Discrete Structure of Music and Lyrics

2018 Fiscal Year Final Research Report

• Project/Area Number: 16KT0136
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 特設分野
• Research Field: Mathematical Sciences in Search of New Cooperation
• Research Institution: Nihon University
• Principal Investigator: SAITO Akira 日本大学, 文理学部, 教授 (90186924)
• Co-Investigator(Kenkyū-buntansha): 北原 鉄朗 日本大学, 文理学部, 准教授 (00454710) ; 韓 東力 日本大学, 文理学部, 准教授 (10365033)
• Project Period (FY): 2016-07-19 – 2019-03-31
• Keywords: 音楽 / グラフ / 辺着色 / サイクル / 弦

Outline of Final Research Achievements

We studied properties of music and lyrics from a viewpoint of discrete mathematics. Observing that accord of the music and the progression of a song can both be represented as an edge-colored graph, we first studied forbidden subgraphs of edge-colored graphs. The properties associated with forbidden subgraphs are often determined by the maximal ones. In order to facilitate the study of these properties, we established a method to identify the maximal forbidden subgraphs in edge-colored graphs.
We also studied chords in graphs. When we interpret a song as a cycle in a graph representing the possible progression of music, a shortcut of a song corresponds to a chord. Hence, we investigated the distribution of the chords. We discovered that if a graph contains sufficiently many edges relative to its order and it contains a cycle of length k, then it contains a chorded cycle of the same length k.

Free Research Field

グラフ理論

Academic Significance and Societal Importance of the Research Achievements

従来の数学による音楽研究は、音を空気の振動と捉えた物理量の研究、あるいは音のパターンに潜む物理量の規則的変化を数列と捉えた初等整数論に基づくものが主流であった。本研究はそうしたアプローチとは別の視点から音楽を捉えた。すなわち、和音の組み合わせや歌詞の乗った楽曲を辺に色がつけられたグラフと解釈し、グラフ理論の立場から音楽研究を行った。この研究は音楽研究に新たな視点を与える。またこのようなアプローチにより、楽曲を音の組み合わせというミクロの視点ではなく、楽曲全体の進行を見渡すマクロな視点が与えられた。