I begin by selecting a favorite trichordal set-class. For me, this has long been 013 (in 12-edo).
I'm drawn to it's tonal associations, and, in many spacings and registrations, it has a
poignant, melancholic character, which seems to be what I am often after in my music.
I want to work in a higher Equal-Temperament, to take advantage of microtonal possibilities
not readily available in standared 12-edo instrumental music. Minor 3rds, (an interval in the 013 set),
are only .2 cents off from Just Intonation in 19 Tone-Equal-Temperament.
Therefore, we elect to work in 19 Tone-Equal-Temperament.
Next, we notice that in terms of sound, there are several trichords in 19tet that
can arguably be said to "sound like" 013 in 12tet. As follows:
- 013 -- C C# D -- very squished in 19-edo
- 014 -- C C# D# -- poignant due to the C-C#
- 015 -- C C# Eb -- (same)
- 025 -- C Db Eb -- Most similar-sounding to 12-edo's 013
We build a 19tet row, saturated with these 013 "sound-alikes":
6 1 4 3 8 5 18 2 16 0 14 17 13 9 12 7 10 15 11
whose saturation is as follows:
- 6 1 4 [0 2 5]
- 1 4 3 [0 1 3]
- 4 3 8 [0 1 5]
- 3 8 5 [0 2 5]
- 18 2 16 [0 2 5]
- 2 16 0 [0 2 5]
- 16 0 14 [0 2 5]
- 14 17 13 [0 1 4]
- 13 9 12 [0 1 4]
- 9 12 7 [0 2 5]
- 12 7 10 [0 2 5]
- 10 15 11 [0 1 5]
Next, from this row, we build a 6-line combinatorial array. (i.e.---Each vertical partition holds all 19 pitch-classes once each.
Each horizontal lyne, going across, holds a form of the row.)
* * * (11 7 12 15 10 13 9) * (5 8 3 6 1 4 17 14 0 18 2 16)
(11 7 12 15 10 13 9) (5 8 3) (6 1 4) (17 14 0) (18 2 16) *
* (11 15 10 7 12 9 13 17 14 0) (16 2 18) (5 8 3) (4 1 6) *
* (16 2 18) (0 14 17) (4 1 6) (3 8 5) (9 13 10 15 12 7 11)
(6 1 4 3 8 5 18 2 16 0 14 17) * (13 9 12 7 10 15 11) * * *
* (6 1 4) (3 8 5) (18 2 16) (0 14 17 13 9 12 7 10 15 11) *
(For these array examples, please scroll across. . . . )
This contains each of the 4 forms (P I R RI) of the "row" at least once.
2 of them are found twice. (The one beginning 6 1 4 3 8. . .
and the one beginning 11 7 12 15 10. . . .)
Thus, if we transpose this array-block 19 times, we will have 19
blocks, each transposed to a different pitch-class, thus containing all
19X4 forms of the row (76). (Plus 19X2
duplicates). This will
happen with any constant transposition-factor from block to block. If we
transpose it by 5 (a minor thrid in 19t-ET) each time, for example, and re-arrange the vertical
positioning of the lines according to a certain scheme (which I'm not going to detail at this point),
from each block to the next, then we will get quite a few swapping
possibilities between blocks.
This enables us to swap around in the giant 114-partition block produced,
and maybe get something approaching the all-partition-array
idea, used by Babbitt in his 12Tet works. To actually produce
all the partitions of 19 items into 6 or fewer parts (lines), would be a
tremendous task, so I first limited myself to producing an
"all-different-partition" array. That is, given the basic 6 block
construct shown above, transposed 19 times, we get 114 blocks
(partitions.) Through swapping we might try to get 114 different
partitional shapes. This also proved an unwieldy task, so I opted for a half-way solution.
The half-way solution was to come up with different partitions for
the first 60 blocks, which gives us:
* * * (11 7 12 15 10 13 9) (5) (8 3 6 1 4 17 14 0 18 2 16) * (11 6 9) (8 13 10 4) (7 2 5 0) (3 18 14 17 12 15 1 16) * * (2 6 1 17 3 0 4 8 5) (10 7 12 9 15) (18) (13 14 11 16) * * (12 17 14) (15 10 13) (0 16 2 18) (4) (1 5 9 6 11 8 3 7) * * * (12 8 13 16 11 14 10) (6 9 4) (7 2 5 18 15 1 0 3 17) (12 7) (10) (9 14 11 5 8) (3) (6 1 4 0 15 18 13 16 2 17) * * (3 7 2 18 4 1 5 9 6 11 8) (13 10 16 0) (14) (15 12 17 13) (18) * (15) (16 11 14) (1 17 3 0) * (5 2 6 10 7 12 9 4) (8) * (13) * (9 14 17 12 15) (11) (7 10 5 8 3 6 0 16 2 1 4 18 13) (8 11) (10 15 12) (6 9 4 7 2 5 1 16 0 17 3 18) * *
(11 7 12 15 10 13 9) (5 8 3) (6 1 4) (17 14 0) (18 2 16) * (11 6 9 8 13 10) (4 7 2 5 0 3) (18 14 17 12 15 1 16) * * * (2 17 3 6 1 4 0) (15 18 13) (16 11 14) (8 5) (10 9 12 7) (2) (16 0 18 4 1 14 17 12 15 10 13) * (9 5 8 3 6 11) (7) * * * (12 8 13 16 11 14 10 6 9 4) (7 2 5 18) * (15 1 0 3 17 12 7) * (10 9 14 11 5 8 3 6 1 4) * (0 15 18 13 16 2 17) * (3) * (18 4 7 2 5 1) (16 0 14) (17 12 15) (9 6 11) (10) (13 8 3) (17 1 0 5 2 15) (18 13 16 11 14) (10 6 9 4 7 12) * (8) (13) (9 14) (17 12 15 11 7 10 5) (8) (3 6) (0 16 2 1 4 18 13 8 11) (10 15 12 6 9 4 7 2 5) * (1 16 0 14 17 3) (18) * * *
* (11 15 10 7 12 9 13 17 14 0) (16 2 18) (5 8) (3 4 1 6) * (2 7 4 5 0 3) * * (9 6 11) (8 13 10) (14 18 15 1 17 12) (16) * * (2 17 3 6 1 4 0) * (15 18 13 16 11 14 8 5 10 9 12) (7 2) (16 0) (18) (4 1 14 17 12) (15 10 13 9 5 8 3 6 11 7 12) * (16 11 8 13 10 14 18 15 1) * (17 3) (0 6 9 4) (5 2) * * (7 3 8 5 6) * (1 4 10 7 12) * (9 14 11 15 0 16 2 18 13 17) * * * (3 18 4 7 2 5 1) (16) (0 14 17 12 15 9 6 11 10) (13) (8 3 17 1) (0 5 2) * (15 18 13 16 11 14 10 6 9 4 7 12) (8) (13 17 12) (9) (14 11 15 0 16 2 18 4 1 7 10) * (5 6 3) (8) * (4 9 6 7 2 5) * (11 8 13 10) (15 12 16 1 17 3 0 14 18) *
* (16 2 18) (0 14 17) (4 1 6 3) (8) (5 9 13 10 15 12 7 11) * * * (16 12 17 1 15 18 14) * (10 13 8 11 6 9 3 0 5 4 7 2) * (16 11 14) (13 18) (15 9 12 7 10) (5 8 4 0 3 17 1 6 2) * * (7 11 6 3 8 5 9 13 10 15) (12 17 14 1 4) * (18 0 16 2) * (17) (3 0) (1 15) (18 5 2 7) * (4 9 6 10 14 11 16 13 8) * (12) * (17 13 18 2 16 0 15) (11 14 9 12) (7 10 4 1 6 5 8) (3) (17 12 15) (14) (10 13 8) (11 6 9 5 1 4 18 2 7 3) * (8 12 7 4 9 6 10 14 11 16) * (13 18 15) * (2 5) (0 1 17 3 18) * (4 1 2 16 0 6) (3) (8 5) (10 7) * (11 15 12 17 14 9) (13) * * * (18 14 0 3 17 1 16 12 15 10 13 8 11 5 2 7 6 9 4)
(6 1 4 3 8 5 18 2 16 0 14 17) * (13 9 12 7 10 15 11) * * * (16 12 17 1 15 18 14) (10 13 8) (11 6 9) (3) (0 5 4 7 2) (16) (11 14 13 18 15 9 12 7 10 5 8) * (4 0 3 17 1 6 2) * * (7) (3 8 11 6 9 5) (1 4 18) (2 16 0) (13 10 15) (14 17) (12) (7 2 5 4 9 6 0 3) (17 1 15 18) (14 10 13 8 11 16 12) * * * * (17 13 18 2 16 0 15 11 14 9) (12 7 10 4 1) (6) (5 8) (3) (17 12 15 14 0 16) (10 13) (8 11 6 9 5 1 4 18 2 7 3) * (8) * * (4 9 12 7 10 6 2 5 0) (3 17 1) (14 11 16 15 18 13 8) * * (3 6 5 10 7 1 4 18 2 16 0 15 11) (14) (9 12 17) (13) * (18 14 0 3 17 1 16) * (12 15 10) (13 8 11) * (5 2 7 6 9 4) *
* (6 1 4) (3 8 5) (18 2 16) (0 14 17 13 9 12 7 10 15 11) * * (16 1 15 12 17 14 18) (3 0 5 2 7) (4 10 13 8) (9 6 11) * * (7 12 9 10) (5 8) (14 11 16 13) (18 15) (0 4 1 6 3 17) * (2) (7) (3 8 11 6 9 5) (1) (4 18 2 16 0 13 10 15 14 17) (12) (7 2 5) (4 9 6 0) (3 17 1 15) (18 14 10 13 8 11 16) (12) (17 2 16 13 18 15 0) (4 1) (6 3) (8 5 11 14 9) (10 7) (12) (8 13 10 11 6 9) * * (0 16 15 12 17) (14 0) (16 1 5 2 7 4) (18 3) * (8) (4 9 12 7 10 6 2 5) (0 3 17 1) (14 11 16 15) * (18 13 8 3) (6 5) (10 7 1 4 18 2 16 0 5 11 14 9 12 17) * (13) * (18) (3 17 14 0 16 1 5 2 7 4 9 6) (12 15) (10 11 8 13) *
Finally, I use not the simple pitch array by itself, but in canonic
counterpoint with itself, as the final pitch super-array.
The lines of the
original (dux) are "in your face", or un-spatialized.
The lines of the new (comes) array are "spatialized"
"reverberated", echoed, or whatever.
Registrally,dux and comes intertwine 0-with-0, 1-with-1,
etc. (It turns out that this particular intertwining will give me (I think) the most unisons between pairs,
which is what I was after.) Thus, the final pitch superarray is:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
* * * (11 7 12 15 10 13 9) (5) (8 3 6 1 4 17 14 0 18 2 16) * (11 6 9) (8 13 10 4) (7 2 5 0) (3 18 14 17 12 15 1 16) * * (2 6 1 17 3 0 4 8 5) (10 7 12 9 15) (18) (13 14 11 16) * * (12 17 14) (15 10 13) (0 16 2 18) (4) (1 5 9 6 11 8 3 7) * * * (12 8 13 16 11 14 10) (6 9 4) (7 2 5 18 15 1 0 3 17) (12 7) (10) (9 14 11 5 8) (3) (6 1 4 0 15 18 13 16 2 17) * * (3 7 2 18 4 1 5 9 6 11 8) (13 10 16 0) (14) (15 12 17 13) (18) * (15) (16 11 14) (1 17 3 0) * (5 2 6 10 7 12 9 4) (8) * (13) * (9 14 17 12 15) (11) (7 10 5 8 3 6 0 16 2 1 4 18 13) (8 11) (10 15 12) (6 9 4 7 2 5 1 16 0 17 3 18) * *
* * * * * * * * * (11 7 12 15 10 13 9) (5) (8 3 6 1 4 17 14 0 18 2 16) * (11 6 9) (8 13 10 4) (7 2 5 0) (3 18 14 17 12 15 1 16) * * (2 6 1 17 3 0 4 8 5) (10 7 12 9 15) (18) (13 14 11 16) * * (12 17 14) (15 10 13) (0 16 2 18) (4) (1 5 9 6 11 8 3 7) * * * (12 8 13 16 11 14 10) (6 9 4) (7 2 5 18 15 1 0 3 17) (12 7) (10) (9 14 11 5 8) (3) (6 1 4 0 15 18 13 16 2 17) * * (3 7 2 18 4 1 5 9 6 11 8) (13 10 16 0) (14) (15 12 17 13) (18) * (15) (16 11 14) (1 17 3 0) * (5 2 6 10 7 12 9 4) (8) * (13) * (9 14 17 12 15) (11) (7 10 5 8 3 6 0 16 2 1 4 18 13) (8 11) (10 15 12) (6 9 4 7 2 5 1 16 0 17 3 18) * *
(11 7 12 15 10 13 9) (5 8 3) (6 1 4) (17 14 0) (18 2 16) * (11 6 9 8 13 10) (4 7 2 5 0 3) (18 14 17 12 15 1 16) * * * (2 17 3 6 1 4 0) (15 18 13) (16 11 14) (8 5) (10 9 12 7) (2) (16 0 18 4 1 14 17 12 15 10 13) * (9 5 8 3 6 11) (7) * * * (12 8 13 16 11 14 10 6 9 4) (7 2 5 18) * (15 1 0 3 17 12 7) * (10 9 14 11 5 8 3 6 1 4) * (0 15 18 13 16 2 17) * (3) * (18 4 7 2 5 1) (16 0 14) (17 12 15) (9 6 11) (10) (13 8 3) (17 1 0 5 2 15) (18 13 16 11 14) (10 6 9 4 7 12) * (8) (13) (9 14) (17 12 15 11 7 10 5) (8) (3 6) (0 16 2 1 4 18 13 8 11) (10 15 12 6 9 4 7 2 5) * (1 16 0 14 17 3) (18) * * *
* * * * * * (11 7 12 15 10 13 9) (5 8 3) (6 1 4) (17 14 0) (18 2 16) * (11 6 9 8 13 10) (4 7 2 5 0 3) (18 14 17 12 15 1 16) * * * (2 17 3 6 1 4 0) (15 18 13) (16 11 14) (8 5) (10 9 12 7) (2) (16 0 18 4 1 14 17 12 15 10 13) * (9 5 8 3 6 11) (7) * * * (12 8 13 16 11 14 10 6 9 4) (7 2 5 18) * (15 1 0 3 17 12 7) * (10 9 14 11 5 8 3 6 1 4) * (0 15 18 13 16 2 17) * (3) * (18 4 7 2 5 1) (16 0 14) (17 12 15) (9 6 11) (10) (13 8 3) (17 1 0 5 2 15) (18 13 16 11 14) (10 6 9 4 7 12) * (8) (13) (9 14) (17 12 15 11 7 10 5) (8) (3 6) (0 16 2 1 4 18 13 8 11) (10 15 12 6 9 4 7 2 5) * (1 16 0 14 17 3) (18) * * *
* (11 15 10 7 12 9 13 17 14 0) (16 2 18) (5 8) (3 4 1 6) * (2 7 4 5 0 3) * * (9 6 11) (8 13 10) (14 18 15 1 17 12) (16) * * (2 17 3 6 1 4 0) * (15 18 13 16 11 14 8 5 10 9 12) (7 2) (16 0) (18) (4 1 14 17 12) (15 10 13 9 5 8 3 6 11 7 12) * (16 11 8 13 10 14 18 15 1) * (17 3) (0 6 9 4) (5 2) * * (7 3 8 5 6) * (1 4 10 7 12) * (9 14 11 15 0 16 2 18 13 17) * * * (3 18 4 7 2 5 1) (16) (0 14 17 12 15 9 6 11 10) (13) (8 3 17 1) (0 5 2) * (15 18 13 16 11 14 10 6 9 4 7 12) (8) (13 17 12) (9) (14 11 15 0 16 2 18 4 1 7 10) * (5 6 3) (8) * (4 9 6 7 2 5) * (11 8 13 10) (15 12 16 1 17 3 0 14 18)
* * * * * * * (11 15 10 7 12 9 13 17 14 0) (16 2 18) (5 8) (3 4 1 6) * (2 7 4 5 0 3) * * (9 6 11) (8 13 10) (14 18 15 1 17 12) (16) * * (2 17 3 6 1 4 0) * (15 18 13 16 11 14 8 5 10 9 12) (7 2) (16 0) (18) (4 1 14 17 12) (15 10 13 9 5 8 3 6 11 7 12) * (16 11 8 13 10 14 18 15 1) * (17 3) (0 6 9 4) (5 2) * * (7 3 8 5 6) * (1 4 10 7 12) * (9 14 11 15 0 16 2 18 13 17) * * * (3 18 4 7 2 5 1) (16) (0 14 17 12 15 9 6 11 10) (13) (8 3 17 1) (0 5 2) * (15 18 13 16 11 14 10 6 9 4 7 12) (8) (13 17 12) (9) (14 11 15 0 16 2 18 4 1 7 10) * (5 6 3) (8) * (4 9 6 7 2 5) * (11 8 13 10) (15 12 16 1 17 3 0 14 18) *
* (16 2 18) (0 14 17) (4 1 6 3) (8) (5 9 13 10 15 12 7 11) * * * (16 12 17 1 15 18 14) * (10 13 8 11 6 9 3 0 5 4 7 2) * (16 11 14) (13 18) (15 9 12 7 10) (5 8 4 0 3 17 1 6 2) * * (7 11 6 3 8 5 9 13 10 15) (12 17 14 1 4) * (18 0 16 2) * (17) (3 0) (1 15) (18 5 2 7) * (4 9 6 10 14 11 16 13 8) * (12) * (17 13 18 2 16 0 15) (11 14 9 12) (7 10 4 1 6 5 8) (3) (17 12 15) (14) (10 13 8) (11 6 9 5 1 4 18 2 7 3) * (8 12 7 4 9 6 10 14 11 16) * (13 18 15) * (2 5) (0 1 17 3 18) * (4 1 2 16 0 6) (3) (8 5) (10 7) * (11 15 12 17 14 9) (13) * * * (18 14 0 3 17 1 16 12 15 10 13 8 11 5 2 7 6 9 4)
* * * * * * * (16 2 18) (0 14 17) (4 1 6 3) (8) (5 9 13 10 15 12 7 11) * * * (16 12 17 1 15 18 14) * (10 13 8 11 6 9 3 0 5 4 7 2) * (16 11 14) (13 18) (15 9 12 7 10) (5 8 4 0 3 17 1 6 2) * * (7 11 6 3 8 5 9 13 10 15) (12 17 14 1 4) * (18 0 16 2) * (17) (3 0) (1 15) (18 5 2 7) * (4 9 6 10 14 11 16 13 8) * (12) * (17 13 18 2 16 0 15) (11 14 9 12) (7 10 4 1 6 5 8) (3) (17 12 15) (14) (10 13 8) (11 6 9 5 1 4 18 2 7 3) * (8 12 7 4 9 6 10 14 11 16) * (13 18 15) * (2 5) (0 1 17 3 18) * (4 1 2 16 0 6) (3) (8 5) (10 7) * (11 15 12 17 14 9) (13) * * * (18 14 0 3 17 1 16 12 15 10 13 8 11 5 2 7 6 9 4)
(6 1 4 3 8 5 18 2 16 0 14 17) * (13 9 12 7 10 15 11) * * * (16 12 17 1 15 18 14) (10 13 8) (11 6 9) (3) (0 5 4 7 2) (16) (11 14 13 18 15 9 12 7 10 5 8) * (4 0 3 17 1 6 2) * * (7) (3 8 11 6 9 5) (1 4 18) (2 16 0) (13 10 15) (14 17) (12) (7 2 5 4 9 6 0 3) (17 1 15 18) (14 10 13 8 11 16 12) * * * * (17 13 18 2 16 0 15 11 14 9) (12 7 10 4 1) (6) (5 8) (3) (17 12 15 14 0 16) (10 13) (8 11 6 9 5 1 4 18 2 7 3) * (8) * * (4 9 12 7 10 6 2 5 0) (3 17 1) (14 11 16 15 18 13 8) * * (3 6 5 10 7 1 4 18 2 16 0 15 11) (14) (9 12 17) (13) * (18 14 0 3 17 1 16) * (12 15 10) (13 8 11) * (5 2 7 6 9 4) *
* * * * * * (6 1 4 3 8 5 18 2 16 0 14 17) * (13 9 12 7 10 15 11) * * * (16 12 17 1 15 18 14) (10 13 8) (11 6 9) (3) (0 5 4 7 2) (16) (11 14 13 18 15 9 12 7 10 5 8) * (4 0 3 17 1 6 2) * * (7) (3 8 11 6 9 5) (1 4 18) (2 16 0) (13 10 15) (14 17) (12) (7 2 5 4 9 6 0 3) (17 1 15 18) (14 10 13 8 11 16 12) * * * * (17 13 18 2 16 0 15 11 14 9) (12 7 10 4 1) (6) (5 8) (3) (17 12 15 14 0 16) (10 13) (8 11 6 9 5 1 4 18 2 7 3) * (8) * * (4 9 12 7 10 6 2 5 0) (3 17 1) (14 11 16 15 18 13 8) * * (3 6 5 10 7 1 4 18 2 16 0 15 11) (14) (9 12 17) (13) * (18 14 0 3 17 1 16) * (12 15 10) (13 8 11) * (5 2 7 6 9 4) *
* (6 1 4) (3 8 5) (18 2 16) (0 14 17 13 9 12 7 10 15 11) * * (16 1 15 12 17 14 18) (3 0 5 2 7) (4 10 13 8) (9 6 11) * * (7 12 9 10) (5 8) (14 11 16 13) (18 15) (0 4 1 6 3 17) * (2) (7) (3 8 11 6 9 5) (1) (4 18 2 16 0 13 10 15 14 17) (12) (7 2 5) (4 9 6 0) (3 17 1 15) (18 14 10 13 8 11 16) (12) (17 2 16 13 18 15 0) (4 1) (6 3) (8 5 11 14 9) (10 7) (12) (8 13 10 11 6 9) * * (0 16 15 12 17) (14 0) (16 1 5 2 7 4) (18 3) * (8) (4 9 12 7 10 6 2 5) (0 3 17 1) (14 11 16 15) * (18 13 8 3) (6 5) (10 7 1 4 18 2 16 0 5 11 14 9 12 17) * (13) * (18) (3 17 14 0 16 1 5 2 7 4 9 6) (12 15) (10 11 8 13) *
* * * * * * * (6 1 4) (3 8 5) (18 2 16) (0 14 17 13 9 12 7 10 15 11) * * (16 1 15 12 17 14 18) (3 0 5 2 7) (4 10 13 8) (9 6 11) * * (7 12 9 10) (5 8) (14 11 16 13) (18 15) (0 4 1 6 3 17) * (2) (7) (3 8 11 6 9 5) (1) (4 18 2 16 0 13 10 15 14 17) (12) (7 2 5) (4 9 6 0) (3 17 1 15) (18 14 10 13 8 11 16) (12) (17 2 16 13 18 15 0) (4 1) (6 3) (8 5 11 14 9) (10 7) (12) (8 13 10 11 6 9) * * (0 16 15 12 17) (14 0) (16 1 5 2 7 4) (18 3) * (8) (4 9 12 7 10 6 2 5) (0 3 17 1) (14 11 16 15) * (18 13 8 3) (6 5) (10 7 1 4 18 2 16 0 5 11 14 9 12 17) * (13) * (18) (3 17 14 0 16 1 5 2 7 4 9 6) (12 15) (10 11 8 13) *