Recursive Superimposed Metric Modulation (Draft)

Update: This post inspired its own repository on GitHub. Visit the project for an evolution of these ideas.

Abstract

It is my intention to posit “recursive superimposed metric modulation” as a musical device used to alter the substructure upon which commonly used rhythmic devices are based.

Metric modulation is a change in meter based on a note value from an existing meter. Recursion is a repeating process whose output at each stage is applied as input in the succeeding stage. Thus, through subsequent modulations to closely related meters, distantly related meters can be accessed while maintaining a relationship to the original. The superimposition of these modulations over one or more of the preceding iterations can be utilized to create rhythmic dissonance.

Rhythm Basics

For sake of clarity in the terms used throughout, a brief overview of basic rhythmic concepts follows.

Organization of Time

The organization of time in music can be reduced fundamentally to pulse, tempo, and meter. ¹

Pulse, or beat, is the regularly recurring underlying pulsation we perceive that compels music to progress through time. Pulse makes us react kinesthetically to music. We tap our feet, we dance, we march, or we may just “feel” the pulse internally. In a piece of music, some durational value is assigned to be the pulse and all other durations are proportionally related.

Tempo is the relative speed or rate at which metrical pulses occur over time. Tempo is expressed as either a descriptive term or a metronomic value.

Meter is a ratio that determines which durational value is assigned to represent the fundamental background pulse, how these pulses are grouped together in discrete segments, how these pulses naturally subdivide into lesser durational values, and the relative strength of perceived accents within segments or groupings of pulses.

Each occupies a specific role in the way time is perceived with varying degrees of correlation to one another. Meter appears to be the most autonomous of the three as it can exist as data devoid of tempo and pulse. Thus, meter will be our primary focus of manipulation to imply the modification of pulse or tempo.

Metric Levels

Meter can exist at various “levels”: beat level (the base unit to which operations are applied), product levels, and quotient levels. In relation to the beat level, multiplication yields a more frequent durational value—product—and division yields a less frequent durational value—quotient.

An elementary example being that a value of 1/4 (quarter note) at the beat level multiplied by 2 results in 1/2 (half note), a value at a product level; 1/4 divided by 2 results in 1/8 (eighth note), a value at a quotient level.

Metric Modulation

…metric modulation is a technique in which a rhythmic pattern is superposed on another, heterometrically, and then supersedes it and becomes the basic meter.² Nicolas Slonimsky

What differentiates a metric modulation from any other tempo change is that the superposed rhythm is based on a note value from the preceding meter, making the duration of the minimal denominator consistent. For example, the superposition of 6/8 over 4/4 yields a consistent 1/8 note value.

The formula for a modulation is:

newTempo / oldTempo = newNumberOfMinDenominator / oldNumberOfMinDenominator

Recursion

A process exhibits recursive behavior when the procedure refers to itself. So a distinction must be made between the process and the execution of the process. The Fibonacci sequence is a classic example of recursion as each step requires input from a previous step’s output to be executed.

Application

The process to be iterated follows:

1. Identify the metric levels of a given meter
2. Designate a metric level as the basis for a new grid (beat level)
3. Apply an operation to the beat level

Identifying Metric Levels

Identify the durational values that make up a given meter.

Designating a Beat Level

Choose any durational value to serve as the basis for the modulation. This is referred to as the beat level value.

Applying Basic Operations

There are three basic operations that can be applied to the beat level value.

• Group
• Multiply
• Divide

Grouping is the process of segmenting the beat level values to imply a new pulse.

Multiplying the beat level value renders larger segments of time.

Dividing the beat level value renders smaller segments of time.

Example 1

Here is a simple example to show how a complex rhythmical grid can be achieved with just one iteration of recursivity.

Iteration 1

1. From a 4/4 meter
2. Take the quarter note
3. Regroup from 4 to 5

What results is a 5-beat phrase superimposed over 4.

Iteration 2

1. From the new 5/4 meter
2. Take the full length of the 5-beat phrase
3. Divide by 2

What results can simply be thought of as 2.5 quarter notes that split a 5/4 measure over 4/4. Even though this new structure is “felt” as something seemingly unrelated to the original 4/4 meter, it is directly derived from 4/4 and can therefore return to any previous level of recursion including the original meter.

Example 1 - Video

Step by Step Abstraction

Even simple operations can create perceived complexity. This example shows the step by step abstraction from a common meter in small increments. In western music, we are typically comfortable with operations based on 2 and 3, so all it takes is one unfamiliar prime (5 in this case) to detach a listener’s ear from what is expected and create rhythmic tension.

• Original meter is 4/4
• Divide the entire measure by 5 (1/5 is the beat level value)
• Multiply 1/5 by 2 (2/5 is the beat level value)
• Multiply 1/5 by 3 (3/5 is the beat level value)
• Divide 3/5 by 2 (3/10 is the beat level value)

Step by Step Abstraction - Video

Apologies. The 3/5 section is incorrectly performed in this video. New video soon.

References