音樂與數(shù)學

目錄
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(xié)和音程與不協(xié)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(tài)與泛音...........................................27
x2.4 達朗貝爾行波解...........................................29
x2.5 傅里葉級數(shù)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(shù)..................................................56
第四章調式、音階與和弦.........................................61
x4.1 調式與音階..............................................61
x4.2 和弦...................................................65
x4.3 調式中的和弦............................................68
第五章音樂與對稱..............................................76
x5.1 等價關系與音類...........................................76
x5.2 旋律的移調變換...........................................81
viii 目錄
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(shù)..............................................101
第六章節(jié)奏...................................................109
x6.1 固定節(jié)奏型..............................................109
x6.2 節(jié)奏型的影子與輪廓.......................................113
x6.3 比約克倫德算法與歐幾里得節(jié)奏..............................116
x6.4 相移與《拍掌音樂》........................................118
第七章和弦與音網..............................................127
x7.1 和弦的幾何..............................................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 音樂骰子游戲............................................160
x8.2 隨機音樂................................................166
x8.3 馬爾科夫鏈..............................................170
x8.4 有色彩的噪聲, 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