Chain-of-major-thirds tunings (was: periodic table/"lousy thirds")

Since my last post on this topic. I've been having fun learning about
dieses I always wondered what they were good for. :-)

E# Cb
C# G#
A E
C G
Ab Eb
E# Cb

dBddBdddBd

So far I've found that this 10-note 10-triad scale only exists in 19-tET,
22-tET, 41-tET and other (much larger) N-tETs where N = 3B + 7d, d > 0, 4d
<= B <= 5d.

I've "discovered" diesis-pump chord sequences, analogous to the comma-pump
sequences we talked about earlier in regard to adaptive JI.In this case the
root of one chord becomes the major third of the next or vice versa.

For example, I quite like this one: C, Ab, Fb, Dbb (all major triads). In
12-tET this is just C, Ab, E, C but in 19-tET it is C, Ab, Fb(=E#), C#.
i.e. it finishes a whole 63c semitone higher than it started, and can then
resolve back to C. Lovely.

If you want to try it, here's a keyboard mapping with offsets in cents from
12-tET:

19-tET
Note Where Ideal Usable
name on kbd offset offset
----------------------------
C C -40 -46
C# C# -82 -64
Eb D +82 +63
E D# +40 +46
E# E -1 +5
F F -43 -49
G F# +62 +56
G# G +21 +27
Ab G# -21 -27
A A -62 -56
B A# +43 +49
Cb B +1 -5

The F and B (on the A# key) are not part of the decatonic scale but allow
it to be modulated up or down by one major third. The "ideal" offsets are
based on 41-tET where the fifths have about a 0.5c error and the thirds
about 6c. The "usable" offsets let you acheive a tolerable approximation if
you are using a tuning table that can only do a -64 to +63c offset (like I
am).

With the above keyboard mapping, the above-mentioned diesis pump (plus
reloution) can be rendered in four voices as:

F# G# G# G F#
D# D E E D#
C C B C# C

C G# E C# C root bass

Note: these are the keys you play, not the actual notes that sound.

I had to phone my brother, who can work out cord progressions by ear in a
flash, and play him this one. :-) I told him it started and ended on a C
chord since he doesn't have absolute pitch. After listening, he had me
confirm that it ended on the same chord it started on. He then guessed it
started C, G# and ended C#, C but the chord in the middle (call it X) had
him stumped. Presumably because he could hear that there was a sustained
note from the G# triad to X and another from X to the C# triad, but X was
clearly a major triad. Impossible in 12-tET (and many others).

Remember that Moodie Blues album "Search for the lost chord". :-)

I eventually put him out of his misery by telling him that X was "half way
between an E and an F chord" and that the G# and C# were shifted a little
too, to distribute the "stretch". But really, this is closer to Just. It's
12-tET that's "compressed".

Surely someone's investigated these scales and these chord sequences before?

Regards,
-- Dave Keenan
http://dkeenan.com