AMATH 390

数学与音乐

Comprehensive mathematical companion for AMATH 390: Mathematics and Music

这是 AMATH 390《数学与音乐》视频系列的配套数学参考网站。每集对应一个完整的数学指南，包含严格的定义、定理证明和音乐应用。

数学基础篇 EP1–9

弦振动、图论、群论、信息论、Shepard悖论、组合音——数学工具箱

EP01 EP01: Chord Space and the Torus EP02 EP02: String Vibration and the Wave Equation EP03 EP03: Bach and Graph Connectivity EP04 EP04: All-Interval Rows and ℤ₁₂ EP05 EP05: Turán's Theorem and Counterpoint EP06 EP06: Graph Coloring and Voice Assignment EP07 EP07: Information Entropy and All-Interval Rows EP08 EP08: Shepard Tones and the Tritone Paradox EP09 EP09: Combination Tones and Nonlinear Acoustics

Jacob Collier 系列 EP10–13

负和声与D₁₂、音差不可能定理、五度圈螺旋、节奏-音高对偶

EP10 EP10: Negative Harmony and the Dihedral Group D₁₂ EP11 EP11: Comma Drift and the Impossibility of Perfect Tuning EP12 EP12: The Circle of Fifths as a Folded Spiral EP13 EP13: Polyrhythm and Rhythm-Pitch Duality

拓扑数学专篇 EP14–14

Tonnetz 单纯复形 · 链复形 · Hodge ★算子 · 调性同调

EP14 EP14: Tonnetz Hodge Duality

歌剧系列 EP15–20

歌手声学、Rossini CFG、Wagner 主导动机、博弈论、FM颤音、人物网络

EP15 EP15: Opera Acoustics and the Singer's Formant EP16 EP16: Rossini's Code — Budden, Gossett, and the Convention System EP17 EP17: Leitmotif Networks — Wagner's Ring Cycle EP18 EP18: Game Theory in Operatic Duets EP19 EP19: Vibrato as FM Synthesis EP20 EP20: Character Interaction Networks in Opera

AI 音乐三部曲 EP21–23

Markov→扩散模型、EnCodec RVQ、潜空间几何与信息瓶颈

EP21 EP21: From Markov to Diffusion — Sixty Years of AI Composition EP22 EP22: How AI Writes Music — EnCodec and RVQ EP23 EP23: Can AI Understand Music? Latent Space and Information Bottleneck

流行音乐 × 数学 EP24–27

Shannon熵、预测误差与鸡皮疙瘩、1/f法则、用数学回应爆款批评

EP24 EP24: The Information Code of Pop Music EP25 EP25: Why Music Gives You Goosebumps — Prediction Error and Frisson EP26 EP26: The 1/f Law of Music EP27 EP27: Answering Viral Critics with Mathematics

音乐技术系列 EP28–31

麦克风ODE、QRD驻波、采样定理、Moog滤波器、FM/波导/声码器、混响FDN、LUFS、MP3

EP28 EP28: Your Microphone Is a Damped Oscillator EP29 EP29: Primes in the Studio — Standing Waves and QRD Diffusers EP30 EP30: The Sampling Theorem Is Not What You Think EP31 EP31: The Warmth of Moog Has an Equation