📚 CCEA A2 Music Formula & Theorem Quick-Reference Handbook | CCEA A2 音乐公式定理速查手册
For Year 13 CCEA Music students, the ability to recall theoretical 'formulas' and compositional 'theorems' quickly is as essential as knowing equations in maths. This handbook gathers the key structural principles of melody, harmony, rhythm, texture, and form into a concise, at-a-glance format, treating them like reliable formulas that underpin both analysis and composition in the Western Classical Tradition and beyond.
对于 Year 13 CCEA 音乐学生来说，迅速回忆理论“公式”和作曲“定理”的能力，就像数学中熟记方程一样至关重要。本手册将旋律、和声、节奏、织体和曲式的关键结构原则，整理成简洁、一目了然的格式，把它们视为支撑西方古典传统及其他风格分析与创作的可靠公式。
1. Interval Formulas & Semitone Counting | 音程公式与半音计数
An interval is the distance between two pitches, measured in semitones. Use these formulas to identify or construct any simple interval within an octave.
音程是两个音高之间的距离，以半音计量。使用以下公式识别或构建八度以内的任何单音程。
Unison = 0 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; Perfect Octave = 12.
纯一度 = 0 半音；小二度 = 1；大二度 = 2；小三度 = 3；大三度 = 4；纯四度 = 5；增四度 / 减五度（三全音） = 6；纯五度 = 7；小六度 = 8；大六度 = 9；小七度 = 10；大七度 = 11；纯八度 = 12。
To find the inversion formula: the semitone sum always equals 12. A Major 3rd (4 semitones) inverts to a Minor 6th (8 semitones). Qualities also invert: major becomes minor, augmented becomes diminished, perfect remains perfect.
求转位音程公式：半音总和始终为 12。大三度（4 半音）转位为小六度（8 半音）。性质亦转位：大变小，增变减，纯不变。
2. Chord Construction: Triad Formulas | 和弦构建：三和弦公式
Triads are built by stacking two thirds above a root. The quality depends on the semitone pattern between root, third, and fifth.
三和弦由根音上方叠加两个三度构成。性质取决于根音、三音、五音之间的半音模式。
- Major triad: Root + 4 semitones (major 3rd) + 3 semitones (minor 3rd) → R–4–3. Example: C–E–G.
- 大 三和弦：根音 + 4 半音（大三度）+ 3 半音（小三度）→ R–4–3。例如：C–E–G。
- Minor triad: R + 3 (minor 3rd) + 4 (major 3rd) → R–3–4. Example: C–E♭–G.
- 小 三和弦：R + 3（小三度）+ 4（大三度）→ R–3–4。例如：C–E♭–G。
- Diminished triad: R + 3 + 3 → R–3–3. Example: C–E♭–G♭.
- 减 三和弦：R + 3 + 3 → R–3–3。例如：C–E♭–G♭。
- Augmented triad: R + 4 + 4 → R–4–4. Example: C–E–G♯.
- 增 三和弦：R + 4 + 4 → R–4–4。例如：C–E–G♯。
The same logic extends to seventh chords: a major 7th chord adds a major 3rd on top of a major triad (R–4–3–4); a dominant 7th adds a minor 3rd (R–4–3–3); a minor 7th adds a minor 3rd (R–3–4–3); a half-diminished 7th uses R–3–3–4; a diminished 7th uses R–3–3–3.
同样逻辑延伸至七和弦：大七和弦在大三和弦上加一个大三度（R–4–3–4）；属七和弦加一个小三度（R–4–3–3）；小七和弦加一个小三度（R–3–4–3）；半减七和弦为 R–3–3–4；减七和弦为 R–3–3–3。
3. Circle of Fifths & Key Relationships | 五度圈与调性关系
The circle of fifths is a theorem-like diagram ordering keys by ascending perfect fifths (clockwise) and ascending fourths (anticlockwise). It predicts closely related keys: those adjacent in the circle share all but one scale degree; relative majors/minors are three steps apart on the circle.
五度圈是一个以纯五度上行（顺时针）和纯四度上行（逆时针）排列各调的定理式图表。它可预测近关系调：相邻调只差一个音级；关系大调和小调在圈上相距三步。
Formula for key signature: each clockwise step adds one sharp; each anticlockwise step adds one flat. Order of sharps: F♯ C♯ G♯ D♯ A♯ E♯ B♯. Order of flats: B♭ E♭ A♭ D♭ G♭ C♭ F♭.
调号公式：顺时针每一步增加一个升号；逆时针每一步增加一个降号。升号调顺序：F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号调顺序：B♭ E♭ A♭ D♭ G♭ C♭ F♭。
As a theorem, the circle also governs modulation distances: moving to a key 2 steps away on the circle (e.g., C major to D major) gives a bright, 'sharp' lift; moving to the subdominant side (C to F) gives a softer turn. These are 'stock' harmonic moves in A2 composition.
作为定理，五度圈还支配转调距离：在圈上移动两步（如 C 大调到 D 大调）会产生明亮、升高的转折；移到下属方向（C 到 F）则带来柔和的转折。这些是 A2 作曲中的“常用”和声动作。
4. Cadence Formulas: The Four Essential Ending Theorems | 终止式公式：四种基本结束定理
Cadences are harmonic punctuation marks. Each follows a fixed chord-pattern formula that signals closure or pause. In Roman numerals:
终止式是和声的标点符号。每一种都遵循固定的和弦模式公式，表示结束或停顿。用罗马数字表示：
| Cadence | 终止式 | Formula | 公式 | Effect | 效果 |
|---|---|---|
| Perfect (authentic) | V → I (both in root position, top voice ends on tonic) | Full stop, conclusive |
| 完美终止 | V → I（均为原位，高音部结束在主音） | 句号，确定 |
| Plagal | IV → I | 'Amen', gentler close |
| 变格终止 | IV → I | “阿门”，较柔和的结束 |
| Imperfect | I → V, ii → V, IV → V (etc.) | Comma, expects continuation |
| 不完全终止 | I → V、ii → V、IV → V 等 | 逗号，期待继续 |
| Interrupted (deceptive) | V → vi (or V → ♭VI in minor) | Surprise, deflection |
| 阻碍终止 | V → vi（或小调 V → ♭VI） | 意外，转向 |
Always check the soprano movement: a perfect cadence is only fully authentic when the leading note (7) resolves up to tonic (8) and the bass leaps V–I.
务必检查高音声部进行：完美终止仅在导音（7）上行解决到主音（8）且低音跳进 V–I 时才为完全正格终止。
5. Chord Function & Substitution Theorems | 和弦功能与替代定理
In functional harmony, chords are classified by their role: tonic (I, vi), subdominant (IV, ii), and dominant (V, vii°). Substitution is possible within each family because they share common tones.
在功能性和声中，和弦按角色分类：主功能（I、vi）、下属功能（IV、ii）和属功能（V、vii°）。因共同音的存在，家族内部可互相替代。
Tonic substitutes: vi for I (a deceptive motion); iii occasionally, but weak. Subdominant substitutes: ii for IV; the supertonic seventh (ii⁷) intensifies the subdominant colour. Dominant substitutes: vii° for V; often used in first inversion as a passing chord.
主功能替代：vi 代 I（阻碍进行）；iii 偶尔可用，但较弱。下属功能替代：ii 代 IV；上主七和弦（ii⁷）强化下属色彩。属功能替代：vii° 代 V；常以第一转位用作经过和弦。
The ii–V–I progression is a fundamental formula in tonal music, functioning like a mathematical axiom: ii (subdominant) moves to V (dominant) then resolves to I. In A2 chorales, extend it to vi–ii–V–I or I–vi–IV–V–I.
ii–V–I 进行是调性音乐的基本公式，犹如数学公理：ii（下属）进行到 V（属），再解决到 I。在 A2 众赞歌作业中，可扩展为 vi–ii–V–I 或 I–vi–IV–V–I。
6. Non-Harmony Notes: Identification Equations | 和弦外音：识别方程
Non-chord tones are like variables inserted into a harmonic equation. Each type has a specific metric and intervallic fingerprint: approached by step/leap, left by step/leap, and occurring on a strong or weak beat.
和弦外音犹如插入和声方程中的变量。每种类型都有特定的节拍和音程指纹：由级进还是跳进进入，由级进还是跳进离开，出现强拍还是弱拍。
| Type | 类型 | Approach | 进入 | Departure | 离开 | Metric position | 拍位 |
|---|---|---|---|
| Passing note | Step | Step (same direction) | Weak (unaccented) |
| 经过音 | 级进 | 级进（同向） | 弱拍（非强拍） |
| Neighbour note | Step | Step (opposite direction) | Weak, occasionally strong |
| 辅助音 | 级进 | 级进（反向） | 弱拍，偶有强拍 |
| Suspension | Same note (tied or repeated) | Step down (usually) | Strong (accented) |
| 延留音 | 同音（延音线或重复） | 下行级进（通常） | 强拍 |
| Appoggiatura | Leap | Step (opposite) | Strong |
| 倚音 | 跳进 | 级进（反向） | 强拍 |
| Escape tone | Step | Leap (opposite) | Weak |
| 逃逸音 | 级进 | 跳进（反向） | 弱拍 |
By mentally applying the 'approach + departure + beat' equation, you can quickly label every non-harmony note in a CCEA analysis question.
通过在脑中套用“进入方式 + 离开方式 + 拍位”方程式，你能迅速标记 CCEA 分析题中的每个和弦外音。
7. Voice Leading & Spacing Fundamental Laws | 声部进行与排列基本法则
In four-part chorale-style writing, treat voice leading as a set of inviolable laws (unless a specific exemption applies). These are the core formulas for smooth part writing:
在四部和声众赞歌式写作中，将声部进行视为一套不可违背的法则（除非有特定例外）。以下是平稳分部写作的核心公式：
- Law of the shortest path: keep common tones in the same voice; move other voices by the smallest possible step.
- 就近进行法则：共同音保持在同一声部；其余声部以尽可能最小的音程移动。
- No parallel perfect 5ths or 8ves: two voices moving in similar motion to a perfect fifth or octave is forbidden. Check between all six voice pairs.
- 禁止平行纯五、八度：两个声部以同向进行至纯五度或纯八度是被禁止的。检查所有六对声部组合。
- Spacing rule: between tenor and alto, maximum distance is an octave; between alto and soprano, also less than an octave. Bass and tenor may span beyond an octave.
- 排列规则：男高音与女中音之间最大距离为八度；女中音与女高音之间也不超过八度。男低音与男高音可超过八度。
- Leading note resolution: the 7th scale degree must resolve up to tonic when in an outer voice; in an inner voice, it may fall to the 5th if the tonic appears elsewhere.
- 导音解决：第七级音在外声部时必须上行解决到主音；在内声部时，若主音在其他声部出现，可下行至第五级。
Apply the 'common-tone-reduction' formula: when moving between chords, determine which notes are shared, hold them, and minimally move the rest. This automatically avoids most parallel error traps.
运用“共同音减省”公式：在不同和弦间移动时，确定哪些音是共有的，保持它们，然后最小幅度地移动其余声部。这样可自动避免多数平行错误陷阱。
8. Musical Form & Phrase Structure Formulae | 曲式与乐句结构公式
Musical forms can be expressed as structural equations. In CCEA A2, you must recognize and deploy several standard patterns:
音乐曲式可表示为结构方程式。在 CCEA A2 中，你必须识别并运用几种标准模式：
- Binary form: A (I → V or related key) ∷ B (modulating back to I). Equation: Statement → Contrast → Return via tonic.
- 二段体：A（I → V 或关系调）∷ B（转回 I）。等式：呈示 → 对比 → 通过主调返回。
- Ternary form (fully closed in tonic) → B (contrasting key/material) → A (exact or varied return). ABA.
- 三段体：A（完全结束在主调）→ B（对比调性/素材）→ A（精确或变化再现）。ABA。
- Sonata form 'theorem' Development (fragments, sequences, remote keys) → Recapitulation (all material in tonic).
- 奏鸣曲式“定理”：呈示部（第一主题主调，连接段转调，第二主题属调或关系大调）→ 展开部（片段、模进、远关系调）→ 再现部（所有素材回归主调）。
- Period structure: antecedent phrase (ends on V, imperfect cadence) + consequent phrase (ends on I, perfect cadence). A question-answer pair.
- 乐段结构：前句（结束在 V 级，不完全终止）+ 后句（结束在 I 级，完美终止）。一个问答对。
In analysis, treat these as templates. Overlay the bars onto the formula to spot divergences and label sections (exposition, transition, etc.).
在分析中，将这些视为模板。将小节套用到公式上，找出偏离之处并标记各部分（呈示部、连接段等）。
9. Rhythm & Metre: Subdivision Equations | 节奏与拍子：细分方程
Rhythm can be quantified through subdivision ratios. In simple time, a beat divides into 2 equal parts; in compound time, a beat divides into 3. The accent patterns produce distinctive time-signature formulas.
节奏可以通过细分比率量化。在单拍子中，一拍分为 2 等份；在复拍子中，一拍分为 3 等份。重音模式产生独特的拍号公式。
| Time signature | 拍号 | Beat subdivision | 拍子细分 | Strong-beat pattern | 强拍模式 |
|---|---|---|
| 2/4, 2/2 | Beat = ♩, divides into 2 ♪ | Strong–weak |
| 3/4 | Beat = ♩, divides into 2 ♪ | Strong–weak–weak |
| 4/4 | Beat = ♩, divides into 2 ♪ | S–w–s–w (secondary accent on beat 3) |
| 6/8 | Beat = ♩., divides into 3 ♪ | S–w (two dotted beats per bar) |
| 9/8 | Beat = ♩. | S–w–w (three dotted beats) |
| 12/8 | Beat = ♩. | S–w–s–w (four dotted beats) |
Syncopation formula: displace the natural accent by tying a weak beat/part to the following strong beat. A2 examples test recognition of cross-rhythms like 2 against 3 (hemiola) and regular syncopation.
切分音公式：通过将弱拍或弱部分延至下一个强拍，从而移置自然重音。A2 考题常测试对交叉节奏（如二对三，即赫米奥拉节奏）和常规切分的识别。
10. Melodic Construction Devices | 旋律构建手法
Melody can be generated using transformation 'functions'. The most frequently examined devices are sequence, imitation, inversion, and retrograde.
旋律可用转换“函数”生成。最常考的构建手法有模进、模仿、倒影和逆行。
- Sequence: a melodic pattern repeated at a higher or lower pitch. Equation: Motif A transposed by interval x. Real sequence keeps the interval exact; tonal sequence adjusts for the key.
- 模进：一个旋律型在高或低音区重复。方程式：动机 A 以 x 音程移位。严格模进保持精确音程；调性模进则调整以符合调性。
- Imitation: a voice repeats the melody of a preceding voice, like a canon at the unison/octave. Counterpoint formula: the second entry overlaps the first.
- 模仿：一个声部重复此前另一声部的旋律，如同度或八度卡农。对位公式：第二声部进入时与第一声部重叠。
- Inversion: interval directions are flipped. An ascending 3rd becomes a descending 3rd. Chromatic inversion may alter qualities.
- 倒影：音程方向翻转。上行三度变为下行三度。半音倒影可能改变音程性质。
- Retrograde: the melody is played backwards. Used in serial music; less common in tonal CCEA excerpts except in analyses of 20th-century works.
- 逆行：将旋律反向演奏。用于序列音乐；在 CCEA 调性音乐选段分析中较少见，除非是 20 世纪作品。
) → Verse → Chorus → Bridge (C) → Chorus → Outro. Recognise this as AABA, ABABCB variants.
对于 32 小节流行歌曲曲式，主歌-副歌结构可表达为：前奏 → 主歌（A）→ 副歌（B）→ 主歌 → 副歌 → 桥段（C）→ 副歌 → 尾奏。识别为 AABA、ABABCB 等变体。
11. 20th-Century Techniques Quick Reference | 20 世纪技法速查
For the CCEA compulsory set works and unfamiliar listening, you need rapid recall of non-tonal vocabulary. Treat each as a theorem departing from traditional harmonic rules.
对于 CCEA 必考作品和不熟悉听力，你需要迅速回忆非调性语汇。将每一种技法视为偏离传统和声规则的定理。
- Whole-tone scale: T–T–T–T–T–T (all steps a whole tone). Gives dominant 9th with raised 5th harmony.
- 全音音阶：全–全–全–全–全–全（每步均全音）。产生升五音的属九和弦音响。
- Octatonic (diminished) scale: alternating T–S or S–T. Formula: tone–semitone–tone–semitone… or semitone–tone… Produces symmetrical diminished chords.
- 八声音阶（减音阶）：全音半音交替。公式：全–半–全–半…… 或 半–全…… 产生对称的减和弦。
- Quartal harmony: chords built in stacked 4ths instead of 3rds. Often produces an open, indeterminate sound.
- 四度和声：和弦以叠置四度而非三度构建。常产生空旷、不确定的音响。
- Tone clusters: adjacent semitones played simultaneously; notated as a solid black bar or cluster markings.
- 音簇：相邻半音同时演奏；记谱为实心黑条或音簇标记。
- Bitonal/polytonal: two keys simultaneously, e.g., melody in C major over accompaniment in D♭ major. Formula: superimposition of two tonal functions.
- 二重调性/多调性：同时运用两个调，如旋律在 C 大调而伴奏在 D♭ 大调。公式：两种调性功能的叠加。
- Serialism: prime row (P) and its transformations: Retrograde (R), Inversion (I), Retrograde-Inversion (RI). All 12 pitches must sound before any repeats.
- 序列主义：原型序列（P）及其变形：逆行（R）、倒影（I）、逆行倒影（RI）。所有 12 个音高必须出现后才可以重复。
12. Quick-Check Equations for Analysis & Composing Tasks | 分析与作曲任务速验方程
Before finalising an analysis or composition answer, run through this checklist logic:
在完成分析或作曲答题前，按此清单逻辑快速检验：
- Key confirmation equation: check the last bass note (usually tonic), leading-note accidentals, and final cadence formula. If a minor piece ends with a raised 3rd, identify the Picardy third.
- 确认调性方程：检查最后一个低音（常为主音）、导音临时记号及最后终止式。若小调作品以升高三音结束，须认出辟卡迪三度。
- Part-writing sandbox: for each chord pair, scan for parallel 5ths/8ves, hidden 5ths/8ves (similar motion to perfect interval in outer voices), and check leading-note treatment.
- 分部写作检查区：对每对和弦，扫描平行五、八度，隐伏五、八度（外声部同向进行到纯音程），并检查导音处理。
- Harmonic rhythm mapping: count the rate of chord change (chords per bar). Rate changes often signal structural points (e.g., faster harmonic rhythm at a cadence approach).
- 和声节奏映射：计算和弦变换速率（每小节和弦数）。速率变化常标志结构点（如终止式前的和声节奏加快）。
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