Yahoo Tuning Groups Ultimate Backup tuning Undertones/subharmonics esp. vocal

[Jim Cole <jimcole@compsol.net> TD 196.10]
>... I am interested to learn about
>subharmonics - we're having a discussion about them over on the new age
>and space music lists and we need help - so I seek your knowledge and
>assistance.

Thanks Jim. I'm interested to learn from you about what may be physically
possible.

Pat Pagano's response [TD 197.6] perpetuates the common confusion between
subharmonic chords and scales versus subharmonic partials of a single note.
The former are quite common, the latter essentially non-existent and
definitely non-existent in continuously driven oscillating physical systems
live the vocal apparatus.

The appearance of (subsets of) the subharmonic series (1/1, 1/2, 1/3, 1/4,
1/5, ...) in the ratios of the frequencies of (the fundamentals of) notes
in many chords and scales is quite sensible and often deliberate.

In the case of partials of a single note, IMHO Dave Hill [TD 197.14] has
hit the nail on the head. The low note is the true fundamental. The
apparent fundamental is simply one of its harmonics which has been strongly
selected (probably by some resonant cavity) supressing the true fundamantal
and other harmonics, and then through some nonlinearity the selected
harmonic has generated its own harmonic series (which are still in the
harmonic series of the true fundamental). In cases where the true
fundamental and the other harmonics are very low in amplitude it makes
sense to call it a "sub-fundamental". But it is definitely not a sub-harmonic.

Two partials do not make a sub-harmonic series (since they can be
interpreted either way with equal ease). One would need at least three
simultaneously, such as an apparent fundamental, a partial an octave lower
and another a fifth lower again. i.e. 1, 1/2, 1/3 (as you describe). But
notice that these could still be a subset of the harmonic series of a true
fundamental with a frequency 1/6 of the apparent fundamantal. 1:1/2:1/3:1/6
= 6:3:2:1. But if no trace of the 1/6 partial could be found, and no trace
of any 4th, 5th, 7th etc. harmonics of it, a subharmonic interpretation
would be tempting. Then there'd be some work for the physicists. Of course
as you add more "subharmonics" a filtered harmonic interpretation gets
rapidly less likely since the true fundamental would have to be extremely low.

>I have been practicing harmonic overtone singing for several years and
>am keenly interested in "undertone" production that I seem to hear in my
>voice and in others (like Tibetan monks' chant/Tuvan throat singers'
>"Kargiraa"). If I start with a particular fundamental I can instantly
>access the octave below by adjusting throat and abdominal areas.

Ok. Dropping the true fundamental an octave but maintaining filtering that
suppresses that fundamental and reinforces its second harmonic (the
continuing apparent fundamantal).

>With
>more relaxation in the throat I hear the fifth below that appear

Ok. Dropping the fundamental a fifth and now selecting its 3rd harmonic.

>while
>the original tone and octave below continue to sound.

Now, as I mentioned above, this is the tough part - the octave below
continuing!

>(at this point the harmonic overtones tend to be rather weak usually)

What do you mean by harmonic overtones here? Do you mean the apparent
fundamental (1) or do you mean 2, 3, 4, 5 etc relative to it.

>There are times
>when I can hear a tone two octaves and even two octaves and a major
>third below the fund. I started with

Are all these simultaneous? How high is your apparent fundamental, or
rather how low can it be for you to still be able to generate the 1/5?

>- this structure corresponds to an
>inverted harmonic series - am I really producing undertones with the
>voice or is something else going on to explain this phenomenon (or my
>perception of hearing them)?

>What about undertones in general - do they
>"exist"

Not as far as I know, in continuously driven systems. But then I'm no expert.

> and how are they generated? What's going on in the vocal
>apparatus to allow these?

Beats the hell out of me. As you suggest, the question is more likely
"what's going on in the ear/brain to make it seem this way". As a first
step, someone on this list might volunteer to analyse a sample if you can
provide one (say on your website). To see what's really there.

Regards,
-- Dave Keenan
http://dkeenan.com

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

Dave
I was by no means trying to perpetuate any misundersatnding
I mentioned nothing about chords but gave a simple
subharmonic series from C using Denny and Kaysers arrangements
I was simply replying to the existence of a series
I agree that the Low note is probably the fundamental and Jim is hearing probably
the harmonics of it
I am not claiming to be an expert on the subharmonic series nor overtone vocals
but i consistently feel you seem to try to perpetuate confusion by construing my
posts however you see fit
Pat

Dave Keenan wrote:

> From: Dave Keenan <d.keenan@uq.net.au>
>
> [Jim Cole <jimcole@compsol.net> TD 196.10]
> >... I am interested to learn about
> >subharmonics - we're having a discussion about them over on the new age
> >and space music lists and we need help - so I seek your knowledge and
> >assistance.
>
> Thanks Jim. I'm interested to learn from you about what may be physically
> possible.
>
> Pat Pagano's response [TD 197.6] perpetuates the common confusion between
> subharmonic chords and scales versus subharmonic partials of a single note.
> The former are quite common, the latter essentially non-existent and
> definitely non-existent in continuously driven oscillating physical systems
> live the vocal apparatus.
>
> The appearance of (subsets of) the subharmonic series (1/1, 1/2, 1/3, 1/4,
> 1/5, ...) in the ratios of the frequencies of (the fundamentals of) notes
> in many chords and scales is quite sensible and often deliberate.
>
> In the case of partials of a single note, IMHO Dave Hill [TD 197.14] has
> hit the nail on the head. The low note is the true fundamental. The
> apparent fundamental is simply one of its harmonics which has been strongly
> selected (probably by some resonant cavity) supressing the true fundamantal
> and other harmonics, and then through some nonlinearity the selected
> harmonic has generated its own harmonic series (which are still in the
> harmonic series of the true fundamental). In cases where the true
> fundamental and the other harmonics are very low in amplitude it makes
> sense to call it a "sub-fundamental". But it is definitely not a sub-harmonic.
>
> Two partials do not make a sub-harmonic series (since they can be
> interpreted either way with equal ease). One would need at least three
> simultaneously, such as an apparent fundamental, a partial an octave lower
> and another a fifth lower again. i.e. 1, 1/2, 1/3 (as you describe). But
> notice that these could still be a subset of the harmonic series of a true
> fundamental with a frequency 1/6 of the apparent fundamantal. 1:1/2:1/3:1/6
> = 6:3:2:1. But if no trace of the 1/6 partial could be found, and no trace
> of any 4th, 5th, 7th etc. harmonics of it, a subharmonic interpretation
> would be tempting. Then there'd be some work for the physicists. Of course
> as you add more "subharmonics" a filtered harmonic interpretation gets
> rapidly less likely since the true fundamental would have to be extremely low.
>
> >I have been practicing harmonic overtone singing for several years and
> >am keenly interested in "undertone" production that I seem to hear in my
> >voice and in others (like Tibetan monks' chant/Tuvan throat singers'
> >"Kargiraa"). If I start with a particular fundamental I can instantly
> >access the octave below by adjusting throat and abdominal areas.
>
> Ok. Dropping the true fundamental an octave but maintaining filtering that
> suppresses that fundamental and reinforces its second harmonic (the
> continuing apparent fundamantal).
>
> >With
> >more relaxation in the throat I hear the fifth below that appear
>
> Ok. Dropping the fundamental a fifth and now selecting its 3rd harmonic.
>
> >while
> >the original tone and octave below continue to sound.
>
> Now, as I mentioned above, this is the tough part - the octave below
> continuing!
>
> >(at this point the harmonic overtones tend to be rather weak usually)
>
> What do you mean by harmonic overtones here? Do you mean the apparent
> fundamental (1) or do you mean 2, 3, 4, 5 etc relative to it.
>
> >There are times
> >when I can hear a tone two octaves and even two octaves and a major
> >third below the fund. I started with
>
> Are all these simultaneous? How high is your apparent fundamental, or
> rather how low can it be for you to still be able to generate the 1/5?
>
> >- this structure corresponds to an
> >inverted harmonic series - am I really producing undertones with the
> >voice or is something else going on to explain this phenomenon (or my
> >perception of hearing them)?
>
> >What about undertones in general - do they
> >"exist"
>
> Not as far as I know, in continuously driven systems. But then I'm no expert.
>
> > and how are they generated? What's going on in the vocal
> >apparatus to allow these?
>
> Beats the hell out of me. As you suggest, the question is more likely
> "what's going on in the ear/brain to make it seem this way". As a first
> step, someone on this list might volunteer to analyse a sample if you can
> provide one (say on your website). To see what's really there.
>
> Regards,
> -- Dave Keenan
> http://dkeenan.com
>
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🔗Jim Cole <jimcole@xxxxxxx.xxxx>

(This reply to my post came through the tuning list but I'm cross-posting it to
Music Theory too as there have been responses from there to my wonderings about
subharmonics - thank you to all who have written so far...)

Dave Keenan wrote:

> In the case of partials of a single note, IMHO Dave Hill [TD 197.14] has
> hit the nail on the head. The low note is the true fundamental. The
> apparent fundamental is simply one of its harmonics which has been strongly
> selected (probably by some resonant cavity) supressing the true fundamantal
> and other harmonics, and then through some nonlinearity the selected
> harmonic has generated its own harmonic series (which are still in the
> harmonic series of the true fundamental). In cases where the true
> fundamental and the other harmonics are very low in amplitude it makes
> sense to call it a "sub-fundamental". But it is definitely not a sub-harmonic.

I am glad you interpreted subfundamental and distinguished it from
"sub-harmonic." Here I was getting excited that a researcher had found evidence
that would confirm what I think occurs in the human voice - I realize now that I
need to learn some terminology to understand discussion of this stuff!

>
> Two partials do not make a sub-harmonic series (since they can be
> interpreted either way with equal ease). One would need at least three
> simultaneously, such as an apparent fundamental, a partial an octave lower
> and another a fifth lower again. i.e. 1, 1/2, 1/3 (as you describe). But
> notice that these could still be a subset of the harmonic series of a true
> fundamental with a frequency 1/6 of the apparent fundamantal. 1:1/2:1/3:1/6
> = 6:3:2:1. But if no trace of the 1/6 partial could be found, and no trace
> of any 4th, 5th, 7th etc. harmonics of it, a subharmonic interpretation
> would be tempting. Then there'd be some work for the physicists. Of course
> as you add more "subharmonics" a filtered harmonic interpretation gets
> rapidly less likely since the true fundamental would have to be extremely low.

Yes, I see that the lowest note present (or a missing fundamental) can be
interpreted as the fundamental and the higher notes are harmonics of it - that
it's still basically ambiguous - one can interpret it either way...

> >I have been practicing harmonic overtone singing for several years and
> >am keenly interested in "undertone" production that I seem to hear in my
> >voice and in others (like Tibetan monks' chant/Tuvan throat singers'
> >"Kargiraa"). If I start with a particular fundamental I can instantly
> >access the octave below by adjusting throat and abdominal areas.
>
> Ok. Dropping the true fundamental an octave but maintaining filtering that
> suppresses that fundamental and reinforces its second harmonic (the
> continuing apparent fundamantal).
>
> >With
> >more relaxation in the throat I hear the fifth below that appear
>
> Ok. Dropping the fundamental a fifth and now selecting its 3rd harmonic.
>
> >while
> >the original tone and octave below continue to sound.
>
> Now, as I mentioned above, this is the tough part - the octave below
> continuing!

I realize now why I have been tempted to interpret what goes on in my voice
(sometimes) as subharmonics: in "normal" voice production I can get down to about
a low C# (I forget which subscript this is but it's the one just below the bass
staff) and possibly a Bb on good days (in the early morning!) - I can either
initiate these notes immediately or glide down smoothly to them. With Kargiraa (a
Tuvan vocal style that I think I imitate) I start with a comfortable note (about
C# an octave above) and then grunt into a note an octave below - it's much
"rougher" sounding than the normal bass production of the same pitch. On a good
day with Kargiraa I can slide it down gradually (meaning slide both apparant fund.
and octave below simultaneously) to about a fourth lower than my lowest normal
bass. With still another style of bass voice production I relax the throat much
more than with Kargiraa and this is the style that esp. seduces me into thinking
it's subharmonic because I start with a note, an octave below "kicks in" along
with it, then the fifth below that, two octaves, two octaves and a major third
below, etc. - in other words the bass notes my voice accesses do not come from
gliding but just sort of "pop in" at those fixed intervals - well okay, sometimes
it skips but still conforming to that basic fixed structure in relation to the
original fundamental - and they always come along with the initiated fundamental
(so the original fundamental sounds like a continuous drone throughout this
process of accessing these lower notes - and the lower notes cannot come until the
(orig.) fundamental is established. In this vocal production I can hear bass
notes that get down about a fourth lower (ex: note an octave and a fifth below a
"low" G that is the initiated fundamental) than my lowest Kargiraa note. To get a
note two octaves and a major third below a given fundamental (what I call
"subharmonic 5") I usually must start from a higher note. Almost always the low
notes in this style of production are quite weak - I've yet to find a way to
strengthen them.

> >(at this point the harmonic overtones tend to be rather weak usually)
>
> What do you mean by harmonic overtones here? Do you mean the apparent
> fundamental (1) or do you mean 2, 3, 4, 5 etc relative to it.
>

the latter - I will try to focus more on these to see what I can hear in them (the
relative strength of each)

>
> >There are times
> >when I can hear a tone two octaves and even two octaves and a major
> >third below the fund. I started with
>
> Are all these simultaneous? How high is your apparent fundamental, or
> rather how low can it be for you to still be able to generate the 1/5?
>

Yes, they are usually all simultaneous though often weak - at least, I think I
hear them all there. The F or G near the top of the bass staff is the area where
it's usually easiest to start the fund. but sometimes from lower notes - this is
really variable depending on the day, my health, etc.

>As a first

> step, someone on this list might volunteer to analyse a sample if you can
> provide one (say on your website). To see what's really there.

I've been trying for a year and half now to get new clips on my site (of our
music) in any format (wav, RA, MP3, etc.) and something is always screwed up with
our sound card, or computer, or ___, (or my head!), etc. - I will try again to get
it all working - this experiment is another motivation! BTW, how limited are
these dinky mics that come with the sound cards (and are there ways to connect
higher quality mics)?

I am more than ever interested in this discussion of subharmonics and have heard
many good comments. It seems that even among the researchers I have contacted
(individually that is) there is no agreement. Some say they definitely exist, and
a acoustics researcher in Sweden said he will send me a copy of his colleague's
thesis which focuses on vocal subharmonic production. He mentioned "there is
increased mechanical and possibly aerodynamic coupling between the vocal folds and
the ventricular folds, which causes subperiodic oscillations." Comments?

Anyone know of recent physics/acoustics books that say (and explain how)
subharmonics/undertones can be produced from a fundamental? I am of course eager
to read that thesis but would like to know if it's been substantiated elsewhere
too.

Thanks to Ken (I'll check Google next!), Ray, Pat, Dave and others who have
responded so far - I am glad I asked about this again seven years later - before I
was frustrated by lack of resources on the topic (that I could find at the time -
I wasn't connected at the time and the local library searches I did just didn't
cut it). Seems that I've found a very interested and knowledgable community at
last. I am inspired and even more intrigued by what's been said so far.

...and of course to you too Dave - thank you for the message.

I can't wait to hear more - let the discussion continue...

~Jim Cole
http://www.compsol.net/users/jimcole

🔗Daniel Wolf <DJWOLF_MATERIAL@xxxxxxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

Paul Erlich here -- been very ill since the microthon -- had my appendix removed so
far, but all is not yet well. Anyway . . .

I think a better description would relate to chaos theory. The opening and closing of
the vocal folds experiences a period-doubling, so the apparent fundamental and its
overtones are still prominent (due to the dynamics and not the filtering) but the
true fundamental is an octave lower. The way the "subharmonic" almost "jumps" in and
out of existence as one varies the mode of singing is a hallmark of a nonlinear
bifurcation. I refer Jim to any of the popular books on chaos theory, such as Manfred
Schroeder's _Fractals, Chaos, and Power Laws_.

> >With
> >more relaxation in the throat I hear the fifth below that appear
>
> Ok. Dropping the fundamental a fifth and now selecting its 3rd harmonic.

Or rather, period-tripling. This is quite a bit more difficult to achieve but I've
done it with my own voice.

> >while
> >the original tone and octave below continue to sound.
>
> Now, as I mentioned above, this is the tough part - the octave below
> continuing!

From my own throat-singing experience and my understanding of the physics involved, I
am highly doubtful that the octave below the original tone does indeed continue to
sound. I am willing to bet that it does not actually exist in the sound and that the
original poster was mistakenly hearing it.

> >There are times
> >when I can hear a tone two octaves and even two octaves and a major
> >third below the fund. I started with
>
> Are all these simultaneous?

I would say they can't be.

Also, psychacoustical research has shown that a single pure tone and some
well-distributed noise can evoke the sensation of a missing fundamental that the pure
tone might be, say, the fifth harmonic of. This is simply the fundamental tracking
mechanism at work, always trying to find the best-fit harmonic series to a stimulus.
In the case of supposedly throat-singing the fifth "subharmonic", I would wonder if
some such mechanism is at play (though it's probably that Jim's throat-singing
ability is much more highly cultivated than my own).

> >What about undertones in general - do they
> >"exist"
>
> Not as far as I know, in continuously driven systems. But then I'm no expert.

Continuously driven systems can experience period-multiplying behavior -- it occurs
wherever chaos occurs. It would certainly be accurate to describe the results as
"subharmonics" or "undertones". But it is important to keep in mind that the spectrum
of any of these sounds, at a given point in time, is always a harmonic series above
the "undertone", and never a subharmonic series below the original fundamental.

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

[Pat Pagano, TD 198.4, wrote]
>Dave
>I was by no means trying to perpetuate any misundersatnding
>I mentioned nothing about chords but gave a simple
>subharmonic series from C using Denny and Kaysers arrangements
>I was simply replying to the existence of a series
>I agree that the Low note is probably the fundamental and Jim is hearing
probably
>the harmonics of it
>I am not claiming to be an expert on the subharmonic series nor overtone
vocals
>but i consistently feel you seem to try to perpetuate confusion by
construing my
>posts however you see fit
>Pat

Sorry Pat. I could have been more careful in my wording so as not to imply
that either (a) you were confused about, or (b) you _intended_ your post to
confuse, subharmonic partials versus subharmonic relationships between
notes as a musical resource.

However, I understood Jim Cole's post to be asking about whether
subharmonic partials exist physically, and yours to be talking about the
fact that subharmonic relationships between notes "exist" musically and
mathematically. A very different type of "existence". He's apparently
claiming to produce them simultaneously with only his vocal apparatus, so
that makes them partials in my book.

I'm sorry if I misconstrued your post. You say you mentioned nothing about
chords. This may be true, but it doesn't change my point, since I meant the
confusion was between (subharmonic chords and scales collectively) versus
(subharmonic partials), and you sure seemed to be talking about either a
chord or a scale when you said [TD 197.6]:

"Partch called the subharmonic the true minor"

Jim is apparently claiming to simultaneously hear 1/1, 1/2, 1/3 1/4, 1/5. I
would love to get to the bottom of why. It seems unlikely that he's
actually _generating_ these simultaneously since a
filtered-harmonic-partials explanation would require the true fundamental
to be at 1/60 (nearly 6 octaves down)! We need someone to analyse a sample.
Any volunteers?

Regards,
-- Dave Keenan
http://dkeenan.com

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

Jim Cole wrote:

> a acoustics researcher in Sweden said he will send me a copy of his colleague's
> thesis which focuses on vocal subharmonic production. He mentioned "there is
> increased mechanical and possibly aerodynamic coupling between the vocal folds and
> the ventricular folds, which causes subperiodic oscillations." Comments?

This is exactly the type of thing I was referring to with chaos theory, and I feel pretty
strongly that it must be what's going on (we discusssed this here on the list about a year
ago). A "coupling" is a nonlinear interaction between various modes of vibration. A
nonlinear dynamical system is the kind where chaos can occur, and also where
period-multiplying behavior ("subperiodic oscillations") can occur.

> Anyone know of recent physics/acoustics books that say (and explain how)
> subharmonics/undertones can be produced from a fundamental?

Like I said, any good book on chaos will explain this. Below, I got my information from
Peitgen, Jurgens, and Saupe, _Chaos and Fractals_ (highly recommended!).

> I am of course eager
> to read that thesis but would like to know if it's been substantiated elsewhere
> too.

I'll give it my vote!

Dave Keenan wrote:

> Jim is apparently claiming to simultaneously hear 1/1, 1/2, 1/3 1/4, 1/5. I
> would love to get to the bottom of why. It seems unlikely that he's
> actually _generating_ these simultaneously since a
> filtered-harmonic-partials explanation would require the true fundamental
> to be at 1/60 (nearly 6 octaves down)!

I just thought of something last night: if Jim is actually not in the period-multiplying
regime but in the chaotic regime itself, then the strongest frequencies might indeed be
1/1, 1/2, 1/3, 1/4, 1/5 . . . a subharmonic series! Of course, there would be harmonic
overtones and lots of noise too, and the sound would probably be very unsteady apart from
its inherent roughness and noisiness.

Consider the simplest example of a continous system that exhibits chaos: the Rossler
attractor:

x' = -(y + z)
y' = x + a*y
z' = b+ x*z - c*z

Setting a = 0.2 and b = 0.2, various values of c between 3 and 8 will lead to period-2,
period-3, period-4, period-5, and intervening bands of chaotic behavior. In a chaotic
regime, the system "orbits" a strange attractor in such a way that it returns (roughly) to
a starting point after an unpredictable, but always integer, number of orbits. The period
of one orbit will be one over the frequency of the original fundamental, so the longer
period lengths will contribute various subharmonic components to the sound. The difficulty
here is that chaotic motion is fractal, not steady, so it is tricky to determine what the
appropriate time window for analysis would be, but perhaps that's the only way a
subharmonic series could be characterized anyway!

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

[TD 199.4]
>Paul Erlich here -- been very ill since the microthon -- had my appendix
removed so
>far, but all is not yet well.

Wah! That's serious. I know I speak for everyone on the list in wishing you
a speedy recovery.

[Paul Erlich]
>I think a better description would relate to chaos theory. The opening and
closing of
>the vocal folds experiences a period-doubling, so the apparent fundamental
and its
>overtones are still prominent (due to the dynamics and not the filtering)
but the
>true fundamental is an octave lower. The way the "subharmonic" almost
"jumps" in and
>out of existence as one varies the mode of singing is a hallmark of a
nonlinear
>bifurcation. I refer Jim to any of the popular books on chaos theory, such
as Manfred
>Schroeder's _Fractals, Chaos, and Power Laws_.

Yes. Of course.

[Jim Cole]
>> >With
>> >more relaxation in the throat I hear the fifth below that appear
>> >while
>> >the original tone and octave below continue to sound.

[Paul Erlich]
>From my own throat-singing experience and my understanding of the physics
involved, I
>am highly doubtful that the octave below the original tone does indeed
continue to
>sound. I am willing to bet that it does not actually exist in the sound
and that the
>original poster was mistakenly hearing it.

So why would he hear it as continuing, i.e. as 1/3, 1/2, 1/1 (a subharmonic
series) rather than say 1/3, 2/3, 3/3 (a harmonic series)?

>It would certainly be accurate to describe the results as
>"subharmonics" or "undertones". But it is important to keep in mind that
the spectrum
>of any of these sounds, at a given point in time, is always a harmonic
series above
>the "undertone", and never a subharmonic series below the original
fundamental.

If it is not part of a subharmonic series, how could it be accurate to
describe it as a subharmonic? This just perpetuates confusion. I like Dave
Hill's term better: subfundamental.

Hey Pat, feel better now I've accused Paul Erlich of the same thing? And on
his death-bed too. ;-)

Sorry Paul, I'm sure it will turn out to be nothing too serious. Most
likely an ulcer from the stress of having to explain stuff to us idiots
over and over. :-)

Regards,
-- Dave Keenan
http://dkeenan.com

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

[Paul Erlich, TD 200.3]
>I just thought of something last night: if Jim is actually not in the
period-multiplying
>regime but in the chaotic regime itself, then the strongest frequencies
might indeed be
>1/1, 1/2, 1/3, 1/4, 1/5 . . . a subharmonic series!

Excellent Paul! I think you're right that it's due to period multiplying in
a nonlinear feedback (i.e. chaotic) system. But note that it doesn't have
to _be_ a subharmonic series of partials to _sound_like_ a subharmonic
series of notes. It would be enough to have
1/1, 1/2, 2/3, 3/4, 4/5, 3/5. i.e one need never have any partial actually
lower than 1/2. Note that Jim's 1/5, if it were actually present, would be
about 35Hz. Basso profundo!

>Of course, there would be harmonic
>overtones and lots of noise too, and the sound would probably be very
unsteady apart from
>its inherent roughness and noisiness.

Agreed. Seems to fit Jim's description too. There would be harmonics of
every subharmonic too.

>Consider the simplest example of a continous system that exhibits chaos:
>the Rossler attractor:

I set it up in a spreadsheet and used Excel's 4096-sample FFT. With delta_t
= 0.075 this was only about 60 cycles, but a good number when looking for
all subharmonics down to 1/6. The discrete simulation was unstable with
delta-t much higher than this. I found some initial values (7.5,0,1) and
then tried different values of c as you suggested.

The spectra, when they are strongly periodic, look very-roughly symmetrical
about the fundamental, e.g. for period 5 we get moderate amounts of 4/5 and
6/5 and weaker 3/5 and 7/5 and hardly any 2/5 and 8/5 and essentially no
1/5 or 9/5. Of course these are not line spectra, these are just peaks in
an otherwise noisy spectrum.

I was hoping for a period 2 and 3 combination which should have been clear.
I didn't actually find any period 3. I managed to find one case around
c=5.7 where I thought I could discern a vestige of the period 2 (or maybe
it was period 4) in addition to the period 5.

So my guess now is that we'll find, not a subharmonic spectrum as such, but
noise peaks at those harmonics-of-the-low-numbered-subharmonics which are
near 1/1, say between about 1/3 and 5/3.

Regards,
-- Dave Keenan
http://dkeenan.com

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

Dave Keenan [TD198.1]

>[Jim Cole <jimcole@compsol.net> TD 196.10]
>>... I am interested to learn about
>>subharmonics - we're having a discussion about them over on the new age
>>and space music lists and we need help - so I seek your knowledge and
>>assistance.

...
>Two partials do not make a sub-harmonic series (since they can be
>interpreted either way with equal ease). One would need at least three
>simultaneously, such as an apparent fundamental, a partial an octave lower
>and another a fifth lower again. i.e. 1, 1/2, 1/3 (as you describe). But
>notice that these could still be a subset of the harmonic series of a true
>fundamental with a frequency 1/6 of the apparent fundamantal. 1:1/2:1/3:1/6
>= 6:3:2:1.

I have found this discussion very interesting. Dave, it seems to me
that if the undertones are appearing by a slight vocal adjustment then
it makes a lot of sense to see them as undertones and not some
arrangement of different overtones with a varying fundamental.
The whole original description shouts "undertones" at me.

I agree that the physics of multiple undertones is not so obvious.
However in this case we have a sentient being deliberately shaping
resonant cavities to achieve the effects. This means that if some small
non-linearities are present then a close approximation to cavity shape
required to produce exact undertones could cause some locking in.

Paul Erlich [TD199.4]
>I think a better description would relate to chaos theory. The opening and closing of
>the vocal folds experiences a period-doubling, so the apparent fundamental and its
>overtones are still prominent (due to the dynamics and not the filtering) but the
>true fundamental is an octave lower. The way the "subharmonic" almost "jumps" in and
>out of existence as one varies the mode of singing is a hallmark of a nonlinear
>bifurcation. I refer Jim to any of the popular books on chaos theory, such as Manfred
>Schroeder's _Fractals, Chaos, and Power Laws_.

I disagree as it is stated that notes 1/2, 1/3, 1/4, 1/5 the original
all came in with slight adjustments. That is not chaotic behaviour but
undertones.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
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🔗unidala <JGill99@imajis.com>

This "Brett Barbaro" fellow seems to have
"throat-singing experience", and (appears)
to doubt that his throat (or perhaps other's)
can produce "sub-harmonic series", yet speaks
of "chaotic regimes" where "subharmonic
components may 'coexist'". Perhaps Brett
could explain this theory?

J Gill :)

--- In tuning@y..., Brett Barbaro <barbaro@xxxxxxxxx.xxxx wrote:
> > [Paul Erlich]
> > >From my own throat-singing experience and my understanding of the physics
> > involved, I
> > >am highly doubtful that the octave below the original tone does indeed
> > continue to
> > >sound. I am willing to bet that it does not actually exist in the sound
> > and that the
> > >original poster was mistakenly hearing it.
>
> [Dave Keenan]
>
> > So why would he hear it as continuing, i.e. as 1/3, 1/2, 1/1 (a subharmonic
> > series) rather than say 1/3, 2/3, 3/3 (a harmonic series)?
>
> Possibly a mistake of octave equivalence (having heard the octave below the original fundamental,
> one's perception of it might lose octave-specificity as one adds lower components to the sound). But
> see my later conjecture (in the actual chaotic regime, all subharmonic components may "coexist").
>
> > >It would certainly be accurate to describe the results as
> > >"subharmonics" or "undertones". But it is important to keep in mind that
> > the spectrum
> > >of any of these sounds, at a given point in time, is always a harmonic
> > series above
> > >the "undertone", and never a subharmonic series below the original
> > fundamental.
> >
> > If it is not part of a subharmonic series, how could it be accurate to
> > describe it as a subharmonic? This just perpetuates confusion. I like Dave
> > Hill's term better: subfundamental.
>
> For better or worse, "subharmonic" is prevalent in the physics literature. Search the web for some
> examples.
>
> -Paul Erlich

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > This "Brett Barbaro" fellow seems to have
> > "throat-singing experience", and (appears)
> > to doubt that his throat (or perhaps other's)
> > can produce "sub-harmonic series",
>
> I don't think that's what was stated.

JG: Sorry, I thought

> > >[Paul Erlich]
> > >From my own throat-singing experience and
> > >my understanding of the physics
> > >involved,

meant that.

> > JG: yet speaks
> > of "chaotic regimes" where "subharmonic
> > components may 'coexist'". Perhaps Brett
> > could explain this theory?
>
> Dave Keenan did an FFT of a simulated chaotic signal, the signal
> calculated subject to one of the simplest kinds of non-linear
> recursion rules.

JG: Sounds interesting! Some (sort) of random noise generator?

> He found strong peaks at all the tonality-diamond
> ratios (calling the 'linear regime' frequency 1/1). This would >imply
> a co-existing set of fundamentals forming a Utonal chord, each with
> their own overtone series. However, this was an FFT of the _whole
> signal_, from beginning to end,

JG: How long of a time (window) relative to the frequencies allowed to exist in this "non-linear" recursion algorithm?

so the question of _simultaneity_ of
> the utonally-related fundamentals remains unresolved. If a better,
> say wavelet-based, analysis showed that they could indeed be
> considered simulateneous, then perhaps we could posit the
> equation "chaos = Utonality".

JG: It seems like in the case of no drone (and nothing else)
that the sinusoids 1/6, 1/5, 1/4 when simply pitch shifted
by a factor of 6 upwards, would sound just like the sinusoids
1/1, 6/5, 3/2. So it is the presence of harmonic multiples of
the fundamental frequencies (at 1/6, 1/5, 1/4, etc, and with
what would then be a variable "timbre" which would give birth
to this "chaotic event"?

Curiously, J Gill

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> >
> > > This "Brett Barbaro" fellow seems to have
> > > "throat-singing experience", and (appears)
> > > to doubt that his throat (or perhaps other's)
> > > can produce "sub-harmonic series",
> >
> > I don't think that's what was stated.
>
> JG: Sorry, I thought
>
> > > >[Paul Erlich]
> > > >From my own throat-singing experience and
> > > >my understanding of the physics
> > > >involved,
>
> meant that.

Doubt that subharmonic _chords_ can be produced in this way. You'll
have to read the preceding messages to get the context.

> JG: Sounds interesting! Some (sort) of random noise generator?

No, _deterministic_ chaos.

> > He found strong peaks at all the tonality-diamond
> > ratios (calling the 'linear regime' frequency 1/1). This would
>imply
> > a co-existing set of fundamentals forming a Utonal chord, each
with
> > their own overtone series. However, this was an FFT of the _whole
> > signal_, from beginning to end,
>
> JG: How long of a time (window) relative to the frequencies allowed
> to exist in this "non-linear" recursion algorithm?

Rather long. Dave?

> > so the question of _simultaneity_ of
> > the utonally-related fundamentals remains unresolved. If a
better,
> > say wavelet-based, analysis showed that they could indeed be
> > considered simulateneous, then perhaps we could posit the
> > equation "chaos = Utonality".
>
> JG: It seems like in the case of no drone (and nothing else)
> that the sinusoids 1/6, 1/5, 1/4 when simply pitch shifted
> by a factor of 6 upwards, would sound just like the sinusoids
> 1/1, 6/5, 3/2. So it is the presence of harmonic multiples of
> the fundamental frequencies (at 1/6, 1/5, 1/4, etc, and with
> what would then be a variable "timbre" which would give birth
> to this "chaotic event"?

No, the chaotic system gives birth to this sound.

Undertones/subharmonics esp. vocal

🔗Jim Cole <jimcole@xxxxxxx.xxxx>

🔗Patrick Pagano <ppagano@bellsouth.net>

🔗Patrick Pagano <ppagano@bellsouth.net>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗Jim Cole <jimcole@xxxxxxx.xxxx>

🔗Daniel Wolf <DJWOLF_MATERIAL@xxxxxxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗dkeenanuqnetau <d.keenan@uq.net.au>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>