Yahoo Tuning Groups Ultimate Backup tuning Odd and prime limits (was: No doubt about Lehman's Bach scale)

On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> >> [YA]
> >> {begin OT}
> >> In this context, I don't know what (prime) limit I recognise in
> >> tuning;
> >
> >> [PE]
> >> How about odd limit?
> >
> >> * No.
>
> OK, then there's no constraint on the size of the numbers in the
> ratios, only on the sizes of their largest prime factors.

That does not follow. I answered that I also don't know what
odd limit I recognise; not that there isn't one!

> ... Are you sure this is the kind of 'limit' you want?

I didn't say so; you merely inferred it, wrongly, as it happens.

>> [YA]
>> ... certainly, some very tiny variations in pitch evoke entirely
>> different sensations in various oriental musics I'm familiar with,
>> including Chinese opera, erhu playing, shakuhachi playing,
>> classical
>> Indian playing and singing, the best Bollywood singing. Do I hear
>> the 19-limit, or perhaps the 23-limit? I really couldn't say; nor
>> do
>> I know how to find out.
>
>> [PE]
>> Are you sure the question is even meaningful, Yahya?
>>
>> * Sure? No. But I think it may be. Would a musical tuning that
>> requires 137-limit intervals be meaningful, Paul, if we
>> couldn't
>> hear the difference between that and, say, the 19-limit?
>
> Musical tunings "meaningful"? That's not the question.

You ask your questions; I'll ask mine. Rephrasing mine, since you
misunderstood it: Would there be any point in defining a tuning
that requires distinctions that no one can ever possibly hear?
And clearly the queston is rhetorical.

> ... But the
> question of whether a particular prime limit is or isn't in
> force . . . since there's no limit on the sizes of the numbers in the
> ratios, you can always find more and more ratios within the prime
> limit that are closer and closer to a given ratio.

If your inference above had been correct, this might be relevant.
It wasn't, so it isn't.

> At what point do
> these high-numbered ratios leave the realm of musical relevance? Of
> course it depends on the tuning system, but my point is that a prime
> limit alone doesn't seem to be capable of unambiguously corresponding
> or not corresponding, in general, to an arbitrary tuning system, via
> any aural or even mathematical test, as long as one recognizes a
> finite uncertainty in hearing intervals (or even merely the minimal
> uncertainty allowed by the classical uncertainty
> principle . . .) . . . The exceptions would seem to be cases where an
> odd, integer, or other "finitary" limit is ascertained first, and
> then the prime limit is simply taken as the largest prime not
> exceeding that limit.

I don't disagree, but I was focussing on what prime limits
I can distinguish, rather than on what overall (odd) limit I
can distinguish. I agree that fixing the latter sets an upper
bound on the former. In an earlier message, I found that
a listener's odd limit must lie somewhere near or below 101,
on the unproven assumption that the frequency resolution of
that listener's hearing is at best 2 cents.

But you may recall that what I asked was:
>> ... Do I hear the 19-limit, or perhaps the 23-limit?

I'm still no nearer an answer to this question!

> >> [PE]
> >> For quiet sounds, the nonlinearity is insignificant. The loudness
> >> of
> >> the difference tone and other nonlinear combinational tones
> >> increases
> >> *nonlinearly* (surprise) and generally in an accelerating manner as
> >> one increases the loudness of the original sound.
> >
> > [YA]
> >> If that were so, it [the beat] would be inaudible.
> >
> > [PE]
> >> It *is* inaudible (that is, there is no heard pitch corresponding
> >> to
> >> the difference frequency) under conditions of exactly linear
> >> response.
> >
> > [YA]
> >> ... which don't exist except as an approximation good at very low
> >> power ...
> >
> > [PE]
> >> Low loudness.
> >
> >> * Why is the distinction important here?
>
> Because the nonlinearity of the response happens in the ear, so what
> matters is how loud the sound is when it reaches the ear, not its
> power at the source.

This is true. But I did not specify power "at the source".

> > [PE]
> >> Bill Sethares, in his book, claims that combinational
> >> tones are irrelevant to music and tuning because they are too quiet
> >> to matter at normal music listening levels. I think he needs
> >> to "temper" this assessment somewhat.
> >
> >> * They'd make ALL the difference if you're trying to achieve
> >> a totally inharmonic spectrum!
>
> Bill Sethares seems to disagree. His book is all about
> consonance/dissonance and making music with *inharmonic spectra* and
> he completely dismisses the relevance of combinational tones to this
> enterprise.

He's welcome to disagree. But are there good reasons for such a
dismissal? For example, can one demonstrate that combinational
tones will never be salient enough to produce an impression of
harmonic spectra?

> >> [PE]
> >> The exact frequency ratios, over the entire spectrum of audible
> >> absolute frequencies, that are just large enough to permit two sine
> >> waves to be resolved separately, have been measured in a large body
> >> of experiments on human subjects. They vary greatly over the
> >> spectrum, but never become narrower than a whole tone (nearing it
> >> only an an optimal "middle" absolute frequency range). Thankfully,
> >> we
> >> rarely use a single pair of pure sine waves to make music!
> >
> >> * So what you're saying implies that at all audible frequencies,
> >> we cannot hear a pair of pure sine waves differing only by a
> >> semitone as such - our ears require that we hear them as an
> >> AM
> >> tone at their average frequency?
>
> We may hear aspects of both -- there's a "roughness" region between
> the "beating" region and the "two tones" region where the sound is a
> confused compromise between the two latter sensations. But the
> beating tone at the average frequency doesn't completely disappear
> from our hearing until the sine waves differ by at least a whole tone.

OK, let me write a little essay to test my understanding.
Audible frequencies are conventionally 20 Hz to 20,000 Hz.
"Middle" is presumably around the middle of the *log* frequency
range; say roughly from 440 Hz to 1,760 Hz for convenience.
That would be a range of two octaves upwards from A above
middle C. (You may have more exact data.)

In this range we can reliably distinguish at best a whole tone
difference, never less. At a semitone difference, we may be in
the roughness region where we confusedly hear *both* a
semitone difference and a beating average note. So, for
example, playing the 880 Hz A together with A# above it means
we would be able to distinguish the two notes some of the time,
while the rest of the time we hear a strongly beating A+. (++=#)
Is this picture approximately correct?

I can confirm that playing and holding these two notes with an
"Ocarina" patch, and listening to the sound evolve over several
seconds, I hear mostly a beating A+, but from time to time it
seems to "switch" to an almost pure A then to an almost pure
A#, each for a fraction of a second, before returning to the
beating A+. So I've heard the "roughness" region quite clearly.

Playing the 880 Hz A and the B above it together still produces
a roughly beating note; while playing A and C together produces
(surprise!) a distinct third, with both notes clear and no
noticeable beating.

Now I know why my attempts, many years ago, to produce
harmonious music using chords built on seconds were MUCH
less successful than using chords built on thirds or fourths!
I guess it has to be hard to follow separate melodic voices
when you can't even decide what notes they are singing or
playing ... :-)

This humble worm may yet hope to achieve enlightenment
at thy feet, O Guru!

> > How much does this difference vary from one individual to
> > another, according to the "large body of experiments"?
>
> Not a lot, and the interval in question is called the "critical
> band" -- one of the most familiar, and hence well-documented,
> entities in psychoacoustics.

Paul, telling me that something is well-documented is not
very much help, I fear. I scarcely leave the house, so am
unable to browse suitable libraries. But whenever you can
provide me with a link to an online reference, I eagerly
follow it up.

> ... It seems to relate to the physical
> properties (such as elasticity) of the membrane in the cochlea which
> physically "analyzes" sound into its component frequencies.

This makes perfectly good sense ... uh, is perfectly reasonable.
No matter how important the "post-processing" of sound by
the brain, it can never achieve results beyond the powers of
the mechanical senory apparatus to discriminate.

Regards,
Yahya

--
No virus found in this outgoing message.
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🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> >
> > >> [YA]
> > >> {begin OT}
> > >> In this context, I don't know what (prime) limit I recognise in
> > >> tuning;
> > >
> > >> [PE]
> > >> How about odd limit?
> > >
> > >> * No.
> >
> > OK, then there's no constraint on the size of the numbers in the
> > ratios, only on the sizes of their largest prime factors.
>
> That does not follow. I answered that I also don't know what
> odd limit I recognise; not that there isn't one!

My apologies. I misunderstood your "* No" to mean that you don't want
to rephrase the above in terms of odd limit instead of prime limit.
Terrible.

> > ... Are you sure this is the kind of 'limit' you want?
>
> I didn't say so; you merely inferred it, wrongly, as it happens.

Awful. I am sorry.

> > At what point do
> > these high-numbered ratios leave the realm of musical relevance?
Of
> > course it depends on the tuning system, but my point is that a
prime
> > limit alone doesn't seem to be capable of unambiguously
corresponding
> > or not corresponding, in general, to an arbitrary tuning system,
via
> > any aural or even mathematical test, as long as one recognizes a
> > finite uncertainty in hearing intervals (or even merely the
minimal
> > uncertainty allowed by the classical uncertainty
> > principle . . .) . . . The exceptions would seem to be cases
where an
> > odd, integer, or other "finitary" limit is ascertained first, and
> > then the prime limit is simply taken as the largest prime not
> > exceeding that limit.
>
> I don't disagree, but I was focussing on what prime limits
> I can distinguish, rather than on what overall (odd) limit I
> can distinguish. I agree that fixing the latter sets an upper
> bound on the former. In an earlier message, I found that
> a listener's odd limit must lie somewhere near or below 101,
> on the unproven assumption that the frequency resolution of
> that listener's hearing is at best 2 cents.

For 101-limit, wouldn't it have to be considerably better than 0.17
cents based on your earlier message? Because if you can't distinguish
a 'dissonant' or 'non-just' region *in between* the interval 100/99
and the interval 101/100, then it's pretty hard to claim that you're
hearing these two ratios per se as the JI intervals they represent,
rather than simply hearing them as similar but slightly different
intervals.

> > > [PE]
> > >> Bill Sethares, in his book, claims that combinational
> > >> tones are irrelevant to music and tuning because they are too
quiet
> > >> to matter at normal music listening levels. I think he needs
> > >> to "temper" this assessment somewhat.
> > >
> > >> * They'd make ALL the difference if you're trying to achieve
> > >> a totally inharmonic spectrum!
> >
> > Bill Sethares seems to disagree. His book is all about
> > consonance/dissonance and making music with *inharmonic spectra*
and
> > he completely dismisses the relevance of combinational tones to
this
> > enterprise.
>
> He's welcome to disagree. But are there good reasons for such a
> dismissal?

You'll have to take that up with him! He's here . . . I don't happen
to think he's right on this . . .

> For example, can one demonstrate that combinational
> tones will never be salient enough to produce an impression of
> harmonic spectra?

Well, interestingly when one attempts to do so using an inharmonic
series of partials, a different subjective sensation (that
of 'virtual pitch'), which tries to comprimise out the
inharmonicities and derive a most plausible fundamental frequency for
the complex, seems to dominate over the combinational tones. But
that's far from the kind of demonstration you're asking about.

> > >> [PE]
> > >> The exact frequency ratios, over the entire spectrum of audible
> > >> absolute frequencies, that are just large enough to permit two
sine
> > >> waves to be resolved separately, have been measured in a large
body
> > >> of experiments on human subjects. They vary greatly over the
> > >> spectrum, but never become narrower than a whole tone (nearing
it
> > >> only an an optimal "middle" absolute frequency range).
Thankfully,
> > >> we
> > >> rarely use a single pair of pure sine waves to make music!
> > >
> > >> * So what you're saying implies that at all audible
frequencies,
> > >> we cannot hear a pair of pure sine waves differing only
by a
> > >> semitone as such - our ears require that we hear them as
an
> > >> AM
> > >> tone at their average frequency?
> >
> > We may hear aspects of both -- there's a "roughness" region
between
> > the "beating" region and the "two tones" region where the sound
is a
> > confused compromise between the two latter sensations. But the
> > beating tone at the average frequency doesn't completely
disappear
> > from our hearing until the sine waves differ by at least a whole
tone.
>
> OK, let me write a little essay to test my understanding.
> Audible frequencies are conventionally 20 Hz to 20,000 Hz.
> "Middle" is presumably around the middle of the *log* frequency
> range; say roughly from 440 Hz to 1,760 Hz for convenience.
> That would be a range of two octaves upwards from A above
> middle C. (You may have more exact data.)
>
> In this range we can reliably distinguish at best a whole tone
> difference, never less. At a semitone difference, we may be in
> the roughness region where we confusedly hear *both* a
> semitone difference and a beating average note. So, for
> example, playing the 880 Hz A together with A# above it means
> we would be able to distinguish the two notes some of the time,
> while the rest of the time we hear a strongly beating A+. (++=#)
> Is this picture approximately correct?

Yes.

> I can confirm that playing and holding these two notes with an
> "Ocarina" patch, and listening to the sound evolve over several
> seconds, I hear mostly a beating A+, but from time to time it
> seems to "switch" to an almost pure A then to an almost pure
> A#, each for a fraction of a second, before returning to the
> beating A+. So I've heard the "roughness" region quite clearly.

Excellent.

> Playing the 880 Hz A and the B above it together still produces
> a roughly beating note; while playing A and C together produces
> (surprise!) a distinct third, with both notes clear and no
> noticeable beating.
>
> Now I know why my attempts, many years ago, to produce
> harmonious music using chords built on seconds were MUCH
> less successful than using chords built on thirds or fourths!
> I guess it has to be hard to follow separate melodic voices
> when you can't even decide what notes they are singing or
> playing ... :-)

Well, the harmonic overtones help you out *a lot* so that this is
much less of an issue with human voices . . . also we have many aural
cues, particularly during the onset of each note, that can help us
distinguish indivicual melodic lines . . .

> > > How much does this difference vary from one individual to
> > > another, according to the "large body of experiments"?
> >
> > Not a lot, and the interval in question is called the "critical
> > band" -- one of the most familiar, and hence well-documented,
> > entities in psychoacoustics.
>
> Paul, telling me that something is well-documented is not
> very much help, I fear. I scarcely leave the house, so am
> unable to browse suitable libraries. But whenever you can
> provide me with a link to an online reference, I eagerly
> follow it up.

I thought you would naturally "Google" the term or something. No time
now for me to scan through different online references in order to
recommend the best ones. Of course Sethares's book covers this topic
very well.

> > ... It seems to relate to the physical
> > properties (such as elasticity) of the membrane in the cochlea
which
> > physically "analyzes" sound into its component frequencies.
>
> This makes perfectly good sense ... uh, is perfectly reasonable.
> No matter how important the "post-processing" of sound by
> the brain, it can never achieve results beyond the powers of
> the mechanical senory apparatus to discriminate.

Strangely, it does seem to do so. There's evidence that the brain
uses a combination of this cochlear analysis information with
information about *timing* to be able to acheive perceptual feats
that are beyond the ken of any artifical system yet designed . . .
You might want to search for some of Peter Cariani's papers . . .

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi Paul,

On Wed, 26 Oct 2005 "wallyesterpaulrus" wrote:
>
>--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
><yahya@m...> wrote:
>>
>> On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote:
>> >
>> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
>> > <yahya@m...> wrote:
>> >
>> > >> [YA]
>> > >> {begin OT}
>> > >> In this context, I don't know what (prime) limit I recognise in
>> > >> tuning;
>> > >
>> > >> [PE]
>> > >> How about odd limit?
>> > >
>> > >> * No.
>> >
>> > OK, then there's no constraint on the size of the numbers in the
>> > ratios, only on the sizes of their largest prime factors.
>>
>> That does not follow. I answered that I also don't know what
>> odd limit I recognise; not that there isn't one!
>
>My apologies. I misunderstood your "* No" to mean that you don't want
>to rephrase the above in terms of odd limit instead of prime limit.
>Terrible.

Well, I thought to give a short reply for once; seems I chose
a bad time! I will continue my lifelong pursuit of clarity in
communication, aware that I'm chasing a chimaera ...

>> > ... Are you sure this is the kind of 'limit' you want?
>>
>> I didn't say so; you merely inferred it, wrongly, as it happens.
>
>Awful. I am sorry.

I'm sorry we misunderstood each other.

>> > At what point do
>> > these high-numbered ratios leave the realm of musical relevance?
>Of
>> > course it depends on the tuning system, but my point is that a
>prime
>> > limit alone doesn't seem to be capable of unambiguously
>corresponding
>> > or not corresponding, in general, to an arbitrary tuning system,
>via
>> > any aural or even mathematical test, as long as one recognizes a
>> > finite uncertainty in hearing intervals (or even merely the
>minimal
>> > uncertainty allowed by the classical uncertainty
>> > principle . . .) . . . The exceptions would seem to be cases
>where an
>> > odd, integer, or other "finitary" limit is ascertained first, and
>> > then the prime limit is simply taken as the largest prime not
>> > exceeding that limit.
>>
>> I don't disagree, but I was focussing on what prime limits
>> I can distinguish, rather than on what overall (odd) limit I
>> can distinguish. I agree that fixing the latter sets an upper
>> bound on the former. In an earlier message, I found that
>> a listener's odd limit must lie somewhere near or below 101,
>> on the unproven assumption that the frequency resolution of
>> that listener's hearing is at best 2 cents.
>
>For 101-limit, wouldn't it have to be considerably better than 0.17
>cents based on your earlier message? Because if you can't distinguish
>a 'dissonant' or 'non-just' region *in between* the interval 100/99
>and the interval 101/100, then it's pretty hard to claim that you're
>hearing these two ratios per se as the JI intervals they represent,
>rather than simply hearing them as similar but slightly different
>intervals.

Yep. Maybe we'd have to divide the resolution required
by three or four, in order to have a band of dissonance of
reasonable width; along the lines of the following sketch:
--::|::--::|::----::|::--::|::---------::|::--::|::---::|::--
which divides the audio frequency spectrum up by equal
ticks, with values - for the sensation of dissonance, : for
the sensation of a JI interval, and | for the exact location
of the JI ratio. If you didn't have the -- regions, and the
:: regions (say + or - 2c, or 1.5c, or whatever, depending on
the resolution of your hearing) overlapped, you couldn't
say which of the two ratios you were hearing. This is
exactly analogous to trying to tune in distant short-wave
radio stations in a crowded band. Mind you, it might still
be possible to say - as people completely untrained in
music do say - "that sounds sweet" - without ever being
able to put numbers to the sound, even when the band of
dissonance is vanishingly small.

Still, I take your argument to imply that the frequency
resolution of the listener's hearing for 101-limit, would
have to be *at least a bit* (not considerably) better
than 0.17 cents.

>> > > [PE]
>> > >> Bill Sethares, in his book, claims that combinational
>> > >> tones are irrelevant to music and tuning because they are too
>quiet
>> > >> to matter at normal music listening levels. I think he needs
>> > >> to "temper" this assessment somewhat.
>> > >
>> > >> * They'd make ALL the difference if you're trying to achieve
>> > >> a totally inharmonic spectrum!
>> >
>> > Bill Sethares seems to disagree. His book is all about
>> > consonance/dissonance and making music with *inharmonic spectra*
>and
>> > he completely dismisses the relevance of combinational tones to
>this
>> > enterprise.
>>
>> He's welcome to disagree. But are there good reasons for such a
>> dismissal?
>
>You'll have to take that up with him! He's here . . . I don't happen
>to think he's right on this . . .
>
>> For example, can one demonstrate that combinational
>> tones will never be salient enough to produce an impression of
>> harmonic spectra?
>
>Well, interestingly when one attempts to do so using an inharmonic
>series of partials, a different subjective sensation (that
>of 'virtual pitch'), which tries to comprimise out the
>inharmonicities and derive a most plausible fundamental frequency for
>the complex, seems to dominate over the combinational tones. But
>that's far from the kind of demonstration you're asking about.

What is the mechanism behind virtual pitch? Is it brain
or bone that produces the sensation of a non-existent
fundamental tone?

[snip lots]

>> Playing the 880 Hz A and the B above it together still produces
>> a roughly beating note; while playing A and C together produces
>> (surprise!) a distinct third, with both notes clear and no
>> noticeable beating.
>>
>> Now I know why my attempts, many years ago, to produce
>> harmonious music using chords built on seconds were MUCH
>> less successful than using chords built on thirds or fourths!
>> I guess it has to be hard to follow separate melodic voices
>> when you can't even decide what notes they are singing or
>> playing ... :-)
>
>Well, the harmonic overtones help you out *a lot* so that this is
>much less of an issue with human voices . . . also we have many aural
>cues, particularly during the onset of each note, that can help us
>distinguish indivicual melodic lines . . .

I see - these might include differences in the timing
of the onsets, their stereo location in space, the
different transient formants that characterise the
different timbres of different people's voices, etc.

>> > > How much does this difference vary from one individual to
>> > > another, according to the "large body of experiments"?
>> >
>> > Not a lot, and the interval in question is called the "critical
>> > band" -- one of the most familiar, and hence well-documented,
>> > entities in psychoacoustics.
>>
>> Paul, telling me that something is well-documented is not
>> very much help, I fear. I scarcely leave the house, so am
>> unable to browse suitable libraries. But whenever you can
>> provide me with a link to an online reference, I eagerly
>> follow it up.
>
>I thought you would naturally "Google" the term or something. No time
>now for me to scan through different online references in order to
>recommend the best ones. Of course Sethares's book covers this topic
>very well.

Naturally, I should have used "google" or "kartoo" to chase
up "critical band". I'll try to remember that next time.

[snip]

>> No matter how important the "post-processing" of sound by
>> the brain, it can never achieve results beyond the powers of
>> the mechanical sen[s]ory apparatus to discriminate.
>
>Strangely, it does seem to do so. There's evidence that the brain
>uses a combination of this cochlear analysis information with
>information about *timing* to be able to acheive perceptual feats
>that are beyond the ken of any artifical system yet designed . . .
>You might want to search for some of Peter Cariani's papers . . .

Thanks, will do. His name's familiar from a few months back
on this list.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> >> > At what point do
> >> > these high-numbered ratios leave the realm of musical
relevance?
> >Of
> >> > course it depends on the tuning system, but my point is that a
> >prime
> >> > limit alone doesn't seem to be capable of unambiguously
> >corresponding
> >> > or not corresponding, in general, to an arbitrary tuning
system,
> >via
> >> > any aural or even mathematical test, as long as one recognizes
a
> >> > finite uncertainty in hearing intervals (or even merely the
> >minimal
> >> > uncertainty allowed by the classical uncertainty
> >> > principle . . .) . . . The exceptions would seem to be cases
> >where an
> >> > odd, integer, or other "finitary" limit is ascertained first,
and
> >> > then the prime limit is simply taken as the largest prime not
> >> > exceeding that limit.
> >>
> >> I don't disagree, but I was focussing on what prime limits
> >> I can distinguish, rather than on what overall (odd) limit I
> >> can distinguish. I agree that fixing the latter sets an upper
> >> bound on the former. In an earlier message, I found that
> >> a listener's odd limit must lie somewhere near or below 101,
> >> on the unproven assumption that the frequency resolution of
> >> that listener's hearing is at best 2 cents.
> >
> >For 101-limit, wouldn't it have to be considerably better than
0.17
> >cents based on your earlier message? Because if you can't
distinguish
> >a 'dissonant' or 'non-just' region *in between* the interval
100/99
> >and the interval 101/100, then it's pretty hard to claim that
you're
> >hearing these two ratios per se as the JI intervals they
represent,
> >rather than simply hearing them as similar but slightly different
> >intervals.
>
> Yep. Maybe we'd have to divide the resolution required
> by three or four, in order to have a band of dissonance of
> reasonable width; along the lines of the following sketch:
> --::|::--::|::----::|::--::|::---------::|::--::|::---::|::--
> which divides the audio frequency spectrum up by equal
> ticks, with values - for the sensation of dissonance, : for
> the sensation of a JI interval, and | for the exact location
> of the JI ratio. If you didn't have the -- regions, and the
> :: regions (say + or - 2c, or 1.5c, or whatever, depending on
> the resolution of your hearing) overlapped, you couldn't
> say which of the two ratios you were hearing. This is
> exactly analogous to trying to tune in distant short-wave
> radio stations in a crowded band. Mind you, it might still
> be possible to say - as people completely untrained in
> music do say - "that sounds sweet" - without ever being
> able to put numbers to the sound, even when the band of
> dissonance is vanishingly small.
>
> Still, I take your argument to imply that the frequency
> resolution of the listener's hearing for 101-limit, would
> have to be *at least a bit* (not considerably) better
> than 0.17 cents.

If it's only a bit better than 0.17 cents, you wouldn't have room for
the -- and : regions, would you?

In any case, I'm glad you basically agree with this.

> >Well, interestingly when one attempts to do so using an inharmonic
> >series of partials, a different subjective sensation (that
> >of 'virtual pitch'), which tries to comprimise out the
> >inharmonicities and derive a most plausible fundamental frequency
for
> >the complex, seems to dominate over the combinational tones. But
> >that's far from the kind of demonstration you're asking about.
>
> What is the mechanism behind virtual pitch? Is it brain
> or bone that produces the sensation of a non-existent
> fundamental tone?

Brain: it happens just as strongly when the "harmonics" are presented
separately to each ear (but at the same time) . . .

> [snip lots]
>
>
> >> Playing the 880 Hz A and the B above it together still produces
> >> a roughly beating note; while playing A and C together produces
> >> (surprise!) a distinct third, with both notes clear and no
> >> noticeable beating.
> >>
> >> Now I know why my attempts, many years ago, to produce
> >> harmonious music using chords built on seconds were MUCH
> >> less successful than using chords built on thirds or fourths!
> >> I guess it has to be hard to follow separate melodic voices
> >> when you can't even decide what notes they are singing or
> >> playing ... :-)
> >
> >Well, the harmonic overtones help you out *a lot* so that this is
> >much less of an issue with human voices . . . also we have many
aural
> >cues, particularly during the onset of each note, that can help us
> >distinguish indivicual melodic lines . . .
>
> I see - these might include differences in the timing
> of the onsets, their stereo location in space, the
> different transient formants that characterise the
> different timbres of different people's voices, etc.

And my first point here was that the non-overlapping overtones help
us to resolve the separate fundamentals even when those themselves
overlap.