The idea of musical theorems is all about drawing simple conclusions from your basic knowledge about notes, and putting them to use in your playing.

An example. You know every major scale contains three notes from one wholetone scale and four notes from the other. Therefore, any wholetone scale playing over a major chord will contain three or possibly four “right” notes. In fact since playing the fourth (subdominant) over a major chord is a no-no, let’s just keep it at three “right” notes.

That’s half of your wholetone scale! So just keep going until you find your feet.

Here’s another one, that I suspect was one of Miles Davis’s favorites. The {+1 -2} pattern (see below) that he liked to use, especially in his early days,  is based on sound thinking.

Given that there are altogether 12 notes and only six of them can sound “right” over a major chord (still keeping that subdominant out of the equation), you have only a 50% chance of blind-hitting a good one.

I read in one of David Baker’s early guitar manuals, that any chromatic pattern will work so long as you end at the right place. Or maybe he said you have to start in the right place too.

Miles was known for his cavalier attitude to rules and his love of the right “wrong” note.

Now, the longest distance between two “right” notes over a major chord is the 3-semitone gap between the third and the fifth. Therefore you never have to loop the {+1 -2} pattern more than three times to hit a right note from below. And it sounds like be-bop to boot!

We can keep collecting theorems by asking questions such as: How many major thirds/ perfect fourths are in a major scale? Using the answers (three and six respectively) can help us design all-purpose patterns that we can insert and use to play “outside” in an intelligent way, and be sure to land safely on terra firma.

An awareness of linksums can also be a boon for improvisers. (Check out the MOVES page if you don’t know what a linksum is). A pattern with linksum (transposition) of +3 when repeated four times gets you back into the original key. One with a linksum of +8 will come out in the same place two octaves higher when repeated three times.

Do you have a favorite musical theorem you would like to share? Add a comment!