Neo Riemannian, Neo Riemannian

Following up from the last blog on Neo Riemannian theory especially in the area of Triad Transformation I've played around with extending these essential moves because each new transformation has these 3 possibilities and so on. These examples could be done in various other ways... It is probably predicted in my friend CKM equations that are for another day.

To review: RPL—remember that major triads transform to minor triads and vice-versa.

In R:  C major (triad) transforms to Ami by raising the 5th 'G' to 'A' creating Ami triad.
In P:  C major (triad) transforms to Cmi by lowering the third 'E' to 'Eb' creating a Cmi triad.
In L:  Cmajor (triad) transforms to Emi by lowering the root 'C' to 'B' creating an Emi triad'

In R: Cmi (triad) transforms to Ebma by lowering the root 'C' to 'Bb' creating an Ebma triad.
In P:  Cmi triad transforms to Cma by raising the minor 3rd 'Eb' to 'E' creating a Cma triad.
In L:  Cmi triad transforms to Abma by raising the 5th 'G' to an 'Ab' creating an Abma triad.

Here's a few examples starting in C major:

C........Ami......C.......Cmi..........Ab............Abmi..........E...........Emi.........C ||


R      R          P L P L           P        L
CEG —CEA - ceg-  CEbG  – CEbAb —CbEbAb— BEG# — BEG —CEG etc.  


I didn't use R very much so here's another:

C.........Ami.....C.....Cmi..........Ab (Ab/C)....Fmi............Db...........Dbmi.
        

R       R       P L R L                P

CEG——CEA -ceg- CEbG——CEbAb——CFAb——DbFAb——DbFbAb

My super math friend CKM says:  ..... "Ok.  All of these are "kosher" R, L and P transforms. 
My only point ........(which you got) was that it is IMPOSSIBLE ever get anything except a major and minor triad using R, L, and P in any combination.  For that, new transformations and/or new "rules" of transformation need to be invoked.  But that's okay.  All RLP can ever do is move to one of the adjacent triangles on the Tonnetz Torus or (equivalently) a neighboring point in Dancing Cubes." 


Dancing Cubans I've seen, but "dancing cubes" sound like they might be electron microscopic images:)....  But makes a little more sense when the Revolving Torus from the last blog is viewed. Some links available to me explain this more but, they are quite involved and lengthy with lots of side links to both educate and distract so I will be following this up later.


I'll try a couple more:


  C..........Ami.......A..........C#mi...........Db........Bbmi..........Gb.........Ebmi..............B............Abmi              


                    R            P            L              P            R              L                 R             L              R                
|| CEG—CEA—C#EA—C#EG#—DbFAb—DbFBb—DbGbBb—EbGbBb—D#F#B—EbAbCb—


...E............Emi.....C...


  L          P        L
EG#B—EGB—CEG ||


So one could spend hours just making little moves and learning them in all keys and inversion starts: that's the trick. I've tried one move at a time through all keys and then just tried to improvise using them in whatever progression it lead to.


I've found an interesting aspect of the Transformational procedures can be imposed on major or minor triads use as extension color changing on the fly. Some are very obvious and are worth pursuing.

For example on a Cma7 chord Ctriad transforms to Emi (through L). If you hold on to the root of C in the left hand it becomes Emi/C = Cma7. The same for Ami/C (through R) = C6.

C/C—Emi/C—C/C—Ami/C creates a Cma7 and C6. With the pedal held down it ends up sounding Cma13 'ish'.

In the same vein, with a Gtriad/ Ctriad the result is Cma9. If the 'G' triad transforms in the way proscribed above then other colors/extensions are generated i.e.

G/C = Cma9 || by imposing transformations on 'G' (/C) : G—Emi—G—Bmi /// all over the C triad the potential result is Cma13(#11) which can emphasis the extensions of the moment created by the sliding transformations of the G triad / C especially with the piano damper pedal discretely used. I'll outline these sorts of things further in the next blog.