音乐和数学：一对隐藏的好朋友
叮咚——你有没有试过一边跟着《小星星》的旋律晃脑袋，一边跟着拍手？当你喊着“1、2、3、4”打节拍的时候，有没有觉得这和数学课上的“找规律”题目特别像？别惊讶，音乐和数学本来就是一对手拉手的好朋友，只是它们的秘密藏得太深，很少有人发现。Ding-dong~ Have you ever hummed along to *Twinkle Twinkle Little Star* while clapping your hands? When you count "1,2,3,4" to the beat, do you feel it's just like the "find the pattern" questions in math class? Don't be surprised, music and math are best friends holding hands, they just hide their secrets too deep for most people to find.
现在我们来玩一个小游戏：先听三个15秒的音乐片段，不用记名字，只要认真听就好： 1. 第一个片段：舒缓的、你肯定听过的、讲星星的儿歌 2. 第二个片段：活泼的、讲老虎的儿歌 3. 第三个片段：气势很足、经常在运动会结尾响起的进行曲
听完之后回答三个小问题： ① 这三个片段里有没有你特别熟悉的部分？ ② 你有没有发现里面有重复出现的声音、歌词或者节奏？ ③ 你觉得这些重复的部分有什么规律？
如果你发现了“有重复”“重复的部分差不多”，那你已经摸到音乐和数学的第一个共同秘密了——模式（Pattern）。数学里的模式就是“重复出现的规律”，音乐里的模式也是一模一样：重复的旋律、重复的节奏、重复的歌词，全都是模式的体现。你看《小星星》的前四个音是“1、1、5、5”，就是两个相同的音各出现两次；《两只老虎》的歌词是“两只老虎两只老虎跑得快，跑得快”，“两只老虎”重复两次，“跑得快”也重复两次，就像数学里的A、A、B、B结构；《拉德斯基进行曲》的鼓点是“哒、哒、哒、哒”每四拍循环一次，就像你走路的时候每走四步换一次脚，规律特别明显。If you found "there is repetition" and "the repeated parts are similar", then you've touched the first common secret of music and math — Pattern. The pattern in math is "the rule that repeats", and it's exactly the same in music: repeated melodies, repeated rhythms, repeated lyrics are all patterns. The first four notes of *Twinkle Twinkle Little Star* are "1,1,5,5", two identical notes appear twice each; the lyrics of *Two Tigers* are "two tigers, two tigers, run fast, run fast", "two tigers" repeats twice, "run fast" also repeats twice, just like the A,A,B,B structure in math; the drumbeat of *Radetzky March* is "da, da, da, da" repeating every four beats, just like you change feet every four steps when walking, the rule is very obvious.
我们的15章学习之旅 | Our 15-Chapter Learning Journey
这本书一共有15章，我们会从最简单的规律开始，一步步挖出音乐里藏的所有数学秘密，每一章都有好听的音乐、好玩的游戏，还有中英文对照的知识点，学完之后你不仅能看懂音乐
现在我们来玩一个小游戏：先听三个15秒的音乐片段，不用记名字，只要认真听就好： 1. 第一个片段：舒缓的、你肯定听过的、讲星星的儿歌 2. 第二个片段：活泼的、讲老虎的儿歌 3. 第三个片段：气势很足、经常在运动会结尾响起的进行曲
听完之后回答三个小问题： ① 这三个片段里有没有你特别熟悉的部分？ ② 你有没有发现里面有重复出现的声音、歌词或者节奏？ ③ 你觉得这些重复的部分有什么规律？
如果你发现了“有重复”“重复的部分差不多”，那你已经摸到音乐和数学的第一个共同秘密了——模式（Pattern）。数学里的模式就是“重复出现的规律”，音乐里的模式也是一模一样：重复的旋律、重复的节奏、重复的歌词，全都是模式的体现。你看《小星星》的前四个音是“1、1、5、5”，就是两个相同的音各出现两次；《两只老虎》的歌词是“两只老虎两只老虎跑得快，跑得快”，“两只老虎”重复两次，“跑得快”也重复两次，就像数学里的A、A、B、B结构；《拉德斯基进行曲》的鼓点是“哒、哒、哒、哒”每四拍循环一次，就像你走路的时候每走四步换一次脚，规律特别明显。If you found "there is repetition" and "the repeated parts are similar", then you've touched the first common secret of music and math — Pattern. The pattern in math is "the rule that repeats", and it's exactly the same in music: repeated melodies, repeated rhythms, repeated lyrics are all patterns. The first four notes of *Twinkle Twinkle Little Star* are "1,1,5,5", two identical notes appear twice each; the lyrics of *Two Tigers* are "two tigers, two tigers, run fast, run fast", "two tigers" repeats twice, "run fast" also repeats twice, just like the A,A,B,B structure in math; the drumbeat of *Radetzky March* is "da, da, da, da" repeating every four beats, just like you change feet every four steps when walking, the rule is very obvious.
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我们的15章学习之旅 | Our 15-Chapter Learning Journey
这本书一共有15章，我们会从最简单的规律开始，一步步挖出音乐里藏的所有数学秘密，每一章都有好听的音乐、好玩的游戏，还有中英文对照的知识点，学完之后你不仅能看懂音乐里的数学，还能自己创作带规律的小曲子哦！This book has 15 chapters, we will start from the simplest rules, step by step dig out all the math secrets hidden in music, each chapter has nice music, fun games, and Chinese-English bilingual knowledge points. After learning, you will not only understand the math in music, but also create your own small pieces with patterns!
每一章的学习内容都像闯关游戏，难度一点点提升，你完全不用担心跟不上： Each chapter's learning content is like a level-breaking game, the difficulty increases little by little, you don't have to worry about not keeping up: 1. 第1章：音乐和数学：一对隐藏的好朋友：发现音乐里的重复、对称、分组规律，学会基础双语词汇，玩找规律游戏（就是你现在正在学的这一章哦！） Chapter 1: Music and Math: A Pair of Hidden Best Friends: Discover the repetition, symmetry and grouping rules in music, learn basic bilingual vocabulary, play pattern-finding games (this is the chapter you are learning now!) 2. 第2章：打节拍的小秘密：把节拍和数学里的分数联系起来，学会数拍子，用分数表示不同音符的长度 Chapter 2: The Secret of Beats: Connect beats with fractions in math, learn to count beats, use fractions to represent the length of different notes 3. 第3章：音高的小阶梯：发现音高和数轴的对应关系，知道为什么音高会越来越高或者越来越低 Chapter 3: The Ladder of Pitch: Discover the corresponding relationship between pitch and number line, know why pitch gets higher and higher or lower and lower 4. 第4章：音符的小算盘：学会用加法和除法计算音符的总时长，知道为什么有的音符长、有的音符短 Chapter 4: The Abacus of Notes: Learn to use addition and division to calculate the total length of notes, know why some notes are long and some are short 5. 第5章：和弦的小密码：发现和弦和质数的关系，知道为什么有的和弦听起来很和谐，有的却很刺耳 Chapter 5: The Password of Chords: Discover the relationship between chords and prime numbers, know why some chords sound very harmonious, but some are harsh 6. 第6章：调式的小循环：学会用模运算理解调式的循环规律，知道为什么唱到“si”就会回到“do” Chapter 6: The Cycle of Scales: Learn to use modular arithmetic to understand the cycle rule of scales, know why you will return to "do" after singing "si" 7. 第7章：节奏的小组合：学会用排列组合拼出不同的节奏型，知道为什么同样的旋律可以有不同的节奏 Chapter 7: The Combination of Rhythms: Learn to use permutation and combination to piece together different rhythm patterns, know why the same melody can have different rhythms 8. 第8章：音程的小距离：学会用数轴上的距离计算音程，知道为什么两个音之间的距离不一样，听起来的感觉也不一样 Chapter 8: The Distance of Intervals: Learn to use the distance on the number line to calculate intervals, know why the distance between two notes is different, and the feeling of listening is also different 9. 第9章：复调的小拼图：学会用集合的交集、并集理解复调音乐，知道为什么多个人一起唱不同的旋律反而更好听 Chapter 9: The Puzzle of Polyphony: Learn to use the intersection and union of sets to understand polyphonic music, know why different melodies sung by multiple people sound better 10. 第10章：钢琴键上的无理数：发现十二平均律里的无理数，知道为什么钢琴的音准是“不准”的，但是却最好听 Chapter 10: Irrational Numbers on Piano Keys: Discover the irrational numbers in the twelve-tone equal temperament, know why the piano tuning is "inaccurate", but it sounds the best 11. 第11章：音乐里的对称美：学会找音乐里的对称结构，知道为什么对称的音乐听起来特别舒服 Chapter 11: The Symmetrical Beauty in Music: Learn to find symmetrical structures in music, know why symmetrical music sounds particularly comfortable 12. 第12章：循环播放的小秘密：学会用周期函数理解音乐的循环规律，知道为什么流行歌的副歌总是重复好几遍 Chapter 12: The Secret of Loop Playback: Learn to use periodic functions to understand the cycle rule of music, know why the chorus of pop songs is always repeated several times 13. 第13章：旋律的小数列：学会用数列和递推理解旋律的走向，知道为什么有的旋律听起来特别顺耳 Chapter 13: The Number Sequence of Melodies: Learn to use number
接上文的15章学习路径，我们把剩下的第14、15章也补全吧：第14章会教你用概率理解流行歌为什么“听起来都差不多”，第15章是终极闯关——用前面学的所有数学知识自己创作一首小曲子，表演给爸爸妈妈看！这15章就像15个藏宝关卡，每过一关你就能拿到一个音乐数学的宝藏，最后就能自己创作音乐啦！The remaining Chapter 14 and 15 are also completed: Chapter 14 will teach you to use probability to understand why pop songs "sound similar", and Chapter 15 is the ultimate level-breaking — use all the math knowledge you learned before to create your own small piece and perform it for your mom and dad! These 15 chapters are like 15 treasure-hunting levels, every level you pass, you can get a music-math treasure, and finally you can create music yourself!
你已经找到了音乐和数学的第一个共同秘密：模式（Pattern），接下来我们再深入挖一挖，音乐里到底藏了多少数学规律？其实你平时听到的所有音乐，不管是中国儿歌、英文童谣还是动画片主题曲，都逃不开三个最基本的数学特征：重复（Repetition）、对称（Symmetry）、分组（Grouping），这三个特征就像音乐的“骨架”，所有的好听旋律都是搭在这个骨架上的。 You have already found the first common secret of music and math: Pattern, next let's dig deeper, how many math rules are hidden in music? In fact, all the music you usually hear, whether it is Chinese nursery rhymes, English nursery rhymes or cartoon theme songs, can't escape three most basic math characteristics: Repetition, Symmetry, Grouping, these three characteristics are like the "skeleton" of music, all the nice melodies are built on this skeleton.