Expert: Scotto - 10/20/2008

Question
Hello, I am a jazz musician searching for a method to eliminate contrafacts in a particular subset of pentatonic scales that meet some conditions I've placed on them. I'll show my work so far to better explain:

12 different scale tones: A,A#,B,C,C#,D,D#,E,F,F#,G,G#, and then of course it repeats with the next octave.

From that, the total number of combinations of possible pentatonic scales (hemitonic and anhemitonic) would I think be 12!/(5! x 7!)= 792

Then, because I want to eliminate contrafacts, I started by getting rid of ones that would have the same interval relationships but be merely a transposition into another key. So, 792/12= 66

Then, I further limited the set to only hemitonic pentatonics (those containing at least 1 semitone, i.e. 2 consecutive numbers), so by trial and error, I found the set of anhemitonic pentatonics (in one key) to be: {A,B,D,E,F#}, {A,C,D#,F,G} and {A,B,C#,D#,F} or 3 different pents.
So, 66-3= 63 hemitonic pentatonics

Now, from this subset, I want to find all the subsets that satisfy the following rules:

A.) No consecutive semitones. So, if each tone were a number 1-12, it would be okay if 1 & 2 were in the same set (as defined by hemitonic), but avoid 1,2,3 or 2,3,4 or 12,1,2. 1,2,5,6,10 would be okay.

B.) I want the largest gap in the scale to be a major third or 4 semitones. Anything larger than a 4 semitone spread must include a tone in between that does not violate rule A.

Note that 12 is adjacent to 1, and would be considered a semitone if both are included.

So, my question is assuming I generate this set of 66 by hand, then remove the 3 with no semitone, is there a quicker and easier way I can eliminate contrafacts short of staring at the keyboard and trying to generate 2 pentatonics containing the same semitone, where one is merely a transposition of the other? Or, did this all get handled when I divided by 12? Sorry so lengthy! What would you do if there were 1,000,000,000 notes to the octave?  

Answer
I have never heard of a hemitonic or pentatonic scale.  However, lets consider being in the key of A.  As far as I can guess, there is only one hemitonic and pentatonic scale in the key of A (just like there is only one major and one minor scale in A).

If you take the regular scale, the hemitonic scale, and the pentatonice scale, it seems like there should be three scales in A.
Since there are 12 possible notes, there would be 3*12=36 possible scales.

Do these scales have anything to do with chords being augmented, major, minor (natural, melodic, and harmonic), diminished, diminished seventh, seventh, or augmented seventh?  Note that there are 15 major and 15 minor key signatures.

By the way, there are only three diminished minor sevenths since, but they are named by any of the four notes in the chord which are all 3 1/2 steps apart.