📚 IGCSE OCR Music: Formula & Theorem Quick Reference | IGCSE OCR 音乐：公式定理速查手册
Music theory can be thought of as a collection of rules, patterns and mathematical relationships that govern how sounds are organised. This quick reference handbook sets out the essential 'formulae' and 'theorems' you need to memorise for the IGCSE OCR Music examination – from interval construction and scale patterns to chord building, cadences and rhythmic ratios. Use it for rapid revision and to check your understanding before tackling listening questions, composition tasks or performance analysis.
音乐理论可以看作是一系列规则、模式和数学关系，它们支配着声音的组织方式。本速查手册列出了 IGCSE OCR 音乐考试需要熟记的关键「公式」与「定理」——从音程构建、音阶模式到和弦构成、终止式及节奏比值。用它来快速复习，并在应对听力题目、作曲任务或演奏分析前检测自己的理解。
1. Pitch Notation & Staff Basics | 音高记谱与五线谱基础
The treble clef (G clef) fixes the second line from the bottom as G above middle C (G₄). The bass clef (F clef) fixes the fourth line from the bottom as F below middle C (F₃). Ledger lines extend the staff beyond its five lines. Accidentals applied: sharp (♯) raises a note by one semitone, flat (♭) lowers by one semitone, natural (♮) cancels a previous accidental. These are absolute rules in staff notation.
高音谱号（G 谱号）将下数第二线确定为中央 C 上方的 G（G₄）。低音谱号（F 谱号）将下数第四线确定为中央 C 下方的 F（F₃）。加线将谱表超出五条线。升降符号规则：升号（♯）将音符升高一个半音，降号（♭）降低一个半音，还原号（♮）取消之前的升降。这些都是五线谱记谱中的绝对规则。
2. Interval Formulae | 音程公式
An interval is measured in two parts: the numerical distance (2nd, 3rd, 4th, etc.) and the quality (perfect, major, minor, augmented, diminished). Key formula to identify an interval: count the letter names inclusive of the starting note to determine the number, then count the semitones to determine the quality. For unisons, 4ths, 5ths, and octaves, the perfect quality occurs when the upper note matches the key of the lower note; a perfect 5th is 7 semitones, a perfect 4th is 5 semitones. For 2nds, 3rds, 6ths, and 7ths, major occurs when the upper note belongs to the major scale built on the lower note; a major 2nd is 2 semitones, a major 3rd is 4 semitones, a major 6th is 9 semitones, a major 7th is 11 semitones. Reduce a major interval by one semitone to get minor, enlarge a perfect or major interval by one semitone to get augmented.
音程由两部分衡量：数字距离（二度、三度、四度等）和性质（纯、大、小、增、减）。识别音程的关键公式：从起始音开始数音名个数（包含起始音）以确定度数，然后数半音数以确定性质。对于同度、四度、五度和八度，当上方音属于下方音的主调时，为纯音程；纯五度为 7 个半音，纯四度为 5 个半音。对于二度、三度、六度和七度，当上方音属于下方音构成的大调音阶时，为大音程；大二度为 2 个半音，大三度为 4 个半音，大六度为 9 个半音，大七度为 11 个半音。将大音程减少一个半音得到小音程，将纯音程或大音程增加一个半音得到增音程。
3. Scale Construction Formulae | 音阶构成公式
Major scale formula: W‑W‑H‑W‑W‑W‑ Starting from any tonic, apply this sequence to produce the major scale. Natural minor scale formula: W‑H‑W‑W‑H‑W‑ ee of the natural minor by one semitone, creating an augmented 2nd step between the 6th and 7th degrees. Melodic minor ascending: raise the 6th and 7th degrees of the natural minor; descending reverts to natural minor. Chromatic scale: movement entirely by semitones; notation rules – nding, respecting the key signature. Pentatonic scale: omit the 4th and 7th degrees from the major scale (major pentatonic), or omit the 2nd and 6th from the natural minor (minor pentatonic). Whole‑tone scale: W‑W‑W‑W‑W‑ six notes, completely symmetrical). Blues scale: minor pentatonic with an added flat 5th (♭5).
大调音阶公式：全‑全‑半‑全‑全‑全‑半（全=全音，半=半音）。从任何主音出发，应用此序列即可得到大调音阶。自然小调音阶公式：全‑半‑全‑全‑半‑全‑全。和声小调：将自然小调的第 7 级音升高一个半音，从而在第 6 与第 7 级之间形成增二度。旋律小调上行：升高自然小调的第 6 和第 7 级；下行恢复为自然小调。半音音阶：完全按半音移动；记谱规则——通常上行用升号、下行用降号，同时遵守调号。五声音阶：从大调音阶中去掉第 4 和第 7 级（大调五声音阶），或从自然小调中去掉第 2 和第 6 级（小调五声音阶）。全音音阶：全‑全‑全‑全‑全‑全（六个音，完全对称）。布鲁斯音阶：小调五声音阶加上一个降五级音（♭5）。
4. Key Signatures & Circle of Fifths | 调号与五度圈
The order of sharps: F♯, C♯, G♯, D♯, A♯, E♯, B♯. The order of flats: B♭, E♭, A♭, D♭, G♭, C♭, F♭ from a sharp key signature: take the last sharp and go up a semitone (e.g. last sharp is D♯ major). For flat key signatures: the penultimate flat is the name of the major key (e.g. B♭ and E♭ → B♭ ves one flat), moving anticlockwise adds one flat (or removes one sharp). Each step around the circle is a perfect 5th ascending (clockwise) or descending (anticlockwise).
升号调顺序：F♯、C♯、G♯、D♯、A♯、E♯、B♯。降号调顺序：B♭、E♭、A♭、D♭、G♭、C♭、F♭。从升号调号寻找大调主音的公式：取最后一个升号再升高一个半音（例如最后一个升号是 D♯ → E 大调）。对于降号调号：倒数第二个降号即为大调主音名（例如 B♭ 和 E♭ → B♭ 大调）。关系小调总是位于大调主音下方小三度（3 个半音）处。五度圈定理：顺时针移动增加一个升号（或减少一个降号），逆时针移动增加一个降号（或减少一个升号）。绕圈每一步都是向上的纯五度（顺时针）或向下的纯五度（逆时针）。
5. Chord Construction Formulae (Triads) | 三和弦构成公式
A triad is a three‑note chord built in thirds. Major triad formula: root + major 3rd (4 semitones) + perfect 5th (7 semitones). Minor triad: root + minor 3rd (3 semitones) + perfect 5th. Diminished triad: root + minor 3rd + diminished 5th (6 semitones). Augmented triad: root + major 3rd + augmented 5th (8 semitones). In Roman numeral analysis within a major key: I, IV, V are major; ii, iii, vi are minor; vii° is diminished. In natural minor: i, iv, v are minor; III, VI, VII are major; ii° is diminished. These formulas remain constant regardless of the tonic note.
三和弦是由三个音按三度叠置构成的和弦。大三和弦公式：根音 + 大三度（4 个半音）+ 纯五度（7 个半音）。小三和弦：根音 + 小三度（3 个半音）+ 纯五度。减三和弦：根音 + 小三度 + 减五度（6 个半音）。增三和弦：根音 + 大三度 + 增五度（8 个半音）。在大调中用罗马数字分析：I、IV、V 级为大三和弦；ii、iii、vi 级为小三和弦；vii° 级为减三和弦。在自然小调中：i、iv、v 级为小三和弦；III、VI、VII 级为大三和弦；ii° 级为减三和弦。这些公式不受主音变化的影响。
6. Seventh Chords & Added‑Note Chords | 七和弦与附加音和弦
r 7th (11 semitones). Minor 7th: minor triad + minor 7th. Diminished 7th: diminished triad + diminished 7th (9 semitones). Half‑diminished 7th: diminished triad + minor 7th. Added‑ 9 semitones). The chord quality is determined by the exact intervals stacked above the root.
属七和弦公式：大三和弦 + 根音上方小七度（距离根音 10 个半音）。大七和弦：大三和弦 + 大七度（11 个半音）。小七和弦：小三和弦 + 小七度。减七和弦：减三和弦 + 减七度（9 个半音）。半减七和弦：减三和弦 + 小七度。附加音和弦：挂四和弦（sus4）用纯四度替代三度音；加六和弦 = 大三和弦 + 大六度（9 个半音）。和弦性质完全由根音上方叠置的准确音程决定。
7. Chord Progressions & Cadence Theorems | 和弦进行与终止式定理
V → I (Amen cadence). Imperfect cadence: ends on V, typically I → V, ii → V, or IV → V. Interrupted cadence: V → vi (major) or V → VI (minor); the listener expects I but receives vi instead. Standard 12‑bar blues progression: I – I – I – I – IV – IV – I – I – V – IV – I – roots moving by descending 5ths (or ascending 4ths) create a strong harmonic pull, e.g. vi – ii – V – I.
终止式是收束乐句的公式化和弦配对。正格终止：V → I（大调）或 V → i（小调）；最强收束。变格终止：IV → I（阿们终止）。半终止：结束在 V 级，通常进行为 I → V、ii → V 或 IV → V。阻碍终止：V → vi（大调）或 V → VI（小调）；听众期待 I 却得到 vi。标准 12 小节布鲁斯进行：I – I – I – I – IV – IV – I – I – V – IV – I – V（可能伴有节奏变化）。循环进行定理：根音以下行五度（或上行四度）移动会产生强烈的和声引力，例如 vi – ii – V – I。
8. Rhythm & Time Signature Ratios | 节奏与拍号公式
uaver/eighth note, 2 = minim/half note). Simple time: each beat divides naturally into two equal parts (e.g. 2/4, 3/4, 4/4). Compound time: each beat divides into three equal parts; the upper number is a multiple of 3 and the beat is a dotted note (e.g. 6/8 = two dotted crotchet beats per bar, 9/8 = three, 12/8 = four). Dot rule: a dot adds half the value of the note (dotted minim = 3 crotchets). Duplet theorem: in compound time, two notes in the time of three are duplets; triplet theorem: in simple time, three notes in the time of two are triplets. The tie formula: tie combines two notes of the same pitch, adding their durations.
拍号公式：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（4 = 四分音符，8 = 八分音符，2 = 二分音符）。单拍子：每一拍自然分成两个均等部分（例如 2/4、3/4、4/4）。复拍子：每一拍分成三个均等部分；上方数字是 3 的倍数，拍子本身是附点音符（例如 6/8 = 每小节两个附点四分拍，9/8 = 三个，12/8 = 四个）。附点规则：附点增加原音符时值的一半（附点二分音符 = 3 个四分音符）。二连音定理：在复拍子中，两音占三音时值即为二连音；三连音定理：在单拍子中，三音占两音时值即为三连音。延音线公式：延音线将两个相同音高的音符连在一起，时值相加。
9. Dynamics, Tempo & Expression Markings | 力度、速度与表情记号
Dynamics follow a graduated scale: pp (pianissimo, very soft), p (piano, soft), mp (mezzo‑piano, moderately soft), mf (mezzo‑forte, moderately loud), f (forte, loud), ff (fortissimo, very loud). Gradual change: cresc. (crescendo, gradually getting louder), dim. (diminuendo, gradually getting softer). Tempo markings are exact or approximate: Largo (broadly/sehr langsam), Adagio (slowly), Andante (at a walking pace), Moderato (moderately), Allegro (fast/lively), Presto (very fast). Articulation formulae: staccato (detached, shortening the note), legato (smoothly connected), accent (> or sfz) stresses the attack. All these are vital for accurate performance and score analysis in the OCR exam.
力度呈阶梯式：pp（很弱）、p（弱）、mp（中弱）、mf（中强）、f（强）、ff（很强）。渐变术语：cresc.（渐强）、dim.（渐弱）。速度标记有精确或近似值：Largo（广板）、Adagio（柔板）、Andante（行板）、Moderato（中板）、Allegro（快板）、Presto（急板）。奏法公式：断奏（staccato，分离、缩短音符），连奏（legato，圆滑连接），重音（> 或 sfz）强调音头。所有这些对于 OCR 考试的准确演奏和乐谱分析都至关重要。
10. Structural Forms & Devices | 结构形式与创作手法
Binary form: AB – two contrasting sections, each usually repeated. Ternary form: ABA – first section returns after a contrasting middle section. Rondo form: ABACA (or ABACABA) – recurring principal theme alternating with episodes. Theme and variations: A A' A'' A''' ... – theme presented then altered in successive variation techniques (melodic, harmonic, rhythmic, textural). Strophic form: same music repeated for each verse (often in songs). Through‑composed: continuous new music throughout, no large‑scale repetition. Ground bass: a repeating bass line over which upper parts vary (ostinato theorem). Canon: one part imitates another strictly after a set distance. Fugue: contrapuntal work based on a subject that enters in different voices following tonic‑dominant answer rules. Sonata form (exposition, development, recapitulation) is also frequently referenced; the exposition introduces first subject (tonic) and second subject (usually dominant or relative major), development explores both, recapitulation states both subjects in the tonic. These structural formulas help identify piece architecture in listening papers.
二部曲式：AB——两个对比乐段，各自通常反复。三部曲式：ABA——第一段在对比中段后回归。回旋曲式：ABACA（或 ABACABA）——反复出现的主部与插部交替。主题与变奏：A A' A'' A''' ……——先呈示主题，然后通过旋律、和声、节奏、织体等变奏手法依次改变。分节歌形式：各段歌词重复同一音乐（常见于歌曲）。通谱体：贯穿全新音乐，没有大规模重复。固定低音：重复的低音线条，上方声部变化（固定音型定理）。卡农：一个声部在固定距离后严格模仿另一声部。赋格：以一个主题为基础的复调作品，主题在不同声部遵循主‑属答题规则进入。奏鸣曲式（呈示部、展开部、再现部）也常被提及；呈示部引入主部主题（主调）和副部主题（通常属调或关系大调），展开部发展两者，再现部将两个主题都置于主调。这些结构公式有助于在听力考试中识别作品框架。
11. Vocal, Instrumental & World Music Rules | 声乐、器乐与世界音乐规则
Vocal ranges from highest to lowest: soprano, mezzo‑soprano, alto (female/boys); tenor, baritone, bass (male). Instrument families: strings (violin, viola, cello, double bass, harp), woodwind (flute, oboe, clarinet, bassoon), brass (trumpet, French horn, trombone, tuba), percussion (tuned – timpani, xylophone; untuned – snare drum, cymbals). Commonly required world music templates: Indian raga – based on a scale (that) with ascending (arohana) and descending (avarohana) rules, often with drone and tala rhythm cycle; Gamelan – metallophones and gongs organized in colotomic structure, using slendro or pelog scales; African drumming – interlocking polyrhythms, call and response, master drummer signals; Samba – polyrhythmic layers built on the surdo pulse, with call and response between leader and ensemble. These conventions act as ‘formulae’ for describing unfamiliar extracts in the exam.
人声范围从高到低：女高音、女中音、女低音（女性/男童）；男高音、男中音、男低音。乐器家族：弦乐（小提琴、中提琴、大提琴、低音提琴、竖琴），木管（长笛、双簧管、单簧管、大管），铜管（小号、圆号、长号、大号），打击乐（有音高——定音鼓、木琴；无音高——小军鼓、钹）。常考的世界音乐模板：印度拉格——基于特定音阶（that），包含上行（arohana）和下行（avarohana）规则，通常辅以持续音和塔拉节奏圈；甘美兰——金属琴和锣以分句结构组织，使用斯连德罗或佩洛格音阶；非洲鼓乐——交错的多重节奏、呼应式、领鼓信号；桑巴——建立在苏尔多脉动上的多重节奏层，领唱与群唱之间呼应。这些惯例可作为描述考试中陌生选段的「公式」。
12. Common Listening Terms & Italian Vocabulary | 常见听力术语与意大利术语
Texture terms: monophonic (single melody line), homophonic (melody with chordal accompaniment), polyphonic (two or more independent melodic lines), heterophonic (simultaneous variations of a single melody). Melodic contours: conjunct (stepwise), disjunct (leaping), triadic, chromatic. Unity and contrast are obtained through repetition, sequence, imitation, inversion, retrograde. Decoration: trill, turn, mordent, appoggiatura, acciaccatura – each has a precise execution rule. Essential tempo modifiers: molto (very), meno (less), più (more), non troppo (not too much). Other directional words: da capo (from the beginning), dal segno (from the sign), fine (end), coda (tail section). These vocabulary items must be memorised as fixed definitions.
织体术语：单声部（单一旋律线），主调（旋律加和弦伴奏），复调（两条或以上独立旋律线），支声复调（同一旋律的多重变体同时进行）。旋律轮廓：级进（步进式），跳进（大跳跃），分解和弦式，半音式。统一与对比通过重复、模进、模仿、倒影、逆行等手法实现。装饰音：颤音、回音、波音、倚音、短倚音——每种都有精确的演奏规则。必备速度修饰词：molto（非常），meno（较少），più（更多），non troppo（不过分）。其他指示词：da capo（从头反复），dal segno（从记号处反复），fine（曲线），coda（尾声）。这些术语必须作为固定定义牢记。
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