The Cognitive Neuroscience of Music

Representation of input

The pitches of the chromatic scale are numbered consecutively. The onset times of tones
having pitch kare denoted by tkin,i1,...,nk, and the offset times by,tkif,i1,...,nk,
where nkis the total number of times the kth pitch occurs.

Pitch vector p(t)(pk(t))k

Each component of the pitch vector has nonzero value whenever a tone with the respective
pitch is sounding. It has the value of 1 at each onset at the respective pitch, decays expo-
nentially after that, and is set to zero at the tone offset. The time evolution ofpis governed
by the equation

where p·kdenotes the time derivative ofpkand (.) the Dirac delta function (unit impulse
function). The time constant phas the value ofp0.5 s. With this value, the integral of
pksaturates at about 1 s after tone onset, thus approximating the durational accent as a
function of tone duration.^16

Pitch memory vector m(t)(mk(t))k

The pitch memory vector provides a measure of both the perceived durational accent and
the recency of notes played at each pitch. In other words, a high value ofmkindicates that
a tone with pitch kand a long duration has been played recently. The dynamics ofmare
governed by the equation

m·pm/m, (3)

The time constant mdetermines the dependence of transition strength on the temporal
distance between the tones. In the simulations, the value ofm3 s has been used, corres-
ponding to typical estimates of the length of the auditory sensory input.^17 –^19

Transition strength matrix S(t)(skl(t))kl

The transition strength matrix provides a measure of the instantaneous strengths of tran-
sitions between all pitch pairs. More specifically, a high value ofsklindicates that a long tone
with pitch khas been played recently and a tone with pitch lis currently sounding. The
temporal evolution ofSis governed by the equation

In this equation, the nonlinear term (1sgn(plpk))/2 is used for distinguishing between
simultaneously and sequentially sounding pitches. This term is nonzero only when plpk,
that is, when the most recent onset of pitch lhas occurred more recently than that of pitch k.

104     

.

ppkkptt p tt,

i

k

ki

n

k

i

k

ki

f

nn

=−+−



 



−−



 





==

τδδ∑∑

11

(2)

sm

pp

kl= k ikpl kl

1 +−−−

2

sgn( )^22

e()/α, (4)