4D3 Sound and Pitch | 4D3 声音与音高

📚 4D3 Sound and Pitch | 4D3 声音与音高

Sound and pitch are fundamental concepts that can be modelled mathematically using trigonometric functions, logarithms, and sequences. In the context of Further Mathematics, the 4D3 unit explores how pure tones are represented as sine waves, how pitch perception follows a logarithmic scale, and how musical scales are constructed using geometric progressions. This article walks through the key mathematical principles linking sound waves to pitch, from simple harmonic motion to Fourier series, providing a rigorous yet accessible revision guide.

声音和音高是可以用三角函数、对数以及数列进行数学建模的基本概念。在进阶数学中，4D3 单元探究纯音如何用正弦波表示、音高感知为何遵循对数尺度，以及音阶如何通过等比数列构建。本文梳理了连接声波与音高的核心数学原理，从简谐运动到傅里叶级数，提供严谨而易于理解的复习指南。

1. The Nature of Sound Waves | 声波的本质

Sound travels through a medium as a longitudinal pressure wave. The alternating compressions and rarefactions can be modelled as a sinusoidal variation in air pressure over time. Mathematically, a pure tone corresponds to a single-frequency sine wave, which is the foundation for all further analysis of pitch.

声音以纵波的形式在介质中传播。疏密相间的气压变化可以随时间建模为正弦波动。数学上，一个纯音对应一个单频正弦波，这是所有音高分析的基础。

In terms of particle displacement or pressure deviation, the simplest model for a sound wave at a fixed point is x(t) = A sin(2πft + φ), where A is amplitude, f is frequency, and φ is phase. This is identical to the form studied in simple harmonic motion.

在粒子位移或压力偏差方面，一个固定点的声波最简单的模型是 x(t) = A sin(2πft + φ)，其中 A 为振幅，f 为频率，φ 为初相。这与简谐运动中所学的形式完全一致。

2. Simple Harmonic Motion and the Sine Function | 简谐运动与正弦函数

A sound wave can be produced by a vibrating object undergoing simple harmonic motion (SHM), such as a tuning fork or a loudspeaker diaphragm. The displacement y at time t satisfies y = A sin(ωt + φ), with angular frequency ω = 2πf. Differentiating twice gives acceleration a = -ω² y, which confirms the restoring force characteristic of SHM.

声波可由进行简谐运动的振动物体产生，比如音叉或扬声器振膜。位移 y 满足 y = A sin(ωt + φ)，角频率 ω = 2πf。对其两次求导得到加速度 a = -ω² y，验证了简谐运动恢复力的特征。

The period T = 1/f is the time taken for one complete oscillation. In the context of sound, this period corresponds to the time for one full pressure cycle. The concepts of frequency and period are reciprocals, central to linking the physical wave to perceived pitch.

周期 T = 1/f 是一次完整振动所需的时间。在声音情境下，该周期对应一个完整压力循环的时间。频率与周期互为倒数，是连接物理波动与音高感知的核心概念。

3. Frequency, Period and Wavelength | 频率、周期与波长

For a sound wave travelling at speed v, the relationship v = fλ links frequency f and wavelength λ. In air at room temperature, v ≈ 343 m/s. If a tuning fork vibrates at 440 Hz, the resulting sound wave has a wavelength λ = v/f ≈ 0.78 m. This equation enables the conversion between physical measurements and the frequency that determines pitch.

对于以速度 v 传播的声波，关系式 v = fλ 将频率 f 与波长 λ 联系在一起。室温下空气中 v ≈ 343 m/s。若一只音叉以 440 Hz 振动，产生的声波波长 λ = v/f ≈ 0.78 m。该公式可实现物理测量与决定音高的频率之间的换算。

Frequency is measured in hertz (Hz). The human ear can typically detect frequencies from about 20 Hz to 20 000 Hz. The upper limit decreases with age. In mathematical modelling, we often work with the angular frequency ω = 2πf to simplify differentiation and integration of trigonometric functions.

频率以赫兹 (Hz) 为单位。人耳通常能感知大约 20 Hz 到 20 000 Hz 的频率，上限随年龄增长而降低。在数学建模中，我们常使用角频率 ω = 2πf 来简化三角函数的微分与积分运算。

4. Pitch Perception and Logarithmic Scaling | 音高感知与对数尺度

Pitch is the perceptual correlate of frequency, but the relationship is not linear. A doubling of frequency (an octave) is perceived as the same pitch interval regardless of the starting frequency. This means our perception of pitch is logarithmic: equal ratios of frequencies correspond to equal pitch steps. Mathematically, the pitch interval between two frequencies f₁ and f₂ can be expressed in cents or semitones using the formula n = 1200 · log₂(f₂/f₁).

音高是频率的感知对应量，但这种关系并非线性。频率加倍（一个八度）无论在哪个起始频率上都被感知为相同的音高间隔。这意味着音高感知是对数的：相等的频率比对应相等的音高步长。数学上，两个频率 f₁ 与 f₂ 之间的音高间隔可用音分或半音表示，公式为 n = 1200 · log₂(f₂/f₁)。

This logarithmic law is similar to the Weber-Fechner law in psychophysics. In terms of further mathematics, we use the logarithm base 2 to model musical intervals. For example, the interval between 440 Hz and 880 Hz is exactly 12 semitones, because 880/440 = 2 and log₂(2) = 1, with 1 octave = 12 semitones.

这一对数定律与心理物理学中的韦伯-费希纳定律相似。在进阶数学中，我们使用以 2 为底的对数来为音程建模。例如，440 Hz 与 880 Hz 之间的音程恰好是 12 个半音，因为 880/440 = 2，log₂(2) = 1，而 1 个八度 = 12 个半音。

5. The Octave and Frequency Ratios | 八度与频率比

An octave corresponds to a frequency ratio of 2:1. Other consonant intervals are also built on simple integer ratios: a perfect fifth is 3:2, a perfect fourth is 4:3, and a major third is 5:4. These ratios emerge from the harmonic series and underpin just intonation. From a mathematical standpoint, they are rational numbers that create standing wave patterns with minimal beats.

一个八度对应 2:1 的频率比。其他协和音程也建立在简单整数比之上：纯五度为 3:2，纯四度为 4:3，大三度为 5:4。这些比率源于泛音列，是纯律的基础。从数学角度看，它们是有理数，能产生拍频最小的驻波模式。

In terms of a geometric sequence, moving up by octaves multiplies the frequency by successive powers of 2. If the reference pitch is A₄ = 440 Hz, then A₅ = 880 Hz, A₃ = 220 Hz, and so on. The nth octave above a base frequency f₀ is given by fₙ = f₀ · 2ⁿ, where n is an integer.

在等比数列中，向上移动八度相当于频率乘以 2 的连续幂次。如果基准音高为 A₄ = 440 Hz，那么 A₅ = 880 Hz，A₃ = 220 Hz，以此类推。一个基准频率 f₀ 上方的第 n 个八度由 fₙ = f₀ · 2ⁿ 给出，其中 n 为整数。

6. Equal Temperament as a Geometric Progression | 十二平均律作为等比数列

Equal temperament divides the octave into 12 equal semitone steps. Because the octave ratio is 2, each semitone multiplies the frequency by the factor 2^(1/12), the twelfth root of 2. This is a constant ratio, so the frequencies of successive semitones form a geometric progression with common ratio r = 2^(1/12) ≈ 1.05946.

十二平均律将一个八度均分为 12 个相等的半音步长。由于八度比率为 2，每个半音将频率乘以因子 2^(1/12)，即 2 的十二次方根。这是一个恒定比值，因此连续半音的频率构成一个公比 r = 2^(1/12) ≈ 1.05946 的等比数列。

Starting from A₄ = 440 Hz, the frequency of the next semitone A♯₄ is 440 × 2^(1/12) ≈ 466.16 Hz. B₄ is 440 × 2^(2/12) ≈ 493.88 Hz, and so on up to A₅ = 440 × 2 = 880 Hz. The general term for the kth semitone above a reference frequency f₀ is fₖ = f₀ · 2^(k/12), where k is an integer counting semitones.

以 A₄ = 440 Hz 为起点，下一个半音 A♯₄ 的频率为 440 × 2^(1/12) ≈ 466.16 Hz。B₄ 为 440 × 2^(2/12) ≈ 493.88 Hz，依此类推直至 A₅ = 440 × 2 = 880 Hz。基准频率 f₀ 上方第 k 个半音的通项公式为 fₖ = f₀ · 2^(k/12)，其中 k 为半音计数的整数。

Semitone (k)	Note	Frequency (Hz)	Ratio
0	A₄	440.00	2^(0/12) = 1
1	A♯₄	466.16	2^(1/12) ≈ 1.0595
2	B₄	493.88	2^(2/12) ≈ 1.1225
12	A₅	880.00	2^(12/12) = 2

Because the frequency ratios are irrational, equal temperament sacrifices the perfect purity of integer ratios for the ability to play in any key. The mathematical beauty is that the scale is a continuous geometric sequence, enabling modulation without retuning.

由于频率比为无理数，十二平均律牺牲了整数比的完美纯度，从而换取了可以在任何调性上演奏的能力。其数学美感在于音阶构成连续的等比数列，使得转调无需重新调律。

7. Superposition of Waves: Beats and Harmonics | 波的叠加：拍频与泛音

When two sound waves of slightly different frequencies interfere, the superposition principle gives rise to beats. The resultant wave can be expressed as sin(2πf₁t) + sin(2πf₂t). Using the trigonometric identity, this equals 2 cos(2π((f₁ – f₂)/2)t) · sin(2π((f₁ + f₂)/2)t). This represents a wave of mean frequency modulated by a slowly varying amplitude envelope.

当两个频率略有差异的声波干涉时，叠加原理产生拍频。合成波可表示为 sin(2πf₁t) + sin(2πf₂t)。利用三角恒等式，这等于 2 cos(2π((f₁ – f₂)/2)t) · sin(2π((f₁ + f₂)/2)t)。它代表一个频率为平均频率的波，被一个缓慢变化的振幅包络所调制。

The beat frequency, which is the rate at which loudness fluctuates, is |f₁ – f₂|. This provides a mathematical tool for tuning instruments: when two strings are slightly out of tune, the beat frequency indicates the difference. When it falls to zero, they are in unison.

拍频，即响度波动的频率，为 |f₁ – f₂|。这为乐器调音提供了一个数学工具：当两根弦略微走调时，拍频指示其差异。当拍频降至零，它们就达到同音。

Harmonics, or overtones, occur because real musical instruments produce not just a single frequency but a series of integer multiples of the fundamental frequency. A note with fundamental f₀ contains harmonic frequencies 2f₀, 3f₀, 4f₀… whose amplitudes shape the timbre. The mathematical analysis of these components leads to Fourier series.

泛音或谐波的出现是因为真实乐器不仅发出单一频率，还发出一系列基频的整数倍。一个基频为 f₀ 的音包含谐波频率 2f₀、3f₀、4f₀……它们的振幅塑造了音色。对这些成分的数学分析引出了傅里叶级数。

8. Fourier Series: Decomposing Complex Sounds | 傅里叶级数：分解复杂声音

Any periodic sound wave, no matter how complex, can be expressed as a sum of sine and cosine waves whose frequencies are integer multiples of a fundamental frequency. This is the Fourier series theorem, central to Further Mathematics. For a periodic function f(t) with period T = 1/f₀, the series is f(t) = a₀ + Σ [aₙ cos(2πn f₀ t) + bₙ sin(2πn f₀ t)].

任何周期声波，无论多么复杂，都可以表示为一组正弦和余弦波之和，其频率为基频的整数倍。这就是傅里叶级数定理，是进阶数学的核心内容。对于一个周期为 T = 1/f₀ 的周期函数 f(t)，其级数为 f(t) = a₀ + Σ [aₙ cos(2πn f₀ t) + bₙ sin(2πn f₀ t)]。

The coefficients aₙ and bₙ determine the amplitude of each harmonic. For example, a square wave (which approximates the sound of a clarinet) contains only odd harmonics with amplitudes decreasing as 1/n. A sawtooth wave (brassy sound) contains all harmonics with amplitudes decreasing as 1/n. These representations link sound synthesis to pure mathematics.

系数 aₙ 与 bₙ 决定了每一个谐波的振幅。例如，方波（近似单簧管的音色）仅包含奇次谐波，振幅以 1/n 递减；锯齿波（铜管乐音色）包含所有谐波，振幅同样以 1/n 递减。这些表示将声音合成与纯数学紧密联系起来。

In music technology, digital synthesizers use additive synthesis, building sounds by combining sine waves according to Fourier coefficients. This is a direct application of Further Mathematics, where understanding the harmonic structure allows the recreation of natural timbres.

在音乐技术中，数字合成器使用加法合成，通过根据傅里叶系数组合正弦波来构建声音。这是进阶数学的直接应用，理解谐波结构使得自然音色得以重现。

9. Decibels and Logarithmic Intensity | 分贝与对数强度

Sound intensity, or loudness, is measured on a logarithmic scale in decibels (dB). The decibel level L is defined as L = 10 log₁₀(I / I₀), where I is the sound intensity in W/m² and I₀ = 10⁻¹² W/m² is the threshold of human hearing. This matches the logarithmic nature of perception, similar to pitch.

声音的强度或响度以对数尺度的分贝 (dB) 度量。分贝水平 L 定义为 L = 10 log₁₀(I / I₀)，其中 I 为声强，单位 W/m²，I₀ = 10⁻¹² W/m² 是人耳听觉的阈值。这与感知的对数特性相符，与音高类似。

An increase of 10 dB corresponds to a tenfold increase in intensity. For instance, normal conversation at about 60 dB has intensity 10⁶ times I₀, whereas a rock concert at 120 dB has intensity 10¹² times I₀. The logarithmic transformation compresses a vast range of intensities into a manageable scale, employing the same mathematical laws used for pitch intervals.

增加 10 dB 对应强度增至 10 倍。例如，约 60 dB 的正常谈话其强度为 I₀ 的 10⁶ 倍，而 120 dB 的摇滚音乐会强度为 I₀ 的 10¹² 倍。对数变换将宽广的强度范围压缩到一个可处理的标尺，其数学定律与音高间隔相同。

This teaches us that both pitch and loudness perception are logarithmic, which in Further Mathematics highlights the importance of logarithmic functions and their inverses, exponentials. These functions appear repeatedly in models of human sensation.

这告诉我们，音高和响度的感知都是对数的，这在进阶数学中突出了对数函数及其反函数指数函数的重要性。这些函数在人类感觉模型中反复出现。

10. Doppler Effect for Sound | 声音的多普勒效应

When a sound source moves relative to an observer, the perceived frequency changes. This is the Doppler effect, which can be expressed mathematically. For a source moving at speed v_s towards a stationary observer, the observed frequency f’ is given by f’ = f (v / (v – v_s)), where v is the speed of sound. If the source moves away, the formula becomes f’ = f (v / (v + v_s)).

当声源相对于观察者运动时，感知到的频率会改变。这就是多普勒效应，可以用数学表示。对于以速度 v_s 朝向静止观察者运动的声源，观测频率 f’ 由 f’ = f (v / (v – v_s)) 给出，其中 v 为声速。如果声源远离，公式变为 f’ = f (v / (v + v_s))。

Similarly, if the observer moves towards a stationary source at speed v_o, the frequency becomes f’ = f ((v + v_o) / v). These equations are rational functions, and they model real-world phenomena such as the changing pitch of a passing ambulance siren. In Further Mathematics, we analyse how f’ changes with v_s or v_o, using limits and derivatives.

类似地，如果观察者以速度 v_o 朝向静止声源运动，频率变为 f’ = f ((v + v_o) / v)。这些方程是有理函数，它们为真实世界现象建模，比如驶过的救护车警报器音高的变化。在进阶数学中，我们利用极限和导数分析 f’ 如何随 v_s 或 v_o 变化。

It is essential to note that the Doppler effect for sound involves the relative motion through the medium, unlike the relativistic Doppler effect for light. The mathematical models are algebraic and demonstrate how frequency, wavelength, and speed interrelate.

务必注意，声音的多普勒效应涉及通过介质的相对运动，不同于光的相对论性多普勒效应。数学模型是代数的，展示了频率、波长和速度如何相互关联。

11. Differential Equations of Sound Propagation | 声传播的微分方程

The wave equation ∂²p/∂t² = c² ∇²p governs sound pressure p in three dimensions, but for a one-dimensional plane wave it simplifies to ∂²p/∂t² = c² ∂²p/∂x². Solutions are functions of the form p(x,t) = f(x – ct) + g(x + ct), representing travelling waves. In Further Mathematics, we verify that sinusoidal solutions satisfy the equation and determine c (speed of sound) from the properties of the medium.

波动方程 ∂²p/∂t² = c² ∇²p 主导着三维空间的声压 p，但对于一维平面波，它简化为 ∂²p/∂t² = c² ∂²p/∂x²。解的形式为 p(x,t) = f(x – ct) + g(x + ct)，代表了行波。在进阶数学中，我们验证正弦解满足该方程，并从介质属性确定 c（声速）。

For a standing wave inside a tube (like an organ pipe), boundary conditions lead to discrete frequencies. For a pipe open at both ends, the resonant frequencies are f_n = n v / (2L), with n = 1, 2, 3… For a pipe closed at one end, f_n = n v / (4L), with n odd. These are applications of trigonometric and algebraic solutions to the wave equation, linking pitch to the geometry of the instrument.

对于管内（如管风琴）的驻波，边界条件引出了离散频率。两端开口管，共振频率为 f_n = n v / (2L)，n = 1, 2, 3…；一端封闭管，f_n = n v / (4L)，n 为奇数。这些是波动方程的三角和代数解的应用，将音高与乐器几何形状联系起来。

The mathematics of partial differential equations in this context shows how standing waves, nodes, and antinodes emerge from sinusoidal functions. It deepens our understanding of why certain lengths produce specific pitches, completing the circle from sound production to musical scales.

在这种背景下，偏微分方程的数学展示了驻波、波节和波腹如何从正弦函数中产生。它加深了我们对为什么特定长度产生特定音高的理解，完成了从声音产生到音阶的循环。

12. Summary of Mathematical Takeaways | 数学要点总结

The study of sound and pitch in Further Mathematics unites trigonometry, logarithms, geometric sequences, Fourier analysis, and differential equations. Key takeaways include: pure tones map to sine functions; pitch is a logarithmic perception best modelled with base-2 logarithms; equal temperament is a geometric progression with ratio 2^(1/12); beats demonstrate trigonometric superposition; and Fourier series decompose any periodic sound into harmonic components. These concepts not only explain how we hear but also provide the mathematical engine behind modern music technology.

进阶数学中对声音与音高的研究将三角学、对数、等比数列、傅里叶分析和微分方程融为一体。核心要点包括：纯音与正弦函数对应；音高是一种对数感知，最好用以 2 为底的对数建模；十二平均律是公比为 2^(1/12) 的等比数列；拍频演示了三角叠加；傅里叶级数可将任何周期声音分解为谐波分量。这些概念不仅解释了我们如何听到声音，也为现代音乐技术提供了数学引擎。

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4D3 Sound and Pitch | 4D3 声音与音高