Perfectly Even Rhythms

Perfectly Even Rhythm: A rhythm where the distance between every pair of adjacent onsets is the same as any other pair of adjacent onsets.

Perfectly even rhythms divide a pulse cycle into equal parts. Geometrically, they can be represented as regular polygons. Examples include:

  • 2 onsets over 4 pulses
  • 4 onsets over 8 pulses
  • 4 onsets over 12 or 16 pulses

2 onsets, 4 pulses

Perfectly even: interval pattern [2-2]

4 onsets, 8 pulses

Perfectly even: interval pattern [2-2-2-2]

4 onsets, 16 pulses

Perfectly even: interval pattern [4-4-4-4]

Stability and Regularity

  • Perfectly even rhythms are the most stable rhythms
  • Geometrically, all perfectly even rhythms are represented by regular polygons

3 onsets, 9 pulses

Perfectly even: interval pattern [3-3-3]
Regular triangle

4 onsets, 12 pulses

Perfectly even: interval pattern [3-3-3-3]
Regular square

5 onsets, 15 pulses

Perfectly even: interval pattern [3-3-3-3-3]
Regular pentagon

Examples

  • If the number of onsets (k) divides evenly into the number of pulses (n), then perfectly even rhythms corresponding to regular polygons are trivial to construct.

n = 16, k = 4, distance = 4 (4/4 or 4/8 meter)

n = 12, k = 4, d = 3 (12/8 or 12/16 meter)

n = 4, k = 2, d = 2 (2/2 or 2/4 meter)

Meter and Rhythm

Regular meters can be defined as the simplest expression of rhythm. A regular pulse with reguarly occuring accents.

Perfectly even rhythms satisfy this definition of meter. All non-complex meters, are perfectly even rhythms.

Not all rhythms are perfectly even

In most rhythms and timelines the number of onsets (k) does not divide the number of pulses (n) evenly. Also, not every organization of the onsets will result in a good or meaningful rhythm.

Here are some "perfectly uneven" rhythms - how many of these are good?

4 onsets, 12 pulses (clustered)

12 pulses, 4 onsets at [0, 1, 2, 3]
Interval pattern: [1-1-1-9]

4 onsets, 8 pulses (clustered)

8 pulses, 4 onsets at [0, 1, 2, 3]
Interval pattern: [1-1-1-5]

2 onsets, 4 pulses (clustered)

4 pulses, 2 onsets at [0, 1]
Interval pattern: [1-3]

4 onsets, 16 pulses (clustered)

16 pulses, 4 onsets at [0, 1, 2, 3]
Interval pattern: [1-1-1-13]

Maximal Evenness

Maximal Evenness: A distribution of onsets that is most spread out.

Many popular rhythms and timelines can be created through maximally even distribution of onsets.

For example, the Bossa Nova is maximally even [3-3-4-3-3]

Important principle: Any rotation of any maximally even rhythm will also be maximally even.

Bossa Nova

16 pulses, 5 onsets at [0, 3, 6, 10, 13]
Interval pattern: [3-3-4-3-3]

Bossa Nova (Rotated)

16 pulses, 5 onsets at [1, 4, 7, 11, 14]
Same pattern, rotated one position

Fume-Fume Timeline

The Fume-Fume timeline shows the maximally even distribution of 5 onsets across 12 pulses. Below are three rotational variations, each displaying different onset positions while maintaining the same spacing relationships.

Fume-Fume (original)

12 pulses, 5 onsets at [0, 2, 4, 7, 9]
Interval pattern: [2-3-2-3-2]

Fume-Fume (Rotation 1)

12 pulses, 5 onsets at [1, 3, 5, 8, 10]
Same pattern, different rotation

Fume-Fume (Rotation 2)

12 pulses, 5 onsets at [2, 4, 6, 8, 11]
Same pattern, different rotation

Bembé Timeline

The Bembé timeline represents 7 onsets distributed maximally across 12 pulses, illustrating another culturally significant rhythmic pattern.

Bembé

12 pulses, 7 onsets at [0, 2, 4, 5, 7, 9, 11]
Interval pattern: [2-2-1-2-2-2-1]

Nearly Even Rhythms

Nearly Even Rhythms: Rhythms that are close to maximally even but have slightly different interval distributions. These rhythms are still musically effective because they maintain a relatively balanced spread of onsets.

We looked earlier in the course at 6 timelines of 5 onsets in 16 pulses. Only the Bossa Nova is maximally even — the other 5 are nearly even:

Bossa Nova (Maximally Even)

Onsets: [0, 3, 6, 10, 13]
Interval pattern: [3-3-4-3-3]

Son (Nearly Even)

Onsets: [0, 3, 6, 10, 12]
Interval pattern: [3-3-4-2-4]

Shiko (Nearly Even)

Onsets: [0, 4, 6, 10, 12]
Interval pattern: [4-2-4-2-4]

Rumba (Nearly Even)

Onsets: [0, 3, 7, 10, 12]
Interval pattern: [3-4-3-2-4]

Soukous (Nearly Even)

Onsets: [0, 3, 6, 10, 11]
Interval pattern: [3-3-4-1-5]

Gahu (Nearly Even)

Onsets: [0, 3, 6, 10, 14]
Interval pattern: [3-3-4-4-2]

Observation: Notice how the Bossa Nova has the most balanced interval pattern [3-3-4-3-3], while the other timelines have more variation in their intervals. Despite not being maximally even, these nearly even rhythms have been used across cultures for centuries, demonstrating that musical effectiveness doesn't require mathematical perfection.

Key Insights

  • Regular polygons represent perfectly even rhythms - equal spacing between all onsets
  • Maximal evenness creates the most spread-out distribution when perfect evenness isn't possible
  • Rotation invariance: All rotations of a maximally even rhythm maintain maximal evenness
  • Cultural significance: Many world music traditions have independently discovered these mathematically optimal distributions
  • Most interesting rhythms are maximally even but not perfectly even - they have two different interval sizes that are as evenly distributed as possible
  • Nearly even rhythms are close to maximally even and remain musically effective - small deviations from perfect balance can create distinctive character while preserving rhythmic coherence

Clarification: Polyrhythms vs. Timelines

Polyrhythm: The simultaneous sounding of two or more perfectly even rhythms within the same pulse cycle. The cycle must be divisible into at least two different numbers that are not a multiples of each other for a polyrhythm to exist.

POLYRHYTHMS (Simultaneous Perfectly Even Groupings)

Example 1: Hemiola (6 pulses grouped as 3 AND as 2)

Cycle of 6 pulses:  1  2  3  4  5  6
Grouped by 3:       X  .  .  X  .  .    (two groups of 3)
Grouped by 2:       X  .  X  .  X  .    (three groups of 2)
✓ Both are perfectly even groupings, heard simultaneously

Example 2: 4 against 3 (12 pulses grouped as 4 AND as 3)

Cycle of 12 pulses: 1  2  3  4  5  6  7  8  9  10 11 12
Grouped by 4:       X  .  .  .  X  .  .  .  X  .  .  .   (three groups of 4)
Grouped by 3:       X  .  .  X  .  .  X  .  .  X  .  .   (four groups of 3)
✓ Both are perfectly even groupings, heard simultaneously

NOT POLYRHYTHMS
(Timelines Against Perfectly Even Rhythms or Other Timelines)

Example 3: Tresillo Against Steady Beat (8 pulses)

8 pulses:        0 1  2  3  4  5  6  7
Steady beat:     X  .  X  .  X  .  X  .   (groups of 2 - perfectly even rhythm)
Tresillo:        X  .  .  X  .  .  X  .   (groups of 3, 3, 2 - maximally even)
✗ Tresillo is a maximally even rhythm not a perfectly even rhythm

Example 4: Bossa Nova Clave Against Steady Pulse (16 pulses)

16 pulses:       0 1  2   3  4  5  6  7  8  9  10 11 12 13 14 15    
Steady pulse:    X  .  .  .  X  .  .  .  X  .  .  .  X  .  .  .  (groups of 4 - perfectly even rhythm)
Bossa nova:      X  .  .  X  .  .  X  .  .  .   X  .  .  X  . .
✗ Bossa nova pattern is maximally even, not a perfectly even rhythm

Example 5: Son Clave (3-2) Against Steady Pulse (16 pulses)

16 pulses:       0 1  2   3  4  5  6  7  8  9  10 11 12 13 14 15
Steady pulse:    X  .  .  .  X  .  .  .  X  .  .  .  X  .  .  .  (groups of 4 - perfectly even rhythm)
Son clave:       X  .  .  X  .  .  X  .  .  .  X  .  X  .  .  .
✗ Son is a nearly even rhythm not a perfectly even rhythm

The Key Distinction

Polyrhythm: Both patterns use perfectly even groupings of the cycle

  • 6 pulses: can be grouped evenly by 2 (three groups of 2)
  • 6 pulses: can be grouped evenly by 3 (two groups of 3)
  • Hear both perfectly even rhythms simultaneously
  • The complete pulse cycle is evenly divided in two distinct (not overlapping) ways
  • Geometrically, the two rhythms form two different regular polygons within the same circle representation
  • Anything else is not a polyrhythm