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。
关于这个系列的几点说明:小红书的推流算法对这类认真科普的内容并不友好,用户群体的反馈也比较稀少。之后的视频将保持周更节奏,但不会仔细核对时间轴对齐和内容细节的严谨性。这个网站的内容也会略去很多证明。之前使用的克隆音色将停止使用,后续统一采用 edge-tts 配音。
这个系列在深度和广度上并不适合小红书这类短视频平台。短视频平台的算法和用户普遍偏好猎奇、争议性内容,而非严肃的数学科普。一些内容粗糙、以猎奇甚至有误导倾向的博主,反而能获得更好的推流。这是平台生态的现实,这个系列选择忽略它,继续做自己认为值得做的东西。
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
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
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
音乐技术系列 EP28–40
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
EP32
EP32: Chebyshev Waveshaping — Buchla and West Coast Synthesis
EP33
EP33: One Delay Line, One String — Physical Modeling
EP34
EP34: The Bessel Secret of the DX7 — FM Synthesis Mathematics
EP35
EP35: The Phase Vocoder — Mathematics of Pitch Shifting
EP36
EP36: Quanta of Sound — Gabor and the Uncertainty Principle
EP37
EP37: Your EQ Moves Poles — Biquad Filters
EP38
EP38: Reverberation Is Matrix Multiplication
EP39
EP39: How Spotify Knows You Are Too Loud — The LUFS Algorithm
EP40
EP40: What MP3 Deletes — Psychoacoustics and MDCT
分离与识别 EP41–42
NMF人声分离 · Wiener掩模 · HTDemucs · HMM和弦识别 · Viterbi
节奏数学篇 EP43–46
Euclidean节奏 · 微时值Groove · 极简主义相位 · 节奏卡农与整数平铺
算法作曲篇 EP47–52
对位CSP · 谱乐配器 · L系统 · 随机作曲 · 十二平均律 · 最优传输
EP47
EP47: Counterpoint as Constraint Satisfaction
EP48
EP48: Spectral Orchestration — Grisey and the Roughness Curve
EP49
EP49: L-Systems & Algorithmic Composition
EP50
EP50: Stochastic Composition — Xenakis and Probabilistic Music
EP51
EP51: Why Twelve Notes? — Continued Fractions and Tuning Optimization
EP52
EP52: Optimal Transport & Melody