Rothenberg Proprietary Scales

I have analyzed the 5 Rothenberg Proprietary Scales.

Diatonic - CDEFGAB
Mel Minor - CDEbFGAB
Harm Major - CDEFGAbB
Harm Minor - CDEbFGBbB
Locrian Major -CDEFGgAbBb

Using the Hexachord lettering of My Hexachord System, here is an
examination of the seven subsets of each scale

N5 S5 N5 M5 R5 R5 M5
V5 V5 N5 T5 K T5 N5
H D J5 B5 G R5 S5
G B5 J5 D H T5 R5
V1 W V5 U V5 W V1

Some facts - only the 5 side of the M5 symmetry is used, except
for V1 V5. Also, O1/O5 is the only M5 couple not represented.

In terms of the basic letters (D4 X S3 X S2) where S2 is
complementation, and arranged by tritone count

1
6
6
1

or 7 in the Upper Realm and 7 in the Lower Relam, where the total
is 13 Upper and 13 Lower of the 26 Letters.

Also, in terms of the 35 sets, arranged by Sisters groupings
and tritone counts (my I-CHING system)

M5

B5N5T5, DR5, S5(S5)

J5V1V5, GK, HW

U

Notice the symmetry in the Sisters groups and in the ATTIC and CELLAR

1

3, 2, 2
3, 2, 2

1

Now both sides of S5 (Z related set with complement) are used,
otherwise only one side of a Z relation are used

T5, DR5
J5, GK

Note the symmetry here also

1,2
1,2

Now of course any hexachord could be found as the subset of some
scale, the M1 sides are of course more chromatic. Adding some
non proprietary scales, like the Hungarian, could probably help
to canvass all 35 hexachord types, and more Z related complements.

Now the fun is that in MathieuGroup(12), or M12, based on Steiner
System (5,6,12) a block design, we can examine pentads and hexads
and thus in music, pentachords and hexachords. Now of course
septachords are merely complements of pentachords, thereby
we consider seven notes scales and their subsets in the same manner
as the Steiner system (5,6,12) just complement 5->7 and 6->6,
which may be Z-related, and of course, that is the purpose of
S2 in the D4 X S3 X S2 or D12 X S2 Group.

Now construction of M12 from Outer(Aut(S6)-> M12 uses these
hexad types B5CDEFGJ1J5KLM1N5QR1R5S1S5T1T5V1WYZ or 23 of the 35
hexachord types, and BCDEFGJKLMNQRSTVWYZ Letters, so only
IPU in the CELLAR and A in the ROOF are missing and also
X and H thus:

A
X
H
UPI

1
1
1
3

Filled out more

AM5
B1N1, X
O1O5, H
UPI

Once again note the Symmetry!

2
2,1
2,1
3

with the CELLAR a little bottom heavy in both cases.

Used

EMY
BNT, DLR, CS
JV, FZGK, QW

(None)

Filled out

EM1Y

B5N5T1T5, DLR1R5, CS1S5
J1J5V1, FZGK, QW

Excluding the ROOF there is symmetry between Picture A and Pictures
B and C from the construction based on PG(3,2).

(12)
40+20
40+20

producing the 132 Steiner Hexads and many amusing patterns.

PGH

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> I have analyzed the 5 Rothenberg Proprietary Scales.
>
> Diatonic - CDEFGAB
> Mel Minor - CDEbFGAB
> Harm Major - CDEFGAbB
> Harm Minor - CDEbFGBbB
> Locrian Major -CDEFGgAbBb
>
> Using the Hexachord lettering of My Hexachord System, here is an
> examination of the seven subsets of each scale
>
> N5 S5 N5 M5 R5 R5 M5
> V5 V5 N5 T5 K T5 N5
> H D J5 B5 G R5 S5
> G B5 J5 D H T5 R5
> V1 W V5 U V5 W V1
>
> Some facts - only the 5 side of the M5 symmetry is used, except
> for V1 V5. Also, O1/O5 is the only M5 couple not represented.
>
> In terms of the basic letters (D4 X S3 X S2) where S2 is
> complementation, and arranged by tritone count
>
> 1
> 6
> 6
> 1
>
> or 7 in the Upper Realm and 7 in the Lower Relam, where the total
> is 13 Upper and 13 Lower of the 26 Letters.
>
> Also, in terms of the 35 sets, arranged by Sisters groupings
> and tritone counts (my I-CHING system)
>
> M5
>
> B5N5T5, DR5, S5(S5)
>
> J5V1V5, GK, HW
>
> U
>
> Notice the symmetry in the Sisters groups and in the ATTIC and
CELLAR
>
> 1
>
> 3, 2, 2
> 3, 2, 2
>
> 1
>
> Now both sides of S5 (Z related set with complement) are used,
> otherwise only one side of a Z relation are used
>
> T5, DR5
> J5, GK
>
> Note the symmetry here also
>
> 1,2
> 1,2
>
> Now of course any hexachord could be found as the subset of some
> scale, the M1 sides are of course more chromatic. Adding some
> non proprietary scales, like the Hungarian, could probably help
> to canvass all 35 hexachord types, and more Z related complements.
>
> Now the fun is that in MathieuGroup(12), or M12, based on Steiner
> System (5,6,12) a block design, we can examine pentads and hexads
> and thus in music, pentachords and hexachords. Now of course
> septachords are merely complements of pentachords, thereby
> we consider seven notes scales and their subsets in the same manner
> as the Steiner system (5,6,12) just complement 5->7 and 6->6,
> which may be Z-related, and of course, that is the purpose of
> S2 in the D4 X S3 X S2 or D12 X S2 Group.
>
> Now construction of M12 from Outer(Aut(S6)-> M12 uses these
> hexad types B5CDEFGJ1J5KLM1N5QR1R5S1S5T1T5V1WYZ or 23 of the 35
> hexachord types, and BCDEFGJKLMNQRSTVWYZ Letters, so only
> IPU in the CELLAR and A in the ROOF are missing and also
> X and H thus:
>
> A
> X
> H
> UPI
>
> 1
> 1
> 1
> 3
>
> Filled out more
>
> AM5
> B1N1, X
> O1O5, H
> UPI
>
> Once again note the Symmetry!
>
> 2
> 2,1
> 2,1
> 3
>
> with the CELLAR a little bottom heavy in both cases.
>
> Used
>
> EMY
> BNT, DLR, CS
> JV, FZGK, QW
>
> (None)
>
> Filled out
>
> EM1Y
>
> B5N5T1T5, DLR1R5, CS1S5
> J1J5V1, FZGK, QW
>
> Excluding the ROOF there is symmetry between Picture A and Pictures
> B and C from the construction based on PG(3,2).
>
> (12)
> 40+20
> 40+20
>
> producing the 132 Steiner Hexads and many amusing patterns.
>
> PGH

I should stick to one subject in a post:)

Anyway, I realized that the reason some hexachords don't get
canvassed easily is due to the fact that they are "Echer-like"
scales/chords.

For example, P=(11,0,1,5,6,7) has B,C,E,F part of CM scale,
and C,Db,F,Gb part of DbM scale (other dissections are possible)
so it is like an ambiguous Escher print.

Now while normal scales like harmonic minor can use sharps and
flats together, going to two or more of each produces ambiguities.

The hexad subsets left out of the Rothenberg Proprietary scales are
like this:

1. The Hexatonic hexachord (A)
2. Sets with long chromatic chains (one side of M5 symmetry)
3. Sets that require forcing natural and accidental both

In fact that covers all that remain.

If we add the Gypsy scale (or Hungarian) we pick up a few more
hexachords

0,2,3,6,7,8,11

0,2,3,6,7,8: Z
0,2,3,6,7,11: D
0,2,3,7,8,11: D
0,2,6,7,8,11: F
0,3,6,7,8,11: D
2,3,6,7,8,11: D

Well, picks up FZ anyway.

C,E,I,L,O,Q,P,X,Y have two semitones (item 3 above)

If we blur these out we get pentads which are quite ordinary,
and fit easily into Rothenberg scales. (remove middle note in 0,1,2)

B1,J1,M1,N1,O1,R1,S1,T1 are in item 2 above

Some hexads are complements or Z-relations, these actually
have the same issues, either 2 or 3 above.

PGH

Does anyone know alot about Myhills' Property? Mixing the black
pentachord and its subsets with the white diaton and its subsets in a
controlled way could have mathematical significance....

Paul H wrote:

> Does anyone know alot about Myhills' Property? Mixing the black
> pentachord and its subsets with the white diaton and its subsets in a > controlled way could have mathematical significance....

Myhill's property is another kind of MOS, but definitely assuming period=octave IIRC. There's an obvious link between MOS scales and propriety which I gave proofs for here:

http://x31eq.com/proof.html

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul H wrote:
>
> > Does anyone know alot about Myhills' Property? Mixing the black
> > pentachord and its subsets with the white diaton and its subsets
in a
> > controlled way could have mathematical significance....
>
> Myhill's property is another kind of MOS, but definitely
> assuming period=octave IIRC. There's an obvious link
> between MOS scales and propriety which I gave proofs for here:
>
> http://x31eq.com/proof.html
>
>
> Graham

Thanks. I will study these proofs. Is it true that subsets of the
pentatonic and diatonic also have Myhill's property? Are there
other interesting proprietary scales besides Rothenberg-proper scales?

On another note, I am finding fascinating patterns studying black
and white keys (literally) in the Steiner System (Outer(Aut(S6)->
Mathieu(M12)) which is a S(5,6,12) system based on the
ProjectiveGeometry(3,2) with 15 points, 35 lines and 15 hyperplanes.

I didn't even have to transpose, and many things just fell in place.
I am mapping My Hexachord System onto this and am working to
map the 35 hexachord types against the 350 combinations found
from 3 black and 3 white keys. However, I don't have a complete
construction yet, but I know there is at least one solution.

The 35 lines of the Smallest Projective space might also map
on the 35 hexachords, using my hexachord extension.

Another fun fact is that there are 35 pentachord types also
based on interval vector, and M12 Symmetry group of (S(5,6,12) of
course relate hexads to pentads and is quintuply and sharply
transitive.

PGH

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul H wrote:
>
> > Does anyone know alot about Myhills' Property? Mixing the black
> > pentachord and its subsets with the white diaton and its subsets
in a
> > controlled way could have mathematical significance....
>
> Myhill's property is another kind of MOS, but definitely
> assuming period=octave IIRC. There's an obvious link
> between MOS scales and propriety which I gave proofs for here:
>
> http://x31eq.com/proof.html
>
>
> Graham

Thanks, your proof is a real cracker, Graham

PGH