Pentatonic Scales
Pappmode
4th degree  with open strings
Gyuri Papp...
...one of my former students once came to me with the following question: What kind of a scale do these five notes A  B  D  F  G make? This scale has a surprisingly very weird inner structure:
Full step  low third  low third  full step  full step
I have to admit it; I didn't have a good instant answer to that. But later we figured out we just discovered something completely new. Although I have not been able to find something even similar to this so far, I suppose, since there seems to be a strong musical relation between minor pentatonics and the basic scales, this musical relation between also exists between the new pentatonic scales and the sister basic scales.
Here comes a little analysis. First let's take a look at the minor pentatonic scale structure:
Low third  full step  full step  low third  full step
A remarkable fact: Whereas there is no low third following another in a minor pentatonic scale, the new Papp pentatonics only features them in this structure, which is why this scales is innovative. Now let's see the Papp pentatonic scale structures in all degrees:

full step  low third  low third  full step  full step

low third  low third  full step  full step  full step

low third  full step  full step  full step  low third

full step  full step  full step  low third  low third

full step  full step  low third  low third  full step
Now let's check out the scale structures in mirror images; first all the scales bound together, then each scale degree. To create these mirror images I used my own software called OGRE, which is able to generate scales of any structure and make the respective scale patterns.
I picked the scale degree Mr Papp showed me first starting from note A to be the first degree, as I can't really where the tonal center is.
Mr Papp later showed me a scale of another structure which after a long analysis proved to be the fifth degree of Papp pentatonics.
It is clear that the minor pentatonics involve the variation of three full steps and two low thirds within an octave, with ten possible combinations in total. Out of this ten combinations five are dedicated to the minor pentatonic scale degrees; Mr Papp discovered the remaining five. Now let's do the math!
Using the wellknown combinatorial formula (as already known from Basic scales VI. section)...
...with the following abstraction:

Full step: 0

Low third: 1
Now let's see the number of combinations:

11000

10100

10010 (minor pentatonics, 1^{st} degree)

10001

01100 (Papp pentatonics, 1^{st} degree)

01010

01001

00110

00101

00011
Just for the record, there have already been discoveries of this significance since I started teaching my methodology; the discovery of the sister basic scales and the inverse harmonic minor scales were all remarkable moments of my profession.