DanielFGray.com

On permutations of rudiments

category: music

This post is still a work in progress, so bare with me please. If you spot any errors or have something you want to add, feel free to leave a comment below.

Displacing Accents

Let's begin with some simple exercises and warm-ups, using basic rudiments and moving the placement of an accent note forward by one note.

Single Strokes

1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
>       >       >       >
R-l-r-l-R-l-r-l-R-l-r-l-R-l-r-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
  >       >       >       >
r-L-r-l-r-L-r-l-r-L-r-l-r-L-r-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
    >       >       >       >
r-l-R-l-r-l-R-l-r-l-R-l-r-l-R-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
      >       >       >       >
r-l-r-L-r-l-r-L-r-l-r-L-r-l-r-L-

Double Strokes

1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
>       >       >       >
R-r-l-l-R-r-l-l-R-r-l-l-R-r-l-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
  >       >       >       >
r-R-l-l-r-R-l-l-r-R-l-l-r-R-l-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
    >       >       >       >
r-r-L-l-r-r-L-l-r-r-L-l-r-r-L-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
      >       >       >       >
r-r-l-L-r-r-l-L-r-r-l-L-r-r-l-L-

Single Paradiddles

1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
>       >       >       >
R-l-r-r-L-r-l-l-R-l-r-r-L-r-l-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
  >       >       >       >
r-L-r-r-l-R-l-l-r-L-r-r-l-R-l-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
    >       >       >       >
r-l-R-r-l-r-L-l-r-l-R-r-l-r-L-l-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
      >       >       >       >
r-l-r-R-l-r-l-L-r-l-r-R-l-r-l-L-

I've heard this concept also called practicing "grids", but there's a mathematical term for this, called permutations.

Permutations

Wikipedia has a page on permutations, the introduction right now says:

"In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting."

It then goes on to describe it's relation to set theory and computer algorithm sorting, and then a lot of math.

Permutations are somewhat of an umbrella term for many ideas.

Anagrams are a common form of permutations: "now", "own", and "won" use the same set of letters without repeating or missing any.

This is contrast from combinations, where you can repeat or remove items from a set.

Musical modes are another example of permutations: the steps in the major scale could be expressed as "WWHWWWH". The Dorian mode is "WHWWWHW", which was made by taking the first "W" and moving it to the end. All seven modes of the major scale are as follows:

Ionian    WWHWWWH
Dorian     WHWWWHW
Phrygian    HWWWHWW
Lydian       WWWHWWH
Mixolydian    WWHWWHW
Aeolian        WHWWHWW
Locrian         HWWHWWW

I'm going to exclusively be talking about cyclic permutations, which is what musical modes are, as opposed to unordered permutations, like anagrams. We'll look at rudiments and try to make variations of them by shifting the order around.

Permuting the Single Paradiddle

Fig. 1
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
R-L-R-R-L-R-L-L-R-L-R-R-L-R-L-L-

Now let's take that first R and move it to the end of the pattern

Fig. 2
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
L-R-R-L-R-L-L-R-L-R-R-L-R-L-L-R-

and continue the process

Fig. 3
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
R-R-L-R-L-L-R-L-R-R-L-R-L-L-R-L-
 
Fig. 4
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
R-L-R-L-L-R-L-R-R-L-R-L-L-R-L-R-
 
Fig. 5
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
L-R-L-L-R-L-R-R-L-R-L-L-R-L-R-R-
 
Fig. 6
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
R-L-L-R-L-R-R-L-R-L-L-R-L-R-R-L-
 
Fig. 7
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
L-L-R-L-R-R-L-R-L-L-R-L-R-R-L-R-
 
Fig. 8
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
L-R-L-R-R-L-R-L-L-R-L-R-R-L-R-L-

Now, start displacing the accents over the permuted rudiments!


If we analyze these patterns, we can group them into four variations

ABAA/BABB - figs. 1 and 5
ABBA/BAAB - figs. 2 and 6
AABA/BBAB - figs. 3 and 7
ABAB/BABA - figs. 4 and 8

I prefer this form of "A/B" notation over "R/L" because it forces me to think more about how to apply it.

It could be starting with my dominant hand, or it could be the inversion.

I might not even play it with my hands, maybe I'll play it with my feet, or between the bass and snare drums.

In fact, you can take just about any pattern apply it six different ways:

  • between both hands: right-hand + left-hand
  • between both feet: right-foot + left-foot
  • between both rights: right-hand + right-foot
  • between both lefts: left-hand + left-foot
  • diagonally one way: right-hand + left-foot
  • diagonally the other way: left-hand + right-foot

Now try and play the four variations (and their inversions) across all six limb-combinations!

Permuting the double paradiddle

1-&-a-2-&-a-3-&-a-4-&-a-
R-L-R-L-R-R-L-R-L-R-L-L-

1-&-a-2-&-a-3-&-a-4-&-a-
R-L-R-R-L-R-L-R-L-L-R-L-

1-&-a-2-&-a-3-&-a-4-&-a-
R-R-L-R-L-R-L-L-R-L-R-L-

1-&-a-2-&-a-3-&-a-4-&-a-
R-L-R-L-R-L-L-R-L-R-L-R-

1-&-a-2-&-a-3-&-a-4-&-a-
R-L-R-L-L-R-L-R-L-R-R-L-

1-&-a-2-&-a-3-&-a-4-&-a-
R-L-L-R-L-R-L-R-R-L-R-L-

More exercises

Single-stroked Hertas in three

1-&-a-2-&-a-3-&-a-4-&-a-
RLR-L-RLR-L-RLR-L-RLR-L-
 
1-&-a-2-&-a-3-&-a-4-&-a-
R-L-RLR-L-RLR-L-RLR-L-RL
 
1-&-a-2-&-a-3-&-a-4-&-a-
R-LRL-R-LRL-R-LRL-R-LRL-

Double-stroked Hertas in three

1-&-a-2-&-a-3-&-a-4-&-a-
RRL-R-LLR-L-RRL-R-LLR-L-
 
1-&-a-2-&-a-3-&-a-4-&-a-
L-R-LLR-L-RRL-R-LLR-L-RR
 
1-&-a-2-&-a-3-&-a-4-&-a-
R-LLR-L-RRL-R-LLR-L-RRL-

Single-stroked Hertas in four

1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
RLR-L-R-LRL-R-L-RLR-L-R-LRL-R-L-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
R-L-R-LRL-R-L-RLR-L-R-LRL-R-L-RL
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
L-R-LRL-R-L-RLR-L-R-LRL-R-L-RLR-
 
1-e-&-a-2-e-&-a-3-e-&-a-4-e-&-a-
R-LRL-R-L-RLR-L-R-LRL-R-L-RLR-L-