counting theory and rhythm: the mathematics of multi-pedal orchestrations

Counting theory, a field within discrete mathematics, is a vast area of study and extremely useful in solving many kinds of problems. One particular application of counting theory is rhythmic interpretation. With the advent of many modern instruments, such as the acoustic drum set and electronic drum simulation software, beginners and professionals alike seem to naturally arrive at the question, “How many rhythms can I create?” Fortunately, combinatorial analysis, or combinatorics, offers clear and theoretically precise methods for counting the number of rhythms that can be formed within specific parameters. Once these parameters are set, simple mathematic principles can be applied, and a definite answer to our question may be revealed. This paper explores all steps required to arrive at such an answer, explaining what kinds of variables must be taken into account, how these variables interact within proven combinatorial methods, and how these different methods interact with each other. Furthermore, specific examples are given to clarify potentially confusing concepts. Although, in reality, there are seemingly countless instances of rhythmic interaction, special emphasis is placed on the modern drum set, since its complexity opens up the space for the majority of rhythmical situations to be understood in close detail. Finally, an overview of relevant questions worthy of further study is given.

The setup of Grant Collins, player of the Southern hemisphere's largest kit. Collins is an active clinician and soloist. [12]

Relevant Mathematic Principles

Permutations

An r-permutation reveals the number of ways r number of objects can be selected from a set of n objects [4]. Our notation will be P(n, r).

P(n, r) = n! / (n-r)!

Example 1: Simple R-Permutation Problem

A simple combination lock composed of 5 buttons only opens if the correct series of 3 digits is entered correctly. The lock is designed such that no button may be used more than once in its combination. How many different combinations are possible with such a lock?

Solution

Because the order in which the digits are entered matters, and because our combinations cannot contain the same digit more than once, we know that we are dealing with a permutation problem.

We have 5 digits to choose from, and must choose 3 numbers. Therefore, n=5 and r=3. Thus:

P(5, 3) = 5! / (5-3)!
= (5 x 4 x 3 x 2 x 1) / (2 x 1)
= 60

So, there are 60 possible combinations.

We will use permutations later to calculate rhythmic arrangements.

Combinations

Combinations are used to calculate the number ways to select r objects given a set of n objects without taking order into account. (Finan: 176) While there are many applications of combinations, rhythm always implies an ordered arrangement of sounds, and thus combinations are not particularly useful for our calculations.

The Addition Principle

The addition principle is used to determine the number of distinct elements contained within two or more sets.

Assume k sets containing distinct elements n1 for the first set, n2 for the second set, etc. Then, the total number of elements in the union of the sets is n1 + n2 . . . nk. [4]

Example 2: Simple Addition of Distinct Set Elements

Let set A = {4, 5, 6} and set B = {7, 8, 9, 1}.
Then, the number of distinct elements in A + B is
|A| + |B| = 3 + 4 = 7.

We can see this clearly by simply listing all of the elements in A and B together in a new set C:

C = {1, 4, 5, 6, 7, 8, 9}.

Keep in mind that these digits within the curly braces are the elements, and do not denote the number of elements.

We will use the addition principle in our calculations for multi-instrumental orchestrations.

The Product Principle

Consider a procedure composed of a sequence of k steps. Suppose that the first step can be accomplished n1 ways, and regardless of how the first step is accomplished, the second step can be accomplished n2 ways, the k step nk ways, and so on. Then, the number of ways the procedure may be executed is n1 x n2 x . . . nk.

Example 3: Simple Single-Drum Example Using the Product Principle

Using two hands, how many ways can we hit a snare drum slowly three times in a row, using either the left or right hand one at a time?

Solution

Let L denote a hit of the drum using the left hand, and R denote a hit of the drum with the right hand. Because we are playing slowly, we can use any hand combination we like. For example, we can easily play L L L or R R R. Thus, for each strike of the drum, or note, we have two choices, L or R.

Using the product principle, we have 3 sets (3 notes to be played), and each for each element within the set, we have two options, or “stickings” (L or R). Thus,
2 x 2 x 2 = 2³ = (the number of stickings per note) ^ (number of notes to be played) = 8

Therefore, we have 8 ways to play a three strike sequence using our left or right hand. Graphically, this can be easily represented and verified for such a small combination:

LLL
LRL
LRR
LLR
RRR
RLR
RLL
RRL

The Pigeonhole Principle

Assume a set of pigeons is placed into pigeonholes. If the number of pigeons exceeds the number of pigeonholes, then at least one of the pigeonholes must contain more than one pigeon. [2]

4: Two Limb Single Instrument Percussive Permutations

Using two hands, how many ways can we hit a snare drum slowly 2 times in a row, using either the left or right hand, including hitting with both hands simultaneously?

Solution

Let (LR) denote a single note played with both hands simultaneously. Note that (LR) = (RL). Then, we have 3 possible ways to play a note, L, R, (LR).

Using the product principle, we can calculate our answer immediately as before:
3 stickings ^ 2 notes = 9 combinations.

Applying the Pigeonhole Principle

Notice that, although we have added another sticking option, we are still making the same number of sounds, 3 notes. To apply the pigeonhole principle, we can think like a musician and define our set of interest to be the number of sounds we can make, in this case, the number of notes we play (3). Therefore, because we are using 3 stickings to play 2 notes, we cannot use all of our stickings in our simple 2 note phrase.

Defining Sets and Variables in the Context of Modern Drum Set
Sounds and Stickings

As the previous example shows, we must be very specific about how we define our sets and variables in our calculations. Ultimately, rhythms are combinations of distinct sounds. Our previous examples display instances in which multiple stickings can be made to make the same sound. Note that despite using different stickings, from a musical perspective, the stickings are unimportant because they all create the same sound. However, from a technical perspective, stickings are crucial, because different stickings create more complex coordination issues, which require tedious practice to work out. Thus, sticking combinations limit what can be played.

Example 5: Identical Rhythms Created Through Distinct Stickings

It may be physically possible to play

L R L R

at a certain tempo, but not

(LR) (LR) (LR) (LR)

at that same tempo, despite the fact that both stickings create the same series of sounds, that is, the same rhythm. This is especially true for high tempo music.

The Drum Set

Although this pun, "set," is coincidental, it is important: Rhythmic calculations are entirely dependent on the number of instruments with which we play notes. It is important to define some terms in order to exemplify the differences that arise in multi-instrumental percussive orchestrations.

Sound or Note: Imagine closing your eyes and listening to a drummer play a rhythm. You will hear distinct noises, yet you will remain unaware of the way the drummer is creating these noises. Naked noises, without regard to how they are played, are sounds.

Sticking: Put yourself in the position of a deaf drummer. Without regard to sound, the raw physical movements involved in drumming are important. A set of such raw physical motions is a sticking. For our purposes, stickings will not be dependent upon dynamics (see the definition of dynamics below). Note that, although drum sticks are typically only played with the hands, the raw physical motions of the feet on foot pedals can also be though of as stickings. Also note that, for our purposes, different sounds can be created with the same sticking, using different instruments, different dynamics, or both.

Instrument: The distinct objects upon which stickings are applied, possibly with dynamics, to create distinct sounds are instruments. For our purposes, a single instrument will create different sounds with the same sticking only if dynamics vary. Realize that this is not technically true, since playing the same sticking at the same dynamic level on the same drum in different places on that drum will create different sounds.

Dynamics: The various volumes at which sounds may be created upon an instrument are dynamics. Note that different dynamics may be applied to create distinct sounds upon any distinct instrument.

Beat: Music is counted and written with reference to subdivisions of time. Such a single subdivision is a beat. Normally, beats are thought of in reference to minutes. For example, 60 beats per minutes means that the musician conceives of 60 musical subdivisions per minute, that is, one beat per second.

Bar: In music, it is useful to conceptualize sets of beats. A set of beats is a bar.

Tempo: Tempo necessitates a beginning point and an ending point. As described in the Veda, our experience can be conceived of as the experience of a constant moving from the Wholeness, Ah, to the collapse of Wholeness, Kah, and back again indefinitely. This is described as consciousness becoming aware of itself, or Wholeness moving within itself. Necessarily, rhythm is a reflection of this fundamental structure: Sounds are played within the space between fullness and its collapse. Without a starting point and an ending point, infinite sounds would blend together indefinitely into a meaningless mess. However, once we set a point of reference by defining a starting point and ending point, discrete sets of sounds can be identified, and all possible rhythmic possibilities become distinct, within our realm of experience. A change in the number of beats in the same amount of time is a change in tempo. For example, 50 beats per minute is 10 less beats per minute than 60 beats per minute, and is a slower tempo.

Rhythm: Now, imagine watching and listening to a drummer play the drums. S/he will create rhythms by using various combinations of stickings and dynamics on different instruments. Such a combination is a rhythm.

Example 6: Using Instruments to Distinguish Stickings from Rhythms

Assume the following sticking played at a constant tempo and constant dynamics, with 1 note per beat and 4 notes per bar:

L R L R

If we must play 2 distinct instruments, 1 with each hand, for 1 bar, how many rhythms can we create?

Solution
Within these restrictive parameters, we can only create 1 rhythm, since we maintain the same tempo, the same sticking, and the same dynamics.

Example 7: Multi-Bar Rhythms Played with Multiple Limbs Upon Multiple Instruments

Now, assume that we can play the same sticking for 2 bars. And, for each bar, we may choose to play both hands on the same instrument or 2 instruments, 1 with each hand. Now, how many rhythms are possible?

Solution

Here, we can apply the product principle. However, we must be careful about how we define our variables. In example 3, the number of notes we played was equal to the number of elements within our set. Similarly, the number of stickings was equal to the number of options for each element within our set. From our more refined perspective, we can now see that there is an ambiguity around the term “set” and, thus, we need to take a closer look at the equation for permutations.
In this case, we can apply the product principle and calculate the number of rhythms possible for each bar separately, and then multiply our answers together to come to a final answer. However, first we must take into account the number of instruments we are allowed to play on.

Number of stickings per note ^ number of notes to be played = number of possible rhythms for instrumental arrangement 1

Where “instrumental arrangement 1" refers to playing on one drum.
= 1 ^ 4 = 1

Similarly, for the second instrumental arrangement, that is, playing 2 distinct instruments, 1 with each hand:
= 1 ^ 4 = 1

Now, we use the addition principle to see how many rhythms are possible for each bar:
1 + 1 = 2

This shows us that we have 2 possible rhythms for each bar we play. Finally, because we are playing 2 bars:
Number of rhythms per bar ^ number of bars played
2 ^ 2 = 4

And arrive at our answer: There are four rhythms possible within the given restrictions.

Questions for Further Study

Upon this groundwork, it is easy to calculate the number of possible rhythms in more complex situations, such as rhythms involving 4 limbs playing several instruments at multiple dynamic levels. Drummers such as Grant Collins [3] use large drum sets with dozens of instruments to play extremely complex stickings with both hands and feet. Another strange musical idea, used by Vinnie Colauita in his song “I'm Tweaked / Attack of the 20lb Pizza” [14], is combining multiple bars of inconsistent length, creating the illusion of removing a quarter of a beat several times to create an 8 – bar rhythm.

Furthermore, these concepts can be applied to calculate not only drum rhythms, but combinations of any set of distinct sounds, such as notes on a piano. It would be very interesting to see, for example, how many rhythms a very fast drummer could create in 60 seconds, or to see what kinds of short-bar rhythms may be played with one limb simultaneously underneath long-bar rhythms and calculate how many repetitions of each rhythm is necessary before the first beat of each bar lines up in real time.

Advanced techniques - developed by professionals such as Thomas Lang[7], Johnny Rabb[6], Jaki Leibziet[8], Marco Minnemann[9], Dom Famularo[10], Claus Hessler[11], Grant Collins[12], Terry Bozzio[13], and many others - open a whole new realm of rhythmic combinations by revealing that they are, indeed, physically possible to play. In particular, Rabb's development of the freehand technique makes previously unachievable speeds possible using one hand instead of two.

Johnny Rabb[6] and the freehand technique. Rabb specializes in a high-speed style of drumming called "Jungle" or "Drum 'n' Bass." He is the author of "Jungle/Drum 'n' Bass for the Acoustic Drum Set" and "The Official Freehand Technique."

Similarly, Thomas Lang [7] has displayed the tremendous possibilities created using multi-pedal orchestrations.

Conclusion

Seemingly countless rhythms can be created using the modern drum set. Furthermore, rhythms are seen in other musical situations, as well as in poetry, including ancient Indian poetry [1], which uses combinations of long and short syllable sounds. However, this number, although extremely large, is finite, and within our realm of conscious perception. Thus, we can use mathematic principles to explain the behavior of rhythms. Furthermore, artists using and teaching these techniques are re-defining the limits of modern drumming. These are Acts which philosopher Alain Badiou would describe as artistic "Truth-Events," in which feats which were previously perceived to be unachievable are accomplished. [5]

Works Cited

1. A.K. Bag, Binomal Theorem in Ancient India Indian J. History Sci.,
1 (1966) 68–74.
2. J. Dossey, C. Eynden, A. Otto, L. Spence Discrete Mathematics Addison-Wesley, Reading, Massachusetts, (1997) 354.
3. Drummer World, “Grant Collins,” available at http://drummerworld.com/Videos/grantcollinsperformance.html
4. M. Finan, Lecture Notes in Discrete Mathematics Arkansas Tech. University, (2001) 173-177.
5. Badiou, Alain. On the Truth-Process: On Open Lecture by Alain Badiou. (2002)
http://www.egs.edu/faculty/alain-badiou/articles/on-the-truth-process/
6. Rabb, Johnny. http://www.johnnyrabb.com/
7. Lang, Thomas. http://www.thomaslangdrumcamp.com/www.thomaslangdrumcamp.com/Welcome.html
8. Liebziet, Jaki. http://www.moderndrummer.com/updatefull/200001390/Jaki%20Liebezeit
9. Minnemann, Marco. http://marcominnemann.com/
10. Famularo, Dom. http://www.domfamularo.com/
11. Hessler, Claus. http://www.claushessler.de/en/frameset_home.html
12. Collins, Grant. http://www.grantcollins.com/drumming.php
13. Bozzio, Terry. http://terrybozzio.com/
14. Colaiuta, Vinnie. http://www.vinniecolaiuta.com/Default.aspx

Counting Theory and Rhythm: The Mathematics of Multi-Pedal Orchestrations by Sheldon Paul Kreger is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
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