Music Theory Quick-Reference Handbook | 音乐公式定理速查手册

📚 Music Theory Quick-Reference Handbook | 音乐公式定理速查手册

This handbook gathers essential formulas, theorems, and quick-reference definitions for Year 12 CAIE Music. It is designed to support analysis, composition, and aural skills by presenting core theoretical concepts in a compact, easy-to-review format. Use it as a daily study companion or a last-minute revision checklist before your exams.

本手册汇集了针对Year 12 CAIE音乐课程的核心公式、定理与速查定义。通过简洁易查的格式梳理基础理论概念，帮助你在作品分析、作曲及听觉训练中快速定位关键信息。将其作为日常学习伴侣或考前快速回顾清单。

1. Intervals and Their Quality | 音程及其性质

An interval is the distance between two pitches, measured in semitones and described by a number (2nd, 3rd, etc.) and a quality (perfect, major, minor, augmented, diminished). The simplest method to identify an interval is to count the semitones between the notes and check the spelling according to the major scale of the lower note.

音程是两个音高之间的距离，以半音为单位计量，并用数字（二度、三度等）和性质（纯、大、小、增、减）来描述。最简单的识别方法是数出两音之间的半音数，再根据下方音的大调音阶核对拼写。

The following table lists the most common intervals within an octave, combining semitone counts with letter-name sizes. Use it as a quick lookup when analysing melodies or building chords.

下表列出了八度以内最常见的音程，结合了半音数与音名字母的度数。在分析旋律或构建和弦时，可用作快速查阅。

Semitones	Interval Name	Example (from C)
0	Perfect unison (P1)	C–C
1	Minor 2nd (m2)	C–D♭
2	Major 2nd (M2)	C–D
3	Minor 3rd (m3) / Augmented 2nd	C–E♭ / C–D♯
4	Major 3rd (M3)	C–E
5	Perfect 4th (P4)	C–F
6	Augmented 4th / Diminished 5th (tritone)	C–F♯ / C–G♭
7	Perfect 5th (P5)	C–G
8	Minor 6th (m6)	C–A♭
9	Major 6th (M6)	C–A
10	Minor 7th (m7)	C–B♭
11	Major 7th (M7)	C–B
12	Perfect octave (P8)	C–C

When an interval is inverted, perfect stays perfect, major becomes minor, minor becomes major, augmented becomes diminished, and the numerical sizes add up to 9. For example, a major 3rd (C–E) inverts to a minor 6th (E–C). Always check the clef and any accidentals to avoid misspelling enharmonically equivalent intervals.

音程转位时，纯音程保持为纯，大变小，小变大，增变减，且度数之和为9。例如大三度（C–E）转位后成为小六度（E–C）。务必检查谱号和临时记号，避免将等音音程拼写错误。

2. Major and Minor Scales Formula | 大小调音阶公式

Scales are built from a fixed pattern of tones (whole steps) and semitones (half steps). The major scale pattern is the foundation for all diatonic modes, while the three forms of minor scale each adjust specific degrees for melodic and harmonic purposes. Memorise these patterns as ‘formulas’ that can be applied to any tonic.

音阶由固定的全音和半音模式构成。大调音阶模式是所有自然调式的基础，而三种小调音阶则出于旋律与和声的需要对特定音级进行调整。请将这些模式当作’公式’记忆，可适用于任何主音。

Major Scale Formula: W – W – H – W – W – W – H

大调音阶公式：全 – 全 – 半 – 全 – 全 – 全 – 半

Natural minor uses the same pattern as the Aeolian mode: W – H – W – W – H – W – W. Harmonic minor raises the 7th degree by a semitone, creating an augmented 2nd between the 6th and 7th degrees. Melodic minor raises both the 6th and 7th degrees when ascending, and reverts to natural minor when descending.

自然小调使用与爱奥利亚调式相同的模式：全 – 半 – 全 – 全 – 半 – 全 – 全。和声小调将第VII级升高半音，在VI级与VII级之间形成一个增二度。旋律小调上行时升高VI级与VII级，下行时还原为自然小调。

For exam-style questions, always write the key signature first and then add accidentals for the altered degrees. Count the steps carefully: a harmonic minor scale in D minor requires a C♯, while A melodic minor ascending uses F♯ and G♯, but descending uses G♮ and F♮.

在考试题型中，务必先写出调号，再为变化音级添加临时记号。仔细数清音程：d小调和声小调需要C♯，而a旋律小调上行使用F♯和G♯，下行则使用G♮和F♮。

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

The circle of fifths arranges keys so that each step clockwise adds a sharp (or removes a flat), moving up a perfect fifth. Anticlockwise movement adds a flat or removes a sharp, moving down a fifth. This visual formula helps you instantly recall the number of sharps or flats in any major or minor key.

五度圈将调性按纯五度关系排列，顺时针每移动一步则增加一个升号（或减少一个降号），逆时针移动则增加降号或减少升号。这一图形公式可帮助你瞬间记住任何大调或小调的升降号数量。

Order of sharps: F♯, C♯, G♯, D♯, A♯, E♯, B♯. A useful mnemonic is ‘Father Charles Goes Down And Ends Battle’ for sharps, and ‘Battle Ends And Down Goes Charles Father’ for flats (B♭, E♭, A♭, D♭, G♭, C♭, F♭). The relative minor is always a minor third below its major tonic.

升号顺序：F♯, C♯, G♯, D♯, A♯, E♯, B♯。实用口诀：升号可用’Father Charles Goes Down And Ends Battle’记忆；降号顺序（B♭, E♭, A♭, D♭, G♭, C♭, F♭）可用’Battle Ends And Down Goes Charles Father’。关系小调的主音永远在其大调主音下方小三度。

To find the tonic of a sharp major key, take the last sharp and go up a semitone (e.g., last sharp D♯ → tonic E major). For a flat major key, the tonic is the second-to-last flat (e.g., flats B♭, E♭, A♭ → E♭ major). For minor keys, first identify the relative major, then apply the minor key signature accordingly.

要找出升号大调的主音，取最后一个升号并升高半音（如最后一个升号为D♯ → 主音E大调）。降号大调的主音则是倒数第二个降号（如降号为B♭, E♭, A♭ → E♭大调）。对于小调，首先确定其关系大调，再套用相应调号。

Relative minors: C major – A minor; G major – E minor; D major – B minor; etc.

关系小调：C大调–a小调；G大调–e小调；D大调–b小调；等等

4. Triads and Chord Construction | 三和弦构成

A triad is a three-note chord consisting of a root, a third, and a fifth. The quality of the triad depends on the size of these intervals: a major triad contains a major third and perfect fifth; a minor triad contains a minor third and perfect fifth. The following formulas express each type using Roman numeral notation relative to a major scale.

三和弦是由根音、三音和五音构成的三音和弦。其性质取决于这些音程的大小：大三和弦包含大三度和纯五度；小三和弦包含小三度和纯五度。以下公式用相对于大调音阶的罗马数字表示每种类型。

Major Triad (I) = Root + M3 + P5 (e.g., C–E–G)

大三和弦 (I) = 根音 + 大三度 + 纯五度 (例：C–E–G)

Minor Triad (ii) = Root + m3 + P5 (e.g., D–F–A)

小三和弦 (ii) = 根音 + 小三度 + 纯五度 (例：D–F–A)

Diminished Triad (vii°) = Root + m3 + d5 (e.g., B–D–F)

减三和弦 (vii°) = 根音 + 小三度 + 减五度 (例：B–D–F)

Augmented Triad (III⁺) = Root + M3 + A5 (e.g., C–E–G♯)

增三和弦 (III⁺) = 根音 + 大三度 + 增五度 (例：C–E–G♯)

In a major key, the diatonic triads on each scale degree follow the pattern: I (major), ii (minor), iii (minor), IV (major), V (major), vi (minor), vii° (diminished). In a natural minor key, the pattern becomes i, ii°, III, iv, v, VI, VII. When analysing harmony, label chords with Roman numerals based on the root’s scale degree and the chord quality.

在大调中，各级音上的自然三和弦遵循以下模式：I (大三), ii (小三), iii (小三), IV (大三), V (大三), vi (小三), vii° (减三)。在自然小调中，模式变为 i, ii°, III, iv, v, VI, VII。分析和声时，请根据根音的音级和和弦性质用罗马数字标记。

5. Chord Inversions and Figured Bass | 和弦转位与数字低音

Inversions rearrange the notes of a chord so that a note other than the root becomes the lowest voice. Figured bass symbols indicate the intervals above the bass note, providing a quick shorthand for inversion. Triads have two inversions, while seventh chords have three. Understanding these numbers speeds up score analysis and harmonic dictation.

转位重新排列和弦音，使非根音成为最低声部。数字低音符号通过标记低音上方的音程，为转位提供了简写方式。三和弦有两种转位，七和弦则有三种。理解这些数字能加速乐谱分析与和声听写。

Position	Figured Bass	Example (C major)	Bass note
Root position	5/3 (often omitted)	C–E–G	C
First inversion	6/3 (usually 6)	E–G–C	E
Second inversion	6/4	G–C–E	G

For seventh chords, root position is 7/5/3 (usually 7), first inversion is 6/5/3 (usually 6/5), second inversion is 6/4/3 (usually 4/3), and third inversion is 6/4/2 (usually 4/2). In tonal harmony, second-inversion triads are considered unstable and typically function as passing, cadential, or pedal six-four chords.

七和弦的根音原位为7/5/3（常简写为7），第一转位为6/5/3（常写为6/5），第二转位为6/4/3（常写为4/3），第三转位为6/4/2（常写为4/2）。在调性和声中，三和弦的第二转位被认为是不稳定的，通常用作经过的、终止的或持续音上的四六和弦。

6. Cadences and Harmonic Functions | 终止式与和声功能

Cadences are the punctuation marks of tonal music, formed by two-chord formulas that close phrases. Knowing the standard cadence types allows you to identify phrase endings and predict large-scale formal boundaries. Each cadence has a specific Roman numeral formula and a characteristic effect of finality or continuation.

终止式是调性音乐的标点符号，由收束乐句的双和弦公式构成。了解标准终止式类型有助于识别乐句结尾并预判段落边界。每种终止式都有特定的罗马数字公式，并带来完结或延续的明确效果。

Perfect Authentic Cadence (PAC): V(⁷) – I, both in root position, tonic in soprano

完满全终止 (PAC)：V(⁷)–I，均为原位且高音结束在主音

Imperfect Authentic Cadence (IAC): V – I with inversions or non-tonic soprano

不完满全终止 (IAC)：V–I，但含转位或高音非主音

Plagal Cadence (PC): IV – I (the ‘Amen’ cadence)

变格终止 (PC)：IV–I (‘阿门’终止)

Half Cadence (HC): any chord – V (most commonly I – V, ii – V, or IV – V)

半终止 (HC)：任一和弦–V (常为 I–V, ii–V 或 IV–V)

Interrupted/Deceptive Cadence: V – vi (or V – ♭VI in minor)

阻碍终止：V–vi (小调中常用 V–♭VI)

In functional harmony, chords are grouped into tonic (I, vi, iii), dominant (V, vii°), and pre-dominant (IV, ii) families. A typical phrase progresses from tonic, to pre-dominant, to dominant, and then back to tonic. Recognising these families helps you reduce any chord progression to a functional skeleton.

在功能和声中，和弦被归为主功能组 (I, vi, iii)、属功能组 (V, vii°) 和下属功能组 (IV, ii)。一个典型的乐句从主功能开始，经过下属功能再到属功能，最后回到主功能。识别这些功能组能将任何和弦进行简化为功能骨架。

7. Non-Chord Tones and Embellishments | 和弦外音与装饰音

Non-chord tones (NCTs) add melodic tension and expressive colour by sounding against the prevailing harmony before resolving. Each type follows a specific pattern of approach and departure relative to the chord tones. Memorising these formulas enables accurate melodic and harmonic analysis in CAIE set works and unfamiliar extracts.

和弦外音通过在高音部与当前和声碰撞后获得解决，从而增添旋律张力与表现色彩。每种外音都有相对和弦音的特定进入与离开模式。熟记这些公式有助于在CAIE指定作品与陌生选段中准确分析旋律与和声。

Passing tone (P): approached by step, left by step in the same direction; connects two chord tones a third apart.

经过音 (P)：级进进入，级进同向离开；连接相距三度的两个和弦音。
Neighbour tone (N): approached by step, left by step in the opposite direction; returns to the original chord tone.

辅助音 (N)：级进进入，反向级进离开；回到原和弦音。
Suspension (S): prepared by the same pitch as a chord tone, then held over as a dissonance, and resolved down by step.

延留音 (S)：前一和弦音延续相同音高构成不协和，然后下行级进解决。
Appoggiatura (App): approached by leap, resolved by step, usually on a strong beat.

倚音 (App)：跳进进入，级进解决，通常位于强拍。
Escape tone (E): approached by step, left by leap in the opposite direction.

规避音 (E)：级进进入，反向跳进离开。
Anticipation (Ant): approached by step, then repeats the following chord tone early; weak beat.

先现音 (Ant)：级进进入，提前重复后一和弦音；弱拍。
Pedal point: sustained or repeated note, usually bass, while harmonies change above it.

持续音：在声部（通常是低音）保持或重复某音，上方和声持续变化。

When labelling NCTs, always circle the dissonant note and trace its preparation and resolution. Check whether the note belongs to the current chord; if not, identify the type based on the approach and departure pattern. This skill is frequently tested in ‘describe the melody’ and harmonic analysis questions.

标注和弦外音时，请圈出不协和音并追溯其准备与解决。检查该音是否属于当前和弦；若不属，则根据进入与离开方式判定其类型。这一技能常在’描述旋律’及和声分析题中考查。

8. Modulation and Tonicization | 转调与主音化

Modulation is the process of changing key within a piece, while tonicization temporarily treats a chord other than the tonic as a local key centre without a full cadence. Both are driven by accidentals that imply new leading-tone relationships. Close examination of the circle of fifths reveals the most closely related keys for smooth modulation.

转调是在乐曲内部改变调性的过程，而主音化则是临时将非主和弦作为局部调中心，但无完整终止式。两者均由暗示新导音关系的临时记号驱动。仔细研读五度圈可揭示用于平滑转调的最亲近调。

Key	Closely related keys (major)	Closely related keys (minor)
C major / A minor	G, F, Am, Dm, Em	C major, Dm, Em, F, G

The most common modulations in Baroque and Classical repertoire move to the dominant (V) in major-key pieces, or to the relative major (III) from minor keys. Pivot chord modulation uses a chord that functions diatonically in both the old and new keys, then proceeds with a cadence in the new key. Look for a cadence confirmed by a V–I in the new tonic to identify a full modulation.

巴洛克与古典时期最常见的转调：大调乐曲转向属调 (V)，小调转向关系大调 (III)。共同和弦转调使用一个在旧调和新调中均属自然和弦的和弦作为支点，然后在新调中形成终止。要识别完整转调，可寻找新调中由 V–I 确认的终止式。

9. Melodic and Rhythmic Devices | 旋律与节奏手法

Composers transform melodic material through a set

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Music Theory Quick-Reference Handbook | 音乐公式定理速查手册