Dissecting Rhythms Mathematically

7/30/2019 Dissecting Rhythms Mathematically

1/8

Dissecting Rhythms Mathematically, variations in accent patterns in

number 1 16

Numbers and Universal concepts

Long & short = 3 & 2

Dissecting Rhythms Mathematically, variations in accent patterns in number 1 16

What follows is an in depth analysis of possible accent patterns in a number of beats.

Although this seems very analytical and mathematical, my intention was to open up creative

possibilities in musical composition by allowing composers to see the variety of rhythms avail-

able in a given time signature.

Before I knew about how Eastern European dances are broken up into long and short steps, I

developed a mathematical system that allows one to calculate all the possible combinations of

2s and 3s in a certain number of beats.

With accents placed on the first beat of each 2, and 3, we see that in the lower numbers up to

8, there are just a few options for accents: 2+2+2+2 would represent an non syncopated

rhythm with a beat on each quarter note(4/4 time), as we introduce 3s into the equation, syn-

copation starts occurring, and we arrive at one of the most common syncopations in 4/4 time:

3+3+2 : dotted8th, dotted 8th

, 8th

. So the 2s and 3s can be thought of as groups of two or

three 16th

notes if you want.

As background, the odd meter music that goes with the dances of Eastern Europe are

described by different chains of short and long steps. The short steps are equivalent to 2 beats

of musical time, (8th

notes or 16th

notes) and the long steps are equivalent to 3 beats of musi-

cal time.

In this way a rhythm in 7/8 time signature that has accents on beats 1, 3, and 5 (like this,

ONE two THREE four FIVE six seven

( with accents on the capitalized numbers)

can be described as SHORT SHORT LONG or: ONE two ONE two ONE

twothree

As another example, there is a rhythm from Bulgaria and other nearby countries called Gank-ino or Kopanitsa, which is in 11/8 time signature. In terms of long and short steps, it can be

described as:

SHORT SHORT LONG SHORT SHORT

Or: ONE two ONE two ONE twothree ONE two ONE two

Or using 2 for short, and 3 for long: 2 2 3 2 2 which of course adds up to 11.

This method of counting rhythms basically divides the rhythm into groups of either 2 beats or 3

beats. It can be used as a basic method to describe accent patterns in rhythms of any timesignature.
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2/8

The restrictions are that there is always an accent on beat one, and that the accents after that

must be separated by either one or two beats.

My system started out by analyzing the lower numbers ( One through ten for starters) and see-

ing how many variations of 2s and 3s could be made out them.

Number one, obviously has no variations possible.

Numbers 2 and 3 are the building blocks of this system so lets use a simple system of tapping

with our right and left hands to make it more understandable:

Number 2 will be played as a tap with the right hand (the Accent) followed by a tap with the

left hand, (non accented)

Number 3 will be played as a tap with the right hand (the Accent) followed by 2 taps with the

left hand, (non accented).

Obviously, to be musical, the taps should be evenly spaced to represent steady time.

For practice, tap out the Gankino rhythm described above. The sequence you tap out should

be a steady repetition of the following cycle of taps.

RIGHT left RIGHT left - RIGHT leftleft RIGHT left RIGHT left.

this can be described as 22-32-2, as mentioned above.

Moving along to numbers 4 and above, a system develops where we use the ODD numbers to

assist us in a sort of Rhythmic short hand which helps us count complex divisions of 2s

and 3s

It goes like this:

Number 4 has only one variation in this system, since 1 is not a building block ( I use 3 instead,

as a 2 + 1, which is where 1 fits in.) so using our building blocks, 4 can only be divided into:

2 + 2,

...so 4 would be played R L R L using our tapping system where 2s are played as a tap

(accent) with the right hand followed by a tap with the left hand(non accented).


3/8

Number 5 becomes our first building block because it conveniently replaces a group of 2 and 3.

5 can be broken into two possible accent patterns using our building blocks of 2 and 3,

they are:

2 3 and3 2

Using the tapping system, this would be:

R L R L L Or

R L L R L

heres the number theory part:

Since a string of 2s can never add up to an odd number, there are reasons why we can use

one of these two sequences as a shortcut to represent a possible rhythmic accent pattern for

the number 5.

From now on our system will say that the 2 possible variations for number 5 are:

5 and

3 2

the 23 pattern is now represented by just the number 5, reasons for this will become clearer

as we dissect possible accent patterns for number 7, for now lets just say that any odd number

becomes a building block representing a string of 2s with a 3 after it. Moving to number 6.

Number 6 also only has two possible variations of accent patterns, they are:

2 2 2 and

3 3

R L R L R L and

R L L R L L

Number 7 can be broken down into the following groups of 2s and3s:

2 2 32 3 2

3 2 2

but for our system, the pattern 2 3 is replaced by just saying 5.

Furthermore the pattern 2 2 3 will now be called just 7.

So the 3 possible variations for the number 7 in this system are:

7 .( 2 2 3 )

5 2 .(2 3) 2, and

3 2 2


4/8

Number 8 has several variations:

5 3 ..(2 3) 3

3 5 .3 (2 3)

3 3 2

2 2 2 2

Some of these are among the most common rhythmic patterns in music, tap them out using R

L L for 3s and R L for 2s while counting along with the building blocks 2s 3s and5s.

All odd numbers, when used as building blocks will be represented by the pattern of a group of

2s with a 3 at the end, so

5 is 2 3, .R L R L L

7 is 2 2 3, .R L R L R L L

9 is 2 2 2 3, .R L R L R L R L L

11 is 2 2 2 2 3,and so on.

This is the basic part of the theory: We are breaking up numbers into combinations of odd

numbers and 2s. So when you see an odd number like 7, it represents the accent pattern 2

2 3, played like this R L R L R L L. Number 5 represents the accent pattern 2 3, played like

this: R L R L L.

Moving on to the variations of accent patterns in number 9

Number 9 can be divided into the following accent patterns:

9 ...(2 2 2 3) (R L R L R L R L L)

7 2 ...(2 2 3) 2 (R L R L R L L) R L

5 2 2 ..(2 3) 2 2 (R L R L L) R L R L

3 3 3 ..( no explanation necesary) (R L L)(R L L)(R L L)

3 2 2 2 (R L L)(R L)(R L)(R L)

Aside from the 3 3 3 all these variations of the number 9 are often used in turkey, whose

music uses a lot of 9/8 time signature.

Number 10 can be divided into these accent patterns:

7 3 . (2 2 3) 3

3 7 ...(3 (2 2 3)

5 5 ....(2 3) (2 3) (R L R L L)(R L R L L)

5 3 2 ..(2 3) 3 2 (R L R L L)(R L L)(R L)

3 5 2 ..3 (2 3) 2

3 3 2 2 (mission impossible)

2 2 2 2 2


5/8

By using odd numbers as building blocks, we can keep track of all the possible variations in the

order of 2s and 3s in a number. Each odd number represents the pattern of however many

2s with a 3 at the end.

To analyze any number for rhythmic accents,(patterns of 2s and 3s) start with the highest odd

number that fits in the pattern, if the number is odd, it will be the number itself that is thelargest odd number. Use this number to represent the pattern of a string of 2s with a three

at the end, then start adding the next highest odd numbers and using 2s only at the end of

the last odd number. For number 11 it would go like this:

Number 11 The highest odd number that fits in 11 is itself, so that would be represent the

pattern of all 2s followed by a 3:

11 .. (2 2 2 2 3)

The next highest odd number that fits into 11 is 9, and we must put a 2 at the end to add up to

11. ( we dont have to put a 2 before 11 because that pattern is already represented by the

digit 11 itself)

9 2 .(2 2 2 3) 2

7 2 2 (2 2 3) 2 2 the Gankino,

5 2 2 2 .(2 3) 2 2 2

5 3 3 .(2 3) 3 3

3 5 3 .3 (2 3) 3

3 3 5 .3 3 (2 3)

3 3 3 2

3 2 2 2 2

Number 12 has the following accent patterns ( I have put in the hand tapping patterns inter-

mittently as a reminder of how they can be applied)

9 3 .(2 2 2 3) 3 (R L R L R L R L L)(R L L)

3 9

7 55 7

3 7 2

7 3 2

5 5 2 (2 3) (2 3) 2 (R L R L L)(R L R L L)(R L)

5 3 2 2

3 5 2 2

3 3 3 3

3 3 2 2 2

2 2 2 2 2 2


6/8

Number 13 has these accent patterns

13 (2 2 2 2 2 3) (R L R L R L R L R L R L L)

11 2

9 2 2

7 3 3 (2 2 3) 3 3 (R L R L R L L)(R L L)(R L L)3 7 3

3 3 7

7 2 2 2

5 5 3 .(2 3) (2 3) 3 (R L R L L)(R L R L L)(R L L)

5 3 5 (2 3)3 (2 3) (R L R L L)(R L L)(R L R L L)

3 5 5 ..3 (2 3) (2 3) (R L L)(R L R L L)(R L R L L)

3 5 3 2

5 3 3 2

3 3 5 2

3 3 3 2 2

5 2 2 2 2

3 2 2 2 2 2

Number 14 has these possible accents:

11 3

3 11

9 55 9

9 3 2

3 9 2

7 7

7 5 2

5 7 2

7 3 2 2

3 7 2 2

5 3 3 3

3 5 3 3

3 3 5 3

3 3 3 5

5 5 2 2

5 3 2 2 2

3 5 2 2 2

3 3 3 3 2

3 3 2 2 2 2

2 2 2 2 2 2 2


7/8

Number 15 has these possible accents

15

13 2

11 2 2 .. Greek/Macedonian rhythm pousnitsa

9 2 2 29 3 3

3 9 3

3 3 9

7 5 3

5 7 3

3 7 5

7 3 5

3 5 7

5 3 7

7 3 3 2

3 7 3 2

3 3 7 2

7 2 2 2 2

5 5 5

5 5 3 2

5 3 5 2

3 5 5 2

5 2 2 2 2 2

5 3 3 2 23 5 3 2 2

3 3 5 2 2

3 3 3 3 3

3 3 3 2 2 2

3 2 2 2 2 2 2

(28 variations)


8/8

FINALLY

Number 16, One measure of 4/4 time has 33 possible combinations of 2s and 3s,

or short and long steps.

13 3 ..(2 2 2 2 2 3)(3) (R L R L R L R L R L R L L)(R L L)

3 1311 3 2

3 11 2

9 7 .. Greek/Macedonian Levendikos

7 9

9 5 2...(R L R L R L R L L) (R L R L L)(R L)

5 9 2

9 3 2 2

3 9 2 2

7 7 2

7 5 2 2

5 7 2 2

7 3 3 3

3 7 3 3

3 3 7 3

3 3 3 7

7 3 2 2 2

3 7 2 2 2

5 5 3 3

3 5 5 33 3 5 5

5 3 5 3

3 5 3 5

5 3 2 2 2

3 5 2 2 2

5 3 3 3 2

3 5 3 3 2

3 3 5 3 2

3 3 3 5 2

3 3 3 3 2 2

3 3 2 2 2 2 2

2 2 2 2 2 2 2 2

For number Sixteen, since its usually represented as a bar of 4/4 time, we can tap our foot

every beat, (every four 16th

notes). With this system, the columns of numbers shown above

will divide themselves into downbeats, and upbeats. When the numbers are odd, the frist odd

number will be on downbeats, the second on upbeats, the third = downbeats, 4th

=

upbeatsetc. When the pattern ends in a string of 2s, they will be all downbeats because a 3

is necessary to turn the beat around.!

Dissecting Rhythms Mathematically

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