📚 A-Level CAIE Music: Formula & Theorem Quick Reference Handbook | A-Level CAIE 音乐：公式定理速查手册
This handbook condenses the core theoretical patterns, harmonic formulas, and structural "rules of thumb" that appear consistently in the CAIE A-Level Music syllabus. Use it as a revision companion before exams or as a compositional checklist when writing your own pieces.
本手册提炼了 CAIE A-Level 音乐考纲中反复出现的核心理论模式、和声公式与结构“经验法则”。它既可用于考前复习，也可作为你进行音乐创作时的自查清单。
1. Interval Identification: The Semitone Formula | 1. 音程识别：半音公式
An interval is measured by counting the number of semitones between two pitches. Memorising the following semitone values will help you instantly name any interval in melodic and harmonic dictation.
音程的大小取决于两个音之间的半音数量。牢记以下几个半音数值，可以帮助你在旋律听写与和声听辨中迅速辨认音程。
| Interval 音程 | Semitones 半音数 | Example from C 从 C 出发示例 |
| Perfect Unison 纯一度 | 0 | C–C |
| Minor 2nd 小二度 | 1 | C–D♭ |
| Major 2nd 大二度 | 2 | C–D |
| Minor 3rd 小三度 | 3 | C–E♭ |
| Major 3rd 大三度 | 4 | C–E |
| Perfect 4th 纯四度 | 5 | C–F |
| Tritone / Augmented 4th 三全音/增四度 | 6 | C–F♯ |
| Perfect 5th 纯五度 | 7 | C–G |
| Minor 6th 小六度 | 8 | C–A♭ |
| Major 6th 大六度 | 9 | C–A |
| Minor 7th 小七度 | 10 | C–B♭ |
| Major 7th 大七度 | 11 | C–B |
| Perfect Octave 纯八度 | 12 | C–C' |
2. Chord Construction: Stacking Thirds | 2. 和弦构建：三度叠置法则
In tonal harmony, triads and seventh chords are built by stacking thirds. The quality of the chord depends on the combination of major and minor thirds in its stack.
在调性和声中，三和弦与七和弦均通过三度叠置构成。和弦的性质取决于叠置中大三度与小三度的组合方式。
Triads: Major = M3 + m3; Minor = m3 + M3; Diminished = m3 + m3; Augmented = M3 + M3
三和弦：大三 = 大三度 + 小三度；小三 = 小三度 + 大三度；减三 = 小三度 + 小三度；增三 = 大三度 + 大三度
- Major triad (root, M3, P5) — e.g., C–E–G. 大三和弦（根音、大三度、纯五度）——例如 C–E–G。
- Minor triad (root, m3, P5) — e.g., C–E♭–G. 小三和弦（根音、小三度、纯五度）——例如 C–E♭–G。
- Diminished triad (root, m3, d5) — e.g., B–D–F. 减三和弦（根音、小三度、减五度）——例如 B–D–F。
- Augmented triad (root, M3, A5) — e.g., C–E–G♯. 增三和弦（根音、大三度、增五度）——例如 C–E–G♯。
Seventh chords add a fourth third above the triad. The dominant 7th is the most common: root, M3, P5, m7 (e.g., G–B–D–F).
七和弦在三和弦之上再叠一个三度。属七和弦最为常见：根音、大三度、纯五度、小七度（例如 G–B–D–F）。
3. Key Signatures: The Circle of Fifths | 3. 调号：五度循环圈
The circle of fifths provides a systematic formula for all twelve major and minor keys. Move clockwise by perfect fifths to add one sharp; counter-clockwise to add one flat.
五度循环圈以系统化公式涵盖全部十二个大调与小调。按顺时针方向以纯五度移动，每步增加一个升号；逆时针方向则每步增加一个降号。
Sharps order: F–C–G–D–A–E–B; Flats order: B–E–A–D–G–C–F
升号顺序：F–C–G–D–A–E–B；降号顺序：B–E–A–D–G–C–F
| Number of Accidentals 变音记号数 | Major Key 大调 | Relative Minor 关系小调 |
| 0 | C major | A minor |
| 1♯ | G major | E minor |
| 2♯ | D major | B minor |
| 3♯ | A major | F♯ minor |
| 4♯ | E major | C♯ minor |
| 1♭ | F major | D minor |
| 2♭ | B♭ major | G minor |
| 3♭ | E♭ major | C minor |
4. Scale Formulae: Major, Minor, and Modes | 4. 音阶公式：大调、小调与调式
Every scale has a consistent whole-tone/semitone formula. Apply these patterns to any starting note to construct the correct scale.
每种音阶都有固定的全音/半音公式。将该模式应用于任意起始音，即可构建出正确的音阶。
Major: W–W–H–W–W–W–H; Natural Minor: W–H–W–W–H–W–W
大调：全–全–半–全–全–全–半；自然小调：全–半–全–全–半–全–全
- Harmonic Minor 和声小调：Raises degree 7 — formula: W–H–W–W–H–W+ aug.2nd–H. 将第 VII 级升高半音——公式：全–半–全–全–半–增二度–半。
- Melodic Minor 旋律小调：Raises degrees 6 and 7 ascending; natural minor descending. 上行时升高第 VI、VII 级；下行时采用自然小调。
- Dorian mode 多利亚调式：W–H–W–W–W–H–W (same as natural minor with raised 6th). 全–半–全–全–全–半–全（即自然小调升高第 VI 级）。
- Mixolydian mode 混合利底亚调式：W–W–H–W–W–H–W (same as major with lowered 7th). 全–全–半–全–全–半–全（即大调降低第 VII 级）。
5. Harmonic Functions: Tonic, Subdominant, Dominant | 5. 和声功能：主功能、下属功能、属功能
The three primary harmonic functions in tonal music create a grammatical framework for chord progressions. The dominant's leading-tone urge to resolve to the tonic is a "theorem" of Western harmony.
调性音乐中的三大和声功能构成了和弦进行的语法框架。属和弦中导音向主音解决的倾向，是西方和声学中的一条“定理”。
I (Tonic) ← IV (Subdominant) ← V (Dominant) ← I (Tonic)
I（主功能）← IV（下属功能）← V（属功能）← I（主功能）
- Dominant 7th (V⁷) contains a tritone between the 3rd and 7th, which resolves inward by step to the tonic triad. 属七和弦（V⁷）的三音与七音之间构成三全音，该三全音按半音级进内收解决至主三和弦。
- Perfect cadence 正格终止：V–I (or V⁷–I); Plagal cadence 变格终止：IV–I; Interrupted cadence 阻碍终止：V–VI; Imperfect cadence 半终止：ends on V. 正格终止：V–I（或 V⁷–I）；变格终止：IV–I；阻碍终止：V–VI；半终止：结束于 V。
- II–V–I progression is the "formula" of functional tonality in jazz and classical contexts alike. II–V–I 进行是爵士乐与古典语境中功能调性的“标准公式”。
6. Figured Bass: Interpreting Harmony Numerals | 6. 数字低音：解读和声数字标记
Figured bass numbers tell the accompanist which intervals above the bass note should be sounded. Learning to read them fluently is a tested skill in the CAIE exam.
数字低音中的数字标示了低音上方应当奏出哪些音程。熟练识读数字低音是 CAIE 考试中的重要考查技能。
| Figure 数字 | Meaning 含义 | Standard Realisation 标准实现 |
| 5–3 (or blank 无数字) | Root position triad 原位三和弦 | Bass + 3rd + 5th 低音 + 三音 + 五音 |
| 6 | First inversion 第一转位 | Bass + 3rd + 6th above bass 低音 + 三音 + 六音 |
| 6–4 | Second inversion 第二转位 | Bass + 4th + 6th above bass 低音 + 四音 + 六音 |
| 7 | Root position seventh chord 原位七和弦 | Bass + 3rd + 5th + 7th 低音 + 三音 + 五音 + 七音 |
| 6–5 | First inversion seventh chord 七和弦第一转位 | Bass + 3rd + 5th + 6th 低音 + 三音 + 五音 + 六音 |
| 4–3 | Second inversion seventh chord 七和弦第二转位 | Bass + 3rd + 4th + 6th 低音 + 三音 + 四音 + 六音 |
| 4–2 | Third inversion seventh chord 七和弦第三转位 | Bass + 2nd + 4th + 6th 低音 + 二音 + 四音 + 六音 |
7. Rhythm and Metre: Time Signature Arithmetic | 7. 节奏与节拍：拍号运算规则
Time signatures are effectively fractions: the upper number tells the number of beats per bar, and the lower number tells the note value receiving one beat. Compound metre is "twice-based" — each beat divides into three subdivisions.
拍号本质上是分数：上方数字表示每小节拍数，下方数字表示以何种音符为一拍。复拍子以“二倍”为基底——但每一大拍内部细分为三小拍。
Simple duple/triple/quadruple: 2/4, 3/4, 4/4; Compound: 6/8 (2 beats of dotted crotchet), 9/8 (3 beats), 12/8 (4 beats)
单拍子二/三/四拍：2/4、3/4、4/4；复拍子：6/8（以附点四分音符为两大拍）、9/8（三大拍）、12/8（四大拍）
- Dotted note adds half of its original value: a dotted crotchet equals 1.5 beats. 附点音符增加原时值的一半：附点四分音符等于一拍半。
- Triplet divides a beat into three equal parts; duplet divides a dotted beat into two equal parts. 三连音将一拍均分为三部分；二连音将附点拍均分为两部分。
- o relationships, use proportional notation: half note = half the duration of a whole note. 计算速度关系时采用比例记谱法：二分音符时长是全音符的一半。
8. Musical Form: Structural Blueprints | 8. 曲式：结构蓝图速查
Western classical forms operate like predictable "theorems" of musical architecture. Recognising their templates helps both score reading and listening questions.
西方古典曲式如同可预测的音乐建筑“定理”。掌握其模板有助于乐谱阅读和听力问答。
- Binary Form 二段曲式：A–B, with each section usually repeated. 由 A、B 两段组成，每段通常反复。
- Ternary Form 三段曲式：A–B–A, with the first section returning at the end. 呈示 A 段，中部 B 段，最后返回 A 段。
- Rondo Form 回旋曲式：A–B–A–C–A–D–A … the refrain alternates with contrasting episodes. 主部 A 与不同插部交替出现：A–B–A–C–A–D–A……
- Sonata Form 奏鸣曲式： 1, P2, codetta) → Development (key exploration) → Recapitulation (both subjects in tonic). 呈示部（主部、副部、小结尾）→ 展开部（调性探索）→ 再现部（两个主题均回到主调）。
- Theme and Variations 主题与变奏：A–A¹–A²–A³–…, keeping the melodic skeleton while changing texture, rhythm, or harmony. 主题 A 及其若干变奏，保持旋律骨架不变，但变换织体、节奏或和声。
9. Counterpoint: The Rules of Species | 9. 对位法：分类对位规则
First-species counterpoint against a cantus firmus is governed by simple interval arithmetic. These historical "formulas" still underpin the CAIE two-part writing question.
固定调之上的第一类对位遵循简单的音程运算规则。这些历史性“公式”仍然是 CAIE 二声部写作题的基础。
Begin and end on perfect unison or octave; approach perfect consonances by contrary or oblique motion; use only consonant intervals on strong beats.
首尾必须使用纯一度或纯八度；以反向或斜向进行进入完全协和音程；强拍上只用协和音程。
- Forbidden: parallel perfect fifths, parallel octaves, and hidden (consecutive-by-similar-motion) fifths/octaves. 禁止：平行纯五度、平行八度以及同向进行的隐伏五度/八度。
- Dissonances (2nd, 4th, 7th) may occur only on weak beats, approached and resolved by step. 不协和音程（二度、四度、七度）仅可出现在弱拍，须级进引入并级进解决。
- The leading tone (degree 7) must resolve upward to the tonic in a minor key context. 小调中的导音（第 VII 级）必须上行解决至主音。
10. Compositional Devices: Formulas for Development | 10. 作曲技法的公式化运用
Knowing how to manipulate musical material is essential for the free composition section. These devices act as your "toolkit theorems" for melodic development.
掌握如何操控音乐素材，是自由创作部分的核心能力。以下技法构成了旋律发展的“工具箱定理”。
| Device 技法 | Formula 公式 | Example 示例 |
| Sequence 模进 | Repeat a melodic fragment at a new pitch level 将旋律片段在新音高上重复 | C–D–E then D–E–F 先 C–D–E，再 D–E–F |
| Inversion 倒影 | Mirror intervals in the opposite direction 音程反向镜像 | C–E–G becomes C–A♭–F |
| Retrograde 逆行 | Play the melody backwards 旋律倒置演奏 | C–D–E becomes E–D–C |
| Augmentation 增值 | Double all note durations 所有音符时值加倍 | ♩♩ → 𝅗𝅥𝅗𝅥 |
| Diminution 减值 | Halve all note durations 所有音符时值减半 | 𝅗𝅥𝅗𝅥 → ♩♩ |
| Fragmentation 碎片化 | Extract a short motivic cell for development 提取短小动机细胞进行发展 | Use only the opening 3 notes 仅使用开头三个音 |
11. Melodic Dictation: The Formula for Success | 11. 旋律听写：解题公式
Melodic dictation is a skill best treated as a problem-solving algorithm. Apply these steps in order to maximise accuracy under exam conditions.
旋律听写是一种适合用“解题算法”来训练的技能。在考试条件下按下列步骤依次进行，可以最大化准确率。
- Step 1: Identify the key and time signature from the first played frame. 步骤一：根据开始前的提示音确定调号与拍号。
- Step 2: Mark the position of bar-lines and write in all rhythmic values first. 步骤二：先写出小节线位置，再填写全部节奏型。
- Step 3: Sketch the contour (up/down/same) for each phrase before filling exact pitches. 步骤三：先标记旋律大致走向（上行/下行/同音），再填入精确音高。
- Step 4: Use the tonic triad (I) and dominant seventh (V⁷) as reference anchors. 步骤四：利用主三和弦与属七和弦作为参照锚点。
12. Score Reading: Transposition and Clefs | 12. 读谱：移调乐器与谱号速查
Transposing instruments are written at their sounding pitch plus/minus a fixed interval. Memorise the six most common ones for Section B of the exam.
移调乐器的记谱音高与实际音高相差固定音程。以下是考试 B 部分最常见的六种移调乐器，需要牢固记忆。
| Instrument 乐器 | Written vs Sounding 记谱音 vs 实际音 |
| B♭ Clarinet / B♭ Trumpet 降B调单簧管/小号 | Written C sounds B♭ (down a major 2nd) 记谱 C 实际为 B♭（低大二度） |
| A Clarinet A调单簧管 | Written C sounds A (down a minor 3rd) 记谱 C 实际为 A（低小三度） |
| F Horn F调圆号 | Written C sounds F (down a perfect 5th) 记谱 C 实际为 F（低纯五度） |
| Violin 小提琴 | Treble clef, non-transposing 高音谱号，不移调 |
| Viola 中提琴 | Alto clef (C clef on middle line), non-transposing 中音谱号（C谱号在第三线），不移调 |
| Cello 大提琴 | Bass clef, non-transposing 低音谱号，不移调 |
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