Superparticularity again

I've seen superparticularity as the subject of a lot of
posts here recently, but haven't really followed the thread.
But I'd like to add my thoughts on it.

I believe that the importance of superparticular is that
it represents subsets from the harmonic series, whose
ratios in turn exhibit simple arithmetical numerical
properties whose relationships are very easy to comprehend.
The harmonic series is thus a sort of archtype by which
we understand tonality.

Franz Richter Herf's interesting theory is related to
this. I posted some info about it here:
/tuning/topicId_22968.html#23086?expand=1

Hmm... I wonder, since I'm really nowhere near as familiar
with Erv Wilson's theories as I'd like to be... does
Erv's CPS theory have anything to do with this? If so,
what? Please feel free to elaborate... I'm anxious to
read as much about this as possible.

love / peace / harmony ...

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

_________________________________________________________
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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> I've seen superparticularity as the subject of a lot of
> posts here recently, but haven't really followed the thread.
> But I'd like to add my thoughts on it.
> I believe that the importance of superparticular is that
> it represents subsets from the harmonic series, whose
> ratios in turn exhibit simple arithmetical numerical
> properties whose relationships are very easy to comprehend.
> The harmonic series is thus a sort of archtype by which
> we understand tonality.
> Franz Richter Herf's interesting theory is related to
> this. I posted some info about it here:
> /tuning/topicId_22968.html#23086?expand=1
> Hmm... I wonder, since I'm really nowhere near as familiar
> with Erv Wilson's theories as I'd like to be... does
> Erv's CPS theory have anything to do with this? If so,
> what? Please feel free to elaborate... I'm anxious to
> read as much about this as possible.

J Gill:

Monz, about all I have been able to discover is:

/tuning/topicId_31138.html#31139
Klaus Shmirler:
<< Superparticular ratios - those that are super - are
generated by the harmonic (arithmetic) division of a larger
interval, and for me they combine harmonic and melodic
smoothness. Some list members think differently; I humbly
opine this is the case because they are interested in rigid
systems and fixed numbers of tones - which may be transposed
to extend their scope, but which unlike the division of
intervals do not generate new ones all the time.
I am one of those who believe in mathematical hearing, that
is the ability to divide an interval harmonically and the
inability to divide geometrically (halving the cents) by
ear. >>

/tuning/topicId_31138.html#31187
Paul Erlich:

> JG: Superparticular ratios made up of higher valued integers do
> not appear to
> result in coincident overtones (of the integer multiples of those
> superparticular ratios) to any degree greater than non-
> superparticular
> ratios made up of higher valued integers.

PE: I agree with this statement.

> JG: While a sequence of 3/2, 4/3, 5/4, 6/5, etc. exists as a "sub-
> branch" of
> the Stern-Brocot tree, it is not clear to me what is significant
> about that...

PE: Nothing, in itself . . .

/tuning/topicId_31138.html#31297
Kraig Grady:
<< here is a paper on epimores. page 3 should explain the previous
http://www.anaphoria.com/epimore.pdf >>

/tuning/topicId_31138.html#31318
Paul Erlich:
<< Could you please gently describe what it is we are supposed to be
looking at and what it means -- "the previous"? >>

/tuning/topicId_31138.html#31388
Kraig Grady:
<< Page 2 is the epimores between the intervals on Page 1 Page 3 shows this in one
diagram although
the last line is messed up you can see it on page two. In other words adjacent
figures between
terms on any particular level of the pierce sequence form superparticular
ratios these two can be
arranged into their own tree as seen on page 2 >>

/tuning/topicId_31138.html#31407
Paul Erlich:
<< Thanks Kraig. This shows very nicely what I've been trying to say to J. >>

/tuning/topicId_31641.html#31663
Paul Erlich:
<< what seems to be the case very often, is that when one
comes up with such a scale in the form of a periodicity block, one
has quite a few arbitrary choices to make as to which version of a
particular scale degree one wants (the different versions differing
by a unison vector), and then _one such set_ of arbitrary choices
does lead to a scale with superparticular step sizes. >>

Regards, J Gill

> From: unidala <JGill99@imajis.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, December 28, 2001 1:19 AM
> Subject: [tuning] Re: Superparticularity again
>
>
> J Gill:
>
> Monz, about all I have been able to discover is:
>
> <etc.>

Thanks for taking the time to cull all of this together, J!

> /tuning/topicId_31641.html#31663
> Paul Erlich:
> << what seems to be the case very often, is that when one
> comes up with such a scale in the form of a periodicity block, one
> has quite a few arbitrary choices to make as to which version of a
> particular scale degree one wants (the different versions differing
> by a unison vector), and then _one such set_ of arbitrary choices
> does lead to a scale with superparticular step sizes. >>

Hmmm... given my big interest in periodicity-blocks, *this*
is something worthy of much further study! It seems like there
might be a convergence of "overtone theory" + "periodicity-block
theory" + "superparticularity theory" lurking in here somewhere,
which I suspect would go a long way towards coalescing many of
my currently vague ideas about finity.

-monz

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

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: unidala <JGill99@i...>
> > To: <tuning@y...>
> > Sent: Friday, December 28, 2001 1:19 AM
> > Subject: [tuning] Re: Superparticularity again
> >
> >
> > J Gill:
> >
> > Monz, about all I have been able to discover is:
> >
> > <etc.>
>
>
> Thanks for taking the time to cull all of this together, J!
>
>
> > /tuning/topicId_31641.html#31663
> > Paul Erlich:
> > << what seems to be the case very often, is that when one
> > comes up with such a scale in the form of a periodicity block, one
> > has quite a few arbitrary choices to make as to which version of a
> > particular scale degree one wants (the different versions differing
> > by a unison vector), and then _one such set_ of arbitrary choices
> > does lead to a scale with superparticular step sizes. >>
>
>
> Hmmm... given my big interest in periodicity-blocks, *this*
> is something worthy of much further study! It seems like there
> might be a convergence of "overtone theory" + "periodicity-block
> theory" + "superparticularity theory" lurking in here somewhere,
> which I suspect would go a long way towards coalescing many of
> my currently vague ideas about finity.

___________________________________________

J Gill: Monz, do you understand what Kraig means
to say by presenting the following posts/link?

/tuning/topicId_31138.html#31297
Kraig Grady:
<< here is a paper on epimores. page 3 should explain the previous
http://www.anaphoria.com/epimore.pdf >>

/tuning/topicId_31138.html#31388
Kraig Grady:
<< Page 2 is the epimores between the intervals on Page 1 Page 3 shows this in
one
diagram although
the last line is messed up you can see it on page two. In other words adjacent
figures between
terms on any particular level of the pierce sequence form superparticular
ratios these two can be
arranged into their own tree as seen on page 2 >>

Curiously, J Gill

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: unidala <JGill99@i...>
> > To: <tuning@y...>
> > Sent: Friday, December 28, 2001 1:19 AM
> > Subject: [tuning] Re: Superparticularity again
> >
> >
> > J Gill:
> >
> > Monz, about all I have been able to discover is:
> >
> > <etc.>
>
>
> Thanks for taking the time to cull all of this together, J!
>
>
> > /tuning/topicId_31641.html#31663
> > Paul Erlich:
> > << what seems to be the case very often, is that when one
> > comes up with such a scale in the form of a periodicity block, one
> > has quite a few arbitrary choices to make as to which version of a
> > particular scale degree one wants (the different versions differing
> > by a unison vector), and then _one such set_ of arbitrary choices
> > does lead to a scale with superparticular step sizes. >>
>
>
> Hmmm... given my big interest in periodicity-blocks, *this*
> is something worthy of much further study! It seems like there
> might be a convergence of "overtone theory" + "periodicity-block
> theory" + "superparticularity theory" lurking in here somewhere,
> which I suspect would go a long way towards coalescing many of
> my currently vague ideas about finity.

J Gill: Lurking it may be, but I have yet to find
much of anything on these tuning lists which does
much more than explore "periodicity-block" theory.
Maybe these lists could actually explore something
more than that someday. There may be more than on
way to "skin the cat" (but we may never know)...

Regards, J Gill

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: unidala <JGill99@i...>
> > To: <tuning@y...>
> > Sent: Friday, December 28, 2001 1:19 AM
> > Subject: [tuning] Re: Superparticularity again
> >
> >
> > J Gill:
> >
> > Monz, about all I have been able to discover is:
> >
> > <etc.>
>
>
> Thanks for taking the time to cull all of this together, J!
>
>
> > /tuning/topicId_31641.html#31663
> > Paul Erlich:
> > << what seems to be the case very often, is that when one
> > comes up with such a scale in the form of a periodicity block, one
> > has quite a few arbitrary choices to make as to which version of a
> > particular scale degree one wants (the different versions differing
> > by a unison vector), and then _one such set_ of arbitrary choices
> > does lead to a scale with superparticular step sizes. >>
>
>
> Hmmm... given my big interest in periodicity-blocks, *this*
> is something worthy of much further study! It seems like there
> might be a convergence of "overtone theory" + "periodicity-block
> theory" + "superparticularity theory" lurking in here somewhere,
> which I suspect would go a long way towards coalescing many of
> my currently vague ideas about finity.

J Gill: Monz, what do you think Kraig means to show
from the posts below?

/tuning/topicId_31138.html#31297
Kraig Grady:
<< here is a paper on epimores. page 3 should explain the previous
http://www.anaphoria.com/epimore.pdf >>

/tuning/topicId_31138.html#31388
Kraig Grady:
<< Page 2 is the epimores between the intervals on Page 1 Page 3 shows this in
one
diagram although
the last line is messed up you can see it on page two. In other words adjacent
figures between
terms on any particular level of the pierce sequence form superparticular
ratios these two can be
arranged into their own tree as seen on page 2 >>

Curiously, J Gill

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: unidala <JGill99@i...>
> > To: <tuning@y...>
> > Sent: Friday, December 28, 2001 1:19 AM
> > Subject: [tuning] Re: Superparticularity again
> >
> >
> > J Gill:
> >
> > Monz, about all I have been able to discover is:
> >
> > <etc.>
>
>
> Thanks for taking the time to cull all of this together, J!
>
>
> > /tuning/topicId_31641.html#31663
> > Paul Erlich:
> > << what seems to be the case very often, is that when one
> > comes up with such a scale in the form of a periodicity block,
one
> > has quite a few arbitrary choices to make as to which version of
a
> > particular scale degree one wants (the different versions
differing
> > by a unison vector), and then _one such set_ of arbitrary choices
> > does lead to a scale with superparticular step sizes. >>
>
>
> Hmmm... given my big interest in periodicity-blocks, *this*
> is something worthy of much further study! It seems like there
> might be a convergence of "overtone theory" + "periodicity-block
> theory" + "superparticularity theory" lurking in here somewhere,
> which I suspect would go a long way towards coalescing many of
> my currently vague ideas about finity.

Well, you might want to ask Gene about this observation of mine -- he
seems to have gone deeper into the superparticularity question than
the rest of us . . .

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> J Gill: Monz, do you understand what Kraig means
> to say by presenting the following posts/link?
>
> /tuning/topicId_31138.html#31297
> Kraig Grady:
> << here is a paper on epimores. page 3 should explain the previous
> http://www.anaphoria.com/epimore.pdf >>
>
>
> /tuning/topicId_31138.html#31388
> Kraig Grady:
> << Page 2 is the epimores between the intervals on Page 1 Page 3
shows this in
> one
> diagram although
> the last line is messed up you can see it on page two. In other
words adjacent
> figures between
> terms on any particular level of the pierce sequence form
superparticular
> ratios these two can be
> arranged into their own tree as seen on page 2 >>

I thought I explained this to you already. You should have said
something if my explanation wasn't good enough. What this shows is
that, just like the Farey series and similar sets, the set of ratios
above some layer in the Stern-Brocot tree will have all its _step
sizes_ be superparticular ratios. Didn't I get that across to you
already? Please let me know how I can communicate more effectively.

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> J Gill: Lurking it may be, but I have yet to find
> much of anything on these tuning lists which does
> much more than explore "periodicity-block" theory.

I'm sure a lot of people wish that you couldn't find that either.

> Maybe these lists could actually explore something
> more than that someday.

What do you have in mind?

> There may be more than on
> way to "skin the cat" (but we may never know)...

Indeed there are . . . Between Tanaka, Wurschmidt, Wilson, etc., we
have tons of differently formulated theories, all of which turn out
to reveal different faces of the periodicity block theory. You could
call it a kind of "grand unified theory" -- but that shouldn't
prevent us from looking outside it for other avenues of inquiry.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> Well, you might want to ask Gene about this observation of mine -- he
> seems to have gone deeper into the superparticularity question than
> the rest of us . . .

The first thing to observe is that this can never be true in general, since there are only a finite number of superparticulars in any p-limit; consequently, it will only be the relatively (compared to p) small scales that it will work for.

The second thing to observe is that when looking at smaller scales, which means larger sizes of intervals, the superparticulars are very important, and it is not surprising that we should be able to do this in many cases.

The third thing to observe is that for any p, there are always JI p-limit scales with superparticular scale steps.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Well, you might want to ask Gene about this observation of mine --
he
> > seems to have gone deeper into the superparticularity question
than
> > the rest of us . . .
>
> The first thing to observe is that this can never be true in
>general, since there are only a finite number of superparticulars in
>any p-limit; consequently, it will only be the relatively (compared
>to p) small scales that it will work for.

Yes -- we haven't looked at too many huge ones, but the 13-limit 41-
tone scales I was creating for Justin White, and even Partch's scale,
have all superparticular step sizes! Whether that's important, well,
I just see it as a 'neat curiosity' until someone convinces me
otherwise.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> Yes -- we haven't looked at too many huge ones, but the 13-limit 41-
> tone scales I was creating for Justin White, and even Partch's scale,
> have all superparticular step sizes! Whether that's important, well,
> I just see it as a 'neat curiosity' until someone convinces me
> otherwise.

In theory the superparticular p-limit blocks (in the sense of epimorphic) could be enumerated, since they would be finite in number. That might be interesting to attempt in the 5-limit, at least.
I'll start with with JI pentatonic and the major and minor forms of JI diatonic. :)

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > J Gill: Monz, do you understand what Kraig means
> > to say by presenting the following posts/link?
> >
> > /tuning/topicId_31138.html#31297
> > Kraig Grady:
> > << here is a paper on epimores. page 3 should explain the previous
> > http://www.anaphoria.com/epimore.pdf >>
> >
> >
> > /tuning/topicId_31138.html#31388
> > Kraig Grady:
> > << Page 2 is the epimores between the intervals on Page 1 Page 3
> shows this in
> > one
> > diagram although
> > the last line is messed up you can see it on page two. In other
> words adjacent
> > figures between
> > terms on any particular level of the pierce sequence form
> >superparticular
> > ratios these two can be
> > arranged into their own tree as seen on page 2 >>
>
> I thought I explained this to you already. You should have said
> something if my explanation wasn't good enough. What this shows is
> that, just like the Farey series and similar sets, the set of >ratios
> above some layer in the Stern-Brocot tree will have all its _step
> sizes_ be superparticular ratios. Didn't I get that across to you
> already? Please let me know how I can communicate more effectively.

J Gill: Paul, at www.anaphoria.com/epimore.pdf on page [1] I see what appears to commonly referred to as "Stern-Brocot" tree. It is labeled "Peirce Sequence" (a Google search reveals one link:
http://www.musicalonline.com/scholarly_works.htm where a dead link exists http://www.anaphoria.com/xen3b.html).

On page [2] I see a clearly different sequence in which each the *node* (expressed in lowest terms) appears to be equal to the *product* of the two "nodes" [the nodes which are directly "below", and branching out "from" (geometrically sub-dividing) the (upper, originally decribed) "node"

On page [3] I see these two structures (apparantly) "super-imposed" upon each other. Kraig's statement, "Page 2 is the epimores between the intervals on Page 1".

In message: /tuning/topicId_31138.html#31144

Kraig stated, "Now epimores exist between adjacent terms of the stern brocot tree (also referred to as the scale tree) implying that it is
also the result of using geometric means.

The "epimores" (it seems) refers to such "superparticular" step-sizes existing between (certain?) "Farey-adjacent" (defined as utilized on the tuning list) "nodal-ratios" in the Stern-Brocot tree structure.

Your statement above, "the set of ratios above some layer in the
Stern-Brocot tree will have all its _step sizes_ be
superparticular ratios." (appears) to indicate that this property ceases to exist at some point farther "downwards" in the S-B Tree.
I thought that the "Farey-adjacence" between such S-B "nodal-ratios" continued ad infinitum as one traverses farther and farther down any "sub-branch" of the (typically depicted as "downward" branching tree). Is this untrue?

Curiously, J Gill :)

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > J Gill: Lurking it may be, but I have yet to find
> > much of anything on these tuning lists which does
> > much more than explore "periodicity-block" theory.

>PE:I'm sure a lot of people wish that you couldn't find that either.

JG: In speaking only for myself - while the PB stuff is really slick, the linear algebra is really dazzling, *since* my personal interests are less oriented towards tempering scales [for which these tools are really "cutting edge" (as you once stated)], and are more along the lines of exploring the tonal character of the presence of (and relationships between) the "tonal interplay" (between two or more simultaneously sounded "complex" tones) which results in a given "timbre", as applied to the application of (rational numbered) scale-pitches, PB stuff is powerful, but does not appear to be very descriptive of the complicated "harmonic relationships" which evolve out of the utilization of such (rational numbered) scale-pitches, where the tones played are "complex" (as opposed to sinusoidal) in nature. This speaks more to my interest in modelling and describing patterns and form in the resultant "timbre" of the final "product", as opposed to being any sort of "indictment" of either PB stuff, or the respectable art of tempering scales.

> >JG: Maybe these lists could actually explore something
> > more than that someday.

> PE: What do you have in mind?

JG: (My personal) interest would be more exploration of the harmonic interplay which exists between the scale-pitches ultimately selected within a given scenario (thus further characterizing the "reduction to practice" of a given scale, whether it be JI, RI, or even ETs).

It's great that folks *occaisionally* express their (calculated or suspected) beliefs about "partials or overtones 'locking-in'" in JI/RI systems. I "just" (bad pun) wish that there was more interest and efforts (but who am I to say?) towards attention to the omprehensive (as opposed to the sometimes sparse, and anecdotal more than specific) tracking of these "timbral" implications of combining such "complex" (as opposed to sinusoidal) tones in harmony (as well as in melody, if such theoretical or experiential knowledge exists).

I realize that a tremendous amount of complexity appears (to the observer) to come in to play in such applied situations, and that the whole business (with all of its overtone energies, "undertone constructs", beatings, roughness (sounds more like S&M than music, eh?), "entropy", "coincidences", summations and cancellations, "implied fundamentals", spectral effects of the "complex" tone's amplitude envelopes' resulting in "continuous spectral energies", the physio-neuro-psycho-logical non-linearities measured in human perception, and sum and difference tones which will result from simultaneously sounded sinusoids over the very wide range of 60 dB (a linear factor of 1000) below the "threshold of pain", etc, etc, ... represents a real "sticky wicket" for aspiring
"mathematical analysts"...

The "consonances" found in scales seem to revolve around finding 3/2 intervals between scale-pitches (or triads and tetrads similarly found). Having done some of this, it makes perfect practical sense.

What I (perhaps) dream of, is further definitions of (and descriptions of) "affinities" between all of the pitches within a given scale. I seriously doubt that any subjective agreement is even possible in the subjectives spheres of the perceptions of our (individual) "aural minds". So, I (in a form similar, but not in the same manner as you) endeavor to characterize these numerical relationships in ways which (rather than representing a dynamic spectral time-varying forest which clearly overwhelms the eye) bring to light topological structures which may lend insight into why (my) "aural mind" has found this or that preferable in my playing of music. Whether such systems can successfully *predict* my (or anyone else's) reaction to movements and choices *dictated* by such atomistic and theoretical endeavors remains a far more open question.

I (like you) would love to "sum it all up" with elegant and beautiful mathematical constructs which somehow manage to avoid the very complicated spectral results which clearly accompany the reduction to practice of such implementations of given scales on given instruments.

But I would be fooling myself to think that Nature has kindly hidden away some "golden fleece" just for us, if we only *think* hard enough. A lot of the process of understanding such complicated, interwoven systems and phenomena rests upon endeavoring to *look* at the situation from many different viewpoints (as you point out below).

Whether any of us will, by pure gumption, "unify" all these many and varied things under some "umbrella" conceptualizations, is doubtful.

And, there is always the old "NIH" (not-invented-here) syndrome, which tends to cause us to gravitate to a global-viewpoint seen "in and through" our favorite personal metaphors for the "side of the elephant" which we have happened, in our personal thoughts, to spent a lot of time looking through... Much like the effect of listening to a tape of oneself "improvising" in a particular performance numerous times, until one finds (upon attempting to "improvise" again over the same musical piece) that one has become accustomed to the "sound of their own voice" (so to speak, or one might say, so to *play*), and finds it *harder* to *continue* to "improvise" [which seems to require that (I as a) *player*, as well as the "audience", not "impose" a predictable pattern, or outcome upon one's *future* choices of timing, tone, and interaction with the other "voices".

Sometimes it seems "hack musicians" (like me) aspire to be mathematicians, or "hack mathematicians" (like me) aspire to be musicians. Being a bit of both (and utilizing *entirely different* "areas" of my "mind" in order to approach these
highly different "areas"), I am skeptical that I could (even if I really, really wanted to) ever meaningfully equate the two areas. In the very least, the *same* person has to execute the "doing", as well as the "thinking", about such ethereal relationships in order to (even) adequately develop a (private) "in-dwelling" within the "theory as reduced to practice".

I spent about 5 years designing and developing a professional audio signal processor (developed around guitar, but more generally targeted) with the goal of improving the "sustain/distortion" ratio, and all without degrading the dynamics of the amplitude envelope of the musical signal to be processed. In the non-linear world of electronic signal processing (true for any device which alters the signal path gain or frequency response in time), the electronic designer in me spent *years* endlessly tweaking a large number of complicated and interacting "input variables" (such as transfer functions and frequency responses) in a "perspiration-dominated" process of theorizing, modifying, then actually *using* my creation in the process of ... *pleasing (my own) ear*.

The idea was to, in addition, try to "draw a line" representing a "best fit" through the (inherently subjective) "datapoints" and "preferences" on a "global basis" (the likes/dislikes of others, as well as myself). The goal was a simple, "one-knob", extremely easy to use device which would spectrally enhance the musicians sound, yet not suddenly sound like distorted crap, and would not output peak levels which would cause (the unkown and uncontrollable) overloading of subsequent equipment in a signal path. Not a small order, indeed!

After about 5 years of humbling my "left-lobe" to the poetic demands of my "right-lobe", and attempting to average over a wide range of musical contexts, all the while obsessing over every *detail* of the quality and duration of the *tone* (where *every* detail matters!!!), found that some world-class musicians (in using my processor in live performances) not only *said* (to me and others) that they really liked how they sounded "through it", but they (also) wanted to use it again, regularly (a profound and humbling joy for me).

Lest your visions of "mosh-pits" beneath bare-chested, long-haired swinging metal-heads making peoples ears' bleed through excessively overdriven Marshall Amp's, where the PA doubles as a "fuzz-tone" too... overtake you at this point, I will say these were not people unaquainted with melody and tonality (jazz greats Herb Ellis, Larry Coryell, Howard Roberts; blues greats Buddy Guy, Elvin Bishop, and John Cipollina). What a thrill it was to (like a luthier of sorts) enable them to find the sweet, resonant string vibrations deeper within their instruments, to see such greats find such a useful tool with which to enhance their subtle low-level nuances, while then launching into rapid arpeggiated chords without the penalties to be paid in the area of the creation of (undesired) distortion products.

I would have *never* have even pleased myself, much less please these others, had I not been in a continual "feedback-loop" between the(focal, technical, calculating) portion of "mind" endlessly adapting to the "mental rubber" meeting the "sonic road" of *listening*. I still cannot *really* define the *reasons* why *others* may have found a friend of ringing strings in my silicon hardware, I still only have [1] My "aural minds" view; [2] The smiles on their face, and the beautiful sounds *they* created; and [3] last and, as well, least, my minds view of the systemic topology and parameters.

It could not have evolved without [3], but [2] in conjunction with [1] still makes the entire experience something much sweeter and real (from the context of music *followed* by thought, and not the other way around).

> > There may be more than on
> > way to "skin the cat" (but we may never know)...
>
> PE:Indeed there are . . . Between Tanaka, Wurschmidt, Wilson, etc., >we
> have tons of differently formulated theories, all of which turn out
> to reveal different faces of the periodicity block theory. You >could
> call it a kind of "grand unified theory" -- but that shouldn't
> prevent us from looking outside it for other avenues of inquiry.

JG: I cannot help but listen the "hardest" to those who have made such sounds, with various timbres, and actuality explored and developed systems of tonal choice. The (subjective) vernaculars which folks develop in order to help communicate the character of musical sound(s) are (in themselves) amazing to attempt to relate to as a removed, analytical party.

Clearly, we may (doubly) lose reference by "thinking" (conceptually)about *other* players (post-performance-perceptions). Only those who master both the analytical as well presentational aspects of music can hope to, having narrowed it down to half as many uncertainties by actually being the one who *does* it, as well as (then) pondering...

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

>
> J Gill: Paul, at www.anaphoria.com/epimore.pdf on page [1] I see
what appears to commonly referred to as "Stern-Brocot" tree. It is
labeled "Peirce Sequence" (a Google search reveals one link:
> http://www.musicalonline.com/scholarly_works.htm where a dead link
exists http://www.anaphoria.com/xen3b.html).
>
> On page [2] I see a clearly different sequence in which each the
*node* (expressed in lowest terms) appears to be equal to the
*product* of the two "nodes" [the nodes which are directly "below",
and branching out "from" (geometrically sub-dividing) the (upper,
originally decribed) "node"
>
> On page [3] I see these two structures (apparantly) "super-imposed"
upon each other. Kraig's statement, "Page 2 is the epimores between
the intervals on Page 1".
>
> In message: /tuning/topicId_31138.html#31144
>
> Kraig stated, "Now epimores exist between adjacent terms of the
stern brocot tree (also referred to as the scale tree) implying that
it is
> also the result of using geometric means.
>
> The "epimores" (it seems) refers to such "superparticular" step-
sizes existing between (certain?) "Farey-adjacent" (defined as
utilized on the tuning list) "nodal-ratios" in the Stern-Brocot tree
structure.

Sounds about right . . . the set of ratios forming the sequence in
question, in which all pairs of consecutive terms are "Farey-
adjacent", is the set of all ratios in the Stern-Brocot tree above
any given level.

> Your statement above, "the set of ratios above some layer in the
> Stern-Brocot tree will have all its _step sizes_ be
> superparticular ratios." (appears) to indicate that this property
ceases to exist at some point farther "downwards" in the S-B Tree.

No, that's not what I meant. I meant you can choose any cutoff level
you want, no matter how far dowm it is, and the set of all ratios
_above_ that level, when brought into a consecutive sequence, will
have the property that every pair of consecutive ratios is Farey-
adjacent; i.e., a superparticular ratio will separate the members of
each consecutive pair.

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> JG: In speaking only for myself - while the PB stuff is really
>slick, the linear algebra is really dazzling, *since* my personal
>interests are less oriented towards tempering scales [for which
>these tools are really "cutting edge" (as you once stated)]

I don't know . . . I see PBs as describing, in the most unified
possible way, the vast majority of JI, untempered systems that have
ever been devised.

>, and are more along the lines of exploring the tonal character of
>the presence of (and relationships between) the "tonal interplay" >
(between two or more simultaneously sounded "complex" tones) which
>results in a given "timbre", as applied to the application of >
(rational numbered) scale-pitches, PB stuff is powerful, but does not
>appear to be very descriptive of the complicated "harmonic
>relationships" which evolve out of the utilization of such (rational
>numbered) scale-pitches, where the tones played are "complex" (as
>opposed to sinusoidal) in nature.

I don't see why you say that, but I'd love to help you develop
whatever you need to understand whatever you want to understand.

> > >JG: Maybe these lists could actually explore something
> > > more than that someday.
>
> > PE: What do you have in mind?
>
> JG: (My personal) interest would be more exploration of the
>harmonic interplay which exists between the scale-pitches ultimately
>selected within a given scenario (thus further characterizing
>the "reduction to practice" of a given scale, whether it be JI, RI,
>or even ETs).

I don't see this as foreign to the kind of things we do on this list
or on tuning-math at all. Care to put forward an example?

> It's great that folks *occaisionally* express their (calculated or
>suspected) beliefs about "partials or overtones 'locking-in'" in
>JI/RI systems. I "just" (bad pun) wish that there was more interest
>and efforts (but who am I to say?) towards attention to the
>omprehensive (as opposed to the sometimes sparse, and anecdotal more
>than specific) tracking of these "timbral" implications of combining
>such "complex" (as opposed to sinusoidal) tones in harmony (as well
>as in melody, if such theoretical or experiential knowledge exists).

Again, I'd love to help with the theory of this. It really doesn't
seem to imply anything outside our usual realms of investigation,
though I'd be happy to be proved wrong.

> I realize that a tremendous amount of complexity appears (to the
>observer) to come in to play in such applied situations, and that
>the whole business (with all of its overtone energies, "undertone
>constructs", beatings, roughness (sounds more like S&M than music,
>eh?), "entropy", "coincidences", summations and
>cancellations, "implied fundamentals", spectral effects of
>the "complex" tone's amplitude envelopes' resulting in "continuous
>spectral energies", the physio-neuro-psycho-logical non-linearities
>measured in human perception, and sum and difference tones which
>will result from simultaneously sounded sinusoids over the very wide
>range of 60 dB (a linear factor of 1000) below the "threshold of
>pain", etc, etc, ... represents a real "sticky wicket" for aspiring
> "mathematical analysts"...

Perhaps . . . music theorists tend to make simplifying assumptions,
aided by lots of empirical validation.

> The "consonances" found in scales seem to revolve around finding
>3/2 intervals between scale-pitches (or triads and tetrads similarly
>found). Having done some of this, it makes perfect practical sense.

That's why latticing is so valuable.
>
> What I (perhaps) dream of, is further definitions of (and
>descriptions of) "affinities" between all of the pitches within a
>given scale. I seriously doubt that any subjective agreement is even
>possible in the subjectives spheres of the perceptions of our >
(individual) "aural minds". So, I (in a form similar, but not in the
>same manner as you) endeavor to characterize these numerical
>relationships in ways which (rather than representing a dynamic
>spectral time-varying forest which clearly overwhelms the eye) bring
>to light topological structures which may lend insight into why >
(my) "aural mind" has found this or that preferable in my playing of
>music.

And I've participated in some discussions with you in which I
critiqued your methods here and proposed others. Where are we with
that?
>>
> After about 5 years of humbling my "left-lobe" to the poetic
>demands of my "right-lobe", and attempting to average over a wide
>range of musical contexts, all the while obsessing over every
>*detail* of the quality and duration of the *tone* (where *every*
>detail matters!!!), found that some world-class musicians (in using
>my processor in live performances) not only *said* (to me and
>others) that they really liked how they sounded "through it", but
>they (also) wanted to use it again, regularly (a profound and
>humbling joy for me).

This sounds like something I'd be interested in! Send me info.
>
> I would have *never* have even pleased myself, much less please
>these others, had I not been in a continual "feedback-loop" between
>the(focal, technical, calculating) portion of "mind" endlessly
>adapting to the "mental rubber" meeting the "sonic road" of
>*listening*.

I'm exactly the same way when it comes to music theory.

> JG: I cannot help but listen the "hardest" to those who have made
>such sounds, with various timbres, and actuality explored and
>developed systems of tonal choice.

Well, here we are!

> Clearly, we may (doubly) lose reference by "thinking" (conceptually)
>about *other* players (post-performance-perceptions). Only those who
>master both the analytical as well presentational aspects of music
>can hope to, having narrowed it down to half as many uncertainties
>by actually being the one who *does* it, as well as (then)
>pondering...

AGREED 100%!!!