Yahoo Tuning Groups Ultimate Backup tuning-math My Hexachord Grid

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
A F G H I K P U Y Z
A C E G K P U W X Y
A G J K M P Q S U Y
A E M P U Y
A C P Q U W X Y
A D E H I L P Q U Y
A B C H I M N O P U W X Y

Letters line up - Hit "Reply" to see the grid correctly.

Multiply each letter by its expressions (from hexachords2.xls on
tuning-math Files - Paul Hj's Stuff)

80 + 16 + 20 + 20 + 8 + 20 + 12 + 32 / 8 = 26 classes of my system

Each row is a group-theoretical operation. (C12, S3, D4, D12, S2-
complement X (C12, S3, D4, D12)) D4 is D4 acting on Z12 and not
the same as D4 X C3. The top four are the four pieces of D4 X S3
and the bottom four are the four pieces of S2-comp X D4 X S3.

I apologize to any real mathematicians for any unorthodoxy in
this presentation. The rest of you, I apologize to also.

Each hexachord maps into itself based on this grid. D4 partners,
assymetrical partners and complements have exactly the same behavior
under all these operations, so they can be expressed by the single
letter. Of course adding everything up and dividing by 8 leaves
A-Z exactly. (The 26 hexachord types)

I can also break this all out by transposition. Another grid
will give numbers for each transpose in each row. (I can also
associate each transpose with the different hexachords, but that is
kind of going overboard)

(This might have application to cryptography, too, at least for
developing ciphers on the 26 letters of the alphabet)

Just for fun:

Music Theory -> Math -> Physics -> Cosmology.

They are up to 26 dimensions in String Theory now too, based on the
Leech Lattice, the character table ot M24, etc. I certainly don't
profess to understand it all!)

Hi Dr. Wild if you are reading this.

Paul Hjelmstad

🔗Carl Lumma <ekin@lumma.org>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >Each row is a group-theoretical operation. (C12, S3, D4, D12,
S2-
> >> >complement X (C12, S3, D4, D12)) D4 is D4 acting on Z12 and not
> >> >the same as D4 X C3. The top four are the four pieces of D4 X S3
> >> >and the bottom four are the four pieces of S2-comp X D4 X S3.
> >>
> >> So this is 12-based?
> >>
> >> -Carl
> >
> >Yes. Just Necklace/Polya theory.
>
> You're just finding all the unique 6-tone scales in 12?
>
> -Carl
>
Yes, in this case, reduced for transposition, mirror-image, D4 and S3,
and complementability.

My system is in Files - Paul Hj's Stuff hexachords2.xls

One Note: I list B1 and B5 separately, to show the sets. But
the expressions are the total number for "B" (B1, B5, -B1, -B5) etc.

A Z-relation is merely when a hexachords complement is not of the
same Tn/TnI type. (80 + 20 + 8 + 32)/4 gives the 35 listed in the
table. This is rows 1,4,5,8 of my hexachord grid.

I like stopping at D4 X S3 instead of going to S4 X S3 because
it "doesn't mess with tritones" and that works into some of my
other 7-limit theories which I have no business talking about
until I master lattices like the 7-limit. Making progress.

Paul Hj
- Paul

🔗Carl Lumma <ekin@lumma.org>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >> So this is 12-based?
> >> >
> >> >Yes. Just Necklace/Polya theory.
> >>
> >> You're just finding all the unique 6-tone scales in 12?
> >>
> >Yes, in this case, reduced for transposition,
>
> OK.
>
> >mirror-image,
>
> About the octave? So, otonal and utonal chords come out
> the same?

Yes
>
> >D4 and S3, and complementability.
>
> What's the musical idea behind these?

It's kind of a work in progress. The "M5" symmetry is based
on D4, (S3 if you take the inverse of the M5 relation). It
merely maps semitones to perfect fifths. So you can use
these generators (1,7)(3,9)(5,11) to find any M5 partner.

Complementary set: Recursive behavior (Think GEB, especially Escher)

See Message 15945 for all the math if you are interested.
>
> -Carl
>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >> >> So this is 12-based?
> >> >> >
> >> >> >Yes. Just Necklace/Polya theory.
> >> >>
> >> >> You're just finding all the unique 6-tone scales in 12?
> >> >>
> >> >Yes, in this case, reduced for transposition,
> >>
> >> OK.
> >>
> >> >mirror-image,
> >>
> >> About the octave? So, otonal and utonal chords come out
> >> the same?
> >
> >Yes
> >
> >> >D4 and S3, and complementability.
> >>
> >> What's the musical idea behind these?
> >
> >It's kind of a work in progress. The "M5" symmetry is based
> >on D4, (S3 if you take the inverse of the M5 relation). It
> >merely maps semitones to perfect fifths. So you can use
> >these generators (1,7)(3,9)(5,11) to find any M5 partner.
>
> Do you think these symmetries are audible?
>
> -Carl

That's a matter of debate. There is of course a lot that is beautiful
in music that can't be quantified (not easily) and lot that is
beautiful in math that doesn't sound like anything. The M5
relation does come into play in jazz, when you consider
what is called tritone substitution. (For example replacing
G with Db in C major). I am not even sure that mirror-image
means much, but of course, that's the basis of major versus
minor etc. Complementability: This is even more tenuous: But
consider the blues scale in C. Take it's complement (which is
not complementary, but Z-related) and you obtain E7+Amaj together!
(C,Eb, F, F#, G, Bb, C) -> (E, G#, B, D, A, C#, E)

So having faith in Goedel Escher Bach, and I guess in atonal
serial techniques, I think there is some basis to this.

For example, with hexachords, you can do the retrograde, inversion,
retrograde-inversion, and with my stuff you can have
hexachord + hexachord complement to contruct a row, so you
have yet another manipulation of a tone row. Not that I am
claiming it, of course, I am sure Schoenberg beat me to it.

Paul Hj

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
> A F G H I K P U Y Z
> A C E G K P U W X Y
> A G J K M P Q S U Y
> A E M P U Y
> A C P Q U W X Y
> A D E H I L P Q U Y
> A B C H I M N O P U W X Y
>
>
> Letters line up - Hit "Reply" to see the grid correctly.
>
> Multiply each letter by its expressions (from hexachords2.xls on
> tuning-math Files - Paul Hj's Stuff)
>
> 80 + 16 + 20 + 20 + 8 + 20 + 12 + 32 / 8 = 26 classes of my system
>
> Each row is a group-theoretical operation. (C12, S3, D4, D12, S2-
> complement X (C12, S3, D4, D12)) D4 is D4 acting on Z12 and not
> the same as D4 X C3. The top four are the four pieces of D4 X S3
> and the bottom four are the four pieces of S2-comp X D4 X S3.
>
> I apologize to any real mathematicians for any unorthodoxy in
> this presentation. The rest of you, I apologize to also.
>
> Each hexachord maps into itself based on this grid. D4 partners,
> assymetrical partners and complements have exactly the same
behavior
> under all these operations, so they can be expressed by the single
> letter. Of course adding everything up and dividing by 8 leaves
> A-Z exactly. (The 26 hexachord types)
>
> I can also break this all out by transposition. Another grid
> will give numbers for each transpose in each row. (I can also
> associate each transpose with the different hexachords, but that
is
> kind of going overboard)
>
> (This might have application to cryptography, too, at least for
> developing ciphers on the 26 letters of the alphabet)
>
> Just for fun:
>
> Music Theory -> Math -> Physics -> Cosmology.
>
> They are up to 26 dimensions in String Theory now too, based on
the
> Leech Lattice, the character table ot M24, etc. I certainly don't
> profess to understand it all!)
>
> Hi Dr. Wild if you are reading this.
>
> Paul Hjelmstad
>
This isn't much compared to the Riemann Zeta thread, (which is how
I got involved in the newsgroup, BTW) but just to finish off my
Hexachord Grid post here is the full count: (I will put this
in my spreadsheet and also break out by hexachords someday)

The order in the above grid is actually C12, D4, S3, D12, S2C,
D4XS2C, S3XS2C and D12 X S2C, so that is what I will use below:

This is hexachords that map into themselves under the above
symmetries with transpositions. (Also, the reason my "D4" is
not the same as C3 X D4, is because I have D4 acting on Z12,
which is really just S3 forwards (C3) and D4 backwards, whereas
C3 X D4 is S3 forwards (C3) and D4 both forwards and backwards)
That's how I think about it, if anyone wants to set me straight,
or give me better names, please do so! Plus, S2C is S2
complementability (2 colors, black and white beads)

Here it is, I didn't try to line up the numbers:

T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T0

0,2,0,6,0,20,0,6,0,2,0,924
2,2,20,2,2,92,2,2,20,2,2,92
0,12,0,36,0,36,0,36,0,12,0,60
20,20,20,20,20,20,20,20,20,20,20,20

2,4,8,0,2,64,2,0,8,4,2,0
4,0,61,0,4,3,4,0,61,0,4,3
8,64,8,0,8,64,8,0,8,64,8,0
64,0,64,0,64,0,64,0,64,0,64,0

So there it is. I'm not sure why "D4" and "D4 X SC2" are so
irregular. I will doublecheck my math. Everything except rows
6 and 7 can be found with formulas. (Polya or Gilbert/Riordan)
I did 6 and 7 by inspection. If I could find a way to combine
Polya with G/R I would!

I got some interesting patterns looking at this. Transpositions in D4
are always (0,6) or (3,9) except for "A" which ends up being all the
odds. In S3, They are always (0,4,8) or (2,6,10). D4XS2C is always
(3,9) except for "A" which is all the odds again. S3XS2C is always
(2,6,10) except for "I" which is all the odds, and "P" which is also
all the odds. I forgot to say "U" is all the evens in D4 and S3 and
all the odds in D4XS2C and S3XS2C.

These are the hexachords of limited transposition....

When I get better with GAP I hope to have something a little more
worth everyone's time.

Thanks

Paul H

🔗Paul G Hjelmstad <phjelmstad@msn.com>

Fixed up some errors/alignments in this grid:

> > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
> > A F G H I K P U Y Z
> > A C F G K P U W X Y
> > A G J K M P Q S U Y
> > A F M P U Y
> > A C P Q U W X Y
> > A D E H I L P Q U Y
> > A B C H I M N O P U W X Y
> >
> >
> > Letters line up - Hit "Reply" to see the grid correctly.
> >
> > Multiply each letter by its expressions (from hexachords2.xls on
> > tuning-math Files - Paul Hj's Stuff)
> >
> > 80 + 16 + 20 + 20 + 8 + 20 + 12 + 32 / 8 = 26 classes of my
system

Corrected this grid too: (Row 6, D4)
>
> T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T0
>
> 0,2,0,6,0,20,0,6,0,2,0,924
> 2,2,20,2,2,92,2,2,20,2,2,92
> 0,12,0,36,0,36,0,36,0,12,0,60
> 20,20,20,20,20,20,20,20,20,20,20,20
>
> 2,4,8,0,2,64,2,0,8,4,2,0
> 4,0,64,0,4,0,4,0,64,0,4,0
> 8,64,8,0,8,64,8,0,8,64,8,0
> 64,0,64,0,64,0,64,0,64,0,64,0

Paul Hj

🔗Paul G Hjelmstad <phjelmstad@msn.com>

Yikes, found even more errors. I've uploaded my new Hexachord
spreadsheet which has three tabs.

Sheet1 is the main key, also shows expressions. Note that totals
for B1 and B5 are really the totals for "B" for example

Sheet2 is the hexachord grid - pan far right for row headings

Sheet3 shows numerical totals for each transpose - headings at right
again

Sheet4 will break out each hexachord and show tranposes under each
group operation

Gene, (or anyone else qualified): I am not happy with my names
for these group identity mappings. D4 is really C3 X D4(neg)
and S3 is really S3(neg) X C4. S2C stands for the S2-complement
(reversal of black and white beads). Are there any other names
that would be better?

All these tables show is which hexachords in my system map into
themselves under which operations, and at which transpose.

Notice that all the complementability mappings are powers of 2.

Paul Hjelmstad

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> Fixed up some errors/alignments in this grid:
>
>
> > > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
> > > A F G H I K P U Y Z
> > > A C F G K P U W X Y
> > > A G J K M P Q S U Y
> > > A F M P U Y
> > > A C P Q U W X Y
> > > A D E H I L P Q U Y
> > > A B C H I M N O P U W X Y
> > >
> > >
> > > Letters line up - Hit "Reply" to see the grid correctly.
> > >
> > > Multiply each letter by its expressions (from hexachords2.xls
on
> > > tuning-math Files - Paul Hj's Stuff)
> > >
> > > 80 + 16 + 20 + 20 + 8 + 20 + 12 + 32 / 8 = 26 classes of my
> system
>
> Corrected this grid too: (Row 6, D4)
> >
> > T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T0
> >
> > 0,2,0,6,0,20,0,6,0,2,0,924
> > 2,2,20,2,2,92,2,2,20,2,2,92
> > 0,12,0,36,0,36,0,36,0,12,0,60
> > 20,20,20,20,20,20,20,20,20,20,20,20
> >
> > 2,4,8,0,2,64,2,0,8,4,2,0
> > 4,0,64,0,4,0,4,0,64,0,4,0
> > 8,64,8,0,8,64,8,0,8,64,8,0
> > 64,0,64,0,64,0,64,0,64,0,64,0
>
> Paul Hj
>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:

> All these tables show is which hexachords in my system map into
> themselves under which operations, and at which transpose.
>
> Notice that all the complementability mappings are powers of 2.
>
> Paul Hjelmstad

I've decided to call the backwards permutations of D4 "D4-".
Also, S3 backwards is S3-

Here are some interesting patterns:

Hexachords that map into themselves as D4- complement at T3:

CX AY PU QW. Totals are 24 + 8 + 8 + 24. CX + PU are Weight 0.
AY + QW are Weight 3.

AY has 0 tritones, CX has 1, QW has 2 and PU has 3.

Adding up weights gives 32, 32. Even/Odd Tritones is 32, 32

CX AY
PU QW

Here is transposes per hexachord, that map at T3:

C 12
X 12
A 2
Y 6
P 6
U 2
Q 12
W 12

Now S3 - complements that map at T2 aren't quite as nice

LD AEY H Q Totals are 32, 16, 8, 8. LD + H are Weight 1.
AEY + Q are Weight 3. LD has 1 Tritone, AEY has zero, H has 2
and Q has 2. Weights 40, 24. All are odd Weight: 64. Based
on Tritones 16, 32, 16, 0, So Tritones Odd/Even are 32, 32

Transposes per hexachord (this gives Totals above)

L 16
D 16
A 4
E 8
Y 4
H 8
Q 8

Mappings at the other transposes are less interesting.

- Paul Hj