音乐与数学

目录
第一章音乐基础知识............................................1
x1.1 声音...................................................1
x1.2 乐音体系................................................6
x1.3 唱名...................................................11
x1.4 音乐的坐标系||五线谱....................................13
x1.5 音程...................................................16
x1.6 协和音程与不协和音程.....................................20
第二章弦的振动................................................24
x2.1 一维振动方程............................................24
x2.2 一维振动方程的解.........................................25
x2.3 振动模态与泛音...........................................27
x2.4 达朗贝尔行波解...........................................29
x2.5 傅里叶级数与拨弦振动.....................................31
第三章乐律||乐音体系的生成...................................38
x3.1 三分损益法..............................................38
x3.2 毕达哥拉斯五度相生律.....................................41
x3.3 纯律与中庸律............................................44
x3.4 平均律..................................................49
x3.5 音分...................................................54
x3.6 连分数..................................................56
第四章调式、音阶与和弦.........................................61
x4.1 调式与音阶..............................................61
x4.2 和弦...................................................65
x4.3 调式中的和弦............................................68
第五章音乐与对称..............................................76
x5.1 等价关系与音类...........................................76
x5.2 旋律的移调变换...........................................81
viii 目录
x5.3 旋律的逆行与倒影.........................................86
x5.4 旋律的变换群............................................89
x5.5 十二音技术..............................................95
x5.6 音列的计数..............................................101
第六章节奏...................................................109
x6.1 固定节奏型..............................................109
x6.2 节奏型的影子与轮廓.......................................113
x6.3 比约克伦德算法与欧几里得节奏..............................116
x6.4 相移与《拍掌音乐》........................................118
第七章和弦与音网..............................................127
x7.1 和弦的几何..............................................127
x7.2 音阶...................................................136
x7.3 三和弦之间的变换.........................................142
x7.4 音网...................................................146
x7.5 新黎曼群................................................153
第八章音乐与随机性............................................160
x8.1 音乐骰子游戏............................................160
x8.2 随机音乐................................................166
x8.3 马尔科夫链..............................................170
x8.4 有色彩的噪声, 1=f 音乐.....................................174
附录A 集合与映射..............................................180
附录B 与书中内容有关的群论知识..................................185
参考文献.......................................................195
名词索引.......................................................200
人名索引.......................................................206