mirrored symmetry and a centralized tonic

Hi Jacky (et al),

I thought the 9-tone scale template I used when I was looking at
bimodal triads might be interesting as it generates scales with both
mirrored symmetry and a centralized tonic.

The idea was basically this: Let the 1-b3-3-5 "bimodal triad" be,

b c c
/|\ /|\ /|\
/ | \ / | \ / | \
a--+--c or, a--+--d or, a--+--c-b
\ | / \ | / \ | /
\|/ \|/ \|/
c-b b b

In the first example the simplest a:b:c that makes the square a
bimodal chord is the familiar 4:5:6. And this would give a traditional
lattice -- albeit, one with 24:25 "connections" -- of:

25/24
/|\
/ | \
/ | \
/ | \
/ | \
5/3----+----5/4
/|\ | /|\
/ | \ | / | \
/ | \ | / | \
/ | \ | / | \
/ | \|/ | \
4/3----+----1/1----+----3/2
\ | /|\ | /
\ | / | \ | /
\ | / | \ | /
\ | / | \ | /
\|/ | \|/
8/5----+----6/5
\ | /
\ | /
\ | /
\ | /
\|/
48/25

Another example of this sort where the lattice more clearly becomes a
template (in this case a subset of a sequence of squares -- "centered
squares" I guess you could call them) would be 14:18:21 which would
give a scale of:

1/1 54/49 7/6 9/7 4/3 3/2 14/9 12/7 49/27 2/1

The second example goes a way towards legitimizing the total
connectivity as it is a full (bimodal chord) identity.

One of my very favorites of this type was the 11:13:14:16.

14/13
/|\
/ | \
/ | \
/ | \
/ | \
22/13---+---14/11
/|\ | /|\
/ | \ | / | \
/ | \ | / | \
/ | \ | / | \
/ | \|/ | \
11/8----+----1/1----+---16/11
\ | /|\ | /
\ | / | \ | /
\ | / | \ | /
\ | / | \ | /
\|/ | \|/
11/7----+---13/11
\ | /
\ | /
\ | /
\ | /
\|/
13/7

Some other examples of this sort would be 12:14:15:18 and 16:19:20:24.

The last example I have posted about before using an 18:21:23.

23/21
/|\
/ | \
/ | \
/ | \
/ | \
12/7----+---23/18
/|\ | /|\
/ | \ | / | \
/ | \ | / | \
/ | \ | / | \
/ | \|/ | \
216/161--+----1/1----+--161/108
\ | /|\ | /
\ | / | \ | /
\ | / | \ | /
\ | / | \ | /
\|/ | \|/
36/23---+----7/6
\ | /
\ | /
\ | /
\ | /
\|/
42/23

The idea of those posts was that if you were to lattice out the
18:21:23 scale in 14 or 23-tET you'd have a situation that was
somewhat analogous to the diatonic scale where the comma -- a
7889/7776 in the following 14-tET example -- is absorbed by the
temperament and an additional consonant chord results. (In this case
an additional bimodal chord.)

Here's the 14-tET example with the additional consonant chord:

8-----------2
/|\ /|\
/ | \ / | \
/ | \ / | \
/ | \ / | \
/ | \ / | \
3-----+----11-----+-----5
\ | /|\ | /|\
\ | / | \ | / | \
\ | / | \ | / | \
\ | / | \ | / | \
\|/ | \|/ | \
6-----+-----0-----+-----8
\ | /|\ | /|\
\ | / | \ | / | \
\ | / | \ | / | \
\ | / | \ | / | \
\|/ | \|/ | \
9-----+-----3-----+----11
\ | / \ | /
\ | / \ | /
\ | / \ | /
\ | / \ | /
\|/ \|/
12-----------6

Anyway, this type of a "centered squares" template seems like a nice
way to generate scales with mirrored symmetry based on full or partial
identities.

--Dan Stearns

I wrote,

The idea was basically this: Let the 1-b3-3-5 "bimodal triad" be,

b c c
/|\ /|\ /|\
/ | \ / | \ / | \
a--+--c or, a--+--d or, a--+--c-b
\ | / \ | / \ | /
\|/ \|/ \|/
c-b b b

The third example should've read:

c
/|\
/ | \
a--+--c+b
\ | /
\|/
b

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Jacky (et al),
>
> I thought the 9-tone scale template I used when I was looking at
> bimodal triads might be interesting as it generates scales with both
> mirrored symmetry and a centralized tonic.

Dan,

This is really nice and thanks for the refresher.

Some of my favorite modes are of eight tones - and they commonly
contain both a major and minor degrees. Looks like these structures
here would have some nice melodic possibilities. These properties
are "native" to the JI symmetries I enjoy.

I want to ask if it is your experience that symmetry of this nature,
is melodically valuable in your compositions?

Why do my ears tell me the inversional symmetry is melodically
useful? Do you know of any acoustical reason why this might be so?

>
> The idea was basically this: Let the 1-b3-3-5 "bimodal triad" be,
>
> b c c
> /|\ /|\ /|\
> / | \ / | \ / | \
> a--+--c or, a--+--d or, a--+--c+b
> \ | / \ | / \ | /
> \|/ \|/ \|/
> c-b b b
>
> In the first example the simplest a:b:c that makes the square a
> bimodal chord is the familiar 4:5:6. And this would give a
traditional
> lattice -- albeit, one with 24:25 "connections" -- of:
>
> 25/24
> /|\
> / | \
> / | \
> / | \
> / | \
> 5/3----+----5/4
> /|\ | /|\
> / | \ | / | \
> / | \ | / | \
> / | \ | / | \
> / | \|/ | \
> 4/3----+----1/1----+----3/2
> \ | /|\ | /
> \ | / | \ | /
> \ | / | \ | /
> \ | / | \ | /
> \|/ | \|/
> 8/5----+----6/5
> \ | /
> \ | /
> \ | /
> \ | /
> \|/
> 48/25
>
> Another example of this sort where the lattice more clearly becomes
a
> template (in this case a subset of a sequence of squares --
"centered
> squares" I guess you could call them) would be 14:18:21 which would
> give a scale of:
>
> 1/1 54/49 7/6 9/7 4/3 3/2 14/9 12/7 49/27 2/1
>
> The second example goes a way towards legitimizing the total
> connectivity as it is a full (bimodal chord) identity.
>
> One of my very favorites of this type was the 11:13:14:16.

I can see why:

1/1 0.000
14/13 128.298
13/11 289.210
14/11 417.508
11/8 551.318
16/11 648.682
11/7 782.492
22/13 910.790
13/7 1071.702
2/1 1200.000

Very tritone oriented, without the fifths. A Stearns' signature!

I have been greatly inspired by your past comments about how we
should have an open mind about varieties of approximate fifths - and
this is a shining example. I love exotic fifths that stretch the
boundaries of what we may call a "Fifth"! As we know, musical
context is where this becomes feasible.

>
> 14/13
> /|\
> / | \
> / | \
> / | \
> / | \
> 22/13---+---14/11
> /|\ | /|\
> / | \ | / | \
> / | \ | / | \
> / | \ | / | \
> / | \|/ | \
> 11/8----+----1/1----+---16/11
> \ | /|\ | /
> \ | / | \ | /
> \ | / | \ | /
> \ | / | \ | /
> \|/ | \|/
> 11/7----+---13/11
> \ | /
> \ | /
> \ | /
> \ | /
> \|/
> 13/7
>
> Some other examples of this sort would be 12:14:15:18 and
16:19:20:24.
>
> The last example I have posted about before using an 18:21:23.
>
> 23/21
> /|\
> / | \
> / | \
> / | \
> / | \
> 12/7----+---23/18
> /|\ | /|\
> / | \ | / | \
> / | \ | / | \
> / | \ | / | \
> / | \|/ | \
> 216/161--+----1/1----+--161/108
> \ | /|\ | /
> \ | / | \ | /
> \ | / | \ | /
> \ | / | \ | /
> \|/ | \|/
> 36/23---+----7/6
> \ | /
> \ | /
> \ | /
> \ | /
> \|/
> 42/23
>
> The idea of those posts was that if you were to lattice out the
> 18:21:23 scale in 14 or 23-tET you'd have a situation that was
> somewhat analogous to the diatonic scale where the comma -- a
> 7889/7776 in the following 14-tET example -- is absorbed by the
> temperament and an additional consonant chord results. (In this case
> an additional bimodal chord.)

23 Limit! Yes!!!

>
> Here's the 14-tET example with the additional consonant chord:
>
> 8-----------2
> /|\ /|\
> / | \ / | \
> / | \ / | \
> / | \ / | \
> / | \ / | \
> 3-----+----11-----+-----5
> \ | /|\ | /|\
> \ | / | \ | / | \
> \ | / | \ | / | \
> \ | / | \ | / | \
> \|/ | \|/ | \
> 6-----+-----0-----+-----8
> \ | /|\ | /|\
> \ | / | \ | / | \
> \ | / | \ | / | \
> \ | / | \ | / | \
> \|/ | \|/ | \
> 9-----+-----3-----+----11
> \ | / \ | /
> \ | / \ | /
> \ | / \ | /
> \ | / \ | /
> \|/ \|/
> 12-----------6
>
> Anyway, this type of a "centered squares" template seems like a nice
> way to generate scales with mirrored symmetry based on full or
partial
> identities.

It a great and useful concept that I plan to explore.

Thanks,

Jacky Ligon

I wrote,

The idea was basically this: Let the 1-b3-3-5 "bimodal triad" be,

b c c
/|\ /|\ /|\
/ | \ / | \ / | \
a--+--c or, a--+--d or, a--+--c-b
\ | / \ | / \ | /
\|/ \|/ \|/
c-b b b

The third example should've read:

c
/|\
/ | \
a--+--c+b
\ | /
\|/
b

--Dan Stearns

Jacky Ligon wrote,

<<Why do my ears tell me the inversional symmetry is melodically
useful?>>

Interestingly, maximally even subsets, or L-out-of-M Fibonacci scales,
tend towards inversional symmetry and repeating tetrachord like
structures. I find this useful, and in fact would have to say that
maximally even subsets seem to be the most "natural" rotation for
two-term scales (inasmuch as this structure results directly from
their constitution).

Perhaps tetrachordal similarity has something to do with what your
ears are telling you here... ?

<<It a great and useful concept that I plan to explore.>>

The next "centered square" after three would be five squared. This
would give a pretty massive 25-tone thing. I haven't really looked at
this, so I'm not sure what to do with it or even how it might work...
simply filling it out with bimodal triads won't do the trick, it'll
take a different type of identity (or partial identity, etc.) to make
it work -- I think a 6:7:8 or an 11:13:15 square would work in the
form of

b
/ \
/ \
a-----c
\ /
\ /
c-b

The regular sequence of squares could be used to generate other
symmetrical structures as well I would imagine (four squared could
make use of the 0-4-5-9 bimodal triad in 16 for example), but then
again I haven't really looked at these either.

--Dan Stearns

Wierd, just as soon as I hit send on that last post it occurred to me
that the sequence of squares are mimicking a sequence of otonal and
utonal identities just as "the tooth fairies algorithm" I posted quite
a while back did!

So a sequence of squares

0,1,4,9,16,25,36,...

is mimicked by a sequence of identities

1:2:3, 1/(3:2:1)
2:3:4, 1/(4:3:2)
3:4:5, 1/(5:4:3)
4:5:6, 1/(6:5:4)
5:6:7, 1/(7:6:5)
6:7:8, 1/(8:7:6)
7:8:9, 1/(9:8:7)

etc.

if the square is taken in the form of

b
/ \
/ \
a-----c
\ /
\ /
c-b

--Dan Stearns