# Subdivisions - Duple & Triple Multiples

1. Pulse time is a common choice in most but not all musical composition. An interval of time can me measured and given a delineation as a tempo expressed in beats per minute. It is the organization, manipulation and permutation of pulse that forms the vocabulary of a rhythmic language.

Once a pulse has been established, simple arithmetical ratios can change the way these primary pulses "feel."

2. From a primary pulse, multiples and divisions can be derived and expressed as ratios. A division of one pulse into two equal parts can be expressed by the ratio 1:2. For every one primary pulse, two pulses are emphasized. The ratio 2:1 yields a multiple of the primary pulse that is exactly twice the length. Thus, for every two primary pulses, one pulse is emphasized.

3. Division ratios yield new pulses smaller than the primary pulse and therefore occur more frequently in time. The simplest arithmetical ratio of 1:1 is in unison with the primary pulse and therefore implements no rhythmic deviation.

4. To imply a new rhythmic value, the next simplest ratio must be taken; that is, 1:2. Each primary pulse is divided into 2 equal parts resulting in a duple multiple of the original pulse. If a quarter note is established as the primary pulse, the duple multiple is expressed as an eighth note.

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5. Taking the next simplest mathematical ratio of 1:3, each primary pulse is divided into 3 equal parts resulting in a triple multiple of the original pulse. If a quarter note is established as the primary pulse, the triple multiple is expressed as an eighth note triplet.

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6. Moving in the opposite direction, multiple ratios yield new pulses larger than the primary pulse and therefore occur less frequently in time. 1:2 and thus expressed as 2:1. than can be subdivided. 1:2 yields quarter notes; 1:3 yields quarter note triplets.

?. Specifying the primary pulse is a habit we must create early in these rhythmic studies. Arithmetical ratios can only be applied to intervals that have been agreed upon. In exploring the application of duple and triple multiples, we can interpret larger primary pulses across the grid we have already created with quarter notes. By grouping two quarter notes together as one pulse, we have created a half note pulse than can be subdivided. 1:2 yields quarter notes; 1:3 yields quarter note triplets.

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?. Grouping three quarter notes together as one pulse crates new subdivision values. 1:2 yields a dotted quarter note (three eighth notes); 1:3 yields quarter notes.

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?. Lastly, by grouping four quarter notes together as one pulse, the ratio 1:2 yields half notes; 1;3 yields half note triplets.

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?. Seeing how ratios can be applied to larger groupings of pulse, the same concept can be applied to smaller groupings. It is apparent that a duple multiple of the primary pulse yields two subdivisions per pulse. Applying the ratio 1:2 to these subdivisions now yields four subdivisions per pulse. The ratio 1:3 yields 6. As applied to the quarter note grid established,