Entropy, Uncertainty, and the Depth of Implicit Knowledge on Musical Creativity: Computational Study of Improvisation in Melody and Rhythm.

Tatsuya Daikoku

doi:10.3389/fncom.2018.00097

Abstract

Recent neurophysiological and computational studies have proposed the hypothesis that our brain automatically codes the nth-order transitional probabilities (TPs) embedded in sequential phenomena such as music and language (i.e., local statistics in nth-order level), grasps the entropy of the TP distribution (i.e., global statistics), and predicts the future state based on the internalized nth-order statistical model. This mechanism is called statistical learning (SL). SL is also believed to contribute to the creativity involved in musical improvisation. The present study examines the interactions among local statistics, global statistics, and different levels of orders (mutual information) in musical improvisation interact. Interactions among local statistics, global statistics, and hierarchy were detected in higher-order SL models of pitches, but not lower-order SL models of pitches or SL models of rhythms. These results suggest that the information-theoretical phenomena of local and global statistics in each order may be reflected in improvisational music. The present study proposes novel methodology to evaluate musical creativity associated with SL based on information theory.

Highlights

The notion of statistical learning (SL) (Saffran et al, 1996), which includes both informatics and neurophysiology (Harrison et al, 2006; Pearce and Wiggins, 2012), involves the hypothesis that our brain automatically codes the nth-order transitional probabilities (TPs) embedded in sequential phenomena such as music and language (Daikoku et al, 2016, 2017b,c; Daikoku and Yumoto, 2017), grasps the entropy/uncertainty of the TP distribution (Hasson, 2017), predicts the future state based on the internalized nth-order statistical model (Daikoku et al, 2014; Yumoto and Daikoku, 2016), and continually updates the model to adapt to the variable external environment (Daikoku et al, 2012, 2017d)
In pitch sequence with temporal information, 1st, 4th, and 5th-order models showed that the conditional entropies of the TP distributions were moderately (0.4 ≦ |r|
In temporal sequence with pitches, 0th−5th-order models showed that the conditional entropies of the TP distributions were moderately (0.4 ≦ |r|

Summary

Introduction

Statistical Learning in the Brain: Local and Global StatisticsThe notion of statistical learning (SL) (Saffran et al, 1996), which includes both informatics and neurophysiology (Harrison et al, 2006; Pearce and Wiggins, 2012), involves the hypothesis that our brain automatically codes the nth-order transitional probabilities (TPs) embedded in sequential phenomena such as music and language (i.e., local statistics in nth-order levels) (Daikoku et al, 2016, 2017b,c; Daikoku and Yumoto, 2017), grasps the entropy/uncertainty of the TP distribution (i.e., global statistics) (Hasson, 2017), predicts the future state based on the internalized nth-order statistical model (Daikoku et al, 2014; Yumoto and Daikoku, 2016), and continually updates the model to adapt to the variable external environment (Daikoku et al, 2012, 2017d). The concept of brain nth-order SL is underpinned by information theory (Shannon, 1951) involving n-gram or Markov models. The nth-order Markov model is a mathematical system based on the conditional probability of sequence in which the probability of the forthcoming state is statistically defined by the most recent n state (i.e., nth-order TP). A recent neurophysiological study suggested that sequences with higher entropy are learned. Uncertainty in Musical Creativity based on higher-order TP whereas those with lower entropy are learned based on lower-order TP (Daikoku et al, 2017a). Another study suggested that certain regions or networks perform specific computations of global statistics (i.e., entropy) that are independent of local statistics (i.e., TP) (Hasson, 2017). It is important to examine the entire process of brain SL in both computational and neurophysiological areas (Daikoku, 2018b)

Methods

Results

Discussion

Conclusion