some continued experiments inspired by Enrique Ubieta's "bimodal chord"

Hello all,

Hope everyone has been having an enjoyable and productive first summer
of the new millennium! These are some continued experiments that were
inspired by the Cuban composer Enrique Ubieta's bimodal chord (i.e., a
1-b3-3-5, four note chord that combines the major and minor triad)
that I was looking at before I took off earlier in the summer to do
some vacationing and composing.

If you let "a" and "b" represent a major and minor third respectively,
then the following (where the "i" is meant to indicate inversion; or
"inverted," and "t" any given pitch; or "tonic") can be seen as a
generalized template for generating nine tone scales where each scale
degree is connected to a centralized tonic:

a-b
|
|
|
|
|
bi | a
\ | /
\ | /
\ | /
\ | /
\|/
(a+b)i----------t---------a+b
/|\
/ | \
/ | \
/ | \
/ | \
ai | b
|
|
|
|
|
(a-b)i

While this template can be used to generate an unusual array of JI
scales (or rather scales based on JI ratios), in an earlier post I had
mentioned that the bimodal "triads" that I most gravitated to by ear
were the ones where the spacing between the thirds was closer to a
major second than it was to a minor second (i.e., the ones with small,
or narrow minor thirds, and large, or wide major thirds), and I've
been most interested in tempering those with a 5L & 4s step structure
by stacking a linear interval that falls between 4/5ths and 7/9ths of
an octave:

4/5 7/9
11/14
15/19 18/23
19/24 26/33 29/37 25/32

(etc.)

There are obviously several ways to look at this as far as what's
being tempered out, but one of the simplest would seem to me to be
looking at this nine note composite as a tempered cross set of a
6:7:8:9 where the 49/48 disappears:

14/9-----7/6-----7/4
,' `. ,' = `. ,' = `.
16/9-----4/3--=--1/1--=--3/2-----9/8
`. = ,' `. = ,' `. ,'
8/7----12/7-----9/7

However, I have also tried an assortment of different 'bimodal
identities,' and one of those that I liked quite a bit was a
18:21:23:26&5/6ths. Plugging this into the 9-tone template, and you
get the following:

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

Now if you were to lattice this out in say 14-tET, you have a
situation somewhat analogous to the standard diatonic scale where the
comma -- a 7889/7776 in this case -- is absorbed by the temperament
and an additional consonant identity or chord results:

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

While I've also used the 9-tone scales of 19 and 23-tET, I've mainly
been focusing on 14. This is a 14-tET, eight letter notation that I've
been experimenting with that is based on the 9-tone scale template:

C
A 3 E 11
H 6 G 8
F 9 I 5
D 12 B 2
Bb 1 D# 13
Ib 4 F# 10
Gb 7 H# 7
Eb 10 A# 4
Cb 13 C# 1
Ab 2 E# 12
Hb 5 G# 9
Fb 8 I# 6
Db 11 B# 3
Bbb 14 Dx 0

I'm using the symmetrical, or rather alternating L s step structure
(as opposed to a more diatonic like tetrachordal type structure), as I
wanted the unaltered note spellings to coincide with the 9-tone form
where each scale degree is connected to the centralized tonic (that
being C here):

A#
/|\
/ | \
D#-+--H#
/|\ | /|\
/ | \|/ | \
G--+--B--+--F#
/|\ | /|\ | /|\
/ | \|/ | \|/ | \
A--+--E--+--I--+--D#
/|\ | /|\ | /|\ | /|\
/ | \|/ | \|/ | \|/ | \
D--+--H--+--C--+--G--+--B
\ | /|\ | /|\ | /|\ | /
\|/ | \|/ | \|/ | \|/
Bb-+--F--+--A--+--E
\ | /|\ | /|\ | /
\|/ | \|/ | \|/
Ib-+--D--+--H
\ | /|\ | /
\|/ | \|/
Gb-+--Bb
\ | /
\|/
Eb

C 0 1st
C#/Bb 86
B 171 2nd
A 257 3rd
A#/Ib 343
I 429 4th
H 514 5th
H#/Gb 600
G 686 6th
F 771 7th
F#/Eb 857
E 943 8th
D 1029 9th
D#/Cb 1114
C 1200 10th

While the 18:21:23:26&5/6ths 'bimodal identity' that I showed earlier
is strictly proper,

0 157 267 424 509 691 776 933 1043 1200
0 110 267 352 534 619 776 886 1043 1200
0 157 242 424 509 666 776 933 1090 1200
0 85 267 352 509 619 776 933 1043 1200
0 182 267 424 534 691 848 958 1115 1200
0 85 242 352 509 666 776 933 1018 1200
0 157 267 424 581 691 848 933 1115 1200
0 110 267 424 534 691 776 958 1043 1200
0 157 314 424 581 666 848 933 1090 1200
0 157 267 424 509 691 776 933 1043 1200

this scale as it sits in 14-tET is a proper scale as it has shared (or
"ambiguous") augmented and diminished 3rds & 4ths, 5ths & 6ths, and
7ths & 8ths:

0 171 257 429 514 686 771 943 1029 1200
0 86 257 343 514 600 771 857 1029 1200
0 171 257 429 514 686 771 943 1114 1200
0 86 257 343 514 600 771 943 1029 1200
0 171 257 429 514 686 857 943 1114 1200
0 86 257 343 514 686 771 943 1029 1200
0 171 257 429 600 686 857 943 1114 1200
0 86 257 429 514 686 771 943 1029 1200
0 171 343 429 600 686 857 943 1114 1200
0 171 257 429 514 686 771 943 1029 1200

For more on Ubieta's bimodal chord, see:

<http://www.ubieta.com/bimodalism/BimodalHarmony.htm>

Dan Stearns

Earlier I wrote,

> This is a 14-tET, eight letter notation that I've been experimenting
with that is based on the 9-tone scale template

That obviously should have read a nine letter notation, and not an
"eight letter notation." I also noticed that I had a different layout
for this that put the three,
______ _____
/ \ / \
6 9 11 14 3 5 8 11 13 2
\___/

V-I-VI bimodal chord coverage in a more usual place:

A
H 3 C 11
F 6 E 8
D 9 G 5
Bb 12 I 2
Ib 1 B 13
Gb 4 D# 10
Eb 7 F# 7
Cb 10 H# 4
Ab 13 A# 1
Hb 2 C# 12
Fb 5 E# 9
Db 8 G# 6
Bbb 11 I# 3
Ibb 14 B# 0

H#
/|\
/ | \
B--+--F#
/|\ | /|\
/ | \|/ | \
E--+--I--+--D#
/|\ | /|\ | /|\
/ | \|/ | \|/ | \
H--+--C--+--G--+--B
/|\ | /|\ | /|\ | /|\
/ | \|/ | \|/ | \|/ | \
Bb-+--F--+--A--+--E--+--I
\ | /|\ | /|\ | /|\ | /
\|/ | \|/ | \|/ | \|/
Ib-+--D--+--H--+--C
\ | /|\ | /|\ | /
\|/ | \|/ | \|/
Gb-+--Bb-+--F
\ | /|\ | /
\|/ | \|/
Eb-+--Ib
\ | /
\|/
Cb

A 0 1st
A#/Ib 86
I 171 2nd
H 257 3rd
H#/Gb 343
G 429 4th
F 514 5th
F#/Eb 600
E 686 6th
D 771 7th
D#/Cb 857
C 943 8th
C#/Bb 1029
B 1114 9th
A 1200 10th

I. 1 2 3 4 5 6 7 8 9 10
II. 1 b2 3 b4 5 b6 7 8 b9 10
III. 1 2 3 4 5 6 #7 8 9 10
IV. 1 b2 3 b4 5 6 7 8 b9 10
V. 1 2 3 4 #5 6 #7 8 9 10
VI. 1 b2 3 4 5 6 7 8 b9 10
VII. 1 2 #3 4 #5 6 #7 8 9 10
VIII. 1 2 3 4 5 6 7 8 b9 10
IX. 1 b2 3 b4 5 b6 7 b8 b9 10
X. 1 2 3 4 5 6 7 8 9 10

Here's a bit of how this would sit on a modified staff:

---------------------------I-------------------------------
A
-------------------------B---------------------------------
C
-----------------------D-----------------------------------
E
---------------------F-------------------------------------
G
-------------------H---------------------------------------
I
-----------------A-----------------------------------------
B
--C--
D
-------------E---------------------------------------------
F
-----------G-----------------------------------------------
H
---------I-------------------------------------------------
A
-------B---------------------------------------------------
C
-----D-----------------------------------------------------
E
---F-------------------------------------------------------

Dan