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Pre-U OCR Music: Formula & Theorem Quick Reference Handbook | Pre-U OCR 音乐：公式定理速查手册

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📚 Pre-U OCR Music: Formula & Theorem Quick Reference Handbook | Pre-U OCR 音乐：公式定理速查手册

This handbook compiles the essential formulas, rules, and theorems required for the OCR Pre-U Music specification, covering topics in music theory, harmony, and analysis. It is designed as a quick revision guide for students preparing for examinations, offering clear and concise summaries paired with practical applications. Each section presents the core concept in English first, followed immediately by its Chinese counterpart to support bilingual learners.

本手册汇编了 OCR Pre-U 音乐课程所需的核心公式、规则与定理，涵盖音乐理论、和声与分析等范畴，专为学生备考提供快速复习指引，以清晰简明的双语对照方式呈现。每一节首先给出英文核心概念，紧接着提供对应的中文解释，便于双语学习者理解与记忆。

1. Interval Formulas | 音程公式

An interval is the distance between two pitches, measured in semitones and by its quality (perfect, major, minor, augmented, diminished). The number of letter names determines the interval number. To calculate semitones between two notes, count the steps on a keyboard, remembering that E–F and B–C are natural semitones. The formula for the size of a perfect interval (unison, fourth, fifth, octave) is always a fixed number of semitones: P1 = 0, P4 = 5, P5 = 7, P8 = 12. Major intervals (second, third, sixth, seventh) each have a smaller ‘minor’ form one semitone narrower and a larger ‘augmented’ form one semitone wider. The relationship is: minor = major – 1 semitone; augmented = major or perfect + 1 semitone; diminished = perfect or minor – 1 semitone.

音程是两个音高之间的距离，以半音数和性质（纯、大、小、增、减）表示。音名数目决定音程的度数。计算两音间的半音数时，可在键盘上数步数，同时牢记 E–F 和 B–C 之间为自然半音。纯音程（同度、四度、五度、八度）的半音数固定不变：纯一度 0，纯四度 5，纯五度 7，纯八度 12。大音程（二度、三度、六度、七度）各有一个小半音的小音程形式，以及宽一个半音的增音程形式。关系为：小音程 = 大音程 − 1 半音；增音程 = 大音程或纯音程 + 1 半音；减音程 = 纯音程或小音程 − 1 半音。

  • Perfect intervals: P1 (0), P4 (5), P5 (7), P8 (12) | 纯音程：纯一度(0)、纯四度(5)、纯五度(7)、纯八度(12)

  • Major intervals: M2 (2), M3 (4), M6 (9), M7 (11) | 大音程：大二度(2)、大三度(4)、大六度(9)、大七度(11)

  • Minor intervals: m2 (1), m3 (3), m6 (8), m7 (10) | 小音程：小二度(1)、小三度(3)、小六度(8)、小七度(10)

  • Inversion theorem: When an interval is inverted, the new interval number = 9 – original number; quality flips (major ↔ minor, perfect stays perfect, augmented ↔ diminished). | 转位定理：音程转位后，新度数 = 9 − 原度数；性质互换（大 ↔ 小，纯保持纯，增 ↔ 减）。

2. Major Scale Formula | 大调音阶公式

A major scale is constructed using a fixed pattern of tones (T) and semitones (S): T–T–S–T–T–T–S. Starting from any tonic note, apply this sequence to generate the seven diatonic pitches. In terms of semitone steps, the pattern is 2–2–1–2–2–2–1. The scale degrees are numbered 1 to 7, with the 8th degree being the octave. Scale-degree names are tonic (I), supertonic (II), mediant (III), subdominant (IV), dominant (V), submediant (VI), leading note (VII). Accidentals are added according to the key signature to maintain the interval pattern without altering the note letter names.

大调音阶按固定的全音 (T) 与半音 (S) 模式构建：全–全–半–全–全–全–半。从任意主音出发，依此序列生成七个自然音级。以半音阶数表示，该模式为 2–2–1–2–2–2–1。音阶各音级编号为 1 至 7，第八级为八度。音级名称为主音 (I)、上主音 (II)、中音 (III)、下属音 (IV)、属音 (V)、下中音 (VI)、导音 (VII)。临时记号依据调号添加，以维持音程模式而不改变音名字母。

The whole-tone/semitone formula for any major scale can be remembered with the mnemonic: T T S T T T S. | 任何大调音阶均可借助“全全半全全全半”的口诀记忆。

  • Example: C major: C–D (T), D–E (T), E–F (S), F–G (T), G–A (T), A–B (T), B–C (S) | 示例：C 大调：C–D (全), D–E (全), E–F (半), F–G (全), G–A (全), A–B (全), B–C (半)
  • Harmonised major scale triads: I–ii–iii–IV–V–vi–vii° | 大调音阶上的三和弦：I–ii–iii–IV–V–vi–vii° (大,小,小,大,大,小,减)

3. Natural Minor Scale Formula | 自然小调音阶公式

The natural minor scale follows the pattern T–S–T–T–S–T–T. This is equivalent to the Aeolian mode and is the relative minor of the major scale starting on the sixth degree. The interval pattern in semitones is 2–1–2–2–1–2–2. Scale-degree names are the same, but the third (mediant), sixth (submediant) and seventh (subtonic) degrees are lowered by a semitone compared with the parallel major. The subtonic (♭VII) replaces the leading note, causing a weaker dominant-to-tonic pull. The relative minor key signature is identical to that of its relative major.

自然小调音阶的模式为全–半–全–全–半–全–全。它相当于爱奥里亚调式，也是大调的六级开始的平行小调（关系小调）。以半音数表示为 2–1–2–2–1–2–2。音级名称相同，但三级（中音）、六级（下中音）和七级（下主音）比同主音大调低一个半音。下主音 (♭VII) 取代导音，使得属到主的倾向性较弱。关系小调调号与其关系大调完全相同。

  • Natural minor harmonised triads: i–ii°–♭III–iv–v–♭VI–♭VII | 自然小调三和弦：i–ii°–♭III–iv–v–♭VI–♭VII
  • Relative minor theorem: To find the relative minor key, count down a minor third from the major tonic, or find the sixth degree of the major scale. | 关系小调定理：从大调主音向下数一个小三度，或找出大调音阶六级音，即为关系小调主音。

4. Chord Construction Theorems | 和弦构成定理

Chords are built by stacking thirds. A triad consists of a root, third and fifth. The quality of the triad is determined by the intervals between these notes. The formula for the four basic triads is as follows (all intervals measured in semitones from the root, then the third to fifth):

和弦由三度叠置构成。三和弦包含根音、三音与五音，其性质取决于音之间的音程。四种基本三和弦的构成公式如下（音程均以半音数为单位，依次为根音到三音、三音到五音）：

  • Major triad: root + 4 semitones (major 3rd) + 3 semitones (minor 3rd) = intervals of 4 and 3. | 大三和弦：根音 + 4 半音（大三度）+ 3 半音（小三度）→ 音程组合 4+3。
  • Minor triad: root + 3 semitones (minor 3rd) + 4 semitones (major 3rd) = 3+4. | 小三和弦：根音 + 3 半音（小三度）+ 4 半音（大三度）→ 3+4。
  • Augmented triad: root + 4 + 4 = 4+4. | 增三和弦：根音 + 4 + 4 → 4+4。
  • Diminished triad: root + 3 + 3 = 3+3. | 减三和弦：根音 + 3 + 3 → 3+3。

Seventh chords add a further third above the triad fifth. The common formulas are:

七和弦在三和弦五音上方再叠置一个三度。常见七和弦的公式如下：

  • Dominant 7th: major triad + minor 3rd → 4+3+3 (e.g. G7: G–B–D–F). | 属七和弦：大三和弦 + 小三度 → 4+3+3（如 G–B–D–F）。
  • Major 7th: major triad + major 3rd → 4+3+4. | 大七和弦：大三和弦 + 大三度 → 4+3+4。
  • Minor 7th: minor triad + minor 3rd → 3+4+3. | 小七和弦：小三和弦 + 小三度 → 3+4+3。
  • Half-diminished 7th: diminished triad + major 3rd → 3+3+4. | 半减七和弦：减三和弦 + 大三度 → 3+3+4。
  • Fully diminished 7th: diminished triad + minor 3rd → 3+3+3. | 减七和弦（全减七）：减三和弦 + 小三度 → 3+3+3。

5. Circle of Fifths & Key Signature Formula | 五度圈与调号公式

The circle of fifths arranges keys by ascending fifths (clockwise) and descending fifths (anticlockwise). Moving clockwise adds one sharp to the key signature; moving anticlockwise adds one flat. The order of sharps is F♯, C♯, G♯, D♯, A♯, E♯, B♯. The order of flats is B♭, E♭, A♭, D♭, G♭, C♭, F♭ – the reverse of the sharps order. For major keys: the last sharp is the leading note (a semitone below the tonic), so the tonic is one semitone above the last sharp. For flat keys: the penultimate flat in the key signature is the tonic (except F major with one flat). Relative minor keys share the same key signature and are found a minor third below the major tonic.

五度圈按上行五度（顺时针）和下行五度（逆时针）排列各调。顺时针方向每步增加一个升号，逆时针每步增加一个降号。升号顺序为 F♯, C♯, G♯, D♯, A♯, E♯, B♯。降号顺序为其逆行：B♭, E♭, A♭, D♭, G♭, C♭, F♭。对大调而言：最后一个升号是导音（比主音低半音），故主音位于最后一个升号上方半音；降号调中，倒数第二个降号即为主音（一个降号的 F 大调除外）。关系小调共享相同调号，位于大调主音下方小三度。

  • Key signature formula for major sharp keys: new sharp = raise the previous sharp by a perfect fifth. | 升号大调调号公式：新升号将前一升号升高纯五度。
  • Key signature formula for flat keys: new flat = raise the previous flat by a perfect fourth (or lower by a perfect fifth). | 降号调公式：新降号将前一降号升高纯四度（或降低纯五度）。
  • Fast key identification: For sharps, the last sharp + 1 semitone = tonic; for flats, the last flat is the subdominant (except if only one flat). | 快速辨调：升号调，最后升号 + 1 半音 = 主音；降号调，最后降号为下属音（除单降号外）。

6. Modal Scale Formulas | 调式音阶公式

The seven diatonic modes can be derived by playing the white notes from C to C (Ionian), D to D (Dorian), E to E (Phrygian), F to F (Lydian), G to G (Mixolydian), A to A (Aeolian), B to B (Locrian). Each mode has a distinctive whole-tone/semitone pattern. The formulas relative to the parallel major scale (or relative to the Ionian mode) are shown below. ‘♭’ denotes a lowered degree, ‘♯’ a raised degree.

七个自然调式可由白键依次开始推演：C–C (伊奥尼亚)、D–D (多利亚)、E–E (弗里吉亚)、F–F (利底亚)、G–G (混合利底亚)、A–A (爱奥里亚)、B–B (洛克里亚)。每个调式均有独特的全半音模式。以下给出相对于同主音大调（或伊奥尼亚调式）的公式，其中 ♭ 表示降低的级数，♯ 表示升高的级数。

Mode / 调式 Formula relative to major / 相对大调公式 Semitone pattern / 半音模式
Ionian (Major) 1 2 3 4 5 6 7 2 2 1 2 2 2 1
Dorian 1 2 ♭3 4 5 6 ♭7 2 1 2 2 2 1 2
Phrygian 1 ♭2 ♭3 4 5 ♭6 ♭7 1 2 2 2 1 2 2
Lydian 1 2 3 ♯4 5 6 7 2 2 2 1 2 2 1
Mixolydian 1 2 3 4 5 6 ♭7 2 2 1 2 2 1 2
Aeolian (Natural minor) 1 2 ♭3 4 5 ♭6 ♭7 2 1 2 2 1 2 2
Locrian 1 ♭2 ♭3 4 ♭5 ♭6 ♭7 1 2 2 1 2 2 2

A quick formula: the Dorian mode is a natural minor with a raised sixth; Phrygian is natural minor with a lowered second; Lydian is major with a raised fourth; Mixolydian is major with a lowered seventh; Locrian is natural minor with a lowered second and fifth. | 快速记忆：多利亚为升高六级的小调；弗里吉亚为降低二级的小调；利底亚为升高四级的大调；混合利底亚为降低七级的大调；洛克里亚为降低二级和五级的小调。

7. Rhythmic Value Ratios & Time Signature Formula | 节奏时值比值与拍号公式

Note durations are related by powers of two: semibreve (whole note) = 2 minims = 4 crotchets = 8 quavers = 16 semiquavers. A dot adds half the original note’s value: dotted note = original duration × 1.5. A double dot adds another half of the dot’s value, totalling 1.75 times the original. The time signature consists of a top number indicating beats per bar and a bottom number indicating the beat unit (2: minim, 4: crotchet, 8: quaver). Simple time signatures have a top number of 2, 3, or 4; compound time signatures have a top number of 6, 9, or 12, where each beat divides into three. The beat formula for compound time: number of beats = top number ÷ 3; beat unit = dotted note of the bottom number’s value.

音符时值以 2 的幂次相关联：全音符（四拍）= 2 个二分音符 = 4 个四分音符 = 8 个八分音符 = 16 个十六分音符。附点增加原时值的一半：附点音符 = 原时值 × 1.5。复附点再增加附点时值的一半，总时长为原时值的 1.75 倍。拍号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（2: 二分音符，4: 四分音符，8: 八分音符）。单拍子拍号上方为 2、3 或 4；复拍子拍号为 6、9 或 12，每拍均分为三等份。复拍子拍数公式：拍数 = 上方数字 ÷ 3；拍位单位等于下方数字对应音符的附点音符。

  • Time signature conversion: simple duple 2/4, triple 3/4, quadruple 4/4; compound duple 6/8, triple 9/8, quadruple 12/8. | 拍号换算：单二拍 2/4，单三拍 3/4，单四拍 4/4；复二拍 6/8，复三拍 9/8，复四拍 12/8。
  • Tuplet formulas: a triplet in simple time fits three notes into the space of two; in compound time, a duplet fits two notes into the space of three. | 连音公式：单拍子中的三连音以三个音代替两个时值；复拍子中的二连音以两个音代替三个时值。

8. Cadence Types & Chord Progression Formulas | 终止式类型与和弦进行公式

Cadences are harmonic formulas that articulate the end of a phrase. The four principal cadences are perfect (V–I), imperfect (I–V, ii–V, IV–V, etc., ending on V), plagal (IV–I), and interrupted (V–vi, or rarely V–♭VI). In minor keys, the V chord is almost always major (with raised leading note) for a perfect cadence; a minor v–i produces a modal cadence. Cadence formulas can include inversions: perfect cadence Ic–V–I is a strong, conventional closing progression where the dominant is approached from a second-inversion tonic. An interrupted cadence creates a deceptive resolution, typically V–vi, but in minor may be V–VI. The plagal cadence (IV–I) is often called the ‘Amen’ cadence.

终止式是明确乐句结束的和声公式。四种主要终止式为：正格终止 (V–I)，半终止 (I–V, ii–V, IV–V 等，结束于 V)，变格终止 (IV–I) 和阻碍终止 (V–vi，少见 V–♭VI)。在小调中，V 和弦几乎总是大三和弦（升高导音）以构成正格终止；小调 v–i 则为调式终止。终止式公式可包含转位：正格终止中的 Ic–V–I 是强有力的传统收束，其中属和弦前面是一个第二转位的主和弦。阻碍终止产生伪终止效果，通常为 V–vi，小调中可为 V–VI。变格终止 (IV–I) 常被称作“阿们终止”。

  • Perfect cadence: V(7)–I; in minor V–i. | 正格终止：V(7)–I；小调 V–i。
  • Plagal cadence: IV–I; minor iv–i. | 变格终止：IV–I；小调 iv–i。
  • Imperfect cadence: any chord ending on V (I–V, ii–V, IV–V). | 半终止：任何结束在 V 的和声进行 (I–V, ii–V, IV–V)。
  • Interrupted cadence: V–vi (major) or V–VI (minor). | 阻碍终止：V–vi (大调) 或 V–VI (小调)。

9. Harmonic Function & Chord Substitution Theorems | 和声功能与和弦替代定理

In functional harmony, chords are grouped into three primary functions: tonic (I, vi, occasionally iii), subdominant (IV, ii, vi, ♭VI, Neapolitan etc.), and dominant (V, vii°, ♭VII). Substitution theorems state that chords sharing two or more common tones can often replace each other while preserving the harmonic direction. The tonic family prolongs the sense of rest; the subdominant family leads away from tonic, preparing the dominant; the dominant family creates tension that resolves back to tonic. The strongest chord progressions move through these functions in the order T–S–D–T or T–D–T. The use of secondary dominants (V/V, V/IV, etc.) tonicises non-tonic chords, creating temporary key centres. The formula for a secondary dominant is the V(7) of the target chord; it must contain the leading note of that temporary key. Chromatic mediants (e.g., I–♭III, I–♭VI) are also possible substitutions that preserve the common tone of the third or sixth degree.

在功能和声中，和弦被分为三种主要功能：主功能 (I, vi，偶有 iii)、下属功能 (IV, ii, vi, ♭VI, 那不勒斯和弦等) 和属功能 (V, vii°, ♭VII)。替代定理指出，共享两个或更多共同音的和弦往往可彼此替换，而保持和声方向不变。主功能族延续了稳定感；下属功能族离开主和弦，预备属和弦；属功能族制造张力，最终解决到主和弦。最强劲的和弦进行通常遵循 T–S–D–T 或 T–D–T 的顺序。副属和弦（V/V、V/IV 等）将非主和弦临时主音化，形成短暂的调性中心。副属和弦公式为目标和弦的 V(7) 和弦；它必须包含该临时调的导音。变化中音和弦（如 I–♭III, I–♭VI）也是可能的替代，它们保留了三音或六音的共同音。

  • Predominant substitutes: IV can be substituted by ii, ii6, or the Neapolitan ♭II6 (first-inversion). | 下属功能替代：IV 可用 ii、ii6 或那不勒斯 ♭II6（第一转位）替代。
  • Dominant substitutes: V7 can be replaced by vii°7 or an augmented sixth chord, providing similar voice-leading tendencies. | 属功能替代：V7 可由 vii°7 或增六和弦替代，提供相似的声部倾向。

10. Voice Leading Rules & Theorems | 声部进行规则与定理

Common-practice voice leading aims for smoothness and independence of parts. Fundamental rules include: avoid parallel perfect fifths and octaves; avoid hidden (direct) fifths and octaves between outer voices unless the soprano moves by step; avoid augmented melodic intervals; leading notes should resolve up by step to the tonic when in an outer voice; chords should be spaced with the largest intervals in the lower voices; the bass and soprano should move in contrary motion wherever possible; common tones are retained in the same voice when possible. The law of the shortest way states that each voice should move the smallest possible interval to the next chord tone. In four-part writing, doubled notes are usually the root of the chord; never double the leading note or the seventh of a chord. Secondary seventh chords resolve by the seventh falling by step and the leading note rising.

共性实践声部进行追求平稳与声部独立。基本规则包括：避免平行纯五度与纯八度；避免外声部间的隐伏（直接）纯五度、纯八度，除非 soprano 为级进；避免增音程的旋律进行；导音在外声部时应向上级进解决到主音；和弦排列时应使低音区间距最大；男低声部与女高声部尽可能反向进行；共同音尽可能保留在同一声部。“最短路径法则”要求每个声部移动到下一和弦音时选择最小的音程。在四部和声写作中，重复音通常为和弦根音；不可重复导音或七音。副属七和弦的解决要求七音下行级进，导音上行。

  • The resolution formula for V7–I: the leading note (3rd of V7) resolves up; the seventh (7th of V7) resolves down by step; the common tone (5th of V7 or root) stays still or moves stepwise. | V7–I 的解决公式：导音 (V7 的三音) 上行解决；七音下行级进；共同音 (V7 的五音或根音) 保持不动或级进。
  • The approach to augmented sixth chords: the Italian sixth (+6) contains ♭6, 1, and ♯4 degrees and resolves outward to the dominant or dominant octave. | 增六和弦的进行：意大利增六和弦包含 ♭6, 1, ♯4 音级，向外解决至属音或属八度。

11. Inversion of Intervals & Chords Theorems | 转位音程与转位和弦定理

Inversion of intervals follows the rule: interval number (inverted) = 9 – original number. Quality changes as major ↔ minor, augmented ↔ diminished, perfect remains perfect. For example, a major 3rd inverts to a minor 6th (9–3=6). Chord inversions are indicated by the bass note: root position has the root in the bass; first inversion has the third in the bass; second inversion has the fifth in the bass; third inversion (for seventh chords) has the seventh in the bass. The figured bass notation summarises the intervals above the bass: root-position triad is ⁵₃ (often abbreviated to nothing), first inversion ⁶₃ (often just 6), second inversion ⁶₄. For seventh chords: root position ⁷ (implied ⁷₅₃), first inversion ⁶₅, second inversion ⁴₃, third inversion ⁴₂ (or simply 2). In analysis, pop and jazz use slash notation: C/E for first inversion of C major.

音程转位遵循“9”定理：转位后度数 = 9 − 原度数。性质互换：大 ↔ 小，增 ↔ 减，纯保持纯。例如大三度转位为小六度（9−3=6）。和弦转位由低音决定：原位低音为根音；第一转位低音为三音；第二转位低音为五音；第三转位（七和弦）低音为七音。数字低音标注低音上方音程：原位三和弦为 ⁵₃（常略去），第一转位为 ⁶₃（常记作 6），第二转位为 ⁶₄。七和弦中：原位 ⁷（含 ⁷₅₃），第一转位 ⁶₅，第二转位 ⁴₃，第三转位 ⁴₂（或简记 2）。流行音乐与爵士乐分析使用斜线标记，如 C/E 表示 C 大三和弦第一转位。

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