Year 12 Cambridge Music: Formula & Theorem Quick Reference | 剑桥12年级音乐：公式定理速查手册

📚 Year 12 Cambridge Music: Formula & Theorem Quick Reference | 剑桥12年级音乐：公式定理速查手册

While music is a profoundly expressive art form, its internal logic often rests on mathematical relationships, rules of tension and release, and repeatable structural patterns. This quick reference handbook distils the essential “formulas” and “theorems” of music theory – from interval construction and scale patterns to frequency ratios and cadential formulae – that every Year 12 Cambridge Music student should have at their fingertips. Whether you are analysing a set work, composing in a given style, or preparing for aural dictation, these principles will form the bedrock of your musical understanding.

尽管音乐是一种极具表现力的艺术形式，但其内部逻辑往往建立在数学关系、张弛规则和可重复的结构模式之上。这本速查手册浓缩了音乐理论中最核心的“公式”与“定理”——从音程构建与音阶模式到频率比与终止进行——是每一位12年级剑桥音乐学生都应掌握的要点。无论你是在分析指定作品、按某种风格作曲，还是准备听觉听写，这些基本原理都将成为你音乐理解的基石。

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1. Interval Formulae | 音程计算公式

An interval is defined by the number of semitones between two notes. The basic interval names and their semitone counts in an octave are as follows:

音程由两个音之间的半音数量定义。在一个八度内，基本音程名称及其半音数如下：

Interval / 音程	Semitones / 半音
Perfect unison / 纯一度	0
Minor 2nd / 小二度	1
Major 2nd / 大二度	2
Minor 3rd / 小三度	3
Major 3rd / 大三度	4
Perfect 4th / 纯四度	5
Augmented 4th / Diminished 5th (Tritone) / 增四度/减五度（三全音）	6
Perfect 5th / 纯五度	7
Minor 6th / 小六度	8
Major 6th / 大六度	9
Minor 7th / 小七度	10
Major 7th / 大七度	11
Perfect octave / 纯八度	12

The general formula for modifying an interval is: diminished → minor → major → augmented, with the semitone count decreasing or increasing by one. Perfect intervals (unison, 4th, 5th, octave) follow the chain: diminished → perfect → augmented.

修改音程性质的一般公式为：减 → 小 → 大 → 增，半音数随之减少或增加一个。纯音程（一度、四度、五度、八度）遵循的规律是：减 → 纯 → 增。

Inversion of intervals: The sum of the interval number and its inversion is always 9. A major interval inverts to minor, augmented to diminished, and perfect to perfect.

音程转位：原位音程数与转位音程数之和恒为9。大音程转位后为小音程，增音程转位为减音程，纯音程转位后仍为纯音程。

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2. Major and Minor Scale Formulae | 大小调音阶公式

The major scale follows a fixed pattern of tones (T) and semitones (S): T T S T T T S. Starting from any tonic, this sequence yields the key signature automatically.

大调音阶遵循固定的全音（T）与半音（S）模式：全全半全全全半。从任何主音开始，按此序列上行即可自动得出相应调号。

Major: T T S T T T S

The natural minor scale uses the pattern: T S T T S T T. This can also be expressed in terms of scale degrees relative to the major scale: 1 2 ♭3 4 5 ♭6 ♭7.

自然小调的模式为：全半全全半全全。这也可以通过与同主音大调的音级关系来记忆：1 2 ♭3 4 5 ♭6 ♭7。

Natural minor: T S T T S T T

The harmonic minor scale raises the 7th degree by a semitone to create a leading note, resulting in an augmented 2nd between ♭6 and ♮7. The melodic minor scale raises the 6th and 7th degrees when ascending and reverts to the natural minor when descending.

和声小调将第Ⅶ级音升高半音，形成导音，并在♭6与♮7之间产生一个增二度。旋律小调上行时升高第Ⅵ和第Ⅶ级，下行时还原为自然小调。

Degree-based formula for harmonic minor: 1 2 ♭3 4 5 ♭6 7; ascending melodic minor: 1 2 ♭3 4 5 ♮6 ♮7 (descending: ♭7 ♭6 5 4 ♭3 2 1).

和声小调音级公式：1 2 ♭3 4 5 ♭6 7；上行旋律小调：1 2 ♭3 4 5 ♮6 ♮7（下行：♭7 ♭6 5 4 ♭3 2 1）。

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3. Circle of Fifths and Key Signatures | 五度圈与调号规律

The circle of fifths arranges keys in order of ascending fifths (clockwise) and descending fifths (anticlockwise). Each step clockwise adds one sharp, while each step anticlockwise adds one flat.

五度圈以向上纯五度（顺时针）和向下纯五度（逆时针）的顺序排列各调。顺时针每移动一步增加一个升号，逆时针每移动一步增加一个降号。

The order of sharps: F♯ C♯ G♯ D♯ A♯ E♯ B♯. The order of flats is the reverse: B♭ E♭ A♭ D♭ G♭ C♭ F♭. Key signature of a major key is shared by its relative minor (a minor 3rd below).

升号出现的顺序为：F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号顺序与之相反：B♭ E♭ A♭ D♭ G♭ C♭ F♭。大调的调号与其关系小调（低小三度）共用。

Formula to identify a major key with sharps: the last sharp is the leading note (semitone below tonic). With flats: the penultimate flat is the tonic.

识别带升号的大调公式：最后一个升号是导音（低于主音半音）。带降号的大调：倒数第二个降号即为主音。

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4. Triad Construction Formulae | 三和弦构成公式

A triad is a three-note chord built in thirds. The quality is determined by stacking intervals above the root.

三和弦是由三度叠置的三音和弦。其性质由根音上方的叠加音程决定。

Triad type / 和弦类型	Intervals (in semitones) / 音程（半音数）	Chord symbol (C root) / C为根音标记
Major / 大三和弦	Root + 4 + 3 (M3 + m3)	C
Minor / 小三和弦	Root + 3 + 4 (m3 + M3)	Cm
Augmented / 增三和弦	Root + 4 + 4 (M3 + M3)	Caug or C⁺
Diminished / 减三和弦	Root + 3 + 3 (m3 + m3)	Cdim or C°

The same logic can be applied using degrees of the scale: I, IV, V are major in a major key; ii, iii, vi are minor; vii° is diminished.

同样的逻辑也可以用于调式内的音级：在大调中，Ⅰ、Ⅳ、Ⅴ级为三和弦；ⅱ、ⅲ、ⅵ级为小三和弦；ⅶ°为减三和弦。

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5. Seventh Chord Formulae | 七和弦构成公式

Seventh chords add a seventh above the root to a triad, creating four-note structures. The semitone stacking patterns are:

七和弦在三和弦上方叠加一个七度音，形成四个音的纵向结构。半音堆叠模式如下：

• Major 7th: R + 4 + 3 + 4 (Major triad + Major 3rd)

  大七和弦：根音 + 4 + 3 + 4（大三和弦 + 大三度）

• Minor 7th: R + 3 + 4 + 3 (Minor triad + Minor 3rd)

  小七和弦：根音 + 3 + 4 + 3（小三和弦 + 小三度）

• Dominant 7th: R + 4 + 3 + 3 (Major triad + Minor 3rd)

  属七和弦：根音 + 4 + 3 + 3（大三和弦 + 小三度）

• Half-diminished 7th (m7♭5): R + 3 + 3 + 4 (Diminished triad + Major 3rd)

  半减七和弦（小七降五）：根音 + 3 + 3 + 4（减三和弦 + 大三度）

• Diminished 7th: R + 3 + 3 + 3 (Diminished triad + Minor 3rd)

  减七和弦：根音 + 3 + 3 + 3（减三和弦 + 小三度）

In tonal harmony, the dominant 7th (V7) is built on scale degree V and almost always resolves to I (authentic cadence).

在调性和声中，属七和弦（V₇）建立于第Ⅴ级音，几乎总是解决到Ⅰ（正格终止）。

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6. Chord Inversion Formulae | 和弦转位公式

Inversions are defined by which chord member is in the bass. Figured bass symbols summarise the intervals above the bass note.

转位由低音位置的和弦音决定。数字低音符号概括了低音上方的音程关系。

For triads:

三和弦：

• Root position: bass is the root; figures ⁵₃ (often omitted).

  原位：低音为根音；数字标记 ⁵₃（常省略）。

• First inversion: bass is the 3rd; figures ⁶₃ or simply ⁶.

  第一转位：低音为三音；标记 ⁶₃ 或简写为 ⁶。

• Second inversion: bass is the 5th; figures ⁶₄.

  第二转位：低音为五音；标记 ⁶₄。

For seventh chords, third inversion adds the 7th in the bass, with figures ⁴₂ or ².

七和弦的第三转位将七音置于低音，标记为 ⁴₂ 或 ²。

Analysing inversions in a score quickly: convert the bass note and the chord tone stack back to close root position, then identify the root and quality.

在乐谱中快速分析转位：将低音与上方和弦音还原为密集排列的原位形式，再确定根音与性质。

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7. Equal Temperament Frequency Formula | 十二平均律频率公式

In equal temperament, the octave is divided into 12 equal semitones. The frequency of any note can be calculated from a reference pitch using the formula:

在十二平均律中，八度被等分为12个半音。任何音高的频率都可以通过参考音高用以下公式计算：

fₙ = f₀ × 2^(n/12)

where f₀ is the reference frequency (e.g. A₄ = 440 Hz), n is the number of semitones away from the reference (positive for higher, negative for lower).

其中 f₀ 为参考频率（如 A₄ = 440 Hz），n 为与参考音相距的半音数（向上为正，向下为负）。

This formula explains why the ratio of a perfect 5th is not exactly 3:2 but 2^(7/12) ≈ 1.4983, a slight tempering that makes all keys equally usable.

这一公式解释了为何纯五度的频率比并非精确的 3:2，而是 2^(7/12) ≈ 1.4983，稍微的失协使得所有调都同样可用。

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8. Harmonic Series Theorem | 泛音列定理

The harmonic series is a naturally occurring sequence of overtones whose frequencies are integer multiples of a fundamental frequency f. It provides the physical basis for consonance and the overtone-rich timbre of acoustic instruments.

泛音列是自然界中存在的泛音序列，其频率为基频 f 的整数倍。它为协和感以及原声乐器丰富的泛音音色提供了物理基础。

Harmonic n: frequency = n × f

The first five harmonics and their musical intervals above the fundamental are:

前五个泛音及其与基音的音程关系为：

• 1st harmonic (fundamental) – unison

  第一泛音（基音）– 同度

• 2nd harmonic – octave

  第二泛音 – 八度

• 3rd harmonic – perfect 12th (octave + perfect 5th)

  第三泛音 – 纯十二度（八度加纯五度）

• 4th harmonic – 2 octaves

  第四泛音 – 两个八度

• 5th harmonic – 2 octaves + major 3rd

  第五泛音 – 两个八度加大三度

Important theorem: The equal-t

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