IGCSE Cambridge Music: Formula & Theorem Quick Reference | IGCSE 剑桥音乐：公式定理速查手册

📚 IGCSE Cambridge Music: Formula & Theorem Quick Reference | IGCSE 剑桥音乐：公式定理速查手册

This handbook compiles the essential formulas, rules, and theoretical principles you need to recall quickly for the IGCSE Cambridge Music examination. From note‑length calculations and interval semitone counts to chord‑construction templates, cadence patterns, and modulation pathways, every concept is presented as a clear “formula” with concise English explanations followed immediately by their Chinese equivalents. Use this guide for last‑minute revision or as a structured reference while practicing past papers.

本手册汇编了 IGCSE 剑桥音乐考试中需要快速回忆的核心公式、规则和理论原理。从音符时值计算和音程半音数，到和弦构建模板、终止式模式和转调路径，每个概念都以清晰的“公式”形式呈现，并配有简洁的英文说明和紧随其后的中文对应。请将本指南用于考前冲刺复习或作为练习历年真题时的结构化参考资料。

1. Note‑Length & Rhythm Formulas | 音符时值与节奏公式

A semibreve (whole note) is the reference value. All other note durations are defined as fractions of a semibreve. The table below summarises the fundamental rhythm formulas in simple time.

全音符（semibreve）是参考时值，所有其他音符的时值都定义为全音符的分数。下表总结了单拍子中的基本节奏公式。

Note/Rest (English)	音符/休止符	Duration Relative to Semibreve	相对时值公式
Semibreve	全音符	1	= 4 crotchets
Minim	二分音符	1/2	= 2 crotchets
Crotchet	四分音符	1/4	= 1 beat in 4/4
Quaver	八分音符	1/8	= ½ crotchet
Semiquaver	十六分音符	1/16	= ¼ crotchet

The dotted rhythm formula adds half the original value: dotted note = original duration + (original duration ÷ 2). For example, a dotted crotchet = 1 + ½ = 1½ beats.

附点节奏公式：附点音符时值 = 原时值 + (原时值 ÷ 2)。例如，附点四分音符 = 1 + ½ = 1½ 拍。

In compound time (e.g. 6/8), the beat is a dotted crotchet. The formula for the number of beats per bar is: top number ÷ 3 (for 6/8, 9/8, 12/8).

在复拍子（如 6/8 拍）中，单位拍是附点四分音符。每小节拍数的公式为：上方数字 ÷ 3（适用于 6/8、9/8、12/8）。

2. Interval Formulas | 音程公式

An interval is measured by counting the number of letter names (the numeric size) and the number of semitones (the quality). The fundamental formula: Quality is determined by the semitone count relative to the major scale built on the lower note.

音程通过计算音名字母的数量（度数）和半音数（性质）来衡量。基本公式：性质由相对于以低音为主音的大调音阶的半音数决定。

For perfect intervals (unison, 4th, 5th, octave), compare with the major scale. If the top note matches the major scale, it is perfect; if it is one semitone smaller, it is diminished; one semitone larger, it is augmented.

对于纯音程（同度、四度、五度、八度），与大调音阶比较。若上方音符合大调音阶，则为纯音程；若小一个半音，则为减音程；大一个半音，则为增音程。

For major/minor intervals (2nd, 3rd, 6th, 7th), the formula: major interval = number of semitones from the major scale; minor interval = major − 1 semitone; diminished = minor − 1 semitone; augmented = major + 1 semitone.

对于大/小音程（二度、三度、六度、七度），公式为：大音程 = 大调音阶中的半音数；小音程 = 大音程 − 1 个半音；减音程 = 小音程 − 1 个半音；增音程 = 大音程 + 1 个半音。

Table of semitone counts for intervals up to an octave:

至八度以内的音程半音数表：

Interval	Semitones
Minor 2nd	1
Major 2nd	2
Minor 3rd	3
Major 3rd	4
Perfect 4th	5
Augmented 4th / Diminished 5th (tritone)	6
Perfect 5th	7
Minor 6th	8
Major 6th	9
Minor 7th	10
Major 7th	11
Octave	12

Compound intervals (greater than an octave) follow the formula: compound interval quality = same as the simple interval; number = simple interval number + 7 (e.g. a major 10th is a major 3rd plus an octave).

复音程（大于八度）的公式为：复音程性质 = 与对应的单音程相同；度数 = 单音程度数 + 7（如大十度是大三度加一个八度）。

3. Scale Formulas | 音阶公式

A scale is a pattern of tones (T) and semitones (S). The major scale formula is the foundation: T – T – S – T – T – T – S.

音阶是由全音（T）和半音（S）组成的模式。大调音阶公式是基础：全 – 全 – 半 – 全 – 全 – 全 – 半。

Natural minor scale formula: T – S – T – T – S – T – T (derived from the major scale by lowering the 3rd, 6th, and 7th degrees).

自然小调音阶公式：全 – 半 – 全 – 全 – 半 – 全 – 全（通过降低大调音阶的第 3、6、7 级音得出）。

Harmonic minor: raise the 7th degree of the natural minor by a semitone, creating an augmented 2nd between the 6th and 7th degrees. Formula: T – S – T – T – S – 1½ T – S.

和声小调：将自然小调的第 7 级音升高半音，在 6、7 级之间形成增二度。公式：全 – 半 – 全 – 全 – 半 – 1½ 全 – 半。

Melodic minor (ascending): raise the 6th and 7th degrees of the natural minor. Formula: T – S – T – T – T – T – S. Decending: same as natural minor.

旋律小调（上行）：升高自然小调的第 6、7 级音。公式：全 – 半 – 全 – 全 – 全 – 全 – 半。下行同自然小调。

The circle of fifths provides a formula for key signatures. Moving clockwise adds sharps: order of sharps: F♯, C♯, G♯, D♯, A♯, E♯, B♯. Moving anticlockwise adds flats: order of flats: B♭, E♭, A♭, D♭, G♭, C♭, F♭.

五度循环提供了调号公式。顺时针方向增加升号：升号顺序：F♯、C♯、G♯、D♯、A♯、E♯、B♯。逆时针方向增加降号：降号顺序：B♭、E♭、A♭、D♭、G♭、C♭、F♭。

Pentatonic scale formula (major pentatonic): use degrees 1, 2, 3, 5, 6 of the major scale. Pattern: T – T – 1½ T – T – 1½ T. Minor pentatonic: 1, ♭3, 4, 5, ♭7 of the major scale.

五声音阶公式（大调五声）：使用大调音阶的 1、2、3、5、6 级。模式：全 – 全 – 1½ 全 – 全 – 1½ 全。小调五声：大调音阶的 1、♭3、4、5、♭7。

4. Chord Construction Formulas | 和弦构成公式

A triad is built by stacking thirds. The formula for a major triad: root + major 3rd (4 semitones) + perfect 5th (7 semitones above root). In terms of intervals above the root: root, 4 semitones, 3 semitones.

三和弦由三度叠置构成。大三和弦公式：根音 + 大三度（4 个半音）+ 纯五度（根音上方 7 个半音）。以根音为基准的音程关系：根音、4 半音、3 半音。

Minor triad: root + minor 3rd (3 semitones) + perfect 5th (7 semitones). Interval stack: 3 semitones + 4 semitones.

小三和弦：根音 + 小三度（3 个半音）+ 纯五度（7 个半音）。音程堆叠：3 半音 + 4 半音。

Diminished triad: root + minor 3rd + diminished 5th (6 semitones). Stack: 3 + 3 semitones.

减三和弦：根音 + 小三度 + 减五度（6 个半音）。堆叠：3 + 3 半音。

Augmented triad: root + major 3rd + augmented 5th (8 semitones). Stack: 4 + 4 semitones.

增三和弦：根音 + 大三度 + 增五度（8 个半音）。堆叠：4 + 4 半音。

Seventh chords add another third on top. Dominant 7th formula: major triad + minor 7th (10 semitones from root). This occurs naturally on the 5th degree of the major scale.

七和弦在顶部再添加一个三度。属七和弦公式：大三和弦 + 小七度（距离根音 10 个半音）。它自然产生于大调音阶的第 5 级。

Major 7th: major triad + major 7th (11 semitones). Minor 7th: minor triad + minor 7th. Diminished 7th: diminished triad + diminished 7th (9 semitones).

大七和弦：大三和弦 + 大七度（11 个半音）。小七和弦：小三和弦 + 小七度。减七和弦：减三和弦 + 减七度（9 个半音）。

Inversion formulas: root position = root in bass; 1st inversion = 3rd in bass; 2nd inversion = 5th in bass; 3rd inversion (for 7th chords) = 7th in bass. The figured bass symbols summarise the intervals above the bass: 5/3 for root, 6/3 (often written 6) for 1st inversion, 6/4 for 2nd inversion.

转位公式：原位 = 低音为根音；第一转位 = 低音为三音；第二转位 = 低音为五音；第三转位（七和弦）= 低音为七音。数字低音符号总结了低音上方的音程关系：原位为 5/3，第一转位为 6/3（常简写为 6），第二转位为 6/4。

5. Cadence Formulas | 终止式公式

Cadences are two‑chord formulas that punctuate phrases. The four primary cadences, defined by the chords and melodic movement, are:

终止式是收束乐句的两种弦公式。四种主要终止式，由和弦及旋律走向定义：

Perfect cadence (V – I): both chords in root position, soprano moves to tonic. Sound: complete, final.

正格终止（V – I）：两个和弦均处于原位，高音声部进行到主音。音响：完满、结束。

Imperfect cadence (I – V, ii – V, IV – V, etc.): any chord moving to V. Sound: incomplete, expects continuation.

半终止（I – V, ii – V, IV – V 等）：任何进行到 V 的和弦。音响：不完满，期待继续。

Plagal cadence (IV – I): both chords in root position, often with the soprano on the tonic for both chords. Sound: “Amen”.

变格终止（IV – I）：两个和弦均为原位，高音声部常保持主音。音响：“阿门”感。

Interrupted cadence (V – vi): the dominant moves to the submediant (minor) instead of the tonic. Sound: surprise, interruption.

阻碍终止（V – vi）：属和弦进行到下中音和弦（小三和弦），而非主和弦。音响：意外、中断。

Formula for identifying a cadence: examine the last two chords of a phrase (often the two chords before a rest or a long note). If the chords are V–I, it’s perfect; if V–vi, interrupted; if ending on V, imperfect; if IV–I, plagal.

识别终止式的公式：检查乐句最后两个和弦（通常是休止或长音前的两个和弦）。若为 V–I，即正格终止；V–vi，即阻碍终止；终止在 V 上，即为半终止；IV–I 为变格终止。

6. Harmonic Progression Rules | 和声进行规则

Functional harmony follows predictable root‑movement formulas. The most common progression pattern: I – IV – V – I (often with vi substituted for I, or ii for IV).

功能性和声遵循可预测的根音运动公式。最常见的进行模式：I – IV – V – I（常以 vi 替换 I，或以 ii 替换 IV）。

Circle‑of‑fifths progression formula: chords move by descending perfect 5ths (or ascending 4ths): iii → vi → ii → V → I. This creates strong, directed motion.

五度循环进行公式：和弦按下行纯五度（或上行纯四度）移动：iii → vi → ii → V → I。这产生强烈、有方向感的运动。

Common chord‑substitution formulas: IV can be replaced by ii (both predominant function); V can be replaced by vii° (dominant function). The formula is based on sharing two common tones.

常见的和弦替换公式：IV 可被 ii 替换（同属下属功能）；V 可被 vii° 替换（属功能）。公式基于共用两个共同音。

Part‑writing rules (vocal style) must be memorised:

No parallel perfect 5ths or octaves between any two voices.
No voice crossing; maintain correct doubling (usually double the root in root‑position chords).
Leading note (7th degree) resolves up to the tonic.

声部写作规则（声乐风格）必须牢记：

任何两个声部之间不得出现平行纯五度或平行八度。
声部不得交叉；保持正确的重复音（原位和弦通常重复根音）。
导音（第 7 级音）向上解决到主音。

These rules can be summarised as “avoid parallels, resolve tendency tones, and maintain smooth voice‑leading.”

这些规则可总结为“避免平行进行，解决倾向音，保持平稳的声部进行”。

7. Modulation Formulas | 转调公式

Modulation is a change of key within a piece. The simplest modulation formula uses a pivot chord — a chord shared by both the old key and the new key — then a cadence in the new key.

转调是作品中调性的变化。最简单的转调公式是使用一个共同和弦——旧调和新调共有的和弦——然后在新调上建立终止式。

Modulation to the dominant (most common in the Classical style): from a major key, the pivot is often I of the old key = IV of the dominant key. Use V–I in the dominant to confirm.

向属调转调（古典风格中最常见）：来自大调的转调，共同和弦常为旧调的 I = 属调的 IV。使用属调中的 V–I 加以确认。

Modulation to the relative minor: from a major key to its relative minor (vi), the pivot can be ii or IV in the major key, which become iv or VI in the minor. The cadence uses V–i in the minor.

向关系小调转调：从大调转到其关系小调（vi），共同和弦可以是大调的 ii 或 IV，在小调中变为 iv 或 VI。终止式在小调中使用 V–i。

Modulation to the subdominant: I in the old key = V in the subdominant key is a common pivot. Follow with a I–IV–V–I in the new key.

向下属调转调：旧调的 I = 下属调的 V 是常用的共同和弦。然后在新调中使用 I–IV–V–I。

Chromatic modulation uses a chord that is not shared diatonically; it arises through chromatic alteration and enharmonic reinterpretation, e.g. a German augmented 6th chord resolving as V7 of a distant key.

半音转调使用非自然音阶共有的和弦；它通过半音变化和等音重新解释产生，例如德国增六和弦作为远关系调的 V7 解决。

8. Melody Writing Principles | 旋律写作原理

A well‑formed melody follows balance and contour formulas. Phrase structure often uses question‑and‑answer (antecedent–consequent) pairs, each typically 4 bars long. The antecedent often ends on V (imperfect cadence) and the consequent on I (perfect cadence).

优美的旋律遵循平衡和轮廓公式。乐句结构常使用问答式（前句–后句）乐对，每句通常 4 小节。前句常结束于 V 级（半终止），后句结束于 I 级（正格终止）。

Melodic contour formula: an arch shape — rising to a climax about two‑thirds through, then falling. Avoid static, repetitive lines; use conjunct (stepwise) motion mainly, with occasional disjunct leaps followed by a step in the opposite direction.

旋律轮廓公式：拱形——约在三分之二处上升至高潮，然后下行。避免静态、重复的线条；主要使用级进（音阶式）进行，偶尔使用跳进，其后需反向级进。

Motivic development formulas: sequence (transpose a short idea up or down by step), inversion (mirror the intervals), augmentation/diminution (lengthen or shorten the note values). A good melody often combines rhythmic and melodic motifs.

动机发展公式：模进（将短小动机按级进上下移位）、倒影（音程反向）、增值/减值（加长或缩短音符时值）。好的旋律通常结合节奏和旋律动机。

9. Musical Form Formulas | 曲式公式

Binary form (AB): A ends on V or relative major; B modulates back to tonic and ends on I. Each section is repeated.

二段体 (AB)：A 段终止在 V 或关系大调；B 段转回主调并终止于 I。每段反复。

Ternary form (ABA): A ends in tonic; B provides contrast (different key, texture, or mood); A returns identical or varied. The overall structure is A (tonic) – B (new key) – A (tonic).

三段体 (ABA)：A 段终止于主调；B 段形成对比（不同调性、织体或情绪）；A 段原样返回或变化再现。整体结构为 A（主调）– B（新调）– A（主调）。

Rondo form: A – B – A – C – A – (etc.) , where A is the refrain in the tonic and episodes contrast. The formula is alternating refrain with digression.

回旋曲式：A – B – A – C – A – （等），其中 A 是主调中的叠句，插部形成对比。公式为叠句与离题的交替。

Sonata form (first‑movement form): Exposition (first subject in tonic, second subject in dominant/relative major) – Development (modulates and develops motives) – Recapitulation (both subjects in tonic). The formula rests on key relationships: tension created by contrasting key areas resolved by returning home.

奏鸣曲式（第一乐章曲式）：呈示部（第一主题在主调，第二主题在属调或关系大调）– 展开部（转调并发展动机）– 再现部（两个主题均回到主调）。公式基于调性关系：通过调性对比制造张力，通过回归主调获得解决。

10. Instrument Transposition Formulas | 乐器移调公式

Transposing instruments sound at a different pitch from written notation. The transposition formula tells the interval between written note (C) and actual sounding pitch.

移调乐器的发音与实际记谱音高不同。移调公式说明了记谱音 C 与实际发音之间的音程关系。

Clarinet in B♭: sounds a major 2nd lower than written. Written C = sounding B♭. Transpose the part down a major 2nd (or the concert pitch up a major 2nd).
B♭ 调单簧管：发音比记谱低大二度。记谱 C = 实际 B♭。将分谱移低大二度（或将实际音高移高大二度）。
Trumpet in B♭: same as clarinet: sounding pitch = written C is B♭.
B♭ 调小号：同单簧管：实际音高 = 记谱 C 为 B♭。
Horn in F: sounds a perfect 5th lower than written. Written C = sounding F. Transpose down a perfect 5th.
F 调圆号：发音比记谱低纯五度。记谱 C = 实际 F。移低纯五度。
Cor Anglais (English horn) in F: same transposition as horn.
英国管（F 调）：移调同圆号。
Alto saxophone in E♭: sounds a major 6th lower than written. Written C = sounding E♭.
E♭ 调中音萨克斯：发音比记谱低大六度。记谱 C = 实际 E♭。

Formula for writing a transposed score: if you have a concert pitch melody, write the transposed part up by the same interval that the instrument sounds down. For B♭ instruments, write a major 2nd higher; for F instruments, a perfect 5th higher.

编写移调总谱的公式：若给定实际

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IGCSE Cambridge Music: Formula & Theorem Quick Reference | IGCSE 剑桥音乐：公式定理速查手册