📚 Year 13 OCR Music: Formula & Theorem Quick Reference Handbook | Year 13 OCR 音乐：公式定理速查手册
Welcome to your essential quick-reference guide for OCR A Level Music. This handbook reframes the fundamental building blocks of Western tonal music, acoustics, twentieth-century techniques and technology into memorable ‘formulae’ and ‘theorems’. Use these concise patterns when analysing set works, composing under controlled conditions, or revising for the Listening and Appraising paper. Every concept is expressed as a repeatable recipe, helping you move from description to secure application in performance and composition.
欢迎使用这本 OCR A Level 音乐核心速查手册。本手册将西方调性音乐、声学、二十世纪技法以及音乐科技的基本构件重新梳理成便于记忆的“公式”与“定理”。在分析规定作品、限时作曲或复习听力与鉴赏试卷时，你可以反复查阅这些精炼模式。每个概念都呈现为可重复操作的配方，帮助你把知识从简单的描述转化为演奏和创作中可靠的实践。
1. Interval Formulae | 音程公式
Every interval can be defined by a precise number of semitones and a quality descriptor. Start by counting the diatonic letter names (2nd, 3rd, 4th, etc.), then adjust the quality according to the semitone count. The formula for a perfect 5th is 7 semitones above the tonic; for a major 3rd it is 4 semitones.
每个音程都可以用精确的半音数与性质描述来定义。先数自然音名（二度、三度、四度等），然后依据半音数调整其性质。纯五度的公式是根音上方 7 个半音；大三度的公式是 4 个半音。
Interval semitone map: m2 = 1, M2 = 2, m3 = 3, M3 = 4, P4 = 5, A4/d5 = 6, P5 = 7, m6 = 8, M6 = 9, m7 = 10, M7 = 11, P8 = 12
音程半音图谱：小二度 = 1, 大二度 = 2, 小三度 = 3, 大三度 = 4, 纯四度 = 5, 增四/减五 = 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
Compound intervals simply add 12 semitones per octave. For example, a major 10th equals a major 3rd plus one octave: 4 + 12 = 16 semitones.
复音程只需在每增加一个八度时加上 12 个半音。例如，大十度等于大三度加一个八度：4 + 12 = 16 个半音。
2. Scale Construction Patterns | 音阶构建模式
The major scale follows the fixed pattern of whole tones (T) and semitones (S): T, T, S, T, T, T, S. Starting on any pitch and applying this sequence guarantees a major scale. The natural minor scale uses T, S, T, T, S, T, T.
大调音阶遵循固定的全音（T）与半音（S）模式：全、全、半、全、全、全、半。从任意音高开始套用此序列，即可得到一个大调音阶。自然小调采用 全、半、全、全、半、全、全。
Major: T-T-S-T-T-T-S | Natural minor: T-S-T-T-S-T-T
大调：全-全-半-全-全-全-半 | 自然小调：全-半-全-全-半-全-全
Harmonic minor raises the 7th degree, creating an augmented 2nd step between the 6th and 7th degrees: T, S, T, T, S, 3S, S (where 3S = three semitones). Melodic minor (ascending) raises both 6th and 7th: T, S, T, T, T, T, S. The descending melodic minor reverts to the natural minor pattern.
和声小调升高第Ⅶ级，在Ⅵ级与Ⅶ级之间形成增二度步进：全、半、全、全、半、增二度（3 个半音）、半。旋律小调（上行）升高第Ⅵ级和第Ⅶ级：全、半、全、全、全、全、半。下行旋律小调回到自然小调模式。
3. Triad and Seventh Chord Recipes | 三和弦与七和弦配方
Chords can be constructed by stacking intervals of thirds. A major triad consists of a root, a major 3rd (4 semitones) and a perfect 5th (7 semitones). From the root, the internal stacking formula is: major 3rd + minor 3rd (4+3 semitones).
和弦可以通过叠加三度音程来构建。大三和弦由根音、大三度（4 个半音）与纯五度（7 个半音）构成。从根音看，其内部叠加公式为：大三度 + 小三度（4+3 个半音）。
Triad recipes (semitones from root to 3rd, 3rd to 5th): Major = 4+3, Minor = 3+4, Diminished = 3+3, Augmented = 4+4
三和弦配方（根音-三音、三音-五音半音数）：大三 = 4+3，小三 = 3+4，减三 = 3+3，增三 = 4+4
Seventh chords add another third on top. A dominant 7th (e.g. G⁷) stacks major 3rd, minor 3rd, minor 3rd: 4+3+3. A major 7th uses 4+3+4. A minor 7th is 3+4+3. A half-diminished 7th is 3+3+4, and a fully diminished 7th is 3+3+3. These stackings are vital for recognising and writing chords in any inversion.
七和弦在顶部再加一个三度。属七和弦（如 G⁷）叠加 大三度、小三度、小三度：4+3+3。大七和弦使用 4+3+4。小七和弦是 3+4+3。半减七和弦是 3+3+4，减七和弦是 3+3+3。这些叠加顺序对于识别和写作任何转位的和弦都至关重要。
4. Cadence Formulas | 终止式公式
Cadences are the punctuation marks of tonal music. Each cadence follows a predictable two-chord formula expressed in Roman numerals. A perfect cadence (V–I) sounds final; a plagal cadence (IV–I) gives an ‘Amen’ close. An imperfect cadence ends on V (e.g. I–V, ii–V, IV–V), and an interrupted cadence typically moves V–vi.
终止式是调性音乐的标点符号。每一种终止式都遵循用罗马数字表示的可预测的双和弦公式。正格终止（V–I）听起来最具结束感；变格终止（IV–I）带来“阿门”式的收束。不完全终止停在 V 级上（如 I–V，ii–V，IV–V），阻碍终止通常进行到 V–vi。
Perfect: V → I (or V⁷ → I) | Plagal: IV → I | Imperfect: ? → V | Interrupted: V → vi
正格：V → I（或 V⁷ → I） | 变格：IV → I | 不完全：? → V | 阻碍：V → vi
In the Classical style, the cadential 6/4 (Ic–V–I) precedes a perfect cadence to intensify the arrival on the tonic. Remember that in a minor key the dominant chord is major (V) due to the raised leading note, which affirms the harmonic minor formula.
在古典风格中，终止四六和弦（Ic–V–I）常在正格终止之前出现，以加强到主和弦的解决感。注意在小调中，由于导音升高，属和弦是大三和弦（V），这印证了和声小调的公式。
5. Harmonic Progression Schemas | 和声进行图式
Functional tonality relies on movement away from and back to the tonic. The circle-of-fifths progression is a backbone pattern in which roots descend by perfect 5th (or ascend by perfect 4th). A typical diatonic cycle is: iii → vi → ii → V → I, often extended to vi → ii → V → I as the fundamental ‘turnaround’.
功能和声依赖于离开主和弦再回归的运动。五度圈进行是一个骨架模式，其根音以纯五度下行（或纯四度上行）。一个常见的自然音循环为：iii → vi → ii → V → I，且常被扩展为由 vi → ii → V → I 构成的基础“回转进行”。
Falling 5ths chain: iii – vi – ii – V – I (in major: Em – Am – Dm – G – C)
下行五度链：iii – vi – ii – V – I（大调中：Em – Am – Dm – G – C）
The 12-bar blues uses a set harmonic grid, typically I–I–I–I, IV–IV–I–I, V–IV–I–V (or I). In the soundtrack and Romantic repertoire, chromatic mediants (e.g. C major → E major, formula: root moves by a major 3rd, chord quality stays or changes) and Neapolitan chords (♭II, a major chord built on the lowered 2nd degree, often in first inversion and resolving to V) provide dramatic colour change.
十二小节布鲁斯使用固定的和声网格，典型为 I–I–I–I，IV–IV–I–I，V–IV–I–V（或 I）。在影视配乐和浪漫主义曲目中，变化中音和弦（如 C 大调到 E 大调，公式为：根音移动大三度，和弦性质保持或改变）以及拿波里和弦（♭II，建立在降Ⅱ级上的大三和弦，常用第一转位并解决到 V）带来强烈的色彩变化。
6. Serialism and 12-tone Rules | 序列主义与十二音规则
In dodecaphonic music, a tone row (P, prime) contains all twelve pitch classes in a fixed order, with no note repeated before the row is complete. The row generates three other standard forms: retrograde (R), inversion (I), and retrograde inversion (RI). All four forms can be transposed to any of the twelve starting pitches.
在十二音音乐中，一个音列（P，原型）以固定顺序包含全部十二个音级，且在序列完成之前任何音不得重复。该序列还可生成其他三种标准形式：逆行（R）、倒影（I）与逆行倒影（RI）。所有四种形式都可以移调到十二个起始音高中的任何一个。
Row operations: P = original order; R = P reversed; I = intervals inverted (direction reversed, same size); RI = I reversed
序列操作：P = 原顺序；R = P 反向；I = 音程倒影（方向反转，距离相同）；RI = I 反向
A matrix (12×12 grid) systematically lists all 48 possible rows. When analysing, label each row form with its starting pitch class (e.g. P₀ starts on C, R₀ ends on C). Knowledge of hexachordal combinatoriality (the first six notes of one row form complement the first six of another without duplication) is crucial for understanding the structure of Webern and late Schoenberg works.
矩阵（12×12 网格）系统性地列出全部 48 种可能的序列形式。分析时，用起始音级标记每种形式（如 P₀ 从 C 开始，R₀ 以 C 结束）。了解六音组组合性（一种序列的前六个音与另一种形式的前六个音互补且无重复）对于理解韦伯恩和勋伯格晚期作品的结构至关重要。
7. Motivic Development Techniques | 动机发展技术
A motif is the smallest identifiable musical idea, and composers manipulate it using a toolkit of transformational ‘theorems’. Repetition (exact), sequence (transposed restatement at a different pitch, often following a falling 3rd or rising 2nd pattern), inversion (melodic contour flipped), retrograde (played backwards), augmentation (note values lengthened) and diminution (note values shortened) are the core operations.
动机是最小的可辨识的乐思，作曲家使用一套转化“定理”来操控它。重复（原样）、模进（在不同音高上移调陈述，常遵循下行三度或上行二度的模式）、倒影（旋律轮廓翻转）、逆行（反向演奏）、增值（音符时值变长）和减值（音符时值缩短）是核心操作方式。
Motivic transformation formulas: Sequence = transposed repetition, Inversion = vertical mirror, Augmentation = multiply note lengths, Diminution = divide note lengths
动机转化公式：模进 = 移调重复，倒影 = 纵向镜像，增值 = 音长乘以系数，减值 = 音长除以系数
Intervallic expansion/contraction stretches or compresses the gaps between notes while preserving rhythmic identity. Fragmentation uses only part of the motif as developmental material. In the Romantic period, thematic transformation changes a theme’s character across movements while retaining its core pitch contour, a principle central to programme music analysis.
音程扩展/压缩在保持节奏特征的同时拉伸或挤压音与音之间的距离。分裂则只取动机的一部分作为展开材料。在浪漫主义时期，主题变形在乐章间改变主题的性格但保留其核心音高轮廓，这一原则对于标题音乐的分析尤为关键。
8. Rhythm and Metre Formulas | 节奏与节拍公式
Metre is the regular grouping of beats, defined by a time signature’s formula: upper number = beats per bar, lower number = beat unit (e.g. 4 = crotchet, 8 = quaver). Simple time divides each beat into two equal parts; compound time divides each beat into three. Thus, compound duple time (e.g. 6/8) feels as two dotted-crotchet beats, each subdivided into three quavers.
节拍是按规律组合起来的拍子，其公式由拍号定义：上方数字 = 每小节拍数，下方数字 = 拍子单位（如 4 代表四分音符，8 代表八分音符）。单拍子将每一拍均分为两部分；复拍子将每一拍均分为三部分。因此，复二拍子（如 6/8）感觉是两个附点四分音符拍，每一拍再细分为三个八分音符。
Simple metre: beat = 2 subdivisions; Compound metre: beat = 3 subdivisions | Dotted beat = 3 × subdivision value
单拍子：每拍 = 2 个细分；复拍子：每拍 = 3 个细分 | 附点拍 = 3 × 细分时值
Syncopation disrupts the regular metric hierarchy by accenting a weak beat or off-beat. A common formula ties a weak note over the bar line or places an accent on the ‘and’ of a beat. Cross-rhythms and hemiola (e.g. two beats in the time of three, or 3:2 ratio) generate metrical tension, frequently found in Baroque dance movements and African-influenced minimalism.
切分音通过在小节的弱拍或半拍上加着重记号来扰乱惯常的节拍等级。常见的公式是把一个弱音通过连线延过小节线，或者把重音放在某一拍的“和”上。交叉节奏与赫米奥拉比例（如三对二，3:2 比例）制造节拍张力，广泛存在于巴洛克舞曲乐章和受非洲音乐影响的简约主义作品中。
9. Harmonic Series and Acoustics | 谐波系列与声学公式
Any sounding pitch (fundamental frequency f₁) generates a series of integer-multiple overtones. The harmonic series formula is fₙ = n × f₁, where n = 1, 2, 3, … . The first overtone (n=2) is an octave above, the second overtone (n=3) a perfect 5th above that, and the fourth overtone (n=5) a major 3rd, which already creates a major triad from a single fundamental.
任何发声音高（基频 f₁）都会产生一系列整数倍的泛音。谐波系列公式为 fₙ = n × f₁，其中 n = 1, 2, 3, …。第一泛音（n=2）高一个八度，第二泛音（n=3）再往上纯五度，第四泛音（n=5）形成大三度，这已经从单一基音中构建出大三和弦。
Harmonic series: 1–f₁, 2–2f₁ (8ve), 3–3f₁ (12th), 4–4f₁ (15th), 5–5f₁ (17th/M3), 6–6f₁ (19th/P5), 7–7f₁ (flat 7th), 8–8f₁ (15th)
谐波列：1–基频，2–2倍频（八度），3–3倍频（十二度），4–4倍频（十五度），5–5倍频（十七度/大三度），6–6倍频（十九度/纯五度），7–7倍频（降七度），8–8倍频（十五度）
Interval size in cents connects perception to mathematical ratio. The formula to convert a frequency ratio (r) into cents is: cents = 1200 × log₂(r). An equally tempered semitone = 100 cents, a pure perfect 5th (ratio 3:2) ≈ 702 cents. This explains why piano tuning requires slight adjustments (temperament) and helps analyse microtonal works set by OCR.
音分值将听觉感知与数学比率联系起来。频率比（r）换算为音分的公式为：音分 = 1200 × log₂(r)。一个平均律半音 = 100 音分，纯律纯五度（比率 3:2）约 702 音分。这解释了钢琴调律为何需要微调（律制），也有助于分析 OCR 指定的微分音作品。
10. Synthesis and Envelope Parameters | 合成器与包络参数
In technology-based composition and electronic set works, sound is sculpted using the ADSR envelope: Attack, Decay, Sustain, Release. Each segment is defined by a time or level parameter. The formula for a pad sound is long attack, long release, high sustain; a percussive hit uses instant attack, zero sustain, quick decay.
在基于科技的作曲和电子音乐规定作品中，声音通过 ADSR 包络进行塑形：起音（Attack）、衰减（Decay）、保持（Sustain）、释音（Release）。每一段由一个时长或电平参数定义。铺底音色的公式是长起音、长释音、高保持电平；打击性音头则使用瞬时起音、零保持、快速衰减。
ADSR: Attack time (0 to max), Decay time (max to sustain level), Sustain level (constant while note held), Release time (sustain level to 0 after note off)
ADSR：起音时长（0 到峰值），衰减时长（峰值降至保持电平），保持电平（按住音符时恒定），释音时长（离键后从保持电平降至 0）
Oscillator waveforms provide the raw harmonic content. The fundamental shapes are sine (pure tone, only f₁), triangle (odd harmonics, amplitude falls as 1/n²), sawtooth (all harmonics, amplitude falls as 1/n), and square/pulse (odd harmonics, amplitude falls as 1/n). A filter (low-pass, high-pass, band-pass) subtracts partials following a cutoff frequency and resonance (boost at cutoff). Modulation workflows like LFO → filter cutoff create vibrato or dubstep wobble, following the central synthesis formula: Modulator (rate, depth) × Carrier parameter.
振荡器波形提供原始谐波内容。基本波形有正弦波（纯音，仅有 f₁）、三角波（奇次谐波，幅度按 1/n² 衰减）、锯齿波（全部谐波，幅度按 1/n 衰减）以及方波/脉冲波（奇次谐波，幅度按 1/n 衰减）。滤波器（低通、高通、带通）根据截止频率和共振（在截止点的抬升）来削减泛音。LFO → 滤波器截止频率等调制工作流可以产生颤音或 dubstep 摇摆效果，这遵循核心合成公式：调制源（速率、深度）× 载波参数。
11. Chord Inversion and Figured Bass Shortcuts | 和弦转位与数字低音速记
In Roman numeral analysis, inversions are indicated with subscript figures derived from figured bass. A root-position triad has no figures (or ⁵₃), first inversion uses ⁶ (or ⁶₃), second inversion uses ⁶₄. For seventh chords: root position = ⁷, first inversion = ⁶₅, second inversion = ⁴₃, third inversion = ⁴₂ or simply ². These are essential formulae for harmonising a bass line or decoding a score quickly.
在罗马数字分析中，转位用源自数字低音的下标数字来表示。原位三和弦没有数字（或 ⁵₃），第一转位用 ⁶（或 ⁶₃），第二转位用 ⁶₄。七和弦：原位 = ⁷，第一转位 = ⁶₅，第二转位 = ⁴₃，第三转位 = ⁴₂ 或简写为 ²。这些是为低音线条配和声或快速解码乐谱所必需的公式。
Triads: ⁵₃ (root), ⁶ (1st), ⁶₄ (2nd) | 7ths: ⁷ (root), ⁶₅ (1st), ⁴₃ (2nd), ⁴₂ (3rd)
三和弦：⁵₃（原位），⁶（一转），⁶₄（二转） | 七和弦：⁷（原位），⁶₅（一转），⁴₃（二转），⁴₂（三转）
The chart of acceptable doublings also acts as a ‘rule set’: in root-position triads, double the root (not the 3rd unless necessary for voice-leading). In first inversion, the soprano note is often doubled. In second inversion (cadential ⁶₄), the bass note (5th of the chord) is doubled. Following these doubling conventions ensures idiomatic four-part writing.
可接受的重复音图表同样充当着一套“规则集”：原位三和弦重复根音（除非为声部进行考虑才重复三音）。第一转位常重复高音声部的音。第二转位（终止四六和弦）重复低音声部的音（和弦的五音）。遵循这些重复音惯例可以确保地道的四部和声织体写作。
12. Structural Forms and Formulaic Sections | 结构曲式与公式化段落
Many set works rely on formal ‘shapes’ that can be expressed as patterns of sections. Sonata form = Exposition (first subject–transition–second subject–codetta) || Development (modulating, fragmenting material) || Recapitulation (first subject–transition–second subject in tonic–coda). The tonal formula demands that the second subject in the exposition is in a non-tonic key (dominant in major, relative major in minor) and returns in the tonic in the recapitulation.
许多规定作品依赖于可以用段落模式来表达的形式“形状”。奏鸣曲式 = 呈示部（第一主题–连接段–第二主题–小结尾）|| 展开部（转调、动机分裂）|| 再现部（第一主题–连接段–第二主题回归主调–尾声）。其调性公式要求呈示部的第二主题采用非主调（大调中为属调，小调中为关系大调），并在再现部中回归主调。
Sonata form tonal plan: Exposition: I → V (major) or i → III (minor); Recapitulation: all themes in I/i
奏鸣曲式调性布局：呈示部：I → V（大调）或 i → III（小调）；再现部：所有主题均回到 I/i
Rondo form alternates a recurring refrain (A) with contrasting episodes: A–B–A–C–A (or A–B–A–C–A–B’–A). Ternary form follows A–B–A, where B provides contrast in key and character. Minuet and Trio embodies a compound ternary: Minuet (A–B–A) – Trio (C–D–C) – Minuet da capo. Recognising these formal formulas helps you predict events in unfamiliar extracts during the listening exam.
回旋曲式交替出现反复的主部（A）与对比插部：A–B–A–C–A（或 A–B–A–C–A–B’–A）。三段体遵循 A–B–A，其中 B 在调性和性格上形成对比。小步舞曲与三声中部体现复三部曲式：小步舞曲（A–B–A）– 三声中部（C–D–C）– 小步舞曲返始。识别这些曲式公式能帮助你在听力考试中对陌生选段做出
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