📚 GCSE CIE Music: Formulae & Theorems Quick Reference | GCSE CIE 音乐：公式定理速查手册
In music, ‘formulae’ and ‘theorems’ refer not to mathematical equations, but to the essential rules, patterns and structures that underpin melody, harmony and rhythm. This quick-reference guide brings together the key theoretical building blocks you need for the GCSE CIE Music examination, covering scales, chords, cadences, intervals and more. Think of each section as a proven recipe or theorem—once learned, they can be applied across any piece you analyse or compose.
在音乐中，“公式”与“定理”指的并非数学方程，而是支撑旋律、和声与节奏的基本规则、模式与结构。这份速查手册汇集了 GCSE CIE 音乐考试所需的关键理论构成要素，涵盖音阶、和弦、终止式、音程等内容。你可以将每一小节视为经过验证的配方或定理——一旦掌握，便可应用于你分析或创作的任何作品。
1. Scale Formulae | 音阶公式
A scale is a series of notes ordered by pitch. The formula is expressed as a sequence of tones (T) and semitones (S).
音阶是按音高排列的一系列音符。其公式以全音（T）和半音（S）的序列表示。
Major scale: T T S T T T S (e.g. C–D–T–S pattern).
大调音阶：全 全 半 全 全 全 半（例如 C–D–E–F–G–A–B–C）。
Natural minor scale: T S T T S T T.
自然小调音阶：全 半 全 全 半 全 全。
Harmonic minor scale: T S T T S augmented 2nd S (e.g. A B C D E F G♯ A). The raised 7th degree creates a distinctive gap of three semitones between the 6th and 7th notes.
和声小调音阶：全 半 全 全 半 增二度 半（例如 A B C D E F G♯ A）。升高第7级音使第6与第7级音之间形成一个独特的三半音缺口。
Melodic minor scale (ascending): T S T T T T S (raised 6th and 7th). Descending follows the natural minor: T T S T T S T.
旋律小调音阶（上行）：全 半 全 全 全 全 半（升高第6、第7级）。下行回归自然小调：全 全 半 全 全 半 全。
Pentatonic major: T T minor 3rd T minor 3rd. (e.g. C D E G A C)
大调五声音阶：全 全 小三度 全 小三度。（例如 C D E G A C）
Blues scale: minor 3rd, T, S, S, minor 3rd, T (e.g. C E♭ F G♭ G B♭ C).
布鲁斯音阶：小三度 全 半 半 小三度 全（例如 C E♭ F G♭ G B♭ C）。
2. Interval Theorems & Inversions | 音程定理与转位
An interval is the distance between two notes, measured in semitones. The quality (major, minor, perfect, augmented, diminished) follows strict rules.
音程是两个音符之间的距离，以半音为单位测量。其性质（大、小、纯、增、减）遵循严格的规则。
Interval inversion rule: When an interval is inverted (lower note moved up an octave), the new interval number always sums to 9. A major interval becomes minor, augmented becomes diminished, and perfect remains perfect.
音程转位规则：当音程转位时（低音移高八度），新的音程度数总和恒为9。大音程变为小音程，增音程变为减音程，纯音程保持为纯音程。
| Interval (degrees) | Semitones | Inversion rule |
|---|---|---|
| Perfect unison (1) | 0 | Octave (8) – perfect |
| Major 2nd | 2 | Minor 7th |
| Major 3rd | 4 | Minor 6th |
| Perfect 4th | 5 | Perfect 5th |
| Perfect 5th | 7 | Perfect 4th |
| Major 6th | 9 | Minor 3rd |
| Major 7th | 11 | Minor 2nd |
| Perfect octave (8) | 12 | Perfect unison |
Augmented 4th (tritone, 6 semitones) is its own inversion, remaining an augmented 4th or diminished 5th.
增四度（三全音，6个半音）是自身的转位，仍为增四度或减五度。
3. Chord Construction Formulae | 和弦构成公式
Chords are built by stacking intervals above a root. The formula gives the interval structure.
和弦通过在根音之上堆叠音程构成。公式给出其音程结构。
Major triad: Root + major 3rd (4 semitones) + perfect 5th (7 semitones). Example: C–E–G.
大三和弦：根音 + 大三度（4个半音）+ 纯五度（7个半音）。示例：C–E–G。
Minor triad: Root + minor 3rd (3 semitones) + perfect 5th. Example: A–C–E.
小三和弦：根音 + 小三度（3个半音）+ 纯五度。示例：A–C–E。
Diminished triad: Root + minor 3rd + diminished 5th (6 semitones). Example: B–D–F.
减三和弦：根音 + 小三度 + 减五度（6个半音）。示例：B–D–F。
Augmented triad: Root + major 3rd + augmented 5th (8 semitones). Example: C–E–G♯.
增三和弦：根音 + 大三度 + 增五度（8个半音）。示例：C–E–G♯。
Dominant 7th: Major triad + minor 7th (10 semitones above root). Used on chord V. Example: G–B–D–F.
属七和弦：大三和弦 + 小七度（根音上方10个半音）。用于V级和弦。示例：G–B–D–F。
Major 7th: Major triad + major 7th (11 semitones).
大七和弦：大三和弦 + 大七度（11个半音）。
Seventh chords in minor keys often use the harmonic minor scale, creating a diminished 7th chord on the raised 7th degree (e.g. G♯–B–D–F in A minor).
小调中的七和弦常使用和声小调音阶，在升高的第7级上形成减七和弦（例如a小调中的G♯–B–D–F）。
4. Chord Progressions & Cadence Theorems | 和弦进行与终止式定理
In tonal music, chord progressions follow functional patterns. Cadences are the punctuation marks at phrase ends.
在调性音乐中，和弦进行遵循功能模式。终止式是乐句末尾的标点符号。
Perfect cadence (V–I): Dominant to tonic. Sounds complete and final.
正格终止（完全终止，V–I）：属和弦到主和弦。听感圆满、终结。
Plagal cadence (IV–I): Subdominant to tonic. ‘Amen’ cadence.
变格终止（IV–I）：下属和弦到主和弦。“阿门”终止。
Imperfect cadence (I/V/IV–V): Ending on V, sounds unfinished. Often I–V, ii–V, IV–V.
不完全终止（以V结尾）：停留在属和弦上，听感未结束。常见I–V、ii–V、IV–V。
Interrupted cadence (V–vi): V moves to the submediant (vi) instead of I. A surprise.
阻碍终止（V–vi）：V接往下中音和弦（vi）而非I，营造意外感。
Common progression theorems: I–IV–V–I; I–vi–IV–V; ii–V–I (circle progression). In minor keys: i–iv–V–i.
常见进行定理：I–IV–V–I; I–vi–IV–V; ii–V–I（循环进行）。小调中：i–iv–V–i。
The leading note (7th degree) resolves up by step to the tonic; chord V⁷–I resolution involves the falling 7th and rising leading note.
导音（第7级）上行级进解决到主音；V⁷–I解决包含下行七度音和上行导音。
5. Key Signature Identification Theorem | 调号识别定理
The circle of fifths governs key signatures. Sharps are added in the order FCGDAEB; flats in the reverse order BEADGCF.
五度圈支配调号。升号添加顺序为FCGDAEB；降号顺序为BEADGCF。
Major keys with sharps: To find the tonic from the last sharp, go up a semitone. E.g. F♯, C♯, G♯ → last sharp is G♯, up a semitone = A major.
升号大调：从最后一个升号上行一个半音即为主音。例如F♯、C♯、G♯ → 最后一个升号为G♯，上行半音得A大调。
Major keys with flats: The second-to-last flat is the tonic. E.g. B♭, E♭, A♭ → second-to-last flat is E♭ major. For F major (one flat), just memorise.
降号大调：倒数第二个降号即为主音。例如B♭、E♭、A♭ → 倒数第二个为E♭大调。F大调（一个降号）单独记忆。
Relative minors: The relative minor key shares the same key signature. To find the relative minor, go down three semitones from the major tonic (e.g. C major → A minor).
关系小调：关系小调共享同一调号。从大调主音下行三个半音即得（如C大调 → a小调）。
Application: This theorem is essential for score reading and transposition at sight.
应用：该定理对于读谱和视奏移调至关重要。
6. Transposition Formula | 移调公式
Transposition moves every note by the same interval, maintaining the original melody’s shape.
移调通过相同的音程移动每个音符，保持原旋律形态。
Formula: New pitch = original pitch + interval of transposition. To transpose from clarinet in B♭ (sounding a major 2nd lower) to concert pitch, you must write the part a major 2nd higher than the desired sound.
公式：新音高 = 原音高 + 移调音程。将降B调单簧管（发音低大二度）乐谱移为实际音高时，需将记谱音高大二度。
Common transposing instruments: Trumpet/Clarinet in B♭ (up a major 2nd written), French horn in F (up a perfect 5th written), Alto sax in E♭ (up a major 6th written).
常见移调乐器：降B调小号/单簧管（记谱高大二度），F调圆号（记谱高纯五度），降E调中音萨克斯（记谱高大六度）。
When transposing a key signature, apply the cycle of fifths accordingly: if moving up a major 2nd, add two sharps or remove two flats.
移调调号时，相应运用五度圈：若上行大二度，则加两个升号或去两个降号。
7. Melodic Development Theorems | 旋律发展定理
Composers use specific devices to transform a motif. These can be analysed as operations.
作曲家使用特定手法发展动机。这些可被视作操作分析。
Sequence (real and tonal): Immediate repetition at a higher or lower pitch. Real sequence transposes exactly; tonal sequence stays within the key.
模进（完全模进和调内模进）：在更高或更低音高上立即重复。完全模进精确移调；调内模进保持在原调内。
Inversion: Turning the melody upside down – ascending intervals become descending and vice versa.
倒影：将旋律翻转——上行音程变为下行，反之亦然。
Retrograde: Playing the melody backwards.
逆行：将旋律反向演奏。
Augmentation / Diminution: Lengthening or shortening the note values proportionally.
增值 / 减值：按比例延长或缩短音符时值。
Ostinato: A persistently repeated pattern, often rhythmic or melodic.
固定音型：持续重复的模式，常为节奏型或旋律型。
These devices are frequently examined in CIE Music when analysing set works and composition.
这些手法在CIE音乐分析固定作品和创作中经常考查。
8. Rhythm & Metre Equations | 节奏与节拍公式
Metre indicates how beats are grouped. The time signature acts as a mathematical equation: top number = beats per bar; bottom number = type of beat (4 = crotchet, 8 = quaver, etc.).
节拍表明拍子的分组方式。拍号如同一个数学等式：上方数字 = 每小节拍数；下方数字 = 以何种音符为一拍（4 = 四分音符，8 = 八分音符等）。
Simple time: Each beat divides into two equal parts. e.g. 2/4, 3/4, 4/4.
单拍子：每一拍可均分为两部分。例如2/4、3/4、4/4。
Compound time: Each beat divides into three equal parts. e.g. 6/8 (two dotted crotchet beats), 9/8 (three), 12/8 (four).
复拍子：每一拍可均分为三部分。例如6/8（两个附点四分音符拍）、9/8（三个）、12/8（四个）。
Syncopation theorem: Emphasis on a weak beat or off-beat, often using ties or accents. Regular syncopation alters the perceived metre.
切分音定理：强拍位置转移到弱拍或拍子之间，常通过连线或重音实现。规律性切分会改变感知的节拍。
Hemiola: A temporary shift from 6/8 (two groups of three) to 3/4 (three groups of two), typical in Baroque music.
赫米奥拉比率：在6/8（两组三拍）中短暂转换为3/4（三组二拍）的感知，常见于巴洛克音乐。
9. Musical Structure Theorems | 音乐结构定理
Large-scale forms follow prescriptive patterns. Recognising them helps map out a piece.
大型曲式遵循规定模式。识别它们有助于勾勒作品结构。
Binary form (AB): Two contrasting sections, each usually repeated. ||: A :||: B :||
二段体（AB）：两个对比乐段，每段通常反复。||: A :||: B :||
Ternary form (ABA): A return to the opening material after a contrasting B section. Often in minuet and trio.
三段体（ABA）：在对比B段后回归开头的素材。常见于小步舞曲与中段。
Rondo form (ABACA or ABACABA): A principal theme (A) alternates with contrasting episodes.
回旋曲式（ABACA或ABACABA）：主部主题（A）与对比插部交替出现。
Sonata form (exposition–development–recapitulation): Exposition presents two subjects in contrasting keys, development manipulates motives, recapitulation restates both subjects in the tonic key.
奏鸣曲式（呈示部–展开部–再现部）：呈示部在对比调性上呈现两个主题，展开部发展动机，再现部在主调上重新陈述两个主题。
Theme and variations: A theme followed by altered versions (melodic, harmonic, rhythmic variations).
主题与变奏：主题之后跟随变化的版本（旋律、和声、节奏变奏）。
Each structure has a predictable harmonic trajectory that aids in exam analysis.
每种结构都有可预测的和声轨迹，有助于考试分析。
10. Texture Classification Rules | 织体分类规则
Texture describes how many layers of sound are heard and how they relate.
织体描述听觉中有多少声音层次及其相互关系。
Monophonic: A single melodic line, unaccompanied. E.g. a solo chant or unison singing.
单声织体：单旋律线条，无伴奏。例如素歌独唱或齐唱。
Homophonic: Melody with chordal accompaniment; all parts move in similar rhythm. Most hymns and pop songs are homophonic.
主调织体：旋律配以和弦伴奏；所有声部节奏相似。多数赞美诗与流行歌曲为主调织体。
Polyphonic (contrapuntal): Two or more independent melodic lines woven together. Canon, fugue, and imitative counterpoint are polyphonic textures.
复调织体（对位织体）：两个或以上独立旋律线条交织。卡农、赋格与模仿对位属于复调织体。
Heterophonic: Simultaneous variations of a single melody, common in some world music traditions.
支声织体：同一旋律的同时变体，常见于某些世界音乐传统。
Melody-dominated homophony: Clear prominent melody supported by subordinate accompaniment, typical in classical and romantic music.
旋律支配的主调织体：明确的突出旋律由从属伴奏支撑，典型于古典与浪漫时期音乐。
11. Dynamics & Expressive Terminology Quick Theorems | 力度与表情术语速查定理
Dynamics and expression marks are integral to performing and analysing music. These Italian terms follow a consistent hierarchy.
力度与表情记号对于演奏和分析音乐不可或缺。这些意大利术语遵循一致的层级。
| Term | Meaning | ‘Theorem’ / usage |
|---|---|---|
| pp (pianissimo) | very soft | softer than p |
| p (piano) | soft | baseline quiet |
| mp (mezzo piano) | moderately soft | between p and mf |
| mf (mezzo forte) | moderately loud | default dynamic in many contexts |
| f (forte) | loud | stronger emphasis |
| ff (fortissimo) | very loud | climactic sections |
| cresc. (crescendo) | gradually getting louder | shown as hairpin or text |
| dim. (diminuendo) | gradually getting softer | opposite of crescendo |
Tempo terms: Allegro (fast), Andante (walking pace), Adagio (slow), Accelerando (getting faster), Rallentando (getting slower).
速度术语：快板（Allegro）、行板（Andante）、柔板（Adagio）、渐快（Accelerando）、渐慢（Rallentando）。
Articulation theorems: staccato (detached), legato (smooth), accent (>) marks a stressed attack.
演奏法定理：断奏（staccato，断开）、连奏（legato，圆滑）、重音记号（>）表示强调起奏。
12. Ornaments & Embellishment Rules | 装饰音与润饰规则
Ornaments are small decorative notes that follow precise performance conventions. In CIE music, recognizing and interpreting them correctly is essential.
装饰音是遵循精确演奏法的小型装饰音符。在CIE音乐中，正确识别与诠释它们至关重要。
Trill (tr): Rapid alternation between the written note and the note above. start on the upper note in Baroque/Classical, often on the main note in romantic music.
颤音（tr）：本音与上方音快速交替。巴洛克/古典时期通常从上方音开始，浪漫时期常从主音开始。
Turn: Four-note pattern: upper note, main note, lower note, main note. Inverted turn reverses the order.
回音：四个音的模式：上方音、本音、下方音、本音。逆回音顺序相反。
Mordent (upper): Main note, upper note, main note – very quick. Lower mordent: main note, lower note, main note (with vertical stroke through the sign).
波音（上波音）：本音、上方音、本音——非常快速。下波音：本音、下方音、本音（符号中部加一竖线）。
Appoggiatura: A dissonant note taking half the value of the main note (or more), leaning into the resolution.
倚音：一个不协和音，占本音时值的一半（或更多），倾向解决。
Acciaccatura: A crushed note, played as quickly as possible before the main note, shown with a oblique stroke through the stem and small note.
碎音：一个挤压音，在本音前尽可能快地奏出，使用带斜线的小音符表示。
Memorising these conventions is like learning a set of mini theorems—once internalised, you can apply them to any score.
记住这些惯例就像学习一组微型定理——一旦内化，便可应用于任何乐谱。
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