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:

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:

(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.)

Fourths vs. Standard Tuning

This diagram shows where the notes of C major fall on a guitar fretboard in Fourths Tuning (EADGCF) versus a fretboard in Standard Tuning (EADGBE). I’ve extended the main pattern across the boundaries of the two fretboards so you can see how it repeats in space.  The red dots show how the pattern is pushed up a fret on the two highest strings in Standard Tuning.

Guitar Modes Unified

In this post I’d like to share my way of thinking about the three-note-per-string fingering patterns for the modes of the major scale on guitar.  Take a look at this diagram:

What you’re looking at is a hypothetical fretboard with 30 strings, tuned in All Fourths.  The dots represent the finger positions for the notes of C major, placing exactly three notes on each string, starting from a G on the low E string.  Do you see a pattern?

When I say this guitar is tuned in All Fourths, I mean there is an interval of a perfect fourth between any two consecutive strings.  The standard guitar tuning (EADGBE) is a “mostly fourths” tuning with one aberration: there’s a major third between the G and B strings.  The easiest way to move from standard tuning to a true All Fourths tuning is to raise the top two strings by a half-step each, giving EADGCF.  In the diagram above, I’ve started with the low E string from standard tuning and tuned the rest of the strings in ascending perfect fourths: E A D G C F Bb…

My reason for using All Fourths tuning here is that it reveals the beauty and regularity of the fingering pattern for a major scale and its modes.  Once we’ve understood how the pattern works in All Fourths, we can translate that knowledge to standard tuning with a simple adjustment.

If you take a closer look at the C major pattern on our 30-string guitar, you’ll see that it’s built of very simple components.  In fact, there are only three things that can happen on any given string.  First, we might play three notes spanning two whole tones — let’s call this Shape X:

Second, we might play a semitone followed by a whole tone — let’s call this Shape Y:

And finally, we might play a whole tone followed by a semitone — let’s call this Shape Z:

Now that we’ve given names to the three component shapes of the pattern, let’s take a look at how those components are arranged in our 30-string example.  Starting from G on the lowest string, we have three X’s, then two Y’s, then two Z’s, and then the sequence repeats: XXXYYZZ again!  You’ll also notice that when we cross from one string to another, the new shape usually starts at the same fret as the previous shape, except when we go from X to Y.  In that one case, the Y shape starts one fret higher than the preceding X shape.

In the diagram below I’ve given a different color to each shape, and I’ve used numbers to differentiate the repetitions of each shape (i.e. X1, X2, X3).  I left out the frets because it’s easy to imagine where they should go:

This diagram shows the XXXYYZZ unit that gets repeated every seven strings in our 30-string example–you’ll find four separate instances of it there.  You can also think of this diagram as a “master key” for playing all the modes. To derive any modal fingering, you start at a certain place in this diagram and read left to right until you’ve covered 6 strings, circling back to the beginning if you hit the end.  Again, each shape starts on the same fret as the previous one except when you cross from X to Y.  If you’re playing in Standard Tuning, you’ll also have to shift anything that falls on the top two strings up by one fret — that’s the only difference between how these patterns work in the two tuning systems!

We’ve been playing the notes of C major starting from G on the lowest string, and the first six components in our pattern are: XXXYYZ.  Since we’ve started from G, this is actually the pattern for G Mixolydian, assuming we treat the lowest note in the fingering as the root of the mode we want to establish.  If we want to play C Ionian on our 30-string guitar, we’d use the same sequence starting from the second X.  Looking at the 30-string diagram, you’ll see this second X shape falls on the A string starting at the third fret, giving C as the first note.  Extended across six strings, the pattern would be XXYYZZ.  If we want to play this full pattern a standard guitar, we can’t start on the A string because we’d run out of strings at the end, so we’d have to find our C starting note at the 8th fret of the lowest string (the E string).

Here are the formulas for all the modes (and you’ll find a diagram summarizing this at the end of the post).  If you want to play F# Dorian, for example, find F# on the lowest string and play ZXXXYY — it’s that simple.

 Mode 6-String Fingering Formula Mixolydian XXXYYZ Ionian XXYYZZ Lydian XYYZZX Locrian YYZZXX Phrygian YZZXXX Aeolian ZZXXXY Dorian ZXXXYY

Again, these formulas are the same for All Fourths and standard tuning — to use them in standard tuning you’ve just got to remember to shift the shapes on the top two strings up by one fret.  So, here’s what C Ionian looks like in All Fourths:

And here’s how you’d adapt the All Fourths pattern to standard tuning:

I’ve included the remaining fingerings below in All Fourths.  Check out my post on the math behind the pattern we’ve seen, and my complementary post on Note Neighborhoods.  These posts are the outcome of years of studying and struggling with the fretboard–if I’ve been helpful to you in any way, please pay me back by leaving a comment and sharing this post.

[In response to interest in a vector version of this chart for high quality printing and lamination, I’ve made an SVG version available under a Creative Commons BY-SA license.  Contact me if you plan to adapt or redistribute the work.  If you print it out, please send me a snapshot of how it looks on your music stand!]