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Apparently, poly-meter creates "rhythmic" counterpoint. What does this mean? Nothing online shows anything. What are other effects of poly-meter?

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No need to over-think this. Homophony is when two (or more) musical voices follow the same rhythm. Counterpoint is when they don't. The terms are most often used to describe music where pitches as well as rhythms are involved. But it's an easy leap to the concept of purely rhythmic counterpoint. And poly-meter - like a 3/4 rhythm being played against a 4/4 one - is pretty well guaranteed to produce this 'rhythmic counterpoint'.

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    Isn't "homorhythm" the rhythmic term? Homophony is about harmony. – Michael Curtis Oct 12 '20 at 14:08
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This post comes in three parts:
1) basics
2) techniques
3) examples

Rhythmic counterpoint basics

Rhythmic counterpoint requires one of two conditions:

  1. Pulse onsets occur at different times,
  2. Pulse onsets occur at different intensities.

Example of condition 1

X:1
T:Rhythmic Counterpoint
T:condition 1 example
K:none
M:C
V:P1 K:none clef=perc stafflines=1
V:P2 K:none clef=perc stafflines=1
L:1/4
[V:P1]BzBz :|]
[V:P2]zBzB :|]

Example of condition 2

X:1
T:Rhythmic Counterpoint
T:condition 2 example
K:none
M:C
V:P1 K:none clef=perc stafflines=1
V:P2 K:none clef=perc stafflines=1
L:1/4
[V:P1]"^>"BB"^>"BB :|]
[V:P2]B"^>"BB"^>"B :|]

Rhythmic counterpoint techniques

Polyrhythm

In polyrhythm, two or more concurrent rhythms share the same underlying pulse, but in contrasting subdivisions that don't obviously relate to each other. Polyrhythm satisfies condition 1.

X:1
T:Rhythmic Counterpoint
T:Polyrhythm
K:none
M:C
V:P1 K:none clef=perc stafflines=1
V:P2 K:none clef=perc stafflines=1
L:1/16
[V:P1] (3B2B2B2 (3B2B2B2 (3B2B2B2 (3B2B2B2 :|]
[V:P2] (5BBBBB  (5BBBBB (5BBBBB (5BBBBB :|]

Polymeter

Meter is the presence of "regularly occurring patterns and accents" (Source: Wikipedia). In polymeter, two or more such "regularly occurring patterns and accents" are present simultaneously. Because they share a basic pulse, but have different groupings of strong and weak beats, polymetric music satisfies condition 2.

One must be careful, however, in choosing the meters. If they coincide too often, the sense of polymeter is lost.

Example of ineffective polymeter

This will just sound like 2/4 time.

X:1
T:Rhythmic Counterpoint
T:ineffective polymeter
K:none
V:P1 K:none clef=perc stafflines=1
V:P2 K:none clef=perc stafflines=1
L:1/4
[V:P1][M:2/4]"^>"BBBB | "^>"BBBB :|]
[V:P2][M:4/4]"^>"BB | "^>"BB | "^>"BB | "^>"BB :|]

Example of effective polymeter

Here, the strong pulses of all three "voices" coincide only once every 30 pulses.

X:1
T:Rhythmic Counterpoint
T:effective polymeter
K:none
V:P1 K:none clef=perc stafflines=1
V:P2 K:none clef=perc stafflines=1
V:P3 K:none clef=perc stafflines=1
L:1/4
[V:P1][M:2/4]"^>"BB | "^>"BB  | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB | "^>"BB  :|]
[V:P2][M:3/4]"^>"BBB | "^>"BBB | "^>"BBB | "^>"BBB | "^>"BBB | "^>"BBB | "^>"BBB | "^>"BBB | "^>"BBB | "^>"BBB :|]
[V:P3][M:5/4]| "^>"BBBBB  | "^>"BBBBB | "^>"BBBBB | "^>"BBBBB | "^>"BBBBB | "^>"BBBBB :|]

Phasing

In phase music, two or more instruments play a single rhythm but with one instrument remaining in exact time while the others introduce gradual delays. This also creates a form of rhythmic counterpoint in which the rhythms are sometimes perceived as entirely separate and at other times as forming a single composite rhythm.

Example of phase music

See musical example, below.

Rhythmic tiling / Rhythmic canons

Rhythmic tiling is a mathematical composition technique by which non-overlapping rhythmic patterns fill a metrical space. A rhythmic cannon accomplishes this with only a single, non-trivial rhythm, distributed among pats. The mathematics behind the technique were first developed by Dan Tudor Vuza1 and figure prominently in the music of Tom Johnson.

Example of rhythmic tiling

Derived from Tom Johnson's "Tilework for String Quartet" (see video below).

X:1
T:Rhythmic Counterpoint
T:tiling rhythmic canon
K:none
M:none
V:P1 K:none clef=perc stafflines=1
V:P2 K:none clef=perc stafflines=1
V:P3 K:none clef=perc stafflines=1
V:P4 K:none clef=perc stafflines=1
L:1/8
[V:P1] BBzBBBzzB
[V:P2] xxxxxxBBzBBBzzB
[V:P3] xxxxxxxxxxxxBBzBBBzzB
[V:P4] xxxxxxxxxxxxxxxxxxBBzBBBzzB

Rhythmic counterpoint examples

Polyrhythm

Polymeter

The Cars's "Touch and Go" (opening section: 5/4 bass/drums; 4/4 vocals/keybaord)

Phasing

Steven Reich's "Clapping Music"

Tiling

Tom Johnson's "Tilework for String Quartet" (includes composer discussion)


1Vuza, Dan Tudor. "Supplementary Sets and Regular Complementary Unending Canons (Part One)." Perspectives of New Music 29, no. 2 (1991): 22-49. Accessed October 9, 2020. doi:10.2307/833429; Vuza, Dan Tudor. "Supplementary Sets and Regular Complementary Unending Canons (Part Two)." Perspectives of New Music 30, no. 1 (1992): 184-207. Accessed October 9, 2020. doi:10.2307/833290; Vuza, Dan Tudor. "Supplementary Sets and Regular Complementary Unending Canons (Part Three)." Perspectives of New Music 30, no. 2 (1992): 102-24. Accessed October 9, 2020. doi:10.2307/3090628; Vuza, Dan Tudor. "Supplementary Sets and Regular Complementary Unending Canons (Part Four)." Perspectives of New Music 31, no. 1 (1993): 270-305. Accessed October 9, 2020. doi:10.2307/833054.

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I think what you are asking about must be about polyrhythm not polymeter.

Counterpoint literally means "point against point" as in pitch against pitch, or two melodic lines that follow different pitch contours.

But, another way to describe counterpoint is independence of parts. Different rhythms in different parts is an effective way to achieve independence of parts. (IMO it's more effective that pitch difference.)

If you look at the "points" on a grid of a polyrhythm you will see that the points occur independently, meaning that don't happen at the same time between parts...

enter image description here

...notice that the two parts only strike at the same time at the 1st and 13th boxes (at the first beat of each measure.) The other times they strike independently. With the polyrhythm there is a lot of independence of the parts, the independence of the parts creates rhythmic counterpoint.

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