implicit knowledge embodies the functions of tones and chords in a key,35,42–44the relations
between different keys,45–48and the change in tonal functions depending on the context.49–53
The influence of this internalized representation has been reported for musical memory,54–57
musical expectancies,52,58–61and with electrophysiological and brain imaging methods.62–69
Learning musical structures
The Western tonal system embodies strong regularities to which listeners become sensitive via
mere exposure to musical pieces.35–37We proposed to simulate implicit learning processes of
tonal regularities with SOMs and used the trained network to simulate experimental data on
music perception via activation spreading through the learned representation.^31
In these simulations, a hierarchical network with two SOMs was defined (i.e. a multi-layer
hierarchical self-organizing map HSOM^70 ). The input layer consisted of 12 units coding the
12 pitch classes. Each unit of this layer was connected to a second layer (a self-organizing
map) that in turn was connected to a third layer (a second self-organizing map). Before
learning started, the units of two layers were fully interconnected with connection weights
set to random values. Four learning simulations that differed in training material and rich-
ness of input coding were realized. The first SOM was trained with a set of chords presented
separately. The second SOM was trained with either groups of chords that belonged to the
same key or short chord sequences. The short chord sequences represented more realistic
harmonic material since they respected chord transitions and frequencies of occurrence in
Western tonal music. In addition, each chord was weighted according to recency in order to
mimic memory decay in short-term memory. All training material was coded by either a
sparse input pattern (i.e. a unit is activated (set to ‘1’) when a note is present in the chord,
and set to ‘0’ otherwise) or a richer input pattern that coded a chord as the weighted sum of
subharmonics of the present pitch classes (as proposed in Refs 71 and 72).
Independent of training material and input coding, the network became sensitive to regu-
larities underlying the musical material (i.e. chords, chord sets, chord sequences). The units
of the first SOM became specialized in the detection of chords and the units of the second
SOM in the detection of keys. Both layers showed a topological organization of the special-
ized units. In the chord layer, units representing chords that share tones (or subharmonics)
were located close to each other on the map, but chords not sharing tones were not repres-
ented by neighbouring units. In the key layer, the units specialized in the detection of keys
were organized in a circle: keys sharing numerous chords and tones were represented close
to each other on the map and the distance between keys increased with decreasing number
of shared events. The organization of key units reflects the music theoretic organization of
the cycle of fifths: the more the keys are harmonically related, the closer they are on the cycle
(and on the network map). After training with the rich input coding, also the chord layer
showed a more global organization reflecting harmonic relationships: chords from one side
of the circle of fifths are represented on one half of the map, the other side of the circle on
the other half of the map.
After learning, the connection weights were not at random values any longer, but
reflected the regularities of cooccurrence. With the sparse input coding, a chord unit was
linked to three tone units, tones that are components of this chord, and a key unit was
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