The 7-limit cube and problems

You probably know the 3-5-7 genus,octony,7-limit cube or whatever name you got for it.

D--------A
/| /|
/ | / |
E--------B |
| | | |
| Bb----|--F
| / | /
|/ |/
C--------G

(C=1/1 G=3/2 E=5/4 B=15/8 Bb=7/4 D=35/32 F=21/16 A=105/64)

The basic 7-limit chord as you also probably know is to be read as C-E-G-Bb(or A#,as to signify the approximation of meantones).The utonal version,is obtained by using the four other notes-A B D F.

Both of these basic four-note chords are to be seen as a tonic(C or A)with 3,5 and 7-direction intervals from it(G/D,E/F,Bb/B respectively).In the first version,all three of them are harmonic or"up"versions.In the second versions,they are"down"or utonal.

Now,any combination of up and down intervals for the 3,5 and 7 respectively is possible and,surprise,the 8 possible tetrachords are obtained by using each of the 8 notes in the set as a tonic.

There is another way to obtain similar chords.Consider this basic formation:
A
/
/
B
|
|
|
|
C--------G

Rather than having three parallel sprouts in the 3,5 and 7 directions,here the intervals are added to each other in series.This alternative tetrachord has 8 variants like the first one.They also share the property that they contain one each of the 3,5 and 7 intervals.

Now my question is:
Is there any other chords based on the one-of-each-interval principle?If not,what other forms are to be thought of as being particularly interesting?

What do you think is the most musically usable of the various"transpositions"of the second chord?Of the first one?How can they be tied together?

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Mats �ljare
Eskilstuna,Sweden
http://www.angelfire.com/mo/oljare
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