📚 AS OCR Music: Quick Reference to Formulae and Theorems | AS OCR 音乐：公式定理速查手册
In AS OCR Music, many concepts can be expressed as concise “rules” or “formulae” — from interval construction to chord progressions and voice-leading principles. This quick-reference handbook collects the essential theoretical building blocks, presenting each as a clear formula or theorem to support your analysis, composition, and aural skills.
在 AS OCR 音乐课程中，许多概念可以提炼为简洁的“规则”或“公式”——从音程的构成到和弦进行及声部进行原则。这本速查手册汇集了关键的理论建构块，将每个要点以清晰的公式或定理形式呈现，助力你的分析、作曲和听觉训练。
1. Interval Formulae | 音程计算公式
Formula: Interval quality + number = (number of semitones from tonic) → Interval name. For major and perfect intervals within one octave: minor 2nd = 1 semitone, Major 2nd = 2, minor 3rd = 3, Major 3rd = 4, Perfect 4th = 5, Augmented 4th/Diminished 5th = 6, Perfect 5th = 7, minor 6th = 8, Major 6th = 9, minor 7th = 10, Major 7th = 11, Perfect 8ve = 12. Augmented = widen by a semitone; diminished = narrow by a semitone.
公式：音程性质 + 度数 = （距离主音的半音数）→ 音程名称。一个八度内的常见音程：小二度 = 1 个半音，大二度 = 2，小三度 = 3，大三度 = 4，纯四度 = 5，增四度/减五度 = 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。增音程 = 扩大一个半音；减音程 = 缩小一个半音。
2. Scale Construction Formulae | 音阶构成公式
Major scale: T–T–S–T–T–T–S (T = tone, S = semitone). Natural minor: T–S–T–T–S–T–T. Harmonic minor: raise the 7th degree of natural minor (T–S–T–T–S–T+S–S). Melodic minor (ascending): raise 6th and 7th of natural minor (T–S–T–T–T–T–S); descending is natural minor.
大调音阶：全–全–半–全–全–全–半。自然小调：全–半–全–全–半–全–全。和声小调：将自然小调第VII级升高半音（全–半–全–全–半–增二度–半）。旋律小调（上行）：升高自然小调的第VI、VII级（全–半–全–全–全–全–半）；下行还原为自然小调。
3. Triad and Seventh Chord Formulae | 三和弦与七和弦构成公式
Triads (root, 3rd, 5th): Major = root + M3 + P5 (4 + 3 semitones). Minor = root + m3 + P5 (3 + 4). Diminished = root + m3 + d5 (3 + 3). Augmented = root + M3 + A5 (4 + 4). Seventh chords: Major 7th = major triad + M7; Dominant 7th = major triad + m7; Minor 7th = minor triad + m7; Half-dim 7th = diminished triad + m7; Dim 7th = diminished triad + d7.
三和弦（根音、三音、五音）：大三和弦 = 根音 + 大三度 + 纯五度（4 + 3 个半音）。小三和弦 = 根音 + 小三度 + 纯五度（3 + 4）。减三和弦 = 根音 + 小三度 + 减五度（3 + 3）。增三和弦 = 根音 + 大三度 + 增五度（4 + 4）。七和弦：大七和弦 = 大三和弦 + 大七度；属七和弦 = 大三和弦 + 小七度；小七和弦 = 小三和弦 + 小七度；半减七和弦 = 减三和弦 + 小七度；减七和弦 = 减三和弦 + 减七度。
4. Chord Inversion Figured Bass Symbols | 和弦转位与数字低音标记
Root position: 5/3 (often abbreviated to nothing). First inversion: 6/3 (often just 6). Second inversion: 6/4. Seventh chords: Root: 7; 1st inv: 6/5; 2nd inv: 4/3; 3rd inv: 4/2. These symbols indicate intervals above the bass note.
原位：5/3（常省略不写）。第一转位：6/3（常简写为 6）。第二转位：6/4。七和弦：原位：7；第一转位：6/5；第二转位：4/3；第三转位：4/2。这些符号表示低音上方的音程。
5. Cadence Formulae | 终止式公式
Perfect (authentic) cadence: V → I (or V⁷ → I), both in root position, top voice on tonic. Imperfect cadence: any chord → V, ends on dominant. Plagal cadence: IV → I. Interrupted (deceptive) cadence: V → vi (or V → VI in minor). These formulae define the standard functional endings of phrases.
正格终止：V → I（或 V⁷ → I），二者均为原位，高音结束在主音。半终止：任何和弦进行到 V，结束在属和弦。变格终止：IV → I。阻碍终止：V → vi（小调中为 V → VI）。这些公式定义了乐句的标准功能和声结尾方式。
6. Common Chord Progression Patterns | 常用和弦进行模式
Circle of fifths progression: I–IV–vii°–iii–vi–ii–V–I (or its segments). Pachelbel sequence: I–V–vi–iii–IV–I–IV–V. Blues 12-bar formula: I⁷ (4 bars) – IV⁷ (2) – I⁷ (2) – V⁷ (1) – IV⁷ (1) – I⁷ (2). These patterns serve as templates for tonal harmony.
五度圈进行：I–IV–vii°–iii–vi–ii–V–I（或其片段）。帕赫贝尔进行：I–V–vi–iii–IV–I–IV–V。蓝调十二小节公式：I⁷（4 小节）– IV⁷（2）– I⁷（2）– V⁷（1）– IV⁷（1）– I⁷（2）。这些模式是调性和声的常用模板。
7. SATB Voice-Leading Rules | 四声部进行规则
Parallel perfect consonances rule: Avoid parallel perfect 5ths, parallel perfect octaves, and parallel unisons between any pair of voices. Hidden (exposed) 5ths/8ves: Avoid motion of two voices into a perfect 5th or 8ve by similar motion if the upper voice leaps. Voice crossing/overlap: Voices must not cross (alto below tenor etc.) and must not overlap between successive chords. Leading note resolution: The leading note (7th degree) must resolve up by step to the tonic when in an outer voice.
平行纯音程规则：任何两声部之间须避免平行纯五度、平行纯八度及平行同度。隐伏（暴露）五度、八度：两声部以同向进行到一个纯五度或纯八度时，若上方声部有跳进，须避免。声部交错与超越：声部不得交错（如中音低于男高音等），相邻和弦之间声部不得超越。导音解决：当导音（VII 级）处于外声部时，必须上行级进解决到主音。
8. Modulation Formulae | 转调公式
Pivot chord modulation: Use a chord common to both the old and new keys as a pivot; then proceed to a chord diatonic only in the new key, especially a cadential V–I. Direct modulation: Shift key without pivot, often at a phrase boundary. Chromatic modulation: Alter one or more notes chromatically to reach the new key. Relative modulation: Between relative major and minor (e.g., C major ↔ A minor) using shared chords.
共同和弦转调：使用一个新旧两调共有的和弦作为枢纽和弦，然后进行到仅属于新调的和弦，尤其是能构成终止式 V–I 的进行。直接转调：不使用共同和弦直接切换调性，通常出现在乐句边界。半音转调：通过改变一个或多个音（半音化）抵达新调。关系调转调：在关系大小调之间（例如 C 大调 ↔ a 小调）利用共用和弦转调。
9. Non-Harmony Notes Classification | 非和弦音分类公式
Passing note (P): stepwise between two different chord tones, on weak beat. Neighbour note (N): step away from a chord tone and return, on weak beat. Suspension (S): prepared (held over) from previous chord, then resolved down by step; formula: Prep – Susp – Res. Appoggiatura (App): approached by leap, resolved by step, on strong beat. Anticipation (Ant): approaches new chord tone early, on weak beat. Escape note (E): step from chord tone, leap in opposite direction to next chord tone.
经过音 (P)：在两个不同和弦音之间以级进运动，位于弱拍。助音 (N)：从和弦音级进离开再返回，位于弱拍。延留音 (S)：从前一和弦预备（保持），然后下行级进解决；公式：预备 – 延留 – 解决。倚音 (App)：跳进引入，级进解决，位于强拍。先现音 (Ant)：提早出现下一个和弦音，位于弱拍。规避音 (E)：从和弦音级进，然后反向跳进到下一和弦音。
10. Time Signature and Metre Formulae | 拍号与节拍公式
Simple time: top number = 2, 3, or 4; beat divides into two equal parts. Compound time: top number = 6, 9, or 12; beat divides into three equal parts, dotted note is the beat unit. Irregular metre: time signatures such as 5/4, 7/8, changing between simple and compound grouping. Duple/Triple/Quadruple: depend on the number of beats per bar, not the beat division.
单拍子：分子为 2、3、4；每拍均分为两个等份。复拍子：分子为 6、9、12；每拍均分为三个等份，以附点音符为拍单位。不规则节拍：如 5/4、7/8，单拍子与复拍子分组交替。二拍子/三拍子/四拍子：取决于每小节的拍数，而非拍的分割方式。
11. Ornaments and Their Execution | 装饰音及其演奏公式
Trill (tr): rapid alternation between written note and the note above; may start on upper or main note depending on period. Mordent (upper mordent): main note – upper neighbour – main note, played quickly. Inverted mordent: main note – lower neighbour – main note. Turn: upper neighbour – main note – lower neighbour – main note. Appoggiatura: small note taking half or more of the main note’s value, usually a step above.
颤音 (tr)：在本音与上方音之间快速交替；起始音依时期可为上方音或本音。波音（上波音）：本音 – 上方助音 – 本音，快速演奏。逆波音：本音 – 下方助音 – 本音。回音：上方助音 – 本音 – 下方助音 – 本音。长倚音：以小音符表示，占据本音时值一半或更多，通常为上方级进。
12. Temperament and Tuning Reference | 律制与音准参考
Equal temperament: Octave divided into 12 equal semitones; each semitone ratio = ²√2 : 1. Concert pitch: A₄ = 440 Hz (standard). Harmonic series: fundamental f, then 2f (8ve), 3f (8ve+5th), 4f (2 8ves), 5f (2 8ves+M3), etc. The series influences brass writing and just intonation comparisons.
十二平均律：八度分为 12 个均等的半音；每个半音比值为 ¹²√2 : 1。标准音高：A₄ = 440 Hz（标准）。泛音列：基音 f，之后 2f（八度），3f（八度+五度），4f（两个八度），5f（两个八度+大三度）等。泛音列影响铜管乐器写作及纯律比较。
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