slightly
> decreasing
>> amplitudes, a "rich" timbre as Carl said. (Well less rich than
> many being
>> limited to 9 harmonics.) What we heard when I played a 8:9:10:11
> chord with
>> that "timbre" when I offset the partials upwards was that
> the "resultant"
>> pitch went up too.
>
> I'm confused. Can you explain what you did a little more fully?
> Wouldn't the main thing you hear when 8:9:10:11 is played with
> inharmonic timbres be that it starts to beat?
>
> I would have recommended a different experiment -- construct a
> *timbre* where the only partials are, say, 5, 6, 7, 8, and 9; offset
> them by a constant Hz offset; and observe the effect on the heard
> pitch.

Ok, so you are just suggesting omitting partials 1 through 4? Otherwise
this sounds identical to what I did.

Carl will maybe be over this week and perhaps we can repeat the experiment
with your suggested modification.

>> So we were hearing *mainly* the implied fundamental (virtual
> pitch), *not*
>> the difference tones as I had always suspected.
>
> Hopefully the experiment I'm suggesting would lead you to this
> conclusion too. I'm kind of missing why your experiment led to it.
>
>> And in fact this is the same phenomenon I am very familiar with and
> recently
>> called "implied fundamental" which I then thought was incorrect
> because I
>> had *assumed* I was hearing difference tones. So that was a
> surprising
>> result for me.
>
> Well, I'm glad you learned something, and welcome to (in a sense) the
> main idea behind this list! :)

Still it might be interesting if you could say briefly how harmonic entropy
is related to virtual pitch. To me harmonic entropy (per se) to me has to
do with how the ability to distinguish different harmonics as
individually-heard pitches varies as a function of where the pitch fits into
the harmonic space, because of the "structure" of that space. That's a bit
too brief but maybe you get what I mean. (Admittedly I haven't read much
yet, so you can feel free just to refer me to something.)

-Kurt

--- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> Paul,
>
> Sorry so slow getting back on this. Somehow I just missed your
message
> until now...
>
> on 8/24/04 9:13 PM, wallyesterpaulrus <wallyesterpaulrus@y...>
wrote:
>
> > --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...>
wrote:
> >
> >> Just to report the results...
> >>
> >> Well as it turned out I didn't need to make that enhancement.
The
> > code
> >> already worked that way, but I tweaked it a little to increase
the
> > range of
> >> the frequency offsets, etc. and Carl was here with me and heard
the
> > results.
> >>
> >> I set up a sequence of partials 1 through 9, with slightly
> > decreasing
> >> amplitudes, a "rich" timbre as Carl said. (Well less rich than
> > many being
> >> limited to 9 harmonics.) What we heard when I played a 8:9:10:11
> > chord with
> >> that "timbre" when I offset the partials upwards was that
> > the "resultant"
> >> pitch went up too.
> >
> > I'm confused. Can you explain what you did a little more fully?
> > Wouldn't the main thing you hear when 8:9:10:11 is played with
> > inharmonic timbres be that it starts to beat?

Still hoping for this clarification . . .

> > I would have recommended a different experiment -- construct a
> > *timbre* where the only partials are, say, 5, 6, 7, 8, and 9;
offset
> > them by a constant Hz offset; and observe the effect on the heard
> > pitch.
>
> Ok, so you are just suggesting omitting partials 1 through 4?
Otherwise
> this sounds identical to what I did.

Really? The above seems to suggest you did something different,
namely chords played using complex (non-sine-wave) timbres. I suppose
your clarification will sort this all out . . .

> Still it might be interesting if you could say briefly how harmonic
entropy
> is related to virtual pitch.

Unlike roughness or combinational tone models of discordance,
harmonic entropy actually acknowledges the virtual pitch phenomenon.
It attempts to determine how clear or unclear the choice is between
various virtual pitch possibilities given a certain set of overtones
(which may or may not have overtones of their own -- if they do, then
we're talking about Parncutt's extension of the virtual pitch idea to
explain roots of chords). Concordance is identified with clarity in
this choice; discordance with uncertainty.

> To me harmonic entropy (per se) to me has to
> do with how the ability to distinguish different harmonics as
> individually-heard pitches

Hmm . . . doesn't seem right so far.

> varies as a function of where the pitch fits into
> the harmonic space, because of the "structure" of that space.

Wow, that really seems way off, at least given how I understand
harmonic space.

>That's a bit
> too brief but maybe you get what I mean.

Maybe I don't, because it seems all wrong. But maybe it's just a
matter of wildly differing vocabularies for the same ideas.

on 9/29/04 3:16 PM, wallyesterpaulrus <wallyesterpaulrus@...> wrote:

> --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> Paul,
>>
>> Sorry so slow getting back on this. Somehow I just missed your
> message
>> until now...
>>
>> on 8/24/04 9:13 PM, wallyesterpaulrus <wallyesterpaulrus@y...>
> wrote:
>>
>>> --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...>
> wrote:
>>>
>>>> Just to report the results...
>>>>
>>>> Well as it turned out I didn't need to make that enhancement.
> The
>>> code
>>>> already worked that way, but I tweaked it a little to increase
> the
>>> range of
>>>> the frequency offsets, etc. and Carl was here with me and heard
> the
>>> results.
>>>>
>>>> I set up a sequence of partials 1 through 9, with slightly
>>> decreasing
>>>> amplitudes, a "rich" timbre as Carl said. (Well less rich than
>>> many being
>>>> limited to 9 harmonics.) What we heard when I played a 8:9:10:11
>>> chord with
>>>> that "timbre" when I offset the partials upwards was that
>>> the "resultant"
>>>> pitch went up too.
>>>
>>> I'm confused. Can you explain what you did a little more fully?
>>> Wouldn't the main thing you hear when 8:9:10:11 is played with
>>> inharmonic timbres be that it starts to beat?
>
> Still hoping for this clarification . . .
>
>>> I would have recommended a different experiment -- construct a
>>> *timbre* where the only partials are, say, 5, 6, 7, 8, and 9;
> offset
>>> them by a constant Hz offset; and observe the effect on the heard
>>> pitch.
>>
>> Ok, so you are just suggesting omitting partials 1 through 4?
> Otherwise
>> this sounds identical to what I did.
>
> Really?

Yes, I'm quite sure of it.

> The above seems to suggest you did something different,
> namely chords played using complex (non-sine-wave) timbres. I suppose
> your clarification will sort this all out . . .

Well the above explanation refers to "partials 1 through 9". I don't use
the word partial for something that isn't non-sine. The "timbre" was
constructed from partials. In the software, the frequency of each sine
partial equals the fundamental times the partial number plus a frequency
offset, with the multiplication being done before the addition.

>> Still it might be interesting if you could say briefly how harmonic
> entropy
>> is related to virtual pitch.
>
> Unlike roughness or combinational tone models of discordance,
> harmonic entropy actually acknowledges the virtual pitch phenomenon.
> It attempts to determine how clear or unclear the choice is between
> various virtual pitch possibilities given a certain set of overtones
> (which may or may not have overtones of their own -- if they do, then
> we're talking about Parncutt's extension of the virtual pitch idea to
> explain roots of chords). Concordance is identified with clarity in
> this choice; discordance with uncertainty.
>
>> To me harmonic entropy (per se) to me has to
>> do with how the ability to distinguish different harmonics as
>> individually-heard pitches
>
> Hmm . . . doesn't seem right so far.

Well, I said some things badly and some things wrong there. The ability to
"identify" is probably more to the point, and it is not just about
"harmonics" but about relative pitches that fall on or near certain ratios.
By "individually-heard" I was thinking in terms of notes in a chord, which
tend to be individually identifiable even if there is some tendency for the
chord to fuse into a single timbre.

>> varies as a function of where the pitch fits into
>> the harmonic space, because of the "structure" of that space.
>
> Wow, that really seems way off, at least given how I understand
> harmonic space.

Yes, here "rational space" is more to the point. I was referring to the
structure of rational space, and how this affects the entropy. And I think
the object that has an entropy is in fact an interval.

>> That's a bit
>> too brief but maybe you get what I mean.
>
> Maybe I don't, because it seems all wrong. But maybe it's just a
> matter of wildly differing vocabularies for the same ideas.

A little better this time?

-Kurt

--- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 9/29/04 3:16 PM, wallyesterpaulrus <wallyesterpaulrus@y...>
wrote:
>
> > --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...>
wrote:
> >> Paul,
> >>
> >> Sorry so slow getting back on this. Somehow I just missed your
> > message
> >> until now...
> >>
> >> on 8/24/04 9:13 PM, wallyesterpaulrus <wallyesterpaulrus@y...>
> > wrote:
> >>
> >>> --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...>
> > wrote:
> >>>
> >>>> Just to report the results...
> >>>>
> >>>> Well as it turned out I didn't need to make that enhancement.
> > The
> >>> code
> >>>> already worked that way, but I tweaked it a little to increase
> > the
> >>> range of
> >>>> the frequency offsets, etc. and Carl was here with me and heard
> > the
> >>> results.
> >>>>
> >>>> I set up a sequence of partials 1 through 9, with slightly
> >>> decreasing
> >>>> amplitudes, a "rich" timbre as Carl said. (Well less rich than
> >>> many being
> >>>> limited to 9 harmonics.) What we heard when I played a
8:9:10:11
> >>> chord with
> >>>> that "timbre" when I offset the partials upwards was that
> >>> the "resultant"
> >>>> pitch went up too.
> >>>
> >>> I'm confused. Can you explain what you did a little more fully?
> >>> Wouldn't the main thing you hear when 8:9:10:11 is played with
> >>> inharmonic timbres be that it starts to beat?
> >
> > Still hoping for this clarification . . .
> >
> >>> I would have recommended a different experiment -- construct a
> >>> *timbre* where the only partials are, say, 5, 6, 7, 8, and 9;
> > offset
> >>> them by a constant Hz offset; and observe the effect on the
heard
> >>> pitch.
> >>
> >> Ok, so you are just suggesting omitting partials 1 through 4?
> > Otherwise
> >> this sounds identical to what I did.
> >
> > Really?
>
> Yes, I'm quite sure of it.

So where does 8:9:10:11 come it? I'm really confused.

> > The above seems to suggest you did something different,
> > namely chords played using complex (non-sine-wave) timbres. I
suppose
> > your clarification will sort this all out . . .
>
> Well the above explanation refers to "partials 1 through 9".

Right.

> I don't use
> the word partial for something that isn't non-sine.

Right.

> The "timbre" was
> constructed from partials.

Right.

> In the software, the frequency of each sine
> partial equals the fundamental times the partial number plus a
frequency
> offset, with the multiplication being done before the addition.

Right. But clearly something's missing here. First you have partials
1 through 9, then you have 8:9:10:11 constructed from those partial 1-
9 timbres? That what you seemed to be saying above, but now you're
insisting otherwise. Meanwhile, my experiment would only have one
list of integers involved, while you clearly have two lists (1-9 and
8-11). Where have we lost each other?

> >> Still it might be interesting if you could say briefly how
harmonic
> > entropy
> >> is related to virtual pitch.
> >
> > Unlike roughness or combinational tone models of discordance,
> > harmonic entropy actually acknowledges the virtual pitch
phenomenon.
> > It attempts to determine how clear or unclear the choice is
between
> > various virtual pitch possibilities given a certain set of
overtones
> > (which may or may not have overtones of their own -- if they do,
then
> > we're talking about Parncutt's extension of the virtual pitch
idea to
> > explain roots of chords). Concordance is identified with clarity
in
> > this choice; discordance with uncertainty.
> >
> >> To me harmonic entropy (per se) to me has to
> >> do with how the ability to distinguish different harmonics as
> >> individually-heard pitches
> >
> > Hmm . . . doesn't seem right so far.
>
> Well, I said some things badly and some things wrong there. The
ability to
> "identify" is probably more to the point, and it is not just about
> "harmonics" but about relative pitches that fall on or near certain
ratios.

Relative pitches? You mean intervals?

> By "individually-heard" I was thinking in terms of notes in a
chord, which
> tend to be individually identifiable even if there is some tendency
for the
> chord to fuse into a single timbre.

When you said "harmonics", did you mean "notes of a chord"? Of
course, most chords harmonic entropy will be evaluated for will not
resemble harmonic series, but those that do tend to be local
minimizers of harmonic entropy.

> >> varies as a function of where the pitch fits into
> >> the harmonic space, because of the "structure" of that space.
> >
> > Wow, that really seems way off, at least given how I understand
> > harmonic space.
>
> Yes, here "rational space" is more to the point.

Hmm, the meaning still seems the same to me.

> I was referring to the
> structure of rational space, and how this affects the entropy.

I can't see for the life of me how one could possibly affect the
other.

> And I think
> the object that has an entropy is in fact an interval.

In most of the cases studied so far, yes, but future studies will
focus on larger chords more heavily.

on 10/3/04 5:51 PM, wallyesterpaulrus <wallyesterpaulrus@...> wrote:

>
>
> --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 9/29/04 3:16 PM, wallyesterpaulrus <wallyesterpaulrus@y...>
> wrote:
>>
>>> --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...>
> wrote:
>>>> Paul,
>>>>
>>>> Sorry so slow getting back on this. Somehow I just missed your
>>> message
>>>> until now...
>>>>
>>>> on 8/24/04 9:13 PM, wallyesterpaulrus <wallyesterpaulrus@y...>
>>> wrote:
>>>>
>>>>> --- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...>
>>> wrote:
>>>>>
>>>>>> Just to report the results...
>>>>>>
>>>>>> Well as it turned out I didn't need to make that enhancement.
>>> The
>>>>> code
>>>>>> already worked that way, but I tweaked it a little to increase
>>> the
>>>>> range of
>>>>>> the frequency offsets, etc. and Carl was here with me and heard
>>> the
>>>>> results.
>>>>>>
>>>>>> I set up a sequence of partials 1 through 9, with slightly
>>>>> decreasing
>>>>>> amplitudes, a "rich" timbre as Carl said. (Well less rich than
>>>>> many being
>>>>>> limited to 9 harmonics.) What we heard when I played a
> 8:9:10:11
>>>>> chord with
>>>>>> that "timbre" when I offset the partials upwards was that
>>>>> the "resultant"
>>>>>> pitch went up too.
>>>>>
>>>>> I'm confused. Can you explain what you did a little more fully?
>>>>> Wouldn't the main thing you hear when 8:9:10:11 is played with
>>>>> inharmonic timbres be that it starts to beat?
>>>
>>> Still hoping for this clarification . . .
>>>
>>>>> I would have recommended a different experiment -- construct a
>>>>> *timbre* where the only partials are, say, 5, 6, 7, 8, and 9;
>>> offset
>>>>> them by a constant Hz offset; and observe the effect on the
> heard
>>>>> pitch.
>>>>
>>>> Ok, so you are just suggesting omitting partials 1 through 4?
>>> Otherwise
>>>> this sounds identical to what I did.
>>>
>>> Really?
>>
>> Yes, I'm quite sure of it.
>
> So where does 8:9:10:11 come it? I'm really confused.

That's the chord. Each note in the chord has partials 1 through 9. Still
seems like that's just what were are describing (with partials 5 through 9).
Once you set up the partials surely you weren't just going to play one note,
were you? Or maybe you were.

>>> The above seems to suggest you did something different,
>>> namely chords played using complex (non-sine-wave) timbres. I
> suppose
>>> your clarification will sort this all out . . .
>>
>> Well the above explanation refers to "partials 1 through 9".
>
> Right.
>
>> I don't use
>> the word partial for something that isn't non-sine.
>
> Right.
>
>> The "timbre" was
>> constructed from partials.
>
> Right.
>
>> In the software, the frequency of each sine
>> partial equals the fundamental times the partial number plus a
> frequency
>> offset, with the multiplication being done before the addition.
>
> Right. But clearly something's missing here. First you have partials
> 1 through 9, then you have 8:9:10:11 constructed from those partial 1-
> 9 timbres? That what you seemed to be saying above, but now you're
> insisting otherwise.

No, I insisted on the wrong thing.

> Meanwhile, my experiment would only have one
> list of integers involved, while you clearly have two lists (1-9 and
> 8-11). Where have we lost each other?

Yup, two lists for me, one list for you. The overall sound is what I think
you would call the combination product of the list or partials and the list
of notes.

The whole experiment relates back to what in retrospect is likely to have
been a misinterpretation of what I heard you suggest. I wanted to study a
particular phenomenon and the experiment seemed to clarify what I was
hearing. The phenomenon I wanted to study in the first place was in fact
involving complex harmonic timbres combined into a chord. I wanted to know
whether the resultant tone I was hearing was a clustering of difference
tones or whether it was a virtual pitch. By substituting a timbre over
which I had full control (allowing me to make it inharmonic) I was then able
to apply offsets. Keep in mind I was adjusting a frequency offset applied
to the entire list of partials across all notes in the chord. So you could
ignore the fact that two lists were involved, because the offset was not a
function of the base frequency, but just an absolute value in Hz. So you
could think of it as a larger list rather than two lists, but the larger
list is the combination product of the two lists. The offset is applied to
every element in the combination product. This allowed the experiment to
reflect something like the actual situation that I am used to hearing, and
in fact allowed me to both confirm that the qualitative phenomenon was the
same one I had been hearing, but also allowed me to tweak all the partials.
The result proved it was not a difference tone phenomenon because the
"resultant" pitch did not stay constant when I aplied the offset.

>> Well, I said some things badly and some things wrong there. The
> ability to
>> "identify" is probably more to the point, and it is not just about
>> "harmonics" but about relative pitches that fall on or near certain
> ratios.
>
> Relative pitches? You mean intervals?

That's what I meant.

>> By "individually-heard" I was thinking in terms of notes in a
> chord, which
>> tend to be individually identifiable even if there is some tendency
> for the
>> chord to fuse into a single timbre.
>
> When you said "harmonics", did you mean "notes of a chord"?

When I said harmonics I wasn't thinking clearly. That was 2 messages back
now.

> Of
> course, most chords harmonic entropy will be evaluated for will not
> resemble harmonic series, but those that do tend to be local
> minimizers of harmonic entropy.

Good, that confirms my understanding of things.
>
>>>> varies as a function of where the pitch fits into
>>>> the harmonic space, because of the "structure" of that space.
>>>
>>> Wow, that really seems way off, at least given how I understand
>>> harmonic space.
>>
>> Yes, here "rational space" is more to the point.
>
> Hmm, the meaning still seems the same to me.
>
>> I was referring to the
>> structure of rational space, and how this affects the entropy.
>
> I can't see for the life of me how one could possibly affect the
> other.

Must be semantics, but I don't know if I can clear it up at this moment.

But I'll try this anyway -- here is how I understand things:

You have a harmonic entropy function. This is a function of a real number
representing the interval. The function itself gets its value because of
something like the degree to which low-integer rationals cluster around it.
The more nearby rationals, the more ambiguity, and I guess that means higher
entropy. Lower entropy means less ambiguity, more clear identification of
an interval as a pure ratio.
>
>> And I think
>> the object that has an entropy is in fact an interval.
>
> In most of the cases studied so far, yes, but future studies will
> focus on larger chords more heavily.

Right, that's all familiar.

-Kurt

--- In harmonic_entropy@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > So where does 8:9:10:11 come it? I'm really confused.
>
> That's the chord. Each note in the chord has partials 1 through
9. Still
> seems like that's just what were are describing (with partials 5
through 9).
> Once you set up the partials surely you weren't just going to play
one note,
> were you? Or maybe you were.

Yes, one note is exactly what I was talking about.
>
> > Meanwhile, my experiment would only have one
> > list of integers involved, while you clearly have two lists (1-9
and
> > 8-11). Where have we lost each other?
>
> Yup, two lists for me, one list for you. The overall sound is what
I think
> you would call the combination product of the list or partials and
the list
> of notes.

OK. Well, if you do this, you lose the effect I was trying to focus
on, because the simple question of beating and/or roughness becomes
so prominent in the discordance equation (not that such an equation
has actually been written down, but you know what I mean).

> The whole experiment relates back to what in retrospect is likely
to have
> been a misinterpretation of what I heard you suggest. I wanted to
study a
> particular phenomenon and the experiment seemed to clarify what I
was
> hearing. The phenomenon I wanted to study in the first place was
in fact
> involving complex harmonic timbres combined into a chord. I wanted
to know
> whether the resultant tone I was hearing was a clustering of
difference
> tones or whether it was a virtual pitch.

OK . . . I would say it was probably all three :) (difference tones
create virtual pitches of their own, so it gets hairy)

> By substituting a timbre over
> which I had full control (allowing me to make it inharmonic) I was
then able
> to apply offsets. Keep in mind I was adjusting a frequency offset
applied
> to the entire list of partials across all notes in the chord. So
you could
> ignore the fact that two lists were involved, because the offset
was not a
> function of the base frequency, but just an absolute value in Hz.

So there's no additional beating compared with the harmonic case. I
think I missed that before. Thanks.

> The result proved it was not a difference tone phenomenon because
the
> "resultant" pitch did not stay constant when I aplied the offset.

Aha. And since all you had was a set of offset partials of a given
fundamental, there were no beating effects to complicate things. So
it really would be no different to consider it a single list of
partials, with various amplitudes, and going up as high as the 99th
partial. And you discovered that combinational tones (a general term
for all orders of difference tones and sum tones) were not the sole
source of your resultant pitch impression. So I'd say that you've
successfully demonstrated the importance of virtual pitch, at least
to your own auditory system. Well done!

> But I'll try this anyway -- here is how I understand things:
>
> You have a harmonic entropy function. This is a function of a real
number
> representing the interval. The function itself gets its value
because of
> something like the degree to which low-integer rationals cluster
around it.
> The more nearby rationals, the more ambiguity, and I guess that
means higher
> entropy. Lower entropy means less ambiguity, more clear
identification of
> an interval as a pure ratio.

Right.