9th and 13th chords and scales and S3 x S4

Take a set, on the diatonic side of M5. For example, the common ninth chord C E G Bb D (C7,9)

It's complement is 1,3,5,6,8,9,11

This can be spelled as the 13th chord – septachord scale –

Db7,9,11b13 (nicely spaced 9th chords produce well-spelled 13th chords)

Now with either we can apply the S4 X S3 transform, and pick up all these chords:

0,4,7,10,2
0,4,1,10,2
0,8,5,2,10
0,8,11,2,10

Disregarding chromatic chords (those on the chromatic side of M5) we pick up Fm11,13

Now going to more severe S3 X S4 (beyond S3 X D4) we pick up

0,4,7,11,10
0,4,1,5,10
0,8,5,1,2
0,8,11,7,2

So we pick up CMaj7#13, and DbMaj7b9

So we have 4 good chords in Class H, subclasses 3 and 17

(13 classes of S4 x S3 and 25 classes of D4 x S3) thus

* * *

1) 25 Classes for S3 X D4, 24 reduced for extra Z-relation
2) 13 Classes for S3 X S4
3) 25 Classes with Odd and Even with displacement (same as 1)
4) 13 Classes for Odd and Even without displacement (but not the same as 2)

Staircase

2111, 2210, 3110,3200
410 X
320 X,X
311 X,X,X
221 X,X,X,X
Classes for 1,3

20
14,21
12,13,16
8,22/9,15

7,18
3,17
6,10/2,11,19

5,25
4,23

1,24

In terms of 2

A1

B2 G2

C2+D1 H2 K2

E2+F2 I3 + J2 L2 M2

Staircase again for 1,3

1
2 2
3 2 2
4 5 2 2

Split

1
2 2
2,1 2 2
2,2 3,2 2 2

* * *

Continuing ...

C7,9
Fm11,13
CMaj7#13
DbMaj7b9

In terms of 13th chords:

Db7,9,11b13
EmMaj7,9,11,13
F#mMaj7,9,#11,b13
B7Maj7,11,13

I don't like the last one – splits the seventh. But otherwise good. And there may be other spellings possible, it would be interesting to study the pantonality and enharmonics of these chords also.