1. Finding Downbeats with a Relaxation Oscillator Douglas Eck Presented by Zhenyao Mo

2. Overview  Purpose: to find downbeats in rhythmical patterns  Method: using oscillators to model the behavior of rhythmical patterns, thus find downbeats

3. Problems in downbeats finding  Imperfect timing –Motor noise –Deviations due to expressive timing

4. Why oscillators  It has CYCLES  It could respond to input perturbation, and finally align its cycles with some periodic components –Thinking of musical notes as input perturbation –Thinking of oscillator cycles as downbeats  It can resist noise signals Perfect !!!

5. A Relaxation Oscillator Model  dv/dt = -v(v-  dv/dt = -v(v-θ)(v-1)-ω+Ω   dω/dt = e(v-r ω) (v is voltage, ω is voltage recovery)

6. A Relaxation Oscillator Model  dv/dt = -v(v-  dv/dt = -v(v-θ)(v-1)-ω+Ω (input notes)   dω/dt = e(v-r ω) (v is voltage, ω is voltage recovery)

7. Input note signals  xxxx..xx.x.x…  x is notes,. is a rest  The base interval between notes is 125 time-steps  x is encoded as 10000…0, one 1, 124 0 . is encode as 00000…0, 125 0

8. Input note signals (continue)  Two input signal strength –Low strength: 0.0625~0.08 (mean = 0.07125) –High strength: 0.0625~0.09 (mean = 0.0775)  Three type of noise –No noise –-5%~+5% –-10%~+10% –(all apply to time/interval, not to signal strength)

9. Simulation  20 oscillators are used –Different initial conditions  Run 8 pattern repetions, output 2 following pattern repetions, then recorded  Results are compared to “P&E” Model

10. Discussion  Why high strength input get worse results? –Oscillators are over-sensitive to “early” events –Later events may be more important for downbeat finding –The “strong” voltage in “early” events cause oscillators to “fire”, and unable to respond to subsequent events

11. Discussion (continue)  When it fails –For those patterns where rests play an important role  Future work –Using oscillator network (coupling and synchronizing) –Comparing different models