Nonspecific hebbian neural network model predicts musical scales discreteness and just intonation without using octave-equivalency mapping

Abstract

This study continues investigating the consonance-pattern emerging neural network model introduced in our previous publication, specifically to test if it will reproduce the results using 100-fold finer precision of 1/100th of a semitone (1 cent). The model is a simplistic feed-forward generic Hebbian-learning generic neural network trained with multiple-harmonic complex sounds from the full auditory sound spectrum of 10 octaves. We use the synaptic weights between the neural correlates of each two-tone from the said spectrum to measure the model’s preference to their inter-tonal interval (12,0002 intervals), considering familiarity as a consonance predictor. We analyze all the 12,000 intervals of a selected tone (the tonic), and the results reveal three distinct yet related features. Firstly, Helmholtz’s list of consonant intervals re-emerges from the synaptic weights of the model, although with disordered dissonant intervals. Additionally, the results show a high preference to a small number of selected intervals, mapping the virtually continual input sound spectrum to a discrete set of intervals. Finally, the model's most preferred (most consonant) intervals are from the Just Intonation scales. The model does not need to use cross-octave interval mapping due to octave equivalence to produce the said results.