# Scale Families: heptatonic scales organized by half-steps in their intervallic spelling

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?

• wouldn't the combination you call skip-scale have to be three m2, three M2, and one m3, in order to add 12 halftones? Aug 7, 2016 at 2:00
• and why consider heptatonic and dodecatonic combinations, but not intermediate number of notes (e.g. octatonic - 4 m2 and 4 M2, etc.)? Aug 7, 2016 at 2:09
• Thank you for spotting that; fixed. I'm working from a group theory perspective. In it, the dodecatonic is the parent scale and the heptatonic the children scale; octotonic and other -tonic scales are "cousins" but not part of this "geneological"investigation Aug 7, 2016 at 2:36

I don't feel comfortable verifying the results, just because I'm not that well-trained as a mathematician and I would feel more comfortable going that route to verify something with this many permutations.

However, here are some great sources in the field of music theory that you should check out:

• Carey, Norman and Clampitt, David (1989). "Aspects of Well-Formed Scales", Music Theory Spectrum 29: 249-70.
• Clough, John (1979). "Aspects of Diatonic Sets", Journal of Music Theory 23:45–61.
• Clough, John and Douthett, Jack (1991). "Maximally Even Sets", Journal of Music Theory 35: 93-173.
• Rahn, Jay (1977), "Some Recurrent Features of Scales", In Theory Only 2, no. 11-12: 43-52.

These are all necessary readings for the study of scale theory. (I recommend starting with the Clough/Douthett 1991 article, which is probably the most famous.) You'll also want to familiarize yourself with concepts such as Myhill's property, the deep scale property, and the notion that cardinality equals variety.

Have fun!

• Thanks Richard. I'm familiar with all these works. "My" families are not based on any set theory property nor is it intended to catalog every possible combination of notes; they only catalog the possible combinations of seven intervals that sum up to 12 half-steps. A notable feature of "my" hierarchy is that it puts 2212221 at one end of the spectrum and 611111 at the other; if there were similar hierarchies they would have similar end points and I've not found them yet. But I would hate to take credit for "my" hierarchy if in fact it has been organized this way before. Aug 6, 2016 at 19:16
• @RicardoJRademacher Got it; sounds like an interesting project! I can't think of any systems like the one you're describing, but you may want to check out Howard Hanson's notion of "chemical analysis" from his Harmonic Materials of Modern Music from 1960. It deals with chords, not scales, but it's a vaguely similar approach. (I only mention it as something you should be aware of, not because it overlaps with your work.) Aug 6, 2016 at 19:51
• Fantastic advice, thanks! I'm looking at Hanson more closely (hadn't come across him much) but like others they share the philosophy of including non-heptatonic scales in their organizational model where I'm focused on heptatonic scales from dodecatonic scales exclusively. Aug 6, 2016 at 20:06
• While what is stated above refers to the number and value of each interval in their spelling, it didn't specify how those intervals are organized. Doing so forms "sub-familys". So for example, the skip-scales have 20 members. But this organizes into 4 sub-famlies and, here is the interesting part, we seem to have chosen only ONE scale from each sub-family in conventional western music and disregard the rest! These sub-families are: Harmonic Minor (6 members), Harmonic Major (6 members), Neapolitan Minor (6 members), and Hungarian Major (2 members). Aug 6, 2016 at 20:13

I recently got interested in number of different heptatonic scales and their classification too.

You are apparently missing a five m2, one M2 and one P4 (perfect fourth) family with 6 members. This totals to 66 heptatonic scales.

The result above can be verified with some combinatorics. The reasoning may go like this: there are 12!/(5!7!) = 792 possible ways to choose 7 notes out of 12 chromatic scale notes. Of course some of these are same up to cyclic permutation. In fact, cyclic permutation to 1, 2, .., 12 positions of any particular choice forms collection of length 12 (it can be proven that all cyclic permutations of any choice are distinct from each other due to 7 and 12 be coprime). So these 792 ways break up into 792/12 = 66 collections.