Separating phase and period correction processes

internal clock period correction, it is necessary to clarify exactly the timing of the motor response to the (auditory) stimulus. In the situation where synchronisation is considered to have been achieved, the sensorimotor goal can be taken to be synchronisation of auditory response feedback (the sound of the drum) to the auditory stimulus, with variance within the mean given by experimental studies.

Let:

Equation 6-9

M

*

= M- S

where Mj is the time of the auditory response feedback onset and Sj is the time of the auditory stimulus onset. For a human being, there will be a delay between the clock trigger which initiates the tapping action and the arrival of auditory feedback from the tap in the central representation area, resulting from the cumulative delay of the nerve path from brain to muscles, the time taken to complete the physical action, the time taken for the sound to travel through the air to the ear drum, and finally the delay from ear drum to central representation area. In cases where there is no external driving stimulus, such as when a drummer is playing alone, S can be taken to be the expected time of the response feedback. In that case, M*_{will vary as a result of motor} noise, or any introduced timing perturbation which does not affect the central clock. Where all aspects of the model (including sound generation) are contained within software, there will be a shorter delay produced by the time taken for the clock to trigger a MIDI event, and for the MIDI event to return to the synchronisation process – no more than 2 or 3ms. However, if events were triggered at a MIDI module

external to the computer and a MIDI note-on message was mirrored back, the model would adjust to the additional latency without difficulty, since it works on the actual asynchrony between the stimulus and the note-on as their respective signal arrive at the synchronisation process. A longer latency would in practice result in a correction to increase the asynchrony between the clock trigger and the stimulus time, thus accommodating the latency. The stimulus itself will originate as an external trigger generated either as a MIDI message from eg a keyboard, electronic drum, or drum trigger. The model will be regarded as successful if it succeeds in reducing M* to within the mean for variance, after an appropriate number of beats.

Note that synchronisation of the auditory response to the auditory stimulus allows an unspecified latency between the trigger from the internal clock (the start of the clock interval) and the arrival of the produced auditory response as sensory input in the central representation area.

Since error correction for the motor implementation phase is now divorced from clock error correction, it is feasible to consider motor or implementation phase in relation to

tatum subdivisions, while clock period remains referenced to the tactus. Assuming linear error correction for phase (without deviations for the moment), and first-order correction, the implementationphase correction equation is:

Equation 6-10

M

*j+1

= M

*j-1

– α

1

M

*j

+ Q

Mj+1 Where M*

j is auditory stimulus/response feedback asynchrony for the jth tatum, α1 is the phase correction gain parameter, and QMj is the motor noise for the jth tatum production. First-order correction only is assumed, on the basis that phase correction has been shown (Repp, 2001a; Thaut, Tian et al., 1998) to be a subconscious process, and therefore less likely to utilise remembered information.

In fact, Equation 6-10 is already capable of producing microtiming deviations of a sort, through the interaction of the noise term QMj+1 and the error correction term. However, applying error correction to the noise effects would not produce the systematic patterns found in analysis of musical examples, such Freeman’s (Freeman & Lacey, 2001) analysis of the Funky Drummer break by Clyde Stubblefield on James Brown’s In The Jungle Groove album (Brown, 1986).

Assuming the sine function still provides the clock period, the clock period correction process becomes:

Equation 6-11

θ

n+1

= θ

n

- (α

2

/2π)sin[2πθ

n

] - (β

2

/2π)sin[2πθ

n-1

] + ξ

n where θn = M* n / P, and M* n is the asynchrony between the nth tactus audible response onset and the nth _{tactus auditory stimulus onset. α}

2 and β2 are respectively first- and second-order period correction parameters, and Qcn is the clock part of the total noise.

In accordance with the evidence that the total of the first- and second-order parameters is constant, β2 can be replaced by:

Equation 6-12

where λ is the constant total of α2 and β2 , giving:

Equation 6-13

θ

n+1

= θ

n

- (α

2

/2π)sin[2πθ

n

] - ((λ - α

2

)/2π)sin[2πθ

n-1

] + ξ

n Note that tactus asynchronies M*

n are a subset of tatum asynchronies M*j. There is as yet no definite evidence whether tatum timing perturbations trigger clock period correction, independently of any nested clocks. Restricting period correction to tactus asynchronies simplifies the model.

It is a feature of the VROOGE model that period and phase correction are separate processes, located in separate areas of Wing and Kristofferson’s two-process model and involving a central clock and separate motor implementation (Wing &

Kristofferson, 1973a; Wing & Kristofferson, 1973b). Hence, the two processes are independent, except for the trigger from the clock to the motor implementation process, at the start of each clock interval:

Note that focus of the VROOGE model is the separation of period and phase correction, to allow microtiming deviations while maintaining a stable internal reference clock. No attempt is made to model tatum subdivisions of the tactus in a psychologically plausible way.

In document Microtiming deviations in groove (Page 103-106)