📚 AS AQA Music: Formula and Theorem Quick Reference Handbook | AS AQA 音乐：公式定理速查手册
Although music is an art, a huge body of its inner workings can be expressed as clear formulas and theorems. For AQA AS Music, being able to recall these patterns instantly – from scale construction and chord building to cadence rules and structural blueprints – gives you a decisive edge in listening, analysis, composing and performing. This handbook brings together the essential ‘musical mathematics’ in one place, presented as plug-and-play formulas.
音乐虽是艺术，但其大量内在规律都可以像公式和定理一样被清晰地表达出来。针对 AQA AS 音乐课程，能瞬间回忆起这些模式——从音阶构造、和弦搭建到终止规则与结构蓝图——会让你的听力、分析、作曲和演奏能力获得巨大优势。本手册将所有关键的“音乐数学”汇集一处，以即学即用的公式呈现。
1. Major Scale Formula | 大调音阶公式
The major scale follows a fixed pattern of tones (T) and semitones (S): T – T – S – T – T – T – S. Starting on any note and applying this sequence instantly yields the correct major scale. For example, beginning on C gives C–D–E–F–G–A–B–C; beginning on G gives G–A–B–C–D–E–F♯–G. This formula is the foundation for key signatures, chord building and melody writing.
大调音阶遵循固定的全音(T)与半音(S)模式：T – T – S – T – T – T – S。以任意音为起点套用该序列，就能立刻得出正确的大调音阶。例如从 C 出发得到 C–D–E–F–G–A–B–C；从 G 出发则得到 G–A–B–C–D–E–F♯–G。此公式是调号、和弦构造与旋律写作的基石。
2. Minor Scale Formulas | 小调音阶公式
AQA AS requires fluency in the three minor scale forms. The natural minor uses T – S – T – T – S – T – T (e.g. A–B–C–D–E–F–G–A). The harmonic minor raises the 7th degree by a semitone, creating a distinctive augmented 2nd between 6 and 7: T – S – T – T – S – T+½ – S. The melodic minor raises both 6th and 7th degrees when ascending: T – S – T – T – T – T – S, but reverts to the natural minor when descending. Treat these as three separate formulas to be applied according to context.
AQA AS 要求熟练掌握三种小调形式。自然小调用 T – S – T – T – S – T – T（如 A–B–C–D–E–F–G–A）。和声小调将第 7 级音升高半音，在 6 与 7 级间产生特征性的增二度：T – S – T – T – S – T+½ – S。旋律小调上行时升高第 6、7 级：T – S – T – T – T – T – S，下行则回到自然小调。把它们当作三个独立公式，根据语境选用。
3. Chord Construction Formulas | 和弦构造公式
Build any triad by counting semitones from the root. The major triad formula: root → +4 semitones (major 3rd) → +3 semitones (minor 3rd). Minor triad: root → +3 → +4. Augmented triad: root → +4 → +4. Diminished triad: root → +3 → +3. For seventh chords, common AQA additions include dominant 7th (major triad + minor 7th: root → +4 → +3 → +3), major 7th (major triad + major 7th: root → +4 → +3 → +4) and minor 7th (minor triad + minor 7th: root → +3 → +4 → +3). These formulas let you spell any chord quickly without memorising every single one.
从根音开始数半音就能构建任何三和弦。大三和弦公式：根音 → +4 半音（大三度） → +3 半音（小三度）。小三和弦：根音 → +3 → +4。增三和弦：根音 → +4 → +4。减三和弦：根音 → +3 → +3。七和弦方面，AQA 常见的包括属七和弦（大三和弦 + 小七度：根音 → +4 → +3 → +3）、大七和弦（大三和弦 + 大七度：根音 → +4 → +3 → +4）和小七和弦（小三和弦 + 小七度：根音 → +3 → +4 → +3）。运用这些公式，无需死记每一个和弦就能迅速拼写。
4. Chord Function and Cadence Theorems | 和弦功能与终止式定理
Functional harmony in the AQA AS syllabus can be thought of as a set of theorems. The primary triads I, IV and V form the backbone: IV leads to V, V leads to I. Cadences are the punctuation marks. The perfect cadence (V–I) sounds final; the plagal cadence (IV–I) is the ‘Amen’ movement; the imperfect cadence (I/ii/IV → V) feels unfinished; the interrupted cadence (V → vi) creates a surprise. In minor keys, remember that the dominant chord is almost always major (using the raised leading note from the harmonic minor), so an E major chord resolves to A minor in a perfect cadence in A minor.
AQA AS 大纲中的功能和声可视为一套定理。正三和弦 I、IV、V 构成骨架：IV 倾向进行到 V，V 倾向解决到 I。终止式如同标点符号：完全终止(V–I)听感结束；变格终止(IV–I)是“阿们”进行；不完全终止(I/ii/IV → V)令人感到悬而未决；阻碍终止(V → vi)制造意外。在小调中，记住属和弦几乎总是大和弦（借用和声小调升高的导音），例如 a 小调完全终止中 E 大三和弦解决到 a 小三和弦。
5. Interval Recognition Theorems | 音程识别定理
An interval’s quality and number can be derived using simple counting rules. Count the letter names to determine the number (e.g. C to A is a 6th). Then measure semitones against the major-scale benchmark: for major 2nd (2 semitones), major 3rd (4), perfect 4th (5), perfect 5th (7), major 6th (9), major 7th (11), perfect octave (12). If a perfect interval is reduced by one semitone it becomes diminished; if enlarged it becomes augmented. For 2nd, 3rd, 6th and 7th, a major interval reduced by a semitone becomes minor; reduced by two semitones becomes diminished. Use these theorems to decode interval-based listening questions and score analysis instantly.
音程的性质和度数可用简单计数规则推算。先数音名字母得出度数（如 C 到 A 是六度）。再以大调音阶为基准测量半音数：大二度 2 个半音、大三度 4、纯四度 5、纯五度 7、大六度 9、大七度 11、纯八度 12。纯音程减少一个半音变为减音程，增加一个半音变为增音程。对于二、三、六、七度，大音程减少一个半音变为小音程，减少两个半音变为减音程。运用这些定理可以瞬间破解音程听力题和乐谱分析。
6. Key Signatures and the Circle of Fifths Formula | 调号与五度圈公式
Key signatures are not random; they follow the circle of fifths. The formula for sharp keys: each step clockwise adds one sharp, and the new sharp is always the leading note of the new key (a semitone below the tonic). Thus G major has F♯, D major has F♯ and C♯, and so on. The order of sharps is F♯ C♯ G♯ D♯ A♯ E♯ B♯. For flat keys, each step anticlockwise adds a flat; the new flat is the subdominant of the new key. The order of flats is B♭ E♭ A♭ D♭ G♭ C♭ F♭. To quickly identify a key signature, use these: for sharps, the last sharp is the leading note; for flats, the second-last flat is the tonic.
调号并非随意排列，它们遵循五度圈。升号调的公式：顺时针每步增加一个升号，新增升号总是新调的导音（主音下方半音）。因此 G 大调有 F♯，D 大调有 F♯ 和 C♯，依此类推。升号顺序是 F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号调：逆时针每步增加一个降号，新增降号是新调的下属音。降号顺序是 B♭ E♭ A♭ D♭ G♭ C♭ F♭。快速识别调号的定理：升号调看最后一个升号，它是该大调的导音；降号调看倒数第二个降号，它即为主音。
7. Time Signature and Rhythmic Pattern Formulas | 拍号与节奏模式公式
Time signatures encode a rhythmic formula: the top number tells you how many beats per bar, the bottom number the beat unit. Simple time (2/4, 3/4, 4/4) has beats that divide into two equal parts. Compound time (6/8, 9/8, 12/8) has dotted-crotchet beats, each dividing naturally into three quavers. The AQA AS exam expects you to distinguish these: if the main beat subdivides into threes, it’s compound. Irregular time signatures (5/4, 7/8) break the formula but can be analysed as combinations (2+3, 3+2, etc.). Rhythmic features like syncopation, hemiola and augmentation can also be framed as pattern theorems: syncopation theorem = stress displaced off strong beats; hemiola theorem = two bars of triple metre felt as three bars of duple metre.
拍号内含节奏公式：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。单拍子（2/4、3/4、4/4）每一拍均等地一分为二。复拍子（6/8、9/8、12/8）以附点四分音符为一拍，每拍自然地一分为三个八分音符。AQA AS 考试要求区辨这两者：如果主要拍子的细分是三分的，就是复拍子。不规则拍号（5/4、7/8）打破了常规公式，但可分析为组合（2+3、3+2 等）。切分、赫米奥拉比例、时值增减等节奏特征也可当作模式定理：切分定理 = 重音移出强拍；赫米奥拉定理 = 两小节三拍子听成三小节二拍子。
8. Scale Degree Name Theorem | 音级名称定理
Every note in a scale has a technical name that reflects its function – this is a consistent naming theorem across all major and minor keys. The tonic (1st) is the home note; supertonic (2nd) is one step above; mediant (3rd) sits between tonic and dominant; subdominant (4th) is a fifth below the tonic; dominant (5th) is the most important active scale degree; submediant (6th) is the relative minor-launching point; leading note (7th in major, raised 7th in harmonic/melodic minor) is a semitone below tonic. The flattened seventh in natural minor is called the subtonic. These names are essential for describing modulation, chord progressions and phrase structures in the AQA exam.
音阶中的每个音都有一个反映其功能的专业技术名称——这是一套适用于所有大小调的命名定理。主音（第 1 级）是调性中心；上主音（第 2 级）在主音上方；中音（第 3 级）介于主音和属音之间；下属音（第 4 级）位于主音下方五度；属音（第 5 级）是最重要的活跃音级；下中音（第 6 级）是关系小调的起点；导音（大调第 7 级，和声/旋律小调升高的第 7 级）在主音下方半音。自然小调中降七级称为下主音。这些名称在 AQA 考试中用于描述转调、和弦进行和乐句结构时不可或缺。
9. Modulation Formulas | 转调公式
Modulation in the AQA AS context often follows standard routes that can be memorised as formulas. To the dominant (V) is the most common in major-key music; to the relative major (III) is a default in minor keys. Pivot chords – chords shared by both keys – act as the logical link. A reliable formula for a smooth modulation: identify a pivot chord that functions as a pre-dominant in the new key, then proceed V–I to nail the new tonic. Other common formulaic modulations in the Western classical tradition: to the subdominant (IV) and to the relative minor (vi). Sequencing a passage around the circle of fifths is also a powerful modulatory formula.
AQA AS 语境下的转调通常遵循可背诵的标准路线。在大调作品中转向属调(V)最为常见；在小调中转向关系大调(III)是默认选择。共同和弦——两调共有的和弦——充当逻辑纽带。一个顺畅转调的可靠公式是：找到一个在新调中担任属前功能的共同和弦，然后进行 V–I 确立新调。西方古典传统中其他公式化转调包括：转向下属调(IV)和关系小调(vi)。围绕五度圈模进乐句也是一种强大的转调公式。
10. Texture Classification Theorem | 织体分类定理
Texture can be systematically classified using a few core theorems that cover virtually every AQA set work. Monophony: a single unaccompanied line (theorems of rhythm alone can vary). Homophony: melody plus chordal accompaniment, where all parts move in the same rhythm (homorhythmic) or a melody dominates over supporting chords (melody-dominated homophony). Polyphony: two or more independent melodic lines interweaving; this splits into contrapuntal (imitative) and non-imitative polyphony. Heterophony: simultaneous variations of the same melody – common in some folk traditions. Antiphony: call-and-response between groups. Knowing these theorems lets you precisely label a texture change the instant you hear it.
织体可以通过几个几乎涵盖所有 AQA 固定作品的核心定理来系统分类。单声织体：单一的、无伴奏线条（仅有节奏变化定理）。主调织体：旋律加和弦伴奏，其下所有声部节奏相同（同节奏主调）或旋律占据主导地位以和弦支撑（旋律支配式主调）。复调织体：两条或以上独立旋律线交织；可分为对位式（模仿复调）与非模仿复调。支声织体：同一条旋律的多个变体同时进行——常见于某些民间传统。交替织体：群体间的一问一答。掌握这些定理，让你在听到织体变化的瞬间就能精准标注。
11. Structural Formulas in Music | 音乐结构公式
Musical forms can be reduced to repeatable structural formulas. Binary form (AB) works as a two-part theorem, often with A ending on V and B returning to I. Ternary form (ABA) states a theme, departs, then returns – the reprise theorem. Rounded binary adds a thematic return within the B section. Sonata form is a grand formula: Exposition (first subject in tonic, bridge modulating to dominant, second subject in dominant), Development (modulating, fragmenting themes), Recapitulation (first and second subjects both in tonic), plus an optional Coda. The rondo formula (ABACA or ABACABA) relies on a recurring refrain. Recognising these templates allows you to map out a piece’s architecture during listening.
音乐曲式可以简化为可重复的结构公式。二部曲式(AB)是一个两部定理，通常 A 段结束在 V 调，B 段回到 I 调。三部曲式(ABA)呈示主题，离开，再返回——即再现定理。复二部曲式在 B 段内加入主题再现。奏鸣曲式是一个宏大的公式：呈示部（主部在主调，连接部转调，副部在属调）、展开部（转调、肢解主题）、再现部（主部与副部均在主调），外加可选的尾声。回旋曲式公式(ABACA 或 ABACABA) 依靠一再重现的叠句。认出这些模板，你便能边听边勾勒出作品的结构蓝图。
12. Ornamentation and Articulation Patterns | 装饰音与演奏法公式
Ornaments follow precise notational formulas. A trill (tr) formula: rapid alternation between the written note and the note above. A mordent (upper or lower, squiggle with or without vertical line) is a single rapid alternation – the upper mordent shakes once to the upper auxiliary and back, the lower mordent descends. A turn groups four notes around the main note: upper auxiliary, main, lower auxiliary, main. An appoggiatura leans on a dissonance that resolves by step; an acciaccatura is crushed quickly before the main note. Articulation marks also obey clear semantic theorems: a dot above/below the note (staccato) means short and detached, a horizontal line (tenuto) means held for full value with slight pressure, a wedge (marcato) means accented and separated. These tiny formulas drastically alter character and are a required part of AQA score analysis.
装饰音遵循精确的记谱公式。颤音(tr)公式：在本音与上方邻音之间快速交替。波音（上波音或下波音，分别为无竖线或有竖线的波浪线）是一次快速交替——上波音向上方辅助音波动一次并返回，下波音则向下波动。回音将四个音围绕本音组合：上助音、本音、下助音、本音。倚音由一个需级进解决的不协和音前倚构成；碎音则在本音前被快速“挤”出。演奏法记号同样遵循清晰的语义定理：音符上方或下方的点(断奏)表示短促、分离，横线(保持音)表示奏满时值并稍加压力，楔形(强重音)表示加重突出。这些微小的公式会显著改变音乐性格，是 AQA 乐谱分析的必要内容。
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