Permutations: 3 – 4 – 5 – 6 – 7

At the start of 2020, this website received a well-deserved makeover to better organize all its content. The redesign created dedicated sections for travelling alongside educational materials. Most drum exercises published here rely on permutations, making this post essential for getting started with any of that material. Currently, only the 3, 4, 5, and 6-dotted systems are available as free downloads. In the future, I plan to publish a comprehensive Permutations book covering variations all the way up to 12-dotted systems.

The core principle driving these rhythmical systems involves systematically working through every possible variation between “no colored dots” and “all colored dots filled.” Each exercise begins with a single dot cycling through every position, then methodically builds through combinations of two, three, and more dots until exhausting all cycles. This exhaustive approach captures literally every rhythmical possibility within the chosen cycle length.

What are permutations?

Technically speaking, permutations and combinations share mathematical roots. In these systems, black dots methodically explore every combination possible within a framework of fixed white dots. Take a 3-dotted system: it generates three single-black-dot positions (1, 2, 3), then three double-black-dot combinations (1-2, 1-3, 2-3), with the final all-dots-filled state making seven total combinations (excluding the blank starting point).

Mathematically, permutations differ from random combinations through their strict sequential order. Every cycle below begins at the first white dot, then shifts methodically once exhausting all combinations from that position. This way creating systematic rhythmic evolution rather than chaos.

3-dotted Permutations:

These are very short but still essential permutations. They work perfect as groupings of fourths in 3/4 or eights in a 3/8 settings with flexible subdivisions.

4-dotted Permutations:

These rank as the most essential system, unlocking endless variations within even metric time signatures like four quarter notes, eighth notes, or sixteenth notes—their versatility spans countless musical contexts.

5-dotted Permutations:

Less common but incredibly engaging, these excel in 5-piece time signatures (5/8) or as polyrhythmic overlays within even meters. Players choose subdivisions like 3-2 or 2-3 based on musical intent.

6-dotted Permutations:

Remarkably versatile, perfect as triplet groupings in even time (ideal for jazz) or anchoring 3/4, 6/8 settings with flexible subdivisions beneath an eighth-note foundation.

7-dotted Permutations:

Like 5-dotted systems, these shine in 7/4 (quarters) or 7/8 (eighths) with popular subdivisions including 3-2-2, 2-3-2, or 2-2-3—limited contexts, maximum enjoyment.

Rhythmic Permutations – eBook:

Major Update: Early 2022 marked completion of my comprehensive Rhythmic Permutations eBook! Starting with 3-beat variations and systematically building through 4, 5, 6, 7, 8, 9, 10, 11, and 12-beat systems, it catalogs 8,174 unique rhythms across 107 densely-packed pages. This definitive archive serves as the ultimate resource for rhythmical studies, technical exercises, sight-reading development, and advanced coordination training—laying the strongest foundation for sophisticated rhythmic mastery.

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