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

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

> 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...

Farey adjacence explains some of why superpaticular ratios show up in
music theory. If you take the ratios of the above sequence, you get
(5/3)/(3/2) = 10/9, (7/4)/(5/3) = 21/20, (9/5)/(7/4) = 36/36, and
in general T/(T-1), where T is triangular of even order--that is, the
sum from 1 to an even number. These numbers and superparticulars like
them appear often as scale steps in JI scales. Moreover, you get
superparticular ratios of superparticular ratios, for instance
(9/8)/(10/9) = 81/80, or (15/14)/(16/15) = 225/224; these are of a
common type, having in one case the square of a triangular number,
and in the other case a fourth power (square of a square) as
numerator.

Gene,

Thanks for your excellent description of the mathematical
significance of "superparticularity" and "Farey series adjacence"
found in certain sub-branches of the Stern-Brocot tree structure, and
their inclusion in certain types of JI musical scales (quoted below):

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Farey adjacence explains some of why superpaticular ratios show up
in
> music theory. If you take the ratios of the above sequence, you get
> (5/3)/(3/2) = 10/9, (7/4)/(5/3) = 21/20, (9/5)/(7/4) = 36/36, and
> in general T/(T-1), where T is triangular of even order--that is,
the
> sum from 1 to an even number. These numbers and superparticulars
like
> them appear often as scale steps in JI scales. Moreover, you get
> superparticular ratios of superparticular ratios, for instance
> (9/8)/(10/9) = 81/80, or (15/14)/(16/15) = 225/224; these are of a
> common type, having in one case the square of a triangular number,
> and in the other case a fourth power (square of a square) as
> numerator.

What I would really like to know is the musical *benefits* of
utilizing such ratios in a JI scale (independent from their possibly
being constructed out of low-valued integer values in such a scale).
A genuine (and, it seems ancient) aura of "uniqueness and
desirability" seems attached by some to "superparticular" interval
ratios, yet my search has not located any specific explanation or
justification for such special status. Is the advantage related to
increased coincidence of the overtones of such notes (it seems not
for superparticular ratios which do not, themselves, contain low-
numbered integers)? Do "Farey series adjacent" ratios possess the
same (or similar) musical advantages, and what are such those
advantages?

This is what prompted me to ask in my original post:

<<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 [NON-
LINEAR, ACTUALLY] combinations of such (sinusoidal or complex) tones,
what are the significant sum and/or difference frequencies (relative
to the 1/1)?>>

Still Curious, J Gill

--- In tuning-math@y..., "unidala" <JGill99@i...> wrote:
> Gene,
>
> Thanks for your excellent description of the mathematical
> significance of "superparticularity" and "Farey series adjacence"
> found in certain sub-branches of the Stern-Brocot tree structure,
and
> their inclusion in certain types of JI musical scales (quoted
below):

Actually, superparticular ratios are associated with each branch of
the Stern-Brocot tree, and not confined to any sub-branch. Simply
take the ratio between the node at level n and a branch node at level
n+1, and label the branch connecting them with this superparticular
ratio.

> What I would really like to know is the musical *benefits* of
> utilizing such ratios in a JI scale (independent from their
possibly
> being constructed out of low-valued integer values in such a
scale).

I don't see anything in the old Greek theory that any old
superparticular ratio has benefits, but the ratios connecting branchs
of the Stern-Brocot tree are a different matter, as are the ratios of
second order, between these ratios. If you do a search in the p-limit
for superparticular ratios you get lists such as the one very
recently posted here; if you look at (say) (n+2)/n for odd n you get
nothing like as many. It may be people noticed the things popping up
constantly, and attributed special benefits to them. If they had done
the same with superparticulars with square or triangular or fourth
power, etc. numerators it would have been more to the point, if so.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Actually, superparticular ratios are associated with each branch of
> the Stern-Brocot tree, and not confined to any sub-branch. Simply
> take the ratio between the node at level n and a branch node at
level
> n+1, and label the branch connecting them with this superparticular
> ratio.

Gene,

I see what you mean above (where the branches in the Stern-Brocot
tree, as opposed to the nodes, are concerned). I was addressing the
ratios of the nodes only in my statement (portion of S-B Tree below):

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
/ \ / \ / \ / \

JG: While all nodes branching from a given node are "Farey adjacent"
in terms of their rational values, only some of the nodes values are,
themselves, superparticular in value.

> GWS: I don't see anything in the old Greek theory that any old
> superparticular ratio has benefits, but the ratios connecting
branchs
> of the Stern-Brocot tree are a different matter, as are the ratios
of
> second order, between these ratios. If you do a search in the p-
limit
> for superparticular ratios you get lists such as the one very
> recently posted here; if you look at (say) (n+2)/n for odd n you
get
> nothing like as many.

JG: Your point here is clearly evident by inspection (where
superparticular ratios constructed of higher numbered integers
rapidly decrease as a percentage of the total number of
superparticular ratios at each "level" of the tree).

> GWS: It may be people noticed the things popping up
> constantly, and attributed special benefits to them.

JG: And, it follows, that the percentage of superparticular ratios
(of nodes)"pops up" to a maximal degree in the low-numbered ratio
values. Perhaps, then, it is their attributes of appearing at such
low-numbered ratios (and the corresponding increased coincidence of
overtones of such scale interval ratios), where by "scale interval
ratios", I am describing the chosen tones of a given scale (rather
than the step-sizes between the tones of the scale), which may have
created a perception that their arithmetic "superparticularity" (as
opposed to their low-numbered integer components in the numerator and
denominator, and the resultant harmonic interplay by "harmonic
coincidences") lends a desirable quality to their combinations (with
the 1/1, and with each other).

> GWS: If they had done
> the same with superparticulars with square or triangular or fourth
> power, etc. numerators it would have been more to the point, if so.

JG: Or would it? Can anyone demonstrate an implicit advantage of
utilizing superparticular scale interval ratios with large valued
integers existing in the numerators and/or denominators of such scale
interval ratios? Or am I missing something regarding Gene's points
made regarding "square or triangular or fourth power" numerators?

The "magic", then, if not arising from low-numbered integers existing
in certain superparticular ratios, might exist only as a beneficial
characteristic of a "rational" (bad pun) choice of the subdivision of
the octave which follows from progressively applying the Stern-Brocot
tree?

Still Curious, J Gill

My "Stern-Brocot" tree diagram hacked to bits by formatting gremlins still beyond my aboriginal ASCII diagrammatic skills, I am trying again, via my email program to re-post my message (directly above in the message list):

--- In tuning-math@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > Actually, superparticular ratios are associated with each branch of
> > the Stern-Brocot tree, and not confined to any sub-branch. Simply
> > take the ratio between the node at level n and a branch node at
> level
> > n+1, and label the branch connecting them with this superparticular
> > ratio.
>
> Gene,
>
> I see what you mean above (where the branches in the Stern-Brocot
> tree, as opposed to the nodes, are concerned). I was addressing the
> ratios of the nodes only in my statement (portion of S-B Tree below):
>
>
> 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
> / \ / \ / \ / \
>
> JG: While all nodes branching from a given node are "Farey adjacent"
> in terms of their rational values, only some of the nodes values are,
> themselves, superparticular in value.
>
>
> > GWS: I don't see anything in the old Greek theory that any old
> > superparticular ratio has benefits, but the ratios connecting
> branchs
> > of the Stern-Brocot tree are a different matter, as are the ratios
> of
> > second order, between these ratios. If you do a search in the p-
> limit
> > for superparticular ratios you get lists such as the one very
> > recently posted here; if you look at (say) (n+2)/n for odd n you
> get
> > nothing like as many.
>
> JG: Your point here is clearly evident by inspection (where
> superparticular ratios constructed of higher numbered integers
> rapidly decrease as a percentage of the total number of
> superparticular ratios at each "level" of the tree).
>
> > GWS: It may be people noticed the things popping up
> > constantly, and attributed special benefits to them.
>
> JG: And, it follows, that the percentage of superparticular ratios
> (of nodes)"pops up" to a maximal degree in the low-numbered ratio
> values. Perhaps, then, it is their attributes of appearing at such
> low-numbered ratios (and the corresponding increased coincidence of
> overtones of such scale interval ratios), where by "scale interval
> ratios", I am describing the chosen tones of a given scale (rather
> than the step-sizes between the tones of the scale), which may have
> created a perception that their arithmetic "superparticularity" (as
> opposed to their low-numbered integer components in the numerator and
> denominator, and the resultant harmonic interplay by "harmonic
> coincidences") lends a desirable quality to their combinations (with
> the 1/1, and with each other).
>
> > GWS: If they had done
> > the same with superparticulars with square or triangular or fourth
> > power, etc. numerators it would have been more to the point, if so.
>
> JG: Or would it? Can anyone demonstrate an implicit advantage of
> utilizing superparticular scale interval ratios with large valued
> integers existing in the numerators and/or denominators of such scale
> interval ratios? Or am I missing something regarding Gene's points
> made regarding "square or triangular or fourth power" numerators?
>
> The "magic", then, if not arising from low-numbered integers existing
> in certain superparticular ratios, might exist only as a beneficial
> characteristic of a "rational" (bad pun) choice of the subdivision of
> the octave which follows from progressively applying the Stern-Brocot
> tree?
>
>
> Still Curious, J Gill

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

> > GWS: If they had done
> > the same with superparticulars with square or triangular or
fourth
> > power, etc. numerators it would have been more to the point, if
so.
>
> JG: Or would it? Can anyone demonstrate an implicit advantage of
> utilizing superparticular scale interval ratios with large valued
> integers existing in the numerators and/or denominators of such
scale
> interval ratios? Or am I missing something regarding Gene's points
> made regarding "square or triangular or fourth power" numerators?

I was talking about commas, not intervals. Commas appear as the
ratios between the superparticulars assoicated to branches of the
tree, and hence to nodes of the tree. From 3/2, we have branchs going
to 4/3 and 5/3, labeled by 9/8 and 10/9; the ratio is 81/80, which
has a fourth power as numerator. We might define a comma function in
this way, which maps from fractions to commas; then comma(3/2)=81/80,
comma(4/3)=64/63, comma(5/3)=126/125, and so forth. The numerators of
these involve various polynomial functions.

Thanks for prodding me, I think I'll code "comma".

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> I don't see anything in the old Greek theory that any old
> superparticular ratio has benefits, but the ratios connecting
branchs
> of the Stern-Brocot tree are a different matter, as are the ratios
of
> second order, between these ratios. If you do a search in the p-
limit
> for superparticular ratios you get lists such as the one very
> recently posted here; if you look at (say) (n+2)/n for odd n you
get
> nothing like as many.

JG: I think I do (now) understand what you meant by the above text.

> GWS: It may be people noticed the things popping up
> constantly, and attributed special benefits to them.

JG: Do you mean: aesthetically from a mathematical viewpoint; or
functionally from a perceptual viewpoint?

> GWS: If they had done
> the same with superparticulars with square or triangular or fourth
> power, etc. numerators it would have been more to the point, if so.

JG: I'm not sure if I do understand the "point" to which you refer.
Could you elaborate a bit on that?

Sincerely, J Gill

--- In tuning-math@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > Actually, superparticular ratios are associated with each branch
of
> > the Stern-Brocot tree, and not confined to any sub-branch. Simply
> > take the ratio between the node at level n and a branch node at
> level
> > n+1, and label the branch connecting them with this
superparticular
> > ratio.
>
> Gene,
>
> I see what you mean above (where the branches in the Stern-Brocot
> tree, as opposed to the nodes, are concerned). I was addressing the
> ratios of the nodes only in my statement (portion of S-B Tree
below):
>
>
> 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
> / \ / \ / \ / \
>
> JG: While all nodes branching from a given node are "Farey
adjacent"
> in terms of their rational values, only some of the nodes values
are,
> themselves, superparticular in value.

Similarly, while the "Greek" scales don't have only superparticular
_pitch ratios_, the size of the steps between adjacent ratios are all
superparticular. I think it's a fetish, but a benign one.
>
> JG: And, it follows, that the percentage of superparticular ratios
> (of nodes)"pops up" to a maximal degree in the low-numbered ratio
> values. Perhaps, then, it is their attributes of appearing at such
> low-numbered ratios (and the corresponding increased coincidence of
> overtones of such scale interval ratios), where by "scale interval
> ratios", I am describing the chosen tones of a given scale

In that sense, I really haven't seen a tendency to use
superparticular ratios, either in ancient Greek theory or today.

>(rather
> than the step-sizes between the tones of the scale),

That's where you tend to see superparticulars pop up. Not only in
scales, but in ordered surveys of ratios within a given distance from
1/1 -- and hence also in choices for unison vectors. (See below.)

>
> > GWS: If they had done
> > the same with superparticulars with square or triangular or
fourth
> > power, etc. numerators it would have been more to the point, if
so.
>
> JG: Or would it? Can anyone demonstrate an implicit advantage of
> utilizing superparticular scale interval ratios with large valued
> integers existing in the numerators and/or denominators of such
scale
> interval ratios? Or am I missing something regarding Gene's points
> made regarding "square or triangular or fourth power" numerators?

Gene is primarily talking about the great propensity for
superparticulars among choices for unison vectors.

> The "magic", then, if not arising from low-numbered integers
existing
> in certain superparticular ratios, might exist only as a beneficial
> characteristic of a "rational" (bad pun) choice of the subdivision
of
> the octave which follows from progressively applying the Stern-
Brocot
> tree?

Sounds like part of the answer.