Harmonic Materials of Modern Music: Resources of the Tempered Scale

Slonimsky, Nicolas; Hanson, Howard

doi:10.2307/895273

Cited by 5 publications

(2 citation statements)

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“…The network learns to classify the stimulus as belonging to one of four different triad types: major, minor, diminished, or augmented. Each triad type is defined by the number of semitones separating each of its component pitch‐classes (e.g., Hanson, 1960). For example, in a major triad like C major, C is the tonic, the pitch‐class E is four semitones above the tonic, and the pitch‐class G is seven semitones above the tonic.…”

Section: Methodsmentioning

confidence: 99%

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How Do Artificial Neural Networks Classify Musical Triads? A Case Study in Eluding Bonini's Paradox

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Zawaduk

et al. 2023

Cognitive Science

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How might artificial neural networks (ANNs) inform cognitive science? Often cognitive scientists use ANNs but do not examine their internal structures. In this paper, we use ANNs to explore how cognition might represent musical properties. We train ANNs to classify musical chords, and we interpret network structure to determine what representations ANNs discover and use. We find connection weights between input units and hidden units can be described using Fourier phase spaces, a representation studied in musical set theory. We find the total signal coming through these weighted connection weights is a measure of the similarity between two Fourier structures: the structure of the hidden unit's weights and the structure of the stimulus. This is surprising because neither of these Fourier structures is computed by the hidden unit. We then show how output units use such similarity measures to classify chords. However, we also find different types of units-units that use different activation functionsuse this similarity measure very differently. This result, combined with other findings, indicates that while our networks are related to the Fourier analysis of musical sets, they do not perform Fourier analyses of the kind usually described in musical set theory. Our results show Fourier representations of music are not limited to musical set theory. Our results also suggest how cognitive psychologists might explore Fourier representations in musical cognition. Critically, such theoretical and empirical implications require researchers to understand how network structure converts stimuli into responses.

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Section: Methodsmentioning

confidence: 99%

“…3). Each phase space represents different harmonic qualities (Hanson, 1960). Each phase space relates to a category proposed by Hanson (Quinn, 2006(Quinn, , 2007.…”

Section: Defining Phase Space Projectionsmentioning

confidence: 99%