Just How Many Scales Are There, Anyway?

Scales, scales and more scales...

Phrygian, Hungarian Minor, Blues Pentatonic, Super-Lochrian, Neopolitan Minor, Neutral Pentatonic, Arabian Major, Spanish 8-Tone, Lydian Dominant, -  does it ever end??!!   YES IT DOES - and I'll show you exactly WHERE it ends.

      Assume that all scales contain and start with the note that is included in the NAME of the scale - for example, a C -scale of any kind contains and starts with the note "C".  This leaves 11 notes from which to construct the rest of the scale, since there are only 12 notes. 
   Let's also assume that we will only be creating 7-note scales that start with C ( a C-MAJOR or C-HARMONIC MINOR scale would be examples of  7-note scales that start with C,  and there are many others ).  Every 7-note scale we create is a simply a different "combination" of 6 other notes ( plus C ) chosen from those 11 notes that are not C. 

      Just HOW MANY scales are possible, then , from the 12 notes we are given..

Fortunately, there's an easy way to calculate that number. 
First, reframing the question:  "How many combinations of "11 items, 6 at a time",  expressed as C(11,6),  are possible?"  Using simple Probability Math ( see below),  that number turns out to be 462.

     IT IS POSSIBLE TO PLAY 462 DIFFERENT 7-NOTE SCALES.  ( many of them may not be very useful, but we'll save that investigation for later)

For fun, let's also include all of the 5-note scales, 6-note scales, and 8-note scales, because these are common configurations in Western music and Jazz Music. ( not to be confused with "Country Western Music" which is something else altogether.)

C(11,4)  = 330      5-note scales

C(11,5) =  462     6-note scales

C(11,7)  =  330    8-note scales

That adds up to a Grand Total of 1584 scales,  5,6,7 and 8 note scales.  And that's just in the key of C.   Finally we see an END to the number of scales that are possible from the 12 note Tempered Scale. Better get practicing because many of these scales are pretty interesting and useful. 

Here's one scale, for example, that has a very intriguing quality, a Japanese  pentatonic ( 5-note) scale  called

                                                         kumoi           1  b2  4  5  b6

There are 1583 more.....

HOW TO CALCULATE COMBINATIONS of anything ---


  Imagine that you have five fruits

and that you need to make a fruit salad that only contains 3 fruits. Obviously it doesn't matter what ORDER you put the fruits in the salad, it really only matters which 3 fruits you choose.  Every possible fruit salad would be a different "combination".  If the order mattered, that would be a "permutation" - there are many more permutations than combinations.

One possible  combination of 3 fruits chosen from 5 fruits might be this fruit salad:

ONE OF 10 POSSIBLE "3-FRUIT-SALADS" FROM 5 FRUITS

 There are 10 different fruit salads you could make with 5 fruits, i.e. 10 combinations of 5 things, 3 at a time.

Combinations of 5 fruits, 3 at a time   or    C(5,3)  is calculated using "factorials".

A factorial is the number that results from multiplying a number by a series of numbers that decrease by 1 each time.   The process is denoted with "!".

for example.....
5! = 5 x 4 x 3 x 2 x 1   =  120
            Combinations of "n" things, "r" at a time can be calculated with this formula:

         or, as applied to the above fruit salad problem,

             C(5,3)   =     5x4x3x2x1            =      10
                                3x2x1 (2x1)

Bon appetit and good luck with the scales.......which 20 scales will you spend YOUR life practicing and why?