23h00 near Paris

[Xavier J.-P. CHARLES:]
> then the augmented triad is "12-15-19"

In 12e I used to try out a lot of different augmented triad
extensions:

0 4 8 11
0 4 8 11 3
0 4 8 11 3 6
0 4 8 11 3 6 10

I also tried carrying this out to a 10-note 1 3 #5 7 #9 #11 #13 #15
#17(18) X19(20) #21(22) set... or to put it another way:

GX
/ \
/ \
/ \
G#------D#------A#------E#------B#
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
E-------B-------F#------C#
/
/
/
C

as:

09
/ \
/ \
/ \
08----- 03------10------05------12
/ \ / \ / \ /
/ \ / \ / \ /
/ \/ \/ \/
04------11------06------01
/
/
/
00

And in all of these augmented extrapolations, I tried to see if I
could make some use of the 24/19 and 19/15 major thirds. For example I
tried the 10-note set by using two 19:15:12:10:8 maj7#5#9 chords
separated by an octave and a half (i.e. 1800�)... so in other words a
Cmaj7#5#9 and an F#maj7#5#9 at 19:15:12:10:8. And I also tried linking
two 19:24:30:36 maj7#5 chords separated by an 81/76 (which would be a
major chroma plus a diaschisma plus the 513/512 - does the difference
between a 19/16 and a 32/27 have name?):

2025/1216
/ \
/ \
/ \
/ \
30/19----45/38---135/76---405/304-1215/608
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
/ \/ \/ \/
24'19----36/19----54/38----81/76
'
'
'
'
1'1

I've also tried identities of 19 for augmented chords in much less
obtuse ways, say a progression along the lines of:

E-add9 E-add9 A-add9 Gmaj7#5
| / / / / | / / / / | / / / / | / / / / |

45/38
/ \
/ \
/ \
/ \
32/19----24'19----36/19----27/19
' ' ' '
' ' ' '
' ' ' '
'' ''
1/1------3/2

And you could render, or show the three inversions (or modulation
possibilities) of the augmented triad by stacking 24/19 (I've used
apostrophes to show the 24/19 and 19/16 connections), 5/4, and 19/15
(I've used periods to show the 19/15 and 45/38 connections):

36/19
' '
' ' ' ' ' ' '
' ' ' ' ' ' '
' '' '' ''
4/3------1/1------3/2------9/8
. . . . . . .
. . . . . . .
. . . . . . .
. .. .. ..
20/19----30/19----45/38---135/76
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
/ \/ \/ \/
32/19----24/19----36/19----27/19
' ' ' ' ' ' '
' ' ' ' ' ' '
' ' ' ' ' ' '
' '' '' ''
4/3------1/1------3/2------9/8

Dan

11/6
. .
. .
. .
. .
4/3------1/1------3/2-------9/8
' ' ' '
' ' ' '
' ' ' '
'' ''
14/9------7/6

This is a i-iv-III harmonic minorish variation on a progression (that
I gave a few days ago),

C-add9 C-add9 F-add9 Emaj7#5
| / / / / | / / / / | / / / / | / / / / |

that attempts to make use of the 7:9:11 that lies inside the
12:14:18:22:27. I think that this sort of an augmented (vertical)
tempering, or closing of the grid of thirds (apostrophes show the 9/7,
periods show the 11/9, and asterisks show the 14/11):

--*--
* *
* * * * * * * * *
* * * * * * * * *
* ** ** ** **
--44/27----11/9-----11/6-----11/8-----33/32--
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. .. .. .. ..
--4/3------1/1------3/2------9/8-----27/16--
' ' ' ' ' ' ' ' '
' ' ' ' ' ' ' ' '
' ' ' ' ' ' ' ' '
' '' '' '' ''
--28/27----14/9------7/6------7/4-----21/16--
** ** ** ** **
* * * * * * * * * *
* * * * * * * * * *
* *
--*--

reveals some interesting modulation possibilities.

Dan