Yahoo Tuning Groups Ultimate Backup harmonic_entropy HE of Random Phase Sources?

Yes. Substantial psychoacoustic experiments have shown that phase changes
have nearly no audible effect. In fact, the BBE Aural Enhancement products
work by fiddling with the phase relationships in your music in order to
eliminate large peaks that may be too much for your sound-reproducing
equipment to handle. To my knowledge, no one has complained that their music
sounds "altered" as a result, though many find that it is "clarified" due to
the reduction of clipping, etc.

-----Original Message-----
From: unidala [mailto:JGill99@...]
Sent: Saturday, December 08, 2001 4:22 AM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] HE of Random Phase Sources?

Paul,

Is it your belief that a dyad constructed of two independent complex
tones (each consisting of a fundamental as well as overtone
frequencies of each of those fundamental frequencies, where the
steady state as well as transient frequency spectrum over time of
each of those complex tones from which the dyad is constructed is
actually, or nearly, identical for the duration of the sounding of
those individual notes), being independent (and, hence, uncorrelated)
complex tones arising from separate sources (such as two
independently plucked strings), would possess a "Harmonic Entropy"
which is fully *independent* of the resultant random phase
relationships which would exist between the components of each of
those complex tones from which such a "harmonic dyad" is constructed?

Curiously, J Gill

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🔗unidala <JGill99@...>

--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Yes. Substantial psychoacoustic experiments have shown that phase
changes
> have nearly no audible effect.

JG: Paul, I thought that the experiments to which you refer above
relate to assessing listener reaction to various "phase shift as a
function of frequency" transfer functions where ALL of the summed
components of a single composite signal are present in a single
signal channel [which, in this case, would include the PRIOR
summation of the two complex (fundamental plus overtones) signals
from which the "dyad" in question is constructed]...

My question, instead is intended to relate to the fact that - the
frequency locations of some of the overtones of fundamental frequency
number 1 may well coincide with the frequency locations of some of
the overtones of fundamental frequency number 2. It seems that the
complete absence of such "coinciding" would apply only in the case of
the combinations of two complex (fundamental plus overtones) signals
which (as a result of no interplay of any of the overtones of either
fundamental) might be of little interest, as a result of such
spectral non-relatedness between the spectrums of such complex
signals. It would seem unreasonable to restrict the model to such
conditions.

What I am asking is - given that overtones from fundamental frequency
number 1 may well appear at common frequency(s) with the overtones of
fundamental frequency number 2 - what provision does your "Harmonic
Entropy" concept make for the random SUMMATION and/or CANCELLATION of
such "coinciding" overtones, which will take place as a result of the
random phase relationships BETWEEN the (uncorrelated) complex
frequency number 1 and complex frequency number 2 (as results from
two separate ringing strings, for instance), AT THE POINT OF THEIR
COMBINATION (as opposed to at a point following their combination,
where they have already been combined into a single signal path)?

Do you simply take a "statistical" approach to the resultant random
summation/cancellation of such uncorrelated sources (thus not
factoring in the property of random phase relationships, since they
are not predictable, and are not, by their nature, controllable)?

J Gill

___________________________________________

>
> -----Original Message-----
> From: unidala [mailto:JGill99@i...]
> Sent: Saturday, December 08, 2001 4:22 AM
> To: harmonic_entropy@y...
> Subject: [harmonic_entropy] HE of Random Phase Sources?
>
>
> Paul,
>
> Is it your belief that a dyad constructed of two independent
complex
> tones (each consisting of a fundamental as well as overtone
> frequencies of each of those fundamental frequencies, where the
> steady state as well as transient frequency spectrum over time of
> each of those complex tones from which the dyad is constructed is
> actually, or nearly, identical for the duration of the sounding of
> those individual notes), being independent (and, hence,
uncorrelated)
> complex tones arising from separate sources (such as two
> independently plucked strings), would possess a "Harmonic Entropy"
> which is fully *independent* of the resultant random phase
> relationships which would exist between the components of each of
> those complex tones from which such a "harmonic dyad" is
constructed?
>
>
> Curiously, J Gill

🔗Paul H. Erlich <PERLICH@...>

>JG: Paul, I thought that the experiments to which you refer above
>relate to assessing listener reaction to various "phase shift as a
>function of frequency" transfer functions where ALL of the summed
>components of a single composite signal are present in a single
>signal channel [which, in this case, would include the PRIOR
>summation of the two complex (fundamental plus overtones) signals
>from which the "dyad" in question is constructed]...

Well, by the time you hear it, the summation has already occured, unless the
tones are presented to separate ears. There's probably some specific
experimental data about the latter case, but all I can say is that, to the
best of my knowledge, all available evidence points to the _central_ origin
of virtual pitch.

>My question, instead is intended to relate to the fact that - the
>frequency locations of some of the overtones of fundamental frequency
>number 1 may well coincide with the frequency locations of some of
>the overtones of fundamental frequency number 2.

Ah . . . well let's say I assume a constant Gaussian "noise" in the
interval, so that phase-locking does not occur. This "noise" can be a mere
frequency modulation in one of the tones, only deep enough so that "in
phase" and "out of phase" occur with roughly equal probability. This depth
will typically be far less than the standard deviation s used in the model,
and will be of similar magnitude to what one finds in the most stable and
harmonic of acoustic musical instruments.

>It seems that the
>complete absence of such "coinciding" would apply only in the case of
>the combinations of two complex (fundamental plus overtones) signals
>which (as a result of no interplay of any of the overtones of either
>fundamental) might be of little interest, as a result of such
>spectral non-relatedness between the spectrums of such complex
>signals. It would seem unreasonable to restrict the model to such
>conditions.

I would say the exact opposite. There is always interplay between overtones,
_except_ in such exact JI cases. If you're talking about intervals with
exactly destructively or constructively interfering partials, I'm afraid
that's a condition that can only be realized in computer music with exact
JI, and corresponds generally, according to my experience and esthetic, to a
very unmusical sound (if the condition is maintained over the course of the
piece). But this sound occurs regardless of the phase relations -- it's
primarily a sensation of _lack of interplay_ between the partials because
the usual pattern of in-phase-out-of-phase cycling (occuring in higher
partials _in rhythmic synchronization_) is absent. I'd be happy to be proved
wrong, but for now, my attention hasn't been focused on such phenomena.

I suppose, if one wished, one could treat the partials individually, but
we're slowly coming along here, the three-partial case looming around the
horizon (I'm desparately awaiting Gene's feedback -- I need a
mathematician!!!)

>What I am asking is - given that overtones from fundamental frequency
>number 1 may well appear at common frequency(s) with the overtones of
>fundamental frequency number 2 - what provision does your "Harmonic
>Entropy" concept make for the random SUMMATION and/or CANCELLATION of
>such "coinciding" overtones, which will take place as a result of the
>random phase relationships BETWEEN the (uncorrelated) complex
>frequency number 1 and complex frequency number 2 (as results from
>two separate ringing strings, for instance), AT THE POINT OF THEIR
>COMBINATION (as opposed to at a point following their combination,
>where they have already been combined into a single signal path)?

See above and also higher above about the signal path.

>Do you simply take a "statistical" approach to the resultant random
>summation/cancellation of such uncorrelated sources (thus not
>factoring in the property of random phase relationships, since they
>are not predictable, and are not, by their nature, controllable)?

Sounds about right.

🔗unidala <JGill99@...>

--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> >JG: Paul, I thought that the experiments to which you refer above
> >relate to assessing listener reaction to various "phase shift as a
> >function of frequency" transfer functions where ALL of the summed
> >components of a single composite signal are present in a single
> >signal channel [which, in this case, would include the PRIOR
> >summation of the two complex (fundamental plus overtones) signals
> >from which the "dyad" in question is constructed]...
>
>PE:Well, by the time you hear it, the summation has already occured,
>unless the
>tones are presented to separate ears.

JG: Granted.

>PE: There's probably some specific
>experimental data about the latter case, but all I can say is that,
>to the
>best of my knowledge, all available evidence points to the
>_central_ origin
>of virtual pitch.

JG: By "central" you mean "non-binaural", I presume?

> >JG: My question, instead is intended to relate to the fact that -
> >the
> >frequency locations of some of the overtones of fundamental
>>frequency
> >number 1 may well coincide with the frequency locations of some of
> >the overtones of fundamental frequency number 2.

> PE: Ah . . . well let's say I assume a constant Gaussian "noise" in
> the
> interval, so that phase-locking does not occur.

JG: By "phase-locking" you mean "static phase alignments resulting in
significant degrees of addition and/or subtraction" between frequency
components, I presume?

>PE: This "noise" can be a mere
>frequency modulation in one of the tones, only deep enough so
>that "in
>phase" and "out of phase" occur with roughly equal probability.
>This depth
>will typically be far less than the standard deviation s used in
>the model,

JG: Isn't it the case that the "in phase" and "out of phase"
(time-domain alignments) occur with roughly equal probability in the
case of two (uncorrelated, random phase) overtones existing (without
any FM) at the *same* frequency? You appear (to me) to be stating
that such slight amounts of FM (in one or both overtone frequency
components) ensure that cancellation phenomena will not rear it's
ugly head (or, if so, the "depth" of such cancellation will be
limited, thus removing any concerns regarding it's implications)

>PE:... and will be of similar magnitude to what one finds in the
>most stable and
>harmonic of acoustic musical instruments.

JG: How have you (or others) determined that such "incidental FM" (as
one finds in musical instruments) "saves the day" in regards to
preventing (significant) amplitude cancellation effects?

> >JG: It seems that the
> >complete absence of such "coinciding" would apply only in the case
> >of
> >the combinations of two complex (fundamental plus overtones)
>>signals
> >which (as a result of no interplay of any of the overtones of
>>either
> >fundamental) might be of little interest, as a result of such
> >spectral non-relatedness between the spectrums of such complex
> >signals. It would seem unreasonable to restrict the model to such
> >conditions.

>PE: I would say the exact opposite. There is always interplay
>between overtones,
> _except_ in such exact JI cases.

JG: I don't get it. Are you saying that combinations of JI intervals
have *less* probability of "harmonic coincidence" than, say ET or MT?

>PE: If you're talking about intervals with
> exactly destructively or constructively interfering partials, I'm
>afraid
> that's a condition that can only be realized in computer music with
>exact
> JI,

JG: This seems to contradict your previous statement directly above
about "interplay between overtones" in "exact JI cases".

>PE:... and corresponds generally, according to my experience and
>esthetic, to a
> very unmusical sound (if the condition is maintained over the
>course of the
> piece). But this sound occurs regardless of the phase relations --
>it's
> primarily a sensation of _lack of interplay_ between the partials
>because
> the usual pattern of in-phase-out-of-phase cycling (occuring in
>higher
> partials _in rhythmic synchronization_) is absent.

JG: So you are saying that a lack of the presence of "the usual
pattern of in-phase-out-of-phase cycling (occuring in higher partials
_in rhythmic synchronization_)", rather than a static degree of
harmonic summation/cancellation, is the primary reason why additive
synthesis (where correlation, thus static phase relationships, exist
between the frequency components present) generates "a very unmusical
sound"? I'll buy that.

My concern has been more along the lines of the potential
*disappearance* (as a result of significant depths of amplitude
cancellation potentially taking place) of the existence of energy
located at frequencies where the (uncorrelated, random phase)
overtones of the "harmonic dyad" may coincide. I grant you that there
is little one can do about such effects, which occur (at least to
some extent) in nature. What I have been wondering about is their
potential impact in the (apparently desirable situation) where the
amplitude scale factors as well as the spectral signature of each of
the two complex (fundamental and overtones) components of
such "harmonic dyads" are (in the ideal case) EQUAL, as your replies
to the "Sinusoidal Entropy?" posts appear to indicate. Such a
situation would seem to tend to invite the potential for such
amplitude cancellation of spectral components which coincide in
frequency, and have similarly valued amplitude scale factors.

>PE: I'd be happy to be proved
> wrong, but for now, my attention hasn't been focused on such
>phenomena.
> I suppose, if one wished, one could treat the partials
>individually, but
> we're slowly coming along here, the three-partial case looming
>around the
> horizon (I'm desparately awaiting Gene's feedback -- I need a
> mathematician!!!)

JG: Do you mean three partials, or three complex (fundamental plus
harmonic) tones? I thought that you have been indicating (in
the "Sinusoidal Entropy?" threads) that the "partials" (as meaning,
essentially, overtones present in addition to the two sinusoidal
fundamental frequencies) were allready "factored in"...?

Regards, J Gill

🔗Paul H. Erlich <PERLICH@...>

>>PE: There's probably some specific
>>experimental data about the latter case, but all I can say is that,
>>to the
>>best of my knowledge, all available evidence points to the
>>_central_ origin
>>of virtual pitch.

>JG: By "central" you mean "non-binaural", I presume?

I mean in the brain, after the information from the two ears has been
combined.

>> >JG: My question, instead is intended to relate to the fact that -
>> >the
>> >frequency locations of some of the overtones of fundamental
>>>frequency
>> >number 1 may well coincide with the frequency locations of some of
>> >the overtones of fundamental frequency number 2.

>> PE: Ah . . . well let's say I assume a constant Gaussian "noise" in
>> the
>> interval, so that phase-locking does not occur.

>JG: By "phase-locking" you mean "static phase alignments resulting in
>significant degrees of addition and/or subtraction" between frequency
>components, I presume?

Yes.

>>PE: This "noise" can be a mere
>>frequency modulation in one of the tones, only deep enough so
>>that "in
>>phase" and "out of phase" occur with roughly equal probability.
>>This depth
>>will typically be far less than the standard deviation s used in
>>the model,

>JG: Isn't it the case that the "in phase" and "out of phase"
>(time-domain alignments) occur with roughly equal probability in the
>case of two (uncorrelated, random phase) overtones existing (without
>any FM) at the *same* frequency?

It depends on the means of tone production. With a computer, you can arrange
that certain partials be perfectly "in phase" or "out of phase", even at
particular locations within a room (La Monte Young's Dream House provides an
example of this).

>You appear (to me) to be stating
>that such slight amounts of FM (in one or both overtone frequency
>components) ensure that cancellation phenomena will not rear it's
>ugly head

Right.

>>PE:... and will be of similar magnitude to what one finds in the
>>most stable and
>>harmonic of acoustic musical instruments.

>JG: How have you (or others) determined that such "incidental FM" (as
>one finds in musical instruments) "saves the day" in regards to
>preventing (significant) amplitude cancellation effects?

Because I'm very sensitive to partials and can always hear them moving in
and out of phase, usually with a pattern more resembling an amoeba than a
regular alteration, when I hear musicians playing in tune, as opposed to a
computer playing in tune.

>> >JG: It seems that the
>> >complete absence of such "coinciding" would apply only in the case
>> >of
>> >the combinations of two complex (fundamental plus overtones)
>>>signals
>> >which (as a result of no interplay of any of the overtones of
>>>either
>> >fundamental) might be of little interest, as a result of such
>> >spectral non-relatedness between the spectrums of such complex
>> >signals. It would seem unreasonable to restrict the model to such
>> >conditions.

>>PE: I would say the exact opposite. There is always interplay
>>between overtones,
>> _except_ in such exact JI cases.

>JG: I don't get it. Are you saying that combinations of JI intervals
>have *less* probability of "harmonic coincidence" than, say ET or MT?

No, but "interplay" to me implies something other than static phase
relationships.

>>PE: If you're talking about intervals with
>> exactly destructively or constructively interfering partials, I'm
>>afraid
>> that's a condition that can only be realized in computer music with
>>exact
>> JI,

>JG: This seems to contradict your previous statement directly above
>about "interplay between overtones" in "exact JI cases".

Does it still?

>My concern has been more along the lines of the potential
>*disappearance* (as a result of significant depths of amplitude
>cancellation potentially taking place) of the existence of energy
>located at frequencies where the (uncorrelated, random phase)
>overtones of the "harmonic dyad" may coincide. I grant you that there
>is little one can do about such effects, which occur (at least to
>some extent) in nature.

Not too much -- if two instruments are playing in some phase relationship,
you simply move to a different location and you'll obtain a different phase
relationship. But of course coinciding overtones have nothing to do with
harmonic entropy anyway. You might be thinking more along the lines of
"critical band dissonance" which is the component of discordance studied by
Sethares. Harmonic entropy is an additional component.

>JG: Do you mean three partials, or three complex (fundamental plus
>harmonic) tones? I thought that you have been indicating (in
>the "Sinusoidal Entropy?" threads) that the "partials" (as meaning,
>essentially, overtones present in addition to the two sinusoidal
>fundamental frequencies) were allready "factored in"...?

That should hold to a good approximation. But to get more specific results,
we would want to extend the model. The voronoi diagrams for the 2-d case
already show that otonal triads have less harmonic entropy than utonal
triads, so we know we're explaining a real phenomenon that Sethares isn't.
Beyond that, both Sethares's model and mine are open to a greal deal of
extension and refinement (though this utonal/otonal distinction will always
show up in harmonic entropy and will never show up in critical band sensory
dissonance).

🔗unidala <JGill99@...>

Paul,

Thanks for your response. Something you said has really got me
thinking...

--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

> But of course coinciding overtones have nothing to do with
> harmonic entropy anyway.

JG: If "Harmonic Entropy" minima, like the "One Footed Bride", favors
JI intervals, and JI intervals composed of small valued integers,
when sounded together simultaneously, clearly have the most
mathematical propensity (relative to ET or MT) to have overtone
multiples which will result in such overtones (if present) to
coincide with each other, your statement above appears (to me) to
imply that it *may* be your impression that "JI preference" on the
part of some listeners does not in any significant way derive from
such "coincidences" in the frequencies at which the individually
generated harmonics (if present) will appear (and "interplay", with
or without the presence of *small* amounts of frequency modulation).

Or are you interpreting the term "coincidences" to mean "at exactly
the same pitch", and "interplay" as "within a *tiny* band around the
exact pitch" (from small amounts of FM, vibrato, and tuning drift)? I
am herein using the term "coincidence" as including such small
variations as you describe (without restriction to exact frequency).

If you are using the term "coinciding" above as including such small
variations in pitch around the target pitch as you have described:
Would you say that the validity of "Harmonic Entropy" does not arise
out of such "harmonic coincidences" because "Harmonic Entropy" is
valid in the sinusoidal (no harmonics at all) case, and thus cannot
be influenced by, or derive from, harmonic coincidence" phenomena?

Your "Harmonic Entropy" certainly seems to imply a "JI preference" of
sorts, doesn't it (without addressing operational complications of
instruments tuned in fixed JI tunings)? How much data have you found
which supports a strong preference for sinusoidal dyads in listening
tests? Do such preferences extend beyond the major minima (at 3/2)?

What if the nature of various persons' "JI preference" does [at
least "partially" (bad pun)] derive from such "harmonic coincidence"?
Could a concept, in that case, which does not in any way derive from
such "harmonic coincidences", adequately speak of aural perception?
Or would its validity in the case of a sinusoidal dyad carry the day?

Curiously, J Gill

🔗clumma <carl@...>

Greetings, all;

[J Gill wrote...]
>JG: Do you mean three partials, or three complex (fundamental plus
>harmonic) tones? I thought that you have been indicating (in
>the "Sinusoidal Entropy?" threads) that the "partials" (as meaning,
>essentially, overtones present in addition to the two sinusoidal
>fundamental frequencies) were allready "factored in"...?

I have suggested in the past that the process of virtual pitch
should be viewed is a hierarchical, recursive one, where past
decisions effect future ones. Under this view, one might argue that
spectrum information (coming from both the place and periodicity
mechanisms) is grouped by source location (determined from timing
and intensity differences between ears, and perhaps register), the
virtual pitch algorithm being applied to each source, and then to
the output of the first run as if these were a new set of simple
tones. In larger chords, we've seen examples where it may make
sense to view the chord as existing partially as one large chord
and partially as a pair of smaller chords simultaneously; as a
simple example the 10:12:15 triad.

The same sort of process may be at work in the temporal experience
of listening to music, where, for example, the interval E-Bb sounds
less dissonant after hearing the chord C-E-G-Bb than it does alone.
Granted, I know of no evidence that the virtual pitch hardware we're
talking about has any sort of memory...

[Paul Erlich wrote...]
>But this sound occurs regardless of the phase relations -- it's
>primarily a sensation of _lack of interplay_ between the partials
>because the usual pattern of in-phase-out-of-phase cycling (occuring
>in higher partials _in rhythmic synchronization_) is absent.

Paul, is this possibly the source of periodicity buzz, and if not,
what is?

Thanks guys -- I've read this thread on phase with much interest.
The topic of phase relationships in music is currently one of the
areas I realize I need to understand better.

-Carl

🔗Paul H. Erlich <PERLICH@...>

>> But of course coinciding overtones have nothing to do with
>> harmonic entropy anyway.

>JG: If "Harmonic Entropy" minima, like the "One Footed Bride", favors
>JI intervals, and JI intervals composed of small valued integers,
>when sounded together simultaneously, clearly have the most
>mathematical propensity (relative to ET or MT) to have overtone
>multiples which will result in such overtones (if present) to
>coincide with each other, your statement above appears (to me) to
>imply that it *may* be your impression that "JI preference" on the
>part of some listeners does not in any significant way derive from
>such "coincidences" in the frequencies at which the individually
>generated harmonics (if present) will appear (and "interplay", with
>or without the presence of *small* amounts of frequency modulation).

I don't know if one would go so far as to say the coincidences are
insignificant for some listeners, but I would say that much of the meaning
of harmony is found not in the coincidences, but rather in the (misapplied)
virtual pitch phenomenon.

>Or are you interpreting the term "coincidences" to mean "at exactly
>the same pitch",

No, here I'm being more general than that.

>and "interplay" as "within a *tiny* band around the
>exact pitch" (from small amounts of FM, vibrato, and tuning drift)? I
>am herein using the term "coincidence" as including such small
>variations as you describe (without restriction to exact frequency).

I understand, and it is with that meaning that I intended it.

>If you are using the term "coinciding" above as including such small
>variations in pitch around the target pitch as you have described:
>Would you say that the validity of "Harmonic Entropy" does not arise
>out of such "harmonic coincidences" because "Harmonic Entropy" is
>valid in the sinusoidal (no harmonics at all) case, and thus cannot
>be influenced by, or derive from, harmonic coincidence" phenomena?

Yup.

>Your "Harmonic Entropy" certainly seems to imply a "JI preference" of
>sorts, doesn't it (without addressing operational complications of
>instruments tuned in fixed JI tunings)? How much data have you found
>which supports a strong preference for sinusoidal dyads in listening
>tests? Do such preferences extend beyond the major minima (at 3/2)?

Not for sinusoids in a middle register and low volume. But, despite the
oft-quoted Plomp experiment that found no minima at all besides 1/1 (and
2/1?), there are other experiments, for example by Vos, which tell somewhat
different stories. Pitch accuracy for sinusoids in a middle register is
pretty bad, so one would use a very large value of s in the harmonic entropy
model for cases like this, and end up with very few minima.

>What if the nature of various persons' "JI preference" does [at
>least "partially" (bad pun)] derive from such "harmonic coincidence"?

No doubt it does, as critical band roughness has local minima at the points
of such coincidence.

>Could a concept, in that case, which does not in any way derive from
>such "harmonic coincidences", adequately speak of aural perception?

Again, it's only a component of perceived discordance, and other components
arise from other phenomena.

🔗Paul H. Erlich <PERLICH@...>

🔗clumma <carl@...>

🔗monz <joemonz@...>

🔗Paul H. Erlich <PERLICH@...>

🔗clumma <carl@...>

🔗Paul H. Erlich <PERLICH@...>

Carl, I think you're right. What happens is that intervals like 7:6 and 9:7,
though just, still give you perceptible roughness and beating, because of
their proximity to 6:5 and 5:4 respectively. The rates of beating, though,
in an otonal chord containing these intervals, are all in perfect
synchronization. That's what "periodicity buzz" means to me -- thanks for
reminding me!

-----Original Message-----
From: clumma [mailto:carl@...]
Sent: Wednesday, December 12, 2001 4:06 PM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] Re: HE of Random Phase Sources?

>> I don't know. I know what we've attached the term to, but I
>> don't know what could cause it.
>
> My interpretation of "periodicity buzz" was a certain result of
> _exact_ JI on a computer.

Wow- you coined the term, as far as I know, in response to a midi
version of Joe Monzo's Invisible Haircut. Is your MIDI setup
accurate enough to bring about the effect you're thinking of?

I always thought it meant the sound, present only in otonal
chords, especially in intervals like 7:6 and 9:7, which is
like... the hum of a flourescent light... definitely audible
in B-shop quartets... I'd always thought it may be some
perceptual kickback from period counting (which would only
happen when the period was sufficiently long -- thus no
effect for utonal or 3-limit chords).

Couldn't quite follow your comment, monz. Can you clarify?

-Carl

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