Rhythmical structures

Mobility

Rhythmical mobility is defined
    v(d1,d2,..) = k / AVG(d1,d2,...),
where k is order of rhythmical system and AVG is an average of relative tone lengths d1,d2,...

E.g.: k = 8

Pattern Modality A-average G-average Mobility
1/1 8 8.00 8.00 1.00
1/2+1/2 4(4) 4.00 4.00 2.00
3/4+1/4 6(2) 4.00 3.46 2.31
7/8+1/8 7(1) 4.00 2.65 3.02
1/2+1/4+1/4 42(2) 2.67 2.52 3.17
1/4+1/4+1/4+1/4 222(2) 2.00 2.00 4.00
1/2+1/4+1/8+1/8 421(1) 2.50 1.68 4.76

Geometric average (G-average) seems to be more suitable than arithmetic average (A-average).

Tension

Rhythmical tension is defined
    s(d1,d2,..) = 2* σ(d1,d2,...) /k,
where k is order of rhythmic system, σ is standard deviation. Approximative values of tension:
s(d..) = (max(d..)-min(d..)) /k.

E.g.: k = 8

Pattern Modality Deviation Tension(max-min)/k
1/1 8 0.00 0.00 0.00
1/2+1/2 4(4) 0.00 0.00 0.00
3/4+1/4 6(2) 2.00 0.50 0.50
7/8+1/8 7(1) 3.00 0.75 0.75
1/2+1/4+1/4 42(2) 0.94 0.24 0.25
1/4+1/4+1/4+1/4 222(2) 0.00 0.00 0.00
1/2+1/4+1/8+1/8 421(1) 1.22 0.31 0.38

Entropy

Rhythmical entropy is defined

                   k
    H(d1,d2,..) = -∑(p(i)*ln(p(i))), where p(i) = d(i)/k
                   i=1

E.g.: k = 8

Pattern Modality Entropy
1/1 8 0.00
1/2+1/2 4(4) 0.69
3/4+1/4 6(2) 0.56
7/8+1/8 7(1) 0.38
1/2+1/4+1/4 42(2) 1.04
1/4+1/4+1/4+1/4 222(2) 1.39
1/2+1/4+1/8+1/8 421(1) 1.21

Reducible patterns

Let us compare the syncope [1/4+1/2+1/4] with rhythm [1/2+1/4+1/4]. The both patterns have the same mobility, tension and entropy but their musical meanings differ.

Reducible pattern can be converted to a simpler pattern.
E.g.:

Pattern Modality Entropy
1/2+1/4+1/8+1/8 421(1) 1.21
1/2+1/4+1/4 42(2) 1.04
1/2+1/2 4(4) 0.69

Syncope is an example of non-reducible pattern.

Example of analysis

W.A.Mozart: Alla Turca March, beginning (bar 1.-5.)
k= 16.
Rhythmical patterns:
  |-,-,-,1111|4,1111,4,1111|1111,1111,4,22|22,22,22,22|22,22, 4, -|
Mobility
  |-,-,-,16  | 4, 16, 4, 16|  16, 16, 4, 8| 8, 8, 8,8 | 8, 8, 4, -|
Tension
  |-,-,-,0.0 |  0.15,  0.15|  0.0,   0.12 |  0.0, 0.0 | 0.0, 0.0,-|
Entropy
  |-,-,-,1.39|  1.39,  1.39|  2.77,  1.04 | 1.39, 1.39| 1.39,0.0,-|
Rhytmic characteristics of classes

Harmonic bindings