tetrads in decatonic scales, notated "conventionally"

Dave Keenan wrote,

>Erlich Sym dec
>A/ E/ B/ F#/ C#/
>Eb Bb F C G
>
>Erlich Pent dec
> E/ B/ F#/ C#/ G#/
>Eb Bb F C G

I would think a forward slash should represent a
sharpening and a backslash should represent a lowering.
Rather than reverse your symbols, I'll use + and -.

OK, as requested by Dave, I've re-done TD 221.19 using
this notation. I also noticed a few things that I missed
before (though if you're interested, you should still
read the original posting).

First, re-doing my paper:

Pentachordal decatonic scale:
F#- G G#- Bb B- C C#- Eb E- F (F#-)

The tetrads formed by scalar template 1,4,7,9:

1. F#- Bb C#- E- = F#- A#-- C#- E- (~4:5:6:7)
2. G B- Eb F (dissonant)
3. G#- C E- F#- (dissonant)
4. Bb C#- F G = Bb Db+ F G (~1/6:1/5:1/4:1/7)
5. B- Eb F#- G#- (dissonant)
6. C E- G Bb (~4:5:6:7)
7. C#- F G#- B- = C#- E#-- G#- B- (~4:5:6:7)
8. Eb F#- Bb C = Eb Gb+ Bb C (~1/6:1/5:1/4:1/7)
9. E- G B- C#- (~1/6:1/5:1/4:1/7)
T. F G#- C Eb (dissonant)

Symmetrical decatonic scale:
B- C C#- Eb E- F F#- G A- Bb (B-)

The tetrads formed by scalar template 1,4,7,9:

1. B- Eb F#- A- = B- D#-- F#- A- (~4:5:6:7)
2. C E- G Bb (~4:5:6:7)
3. C#- F A- B- (dissonant)
4. Eb F#- Bb C = Eb Gb+ Bb C (~1/6:1/5:1/4:1/7)
5. E- G B- C#- (~1/6:1/5:1/4:1/7)
6. F A- C Eb (~4:5:6:7)
7. F#- Bb C#- E- = F#- A#-- C#- E- (~4:5:6:7)
8. G B- Eb F (dissonant)
9. A- C E- F#- (~1/6:1/5:1/4:1/7)
T. Bb C#- F G = Bb Db+ F G (~1/6:1/5:1/4:1/7)

Next, the "major sevenths" and "minor sevenths" constructed
from 5-limit intervals:

Pentachordal decatonic scale:
F#- G G#- Bb B- C C#- Eb E- F (F#-)

The tetrads formed by scalar template 1,4,7,10:

1. F#- Bb C#- F = F#- A#-- C#- E#--(~8:10:12:15)
2. G B- Eb F#- (dissonant)
3. G#- C E- G (dissonant)
4. Bb C#- F G#- = Bb Db+ F Ab+ (~10:12:15:18)
5. B- Eb F#- Bb = B- D#-- F#- A#-- (~8:10:12:15)
6. C E- G B- (~8:10:12:15)
7. C#- F G#- C = C#- E#-- G#- B#-- (~8:10:12:15)
8. Eb F#- Bb C#- = Eb Gb+ Bb Db+ (~10:12:15:18)
9. E- G B- Eb (dissonant)
T. F G#- C E- (dissonant)

Symmetrical decatonic scale:
B- C C#- Eb E- F F#- G A- Bb (B-)

The tetrads formed by scalar template 1,4,7,10:

1. B- Eb F#- Bb = B- D#-- F#- A#-- (~8:10:12:15)
2. C E- G B- (~8:10:12:15)
3. C#- F A- C (dissonant)
4. Eb F#- Bb C#- = Eb Gb+ Bb Db+ (~10:12:15:18)
5. E- G B- Eb (dissonant)
6. F A- C E- (~8:10:12:15)
7. F#- Bb C#- F = F#- A#-- C#- E#--(~8:10:12:15)
8. G B- Eb F#- (dissonant)
9. A- C E- G (~10:12:15:18)
T. Bb C#- F A- (dissonant)

For the next set, instead of using scalar template 1,3,7,9,
I'll use the first inversion, 1,5,7,9. That way the two
"consonant" chord types resemble: a 7sus4, which better
expresses the combination of 7-limit and 3-limit implications,
and a 14:18:21:24, which is the form that the second saturated
9-limit tetrad was originally mentioned on this list, by Kami
Rousseau, as being his favorite tuning of a 6th chord. The
"dissonant" chord types are actually not bad, being incomplete
versions of 9-limit pentads.

Pentachordal decatonic scale:
F#- G G#- Bb B- C C#- Eb E- F (F#-)

The tetrads formed by scalar template 1,5,7,9:

1. F#- B- C#- E- (7sus4)
2. G C Eb F (~9:12:14:16)
3. G#- C#- E- F#- (~9:12:14:16)
4. Bb Eb F G (~1/12:1/9:1/8:1/7)
5. B- E- F#- G#- (~1/12:1/9:1/8:1/7)
6. C F G Bb (7sus4)
7. C#- F#- G#- B- (7sus4)
8. Eb G Bb C (~14:18:21:24)
9. E- G#- B- C#- (~14:18:21:24)
T. F Bb C Eb (7sus4)

Symmetrical decatonic scale:
B- C C#- Eb E- F F#- G A- Bb (B-)

The tetrads formed by scalar template 1,5,7,9:

1. B- E- F#- A- (7sus4)
2. C F G Bb (7sus4)
3. C#- F#- A- B- (~9:12:14:16)
4. Eb G Bb C (~14:18:21:24)
5. E- A- B- C#- (~1/12:1/9:1/8:1/7)
6. F Bb C Eb (7sus4)
7. F#- B- C#- E- (7sus4)
8. G C Eb F (~9:12:14:16)
9. A- C#- E- F#- (~14:18:21:24)
T. Bb Eb F G (~1/12:1/9:1/8:1/7)

And finally,

Pentachordal decatonic scale:
F#- G G#- Bb B- C C#- Eb E- F (F#-)

The tetrads formed by scalar template 1,4,6,9:

1. F#- Bb C E- = Gb+ Bb C E- (French 6th)
2. G B- C#- F = G B- C#- E#-- (French 6th)
3. G#- C Eb F#- (dissonant)
4. Bb C#- E- G (diminished 7th)
5. B- Eb F G#- (dissonant)

Since the scalar template is its own second inversion,
the rest of the chords are just the second inversions
of the above.

Symmetrical decatonic scale:
B- C C#- Eb E- F F#- G A- Bb (B-)

The tetrads formed by scalar template 1,4,6,9:

1. B- Eb F A- = Cb+ Eb F A- (French 6th)
2. C E- F#- Bb = C E- F#- A#-- (French 6th)
3. C#- F G B- = C#- E-- G B-- (French 6th)
4. Eb F#- A- C (diminished 7th)
5. E- G Bb C#- (diminished 7th)

same deal with 2nd inversions.