數學與音樂的對話——探討音樂中的數學應用

The Dialogue between Mathematics and Music-A Study of the Application of Mathematics to Music

代表理性的數學，其規律、和諧與秩序所產生的美感，雖無聲音之傳遞，但與音樂是根本相連的；而代表感性的音樂，其音強、音高、音色、節奏、旋律、曲式及風格，雖無明顯之數字表達，但數學的蹤影處處可見。本文藉由數學與音樂的多元對話，探討音樂領域中數學的應用範例；黃金比例在樂曲結構之應用；運用數學推演和計算畢氏音階、純律音階和十二平均律音階的組成；探討五度相生律的應用範例與數學的密切關係；歸納和分析音程和音調的數學二元圖表，在在都顯示出數學與音樂密不可分的關係。期望藉由數學與音樂的多元對話，探討音樂中的數學關係，以提供促進科技與人文藝術之融合的另一種思考的模式。

Mathematics as a representation of rationality produces aesthetics by its properties of order and harmony with music, and is closely connected to music though there is no transmission of sounds. On the other hand, without any number description, mathematics is all around the amplitude、frequency、quality、rhythm、melody、form and style of music. In this study, we try to provide the examples of the application of mathematics to music; the application of golden ratio to musical structure; the calculation and derivation of Pythagoras scale、pure scale and twelve-tone equal temperament scale; the application and analysis of mathematics in fifth-degree co-generation method; and the two-dimension research of the interval and scale. These discussions apparently show the strong correlation of mathematics and music. This study aims to provide a new concept to promote the integration of technologies and humanities from the diversified dialogues between mathematics and music.