Yahoo Tuning Groups Ultimate Backup tuning Prime Rarity?

Do prime numbers (when included in the numerator/denominator of rationally described scale pitches) impart a special characteristic as
differentiated from non-prime numbers (unique to each prime which
constitutes one of the factors within the numerator/denominator of such
scale pitches)? If so, is it true in the case of: small valued primes only;
large valued primes only; or is such a prime status the only factor
determining whether or not such a special characteristic may present
itself to you, the listener, in evaluating the results of listening tests?

Here is (what should prove to be) a very interesting TEST of the *audible
existence* such "prime uniqueness":

(1) Setup your system to play a 1/1 and a 3/2 pitch in unison.

(2) Add to this dyad the following two (approximate) "major 3rd" pitches
(one at a time). The first is 31/24 [consisting of 2^(-3), 3^(-1), 31^(1)].
Now replace 31/24 with a second pitch (which is *only* 1.01568 cents flat
from 31/24) of 71/55 [consisting of 5^(1), 11^(1), 71^(1), a very different
set of higher numbered primes].

(3) Can you hear any difference between these (virtually identical) scale
pitches when played in *unison* with the dyad consisting of 1/1 and 3/2
(or, perhaps, in unison with the 1/1 pitch only)?

Here is a SECOND TEST [where you could add one of the pitches below,
together in unison with the 1/1 and 3/2 "basis dyad" *alone*, or in unison
with *one* of the two previously (above) given "major 3rd" scale pitches of
31/24 or 71/55 to form a (sort of) "dominant 7th" chord]:

(1) Setup your system to play the 1/1 and a 3/2 pitch in unison.

(2) Add to this dyad the following two (approximate) "minor 7th" pitches
(one at a time). The first is "good-old" 16/9 [consisting of 2^(4),
3^(-2)]. Now replace 16/9 with a second pitch (which is *only* 2.92685
cents flat from 16/9) of 197/111 [consisting of 111^(-1), 197^(1), a very
different set of higher numbered primes].

Here is a THIRD TEST using "major 6th" scale pitches, as well:

(1) Setup your system to play the 1/1 and a 3/2 pitch in unison.

(2) Add to this dyad the following two (approximate) "major 6th" pitches
(one at a time). The first is 49/30 [consisting of 2^(-1), 3^(-1), 5^(-1),
7^(2)]. Now replace 49/30 with a second pitch (which is *only* 1.86054
cents flat from 49/30) of 31/19 [consisting of 19^(-1), 31^(1), a very
different set of higher numbered primes].

I don't have the hardware to do a decent "A to B" test of the above
scenarios, but am VERY CURIOUS as to what *audible* results you may find!!!

What I SUSPECT (without having *listened* to the above examples, at
all...), is that these *miniscule* shifts in pitch, and perceived
"character" of these various closely alinged pitches (despite the marked
difference in the size and value of the primes involved) will *not* be
audibly different. Let me know what you find, and what your ear tells
you! :)

I (theoretically, granted) suspect that it is the "small numerical values"
of prime numerators (and denominators) which is the "cogent" characteristic
of their "sweetness" (how's that for a vague term?). Within the set of
integers (of 1, 2, 3, 4, 5, 6, 7), note that only *two* of them are *not*
prime (28.57%). Considering the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14 (which is twice the size of the original set), there are seven which
are not prime (50 %). Considering the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
(four times the size of the original set), there are eighteen which are
*not* prime (64.29 %). At a value of 28 (if not a smaller numerical integer
value than that), we have (I believe) reached values where "harmonic
coincidence in frequency" (as a result of the large valued numerators) and
the "harmonic coincidence in period" (as a result of the large valued
denominators) become (virtually) insignificant to "the ear".

As a result of the above, (I suspect) that the "special" significance of
"primes" *may* be a result of their numerical *predominance* (as a
percentage of the total number of integers within such a set considered) in
the area (in a given set ranging between 1 and N) of the "lower-numbered"
integers on the real number line. This would not preclude the listener
perceiving a unique "character" associated with each of the
"lower-numbered" primes (say, up to 19) , since the perceived "octave
equivalence" of the (harmonic multiples 4, 6, 8, 9, 10, 12, 14, 15, 16, 18,
etc) of the "lower-numbered" primes 2, 3, 5, and 7 (which, themselves,
generate all of the non-prime integers up to and including the integer 21)
might tend to *suggest* (as opposed to be perceived as distinct from),
therefore "perceptually reinforce", the "unique character" of the primes 2,
3, 5, and 7 via such an "octave equivalence" effect...

And, for those disciples of Terhardt (bless their hearts) who (may) believe
that an "implied fundamental" is invariably (without spectral contention or
confusion) discerned by the human "aural mind" when the ear is presented
with complicated "forests" of spectra which may result from the combination
of "complex" (fundamental with harmonics) tones sounded in unison, they
will find nothing in the hypothesis above which would tend to denigrate the
veracity of Terhardt's "virtual pitch" hypothesis! :)

Feedback from those who try some/all of the above "listening tests" invited!!!

Regards, J Gill

🔗clumma <carl@lumma.org>

--- In tuning@y..., J Gill <JGill99@i...> wrote:
> Do prime numbers (when included in the numerator/denominator of
> rationally described scale pitches) impart a "special
> characteristic" as differentiated from non-prime numbers (unique
> to each prime which constitutes one of the factors within the
> numerator/denominator of such scale pitches)?

After much listening, thinking, and discussion, the consensus on
this list has been that they do not.

> Here is (what should prove to be) a very interesting TEST of the
> *audible existence* such "prime uniqueness":
//
> (3) Can you hear any difference between these (virtually identical)
> scale pitches when played in *unison* with the dyad consisting of
> 1/1 and 3/2 (or, perhaps, in unison with the 1/1 pitch only)?

I'm sure I could hear a difference, but it does not follow that
this difference would be due to the prime factorization of the
ratios.

> What I SUSPECT (without having *listened* to the above examples, at
> all...), is that these *miniscule* shifts in pitch, and perceived
> "character" of these various closely alinged pitches (despite the
> marked difference in the size and value of the primes involved)
> will *not* be audibly different. Let me know what you find, and
> what your ear tells you! :)

The pitches are different, so their "character" can be different,
or the same -- it would be completely subjective either way. I
have no doubt that people could learn to differentiate these
intervals by ear. Many people can be trained to identify the
chromatic pitches by ear, even when played in isolation -- "absolute
pitch".

> I (theoretically, granted) suspect that it is the "small numerical
> values" of prime numerators (and denominators)

In my experience, small numerical values in interval ratios are
responsible for a unique, musically important effect, which plays
a role in consonance and dissonance -- but the same cannot be said
of prime numerators and denominators.

> (how's that for a vague term?). Within the set of
> integers (of 1, 2, 3, 4, 5, 6, 7), note that only
> *two* of them are *not* prime (28.57%).

Do they sound different to you? You might argue that this is no
fair, because 6 and 4 are octave-equivalents of prime integers...
in this case the set (1 3 5 7 9 11 15) provides a better test.

> As a result of the above, (I suspect) that the "special"
> significance of "primes" *may* be a result of their numerical
> *predominance* (as a percentage of the total number of integers
> within such a set considered) in the area (in a given set ranging
> between 1 and N) of the "lower-numbered" integers on the real
> number line.

Now this I can agree with. The important tests here are the
identities 9 and 15... do they share a difference relative to
say, 5 7 and 11?

Having said all this, it's worth pointing out that most people
can be trained to differentiate harmonic intervals via "relative
pitch"... which holds regardless of absolute pitch. To what are
they listening? A few have agreed with me that pairs like
(4:3, 3:2) and (5:4, 8:5) are defined by a common sound quality...
which I describe as "square" and "sweet" respectively. And if so,
what are we listening to? Is there anyone who will say that 5:2
does not sound more like 5:4 than 4:3 or 8:3? And if so, what is
special about the octave? (7:4, 7:6, 7:5) seem to share a
quality... while 9:4 seems to belong to the (3:2, 4:3, 3:1) family,
to this listener.

One suggestion is that intervals have a 'period signature'. A
3-againt-anything beat in the virtual pitch processor gives certain
sensation wherever it appears, and the same for 5- and 7-. Here
prime numbers would be important, since they represent points at
which the atomic periods get shorter, ie a 15:8 still has sub-
periods of 3 and 5.

I put all of this in a disclaimer because I do not feel it is
1/10th as flagrant an effect as whatever the prime-number theorists
are talking about. For one thing, it has nothing (directly) to do
with consonance and dissonance; only with a _subtle_ perception of
interval quality along the lines of what is useful for relative
pitch.

-Carl

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "clumma" <carl@l...> wrote:
> --- In tuning@y..., J Gill <JGill99@i...> wrote:

> > Do prime numbers (when included in the numerator/denominator of
> > rationally described scale pitches) impart a "special
> > characteristic" as differentiated from non-prime numbers (unique
> > to each prime which constitutes one of the factors within the
> > numerator/denominator of such scale pitches)?
>
> After much listening, thinking, and discussion, the consensus on
> this list has been that they do not.
>
> > Here is (what should prove to be) a very interesting TEST of the
> > *audible existence* such "prime uniqueness":
> //
> > (3) Can you hear any difference between these (virtually identical)
> > scale pitches when played in *unison* with the dyad consisting of
> > 1/1 and 3/2 (or, perhaps, in unison with the 1/1 pitch only)?
>
> I'm sure I could hear a difference,

JG: Have you tried the experiment yourself?

> but it does not follow that
> this difference would be due to the prime factorization of the
> ratios.

JG: Are you saying that it would, then, be a result of the 1.01568 Cent difference between the two (close to a "major 3rd") pitches?

> > What I SUSPECT (without having *listened* to the above examples, at
> > all...), is that these *miniscule* shifts in pitch, and perceived
> > "character" of these various closely alinged pitches (despite the
> > marked difference in the size and value of the primes involved)
> > will *not* be audibly different. Let me know what you find, and
> > what your ear tells you! :)
>
> The pitches are different,

JG: By barely 1 Cent.

> so their "character" can be different,
> or the same -- it would be completely subjective either way. I
> have no doubt that people could learn to differentiate these
> intervals by ear. Many people can be trained to identify the
> chromatic pitches by ear, even when played in isolation -- "absolute
> pitch".

JG: Resolving a 1 Cent difference?

> > I (theoretically, granted) suspect that it is the "small numerical
> > values" of prime numerators (and denominators)
>
> In my experience, small numerical values in interval ratios are
> responsible for a unique, musically important effect, which plays
> a role in consonance and dissonance -- but the same cannot be said
> of prime numerators and denominators.

JG: Don't prime numbers (as opposed to the product of primes, or as opposed to a prime taken to an integer valued exponent) rather predominate in the "small numerical values" in the range between 1 and 7 (how would you define the phrase "small numerical values")?

> > (how's that for a vague term?). Within the set of
> > integers (of 1, 2, 3, 4, 5, 6, 7), note that only
> > *two* of them are *not* prime (28.57%).
>
> Do they sound different to you? You might argue that this is no
> fair, because 6 and 4 are octave-equivalents of prime integers...
> in this case the set (1 3 5 7 9 11 15) provides a better test.

JG: My point above was to demonstrate the *predominance* (within the range of 1 to 7, anyway) of "stand-alone" primes [those that are *not* (like the integers 4 and 6), "octave-equivalents of prime integers" (as you stated above)].
>
> > As a result of the above, (I suspect) that the "special"
> > significance of "primes" *may* be a result of their numerical
> > *predominance* (as a percentage of the total number of integers
> > within such a set considered) in the area (in a given set ranging
> > between 1 and N) of the "lower-numbered" integers on the real
> > number line.
>
> Now this I can agree with. The important tests here are the
> identities 9 and 15... do they share a difference relative to
> say, 5 7 and 11?

JG: Good question. I would not know how to attempt to determine this (other than find pitch-ratios which are [nearly, or"imperceptibly" (whatever miniscule fraction of a cent which you might contend can *not* be discerned by any listener, "trained" or otherwise)] identical in frequency to each other (as a method for differentiating such an effect via listening tests).

> Having said all this, it's worth pointing out that most people
> can be trained to differentiate harmonic intervals via "relative
> pitch"... which holds regardless of absolute pitch. To what are
> they listening? A few have agreed with me that pairs like
> (4:3, 3:2) and (5:4, 8:5) are defined by a common sound quality...
> which I describe as "square" and "sweet" respectively. And if so,
> what are we listening to? Is there anyone who will say that 5:2
> does not sound more like 5:4 than 4:3 or 8:3? And if so, what is
> special about the octave? (7:4, 7:6, 7:5) seem to share a
> quality... while 9:4 seems to belong to the (3:2, 4:3, 3:1) family,
> to this listener.

JG: Excellent questions, Carl!

> One suggestion is that intervals have a 'period signature'. A
> 3-againt-anything beat in the virtual pitch processor gives certain
> sensation wherever it appears, and the same for 5- and 7-. Here
> prime numbers would be important, since they represent points at
> which the atomic periods get shorter, ie a 15:8 still has sub-
> periods of 3 and 5.

JG: Now, are you speaking (directly above) of something that would work with sin/cos waveforms alone, or are you speaking of phenomena which occur only when "complex" (fundamental along with harmonics) tones are listened to?

> I put all of this in a disclaimer because I do not feel it is
> 1/10th as flagrant an effect as whatever the prime-number theorists
> are talking about.

JG: Then what *are* "the prime theorists" talking about...?

> For one thing, it has nothing (directly) to do
> with consonance and dissonance;
> only with a _subtle_ perception of
> interval quality along the lines of what is useful for relative
> pitch.

JG: But, it seems to have a *lot* to do with whether the utilization of various primes in the numerators/denominators of "pitch-ratios" shows any sort of distinctness *between the different primes*, doesn't it . The question was - do they sound any *different*? My assumption (with which it appears that you may well disagree) was that a 1 or 2 cent shift in pitch would be non-discernable (from "pitch-height" perspective). As a result of that assumption, my premise was that *any* discernable difference would, then, arise out of the utilization of *different* primes ONLY. If it is *not* the case that *different* primes *sound different* when utilized in pitch ratios (at, or close to) *identical* values of frequency ("pitch-height"), the WHAT THE HECK *ARE* "the prime theorists" talking about, anyway??? Would you argue that the larger valued primes are different that the smaller valued primes [thus dismissing the meaningfullness of substituting primes such as 19 and 31 for the primes 2, 3, 5, and 7, as in the case of my third "listening test", involving an scale-pitch close to that of a "major 6th" (ie JI 5/3)]?

In the beginning of your post, in response to my questions:

You state:

> After much listening, thinking, and discussion, the consensus on
> this list has been that they do not.

Your statement (from one perspective) could be seen to imply that you would answer in the negative to *both* parts of my questions above.

Am I correct in assuming (from your statements therein) that your statement (quoted directly above) is answering in the negative only to the *first* part of the questions, which is:

<<Do prime numbers (when included in the numerator/denominator of
rationally described scale pitches) impart a "special
characteristic" as differentiated from non-prime numbers...?>>

Should I assume (from your statements therein) that your statement (quoted directly above) is answering in the negative only to the *second* part of the questions, which is:

<<...unique to each prime which constitutes one of the factors
within the numerator/denominator of such scale pitches?>>

Or, should I assume (from your statements therein) that your statement (quoted directly above) is answering in the *positive* to *both* the first as well as the second parts of my question above?

It's *great* to chat with you about this stuff, J Gill :)

🔗clumma <carl@lumma.org>

>> I'm sure I could hear a difference,
>
> JG: Have you tried the experiment yourself?

Not with these precise intervals. But I have tried
to resolve 1-cent differences before. If I were to
change a single pitch on a piano by 1 cent and then
played music on it, even the best ears would probably
pick up nothing amiss. But those same ears, when
hearing the two different intervals side by side,
could tell the difference. Randomly detuning 50% of
the notes on a piano by 1 cent and then playing music
on it, the best ears would say the tuning was funny,
or maybe that the tone of the piano was odd.

>> but it does not follow that
>> this difference would be due to the prime factorization of the
>> ratios.
>
> JG: Are you saying that it would, then, be a result of the 1.01568
> Cent difference between the two (close to a "major 3rd") pitches?

I don't know what the difference would be due to. I was just
pointing out that it wouldn't necessarily be primeness.

I grant you that 1 cent is pretty small, but if primeness was
really special, would it not extend to tests like 7:4 vs. 9:5
and 16:9?

>> In my experience, small numerical values in interval ratios are
>> responsible for a unique, musically important effect, which plays
>> a role in consonance and dissonance -- but the same cannot be said
>> of prime numerators and denominators.
>
> JG: Don't prime numbers (as opposed to the product of primes, or
> as opposed to a prime taken to an integer valued exponent) rather
> predominate in the "small numerical values" in the range between 1
> and 7

yes.

> (how would you define the phrase "small numerical values")?

The product of numerator and denominator of ratios in lowest
terms has shown itself to be a good rule of thumb for ratios
whose product is less than 100.

> > > (how's that for a vague term?). Within the set of
> > > integers (of 1, 2, 3, 4, 5, 6, 7), note that only
> > > *two* of them are *not* prime (28.57%).
> >
> > Do they sound different to you? You might argue that this is no
> > fair, because 6 and 4 are octave-equivalents of prime integers...
> > in this case the set (1 3 5 7 9 11 15) provides a better test.
>
> JG: My point above was to demonstrate the *predominance* (within
>the range of 1 to 7, anyway) of "stand-alone" primes [those that
>are *not* (like the integers 4 and 6), "octave-equivalents of prime
>integers" (as you stated above)].

Okay.

>> Now this I can agree with. The important tests here are the
>> identities 9 and 15... do they share a difference relative to
>> say, 5 7 and 11?
>
> JG: Good question. I would not know how to attempt to determine
>this (other than find pitch-ratios which are [nearly, or
>"imperceptibly" (whatever miniscule fraction of a cent which you
>might contend can *not* be discerned by any listener, "trained" or
>otherwise)] identical in frequency to each other (as a method for
>differentiating such an effect via listening tests).

Why not just listen to the intervals
9:8, 5:4, 11:8, 7:4, 15:8, 9:4, 5:2, 11:4 ?

>> One suggestion is that intervals have a 'period signature'. A
>> 3-againt-anything beat in the virtual pitch processor gives
>> certain sensation wherever it appears, and the same for 5- and
>> 7-. Here prime numbers would be important, since they represent
>> points at which the atomic periods get shorter, ie a 15:8 still
>> has sub-periods of 3 and 5.
>
>JG: Now, are you speaking (directly above) of something that would
>work with sin/cos waveforms alone, or are you speaking of phenomena
>which occur only when "complex" (fundamental along with harmonics)
>tones are listened to?

An important feature of this suggestion is that it should occur
with pure tones as well as complex tones, though maybe less
intensely because there are fewer things going on. Perhaps an
artificial timbre like a Shepherd tone could also be tried.

>> I put all of this in a disclaimer because I do not feel it is
>> 1/10th as flagrant an effect as whatever the prime-number
>> theorists are talking about.
>
> JG: Then what *are* "the prime theorists" talking about...?

Often, they're just wrong. But there is an effect that happens
when composing in fixed JI -- composing with a pitch set like
1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 rather than an interval set
like 1:1, 6:5, 5:4, 3:2, 2:1. Here, the prime limit of the
_pitch set_ tends to approximate the odd-limit of the interval
set, for certain types of scales at least. I think this explains
how the prime-limit thing has lasted so long. That, a mysticism
seems to be a powerful force in art (LaMonte young, for example,
hardly bothers to make psychoacoustic claims for primes).

>The question was - do they sound any *different*? My assumption
>(with which it appears that you may well disagree) was that a 1
>or 2 cent shift in pitch would be non-discernable (from "pitch-
>height" perspective).

I will try your particular examples. The Justonic Pitch
Palette software has a "wave audio" setting which is suspect
is pretty accurate. Give me a day to dig up my copy and
install it.

>As a result of that assumption, my premise was that *any*
>discernable difference would, then, arise out of the utilization
>of *different* primes ONLY.

Then you can show that pitch height and prime factorization are
the only two properties of musical intervals?

>Would you argue that the larger valued primes are different that
>the smaller valued primes [thus dismissing the meaningfullness
>of substituting primes such as 19 and 31 for the primes 2, 3, 5,
>and 7, as in the case of my third "listening test", involving an
>scale-pitch close to that of a "major 6th" (ie JI 5/3)]?

Yes. For isolated dyads, the product of numerator and denominator
gives a good ranking. For music, the odd-limit of the set of
consonant dyads used gives a good ranking. In either case, big
primes are less concordant than small primes, just as big odd
composite numbers are less concordant than small odd composite
numbers.

> Am I correct in assuming (from your statements therein) that your
>statement (quoted directly above) is answering in the negative only
>to the *first* part of the questions, which is:
>
> <<Do prime numbers (when included in the numerator/denominator of
> rationally described scale pitches) impart a "special
> characteristic" as differentiated from non-prime numbers...?

It answers in the negative to this. Note this isn't quite the same
as my disclaimer, which is that groups of intervals of a given
prime limit may share a *subtle* 'flavor' characteristic. In this
case, composites still have the special characteristic -- they have
the one of their prime factors. I should mention that my disclaimer
is purely speculative, and the scarcity of small composite numbers
(only 9 and 15), as you point out, makes it hard, if not impossible,
to test (because once the numbers get above 15, their discordance
begins to over-power the subtler effect of my disclaimer).

-Carl

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., J Gill <JGill99@i...> wrote:

> Do prime numbers (when included in the numerator/denominator of
rationally
> described scale pitches) impart a "special characteristic" as
> differentiated from non-prime numbers (unique to each prime which
> constitutes one of the factors within the numerator/denominator of
such
> scale pitches)?

A pitch is just a pitch, no matter what ratio, if any, is used to
describe it. What matters is its relationship with other pitches. So
you have to tell me what other pitches are around.

We've discussed this issue at great length on many occasions on this
list. I have a very strong opinion about this issue that I'll defend
to my wit's end -- that's how sure I am about my point of view here.
I'll also insist that my point of view is identical to Harry
Partch's, if he is read _carefully_ and not in the uncritical way
that many people have run with his ideas. When it comes to
_intervals_, primeness or non-primeness is _unimportant_ in
describing the effect of the interval. I agree with Dave Keenan that
the three factors of importance are

(1) COMPLEXITY -- the size of the numbers in the ratio
(2) TOLERANCE -- the inability of the ear to distinguish slight
mistuning

[(1) & (2) are combined in a more sophisticated way in harmonic
entropy]

(3) SPAN -- the actual *size* (as in cents) of the interval.

> Here is (what should prove to be) a very interesting TEST of the
*audible
> existence* such "prime uniqueness":
>
> (1) Setup your system to play a 1/1 and a 3/2 pitch in unison.
>
> (2) Add to this dyad the following two (approximate) "major 3rd"
pitches
> (one at a time). The first is 31/24 [consisting of 2^(-3), 3^(-1),
31^(1)].
> Now replace 31/24 with a second pitch (which is *only* 1.01568
cents flat
> from 31/24) of 71/55 [consisting of 5^(1), 11^(1), 71^(1), a very
different
> set of higher numbered primes].

These are pretty poor tests, since almost no one will hear any
qualitative differences -- one may notice a slight change in the rate
of beating of one or another pair of partials, that's all. The
simplest chord in your tests is 1/1:3/2:16/9, and changing the top
note only changes the 16:9 and 32:27 intervals in the chord,
intervals which, with normal timbres, one cannot tune just by ear
anyway (unless one adds additional pitches to the sound). As for the
other tests, fuggeddaboudditt :)

> I (theoretically, granted) suspect that it is the "small numerical
values"
> of prime numerators (and denominators) which is the "cogent"
characteristic
> of their "sweetness" (how's that for a vague term?). Within the set
of
> integers (of 1, 2, 3, 4, 5, 6, 7), note that only *two* of them are
*not*
> prime (28.57%). Considering the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12,
> 13, 14 (which is twice the size of the original set), there are
seven which
> are not prime (50 %). Considering the set 1, 2, 3, 4, 5, 6, 7, 8,
9, 10,
> 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28
> (four times the size of the original set), there are eighteen which
are
> *not* prime (64.29 %). At a value of 28 (if not a smaller numerical
integer
> value than that), we have (I believe) reached values
where "harmonic
> coincidence in frequency" (as a result of the large valued
numerators) and
> the "harmonic coincidence in period" (as a result of the large
valued
> denominators) become (virtually) insignificant to "the ear".

True, but below that point, it's easy to fall into the trap of
ascribing an acoustical significance to primeness/compositeness . . .
which I hereby warn you against.

> As a result of the above, (I suspect) that the "special"
significance of
> "primes" *may* be a result of their numerical *predominance* (as a
> percentage of the total number of integers within such a set
considered) in
> the area (in a given set ranging between 1 and N) of the "lower-
numbered"
> integers on the real number line. This would not preclude the
listener
> perceiving a unique "character" associated with each of the
> "lower-numbered" primes (say, up to 19) , since the
perceived "octave
> equivalence" of the (harmonic multiples 4, 6, 8, 9, 10, 12, 14, 15,
16, 18,
> etc) of the "lower-numbered" primes 2, 3, 5, and 7 (which,
themselves,
> generate all of the non-prime integers up to and including the
integer 21)
> might tend to *suggest* (as opposed to be perceived as distinct
from),
> therefore "perceptually reinforce", the "unique character" of the
primes 2,
> 3, 5, and 7 via such an "octave equivalence" effect...

How does 9 arise via an "octave equivalence" effect?

> And, for those disciples of Terhardt (bless their hearts) who (may)
believe
> that an "implied fundamental" is invariably (without spectral
contention or
> confusion)

Of course there's spectral contention and confusion!

🔗paulerlich <paul@stretch-music.com>

🔗Dante Rosati <dante.interport@rcn.com>

here is my take on this issue, from my webpage:

http://users.rcn.com/dante.interport//hartheory2.html

Primes
Musical sound can be thought of as audible number relations. How it is that
audible number relations become the medium for the deep and complex musical
languages that exist in every culture is unknown and, perhaps, unknowable.
Harmonic theory rightly investigates the way number is manifest in sound
itself, and in what ways, if any, these relationships are audible.

Just as prime numbers are the building blocks of the natural numbers, so are
they the building blocks of sounds. In the harmonic series, even numbered
partials always generate octave "duplications" of some lower partial. This
feeling of octave equivalence seems to be universal, and a qualitative
difference between even and odd partials is generally acknowledged.
Nevertheless, noone can really say why octave duplication sounds to us like
a projection of the same note at another level. It is as if the ratio 2/1
was in a class by itself, utterly different from all other ratios. I think
it is more likely that the identity we hear in 2/1 is still present, though
to a lesser degree, in 3/2 (even more so in 3/1). The identification with
the fundamental lessens still further with each successive odd number
introduced, while even partials always echo a quality from lower in the
series.

Each partial is also itself the fundamental of its own series. For example,
the ninth partial of a series is coincident with the third partial of its
own third partial (3x3). This shows that the ninth partial is, on one level,
redundant, because it can be derived both from the original series and as a
projection of lower combined first-order and second-order partials. To my
ears this results in nine being heard as a projection (and perhaps
intensification -3^2) of threeness rather than as a new quality. The
seventh partial, on the other hand, cannot be produced from lower partials
upward by any method. It is "new" in a deeper sense than the ninth is, or
the fifteenth. Nine sounds like it belongs with its parent: three. Fifteen
shares a quality of sound with both its parents: five and three. Seven is,
on the other hand, an original and unprecedented emanation from the source,
as are 11 and 13. So primes mark the beginnings of new lineages of number
relations. Once a new prime makes its appearance, it will interact with
lower primes to project hybrids up into the harmonic series. As a bare
measure of consonance, perhaps odd numbers tell the quantitative tale, but
there is another relational dimension to musical sound in which the
phenomena of prime numbers is audibly reflected.

🔗paulerlich <paul@stretch-music.com>

🔗clumma <carl@lumma.org>

🔗paulerlich <paul@stretch-music.com>

🔗clumma <carl@lumma.org>

🔗Dante Rosati <dante.interport@rcn.com>

🔗clumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗dkeenanuqnetau <d.keenan@uq.net.au>

🔗Dante Rosati <dante.interport@rcn.com>

🔗paulerlich <paul@stretch-music.com>

🔗clumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "Dante Rosati" <dante.interport@r...> wrote:
> > --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > > --- In tuning@y..., "clumma" <carl@l...> wrote:
> > > > > If 15 has fiveness and threeness, what does 5:3 have?
> > > >
> > > > It seems to have a 5-ness.
> > >
> > > Does 9/7 have 3-ness, 7-ness, 9-ness, or more than one of the
> above?
> >
> > I would say it has a "7-ness", that being the highest prime
factor.
>
> Going back to prime limit 5 . . to my ears 9:5 and 16:9 have a
> completely different "ness" than any 3- or 5-limit consonant
> interval. Saying otherwise often strikes me as a case of using the
> conclusion to justify the premise. And when you get to intervals
like
> 32:25 and 32:27, proximity to simpler ratios takes over anyway.

Bob Wendell:
I agree wholeheartedly that close intervalic proximity of intervals
containing different prime factors obliterates any subjective
difference in intervalic quality, so the higher primes don't enter
the picture. I think 19 is about the limit and even that's a bit of a
stretch, perhaps. I think it is difficult to define a precise cutoff
point, since it is a fading continuum, but the fade is rapid enough
to at least specify a range somewhere in the teens, I would say.
Certainly up through 13. 17 and 19 get a little fuzzy and then it's a
lost cause in my opinion, due to tuning proximity to nearly identical
intervals with different prime factors.

I do hear a subtle qualitative difference in the just 5:6 and 16:19,
and certainly the qualitative shift is far less subtle when either is
compared with 6:7. But 5:9 and 9:16 also differ qualitatively to my
ear, even though they do not partake of the same qualities as other
intervals containing the same primes, as one might think if too
deeply devoted to the prime theorists' conjectures.

My hypothesis of the moment, based strictly on personal subjective
experience, is that lower primes have some weight in determining the
subjective color or tonal quality of an interval. It seems to me that
there is at least some merit in the association of lower primes with
quality, and that simple pitch span is not enough.

I base this on the observation of the drastically different
subjective color I personally perceive between 6:7 and 5:6, for
example, and I'm speaking melodically as well as harmonically,
although I personally find the harmonic situation even more
contrasting and it is hard to know without some very good research
how much of the melodic perception is preconditioned by previous
harmonic experience.

I will also allow for the possibility that subjective
intervalic "color" could be explained in terms of pitch span taken as
a kind of psychoacoustic spectrum analagous to visual color. This
possibility is admittedly reinforced by the tuning proximity
phenomenon erasing qualitative differences. However, as some have
pointed out already, there is a certain "7-ness" common to several
intervals that contain it as a prime factor.

On the other hand, Gene's excellent question, "Does 9/7 have 3-ness,
7-ness, 9-ness, or more than one of the above?", does make one think
about even that. To me it has "7-ness" as was previously proposed,
possibly because the implied fundamental is the same as for 6:7 or
5:7. I would conjecture that intoducing 7 as prime into harmony lends
a distinctive new flavor that tends to dominate. This leads to a
possible hypothesis that until we reach the upper limit imposed by
problems with tuning resolution, the higher primes dominate the
flavor. When I hear an open octave and perfect fifth without the
major third, I hear "3-ness". When the triad is filled in with the
M3, I hear "5-ness" dominating.

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

> > Going back to prime limit 5 . . to my ears 9:5 and 16:9 have a
> > completely different "ness" than any 3- or 5-limit consonant
> > interval. Saying otherwise often strikes me as a case of using
the
> > conclusion to justify the premise. And when you get to intervals
> like
> > 32:25 and 32:27, proximity to simpler ratios takes over anyway.
>
> Bob Wendell:
> I agree wholeheartedly that close intervalic proximity of intervals
> containing different prime factors obliterates any subjective
> difference in intervalic quality, so the higher primes don't enter
> the picture.

I think you took my remarks in the wrong way. It is only by accident
that close intervallic proximity came into my examples.

> I think 19 is about the limit and even that's a bit of a
> stretch, perhaps.

The prime limit? The odd limit? The integer limit?

> I do hear a subtle qualitative difference in the just 5:6 and
16:19, and certainly the qualitative shift is far less subtle when
either is
> compared with 6:7. But 5:9 and 9:16 also differ qualitatively to my
> ear, even though they do not partake of the same qualities as other
> intervals containing the same primes, as one might think if too
> deeply devoted to the prime theorists' conjectures.

With you so far . . .

> My hypothesis of the moment, based strictly on personal subjective
> experience, is that lower primes have some weight in determining
the
> subjective color or tonal quality of an interval. It seems to me
that
> there is at least some merit in the association of lower primes
with
> quality, and that simple pitch span is not enough.
>
> I base this on the observation of the drastically different
> subjective color I personally perceive between 6:7 and 5:6, for
> example, and I'm speaking melodically as well as harmonically,
> although I personally find the harmonic situation even more
> contrasting and it is hard to know without some very good research
> how much of the melodic perception is preconditioned by previous
> harmonic experience.

But why do you attribute this difference to "primeness"? Isn't it
enough that these intervals are formed from different numbers, or
perhaps even different _odd_ factors?

> I will also allow for the possibility that subjective
> intervalic "color" could be explained in terms of pitch span taken
as
> a kind of psychoacoustic spectrum analagous to visual color. This
> possibility is admittedly reinforced by the tuning proximity
> phenomenon erasing qualitative differences. However, as some have
> pointed out already, there is a certain "7-ness" common to several
> intervals that contain it as a prime factor.

Perhaps you could eliminate "prime" from that statement, and it would
hold just as true in your mind?

> On the other hand, Gene's excellent question, "Does 9/7 have 3-
ness,
> 7-ness, 9-ness, or more than one of the above?", does make one
think
> about even that. To me it has "7-ness" as was previously proposed,
> possibly because the implied fundamental is the same as for 6:7 or
> 5:7.

The implied fundamental is the same? Only in a very contrived set of
circumstances. Once you fall into _assuming_ or _contriving_ such a
set of circumstances, you've fallen into one of the traps that leads
to the fallacy of prime qualities.

> I would conjecture that intoducing 7 as prime into harmony lends
> a distinctive new flavor that tends to dominate.

I think that's true, but only because the music we're used to tends
to avoid ratios of 7, and feature approximate ratios of 3, 5, and 9,
and even 15, that do not involve 7.

🔗robert_wendell <BobWendell@technet-inc.com>

Bob earlier:
To me it has "7-ness" as was previously proposed,
> > possibly because the implied fundamental is the same for 6:7
as for 5:7.
>
Paul Erlich:
> The implied fundamental is the same? Only in a very contrived set
of
> circumstances. Once you fall into _assuming_ or _contriving_ such a
> set of circumstances, you've fallen into one of the traps that
leads
> to the fallacy of prime qualities.

Bob answers:
Well, here's a chance for me to ask a question then. Why are not the
implied fundamentals of 6:7 and 5:7 the same, namely at 1. I'm
assuming harmonic timbres. Let's not introduce Sethares and timbres
customized to the tuning, otherwise the whole issue becomes radically
altered.

Paul on another issue:
> I think you took my remarks in the wrong way. It is only by
accident
> that close intervallic proximity came into my examples.

Bob answers:
Well, I think it is an important and telling phenomenon that
intervallic proximity erases the distinctions among higher primes. It
could be used cogently to argue against the prime qualities
hypothesis. It certainly doesn't reinforce it.
>
Bob earlier:
> > I think 19 is about the limit and even that's a bit of a
> > stretch, perhaps.
>
Paul:
> The prime limit? The odd limit? The integer limit?
>
Bob answers:
I see your point. It may have nothing to do with "primeness", but
simply with what integer, at least odd ones if we invoke octave
equivalence, but even then do tenths really have the same quality as
thirds? Good point, Paul. They do still have "thirdness" to my ear,
but it is a modified quality of "thirdness" even though the
difference is strictly a matter of octave displacement.

Paul:
But why do you attribute this difference to "primeness"? Isn't it
> enough that these intervals are formed from different numbers, or
> perhaps even different _odd_ factors?

Bob answers:
Yep...Same point. Got it.

Bob earlier:
> > I will also allow for the possibility that subjective
> > intervalic "color" could be explained in terms of pitch span
taken
> as
> > a kind of psychoacoustic spectrum analagous to visual color. This
> > possibility is admittedly reinforced by the tuning proximity
> > phenomenon erasing qualitative differences. However, as some have
> > pointed out already, there is a certain "7-ness" common to
several
> > intervals that contain it as a prime factor.
>
Paul:
> Perhaps you could eliminate "prime" from that statement, and it
would
> hold just as true in your mind?

Bob answers:
Probably so...

Bob earlier:
> > I would conjecture that intoducing 7 as prime into harmony lends
> > a distinctive new flavor that tends to dominate.
>
Paul:
> I think that's true, but only because the music we're used to tends
> to avoid ratios of 7, and feature approximate ratios of 3, 5, and
9,
> and even 15, that do not involve 7.

Bob answers:
Well, so you're proposing that this distintive new flavor, whether
from primeness, or perhaps better, just a new odd integer or even
just a new integer, is a matter of conditioning that has led us not
to expect it, and therefore when it's there, the violation of our
expectations is purely and completely responsible for
our "distinctive new flavor" reaction? It seems plausible to me, but
I nonethless have my doubts about that. I suspect, maybe even intuit
that some element of "nature" might be involved, rendering it less
than purely "nurture".

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> Bob earlier:
> To me it has "7-ness" as was previously proposed,
> > > possibly because the implied fundamental is the same for 6:7
> as for 5:7.
> >
> Paul Erlich:
> > The implied fundamental is the same? Only in a very contrived set
> of
> > circumstances. Once you fall into _assuming_ or _contriving_ such
a
> > set of circumstances, you've fallen into one of the traps that
> leads
> > to the fallacy of prime qualities.
>
> Bob answers:
> Well, here's a chance for me to ask a question then. Why are not
the
> implied fundamentals of 6:7 and 5:7 the same, namely at 1.

Well, with that kind of calculation, the implied fundamental of _any_
ratio in lowest terms is 1! So then *any* ratio would have 7-ness??
Or did I miss your meaning?

> Bob earlier:
> > > I would conjecture that intoducing 7 as prime into harmony
lends
> > > a distinctive new flavor that tends to dominate.
> >
> Paul:
> > I think that's true, but only because the music we're used to
tends
> > to avoid ratios of 7, and feature approximate ratios of 3, 5, and
> 9,
> > and even 15, that do not involve 7.
>
> Bob answers:
> Well, so you're proposing that this distintive new flavor, whether
> from primeness, or perhaps better, just a new odd integer or even
> just a new integer, is a matter of conditioning that has led us not
> to expect it, and therefore when it's there, the violation of our
> expectations is purely and completely responsible for
> our "distinctive new flavor" reaction?

Yes.

> It seems plausible to me, but
> I nonethless have my doubts about that. I suspect, maybe even
intuit
> that some element of "nature" might be involved, rendering it less
> than purely "nurture".

Well, I would agree that the "nurture" was itself guided and informed
by "nature", so perhaps unavoidable. When building scales or even
extended chords out of ratios of 3 and ratios of 5, the ratios of 9
and of 15 that result and don't involve any higher primes than 5 will
inevitably become familiar. But this is an _experiential_
familiarity, I claim, not an inborn one.

🔗clumma <carl@lumma.org>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗robert_wendell <BobWendell@technet-inc.com>

Bob earlier:
> Well, here's a chance for me to ask a question then. Why are not
the
> implied fundamentals of 6:7 and 5:7 the same, namely at 1.

Paul:
Well, with that kind of calculation, the implied fundamental of _any_
ratio in lowest terms is 1! So then *any* ratio would have 7-ness??
Or did I miss your meaning?

Bob:
If we hold 7 as a common pitch, we get 1 as the implied fundamental
for both. If we take 16:19 or 5:6 and make the higher integer of each
correspond to a pitch common with the 7s and each other, we get
different implied fundamentals by the same calculation.

As a perhaps more telling example, I feel from my own experimentation
that the flavor of the different m3s resulting from 6:7, 16:19, and
5:6, whether we hold the higher or lower integers as the common
pitch, is associated with the different implied fundamentals in each
case.

The pitch spans are not so terribly different between 16:19 and 5:6,
differing by only about 17 cents if I remember correctly, which is
nevertheless enough to make a significant change (a diatonic half-
step at 112 cents)in the implied fundamental with very different
harmonic implications.

However, many highly touted professional singers make pitch errors of
this magnitude (17 cents) routinely, although I don't particularly
cherish that
fact, specifically since such errors imply a gross insensitivity to
these very differences in flavor. This implies directly that their
pitch resolution is low enough to make even the lower integers fall
into the category of those affected by the tuning proximities that
obliterate the flavor differences of the higher primes (or integers)
as we previously discussed. So where the line between flavors blurs
as we move to higher integers is clearly a function of individual
ability to discriminate pitch finely.

What calculation would you propose for an implied fundamental?

Paul earlier:
> Well, I would agree that the "nurture" was itself guided and
informed
> by "nature", so perhaps unavoidable. When building scales or even
> extended chords out of ratios of 3 and ratios of 5, the ratios of 9
> and of 15 that result and don't involve any higher primes than 5
will
> inevitably become familiar. But this is an _experiential_
> familiarity, I claim, not an inborn one.

Bob:
Yes, I see your point, Paul. However, we run into something of a
philosophical conundrum on the nature vs. nurture issue, since, as
you indicate, nature does indeed always and inevitably guide and
inform nurture. I believe we too often underestimate the degree to
which this is so. Maybe we should ask how sigificant is the issue in
the first place, if the degree to which nature preconditions nurture
in a particular case nearly mandates a particular outcome.

Then there are those cases, such as the one in our discussion, that
are dependent on factors such as the individual development of pitch
discrimination, especially in the context of perceiving harmonic
relationships. Nurture is powerfully involved in the development of
this faculty, yet both the process and the outcome are preconditioned
by a plethora of natural structures, both acoustic and
psychoacoustic.

Since the resulting skill in discrimating pitch is highly prized by
the most discriminating audiences, I don't think we should
necessarily allow ourselves to be guided by the lowest common
denominator or simply write off the whole thing as purely subjective
and arbitrary because of the range of perceptual acuity that exists
among consumers of music.

I get the impression sometimes (not a reference to anything you've
said, Paul) that people who heavily push the nurture side of the
equation are using that to argue that the whole thing is arbitrary
and strictly a matter of personal preference, failing to allow that
there could be natural principles that are indeed universals and not
strictly products of some kind of cultural relativism. I also get the
distinct impression that some people use this line of argument to
justify their own lack of development in whatever area is at stake in
their minds.

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> Bob earlier:
> > Well, here's a chance for me to ask a question then. Why are not
> the
> > implied fundamentals of 6:7 and 5:7 the same, namely at 1.
>
> Paul:
> Well, with that kind of calculation, the implied fundamental of
_any_
> ratio in lowest terms is 1! So then *any* ratio would have 7-ness??
> Or did I miss your meaning?
>
> Bob:
> If we hold 7 as a common pitch, we get 1 as the implied fundamental
> for both.

Right, but your original list was 7:9, 6:7, and 5:7. Isn't it a bit
or circular reasoning to hold 7 as a common pitch for all three of
these, and then claim that, because the implied fundamental is the
same for all three, they all have "7-ness"?

> As a perhaps more telling example, I feel from my own
experimentation
> that the flavor of the different m3s resulting from 6:7, 16:19, and
> 5:6, whether we hold the higher or lower integers as the common
> pitch, is associated with the different implied fundamentals in
each
> case.

But 7:9 would not agree with 6:7 whether you held the higher or lower
integers as the common pitch.

> The pitch spans are not so terribly different between 16:19 and
5:6,
> differing by only about 17 cents if I remember correctly, which is
> nevertheless enough to make a significant change (a diatonic half-
> step at 112 cents)in the implied fundamental with very different
> harmonic implications.

How do you get a diatonic half-step? I get a major third.

> What calculation would you propose for an implied fundamental?

It seems to me that a variety of implied fundamentals is always
potentially implied, though sometimes one of them is much more
salient than the others. The degree of confusion here is what my
harmonic entropy model is meant to measure -- please join

harmonic_entropy@yahoogroups.com

to learn more.

> Maybe we should ask how sigificant is the issue in
> the first place, if the degree to which nature preconditions
nurture
> in a particular case nearly mandates a particular outcome.

It doesn't. There are plenty of other ways in which humans have
decided to create music.

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > Bob earlier:
> > > Well, here's a chance for me to ask a question then. Why are
not
> > the
> > > implied fundamentals of 6:7 and 5:7 the same, namely at 1.
> >
> > Paul:
> > Well, with that kind of calculation, the implied fundamental of
> _any_
> > ratio in lowest terms is 1! So then *any* ratio would have 7-
ness??
> > Or did I miss your meaning?
> >
> > Bob:
> > If we hold 7 as a common pitch, we get 1 as the implied
fundamental
> > for both.
>
> Right, but your original list was 7:9, 6:7, and 5:7. Isn't it a bit
> or circular reasoning to hold 7 as a common pitch for all three of
> these, and then claim that, because the implied fundamental is the
> same for all three, they all have "7-ness"?

Bob:
Not if, as I conjecture, the highest prime, or integer at least,
dominates as long as we don't go so high as to obliterate the
distinctions with the tuning proximity phenomenon. In that case to
state that this is circular reasoning would be tantamount to saying
that because green looks green it therefore can't be reliably said to
be green, since that would be a circular argument.

>
> > As a perhaps more telling example, I feel from my own
> experimentation
> > that the flavor of the different m3s resulting from 6:7, 16:19,
and
> > 5:6, whether we hold the higher or lower integers as the common
> > pitch, is associated with the different implied fundamentals in
> each
> > case.
>
> But 7:9 would not agree with 6:7 whether you held the higher or
lower
> integers as the common pitch.
>
> > The pitch spans are not so terribly different between 16:19 and
> 5:6,
> > differing by only about 17 cents if I remember correctly, which
is
> > nevertheless enough to make a significant change (a diatonic half-
> > step at 112 cents)in the implied fundamental with very different
> > harmonic implications.
>
> How do you get a diatonic half-step? I get a major third.
>

Bob:
Convert to 80:95 and 80:96 and the implied fundamentals can be seen
as having a 15:16 ratio. This is born out by my hearing, to boot. If
we put the lower value as C, I can clearly hear the Ab in the first
case and the G in the other. They're harmonically completely
different animals.

> > What calculation would you propose for an implied fundamental?
>
> It seems to me that a variety of implied fundamentals is always
> potentially implied, though sometimes one of them is much more
> salient than the others. The degree of confusion here is what my
> harmonic entropy model is meant to measure -- please join
>
> harmonic_entropy@y...
>
> to learn more.
>
> > Maybe we should ask how sigificant is the issue in
> > the first place, if the degree to which nature preconditions
> nurture
> > in a particular case nearly mandates a particular outcome.
>
Paul:
> It doesn't. There are plenty of other ways in which humans have
> decided to create music.

Bob:
I meant that to apply in context to the cases in which it does. I'm
quoting from you here, when you stated that nature guides and informs
the nurture for the case in point in our previous post.

Besides, how can any experience, and by implication nurturing, not be
guided and informed by nature, since our very perceptions are
conditioned by the structure of our perceptual apparatus and their
neurological processing in the associated aspects of the nervous
system and brain, not to mention all the natural laws that govern the
nature and behavior of the objects perceived?

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

> Bob:
> Not if, as I conjecture, the highest prime, or integer at least,
> dominates as long as we don't go so high as to obliterate the
> distinctions with the tuning proximity phenomenon. In that case to
> state that this is circular reasoning would be tantamount to saying
> that because green looks green it therefore can't be reliably said
to
> be green, since that would be a circular argument.

I meant the argument you proposed which involves holding the pitch of
one note constant while comparing the implied fundamentals. It
appears you wanted to hold 7 constant regardless of whether it was
the lower or higher pitch in the intervals involved.

> > > The pitch spans are not so terribly different between 16:19 and
> > 5:6,
> > > differing by only about 17 cents if I remember correctly, which
> is
> > > nevertheless enough to make a significant change (a diatonic
half-
> > > step at 112 cents)in the implied fundamental with very
different
> > > harmonic implications.
> >
> > How do you get a diatonic half-step? I get a major third.
> >
>
> Bob:
> Convert to 80:95 and 80:96 and the implied fundamentals can be seen
> as having a 15:16 ratio. This is born out by my hearing, to boot.
If
> we put the lower value as C, I can clearly hear the Ab in the first
> case and the G in the other. They're harmonically completely
> different animals.

Bob, the implied fundamental of a 16:19 C-Eb would be C, not G. What
overtones would this C and Eb be over a G fundamental??

> > > Maybe we should ask how sigificant is the issue in
> > > the first place, if the degree to which nature preconditions
> > nurture
> > > in a particular case nearly mandates a particular outcome.
> >
> Paul:
> > It doesn't. There are plenty of other ways in which humans have
> > decided to create music.
>
> Bob:
> I meant that to apply in context to the cases in which it does.

How conveeeeenient :)

> I'm
> quoting from you here, when you stated that nature guides and
informs
> the nurture for the case in point in our previous post.
>
> Besides, how can any experience, and by implication nurturing, not
be
> guided and informed by nature, since our very perceptions are
> conditioned by the structure of our perceptual apparatus and their
> neurological processing in the associated aspects of the nervous
> system and brain, not to mention all the natural laws that govern
the
> nature and behavior of the objects perceived?

Well, I would argue that the neurological processes involved in
listening to and learning music are often the same ones involved in
listening to and learning language. In that regard, musical systems
can be expected to diverge and become mutually unintelligible much
the way languages do. That's really the kind of point the "nurture"
tack is trying to make, regardless of whether one views all of this
as a "natural" process.

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> > Bob earlier:
> > Not if, as I conjecture, the highest prime, or integer at least,
> > dominates as long as we don't go so high as to obliterate the
> > distinctions with the tuning proximity phenomenon. In that case
to
> > state that this is circular reasoning would be tantamount to
saying
> > that because green looks green it therefore can't be reliably
said
> to
> > be green, since that would be a circular argument.
>

Paul replied:
> I meant the argument you proposed which involves holding the pitch
of
> one note constant while comparing the implied fundamentals. It
> appears you wanted to hold 7 constant regardless of whether it was
> the lower or higher pitch in the intervals involved.
>
Bob now:
If the higher prime or integer dominates as hypopthesized, then it
doesn't matter whether it's the upper or lower one. I simply was
comparing three minor thirds, so naturally the bottom pitch of the
ration was held constant while listening for the differnt minor third
qualities above it. Somewhat arbitrary, but it seems more intuitive
and simple to compare that way to me.

Bob earlier:
> > > > The pitch spans are not so terribly different between 16:19
and
> > > 5:6,
> > > > differing by only about 17 cents if I remember correctly,
which
> > is
> > > > nevertheless enough to make a significant change (a diatonic
> half-
> > > > step at 112 cents)in the implied fundamental with very
> different
> > > > harmonic implications.
> > >
> > > How do you get a diatonic half-step? I get a major third.
> > >
> >
> > Bob:
> > Convert to 80:95 and 80:96 and the implied fundamentals can be
seen
> > as having a 15:16 ratio. This is born out by my hearing, to boot.
> If
> > we put the lower value as C, I can clearly hear the Ab in the
first
> > case and the G in the other. They're harmonically completely
> > different animals.
>
> Bob, the implied fundamental of a 16:19 C-Eb would be C, not G.
What
> overtones would this C and Eb be over a G fundamental??
>
Bob now:
Sorry! Of course. I was confusing myself with the difference tone at
G, which I can easily hear, but which is not the fundamental. I was
letting clear experience cloud my analysis, not thinking beyond it.
My apologies. I should make it clear that I'm not necessarily trying
to promote a particular view, even when I may be admittedly a bit
attached to it, but wishing to subject hypotheses to rigorous
analysis.

Bob ealier:
> > > > Maybe we should ask how sigificant is the issue in
> > > > the first place, if the degree to which nature preconditions
> > > nurture
> > > > in a particular case nearly mandates a particular outcome.
> > >
> > Paul:
> > > It doesn't. There are plenty of other ways in which humans have
> > > decided to create music.
> >
> > Bob replied:
> > I meant that to apply in context to the cases in which it does.
>
> How conveeeeenient :)
>
Bob now:
Well, I'm not trying to win anything. I simply was referring to a
concession you had made earlier to a particular point and applying it
to that case and whatever others might fall in the same category.

Bob had continued:
> > I'm
> > quoting from you here, when you stated that nature guides and
> informs
> > the nurture for the case in point in our previous post.
> >
> > Besides, how can any experience, and by implication nurturing,
not
> be
> > guided and informed by nature, since our very perceptions are
> > conditioned by the structure of our perceptual apparatus and
their
> > neurological processing in the associated aspects of the nervous
> > system and brain, not to mention all the natural laws that govern
> the
> > nature and behavior of the objects perceived?
>
Paul:
> Well, I would argue that the neurological processes involved in
> listening to and learning music are often the same ones involved in
> listening to and learning language. In that regard, musical systems
> can be expected to diverge and become mutually unintelligible much
> the way languages do. That's really the kind of point the "nurture"
> tack is trying to make, regardless of whether one views all of this
> as a "natural" process.

Bob now:
Well, me thinks it's quite a stretch to say that different musical
idioms are as difficult to access and begin to comprehend as
unfamiliar spoken languages. Or are you not saying that, but rather
making a comparison in quality and not in degree?

I happen to be familiar with both and speak a few languages besides
English at a basic level, and in Spanish I am frequently taken for a
native by native speakers. Nevertheless, the ease with which I adapt
to exotic musical idioms as opposed to the acquisition of an
unfamiliar language defies any serious comparison, even though I'm
unusually quick in acquiring new spoken languages. How much less
comparison is there for those musicians, of whom I've known quite a
few, whose linguistic acumen outside of music is quite limited, even
in their own language?

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

Back on the original theme of this thread, and to possibly reopen and
big can of worms, I wish to elaborate a point Joe Monz made earlier.
It intuitively resonates very strongly with me, if you will allow for
that.

I notice that rhythmically, patterns of two and three beats grouped
together by emphasis and/or meter create a different intuitive
feeling or sense of rhythmic flavor in the listener. I feel that this
is not so different in principle from what happens in frequency
relationships among musical tones. The mechanics and subjective
nature of the perception of rhythm and pitch, although quite
different, may have important parallels in terms of perceptual flavor
and classification.

If we follow this model, we might note that although 3s may be
grouped in 2s or multiples thereof, they still maintain
their "threeness". A 9/8 meter has no "twoness", but is
pure "threeness". Even in 6/8 meter, to me there is a subjective
domination of "threeness" inspite of its being at a lower stratum
rhythmically. On the other hand, note that in 3/4 time, where the
next lower strata are grouped by 2s, the dominant flavor is
still "threeness".

One of the things that differentiates jazz and related musics'
rhythmic structure from traditional European meters and rhythmic
style is the "swing" feel, which comes largely from lots
of "threeness" at the lower rhythmic strata, whereas European styles
in three always seem to revert to "twoness" at the lowest strata.
There are, of course, other subtler manipulations of rhythmic
structure that characterize jazz and related styles.

In Indian systems and some of the more experimental western styles,
there are groupings of five and more in rhythmic stuctures. I feel
that five definitely takes a dominant role in giving flavor to those
rhythms that it characterizes. But like pitch, when going to the
higher primes in these groupings, the mind does tend to simplify,
and break these groupings down, excepting only the most experienced
and sophisticated listeners.

I feel that there is no reason to assume out of hand that there is
not a similar differentiation that is natural to pitch perception.
I'm not pretending to prove anything with these arguments, but I
think it is a point worth bearing in mind before we eliminate this
line of thinking out of hand.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> > I'm
> > quoting from you here, when you stated that nature guides and
> informs
> > the nurture for the case in point in our previous post
>
> Well, the whole question of why we would want to construct scales
> with fixed pitches seems to be an additional mystery. Nearly all
> cultures use such scales -- but why? Why didn't music arise in Lou
> Harrison's "free style", for example, or like Bill Sethares's
> recent "ascalar" composition -- even though in both cases all the
> _relationships_ are simple-integer ones?

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

Bob earlier:
> > To venture a guess, I would say that grammatical consistency and
> > simplicity have a lot to do with the ease, or lack of it, with
> which
> > we acquire language of any kind, including musical ones. A
small,
> > fixed vocabulary or small set of them, each with a predefined,
> > stable, grammatical structure (both as intrinsic to the scale
> > structure and implicit in elements of style and cadential
> formulae),
> > would be much more quickly grasped and therefore have a much more
> > immediate and universal appeal than the alternatives, I would
> > speculate. The advantage of that in terms evolutionary economics
> and
> > the consequent potential for cultural replication and survival
> should
> > be evident from there.
>
Paul:
> Sounds reasonable. So you can understand why I'd feel that
> approximations to ratios involving 9 and 15 would quickly
> acquire "grammatical meaning" in a system built on 5-limit harmony,
> while ratios involving 7 would be perceived as "meaningless" by
> someone whose native language was such a system. This is apart from
> any _acoustical_ or _sensual_ qualities that these intervals might
> have.

Bob now:
Yes, initially I can see that. For instance, in the blues, I believe
there should ideally be minor thirds of 6:7 ratio, as well as some 11-
limit stuff. However, I hear white singers singing the "same"
melodies stripped of anything but 5-limit intervals and harmonies.
For me, all the true "guts" of the blues feel are gone. To me these
singers have "white-washed" the blues. I'm white, but I don't
like "white-washed" blues! The best white blues musicians DO NOT DO
THIS to the blues!!! (I'm sure you don't, remembering that you're a
professional blues guitarist interested in microtonal music.)

However, in the blues, there is not really anything deeply
grammatical about the use of 7 and 11-limit harmonies. The flavorings
largely come from what you referred to as the sensual or acoustic
properties of these tonal elements. There is a certain grammatical
imperative that the 1l-limit F# of a C blues scale, almost exactly a
quarter step (47 cents, I think) above an F, to resolve down to the
F. I notice that when an F# goes to the fifth on G in that scale,
though, it is not the 11th anymore. It tends to be a conventional,
western F#.

The point is that as long as there are no strong harmonically
grammatical imperatives to appreciate the 7 prime intervals, they get
ignored. I don't think the same people who "white-wash" the blues
that way could ignore harmonic modulations based on 7-limit harmonic
pivots. It might take them a long time to acquire the musical skill
to sing that way, though.

🔗paulerlich <paul@stretch-music.com>

🔗Robert C Valentine <BVAL@IIL.INTEL.COM>

> From: "robert_wendell" <BobWendell@technet-inc.com>
> Subject: Re: Prime Rarity?
>
> Back on the original theme of this thread, and to possibly reopen and
> big can of worms, I wish to elaborate a point Joe Monz made earlier.
> It intuitively resonates very strongly with me, if you will allow for
> that.
>
> I notice that rhythmically, patterns of two and three beats grouped
> together by emphasis and/or meter create a different intuitive
> feeling or sense of rhythmic flavor in the listener. I feel that this
> is not so different in principle from what happens in frequency
> relationships among musical tones. The mechanics and subjective
> nature of the perception of rhythm and pitch, although quite
> different, may have important parallels in terms of perceptual flavor
> and classification.
>
> If we follow this model, we might note that although 3s may be
> grouped in 2s or multiples thereof, they still maintain
> their "threeness". A 9/8 meter has no "twoness", but is
> pure "threeness". Even in 6/8 meter, to me there is a subjective
> domination of "threeness" inspite of its being at a lower stratum
> rhythmically. On the other hand, note that in 3/4 time, where the
> next lower strata are grouped by 2s, the dominant flavor is
> still "threeness".

I'm really not sure about this. Although there is the "three threes"
Irish jig style 9/8, there is also the very popular Balkan beat that
goes 2 2 2 3. If I think of 6/8, I think it is ambiguous whether its
twoness or threeness dominates. My experimental method is to
superimpose 3 quarters or 4 dotted eights over it. They are both
at about the same degree of comfort (which is one of the magics of
West African rhythms that makes up the rhythms of the black diaspora,
as Steve Coleman put it "the marriage of 2 and 3").

>
> One of the things that differentiates jazz and related musics'
> rhythmic structure from traditional European meters and rhythmic
> style is the "swing" feel, which comes largely from lots
> of "threeness" at the lower rhythmic strata, whereas European styles
> in three always seem to revert to "twoness" at the lowest strata.

I would go even further than this and say that a lot of the great
feels in blues, jazz, etc are ambiguous or perhaps temperring a
rhythm that is somewhere between "dotted_eigth sixteenth" and
"triplet quarter eigth". Particularly in the more "field recordings"
oriented musics, you can hear different players using different
"eighth" notes. A group that really captures this is Bahia Black.

>
> In Indian systems and some of the more experimental western styles,
> there are groupings of five and more in rhythmic stuctures. I feel
> that five definitely takes a dominant role in giving flavor to those
> rhythms that it characterizes. But like pitch, when going to the
> higher primes in these groupings, the mind does tend to simplify,
> and break these groupings down, excepting only the most experienced
> and sophisticated listeners.
>

I think there are a few different ways to look at rhythm. We in the
West usually stand somewhere in the middle, starting with barlines
and then subdividing down to "march" or "waltz" and building up
towards "A section" etc...

The Balkan orientation (where the fast oddball dance rhythms come from)
is much more bottom up. From what I understand, they really do think
of the 9/8 example I gave as sssL (2 2 2 3).

In some ways the Indian orientation is more top down. I don't have my
notes with me here, but the day of study I had with a Tabla master
showed some of the tradition of tabla composition and it really did
start from the macro-form (in this case 128 beats) and work its way
down. When it got to the bottom level, it looked in Western terms
like a very complicated set of barlines repeating strange amounts
of times. (The inner sub-divisions in Indian music are usually just
duple, HOWEVER, it is very common to play something "two times at
twice as fast" or "three times at three times as fast" or "two
and a half times at twoa and a half times as fast").

If I can make a point that is relevant here, it is that there are
many different ways of thinking of rhythm, even if they all seem
to end up dealing with "small numbers".

Considering that the Western tradition managed to boil all of
music down to "major minor march and waltz" it is a real uphill
climb to learn about musics (or try to discover an inner music)
that managed not to fall into these local minima.

> I feel that there is no reason to assume out of hand that there is
> not a similar differentiation that is natural to pitch perception.
> I'm not pretending to prove anything with these arguments, but I
> think it is a point worth bearing in mind before we eliminate this
> line of thinking out of hand.
>

Something that came out of this thread
somewhere that a 3-limit music will naturally explore the
realm of 9, 27, etc... a 5-limit music will naturally bump
into 15, 9/5 etc, just as a matter of continueing what it
does naturally. I do tend to think that, at least for musicians,
there is something about this referentiality that does become
familiar. For instance, I do think 9/8 has a feeling of
stability that 8/7 doesn't. I think 16/9 also has a greater
stability than 9/5. I suppose I should go through lots of
double blind tests to see whether the three pockets of
16/9, 9/5 and 7/4 are all meaningfully different to me...

Back to rhythm, is dividing and multiplying by 2 in rhythm a
suitable analogy to octave equivalence. And is three the next
step? And does 4 and 6 not add new information (due to this
rhythm analog to octave equivalence) but 5, and 7 do? And
9 is a bit oblique because it is built from 3's?

However, some forms of 7 and 15 can be thought of "8 with a
beat missing" or "16 with a beat missing". 9 and 17 are
similar "added beats" to familiar forms (is the analogy
to tuning that sometimes a 7/6 is an extremely bruised minor
third???).

Enough for now, Happy Holidays to all!

Bob Valentine

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

> (which is one of the magics of
> West African rhythms that makes up the rhythms of the black
diaspora,
> as Steve Coleman put it "the marriage of 2 and 3").

Yes, the standard West African and even Cajun rhythms are thus more
intricate than the standard rhythms of either jazz (which is more
flexible though) or classical (straight twos) music.

Or, more commonly by a fast soloist, somewhere between "triplet
quarter eighth" and "eighth eighth".

> Particularly in the more "field recordings"
> oriented musics, you can hear different players using different
> "eighth" notes.

Also in good jazz, where the different players tend to synchronize on
the weak eighth, instead of on the strong eighth. This, I believe,
goes much of the way toward explaining what good "swing" is all
about.>
> > I feel that there is no reason to assume out of hand that there
is
> > not a similar differentiation that is natural to pitch
perception.
> > I'm not pretending to prove anything with these arguments, but I
> > think it is a point worth bearing in mind before we eliminate
this
> > line of thinking out of hand.
> >
>
> Something that came out of this thread
> somewhere that a 3-limit music will naturally explore the
> realm of 9, 27, etc... a 5-limit music will naturally bump
> into 15, 9/5 etc, just as a matter of continueing what it
> does naturally.

Yes.

> I do tend to think that, at least for musicians,
> there is something about this referentiality that does become
> familiar.

And, perhaps even more so, something about the 12-tET grid that does
become familiar.

> For instance, I do think 9/8 has a feeling of
> stability that 8/7 doesn't.

This was the example in Partch's half-page "Enigma of the Multiple-
Number Ratio", but it also reinforces my "12-tET familiarity" bit.

> I think 16/9 also has a greater
> stability than 9/5.

This also reinforces my 12-tET familiarity bit. And, I think the fact
that Bob Wendell is so interested in 19/16 rather than 13/8 also
reinforces my 12-tET familiarity theory.

> However, some forms of 7 and 15 can be thought of "8 with a
> beat missing" or "16 with a beat missing". 9 and 17 are
> similar "added beats" to familiar forms (is the analogy
> to tuning that sometimes a 7/6 is an extremely bruised minor
> third???).

9/8 and 17/16 would have to be considered extremely bruised unisons.

🔗Robert C Valentine <BVAL@IIL.INTEL.COM>

> From: "paulerlich" <paul@stretch-music.com>
> Subject: Re: Prime Rarity?
>
> >
> > I said :
> >
> > Particularly in the more "field recordings"
> > oriented musics, you can hear different players using different
> > "eighth" notes.
>
> Also in good jazz, where the different players tend to synchronize on
> the weak eighth, instead of on the strong eighth. This, I believe,
> goes much of the way toward explaining what good "swing" is all
> about.

Hi Paul,

I think, as you may have alluded to, in jazz there is often a lock,
at least by the rhythm section, on a certain definition of the eighth
note. It may be arrived at in a pretty dynamic manner though. Once I
played bass (guitar) on a jazz gig and the drummer had me 'push' the
quarters while he held them back. It was tiring work for me but the
combined tension 'locked' in a "laid back but propelled" i.e., swinging
manner.

What I was more thinking about was what many would refer to as less
sophisticated players. I mentioned Bahia Black but field recordings
of Yoruba and derivative religions (Santeria, Voudou) will have a
lot of simultaneous dotted_eighth+sixteenth on one player and triplet
feel in another player. On a micro-level, there is this greater
amount of syncopation happenning in all the little "flammy" things
between instruments. What I tend to think also happens is that
players are pulled (tempered) towards a mid-point. Now we're back
in the realm of the swing eighth.

For those interested in playing fun rhythms, there is a metronome
called 'mundobeat' which is cheaper than the Dr Beat and other
polyrhythm supporting metronomes and also is more capable. As an
example of this rhythmic tempering thing, I programmed it with
a very fast five, basically just

low_dotted_quarter high_eighth

so this is just a 3+2 pattern. (fast means that it goes at about
60 bars per minute, or the eighth = 300).

I had played this tune with a drummer and had noticed that we
sometimes slipped into six, with the barlines remaining constant.
I tried improvising with the metronome and the same phenomena
occurred. On the one hand I can try to fight it since expressing
the song should be more in the realm of 'fiveness'. On the other
hand, the fast five-over-six has a tension that is a cool thing
on its own. Basically the metronome feels like a slightly
egg-shaped version of the six.

>
> > However, some forms of 7 and 15 can be thought of "8 with a
> > beat missing" or "16 with a beat missing". 9 and 17 are
> > similar "added beats" to familiar forms (is the analogy
> > to tuning that sometimes a 7/6 is an extremely bruised minor
> > third???).
>
> 9/8 and 17/16 would have to be considered extremely bruised
> unisons.
>

Analogys between rhythmn and harmony do get stretched pretty
easily. I don't know where the exact point is that a bruised
unison occurs (we've conjectured about the pocket being +- 20c
which is suspiciously close to 81/80). On the other hand,
slowed down enough to hear 81 over 80 metrically, you would
hear all sorts of "phase unisons" instead. First it would
sound like simultaneous quarters, later like quarters
displaced by sixteenth, then by tripletized quarter, then
by eighth etc.

The thing I'm pointing out with the egg shaped six on the
metronome and this example is that some sort of rounding
(temperring) takes place in rhythm too. One reason why
analogys break down is that rhythm takes place in a
linear domain and intervals take place in a log domain.
Another thing is that phase relations are very important
in rhythm and practically ignorable in harmony.

Good holidays to you Paul.

Bob

🔗monz <joemonz@yahoo.com>

Hi Bob,

> From: robert_wendell <BobWendell@technet-inc.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 24, 2001 1:19 PM
> Subject: [tuning] Re: Prime Rarity?
>
>
> Back on the original theme of this thread, and to possibly reopen and
> big can of worms, I wish to elaborate a point Joe Monz made earlier.
> It intuitively resonates very strongly with me, if you will allow for
> that.
>
> I notice that rhythmically, patterns of two and three beats grouped
> together by emphasis and/or meter create a different intuitive
> feeling or sense of rhythmic flavor in the listener. I feel that this
> is not so different in principle from what happens in frequency
> relationships among musical tones. The mechanics and subjective
> nature of the perception of rhythm and pitch, although quite
> different, may have important parallels in terms of perceptual flavor
> and classification.
>

Thanks for the vote of support.

> If we follow this model, we might note that although 3s may be
> grouped in 2s or multiples thereof, they still maintain
> their "threeness". A 9/8 meter has no "twoness", but is
> pure "threeness".

Umm... well, not necessarily. Are you familiar with Dave Brubeck's
great tune "Blue Rondo a la Turk"? It's in 9/8 meter, but divided
2 + 2 + 2 + 3. OK, I can still see this pattern as exhibiting
lots of "threeness", but it definitely also contains "twoness"
along with the "threeness", so it's quite different from the
"pure threeness" of 3 + 3 + 3, which is what I think you were
referring to.

> Even in 6/8 meter, to me there is a subjective
> domination of "threeness" inspite of its being at a lower stratum
> rhythmically. On the other hand, note that in 3/4 time, where the
> next lower strata are grouped by 2s, the dominant flavor is
> still "threeness".

Do you know about the metrical device called "hemiola"
(Greek for 3:2). It's a simultaneous combination of 6/8
and 3/4, wherein both meters exhibit both "threeness" and
"twoness", but each in opposite "strata".

> One of the things that differentiates jazz and related musics'
> rhythmic structure from traditional European meters and rhythmic
> style is the "swing" feel, which comes largely from lots
> of "threeness" at the lower rhythmic strata, whereas European styles
> in three always seem to revert to "twoness" at the lowest strata.
> There are, of course, other subtler manipulations of rhythmic
> structure that characterize jazz and related styles.
>
> In Indian systems and some of the more experimental western styles,
> there are groupings of five and more in rhythmic stuctures. I feel
> that five definitely takes a dominant role in giving flavor to those
> rhythms that it characterizes.

Just thought I'd mention that much of Partch's music has good examples of
complex meters containing 5 in different strata.

> But like pitch, when going to the higher primes in these groupings,
> the mind does tend to simplify, and break these groupings down,
> excepting only the most experienced and sophisticated listeners.
>
>
> I feel that there is no reason to assume out of hand that there is
> not a similar differentiation that is natural to pitch perception.
> I'm not pretending to prove anything with these arguments, but I
> think it is a point worth bearing in mind before we eliminate this
> line of thinking out of hand.

Ah, now I think you're really onto something here, because I see an
interesting parallel! Just as listeners hypothetically break down
a 5-meter into a 2- and a 3-, perhaps the brain strives to comprehend
audible 5-limit relationships in terms of 3-limit ones, i.e., the
way of understanding meantones as chains of a generator resembling
the 3:2 unit-size of the 3-limit chain. Hmmm...

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

Hi Paul and Bob,

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 24, 2001 1:48 PM
> Subject: [tuning] Re: Prime Rarity?
>
>
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> > Bob now:
> > Yes, initially I can see that. For instance, in the blues, I
> > believe there should ideally be minor thirds of 6:7 ratio,
> > as well as some 11-limit stuff. However, I hear white singers
> > singing the "same" melodies stripped of anything but 5-limit
> > intervals and harmonies.
>
> If that. I generally hear a lot more 5-limit (as opposed to 12-tET)
> intervals from black singers than from white singers.

I agree with Paul. White singers's performances generally tend
to sound quite 12-EDO-ish.

> > However, in the blues, there is not really anything deeply
> > grammatical about the use of 7 and 11-limit harmonies. The
> > flavorings largely come from what you referred to as the sensual
> > or acoustic properties of these tonal elements. There is a certain
> > grammatical imperative that the 1l-limit F# of a C blues scale,
> > almost exactly a quarter step (47 cents,
>
> 51 cents
>
> > I think) above an F, to resolve down to the F.
>
> Often it seems to skip the F and proceed to a neutral third above C,
> perhaps 11/9, though the expressiveness of the vocalization would
> prevent an exact pinning down of the pitches to these ratios of 11.

Paul, you've seen this long ago, but it may be new for Bob.
Based on what you posted here, you'd be interested in this:
/tuning/topicId_2179.html#2179?expand=1

I'm referring to the first (long) post, by me, in which I
explore some rational and ET implications for some "blue notes"
I found in a MIDI file of an Etta James tune which I got
off the web.

The criteria I used for the ratios were to find the combination
of lowest prime-factors and lowest exponents. ET criteria was
simple pitch-height proximity. The key of the piece is F# major.

Under the section titled "example 2: measure 16 (16.3.004
-- 16.3.156)", I examine a C# (i.e., V7) chord with a melody
note that starts as 12-EDO D# and bends up to a blue E. The
actual file's longest-lasting pitch-bend value clearly implies
a 7:6 ratio or "E<" in my HEWM notation,
http://tonalsoft.com/enc/number/72edo.aspx
which in various EDOs is 2^(16/72) = 2^(8/36) = 2^(4/18)
= ~267 cents.

There was some additional pitch-bend after the NOTE OFF event,
so I extended the note duration and found that 13:11 = 2^(717/72)
= ~283 cents was also very clearly implied by this nice blue note.

I quote:

>> These notes are functioning, simultaneously,
>> harmonically, as 'minor 3rd' or 'sharp 9th' of the C# bass
>> and melodically, as a kind of 'harmonic 7th' of the F# 'tonic'.
>>
>> 7/6, of course, *is* in this case the harmonic 7th of F#.
>> (again, we're ignoring the ~2 cent difference between JI and 12-Eq)
>>
>> 13/11, in relation to F# n^0, is closer to the E 16/9 of F#
>> than to its harmonic '7th'.

Under the next section, "example 3: measure 18 (18.2.472
-- 18.2.316)", I examine a C# (V7) chord with a melody note
that starts as 12-EDO "A" then bends upward to a blue "A#".
A clearly implied ET version is "A#<" = '1/6th-low A#'
= 2^(52/72) = 2^(26/36) = 2^(13/18) = ~866.7 cents. The
best rational implication by my criteria is 33:20, which
has lattice-rungs along the 3, 5, and 11 axes.

And I conclude:

>> Take note of its dual function as botha flattened 'major 6th'
>> over C# and a flattened ('blue') 'major 3rd' over the melodic
>> tonic F#.

The 13:11 and 33:20 would be an example of what Boomsleiter and
Creel refer to as "extended reference".

Also, on my Robert Johnson webpage
http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
I discuss a phrase-ending in Johnson's vocal which very clearly
implies Ab> 23:16 G^ 11:8 F#- 5:4 D 1:1, and which
I hear clearly as blue flavors of flatted 5th, perfect 4th,
major 3rd and tonic respectively.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗hbakshi1 <hareshbakshi@hotmail.com>

🔗clumma <carl@lumma.org>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

> >
> > >>I think 16/9 also has a greater
> > >>stability than 9/5.
> > >
> > >This also reinforces my 12-tET familiarity bit. And, I think the
> > >fact that Bob Wendell is so interested in 19/16 rather than 13/8
> > >also reinforces my 12-tET familiarity theory.
> >
> > I'd point out that:
> >
> > () 13:8 approximates the golden 'least partial aligned' interval.
>
> Whatever, I've seen no signs of anything special at that golden
> interval, either experimentally or theoretically . . . though I
> wouldn't presume to make a final judgment on that.
>
> > () 19:16 approximates a 6:5.
>
> Don't get it. When Bob W. compares 19:16 with 6:5, 19:16 compares
> favorably over 6:5 because . . . ?
>
> >
> > -Carl

Bob W. answers:
Hi, Carl. It's totally context dependent. 5:6 is perfect if it's the
top of a major triad, so C-Eb as 5:6 would imply a triad on Ab.

HOWEVER, if we're wanting an implicit C minor triad, 16:19 is
superior and much more stable harmonically, since its implied
fundamental is a C and not an Ab, and the primary difference tone is
on G, fleshing out the full triad implicitly.

Some people speculate that the romantic composers began writing so
much in minor keys at least partially due to the almost perfect
approximation of 16:19 minor thirds in 12-tET. Interestingly, I
first heard this from Tom Stone, with whom I worked for a time, and
who was the founder and CEO of Novatone. Good luck, Paul!

I used to try to sell those retrofits to famous guitarists and also
electric bass players all over the country who might be interested in
swapping out fretted and fretless fingerboards. Tough sell, though,
to get a guy to relinquish his cherished instrument for a significant
length of time so we could chop the neck up quite a bit and install a
magnetic bed on it in which to cradle the multiple interchangeable
fret boards. I think you can see the difficulty of that and why the
company eventually turned its belly skyward.

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Hi Bob,
>
>
> > From: robert_wendell <BobWendell@t...>
> > To: <tuning@y...>
> > Sent: Monday, December 24, 2001 1:19 PM
> > Subject: [tuning] Re: Prime Rarity?
> >
> >
> > Back on the original theme of this thread, and to possibly reopen
and
> > big can of worms, I wish to elaborate a point Joe Monz made
earlier.
> > It intuitively resonates very strongly with me, if you will allow
for
> > that.
> >
> > I notice that rhythmically, patterns of two and three beats
grouped
> > together by emphasis and/or meter create a different intuitive
> > feeling or sense of rhythmic flavor in the listener. I feel that
this
> > is not so different in principle from what happens in frequency
> > relationships among musical tones. The mechanics and subjective
> > nature of the perception of rhythm and pitch, although quite
> > different, may have important parallels in terms of perceptual
flavor
> > and classification.
> >
>
>
> Thanks for the vote of support.
>
>
>
> > If we follow this model, we might note that although 3s may be
> > grouped in 2s or multiples thereof, they still maintain
> > their "threeness". A 9/8 meter has no "twoness", but is
> > pure "threeness".
>
>
> Umm... well, not necessarily. Are you familiar with Dave Brubeck's
> great tune "Blue Rondo a la Turk"? It's in 9/8 meter, but divided
> 2 + 2 + 2 + 3. OK, I can still see this pattern as exhibiting
> lots of "threeness", but it definitely also contains "twoness"
> along with the "threeness", so it's quite different from the
> "pure threeness" of 3 + 3 + 3, which is what I think you were
> referring to.
>
Bob now:
I was indeed referring only to 3 groups of 3.
>
>
> > Even in 6/8 meter, to me there is a subjective
> > domination of "threeness" inspite of its being at a lower stratum
> > rhythmically. On the other hand, note that in 3/4 time, where the
> > next lower strata are grouped by 2s, the dominant flavor is
> > still "threeness".
>
>
> Do you know about the metrical device called "hemiola"
> (Greek for 3:2). It's a simultaneous combination of 6/8
> and 3/4, wherein both meters exhibit both "threeness" and
> "twoness", but each in opposite "strata".
>
Bob now:
Of course. Hemiola is very characteristic of a lot of early and
middle baroque style, with which I'm intimately familiar from a
performance perspective.

> > One of the things that differentiates jazz and related musics'
> > rhythmic structure from traditional European meters and rhythmic
> > style is the "swing" feel, which comes largely from lots
> > of "threeness" at the lower rhythmic strata, whereas European
styles
> > in three always seem to revert to "twoness" at the lowest strata.
> > There are, of course, other subtler manipulations of rhythmic
> > structure that characterize jazz and related styles.
> >
> > In Indian systems and some of the more experimental western
styles,
> > there are groupings of five and more in rhythmic stuctures. I
feel
> > that five definitely takes a dominant role in giving flavor to
those
> > rhythms that it characterizes.
>
>
> Just thought I'd mention that much of Partch's music has good
examples of
> complex meters containing 5 in different strata.
>
>
> > But like pitch, when going to the higher primes in these
groupings,
> > the mind does tend to simplify, and break these groupings down,
> > excepting only the most experienced and sophisticated listeners.
> >
> >
> > I feel that there is no reason to assume out of hand that there
is
> > not a similar differentiation that is natural to pitch
perception.
> > I'm not pretending to prove anything with these arguments, but I
> > think it is a point worth bearing in mind before we eliminate
this
> > line of thinking out of hand.
>
>
>
> Ah, now I think you're really onto something here, because I see an
> interesting parallel! Just as listeners hypothetically break down
> a 5-meter into a 2- and a 3-, perhaps the brain strives to
comprehend
> audible 5-limit relationships in terms of 3-limit ones, i.e., the
> way of understanding meantones as chains of a generator resembling
> the 3:2 unit-size of the 3-limit chain. Hmmm...
>
>
>
> love / peace / harmony ...
>
> -monz
>
Bob now:
Thanks, Joe! I want to reiterate to everyone participating in this
thread that I see "threeness" at lower rhythmic strata as ONE of the
important contributors to what we percieve as "swing" rhythm. If you
red my original comments duplicated above, you will see this clearly
stated. If you listen to a lot of black gospel, for instance, you
will find the "threeness" at lower strata clearly manifest in
undiluted fashion!

I love polyrhythmic African music, by the way. Sends to me to
paradise. Can't get enough! It's very subtle, powerful stuff! I also
savor the wonderful shifting ambiguity in the way beats are grouped
such as the way "threeness" dissolves into "twoness" and back in
patterns of six, much like hemiola, but infinitely more
sophisticated, and not limited to two and three, but also applied to
higher numbers.

There is also the "bending", not of tempo, but of rhythms against the
backdrop of tempo, their relationship to the steady underlying tempo,
such as "pushing" and "lagging". These phenomena are why I said
the "threeness" at lower strata are ONE of the major contributors to
the swing feel, since I understand DEEPLY that this would be VASTLY
overstated and oversimplified otherwise.
>
But none of these phenomena are directly applicable to frequency,
since we cannot "micromanage" individual cycles of frequency in this
way. Might not the way 3 and 2 cycles define each other when they are
in exact ratio in a perfect P5 to set the "meter" for our perceptual
apparatus in a way that would give it a distinctly different
subjective flavor from other ratios? Just a question, but an
interesting and perhaps important one psychoacoustically.
>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

Whatever research seems to indicate that the difference tones are
audible only if the source frequencies are very loud is bogus in that
it does not take into account trained vs. untrained listeners. This
is true for untrained listeners, including musicians who have not
been specifically trained in this skill, only because of a
psychological phenomenon called masking.

People generally don't hear the individual harmonics that constitute
timbre either, and I found that even some musicians have difficulty
in learning to pick them out individually with the ear. However, it
can be done. Learning to hear differential tones is similar in
nature, but the tones can be quite soft. I know this empirically from
not only my own experience, but from that of many others I have
trained.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> > But none of these phenomena are directly applicable to frequency,
> > since we cannot "micromanage" individual cycles of frequency in
> this
> > way. Might not the way 3 and 2 cycles define each other when they
> are
> > in exact ratio in a perfect P5 to set the "meter" for our
> perceptual
> > apparatus in a way that would give it a distinctly different
> > subjective flavor from other ratios? Just a question, but an
> > interesting and perhaps important one psychoacoustically.
>
> Yes, and there is much psychoacoustical evidence that can be
brought
> to bear on this question. The evidence seems to indicate that two
> sine waves which are at an out-of-tune P5 will still manage to
evoke
> an "implied fundamental" nearly as strongly as an in-tune P5, so
that
> the rhythmic analogy would seem to fall short. Also, the implied
> fundamental will not, in general, be equal to the difference tone,
> which however may also be audible if the tones are loud enough.
John
> Chalmers once referenced a study showing that difference tones can
> actually beat against virtual fundamentals, but I haven't seen that
> study myself -- it seems strange to me that a virtual fundamental
> would have any phase information associated with it.

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

/tuning/topicId_31632.html#31874

> Back on the original theme of this thread, and to possibly reopen
and
> big can of worms, I wish to elaborate a point Joe Monz made
earlier.
> It intuitively resonates very strongly with me, if you will allow
for
> that.
>
> I notice that rhythmically, patterns of two and three beats grouped
> together by emphasis and/or meter create a different intuitive
> feeling or sense of rhythmic flavor in the listener. I feel that
this
> is not so different in principle from what happens in frequency
> relationships among musical tones. The mechanics and subjective
> nature of the perception of rhythm and pitch, although quite
> different, may have important parallels in terms of perceptual
flavor
> and classification.
>
> If we follow this model, we might note that although 3s may be
> grouped in 2s or multiples thereof, they still maintain
> their "threeness". A 9/8 meter has no "twoness", but is
> pure "threeness". Even in 6/8 meter, to me there is a subjective
> domination of "threeness" inspite of its being at a lower stratum
> rhythmically. On the other hand, note that in 3/4 time, where the
> next lower strata are grouped by 2s, the dominant flavor is
> still "threeness".
>
> One of the things that differentiates jazz and related musics'
> rhythmic structure from traditional European meters and rhythmic
> style is the "swing" feel, which comes largely from lots
> of "threeness" at the lower rhythmic strata, whereas European
styles
> in three always seem to revert to "twoness" at the lowest strata.
> There are, of course, other subtler manipulations of rhythmic
> structure that characterize jazz and related styles.
>
> In Indian systems and some of the more experimental western styles,
> there are groupings of five and more in rhythmic stuctures. I feel
> that five definitely takes a dominant role in giving flavor to
those
> rhythms that it characterizes. But like pitch, when going to the
> higher primes in these groupings, the mind does tend to simplify,
> and break these groupings down, excepting only the most experienced
> and sophisticated listeners.
>
> I feel that there is no reason to assume out of hand that there is
> not a similar differentiation that is natural to pitch perception.
> I'm not pretending to prove anything with these arguments, but I
> think it is a point worth bearing in mind before we eliminate this
> line of thinking out of hand.
>

Hello Bob!

Well, this is an interesting discussion and the integration of pitch
and rhythm has been on the mind of many contemporary composers (as
well as, I'm sure many others throughout history...) Of course you
know this...

However, it seems in order to make your analogy truly appropriate in
an *historical* context, we would have to see some kind
of "evolution" in taste from perhaps duple meter to triple meter to
quintal, whatever.

Is this an historically accurate assessment? I'm not sure.

The comparison, of course, would be to the "evolution" or at
least "transformation" of music from 3-limit harmonies to 5-limit
harmonies... maybe later 7...

But, his this kind of evolution in *rhythm* really happened in music
history??

Bartok uses *fives* but I don't think that's going to clinch the
argument... :)

best,

J. Pehrson

🔗robert_wendell <BobWendell@technet-inc.com>

Hi, Joseph! Didn't really intend to imply such a parallel
evolutionary trend. Just thought it interesting the way we perceive
rhythmic flavors and am suggesting we consider there could be a
similar phenomenon occuring with frequency relationships. Nothing
more tight or literal than that. Food for thought.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> /tuning/topicId_31632.html#31874
>
>
> > Back on the original theme of this thread, and to possibly reopen
> and
> > big can of worms, I wish to elaborate a point Joe Monz made
> earlier.
> > It intuitively resonates very strongly with me, if you will allow
> for
> > that.
> >
> > I notice that rhythmically, patterns of two and three beats
grouped
> > together by emphasis and/or meter create a different intuitive
> > feeling or sense of rhythmic flavor in the listener. I feel that
> this
> > is not so different in principle from what happens in frequency
> > relationships among musical tones. The mechanics and subjective
> > nature of the perception of rhythm and pitch, although quite
> > different, may have important parallels in terms of perceptual
> flavor
> > and classification.
> >
> > If we follow this model, we might note that although 3s may be
> > grouped in 2s or multiples thereof, they still maintain
> > their "threeness". A 9/8 meter has no "twoness", but is
> > pure "threeness". Even in 6/8 meter, to me there is a subjective
> > domination of "threeness" inspite of its being at a lower stratum
> > rhythmically. On the other hand, note that in 3/4 time, where the
> > next lower strata are grouped by 2s, the dominant flavor is
> > still "threeness".
> >
> > One of the things that differentiates jazz and related musics'
> > rhythmic structure from traditional European meters and rhythmic
> > style is the "swing" feel, which comes largely from lots
> > of "threeness" at the lower rhythmic strata, whereas European
> styles
> > in three always seem to revert to "twoness" at the lowest strata.
> > There are, of course, other subtler manipulations of rhythmic
> > structure that characterize jazz and related styles.
> >
> > In Indian systems and some of the more experimental western
styles,
> > there are groupings of five and more in rhythmic stuctures. I
feel
> > that five definitely takes a dominant role in giving flavor to
> those
> > rhythms that it characterizes. But like pitch, when going to the
> > higher primes in these groupings, the mind does tend to simplify,
> > and break these groupings down, excepting only the most
experienced
> > and sophisticated listeners.
> >
> > I feel that there is no reason to assume out of hand that there
is
> > not a similar differentiation that is natural to pitch
perception.
> > I'm not pretending to prove anything with these arguments, but I
> > think it is a point worth bearing in mind before we eliminate
this
> > line of thinking out of hand.
> >
>
> Hello Bob!
>
> Well, this is an interesting discussion and the integration of
pitch
> and rhythm has been on the mind of many contemporary composers (as
> well as, I'm sure many others throughout history...) Of course you
> know this...
>
> However, it seems in order to make your analogy truly appropriate
in
> an *historical* context, we would have to see some kind
> of "evolution" in taste from perhaps duple meter to triple meter to
> quintal, whatever.
>
> Is this an historically accurate assessment? I'm not sure.
>
> The comparison, of course, would be to the "evolution" or at
> least "transformation" of music from 3-limit harmonies to 5-limit
> harmonies... maybe later 7...
>
> But, his this kind of evolution in *rhythm* really happened in
music
> history??
>
> Bartok uses *fives* but I don't think that's going to clinch the
> argument... :)
>
> best,
>
> J. Pehrson

🔗jpehrson2 <jpehrson@rcn.com>

🔗monz <joemonz@yahoo.com>

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, December 30, 2001 11:44 AM
> Subject: [tuning] Re: rhythm-pitch analogy
>
>
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> /tuning/topicId_31632.html#32087
>
>
> > Hi, Joseph! Didn't really intend to imply such a parallel
> > evolutionary trend. Just thought it interesting the way we perceive
> > rhythmic flavors and am suggesting we consider there could be a
> > similar phenomenon occuring with frequency relationships. Nothing
> > more tight or literal than that. Food for thought.
> >
>
> This is very interesting, though, Bob, since so many people have
> tried, particularly in recent years, to integrate rhythm-meter and
> pitch. Milton Babbitt, in fact, made his entire career on this
> attempt. (Boulez and Stockhausen, too, to a degree).
>
> Also, Ives, in his Universe symphony uses rhythms based on small
> number ratios much like the overtone series... (well, a bit slower, I
> must admit).
>
> There have been many attempts on this list to show parallels between
> various physical phenomina... light waves, sound waves, pitch,
> rhythm, etc.
>
> Most seem to end up as just that ... *analogies* since, when
> subjected to more rigorous scientific methods, most of the parallels
> don't seem to hold up...
>
> Or at least, that's the was it has seemed to me...

I've mentioned this here before, but it's very pertinent to
this thread... and since you're a fan of Ben Johnston's music,
and you have the CDs of his quartets...

Johnston's rhythmic structures are very often a reflection
of the rational relationships in his harmonies. He'll use
triplets with 3-limit harmonies, quintuplets with 5-limit,
septuplets with 7-limit, etc.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗jpehrson2 <jpehrson@rcn.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗clumma <carl@lumma.org>

31632

>Here is (what should prove to be) a very interesting TEST of the
>*audible existence* such "prime uniqueness":
>
>(1) Setup your system to play a 1/1 and a 3/2 pitch in unison.
>
>(2) Add to this dyad the following two (approximate) "major 3rd"
>pitches (one at a time). The first is 31/24 [consisting of 2^(-3),
>3^(-1), 31^(1)]. Now replace 31/24 with a second pitch (which is
>*only* 1.01568 cents flat from 31/24) of 71/55 [consisting of
>5^(1), 11^(1), 71^(1), a very different set of higher numbered
>primes].

I tried this setup in the Justonic Pitch Palette with "wave audio"
at 44.1Khz and headphones. With the "triangle wave" timbre, I
could _not_ hear a difference in either of two registers. Playing
the two 'thirds' together, I counted one beat every ~2.5 seconds
(1/1= C above middle C at A=440). With the "square wave" timbre,
I could hear a difference in the (rapid) beating between higher
partials. I could also tell the 1/1-31/24 and 1/1-71/55 dyads,
and the 31/24-3/2 and 31/24-3/2 dyads apart by this beating. With
the "sawtooth" timbre, I could no longer distinguish higher-order
beating (just a mess), but there seemed to be a vague sense that
the 71/55 was lower in pitch than the 31/24. Don't know if this
would pan out in a blind test.

>(1) Setup your system to play the 1/1 and a 3/2 pitch in unison.
>
>(2) Add to this dyad the following two (approximate) "minor 7th"
>pitches (one at a time). The first is "good-old" 16/9 [consisting
>of 2^(4), 3^(-2)]. Now replace 16/9 with a second pitch (which is
>*only* 2.92685 cents flat from 16/9) of 197/111 [consisting of
>111^(-1), 197^(1), a very different set of higher numbered primes].

With the "sine" timbre, these two intervals have a slightly
different flavor when the 3/2 is present. Without the 3:2, a vague
sense of size difference is all I pick up. With the "sawtooth"
timbre, the cloud of beating sounds distinctly different, with or
without the 3/2, at 1/1=middle C. At 1/1=C above middle C, the
cloud is too messy with the 3/2 present to tell anything, but still
different between the bare 7ths. With the "triangle" and "square"
timbres, I get a flavor difference no matter what.

Still has nothing to do with primeness, as far as I can tell.

-Carl

🔗robert_wendell <BobWendell@technet-inc.com>

🔗Afmmjr@aol.com

🔗unidala <JGill99@imajis.com>

Carl: Hey, congratulations, you are the only (poster)
to report having tried it out! A handful of comments
below.

Regards, J Gill

--- In tuning@y..., "clumma" <carl@l...> wrote:
> 31632
>
> >Here is (what should prove to be) a very interesting TEST of the
> >*audible existence* such "prime uniqueness":
> >
> >(1) Setup your system to play a 1/1 and a 3/2 pitch
[[simultaneously]].
> >
> >(2) Add to this dyad the following two (approximate) "major 3rd"
> >pitches (one at a time). The first is 31/24 [consisting of 2^(-3),
> >3^(-1), 31^(1)]. Now replace 31/24 with a second pitch (which is
> >*only* 1.01568 cents flat from 31/24) of 71/55 [consisting of
> >5^(1), 11^(1), 71^(1), a very different set of higher numbered
> >primes].
>
>
> I tried this setup in the Justonic Pitch Palette with "wave audio"
> at 44.1Khz and headphones. With the "triangle wave" timbre, I
> could _not_ hear a difference in either of two registers.

JG: The overtones of a triangle wave decrease at a rate of
-12 dB / octave (as opposed to "square and "sawtooth", which
decrease at a rate of -6 dB / octave). This seems to imply
that (had you used sinusoidal waves), the 1-cent difference
would be non-discernable (not surprisingly).

> Playing
> the two 'thirds' together, I counted one beat every ~2.5 seconds
> (1/1= C above middle C at A=440).

JG: I get a difference frequency of about 5 Hz (assuming
880 Hz). But there are *two* (indistinguishable)
events per cycle which would be heard as the "beat rate".
However at 880 Hz * 31/24, I would expect a 3.3 second
"beat rate" for an approximately 1-cent difference
(between 880 * 31/24 and 880 * 71/55)...?

> With the "square wave" timbre,
> I could hear a difference in the (rapid) beating between higher
> partials. I could also tell the 1/1-31/24 and 1/1-71/55 dyads,
> and the 31/24-3/2 and 31/24-3/2 dyads apart by this beating.

JG: You mean "the 31/24-3/2 and the 71/55-3/2 dyads", right?
Sounds like those overtones (and "beats") have a *lot* to do
with one being able to resolve such small (~1-cent) differences!

> With
> the "sawtooth" timbre, I could no longer distinguish higher-order
> beating (just a mess), but there seemed to be a vague sense that
> the 71/55 was lower in pitch than the 31/24. Don't know if this
> would pan out in a blind test.

JG: It's interesting that the "sawtooth" (identical spectrally
to the "squarewave", except for the *even-order* harmonic
content) obscures that perception of "beat rates". Something
to think about!

> >(1) Setup your system to play the 1/1 and a 3/2 pitch
[[simultaneously]].
> >
> >(2) Add to this dyad the following two (approximate) "minor 7th"
> >pitches (one at a time). The first is "good-old" 16/9 [consisting
> >of 2^(4), 3^(-2)]. Now replace 16/9 with a second pitch (which is
> >*only* 2.92685 cents flat from 16/9) of 197/111 [consisting of
> >111^(-1), 197^(1), a very different set of higher numbered primes].
>
> With the "sine" timbre, these two intervals have a slightly
> different flavor when the 3/2 is present.

JG: So, perhaps 3-cents or so *does* work with sinusoidal sources.
You've (no doubt) got a "good ear"!

> Without the 3:2, a vague
> sense of size difference is all I pick up. With the "sawtooth"
> timbre, the cloud of beating sounds distinctly different, with or
> without the 3/2, at 1/1=middle C. At 1/1=C above middle C, the
> cloud is too messy with the 3/2 present to tell anything, but still
> different between the bare 7ths.

JG: I guess those "even-order" harmonics (of "sawtooth" waves)
really "mess" with the ear. Interesting!

>With the "triangle" and "square"
> timbres, I get a flavor difference no matter what.
>
> Still has nothing to do with primeness, as far as I can tell.
>
> -Carl

JG: If you had found these difference indistinguishable using
all-sinusoidal sources, it would seem clear that "primeness"
was likely not involved. Utilizing the overtones of "triangle",
"square", and "sawtooth" waves, can we be so sure? You tell me.

Regards, J Gill

🔗clumma <carl@lumma.org>

>>I tried this setup in the Justonic Pitch Palette with "wave
>>audio" at 44.1Khz and headphones. With the "triangle wave"
>>timbre, I could _not_ hear a difference in either of two
>>registers.
>
>JG: The overtones of a triangle wave decrease at a rate of
>-12 dB / octave (as opposed to "square and "sawtooth", which
>decrease at a rate of -6 dB / octave). This seems to imply
>that (had you used sinusoidal waves), the 1-cent difference
>would be non-discernable (not surprisingly).

Just a note that I'm not sure how much like the ideal waves
these timbres from the Pitch Palette are, coming through
headphones from my Thinkpad.

>> Playing the two 'thirds' together, I counted one beat every
>> ~2.5 seconds (1/1= C above middle C at A=440).
>
> JG: I get a difference frequency of about 5 Hz (assuming
> 880 Hz). But there are *two* (indistinguishable)
> events per cycle which would be heard as the "beat rate".
> However at 880 Hz * 31/24, I would expect a 3.3 second
> "beat rate" for an approximately 1-cent difference
> (between 880 * 31/24 and 880 * 71/55)...?

The thirds were actually at 1/1 a 6:5 above A=440. Sorry
for any confusion. I counted approx. 2.5 seconds per total
cancelation (there were smaller beats inside this).

>> With the "square wave" timbre,
>> I could hear a difference in the (rapid) beating between higher
>> partials. I could also tell the 1/1-31/24 and 1/1-71/55 dyads,
>> and the 31/24-3/2 and 31/24-3/2 dyads apart by this beating.
>
> JG: You mean "the 31/24-3/2 and the 71/55-3/2 dyads", right?

Yes.

> Sounds like those overtones (and "beats") have a *lot* to do
> with one being able to resolve such small (~1-cent) differences!

Yes.

>> With the "sawtooth" timbre, I could no longer distinguish
>> higher-order beating (just a mess), but there seemed to be
>> a vague sense that the 71/55 was lower in pitch than the 31/24.
>> Don't know if this would pan out in a blind test.
>
> JG: It's interesting that the "sawtooth" (identical spectrally
> to the "squarewave", except for the *even-order* harmonic
> content)

You sure? I thought sawtooth waves have rich harmonic content of
both even and odd partials.

>> With the "sine" timbre, these two intervals have a slightly
>> different flavor when the 3/2 is present.
>
> JG: So, perhaps 3-cents or so *does* work with sinusoidal sources.
> You've (no doubt) got a "good ear"!

I don't think I was getting a sine wave, which is why I put it
in quotes. In my experience, it is nearly impossible to deliver
true sines with affordable electronic equipment.

I tune pianos in my spare time (which lately, has been zero), and
you have to get very good at listening to beating overtones to get
the unisons to sound tolerable. It takes me a long time, but I
usually get an excellent result.

I know that I can hear ultrasonics from electronic equipment that
many cannot. This is most likely due to the fact that I'm still
in my 20's, and that I managed to avoid loud rock concerts and
industrial sounds for most of my life, being a nerd raised in a
rural area.

> JG: If you had found these difference indistinguishable using
> all-sinusoidal sources, it would seem clear that "primeness"
> was likely not involved. Utilizing the overtones of "triangle",
> "square", and "sawtooth" waves, can we be so sure? You tell me.

Yes, we can be sure. In the past I've tried similar experiments
irrational intervals, and noticed no difference in the type of
results. One could find all sorts of ratio interpretations for
these irrational intervals, which would likely have different
prime factors, and this only shows that primeness has nothing to
do with anything when the numbers get big.

-Carl

🔗jpehrson2 <jpehrson@rcn.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >>I tried this setup in the Justonic Pitch Palette with "wave
> >>audio" at 44.1Khz and headphones. With the "triangle wave"
> >>timbre, I could _not_ hear a difference in either of two
> >>registers.
> >
> >JG: The overtones of a triangle wave decrease at a rate of
> >-12 dB / octave (as opposed to "square and "sawtooth", which
> >decrease at a rate of -6 dB / octave). This seems to imply
> >that (had you used sinusoidal waves), the 1-cent difference
> >would be non-discernable (not surprisingly).
>
> Just a note that I'm not sure how much like the ideal waves
> these timbres from the Pitch Palette are, coming through
> headphones from my Thinkpad.

JG: Having been the owner of a ThinkPad myself, and being
familiar with the (not) superb sound quality, and not
being familiar with your headphines, I see your point.
>
> >> Playing the two 'thirds' together, I counted one beat every
> >> ~2.5 seconds (1/1= C above middle C at A=440).
> >
> > JG: I get a difference frequency of about 5 Hz (assuming
> > 880 Hz). But there are *two* (indistinguishable)
> > events per cycle which would be heard as the "beat rate".
> > However at 880 Hz * 31/24, I would expect a 3.3 second
> > "beat rate" for an approximately 1-cent difference
> > (between 880 * 31/24 and 880 * 71/55)...?
>
> The thirds were actually at 1/1 a 6:5 above A=440. Sorry
> for any confusion. I counted approx. 2.5 seconds per total
> cancelation (there were smaller beats inside this).

JG: Given further thought, I realized that my calculations
(above) were off by a factor of 2 (there is only *one*,
not two, cancellations per cycle of the difference frequency).
for 528 Hz (6/5*440), I now get about 3.06 "cancellations"/sec.

> >> With the "square wave" timbre,
> >> I could hear a difference in the (rapid) beating between higher
> >> partials. I could also tell the 1/1-31/24 and 1/1-71/55 dyads,
> >> and the 31/24-3/2 and 31/24-3/2 dyads apart by this beating.
> >
> > JG: You mean "the 31/24-3/2 and the 71/55-3/2 dyads", right?
>
> Yes.
>
> > Sounds like those overtones (and "beats") have a *lot* to do
> > with one being able to resolve such small (~1-cent) differences!
>
> Yes.
>
> >> With the "sawtooth" timbre, I could no longer distinguish
> >> higher-order beating (just a mess), but there seemed to be
> >> a vague sense that the 71/55 was lower in pitch than the 31/24.
> >> Don't know if this would pan out in a blind test.
> >
> > JG: It's interesting that the "sawtooth" (identical spectrally
> > to the "squarewave", except for the *even-order* harmonic
> > content)
>
> You sure? I thought sawtooth waves have rich harmonic content of
> both even and odd partials.

JG: That *is* what I was trying to say (above)! Doing some
checking, the phase of the odd harmonics in a square wave
alternate 1, 5, 9, ... are positive, and 3, 7, 11, ... are
negative, etc. Ditto for the "triangle" wave. The "sawtooth"
wave has *all* positive phase harmonics 1, 2, 3, 4, 5, ....
The *magnitude* of "sawtooth" and "square" waves are identical
in the case of the odd-numbered harmonics, with the even-order
harmonics (also) decreasing at a rate of -6 dB/octave. The
"triangle" wave has (odd) harmonics which decrease at -12dB/oct.

> >> With the "sine" timbre, these two intervals have a slightly
> >> different flavor when the 3/2 is present.
> >
> > JG: So, perhaps 3-cents or so *does* work with sinusoidal sources.
> > You've (no doubt) got a "good ear"!
>
> I don't think I was getting a sine wave, which is why I put it
> in quotes. In my experience, it is nearly impossible to deliver
> true sines with affordable electronic equipment.
>
> I tune pianos in my spare time (which lately, has been zero), and
> you have to get very good at listening to beating overtones to get
> the unisons to sound tolerable. It takes me a long time, but I
> usually get an excellent result.
>
> I know that I can hear ultrasonics from electronic equipment that
> many cannot. This is most likely due to the fact that I'm still
> in my 20's, and that I managed to avoid loud rock concerts and
> industrial sounds for most of my life, being a nerd raised in a
> rural area.
>
> > JG: If you had found these difference indistinguishable using
> > all-sinusoidal sources, it would seem clear that "primeness"
> > was likely not involved. Utilizing the overtones of "triangle",
> > "square", and "sawtooth" waves, can we be so sure? You tell me.
>
> Yes, we can be sure. In the past I've tried similar experiments
> irrational intervals, and noticed no difference in the type of
> results.

JG: With sinusoidal (or approximately sinusoidal) tones?

> One could find all sorts of ratio interpretations for
> these irrational intervals, which would likely have different
> prime factors, and this only shows that primeness has nothing to
> do with anything when the numbers get big.

JG: I *do* agree with your proposition.

Regards, J Gill

🔗clumma <carl@lumma.org>

>>Just a note that I'm not sure how much like the ideal waves
>>these timbres from the Pitch Palette are, coming through
>>headphones from my Thinkpad.
>
>JG: Having been the owner of a ThinkPad myself, and being
>familiar with the (not) superb sound quality, and not
>being familiar with your headphines, I see your point.

My headphones are quite good, but the amp in the Thinkpad,
and the method the Pitch Palette uses are suspect.

>> The thirds were actually at 1/1 a 6:5 above A=440. Sorry
>> for any confusion. I counted approx. 2.5 seconds per total
>> cancelation (there were smaller beats inside this).
>
>JG: Given further thought, I realized that my calculations
>(above) were off by a factor of 2 (there is only *one*,
>not two, cancellations per cycle of the difference frequency).
>for 528 Hz (6/5*440), I now get about 3.06 "cancellations"/sec.

That's within the range of accuracy of my counting.

>>>>With the "sawtooth" timbre, I could no longer distinguish
>>>>higher-order beating (just a mess), but there seemed to be
>>>>a vague sense that the 71/55 was lower in pitch than the 31/24.
>>>>Don't know if this would pan out in a blind test.
>>>
>>>JG: It's interesting that the "sawtooth" (identical spectrally
>>>to the "squarewave", except for the *even-order* harmonic
>>>content)
>>
>>You sure? I thought sawtooth waves have rich harmonic content of
>>both even and odd partials.
>
>JG: That *is* what I was trying to say (above)!

Ah, okay.

>Doing some checking, the phase of the odd harmonics in a square
>wave alternate 1, 5, 9, ... are positive, and 3, 7, 11, ... are
>negative, etc.

What does negative phase mean? Maybe a picture would help?

>>>JG: If you had found these difference indistinguishable using
>>>all-sinusoidal sources, it would seem clear that "primeness"
>>>was likely not involved. Utilizing the overtones of "triangle",
>>>"square", and "sawtooth" waves, can we be so sure? You tell me.
>>
>>Yes, we can be sure. In the past I've tried similar experiments
>>irrational intervals, and noticed no difference in the type of
>>results.
>
> JG: With sinusoidal (or approximately sinusoidal) tones?

Uh-huh.

-Carl

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >>Just a note that I'm not sure how much like the ideal waves
> >>these timbres from the Pitch Palette are, coming through
> >>headphones from my Thinkpad.
> >
> >JG: Having been the owner of a ThinkPad myself, and being
> >familiar with the (not) superb sound quality, and not
> >being familiar with your headphines, I see your point.
>
> My headphones are quite good, but the amp in the Thinkpad,
> and the method the Pitch Palette uses are suspect.
>
> >> The thirds were actually at 1/1 a 6:5 above A=440. Sorry
> >> for any confusion. I counted approx. 2.5 seconds per total
> >> cancelation (there were smaller beats inside this).
> >
> >JG: Given further thought, I realized that my calculations
> >(above) were off by a factor of 2 (there is only *one*,
> >not two, cancellations per cycle of the difference frequency).
> >for 528 Hz (6/5*440), I now get about 3.06 "cancellations"/sec.
>
> That's within the range of accuracy of my counting.
>
> >>>>With the "sawtooth" timbre, I could no longer distinguish
> >>>>higher-order beating (just a mess), but there seemed to be
> >>>>a vague sense that the 71/55 was lower in pitch than the 31/24.
> >>>>Don't know if this would pan out in a blind test.
> >>>
> >>>JG: It's interesting that the "sawtooth" (identical spectrally
> >>>to the "squarewave", except for the *even-order* harmonic
> >>>content)
> >>
> >>You sure? I thought sawtooth waves have rich harmonic content of
> >>both even and odd partials.
> >
> >JG: That *is* what I was trying to say (above)!
>
> Ah, okay.
>
> >Doing some checking, the phase of the odd harmonics in a square
> >wave alternate 1, 5, 9, ... are positive, and 3, 7, 11, ... are
> >negative, etc.
>
> What does negative phase mean? Maybe a picture would help?

JG: Carl, the identities are (for unit amplitude and frequency,
where w = 2*pi*f, and f = frequency, in Hz or CPS):

SQUARE = (4/pi)*( sin(wt)-sin(3wt)/3+sin(5wt)/5-sin(7wt)/7+.... )

SAWTOO = (2/pi)*( sin(wt)+sin(2wt)/2+sin(3wt)/3+sin(4wt)/4+.... )

TRIANGL = (8/pi^2)*( sin(wt)-sin(3wt)/(3^2)+sin(5wt)/(5^2)+.... )

So the "phase spectrum" (as opposed to the "magnitude spectrum")
of these waveforms contains harmonics which alternate in the
algebraic *sign* (in the case of the SQUARE and TRIANGLE waves).

The "phase difference" *between* unequal frequencies is a tricky
concept [since "phase" must be referenced to *something*, to some
argument (Kwt) of one of the two (sin/cos) waves]. Such may be
viewed as a "phase precession" (a "ramping" of phase) relative
to the chosen reference wave.

The way to picture it (I think) is to visualize the first
moments in time as an (additive) synthesis of these waves
takes place. The various harmonics will initially begin to
oscillate (at their respective frequencies) in either a
positive or negative direction in (time domain) amplitude
(relative to the "fundamental" frequency's trajectory,
which is in a positive direction in all the examples above).

> >>>JG: If you had found these difference indistinguishable using
> >>>all-sinusoidal sources, it would seem clear that "primeness"
> >>>was likely not involved. Utilizing the overtones of "triangle",
> >>>"square", and "sawtooth" waves, can we be so sure? You tell me.
> >>
> >>Yes, we can be sure. In the past I've tried similar experiments
> >>irrational intervals, and noticed no difference in the type of
> >>results.
> >
> > JG: With sinusoidal (or approximately sinusoidal) tones?
>
> Uh-huh.

JG: You mean "yes" (or "no") by, "Uh-huh"?

J Gill

🔗clumma <carl@lumma.org>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >> What does negative phase mean? Maybe a picture would help?
> >
> > JG: Carl, the identities are (for unit amplitude and frequency,
> > where w = 2*pi*f, and f = frequency, in Hz or CPS):
> >
> > SQUARE = (4/pi)*( sin(wt)-sin(3wt)/3+sin(5wt)/5-sin(7wt)/7+.... )
> >
> > SAWTOO = (2/pi)*( sin(wt)+sin(2wt)/2+sin(3wt)/3+sin(4wt)/4+.... )
> >
> > TRIANGL = (8/pi^2)*( sin(wt)-sin(3wt)/(3^2)+sin(5wt)/(5^2)+.... )
> >
> > So the "phase spectrum" (as opposed to the "magnitude spectrum")
> > of these waveforms contains harmonics which alternate in the
> > algebraic *sign* (in the case of the SQUARE and TRIANGLE waves).
> >
> > The "phase difference" *between* unequal frequencies is a tricky
> > concept [since "phase" must be referenced to *something*, to some
> > argument (Kwt) of one of the two (sin/cos) waves]. Such may be
> > viewed as a "phase precession" (a "ramping" of phase) relative
> > to the chosen reference wave.
> >
> > The way to picture it (I think) is to visualize the first
> > moments in time as an (additive) synthesis of these waves
> > takes place. The various harmonics will initially begin to
> > oscillate (at their respective frequencies) in either a
> > positive or negative direction in (time domain) amplitude
> > (relative to the "fundamental" frequency's trajectory,
> > which is in a positive direction in all the examples above).
>
> Thanks!
>
> >>>>Yes, we can be sure. In the past I've tried similar experiments
> >>>>irrational intervals, and noticed no difference in the type of
> >>>>results.
> >>>
> >>> JG: With sinusoidal (or approximately sinusoidal) tones?
> >>
> >> Uh-huh.
> >
> > JG: You mean "yes" (or "no") by, "Uh-huh"?
>
> I meant yes. I've gotta watch that.

J Gill: Carl, what do you estimate your (sinusoidal tones *only*)
resolution is (in Cents), and at what frequencies (if applicable)?

That is - what is the smallest (Cent) variation which you can
resolve without the presence of overtones to allow the perception
of low-frequency "beat frequencies"?

Curiously, J Gill

🔗clumma <carl@lumma.org>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >J Gill: Carl, what do you estimate your (sinusoidal tones *only*)
> >resolution is (in Cents), and at what frequencies (if applicable)?
>
> I don't know. Probably very accurate... a tenth of a CPS, flat
> across the frequency spectrum.

JG: Wowie zowie! That's 8.63 Cents at 20 Hz, 0.865 Cents at 200 Hz,
.0866 Cents at 2000 Hz, and .00866 Cents at 20,000 Hz. You're good,
you're *very* good!

> > That is - what is the smallest (Cent) variation which you can
> > resolve without the presence of overtones to allow the perception
> > of low-frequency "beat frequencies"?
>
> I would like to reiterate that I do not have access to pure
> tones! How can I make this more clear?

I guess that your statements (originally made in message
#32437), and regarding accuracies in pitch perception
on the order of 1 - 3 Cents, such as:

<< In the past I've tried similar experiments
irrational intervals, and noticed no difference in the type of
results. >>

which caused me to ask of you:

> JG: With sinusoidal (or approximately sinusoidal) tones?

to which you answered:

<< Uh-huh. >>

which I asked be clarified by asking:

> JG: You mean "yes" (or "no") by, "Uh-huh"?

to which you responded:

<< I meant yes. I've gotta watch that. >>

might have caused me to think that you were talking about
"sinusoidal (or approximately sinusoidal) tones"? Dunno.

J Gill :)

Prime Rarity?

🔗J Gill <JGill99@imajis.com>

🔗clumma <carl@lumma.org>

🔗unidala <JGill99@imajis.com>

🔗clumma <carl@lumma.org>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗Dante Rosati <dante.interport@rcn.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗clumma <carl@lumma.org>

🔗paulerlich <paul@stretch-music.com>

🔗clumma <carl@lumma.org>

🔗Dante Rosati <dante.interport@rcn.com>

🔗clumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗dkeenanuqnetau <d.keenan@uq.net.au>

🔗Dante Rosati <dante.interport@rcn.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗clumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗clumma <carl@lumma.org>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗Robert C Valentine <BVAL@IIL.INTEL.COM>

🔗paulerlich <paul@stretch-music.com>

🔗Robert C Valentine <BVAL@IIL.INTEL.COM>

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗hbakshi1 <hareshbakshi@hotmail.com>

🔗clumma <carl@lumma.org>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗monz <joemonz@yahoo.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗clumma <carl@lumma.org>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗Afmmjr@aol.com

🔗unidala <JGill99@imajis.com>

🔗unidala <JGill99@imajis.com>

🔗clumma <carl@lumma.org>

🔗jpehrson2 <jpehrson@rcn.com>

🔗unidala <JGill99@imajis.com>

🔗clumma <carl@lumma.org>