We all think we know what it means for me to play, say, the D Natural Minor scale. The scale contains the notes D, E, F, G, A Bb and C, so if I play D Natural Minor then I play all and only those notes. Simple. Or is it?

This has begun to bug me enough to want to think about it a bit more for a couple of reasons. The main one is what I think is an increasing use of large "parent scales" in jazz pedagogy. This goes back at least to George Russell, but I seem to see ever more claims that 9- or even 10-note scales are in regular use by improvisors in mainstream traditions. I always thought such scales were extremely exotic. Was I wrong? Yes and no.

It still seems correct to me to consider these to be "smaller" scales with added chromatic passing tones. Do people really play the "Bebop Dominant" scale or do they use the Mixolydian and often throw in the major seventh as a passing tone? The latter seems more accurate to me. But once your confidence has been shaken a bit, you start to wonder. Why does it seem that way?

There follow some highly speculative thoughts on this subject. Readers who don't know my book will need to know that when I say "scale" it includes arpeggios, which I take to be small scales. So if you claim to "play chord tones, not scales" all this still applies to you.

Concreteness

One problem everyone acknowledges about music theory is that it can be a bit abstract: it's a good antidote to actually think of what happens in practical playing situations. If you asked me to play the D Natural Minor scale, what would I do? Probably just ascend through the scale from D to D an octave above. So how often do I do that in real-life playing? Never! Does that mean I never play the D Natural Minor scale?

Say you asked me to play a phrase using this scale over a chord you were comping for me. Then imagine you transcribed it and discovered I hadn't once used the note Bb. Did I fail to play a phrase using the D Natural Minor scale? Did I actually use an exotic hexatonic scale that has no well-known name? Have I just discovered that I know a scale I didn't know I knew?

Say I tried again, but this time you caught me adding the C# as a passing note. Again, have I played an esoteric octatonic scale you have to go to music college to understand? Wouldn't I say -- especially if I didn't know I was being put on the spot -- that I'd played the D Natural Minor scale, just in a different way?

The lesson here is that when improvisors use scales in practice we don't just play the scale straight up and down. We use it to exert a sort of gravitational force on all twelve notes. If I'm playing D Natural Minor I'm probably more likely to play C# than F#, yet neither note is in the scale. Whats up with that? Am I "really" playing some kind of parent scale that includes these notes? If so, since I could have used any notes as examples, the parent scale must be the Chromatic. How useful is that, analytically or educationally? Not very.

Scales As Force Fields Or Heat Maps

You hear moderately seasoned players say it all the time: you can play any note over any chord. To which the student says, "Yes, but how do I choose?". How do we answer the student without lying -- that is, without contradicting what we first said, which was true, giving them "scales" to learn and rapping their knuckles when they play "wrong notes"? Let's assume we were right the first time. Does that mean we don't use scales?

I think not. But we don't treat them as rigid sets of notes that must all be played and must not be diverted from. I tend to see a scale as a kind of territory sketched out on the fingerboard that I can move around freely. I might be thinking of the D Natural Minor but actually playing no notes from the scale at all. It still makes sense to say that I'm using the scale, because that's what's structuring my playing.

Think about the time you first started moving beyond pure triads and began to fit in other notes around them to form scales. Didn't the triad notes seem to stand out from the others? Maybe of I was feeling a bit whimsical I might draw a diagram of the minor pentatonic like this:

I dashed this off but the ideas is that the triad tones are "coolest", then the pentatonic notes, then the others are "hotter". This isn't supposed to be scientific, of course, but the idea is that I can be thinking of the Minor Pentatonic and it will do something a bit like this to my picture of the available notes.

So, which notes are in the Minor Pentatonic scale? Looking at the diagram above, it seems like the answer is, "All twelve". But that can't be right, because if it were we wouldn't have any way to tell scales apart. The Chromatic Parent Scale would have absorbed them all like a hungry amoeba. Clearly we can tell our scales apart, so something funky is going on.

Scales As Fuzzy Sets

The problem arises because scales are "crisp sets" -- that is, every note is supposed to be either in a scale or not in it. A way to model this mathematically is to have a list (a "vector") of twelve numbers that correspond to each of the twelve notes in the octave, and put "1" in each place that corresponds to a note in the scale and "0" everywhere else. We may as well use numbered notes because we don't care about transpositions. Here, for example, is the Major scale:

Yet this doesn't seem to match our practice as well as we'd like. There are plenty of approaches one could take to fixing this, but one easy and well-trodden one is to allow notes to be members of a scale to a certain degree. We say the value of "1" only applies to a note that is definitely in (i.e. must always be played), "0" to a note that's definitely out (i.e. must never be played) and a value in between to notes that can be played but need not.

It seems to me that there are no notes that absolutely must or must not be played. But thinking in terms of some particular scale means that some notes show up as being more likely to be played, or more readily-available, than others. Why not represent this with fractional numbers between 0 and 1?

Lack of Definiteness

The only issue I see here is that the actual value you assign to a given note may seem to be arbitrary. There's no way to measure this stuff or calculate it analytically. Notice that we can't use the frequency with which you play a particular note as a guide, for you may often play outside all the notes you consider to be definite scale-members.

This doesn't mean we know nothing. Imagine I make an assignment of numbers like this for the Major scale:

Now, I can order the notes by their degree of membership of (this) Major scale:

If your own assignment of numbers leads to the same ordering then we undoubtedly have something musically meaningful in common. If not, we are using the scale in decidedly different ways. Perhaps it would make real-world sense to say we were playing different scales even though the numbers we assign are, in a sense, made up.

We could enforce a requirement like "the sum of all the entries must be 6", so that the average entry is 0.5. In the Chromatic Scale we would indeed then see 0.5 in every entry; other scales would get their distinctiveness by awarding higher values to some notes by "borrowing from" others.

With a system like this we can do a bit better with comparing assignments of numbers. For example, if two notes for me are ordered the same as for you but my numbers are a lot closer together, it may mean something musical (or at least make us both think harder about what numbers we'd like to assign).

Open Questions

• What work has already been done in this area by scale theorists?
• Can we do better when it comes to assigning values?
• If two people say they're using the same scale but assign different values, how do we make sense of that?
• Is there a connection between frequency of the note in actual playing and its "membership value" in the scale?
• Can it possibly make pedagogic sense to think of scales in this way, except as advice not to be too dogmatic about them?
• The space described is the 12-dimensional unit hypercube with corners at points that are the Cartesian unit basis vectors. We can do calculus and all kinds of other things in this space. It is not, however, a vector space because the unit interval [0, 1] isn't a field (it lacks inverses). What mathematics works in this space and what doesn't? Can it or should it be extended to a "nicer" space such as R¹²?

I'm only half-serious about this "scales as fuzzy sets" idea. I'm not sure it's worth spending time pursuing any of the questions above. But I do think it's worthwhile just thinking about what it means to "play a scale" in a different way from the usual, black-and-white way that doesn't seem to match what we really do except in very artificial situations.