數學與音樂的對話：探討莫札特音樂的數學應用及其效應

The Dialogue between Mathematics and Music: the Mathematical Analysis of Mozart's Music and its Effect on the Listener

數學是推理中的音樂，而音樂則是感覺中的數學。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，但與音樂是根本相通的；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，但數學的蹤影卻處處可見。在音樂的領域中，運用數學來處理相關問題的範例很多，本文乃運用數學的各種運算，探討其在莫札特樂曲結構分析的應用，包括敘述音樂領域中數學的應用範例；莫札特的音樂效應；探討莫札特樂曲結構之黃金比例的應用，而呈現了完美的數學比例關係；及敘述莫札特第三(G大調）、四(D大調）和五號(A大調）小提琴協奏曲第一樂章樂曲結構的數學解析，而展現了莫札特創作之和諧性和規律性，也因此說明了為什麼莫札特音樂不論在創造力、學習力、健康和治療方面，都被證實能發揮其神奇的效應而受到廣大的重視等，在在都顯示出數學與音樂密不可分的關係。

The roots of mathematics are closely connected with those of music. The rational structure or system of mathematics is implicitly aesthetic, given its properties of order and harmony; in this sense it is musical, even though there is no transmission of sound. By the same token, music--even though in it there are no explicit digits or other mathematical signs-is implicitly mathematical in and through its amplitude, frequency, quality, rhythm, melody, form and style. The present investigation sets out to describe Mozart's music mathematically. This structural analysis of the music and its effects on the listener therefore includes: examples of the application of mathematics to music, the measurement of the effects of Mozart's music, the application of the golden ratio to Mozart's musical structure, and an analysis of the application of mathematics to the musical structure of Mozart concertos: specifically the first movements of No. 3（G major）, 4（D major） and 5（A major）. These analyses strongly suggest the close correlation between mathematics and music. This study thus aims to promote the integration of technology and the humanities by opening a dialogue between music and mathematics.