📚 Year 11 Cambridge Music: Formula & Theorem Quick-Reference Handbook | 剑桥11年级音乐：公式定理速查手册
For Cambridge IGCSE or equivalent Year 11 Music students, the ability to recall and apply theoretical ‘formulae’ instantly is essential for both the listening and written papers. This handbook distils the core rules of Western tonal music—from interval construction and chord spelling to cadence identification and transposition—into a compact, exam-ready reference. Use it to check your grasp of the building blocks, sharpen your analysis skills, and ensure you never lose marks for avoidable errors.
对修读剑桥IGCSE或同等程度的11年级音乐学生而言，能否瞬间调取并运用理论“公式”，直接关系到听力卷与笔试的表现。这本手册将西方调性音乐的核心法则——从音程构建、和弦拼写到终止式识别与移调——浓缩为一本临考速查手册，帮助你检视基本功、强化分析能力，并杜绝因疏忽丢分的可能。
1. The Chromatic Formula: Semitone & Whole-Tone Map | 半音与全音地图
The chromatic scale is the raw material of all intervals and chords. Memorise the pattern of half steps (H) and whole steps (W) in various scales, starting with the chromatic alteration of each natural note.
半音阶是所有音程与和弦的原材料。请牢记不同音阶中半音(H)与全音(W)的排列规律，并从每个自然音的升降变化入手。
- Chromatic scale ascending (sharps): C–C♯–D–D♯–E–F–F♯–G–G♯–A–A♯–B–C
上行半音阶（升号）：C–C♯–D–D♯–E–F–F♯–G–G♯–A–A♯–B–C - Chromatic scale ascending (flats): C–D♭–D–E♭–E–F–G♭–G–A♭–A–B♭–B–C
上行半音阶（降号）：C–D♭–D–E♭–E–F–G♭–G–A♭–A–B♭–B–C - 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 scale formula: W–H–W–W–H–W–W
自然小调音阶公式：全–半–全–全–半–全–全 - Harmonic minor scale: raise the 7th degree by a semitone → augmented 2nd between 6th and 7th.
和声小调：将第七级音升高半音 → 第六级与第七级之间形成增二度。 - Melodic minor (ascending): raised 6th and 7th → W–H–W–W–W–W–H; descending = natural minor.
旋律小调（上行）：升高第六、七级 → 全–半–全–全–全–全–半；下行还原为自然小调。
2. Interval Formula: Counting Half Steps and Quality | 音程公式：以半音数和性质定名
Every interval is defined by two components: the numerical distance (2nd, 3rd, 4th, etc.) and the quality (perfect, major, minor, augmented, diminished). The quality is determined by the exact number of half steps (semitones) between the notes.
任何一个音程都由两个要素定义：级数距离（二度、三度、四度等）和性质（纯、大、小、增、减）。性质由两音之间精确的半音数决定。
| Interval 音程 | Half steps 半音数 | Example from C 自C起示例 |
|---|---|---|
| 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 |
| Augmented 4th / Diminished 5th (tritone) 增四度/减五度 | 6 | C–F♯ / C–G♭ |
| 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 (higher) |
Inversion rule: Major intervals invert to minor (and vice versa); perfect intervals remain perfect; augmented ↔ diminished. The sum of half steps always equals 12.
转位规则：大音程转位后变小音程，反之亦然；纯音程转位后仍为纯；增 ↔ 减相互转位。半音数之和恒为12。
3. Triad Construction Theorem | 三和弦构成定理
A triad is built by stacking two thirds (root–3rd, 3rd–5th). The quality—major, minor, diminished, or augmented—follows a fixed pattern of half steps between the chord members.
三和弦由两个三度音程叠置而成（根音–三音，三音–五音）。其性质（大、小、减、增）由和弦音之间的半音数固定模式决定。
| Triad type 三和弦类型 | Root–3rd 根音–三音 | 3rd–5th 三音–五音 | Outer frame 外框架 | Example 示例 |
|---|---|---|---|---|
| Major 大三和弦 | Major 3rd (4 h) | minor 3rd (3 h) | Perfect 5th (7 h) | C–E–G |
| Minor 小三和弦 | minor 3rd (3 h) | Major 3rd (4 h) | Perfect 5th (7 h) | C–E♭–G |
| Diminished 减三和弦 | minor 3rd (3 h) | minor 3rd (3 h) | Diminished 5th (6 h) | C–E♭–G♭ |
| Augmented 增三和弦 | Major 3rd (4 h) | Major 3rd (4 h) | Augmented 5th (8 h) | C–E–G♯ |
The diatonic triads in a major key follow the pattern: I (major), ii (minor), iii (minor), IV (major), V (major), vi (minor), vii° (diminished).
大调内的自然三和弦序列为：I（大）、ii（小）、iii（小）、IV（大）、V（大）、vi（小）、vii°（减）。
4. Seventh Chord Formulae | 七和弦公式
Seventh chords add a fourth pitch above the root, forming a triadic stack with a seventh. The common types in tonal music are:
七和弦在根音上方再加入第四个音，构成三和弦之上再叠置一个七度音。调性音乐中常见的七和弦类型如下：
- Major 7th 大七和弦： Major triad + Major 7th (11 h above root). Formula: M3+m3+M3. Cmaj7: C–E–G–B.
- Dominant 7th 属七和弦： Major triad + minor 7th (10 h). Formula: M3+m3+m3. C7: C–E–G–B♭.
- Minor 7th 小七和弦： Minor triad + minor 7th. Formula: m3+M3+m3. Cm7: C–E♭–G–B♭.
- Half-diminished 7th (ø7) 半减七和弦： Diminished triad + minor 7th. Formula: m3+m3+M3. Cø7: C–E♭–G♭–B♭.
- Diminished 7th 减七和弦： Diminished triad + diminished 7th (9 h). Formula: m3+m3+m3. C°7: C–E♭–G♭–B♭♭ (B double flat, enharmonically A).
In inversions, each 7th chord retains its quality; the figured bass notation (7, 65, 43, 42) identifies the bass note and intervals above it.
转位时，七和弦性质不变；数字低音标记（7、65、43、42）标示了低音与其上方音的音程关系。
5. Circle of Fifths and Key Signatures | 五度圈与调号定理
The circle of fifths is the visual formula linking key signatures, relative minors, and the order of sharps and flats. Moving clockwise adds sharps (or removes flats); anticlockwise adds flats.
五度圈是将调号、关系小调以及升/降号出现顺序联系起来的视觉公式。顺时针方向增加升号（或减少降号）；逆时针方向增加降号。
| Major key 大调 | Key signature 调号 | Relative minor 关系小调 |
|---|---|---|
| C major | 0 ♯/♭ | A minor |
| G major | 1 ♯ (F♯) | E minor |
| D major | 2 ♯ (F♯, C♯) | B minor |
| A major | 3 ♯ (F♯, C♯, G♯) | F♯ minor |
| E major | 4 ♯ (F♯, C♯, G♯, D♯) | C♯ minor |
| B major | 5 ♯ (F♯, C♯, G♯, D♯, A♯) | G♯ minor |
| F♯ major | 6 ♯ | D♯ minor |
| F major | 1 ♭ (B♭) | D minor |
| B♭ major | 2 ♭ (B♭, E♭) | G minor |
| E♭ major | 3 ♭ (B♭, E♭, A♭) | C minor |
| A♭ major | 4 ♭ (B♭, E♭, A♭, D♭) | F minor |
| D♭ major | 5 ♭ | B♭ minor |
Order of sharps (mnemonic): Father Charles Goes Down And Ends Battle – F♯ C♯ G♯ D♯ A♯ E♯ B♯. Order of flats: Battle Ends And Down Goes Charles Father – B♭ E♭ A♭ D♭ G♭ C♭ F♭ (reverse of sharps).
升号顺序（记忆口诀）：Father Charles Goes Down And Ends Battle – F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号顺序：Battle Ends And Down Goes Charles Father – B♭ E♭ A♭ D♭ G♭ C♭ F♭（升号的反向）。
6. Cadence Formulae: The Four Pillars of Phrase Endings | 终止式公式：乐句收束的四大支柱
Cadences are the punctuation of music. Each cadence type has a distinctive harmonic move that defines a phrase’s sense of closure or forward motion.
终止式是音乐里的标点符号。每种终止式都有标志性的和声进行，决定乐句的收束感或继续推进的趋势。
- Perfect cadence (V–I) 正格终止： dominant to tonic, both chords in root position, with the tonic in the soprano of the final chord. Sounds conclusive.
属和弦到主和弦，两和弦皆为原位，且最后主和弦最高音为根音。听感完满终止。 - Imperfect cadence (I/ii/IV–V) 半终止： any chord moving to V, giving an unfinished, expectant feeling. Often found at phrase midpoints.
任何和弦进入属和弦，带来未完成、期待感，常见于乐句中点。 - Plagal cadence (IV–I) 变格终止： subdominant to tonic; the “Amen” cadence. Both chords usually in root position.
下属和弦到主和弦；“阿门”终止。两和弦通常为原位。 - Interrupted (deceptive) cadence (V–vi) 阻碍终止： dominant to submediant (vi) instead of I. Surprising, prolongs the phrase.
属和弦到六级和弦(vi)而非主和弦，产生意外感，延长乐句。
In minor keys, remember that the leading note (raised 7th) is needed for a true dominant function. A V–vi cadence in minor is V–♭VI, still considered interrupted.
在小调中，需注意导音（升高第七级）才能构成真正的属功能。小调中的V–vi为V–♭VI，仍视为阻碍终止。
7. Transposition Rules | 移调法则
Transposition is the process of moving a melody or chord progression to a new pitch level while preserving the exact interval relationships. For exam melody-writing or score reading, the formula is simple: determine the interval of transposition, then apply that interval consistently to every note and adjust accidentals to fit the new key signature.
移调是将旋律或和弦进行移到新的音高，同时保持精确的音程关系。在考试中的旋律写作或读谱环节，公式很简单：确定移调音程，然后将该音程始终应用于每个音符，并根据新调号调整临时记号。
- Steps 步骤： (1) Identify the original key and the target key. (2) Work out the interval between the original tonic and the target tonic (e.g. up a perfect 4th). (3) Transpose each note by that interval. (4) Adjust for new key signature accidentals and any chromatic alterations.
(1) 识别原调与目标调。(2) 计算原调主音与目标调主音之间的音程（例如向上纯四度）。(3) 按照该音程移动每个音符。(4) 根据新调号调整临时记号及半音变化。 - Enharmonic transposition 等音移调： If the new key is enharmonically equivalent (e.g. C♯ major → D♭ major), simply re-spell, preserving pitch but changing notation.
若新调为等音调（如C♯大调→D♭大调），仅重写记谱，保持音高不变。 - Transposing instruments 移调乐器： For B♭ = major 2nd higher. For F horn: sounds a perfect 5th lower; written = perfect 5th higher.
B♭单簧管：实际音高比记谱低大二度；记谱比实际音高高大二度。F调圆号：实际音高低纯五度；记谱高纯五度。
8. Time Signature and Metre Rules | 拍号与节拍规则
Time signatures tell us how many beats per bar and which note value represents one beat. They also imply accent patterns and the division of beats.
拍号告诉我们每小节有几拍、以何种音符为一拍，同时提示重音规律与拍子分割方式。
- Simple time 单拍子： triple, quadruple). Bottom number: 4 = crotchet beat, 8 = quaver beat. E.g. 2/4, 3/4, 4/4.
- Compound time 复拍子： each beat divides into three equal parts. Top number: 6, 9, or 12. Bottom number: usually 8 (dotted crotchet beat). E.g. 6/8 (two dotted crotchet beats), 9/8 (three), 12/8 (four).
- Asymmetrical metres 不对称节拍： irregular groupings, e.g. 5/4 (3+2 or 2+3), 7/8 (3+2+2 etc.).
- Anacrusis 弱起： an upbeat; the incomplete first bar must be balanced by an incomplete final bar so total number of beats equals the metre.
top number ÷ 3. Each beat is a dotted note. For 6/8, the primary accent pattern is strong–weak, with secondary accents on the first of each dotted crotchet group.
复拍子公式：拍数 = 上方数字 ÷ 3。每拍是一个附点音符。6/8的重音模式为强–弱，每组的第一个附点四分音符带有次重音。
9. Chord Progression Voice-Leading Rules | 和弦进行声部进行法则
Smooth voice leading is governed by a set of “laws” that minimise leaps and avoid forbidden parallels. These rules are central to both chorale-style writing and 4-part harmony exam questions.
平稳的声部进行由一系列“法则”支配，旨在最小化跳进并避免禁用的平行进行。这些规则是众赞歌式写作与四部和声考题的核心。
- Keep common tones 保留共同音： When two chords share a pitch, keep that pitch in the same voice.
- Stepwise motion 级进优先： Move each voice the smallest possible distance; contrary motion is preferred between outer voices.
- No parallel perfect 5ths or octaves 禁用平行纯五/八度： Two voices moving from a perfect 5th to another perfect 5th, or an octave to another octave, by similar motion is forbidden.
- Hidden 5ths/octaves 隐伏五/八度： Approaching a perfect 5th or octave by similar motion when the soprano leaps is also to be avoided where possible.
- Leading note resolution 导音解决： The 7th degree (leading note) in a major/minor key must rise to the tonic when it occurs in an outer voice (soprano/bass) unless part of a descending scale.
- Doubling rules 重复音规则： In root-position triads, double the root; in first inversion, double the soprano note (often the 3rd or root); avoid doubling the leading note or the 3rd of a major chord in root position if it weakens the tonality.
10. Ornamentation and Expressive Indications | 装饰音与表情记号速查
Ornaments and expression marks are a shorthand system. Know the symbol, its execution (usually in terms of principal note and auxiliary notes), and its effect on phrasing.
装饰音与表情记号是一套速记系统。应了解其符号、演奏方式（通常涉及主要音与辅助音）以及对乐句表情的影响。
| Ornament 装饰音 | Symbol example 符号示例 | Execution 演奏方式 |
|---|---|---|
| Trill (shake) 颤音 | tr | Rapid alternation between principal note and the note above (diatonic unless modified by an accidental). |
| Mordent 波音 | upper mordent: principal–upper–principal; lower mordent (with vertical line): principal–lower–principal. | Played quickly before the beat or on the beat, depending on style. |
| Turn 回音 | curly S-shape | Upper auxiliary–principal–lower auxiliary–principal; often placed between notes or directly above a note. |
| Appoggiatura 倚音 | Small note, usually slurred, taking half the value of the main note (or two-thirds in triple time). | On the beat, dissonant, resolves by step. |
| Acciaccatura 短倚音 | Small note with a stroke through the stem. | Crushed in before the beat, as short as possible. |
Dynamics: pp (pianissimo), p (piano), mp (mezzo-piano), mf (mezzo-forte), f (forte), ff (fortissimo). Gradual changes: cresc. (crescendo) or hairpin <, dim. (diminuendo) or hairpin >.
力度记号：pp（很弱）、p（弱）、mp（中弱）、mf（中强）、f（强）、ff（很强）。渐强用cresc.或<，渐弱用dim.或>。
11. Modulation Map: Pivot Chords and Common Tones | 转调地图：共同和弦与共同音
Modulation is the process of changing key within a piece. The smoothest method involves a pivot chord—a chord that belongs to both the old key and the new key—followed by a cadence in the new key.
转调是作品中变换调性的过程。最平顺的方法是通过共同和弦——一个同时属于原调与新调的和弦——然后在新调上完结终止式。
- Closely related keys 近关系调： keys whose tonics are a perfect 5th apart, or relative major/minor. They differ by no more than one sharp or flat in the key signature.
- Pivot chord formula 共同和弦公式： In C major modulating to G major, the chord vi (A minor) in C is ii in G; IV in C (F major) is ♭III in G. Choose a chord that functions as a weaker pre-dominant in the new key.
- Common-tone modulation 共同音转调： Sustain one note across the key change, often the dominant or tonic, then reinterpret it as a member of the new key’s dominant 7th or tonic.
- Chromatic modulation 半音转调： Use a chromatic alteration to pivot into distant keys. For instance, F–A–C–E♭ (
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