📚 WJEC Year 13 Music: Quick Reference Formulas & Theorems | WJEC Year 13 音乐公式定理速查手册
In WJEC Year 13 Music, a deep understanding of the structural ‘formulas’ underlying harmony, melody, and rhythm is essential for analysis, composition, and aural perception. This quick-reference handbook presents the core theoretical principles as concise formulas and theorems, bridging the gap between mathematical precision and musical creativity. Whether you are preparing for the listening examination, refining your compositions, or strengthening your analytical writing, these formulas serve as a reliable foundation.
在 WJEC Year 13 音乐课程中，深刻理解潜藏于和声、旋律与节奏背后的结构性“公式”，是进行分析、作曲与听觉辨识的关键。这本速查手册将核心理论原理以简洁的公式与定理形式呈现，在数学般的精确与音乐创意之间搭建桥梁。无论你是在备考听力考试、打磨作曲，还是提升分析写作，这些公式都能成为你坚实可靠的基石。
1. Interval Semitone Formula | 音程半音公式
Every musical interval can be defined by the number of semitones it spans. In a chromatic scale system, the formula maps semitone count to interval name and quality. For any two given pitches, count the semitones from the lower note to the higher (including accidentals). Then apply the following mapping: 0 = perfect unison; 1 = minor 2nd; 2 = major 2nd; 3 = minor 3rd; 4 = major 3rd; 5 = perfect 4th; 6 = augmented 4th/diminished 5th; 7 = perfect 5th; 8 = minor 6th; 9 = major 6th; 10 = minor 7th; 11 = major 7th; 12 = perfect octave. Enharmonic intervals share the same semitone count.
每个音程都可以用所包含的半音数量来定义。在十二平均律体系中，该公式将半音数映射为音程名称与性质。对于任意两个给定的音高，从较低音向较高音（包含变音记号）数半音。然后对应如下映射：0 = 纯一度；1 = 小二度；2 = 大二度；3 = 小三度；4 = 大三度；5 = 纯四度；6 = 增四度/减五度；7 = 纯五度；8 = 小六度；9 = 大六度；10 = 小七度；11 = 大七度；12 = 纯八度。等音音程共享同一个半音数。
| Semitones | Interval Name | Example (C to …) |
|---|---|---|
| 0 | Perfect Unison | C – C |
| 4 | Major 3rd | C – E |
| 7 | Perfect 5th | C – G |
| 10 | Minor 7th | C – B♭ |
2. Major Scale Step Formula | 大调音阶步程公式
The construction of any major scale follows a fixed pattern of whole tones (W) and semitones (H). Starting on any tonic note, the sequence is: W – W – H – W – W – W – H. This yields the familiar diatonic major scale. In scale-degree terms, the semitones fall between degrees 3–4 and 7–8. For example, C major: C (W) D (W) E (H) F (W) G (W) A (W) B (H) C. This formula is the foundation of key signatures and diatonic harmony.
任何大调音阶的构建都遵循固定的全音（W）与半音（H）模式。从任意主音开始，序列为：全 – 全 – 半 – 全 – 全 – 全 – 半。这就产生了熟悉的自然大调音阶。用音阶级数表示，半音出现在第 3–4 级与第 7–8 级之间。例如 C 大调：C（全）D（全）E（半）F（全）G（全）A（全）B（半）C。这个公式是所有调号和自然音和声的基础。
Major Scale Formula: T – T – S – T – T – T – S (or W – W – H – W – W – W – H)
3. Minor Scale Formulae | 小调音阶公式
The minor mode has three important variants in common-practice harmony, each with its own step pattern. The natural minor scale follows the formula: W – H – W – W – H – W – W (or T – S – T – T – S – T – T), which is equivalent to the Aeolian mode. The harmonic minor scale raises the 7th degree by a semitone to create a leading tone; its step pattern between degrees 6–7 becomes an augmented 2nd. The melodic minor scale raises both the 6th and 7th degrees when ascending, and reverts to natural minor when descending. Ascending formula: W – H – W – W – W – W – H. These three forms are used for harmonic analysis, cadences, and melodic writing and must be recognised in scores.
小调在共性实践中拥有三种重要变体，每种都有各自的步程模式。自然小调公式为：全 – 半 – 全 – 全 – 半 – 全 – 全，等同于爱奥利亚调式。和声小调将第七级升高半音，以制造导音；其第 6–7 级之间形成增二度。旋律小调在上行时升高第六和第七级，下行时还原为自然小调。上行公式：全 – 半 – 全 – 全 – 全 – 全 – 半。这三种形式用于和声分析、终止式与旋律写作，并且必须在乐谱中识别出来。
| Type | Formula (Ascending) | A minor example |
|---|---|---|
| Natural minor | T–S–T–T–S–T–T | A-B-C-D-E-F-G-A |
| Harmonic minor | T–S–T–T–S–A2–S | A-B-C-D-E-F-G♯-A |
| Melodic minor (ascending) | T–S–T–T–T–T–S | A-B-C-D-E-F♯-G♯-A |
4. Triad Quality Theorem | 三和弦性质定理
The quality of a triad built on a given root is determined by the intervals above the root. A triad consists of a root, a third, and a fifth. The nature of the third (major 3rd = 4 semitones; minor 3rd = 3 semitones) and the fifth (perfect 5th = 7 semitones; diminished 5th = 6 semitones; augmented 5th = 8 semitones) sets the quality. The four basic triads are: major (major 3rd + perfect 5th), minor (minor 3rd + perfect 5th), diminished (minor 3rd + diminished 5th), and augmented (major 3rd + augmented 5th). This theorem allows you to identify or construct any triad quickly, regardless of context.
三和弦的性质由根音上方的音程决定。三和弦由根音、三音和五音构成。三音的性质（大三度=4 个半音；小三度=3 个半音）和五音的性质（纯五度=7 个半音；减五度=6 个半音；增五度=8 个半音）共同决定和弦性质。四种基本三和弦为：大三和弦（大三度+纯五度）、小三和弦（小三度+纯五度）、减三和弦（小三度+减五度）和增三和弦（大三度+增五度）。这一定理能让你在任何语境中快速识别或构建三和弦。
- Major: Root + 4 semitones + 7 semitones
- Minor: Root + 3 semitones + 7 semitones
- Diminished: Root + 3 semitones + 6 semitones
- Augmented: Root + 4 semitones + 8 semitones
5. Seventh Chord Formula Set | 七和弦公式集
Seventh chords add a diatonic (or chromatic) seventh above the root to the triad structure. The quality of the seventh and the underlying triad create the chord type. In dominant-function harmony, the following formulas are essential: Dominant 7th = major triad + minor 7th (root + 4 + 7 + 10 semitones); Major 7th = major triad + major 7th (4 + 7 + 11); Minor 7th = minor triad + minor 7th (3 + 7 + 10); Half-diminished 7th = diminished triad + minor 7th (3 + 6 + 10); Diminished 7th = diminished triad + diminished 7th (3 + 6 + 9). In WJEC Year 13, you will be expected to recognise these chords in score and analyse their function within tonal harmony.
七和弦在三和弦结构基础上方加上一个自然音（或变音）七度音。七度音的性质与下方三和弦共同决定和弦类型。在属功能性和声中，以下公式至关重要：大小七和弦（属七）= 大三和弦+小七度（根音+4+7+10 个半音）；大七和弦=大三和弦+大七度（4+7+11）；小七和弦=小三和弦+小七度（3+7+10）；半减七和弦=减三和弦+小七度（3+6+10）；减七和弦=减三和弦+减七度（3+6+9）。在 WJEC Year 13 考试中，你需要能在乐谱中辨认这些和弦，并在调性和声中分析它们的功能。
Dominant 7th structure: root → M3 → P5 → m7
6. Key Signature Theorem (Circle of Fifths) | 调号定理（五度圈）
Key signatures follow a cyclical order dictated by the circle of fifths. Sharp keys: each new sharp is added a perfect fifth above the previous, starting with F♯ (G major → D major → A major → E major → B major → F♯ major → C♯ major). The order of sharps is: F♯ C♯ G♯ D♯ A♯ E♯ B♯. Flat keys proceed by descending fifths: B♭, E♭, A♭, D♭, G♭, C♭, F♭. The order of flats is: B♭ E♭ A♭ D♭ G♭ C♭ F♭. To identify a major key from its sharp signature, go one semitone above the last sharp. For flat keys, the second-to-last flat names the major key (except F major, which has one flat). This theorem underpins all modulation and transposition tasks.
调号遵循由五度循环决定的周期性顺序。升号调：每一个新升号都在前一个上方纯五度添加，从 F♯ 开始（G 大调→D 大调→A 大调→E 大调→B 大调→F♯ 大调→C♯ 大调）。升号出现顺序为：F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号调以降序五度进行：B♭、E♭、A♭、D♭、G♭、C♭、F♭。降号顺序为：B♭ E♭ A♭ D♭ G♭ C♭ F♭。要从升号调号识别大调，看最后一个升号上方半音。对于降号调，倒数第二个降号即为该大调名称（F 大调除外，它仅有一个降号）。这一定理是所有转调与移调任务的基础。
Relative minor = Major tonic ↓ minor 3rd (three semitones down)
7. Chord Progression Strength Theorem | 和弦进行力度定理
In common-practice harmony, chords tend to move along specific root-movement patterns that create varying degrees of tension and resolution. The strongest progressions move by descending perfect fifth (or ascending perfect fourth), as in the cycle of fifths: iii → vi → ii → V → I. Movement by step (especially descending second) is also common and smooth. Movement by third is weaker but frequently used for colour. Ascending fifth (or descending fourth) progressions are often retrogressions and should be handled with care. In WJEC analysis, you must be able to label harmonic rhythm and recognise these functional patterns in both chorales and extended works.
在共性实践和声中，和弦倾向于沿着特定的根音进行模式移动，这些模式制造出不同程度的紧张与解决感。最强烈的进行是下行纯五度（或上行纯四度）的移动，正如五度圈序进：iii → vi → ii → V → I。级进（特别是下行二度）同样常见且平滑。三度进行力度较弱，但常用于增添色彩。上行五度（或下行四度）进行常被视为逆行，需要谨慎处理。在 WJEC 分析中，你必须能够标记和声节奏，并在众赞歌与大型作品中识别这些功能性模式。
- Strong progression: down a perfect 5th (e.g. V – I, ii – V)
- Common progression: down a 3rd (e.g. I – vi, IV – ii)
- Stepwise progression: ascending or descending 2nd (e.g. IV – V, V – vi)
8. Cadence Formula | 终止式公式
A cadence is a harmonic formula that marks the end of a phrase. Four principal cadences are defined by the chords preceding the final tonic (or substitute chord). Perfect cadence: V → I (or V⁷ → I), with both chords in root position and the tonic in the soprano for a perfect close. Imperfect cadence: any chord ending on V (e.g. I → V, ii → V). Plagal cadence: IV → I (the ‘Amen’ cadence). Interrupted (deceptive) cadence: V → vi (or V → ♭VI in minor keys). These formulas are vital for phrase analysis and for writing convincing chordal textures in composition tasks.
终止式是标志着乐句结束的和声公式。四种主要终止式由位于最后主和弦（或替代和弦）之前的和弦界定。完全终止：V → I（或 V⁷ → I），两个和弦均应为原位，且高声部结束在主音上以构成完满终止。不完全终止：任何结束在 V 级上的和弦序进（如 I → V、ii → V）。变格终止：IV → I（“阿们”终止）。阻碍终止：V → vi（或小调中的 V → ♭VI）。这些公式对于乐句分析以及在作曲任务中写出令人信服的和弦织体至关重要。
Perfect cadence: V(⁷) – I, both root position, tonic in soprano
9. Interval Inversion Rule | 音程转位规则
When an interval is inverted (the lower note is moved above the upper note or vice versa), its numeric value and quality transform according to a strict rule. The sum of the original interval number and the inverted interval number always equals 9. A perfect interval remains perfect. A major interval becomes minor, minor becomes major, augmented becomes diminished, and diminished becomes augmented. This rule is immensely useful for rapid calculation of compound intervals and for understanding the complementary nature of harmonic intervals in two-part counterpoint.
当音程转位时（将较低音移到较高音上方，或反之），其度数数字和性质将按严格规则变化。原音程度数与转位音程度数之和恒等于 9。纯音程转位后仍为纯音程。大音程变为小音程，小音程变为大音程，增音程变为减音程，减音程变为增音程。这一规则在快速计算复音程以及理解二声部对位中和声音程的互补特性时非常有用。
| Original | Inversion | Example (Original → Inverted) |
|---|---|---|
| M3 | m6 | C–E → E–C (m6) |
| P5 | P4 | C–G → G–C (P4) |
| Augmented 4th | Diminished 5th | C–F♯ → F♯–C (dim5) |
10. Roman Numeral Chord Labelling Theorem | 罗马数字和弦标记定理
In functional harmonic analysis, chords are labelled with Roman numerals that denote the scale degree of the root and the chord quality. Uppercase numerals indicate major or augmented triads; lowercase indicate minor or diminished triads. Chord inversions are indicated by adding figured bass symbols: ‘a’ for root position (often omitted), ‘b’ for first inversion, ‘c’ for second inversion (and ‘d’ for third inversion of seventh chords). For example, in C major: V = G major (root position); Vb = G major in first inversion (B in the bass); ii⁷c = D minor 7th in second inversion (A in the bass). This system allows precise identification of harmonic function across any tonal piece. In WJEC Year 13, you must apply this labelling fluently in both diatonic and chromatic contexts.
在功能性和声分析中，和弦使用罗马数字标记，以表示根音所在音级及和弦性质。大写数字表示大三或增三和弦；小写数字表示小三或减三和弦。和弦转位通过添加数字低音符号表示：’a’ 代表原位（常省略），’b’ 代表第一转位，’c’ 代表第二转位（七和弦的第三转位用 ‘d’ 表示）。例如，在 C 大调中：V = G 大三和弦原位；Vb = G 大三和弦第一转位（低音为 B）；ii⁷c = D 小七和弦第二转位（低音为 A）。这一体系使我们能够精确识别任何调性作品中和声的功能。在 WJEC Year 13 中，你必须能熟练地在自然音和半音语境中运用这种标记。
- Triads: I ii iii IV V vi vii° (major key); i ii° III iv V VI vii° (minor key, harmonic form influences V and vii°)
- Seventh chords: I⁷ ii⁷ iii⁷ IV⁷ V⁷ vi⁷ vii°⁷ (major); i⁷ ii°⁷ III⁷ iv⁷ V⁷ VI⁷ vii°⁷ (minor)
- Inversion symbols: a (root), b (1st inv.), c (2nd inv.), d (3rd inv.) – often subscripted or superscripted
11. Time Signature and Metre Formula | 拍号与节拍公式
Metre is governed by time signatures that indicate the number of beats per bar and the beat unit. The top number indicates how many beats; the bottom number indicates the note value that gets one beat (4 = crotchet, 8 = quaver, 2 = minim). Simple metres have a beat that divides into two equal parts (e.g. 2/4, 3/4, 4/4), while compound metres have a beat that divides into three (e.g. 6/8, 9/8, 12/8). The formula for determining total rhythmic value per bar is: (top number) × (1/bottom number) in terms of a semibreve (1 = semibreve). For aural dictation and score analysis, you must be able to identify whether a metre is simple or compound and accent patterns (duple, triple, quadruple). This theorem helps in rhythmic grouping and beaming rules.
节拍由拍号控制，拍号表示每小节拍数以及拍子单位。上方数字表示拍数；下方数字表示以几分音符为一拍（4=四分音符，8=八分音符，2=二分音符）。单拍子每一拍可分为两个等份（如 2/4、3/4、4/4），而复拍子每一拍可分为三个等份（如 6/8、9/8、12/8）。计算每小节总时值的公式为：（上方数字）×（1/下方数字），以全音符为 1。在听觉听写和乐谱分析中，你必须能够识别节拍是单拍子还是复拍子，以及重音模式（二拍子、三拍子、四拍子）。这一定理有助于节奏分组和符尾连接规则。
Simple metre beat division = 2; Compound metre beat division = 3
12. Modulation Pathway Formula | 转调路径公式
Modulation in tonal music commonly proceeds to closely related keys: those whose key signatures differ by no more than one accidental. From a major tonic, the closely related keys are: the dominant (V), the subdominant (IV), and the relative minor (vi) and its related keys (ii, iii). From a minor tonic: the relative major (III), the dominant minor (v) or major (V), the subdominant minor (iv) or major (IV), and the subtonic (VII). The pivot-chord formula states that modulation occurs via a chord that is diatonic to both the old and new keys. In analysis, label the chord with two Roman numerals connected by an equals sign or bracket, indicating its dual function. Being able to trace modulation pathways is vital for the WJEC score-analysis question and the extended essay.
调性音乐中的转调通常进行至近关系调：即调号相差不超过一个变音记号的调。从大调主音出发，近关系调为：属调（V）、下属调（IV）、关系小调（vi）及其关系调（ii、iii）。从小调主音出发：关系大调（III）、属小调（v）或属大调（V）、下属小调（iv）或下属大调（IV），以及下主音调（VII）。共同和弦转调公式表明，转调通过一个在两个调中均为自然音的和弦来实现。在分析中，用等号或括号连接两个罗马数字标记该和弦，表示其双重功能。能够追踪转调路径对于 WJEC 乐谱分析题与长篇论文至关重要。
Closely related keys: Tonic ± one accidental in key signature difference
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