Consider a note a collection of frequencies, a mode a collection of seven notes, and a scale a collection of seven modes. Hence, C is a collection of frequencies, CDEFGAB a collection of notes, the Ionian mode, and its seven cyclic permutations form a collection called the Major scale.
Now, consider a scale family as a collection of scales that have the same number of half-step in their spelling. Using this definition, we get the following families:
- Step-Scale (3 members): two m2 and five M2.
- Skip-Scale (20 members): three m2, three M2, and one m3.
- Double-Skip-Scale (15 members): four m2, one M2, and two m3.
So these are the 38 scales and 266 modes that can be built by using upwards of a m3 in their intervallic spelling. This method, however, allows you to easily extend this other families with larger intervals than a m3:
- Major-Four-Step-Scale (15 members): four m2, two M2, and one M3.
- Major-Five-Step-Scale (6 members): five m2, one m3, and one M3.
- Tritone-Scale (1 member): six m2 and one TT.
And while not heptatonic, to complete all possible half-step spellings, we must include
- Dodecatonic-Scale (1 member): twelve m2.
Thus the total number of possible heptatonic scales is 60 encompassing a total of 420 modes.
Can anyone verify these result for me and ideally point me to journal or book that has organized scales along similar principles, i.e., set up scale families even if they are not called that?