Fingerings Derived

In my post on the three-note-per-string fingerings for the major modes on guitar, I showed how those fingerings all stem from a pattern we can write as XXXYYZZ.  In this notation, X means that we play two whole tones on a given string, Y means we play a semitone followed by a whole tone, and Z means we play a whole tone followed by a semitone.  If you looked over my post, you had to trust that I put the dots in the right places in my fingering diagrams, or else you had to check the positions for yourself.  Here, I’d like to show that this XXXYYZZ pattern can be obtained directly from the “formula” for a major scale, using some simple variable substitution.  Remember that the pattern of semitones and whole tones in a major scale is whole-whole-semi-whole-whole-whole-semi or 2212221. If we start at the fifth note of the scale and keep ascending over several octaves, we’ll play a pattern like this:

…22122122212212221221222122122212212221221222122122

If we’re playing exactly three notes per string on the guitar, then we’ll always traverse two intervals on one string, and one interval when we cross strings. We can use parentheses to group the intervals that would fall on the same string like this:

…(22)1(22)1(22)2(12)2(12)2(21)2(21)2(22)1(22)1(22)2(12)2(12)2(21)2(21)2(22)1(22)1(22)…

Now we can replace (22), meaning “play two whole tones,” with X, our abbreviation for the two-whole-tones-on-one-string fingering pattern.  Similarly, we can replace (12) with Y and (21) with Z, giving this:

…X1X1X2Y2Y2Z2Z2X1X1X2Y2Y2Z2Z2X1X1X…

Finally, we can reveal the pattern by replacing the 2’s (meaning “play a whole tone when you cross strings”) with a space, and the 1’s with no space, giving this:

…XXX Y Y Z Z XXX Y Y Z Z XXX…

Here it is again:

XXXYYZZDerivedFromMajorScaleFormula

And now a follow-up regarding the Jazz Melodic Minor scale.  If you’ve ever worked out the 3-note-per-string fingerings for melodic minor and its modes, you know those fingerings are not as “nice” as the major mode fingerings.  But how can we characterize the difference?  One way to explain what makes the major mode fingerings “nice” is that the single-string shapes appear in distinct groups: three X’s, then two Y’s, then two Z’s.  If we do a similar derivation for the melodic minor fingerings, starting with 2122221 as our scale formula, the pattern we arrive at is XZ X YX Z Y.  In other words we have to switch from X to Z, then back to X, then to Y, and then to X again (and so on) as we’re playing. This interleaving of single-string shapes is one thing that makes the melodic minor fingerings harder to work with.  Here is the derivation for the melodic minor 3-note-per-string pattern:

XZXYXZY Derived From Melodic Minor Formula

(By the way, I expect this post to make sense to people who are interested in mathematical music theory and who have studied guitar and read my previous post on modal fingerings.  If this doesn’t make sense and you want to know more about it, please ask me a question as I’d be happy to provide more of an explanation.)

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