Yahoo Tuning Groups Ultimate Backup tuning What's so Super about Superparticularity?

I recently did a Google search for the term "superparticular", and (as it appears to me) found no concrete statements in any of the search results as to *WHY* "superparticularity" would impart any "particularly" special quality to a musical interval - when sounded in a dyad with a 1/1 reference tone.

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.

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...

If the significance is one of "Farey adjacence" (where the absolute value of N1*D2 - D1*N2 = 1), then it seems that the sequence 3/2, 5/3, 7/4, 9/5 (which also possesses this characteristic), while not made up of "superparticular" ratios, would also possess similar properties...

What, then, would those "magic" properties be? What desirable harmonic structures (relative to the 1/1) are implied by either "superparticularity" or "Farey adjacence"? If a "magic" property derives from the resulting sum and difference frequencies of linear combinations of such (sinusoidal or complex) tones, what are the significant sum and/or difference frequencies (relative to the 1/1)?

Hoping to hear a diversity of individual opinions, J Gill

🔗klaus schmirler <KSchmir@z.zgs.de>

J Gill schrieb:
>
> I recently did a Google search for the term "superparticular", and (as it
> appears to me) found no concrete statements in any of the search results as
> to *WHY* "superparticularity" would impart any "particularly" special
> quality to a musical interval - when sounded in a dyad with a 1/1
> reference tone.
>
> 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.

I might not understand this (well, I don't), so my answer
will start in a different place:

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.

klaus schmirler

> 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...
>
> If the significance is one of "Farey adjacence" (where the absolute value
> of N1*D2 - D1*N2 = 1), then it seems that the sequence 3/2, 5/3, 7/4, 9/5
> (which also possesses this characteristic), while not made up of
> "superparticular" ratios, would also possess similar properties...
>
> What, then, would those "magic" properties be? What desirable harmonic
> structures (relative to the 1/1) are implied by either "superparticularity"
> or "Farey adjacence"? If a "magic" property derives from the resulting sum
> and difference frequencies of linear combinations of such (sinusoidal or
> complex) tones, what are the significant sum and/or difference frequencies
> (relative to the 1/1)?
>
> Hoping to hear a diversity of individual opinions, J Gill
>
>
>
>
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🔗Afmmjr@aol.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

J Gill!
First you make an assumption which i do not believe is true. That the magic of
superparticulars is in their relationship to a 1/1. To the greeks and many other scale builders,
it's recognition lies in the relationship of adjacent tones not to a single fixed point. It is
usually not done as a deliberate method of construction, but when one constructs a scale and finds
all the terms epimoric (another term for superparticular) one usually finds the results to ones
liking.

I think that the reason they work is because these intervals imply no further subdivisions as
if one has reach a stopping place.
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.

Their magic is that they impose themselves on us (scale builders working in ratios) not the
other way around.
The CPS more often results in epimores (except at the point where the two opposite functions are
adjacent) and the diamond/ lambdomas likewise. in the latter case due to the size of larger
epimores, scale builders have subdivided them into epimoric units.
I have yet to construct a Constant Structure scale that did not produce them yet i will leave it
to others to prove it can be done.

That a mathematical explanation has so far escaped satisfaction shows possibly how little is
understood of even the most basic building blocks used for centuries and as we zoom ahead into a
myriad of complex scales and temperments, i imagine that these tower of Babel's might collapse
when such thing are understood.

J Gill wrote:

> I recently did a Google search for the term "superparticular", and (as it
> appears to me) found no concrete statements in any of the search results as
> to *WHY* "superparticularity" would impart any "particularly" special
> quality to a musical interval - when sounded in a dyad with a 1/1
> reference tone.
>
> 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.
>
> 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...
>
> If the significance is one of "Farey adjacence" (where the absolute value
> of N1*D2 - D1*N2 = 1), then it seems that the sequence 3/2, 5/3, 7/4, 9/5
> (which also possesses this characteristic), while not made up of
> "superparticular" ratios, would also possess similar properties...
>
> What, then, would those "magic" properties be? What desirable harmonic
> structures (relative to the 1/1) are implied by either "superparticularity"
> or "Farey adjacence"? If a "magic" property derives from the resulting sum
> and difference frequencies of linear combinations of such (sinusoidal or
> complex) tones, what are the significant sum and/or difference frequencies
> (relative to the 1/1)?
>
> Hoping to hear a diversity of individual opinions, J Gill

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗unidala <JGill99@imajis.com>

Kraig,

Thanks for your thoughts on this!

JG: I just thought that a (normalized) numerical ratio of 1/1 was a
good place to start (as one position in a possible rotation of a
reference pitch), without precluding a potential significance in the
ratiometric relationships existing between one or more of the other
tones (when sounded) which may also exist within a given scale...

> KG: To the greeks and many other scale builders,
> it's recognition lies in the relationship of adjacent tones not to
> a single fixed point.

JG: This sounds reasonable to me, as we are not restricted to playing
only dyadic chords where one note is always a given reference pitch!

> KG: It is
> usually not done as a deliberate method of construction, but when
> one constructs a scale and finds
> all the terms epimoric (another term for superparticular) one
usually finds the results to ones
> liking.
>
> I think that the reason they work is because these intervals
imply no further subdivisions as
> if one has reach a stopping place.

JG: In the segment of the Stern-Brocot tree branching (upwards,
generally) into the 3/2 node (then following on upwards to the 2/1
node, and the 1/1 node), the 6/5 and 9/7 nodes have a mediant of 5/4,
the 5/4 and 7/5 nodes have a mediant of 4/3, and the 4/3 and 5/3
nodes have a mediant of 3/2. However, in a "mirror-image" pattern
(centered about the 3/2 node), the "Farey adjacent" but
not "superparticular" 9/5 and 12/7 nodes have a mediant of 7/4, the
7/4 and 8/5 nodes have a mediant of 5/3, and (reduntantly), the 5/3
and the 4/3 nodes also have a mediant of 3/2. Can these two "mirror-
image" groups (where both groups contain nodal ratio values which
are "Farey adjacent" to each other traveling upwards or downwards
along the branches involved, but only the *first* group cited above
contains nodal ratio values which are "superparticular") be
considered musically equivalent (from the perspective of desirability
when combined with a 1/1, or other ratios present in a given scale),
or does the superparticularity of such ratios impart a preferable
quality as compared to the other non-superparticular ratios which one
might select to include within a given scale. Or (along the lines of
your original thought), would you refrain from judging the
desirability of a given superparticular ratio in a scale until you
had evaluated its use in conjunction with all *other* scale ratios?

> KG: 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.

JG: Do you mean as 25/24 is equal to the ratio of 5/4 over 6/5, 16/15
is equal to the ratio of 4/3 over 5/4, 9/8 is equal to the ratio of
3/2 over 4/3, and 4/3 is equal to the ratio of 2/1 over 3/2 ?

> KG: Their magic is that they impose themselves on us (scale
builders working in ratios) not the
> other way around.
> The CPS more often results in epimores (except at the point where
the two opposite functions are
> adjacent) and the diamond/ lambdomas likewise. in the latter case
due to the size of larger
> epimores, scale builders have subdivided them into epimoric units.
> I have yet to construct a Constant Structure scale that did not
produce them yet i will leave it
> to others to prove it can be done.

JG: Would you say that ratios such as 5/3, 7/4, and 9/5 (these nodes
occupying a "mirror-image" postion, centered around the 3/2 node, to
the ratios 4/3, 5/4, and 6/5 in the scale tree) are less prevalent in
appearance than the ratios 4/3, 5/4, and 6/5, in your aboved
described situations (relating to CPS and CS)? I am speaking here of
the resultant ratios which describe the pitches of the notes of a
given derived scale (as opposed to the ratiometric steps between such
pitches, or the products of such pitches).

> KG: That a mathematical explanation has so far escaped
satisfaction shows possibly how little is
> understood of even the most basic building blocks used for
centuries and as we zoom ahead into a
> myriad of complex scales and temperments, i imagine that these
tower of Babel's might collapse
> when such thing are understood.

> -- Kraig Grady
> North American Embassy of Anaphoria island
> http://www.anaphoria.com
>
> The Wandering Medicine Show
> Wed. 8-9 KXLU 88.9 fm

JG: Do you believe that ratios which describe the pitch of a given
note of a scale (where superparticular, but comprised of large valued
integers in the numerator and/or denominator, as opposed small valued
integers) are any more (or less) "useful" than non-superparticular
ratios which are similarly comprised of large valued integers in the
numerator and/or denominator? Or is such "utility" only evaluatable
with reference to such a ratio describing the pitch of a given note
of a scale as it inter-relates in practice in conjunction with
some/all of the scale's *other* pitches?

Thank you much for your time and consideration, J Gill
>
__________________________________

> J Gill (previously) wrote:
>
> > I recently did a Google search for the term "superparticular",
and (as it
> > appears to me) found no concrete statements in any of the search
results as
> > to *WHY* "superparticularity" would impart any "particularly"
special
> > quality to a musical interval - when sounded in a dyad with a 1/1
> > reference tone.
> >
> > 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.
> >
> > 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...
> >
> > If the significance is one of "Farey adjacence" (where the
absolute value
> > of N1*D2 - D1*N2 = 1), then it seems that the sequence 3/2, 5/3,
7/4, 9/5
> > (which also possesses this characteristic), while not made up of
> > "superparticular" ratios, would also possess similar properties...
> >
> > What, then, would those "magic" properties be? What desirable
harmonic
> > structures (relative to the 1/1) are implied by
either "superparticularity"
> > or "Farey adjacence"? If a "magic" property derives from the
resulting sum
> > and difference frequencies of linear combinations of such
(sinusoidal or
> > complex) tones, what are the significant sum and/or difference
frequencies
> > (relative to the 1/1)?
> >
> > Hoping to hear a diversity of individual opinions, J Gill

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., J Gill <JGill99@i...> wrote:
> I recently did a Google search for the term "superparticular", and
(as it
> appears to me) found no concrete statements in any of the search
results as
> to *WHY* "superparticularity" would impart any "particularly"
special
> quality to a musical interval - when sounded in a dyad with a 1/1
> reference tone.

Hi J!

I'm shocked that Gene hasn't replied to this -- he did an intensive
study over at tuning-math@yahoogroups.com . . .

I'd recommend checking in with him over there, if you haven't already.

>
> 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.

I agree with this statement.
>
> 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...

Nothing, in itself . . .

>
> If the significance is one of "Farey adjacence" (where the absolute
value
> of N1*D2 - D1*N2 = 1), then it seems that the sequence 3/2, 5/3,
7/4, 9/5
> (which also possesses this characteristic), while not made up of
> "superparticular" ratios, would also possess similar properties...

Hmm . . . it seems to me that the ratio of every pair of neighboring
ratios in a Farey series, or Mann series, or Tenney series, is
superparticular.

But ultimately, I think Gene and I would agree with you that
superparticulars are _not_ in themselves special; that the reason
they appear "special" is that most of the commas in a given JI system
do turn out to be superparticular. Superparticulars have some unique
properties that I've been trying to investigate over at tuning-math.

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> J Gill!
> First you make an assumption which i do not believe is true.
That the magic of
> superparticulars is in their relationship to a 1/1. To the greeks
and many other scale builders,
> it's recognition lies in the relationship of adjacent tones not to
a single fixed point. It is
> usually not done as a deliberate method of construction, but when
one constructs a scale and finds
> all the terms epimoric (another term for superparticular) one
usually finds the results to ones
> liking.

I agree. From my point of view, most interesting scales, when
detempered, can be detempered into a scale with epimoric intervals
between adjacent scale degrees.

> I think that the reason they work is because these intervals
imply no further subdivisions as
> if one has reach a stopping place.

That's correct when, for example, looking at the set of all ratios at
a given harmonic distance from 1/1.
>
> Their magic is that they impose themselves on us (scale
builders working in ratios) not the
> other way around.

Yup.

> I have yet to construct a Constant Structure scale that did not
produce them yet i will leave it
> to others to prove it can be done.

A CS scale that did not have _any_? It could be concocted, but it
would be very contrived.

> That a mathematical explanation has so far escaped satisfaction

Have you checked out tuning-math lately?

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

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

> JG: In the segment of the Stern-Brocot tree branching (upwards,
> generally) into the 3/2 node (then following on upwards to the 2/1
> node, and the 1/1 node), the 6/5 and 9/7 nodes have a mediant of
5/4,
> the 5/4 and 7/5 nodes have a mediant of 4/3, and the 4/3 and 5/3
> nodes have a mediant of 3/2. However, in a "mirror-image" pattern
> (centered about the 3/2 node), the "Farey adjacent" but
> not "superparticular" 9/5 and 12/7 nodes have a mediant of 7/4, the
> 7/4 and 8/5 nodes have a mediant of 5/3, and (reduntantly), the 5/3
> and the 4/3 nodes also have a mediant of 3/2. Can these two "mirror-
> image" groups (where both groups contain nodal ratio values which
> are "Farey adjacent" to each other traveling upwards or downwards
> along the branches involved, but only the *first* group cited above
> contains nodal ratio values which are "superparticular") be
> considered musically equivalent (from the perspective of
desirability
> when combined with a 1/1, or other ratios present in a given
scale),
> or does the superparticularity of such ratios impart a preferable
> quality as compared to the other non-superparticular ratios which
one
> might select to include within a given scale. Or (along the lines
of
> your original thought), would you refrain from judging the
> desirability of a given superparticular ratio in a scale until you
> had evaluated its use in conjunction with all *other* scale ratios?

Hello J . . .

I'd _really_ like to try to answer your questions here, but I'm
having trouble following you at the moment. It seems that there is a
huge amount of infomation and knowledge behind them, and you have a
great number of things to express that you're trying to get across
here . . .

Could I ask you to try to expand on these things in as slow and clear
language as you can? The problem is that we all come at this stuff as
near-isolates so have all come up with our own languages . . . I'd
really like to get a chance to really talk with you about your ideas.

-P

>
> > KG: 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.
>
> JG: Do you mean as 25/24 is equal to the ratio of 5/4 over 6/5,
16/15
> is equal to the ratio of 4/3 over 5/4, 9/8 is equal to the ratio of
> 3/2 over 4/3, and 4/3 is equal to the ratio of 2/1 over 3/2 ?
>
> > KG: Their magic is that they impose themselves on us (scale
> builders working in ratios) not the
> > other way around.
> > The CPS more often results in epimores (except at the point
where
> the two opposite functions are
> > adjacent) and the diamond/ lambdomas likewise. in the latter case
> due to the size of larger
> > epimores, scale builders have subdivided them into epimoric units.
> > I have yet to construct a Constant Structure scale that did not
> produce them yet i will leave it
> > to others to prove it can be done.
>
> JG: Would you say that ratios such as 5/3, 7/4, and 9/5 (these
nodes
> occupying a "mirror-image" postion, centered around the 3/2 node,
to
> the ratios 4/3, 5/4, and 6/5 in the scale tree) are less prevalent
in
> appearance than the ratios 4/3, 5/4, and 6/5, in your aboved
> described situations (relating to CPS and CS)? I am speaking here
of
> the resultant ratios which describe the pitches of the notes of a
> given derived scale (as opposed to the ratiometric steps between
such
> pitches, or the products of such pitches).
>
> > KG: That a mathematical explanation has so far escaped
> satisfaction shows possibly how little is
> > understood of even the most basic building blocks used for
> centuries and as we zoom ahead into a
> > myriad of complex scales and temperments, i imagine that these
> tower of Babel's might collapse
> > when such thing are understood.
>
> > -- Kraig Grady
> > North American Embassy of Anaphoria island
> > http://www.anaphoria.com
> >
> > The Wandering Medicine Show
> > Wed. 8-9 KXLU 88.9 fm
>
> JG: Do you believe that ratios which describe the pitch of a given
> note of a scale (where superparticular, but comprised of large
valued
> integers in the numerator and/or denominator, as opposed small
valued
> integers) are any more (or less) "useful" than non-superparticular
> ratios which are similarly comprised of large valued integers in
the
> numerator and/or denominator? Or is such "utility" only evaluatable
> with reference to such a ratio describing the pitch of a given note
> of a scale as it inter-relates in practice in conjunction with
> some/all of the scale's *other* pitches?
>
>
> Thank you much for your time and consideration, J Gill
> >
> __________________________________
>
>
> > J Gill (previously) wrote:
> >
> > > I recently did a Google search for the term "superparticular",
> and (as it
> > > appears to me) found no concrete statements in any of the
search
> results as
> > > to *WHY* "superparticularity" would impart any "particularly"
> special
> > > quality to a musical interval - when sounded in a dyad with a
1/1
> > > reference tone.
> > >
> > > 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.
> > >
> > > 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...
> > >
> > > If the significance is one of "Farey adjacence" (where the
> absolute value
> > > of N1*D2 - D1*N2 = 1), then it seems that the sequence 3/2,
5/3,
> 7/4, 9/5
> > > (which also possesses this characteristic), while not made up of
> > > "superparticular" ratios, would also possess similar
properties...
> > >
> > > What, then, would those "magic" properties be? What desirable
> harmonic
> > > structures (relative to the 1/1) are implied by
> either "superparticularity"
> > > or "Farey adjacence"? If a "magic" property derives from the
> resulting sum
> > > and difference frequencies of linear combinations of such
> (sinusoidal or
> > > complex) tones, what are the significant sum and/or difference
> frequencies
> > > (relative to the 1/1)?
> > >
> > > Hoping to hear a diversity of individual opinions, J Gill

🔗unidala <JGill99@imajis.com>

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

> Hello J . . .
>
> I'd _really_ like to try to answer your questions here, but I'm
> having trouble following you at the moment. It seems that there is
a
> huge amount of infomation and knowledge behind them, and you have a
> great number of things to express that you're trying to get across
> here . . .

JG: You are too kind. "Just" making some observations...

> Could I ask you to try to expand on these things in as slow and
clear
> language as you can? The problem is that we all come at this stuff
as
> near-isolates so have all come up with our own languages . . . I'd
> really like to get a chance to really talk with you about your
ideas.
>
> -P

JG: With the "new-fangled" brute-force solution to the Yahoo "de-
formatting-machine" below, perhaps my (modified) text will make more
sense:
----------------------3/2
---------------------/--\
--------------------/----\
-------------------/------\
------------------/--------\
-----------------/----------\
----------------/------------\
---------------/--------------\
--------------/----------------\
-------------/------------------\
------------/--------------------\
-----------4/3------------------5/3
----------/---\----------------/---\
---------/-----\--------------/-----\
--------/-------\------------/-------\
-------/---------\----------/---------\
------5/4--------7/5-------8/5--------7/4
-----/--\-------/--\------/--\-------/--\
---6/5--9/7--11/8-10/7-11/7--13/8-12/7--9/5

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

> However, in a "mirror-image" pattern
> (centered about the 3/2 node), the "Farey adjacent" but
> not "superparticular" 9/5 and 12/7 nodes have a mediant of 7/4, the
> 7/4 and 8/5 nodes have a mediant of 5/3, and (reduntantly), the 5/3
> and the 4/3 nodes also have a mediant of 3/2. THE NODAL RATIOS 9/5,
7/4, 5/3, AND 3/2 DO *NOT* FORM A "SUPERPARTICULAR CHAIN" OF NODAL
RATIOS ALONG THE EDGE OF THIS SUB-BRANCH.

> Can these two "mirror-
> image" groups (where BOTH groups contain nodal ratio values which
> are "Farey adjacent" to each other traveling upwards or downwards
> along the branches involved, but only the *FIRST* group cited above
> contains nodal ratio values which are "superparticular") be
> considered musically EQUIVALENT (FROM THE PERSPECTIVE OF
DESIRABILITY AS CHOICES OF VALUES OF "SCALE INTERVALS" THEMSELVES)
> when combined with a 1/1, or other ratios present in a given
scale),
> or does the superparticularity of such ratios impart a PREFERABLE
> quality as compared to the other non-superparticular "SCALE
INTERVAL RATIOS" which one
> might select to include within a given scale.

Or (along the lines of - WHAT I INTERPRETED TO BE KRAIG'S -

original thought), would (KRAIG) refrain from judging the
desirability of a given superparticular ratio in a scale until KRAIG
had evaluated its use in conjunction with all *other* scale ratios?

PAUL - DOES THE ABOVE HELP IN CLARIFYING THIS TEXT?

J Gill

🔗paulerlich <paul@stretch-music.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> Hello J Gill,
>
> I attempted to address some of your points in my previous responses
> to you under this subject heading (either here or at tuning-math, I
> don't remember). You can reply to those posts.
>
> -Paul

Paul,

In post /tuning/topicId_31138.html#31210

you indicated:

Hello J . . .

So, in post /tuning/topicId_31138.html#31225

I (attempted to) clarify my intended questions:

JG: With the "new-fangled" brute-force solution to the Yahoo "de-
formatting-machine" below, perhaps my (modified) text will make more
sense:

----------------------3/2
---------------------/--\
--------------------/----\
-------------------/------\
------------------/--------\
-----------------/----------\
----------------/------------\
---------------/--------------\
--------------/----------------\
-------------/------------------\
------------/--------------------\
-----------4/3------------------5/3
----------/---\----------------/---\
---------/-----\--------------/-----\
--------/-------\------------/-------\
-------/---------\----------/---------\
------5/4--------7/5-------8/5--------7/4
-----/--\-------/--\------/--\-------/--\
---6/5--9/7--11/8-10/7-11/7--13/8-12/7--9/5

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

What do you think about this business of all of the above described
ratios being "Farey adjacent" to each other, but only half of them
being "superparticular" as well as "Farey adjacent" to each other,
and how those differing characteristics may (or may not) imply any
demonstrable status in terms of desirability as interval choices.

Sincerely, J Gill

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> here is a paper on epimores. page 3 should explain the previous
> http://www.anaphoria.com/epimore.pdf
>
> paulerlich wrote:
>
> > Hello J Gill,
> >
> >
>
> -- Kraig Grady
> North American Embassy of Anaphoria island
> http://www.anaphoria.com
>
> The Wandering Medicine Show
> Wed. 8-9 KXLU 88.9 fm

Kraig,

I downloaded and took a look at page 3 of "epimore.pdf".

It looks very interesting, but my relatively uninitiated mind still
swims with some of the questions which I put forth in message post
/tuning/topicId_31138.html#31171

Hoping to hear from you, J Gill

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗unidala <JGill99@imajis.com>

Joe,

http://www.cut-the-knot.com/blue/Stern.html

http://www.cut-the-knot.com/blue/ContinuedFractions.html

and check out:

http://www.cut-the-knot.com/blue/Farey.html

http://www.cut-the-knot.com/ctk/Farey.html

Best Regards, J Gill

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_31138.html#31336
>
> > --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > > here is a paper on epimores. page 3 should explain the previous
> > > http://www.anaphoria.com/epimore.pdf
> > >
> >
> >
> > ----------------------3/2
> > ---------------------/--\
> > --------------------/----\
> > -------------------/------\
> > ------------------/--------\
> > -----------------/----------\
> > ----------------/------------\
> > ---------------/--------------\
> > --------------/----------------\
> > -------------/------------------\
> > ------------/--------------------\
> > -----------4/3------------------5/3
> > ----------/---\----------------/---\
> > ---------/-----\--------------/-----\
> > --------/-------\------------/-------\
> > -------/---------\----------/---------\
> > ------5/4--------7/5-------8/5--------7/4
> > -----/--\-------/--\------/--\-------/--\
> > ---6/5--9/7--11/8-10/7-11/7--13/8-12/7--9/5
> >
> >
>
> I'm sorry... but I've forgotten the simple arithmetic procedure to
> derive the Stern Brocot tree... :(
>
> This is embarassing, but I'm beyond embarassment... I'm not seeing
it
> at the moment :(
>
> Thanks!
>
> JP

🔗paulerlich <paul@stretch-music.com>

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

> JG: With the "new-fangled" brute-force solution to the Yahoo "de-
> formatting-machine" below, perhaps my (modified) text will make
more
> sense:
>
> ----------------------3/2
> ---------------------/--\
> --------------------/----\
> -------------------/------\
> ------------------/--------\
> -----------------/----------\
> ----------------/------------\
> ---------------/--------------\
> --------------/----------------\
> -------------/------------------\
> ------------/--------------------\
> -----------4/3------------------5/3
> ----------/---\----------------/---\
> ---------/-----\--------------/-----\
> --------/-------\------------/-------\
> -------/---------\----------/---------\
> ------5/4--------7/5-------8/5--------7/4
> -----/--\-------/--\------/--\-------/--\
> ---6/5--9/7--11/8-10/7-11/7--13/8-12/7--9/5
>
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > JG: In the segment of the Stern-Brocot tree branching (upwards,
> > generally) into the 3/2 node (then following on upwards to the
2/1
> > node, and the 1/1 node), the 6/5 and 9/7 nodes have a mediant of
> > 5/4,
> > the 5/4 and 7/5 nodes have a mediant of 4/3, and the 4/3 and 5/3
> > nodes have a mediant of 3/2. THE NODAL RATIOS 6/5, 5/4, 4/3, AND
> 3/2 FORM A "SUPERPARTICULAR CHAIN" OF NODAL RATIOS ALONG THE EDGE
OF
> THIS SUB-BRANCH.
>
> > However, in a "mirror-image" pattern
> > (centered about the 3/2 node), the "Farey adjacent" but
> > not "superparticular" 9/5 and 12/7 nodes have a mediant of 7/4,
the
> > 7/4 and 8/5 nodes have a mediant of 5/3, and (reduntantly), the
5/3
> > and the 4/3 nodes also have a mediant of 3/2. THE NODAL RATIOS
9/5,
> 7/4, 5/3, AND 3/2 DO *NOT* FORM A "SUPERPARTICULAR CHAIN" OF NODAL
> RATIOS ALONG THE EDGE OF THIS SUB-BRANCH.
>
> > Can these two "mirror-
> > image" groups (where BOTH groups contain nodal ratio values which
> > are "Farey adjacent" to each other traveling upwards or downwards
> > along the branches involved, but only the *FIRST* group cited
above
> > contains nodal ratio values which are "superparticular") be
> > considered musically EQUIVALENT (FROM THE PERSPECTIVE OF
> DESIRABILITY AS CHOICES OF VALUES OF "SCALE INTERVALS" THEMSELVES)

Yes.

> > when combined with a 1/1, or other ratios present in a given
> scale),
> > or does the superparticularity of such ratios impart a PREFERABLE
> > quality as compared to the other non-superparticular "SCALE
> INTERVAL RATIOS" which one
> > might select to include within a given scale.

No. Again, from the ancient Greeks to today, superparticularity was
considered a desirable feature to have *in the ratios describing the
step sizes in the scale*, NOT *in the ratios describing the pitches
in the scale*.

> What do you think about this business of all of the above described
> ratios being "Farey adjacent" to each other, but only half of them
> being "superparticular" as well as "Farey adjacent" to each other,
> and how those differing characteristics may (or may not) imply any
> demonstrable status in terms of desirability as interval choices.

Really, all "Farey adjacent" (in a Farey series) pairs of ratios have
a superparticular step measuring the distance between them. For
example, in the Farey series of order 10, we find

...6/5--5/4--9/7--4/3--7/5--10/7--...

The intervals between adjecent ratios are:

---25/24--36/35--28/27--21/20--50/49--

This is where the "magic" of superparticulars shows itself.

What's so Super about Superparticularity?

🔗J Gill <JGill99@imajis.com>

🔗klaus schmirler <KSchmir@z.zgs.de>

🔗Afmmjr@aol.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

🔗unidala <JGill99@imajis.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>