The Prime Duodala puzzle

I've uploaded a diagram which I call "The Prime Duodala" to the "Files" section of this group. In a fashion similar to the depiction of covalent chemical bonds between atoms in a molecule, the diagram intends to impart a 3rd dimension of "depth" in viewing the diagram. This is attempted by the three different sizes of the circles (or globes) emblazoned with the JI scale interval ratio associated with that circle (or globe). The (3 dimensional) "inter-ratio bonds" between the JI interval ratios come in two familiar flavors and represent commas (or unison-vectors) : Syntonic (Orange, increasing in pitch from Yellow 3-limit intervals) ; Syntonic (Green, decreasing in pitch from Yellow 3-limit intervals); and Diaschismic (Violet, increasing in pitch as the "bond" widens).

Other than the properties explained herein, as well as the properties within the "legend" (lower-right) section of the diagram, there exists SIX specific and "remarkable" properties of "The Prime Duodala"...

Can you find them?

Those who do will win an all-expense-paid trip to a symmetrical structured "melodic world", which coexists with a separate (and differently structured "harmonic world", both of which simultaneously exist, yet, by their nature, come quite close to (but never exactly reach) complete congruence. Yet both are present and active "in and through" the choices of (and the simultaneous soundings of) certain specific JI scale intervals, in combination with each other as well as in combination with 1/1.

CLUES:

The (octave stretched) JI scale intervals as shown are significant to the structure (there for certain reasons).

Standing in the way of a precise mapping between the 12-tone "melodic" and "harmonic" worlds is our old nemesis (or savior, to some), the syntonic comma (or 81/80). Only an increase to a 14-tone scale (say wha...?) where an extra (comma "bridged") option is available with regards to 2 of the 12 scale intervals.

With our bold and foxy new set of JI intervals firmly in hand, a "non-fuzzy" (yet still "robust") *unified* melodic/harmonic model emerges which encompasses certain JI intervals to the 7-limit.

Dyads, triads, tetrads do not escape this analytical universe...

No compromises need be made in order to unify the melodic symmetries of the scale intervals together with the harmonic similarities of the scale. No ifs/ands/buts... It is The Unidala 2001, a tone/space odyssey fulfilled within a monolithic topology

Enjoy, J Gill :)

Well so far the symmetries in this puzzle remain an enigma to me, but
I've figured out one thing -- your first name is Jeremy!

I've looked at your "Prime Duodala" a bit more. Assuming it's a
scale . . . from the point of view of "spectral coincidence"
or "periodicity" or "consonance", I find the following "consonances":

3:2 between 3/8 and 9/16
3:2 between 8/15 and 4/5
3:2 between 4/5 and 6/5
3:2 between 5/6 and 5/4
3:2 between 6/5 and 9/5
3:2 between 5/4 and 15/8
3:2 between 15/8 and 45/16
3:2 between 8/3 and 4/1

5:1 between 3/8 and 15/8
5:1 between 8/15 and 8/3
5:1 between 9/16 and 45/16
5:1 between 4/5 and 4/1

and precious little in the way of further consonances (9:4 is just
two 3:2s, and 10:3 and 15:2 are the next candidates, showing up as
diagonals of the squares in the lattices that follow).

A lattice of this scale shows that it breaks up into two disconnected
pieces which are mirror-images of one another:

8/3------4/1··················
·|········|···················
·|········|···················
·|········|···················
·|········|···················
8/15-----4/5------6/5------9/5

and

5/6------5/4-----15/8----45/16
···················|········|·
···················|········|·
···················|········|·
···················|········|·
··················3/8------9/16

Your diagram still has me completely mystified, though.

I have revised the portion of the text (CAPITALIZED below) in the
interest of clarifying/qualifying the content of my original post:

--- In tuning@y..., J Gill <JGill99@i...> wrote:
> I've uploaded a diagram which I call "The Prime Duodala" to
>the "Files"
> section of this group. In a fashion similar to the depiction of
>covalent
> chemical bonds between atoms in a molecule, the diagram intends to
>impart a
> 3rd dimension of "depth" in viewing the diagram. This is attempted
>by the
> three different sizes of the circles (or globes) emblazoned with
>the JI
> scale interval ratio associated with that circle (or globe).

>The (3
> dimensional) "inter-ratio bonds" between the JI interval ratios
REPRESENT THE DEPICTION OF AN "OCTAVE-INVARIANT" INTERPRETATION OF
THE (OCTAVE STRETCHED/SHRUNK)SCALE INTERVAL RATIOS PRESENT IN THE
DIAGRAM, AND
> come in two
> familiar flavors and represent commas (or unison-vectors) :
>Syntonic
> (Orange, increasing in pitch from Yellow 3-limit intervals) ;
>Syntonic
> (Green, decreasing in pitch from Yellow 3-limit intervals); and
>Diaschismic
> (Violet, increasing in pitch as the "bond" widens).
>
> Other than the properties explained herein, as well as the
>properties
> within the "legend" (lower-right) section of the diagram, there
>exists SIX
> specific and "remarkable" properties of "The Prime Duodala"...
>
> Can you find them?
>
> Those who do will win an all-expense-paid trip to a symmetrical
>structured
> "melodic world", which coexists with a separate (and differently
>structured
> "harmonic world", both of which simultaneously exist, yet, by their
>nature,
> come quite close to (but never exactly reach) complete congruence.
>Yet both
> are present and active "in and through" the choices of (and the
> simultaneous soundings of) certain specific JI scale intervals, in
> combination with each other as well as in combination with 1/1.
>
>
> CLUES:
>
> The (octave stretched) JI scale intervals as shown are significant ?
> to the
structure (THEY ARE there for certain reasons, BUT ARE NOT
MANDATORILY OCTAVE STRETCHED/SHRUNK AS A CONDITION OF VALIDITY).
OTHER OCTAVE MULTIPLES/SUBMULTIPLES OF THE INTERVALS SHOWN MAY BE
EXPLORED, AS WELL, AND WILL RETAIN MEANING WITHIN THE UNIFIED CONCEPT
(OF THE MELODIC AND HARMONIC "WORLDS" COINCIDING).
>
> Standing in the way of a precise mapping between the 12-
>tone "melodic" and
> "harmonic" worlds is our old nemesis (or savior, to some), the
>syntonic
> comma (or 81/80). Only an increase to a 14-tone scale (say wha...?)
>where
> an extra (comma "bridged") option is available with regards to 2 of
>the 12
scale (WHICH PERHAPS WOULD BE BETTER TERMED "IN THE PLAYABLE GAMUT")
intervals.

TWELVE (WITH TWO ALTERNATE TONES AVAILABLE IN THE "PLAYABLE GAMUT")IS
NOT AN ABSOLUTE. SUCH "GAMUTS" WITH GREATER THAN (14) INTERVALS COULD
BE FORMED, BUT THE RESULTING INTERVALS APPEAR TO BE OF DIMINISHING
USEFULNESS (AS ANYTHING BUT SUB-PARTS OF A MUCH LARGER "N-TONE GAMUT")
>
> With our bold and foxy new set of JI intervals firmly in hand, a
> "non-fuzzy" (yet still "robust") *unified* melodic/harmonic model
>emerges
which encompasses certain JI intervals to the 7-limit (THOUGH THE 7-
LIMIT SCALE INTERVALS ARE NOT INCORPORATED INTO THIS DIAGRAM).
>
> Dyads, triads, tetrads do not escape this analytical universe...
>
> No compromises need be made in order to unify the melodic
>symmetries of the
> scale intervals together with the harmonic similarities of the
>scale. No
> ifs/ands/buts... It is The Unidala 2001, a tone/space odyssey
>fulfilled
> within a monolithic topology.

THE DIAGRAMS DEPICTING SOME OF THE RELATIONSHIPS DISCUSSED ARE STILL
WORKS-IN-PROGRESS (IN THE HOPE OF IN THE FUTURE PROVIDING A
COMPREHENSIVE AND READER-FRIENDLY PRESENTATION OF A NUMBER OF IDEAS
IN A "UNIFIED" FORMAT). THE CONCEPTS THEMSELVES, APPEAR WORTHY OF
PRESENTATION. I HOPE TO BE ABLE TO COMPLETE SUCH A PAPER IN THE
COURSE OF A FAIRLY BUSY SCHEDULE OTHERWISE. THE "UNIDALA 2001", AND
OTHER RELATED DIAGRAMS (INCLUDING "THE PRIME DUODALA") *WILL* BE
PRESENTED IN THEIR ENTIRETY, WHEN I AM ABLE!...

> Enjoy, J Gill :)

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

> I've looked at your "Prime Duodala" a bit more. Assuming it's a
> scale

JG: Indeed, though it is (perhaps) a scale to please "mathematical
topologists" more than it may musicians... It may *also* be
interpreted through an "octave-invariant" viewpoint, at which point
one finds the more familiar JI scale [see diagram (3) below]:

>. . . from the point of view of "spectral coincidence"
> or "periodicity" or "consonance", I find the
following "consonances":
>
> 3:2 between 3/8 and 9/16
> 3:2 between 8/15 and 4/5
> 3:2 between 4/5 and 6/5
> 3:2 between 5/6 and 5/4
> 3:2 between 6/5 and 9/5
> 3:2 between 5/4 and 15/8
> 3:2 between 15/8 and 45/16
> 3:2 between 8/3 and 4/1
>
> 5:1 between 3/8 and 15/8
> 5:1 between 8/15 and 8/3
> 5:1 between 9/16 and 45/16
> 5:1 between 4/5 and 4/1
>
> and precious little in the way of further consonances (9:4 is just
> two 3:2s, and 10:3 and 15:2 are the next candidates, showing up as
> diagonals of the squares in the lattices that follow).
>
> A lattice of this scale shows that it breaks up into two
disconnected
> pieces which are mirror-images of one another:
>
>
> 8/3------4/1··················
> ·|········|···················
> ·|········|···················
> ·|········|···················
> ·|········|···················
> 8/15-----4/5------6/5------9/5
>
> and
>
> 5/6------5/4-----15/8----45/16
> ···················|········|·
> ···················|········|·
> ···················|········|·
> ···················|········|·
> ··················3/8------9/16

JG: Paul (just a note). I like the (extended) ASCII character which
you have utilized in your above diagramms, but (only) when I print-
out your post, they end up a large square printed symbol in each
character location, instead (which makes it a bit harder, though not
impossible, for the eye to see the "stuff" of the diagram).

>PE: Your diagram still has me completely mystified, though.

JG: (1) Have you tried arranging them like this? (below):

5/6------5/4-----15/8-----45/16
-------------------|--------|
-------------------|--------|
-------------------|--------|
------------------3/8------9/16
-------------------|--------|
=======================<>=======================
-------------------|--------|
------------------8/3------4/1
-------------------|--------|
-------------------|--------|
-------------------|--------|
------------------8/15-----4/5------6/5------9/5

(2) Or, like this? (below):

------------------3/8------9/16
-------------------|--------|
-------------------|--------|
-------------------|--------|
5/6------5/4-----15/8-----45/16
-------------------|--------|
=======================<>=======================
-------------------|--------|
------------------8/15-----4/5------6/5------9/5
-------------------|--------|
-------------------|--------|
-------------------|--------|
------------------8/3------4/1

(3) Or, how about this (octave-invariant) interpretation?

------------5/6------5/4-----15/8-----45/16------------
-------------|--------|--------|--------|--------------
-------------|--------|-----===========================
-------------|--------|----=---|--------|--------------
------------8/3------4/1--<>--3/8------9/16------------
-------------|--------|---=----|--------|--------------
==========================-----|--------|--------------
-------------|--------|--------|--------|--------------
------------8/15-----4/5------6/5------9/5-------------

Revealing Ellis' "Duodene" ("octave-restored") below:

1/1--16/15--9/8--6/5--5/4--4/3--45/32--3/2--8/5--5/3--9/5--15/8

Regards, J Gill

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

> JG: (1) Have you tried arranging them like this? (below):
>
> 5/6------5/4-----15/8-----45/16
> -------------------|--------|
> -------------------|--------|
> -------------------|--------|
> ------------------3/8------9/16
> -------------------|--------|
> =======================<>=======================
> -------------------|--------|
> ------------------8/3------4/1
> -------------------|--------|
> -------------------|--------|
> -------------------|--------|
> ------------------8/15-----4/5------6/5------9/5

This makes no sense to me, as the interval between 3/8 and 8/3 is
9:64, and the interval between 9/16 and 4/1 is also 9:64. You're not
suggesting 9:64 is any sort of "consonant" interval, are you? Have
you checked out the pattern of overtones in this interval?

> (2) Or, like this? (below):
>
> ------------------3/8------9/16
> -------------------|--------|
> -------------------|--------|
> -------------------|--------|
> 5/6------5/4-----15/8-----45/16
> -------------------|--------|
> =======================<>=======================
> -------------------|--------|
> ------------------8/15-----4/5------6/5------9/5
> -------------------|--------|
> -------------------|--------|
> -------------------|--------|
> ------------------8/3------4/1

Ditto, but now for 64:225 (?)

> (3) Or, how about this (octave-invariant) interpretation?
>
> ------------5/6------5/4-----15/8-----45/16------------
> -------------|--------|--------|--------|--------------
> -------------|--------|-----===========================
> -------------|--------|----=---|--------|--------------
> ------------8/3------4/1--<>--3/8------9/16------------
> -------------|--------|---=----|--------|--------------
> ==========================-----|--------|--------------
> -------------|--------|--------|--------|--------------
> ------------8/15-----4/5------6/5------9/5-------------

Rewriting that within one octave:

> ------------5/3------5/4-----15/8-----45/32------------
> -------------|--------|--------|--------|--------------
> -------------|--------|-----===========================
> -------------|--------|----=---|--------|--------------
> ------------4/3------1/1--<>--3/2------9/8-------------
> -------------|--------|---=----|--------|--------------
> ==========================-----|--------|--------------
> -------------|--------|--------|--------|--------------
> -----------16/15-----4/5------6/5------9/5-------------

This now maked perfect sense and is of course one of the periodicity
blocks in my "Gentle Introduction".