音楽のコードの幾何学について
1Department of Music, Princeton University, Princeton, NJ 08544, USA. dmitri@princeton.edu
まとめ
音楽のコードは,幾何学的オービフォールドの点であり,線段が音符の変化を示しています. 作曲家は,これらの空間における近対称性を音楽を作るために使用し,共音と不協和音のコードは,異なる用途を示唆しています.
科学分野:
- ミュージック理論,音楽理論
- 計算による音楽学
- ジオメトリック音楽理論
背景:
- 音楽のコードは,オービフォルドと呼ばれる幾何学的な空間内の点として概念化することができます.
- この空間の線段は,コードの間の変換を表し,音符のマッピングを図示しています.
- これらの空間の非ユークリッド的幾何学は,様々なスタイルで作曲家によって利用されています.
研究 の 目的:
- ミュージカル・コードの幾何学的な表現とその変換を探求する.
- 作曲家が音楽作曲において,オービフォールドの幾何学の特性をどのように活用しているのかを調査する.
- 弦対称性と音楽の応用の関係を理解する.
主な方法:
- ミュージカル・コードをオービフォルド・スペースの点として表現する.
- 線段をコード間のマッピングとして分析する.
- アコードの近似対称性 (変換,反射,交配) を識別する.
- 弦対称性を音楽的な用法と相関させる.
主要な成果:
- 音楽的な変化を表す短い線断片は,構造的に似たコードの間に見られます.
- これらの短いセグメントの存在は,コード内の近対称性に依存しています.
- 協和音と不協和音のコードは,明確な近対称性を表している.
- これらの独特の対称性は,異なる潜在的な音楽的応用を示唆しています.
結論:
- オービフォルド幾何学は,音楽の構造と変換を理解するための枠組みを提供します.
- 弦対称性は,音楽の構成において重要な要素であり,変換の選択に影響を与える.
- 弦の幾何学的な性質,特にその対称性は,その音楽的有用性と応用を直接決定する.
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