音樂與數學

第一章音樂基礎知識............................................1
x1.1 聲音...................................................1
x1.2 樂音體系................................................6
x1.3 唱名...................................................11
x1.4 音樂的坐標系||五線譜....................................13
x1.5 音程...................................................16
x1.6 協和音程與不協和音程.....................................20

第二章弦的振動................................................24
x2.1 一維振動方程............................................24
x2.2 一維振動方程的解.........................................25
x2.3 振動模態與泛音...........................................27
x2.4 達朗貝爾行波解...........................................29
x2.5 傅裡葉級數與撥弦振動.....................................31

第三章樂律||樂音體系的生成...................................38
x3.1 三分損益法..............................................38
x3.2 畢達哥拉斯五度相生律.....................................41
x3.3 純律與中庸律............................................44
x3.4 平均律..................................................49
x3.5 音分...................................................54
x3.6 連分數..................................................56

第四章調式、音階與和絃.........................................61
x4.1 調式與音階..............................................61
x4.2 和絃...................................................65
x4.3 調式中的和絃............................................68

第五章音樂與對稱..............................................76
x5.1 等價關係與音類...........................................76
x5.2 旋律的移調變換...........................................81
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x5.3 旋律的逆行與倒影.........................................86
x5.4 旋律的變換群............................................89
x5.5 十二音技術..............................................95
x5.6 音列的計數..............................................101

第六章節奏...................................................109
x6.1 固定節奏型..............................................109
x6.2 節奏型的影子與輪廓.......................................113
x6.3 比約克倫德演算法與歐幾裡得節奏..............................116
x6.4 相移與《拍掌音樂》........................................118

第七章和絃與音網..............................................127
x7.1 和絃的幾何..............................................127
x7.2 音階...................................................136
x7.3 三和弦之間的變換.........................................142
x7.4 音網...................................................146
x7.5 新黎曼群................................................153

第八章音樂與隨機性............................................160
x8.1 音樂骰子遊戲............................................160
x8.2 隨機音樂................................................166
x8.3 瑪律科夫鏈..............................................170
x8.4 有色彩的雜訊, 1=f 音樂.....................................174
附錄A 集合與映射..............................................180
附錄B 與書中內容有關的群論知識..................................185
參考文獻.......................................................195
名詞索引.......................................................200
人名索引.......................................................206