Combinations And Permutations

One experience I had that really got me to see a true connection between math and music was while I was sitting in a math class, and as usual I was drawing pictures of fantasy guitars and dreaming about band practice. The lecture was about combinations and permutations, different ways of combining and mixing up sets of things like numbers. In a combination the order of things doesn't matter, like making stew, is it beef potatoes and carrots or is it carrots, beef and potatoes? It really doesn’t matter. In a permutation, the order of things in the set does matter, as in the pass code to my cell phone, is GUITAR or is it AGIRTU? In this case, it really does matter.

I started to see the similarities between mathematical concepts like sequences, patterns, combinations and permutations and scale usage in virtually any musical setting. Lets face it, when playing lead guitar we are not always agonizing over every note choice, (combination) sometimes we are picking scales and letting them do their thing, making it up as we go along and placing our faith in the chord scale relationship we have decided on (permutation). That type of intuitive scale playing can be enhanced, and stretched by applying concepts that are more less mathematical. By learning and hearing these new scale tricks, you will be a better improviser, better at playing interesting and musical solos.

Working With Scales

All improvising musicians and even composers have applied different organizational, mathematical and pattern oriented approaches to working with scales. Being a lead guitarist therefore involves working with and understanding scale sounds and scale patterns to create melodic material, solos and musical ideas.

The first exercise of this lesson is a 'free form' one designed to make you invent and play scale oriented patterns, permutations and combinations of your own, simply to do this 'math' type of thinking. As you play the exercise use the corresponding scale fingering to play along with the track. When you do the exercise:

  • Stay within the scale.
  • Observe the tempo and feeling of the track.
  • Use the complete 8 note scale, in correct order from lowest to highest only occasionally.
  • Change directions often.
  • Skip notes often.

Just Play Along Please

Experiment heavily with going up, down and back again with three, four or even five notes at a time -letting the notes go where they want to, letting your playing take control. Work and re-work the scales like a mathematician.

Another Pentatonic Perspective

This lesson is about creating and applying organizational techniques and patterns to the scale shapes you've learned in this course as a way of developing your improvisational abilities. As you're working with, creating and recreating these permutations, combinations and patterns you'll be noticing musical moments as well as melodic and expressive possibilities inherent in your musical lines. When you're involved in this creative - inventive process, you're forming a style - by seeking out and developing those bits and pieces that you perceive as having musical value. This will help you avoid an over done, ‘too many notes’, babbling sound and help you to develop a musical, melodic and professional sound.

Pentatonic scales are a great place to start a new type of thinking about and listening to your scale playing because they have only two notes per string. Working with pentatonic permutations, combinations and patterns is an exciting, easy and new way to add a freshness and life to the standard scale patterns lead guitarists know so well and sound so good with. To begin, re-examine the 5th fret A Minor Pentatonic scale pattern by playing the familiar shape on the 5th fret but this time make sure to give each of the notes in the scale an exact letter name as you play them.

Note Skipping: One Note

A pattern of skipping every other note, A - D - C - E and so on gives you an amazing result, click the animation to the right and hear the A minor pentatonic scale being treated to this logical re-ordering of notes: playing every other note, or skipping one note.


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The musical result is wide skips and intervallic leaps causing an angular and broad, or almost outside type of sound. As you play this on guitar you'll obtain an adjacent note type feeling, by rolling across the fingerboard from one string to the next. Remember, having only two notes per string gives rise to many, many interesting thought patterns, playing feels and visualizations.

You're training your ear, mind and fingers to play the minor pentatonic scale in an every other note sort of fashion. Most of us find this to be a little daunting or confusing at first -but don't fret, you can quickly learn this pattern and great sounding music will be under your fingertips.

The musical example below was played with this type of scale treatment in mind. As you listen to, learn and copy the solo below on your guitar you will notice a few tried and true old licks mixed in with the note skipping concept. There is soloing space for you to practice playing along with the backing music right after the recorded solo.

Note Skipping: Two Notes

Here the A minor pentatonic scale is being re-ordering by skipping two scale degrees between notes, and a singing, ringing or soaring sound is achieved by even wider intervallic leaps. Use the little animation below to first get the idea of the sound of the reordered scale in your head.

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The solo below employs this thinking to get a new and fresh sound out of the old reliable A minor pentatonic box pattern shape, and some old reliable blues licks among the 'skipping two notes' scale treatment as I have labeled in the transcription. You are again re-training your ear, mind and fingers to play the minor pentatonic scale, only this time you're leaping over two notes as you build melodies. Once again, this can be a little tricky at first but you will soon play and work in this manner and the common minor pentatonic pattern will yeild some fantastic musical results. Remember, soloing space is provided right after rhe solo for you to copy the slo or play one just like it.


A sequence is taking a small scale passage or grouping of notes, and exactly repeating that grouping beginning with a different scale tone -shifting all of the notes in the passage or grouping by the same amount. The result is that small bits of melodic material are repeated at different pich levels along the scale rising or falling by the same or a similar interval and causing melodic interest. You want to hear, recognize, create and work with sequences as found printed music and in common knowledge scale fingering patterns -look at this as is essential to art of lead guitar playing. The animation at right conceptualizes a 4 note minor pentatonic sequence.

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Musical Examples

The solo below employs a similar minor pentatonic sequence. The graphic at left maps out the thinking strategy I employed: 4 ascending notes, 2 descending notes. The entire idea is then repeated on the next highest scale tone. This type of thinking is very important to and found quite often in contemporary instrumental guitar.

Play Track

Slowly Play Solo Only

Major Scale Techniques

Mathematical thinking and sequencing techniques can also be used to great effectiveness with the major scale. One of the most played and most musical sequence is playing the scale in an every other note fashion, this is called ‘playing in 3rds’ because the interval separating every other note from the next highest note in the sequence is a third.

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Try working with this idea a little bit using a well known major scale fingering pattern. In this scale the root note is located on the 12th fret of the fifth string. When playing that note with your 4th finger, you'll be in position 9, with your pinkie on the root note, the A found on fret XII. This is often referred to as a ‘C type’ because it's a transposition of the most basic, open string C major scale as can be seen in the diagram below. The play along track has the scale as a slow series of quarter notes for you to get the shape in your fingers and the sound in your ears.

The Instant Melody

Next, learn to play the A major scale, in position 9 by thirds, the 'every other note' pattern. I have found this technique to be so musical and generally effective, I call it the ‘instant melody’ because it always makes my major scale based playing work like real music, with nice melodic flow and content. This playing technique makes major scales sound instantly musical and adds depth to your soloing. At first, you may find this pattern to be a little baffling, trying saying the letter name of each note as a way of remembering where you are when you play the exercise. The printed study notes for this lesson will make learning scales in 3rds much easier.

Play Along Track

The track below is a two chord, I Maj7 to IV Maj7 vamp which firmly establishes a strong and sweet A major tonality. The first and most obvious scale choice is of course the A Major Scale.


When you use the backing track, try to play the A major scale in the two ways we’ve studied in this lesson: (1) play scale passages but be sure to know and use the pattern in position IX and (2) paly the major scale up and down in thirds as you try to create nice sounding melodic ideas.

Musical Example

In the final solo in this lesson I am using the ‘instant melody’, or the major scale in thirds, over the same A major vamp in the previous play along exercise. The melodic movement by thirds really gives the track a melodic and profesional sound, the movement is clearly illustrated in the transcription below. The track plays the solo once and the has solo space for you to play along with.

Open this link to find a printer friendly summary of this lesson with additional exercises and diagrams and exercises.