Formula & Theorem Quick-Reference Handbook | 公式定理速查手册

📚 Formula & Theorem Quick-Reference Handbook | 公式定理速查手册

This quick-reference guide collates the essential theoretical ‘formulas’ and rules that Year 13 musicians need for the CAIE A Level Music course. From interval spelling to cadence recipes, harmonic conventions to transposing instrument calculations, these concise statements will support your listening, analysis, and composing tasks—treat them as your musical equations.

本速查手册整理了CAIE A Level音乐课程中 Year 13 学生必须掌握的核心理论“公式”与规则。从音程拼写到终止式配方，从和声惯例到移调乐器的换算，这些精练的陈述将为你的听力、分析与作曲任务提供支持——把它们当作你的音乐方程式吧。

1. Interval Semitone Count | 音程半音数公式

To identify an interval, count the letter names to determine the number (2nd, 3rd, etc.), then count the semitones between the notes. For example, a major 2nd spans 2 semitones (C–D), a minor 3rd spans 3 semitones (C–E♭), a perfect 4th spans 5 semitones (C–F), and an augmented 4th/diminished 5th spans 6 semitones (C–F♯).

识别音程时，先数字母名以确定度数（二度、三度等），再数两个音之间的半音数。例如，大二度跨2个半音（C–D），小三度跨3个半音（C–E♭），纯四度跨5个半音（C–F），增四度/减五度跨6个半音（C–F♯）。

Interval 音程	Semitones 半音数
Minor 2nd	1
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 Octave	12

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2. Triad Construction Formula | 三和弦构成公式

A triad is built by stacking two thirds on a root. Major triad: root + major 3rd (4 semitones) + perfect 5th (7 semitones from root). 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).

三和弦由根音上方堆叠两个三度构成。大三和弦：根音 + 大三度（4个半音） + 纯五度（距根音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）。

Major = R + 4 + 7 | Minor = R + 3 + 7 | Dim = R + 3 + 6 | Aug = R + 4 + 8

大三和弦 = 根 + 4 + 7 | 小三和弦 = 根 + 3 + 7 | 减三 = 根 + 3 + 6 | 增三 = 根 + 4 + 8

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3. Seventh Chord Formulas | 七和弦公式

The four main types of seventh chords are built by adding a seventh above the root to a triad. Dominant 7th: major triad + minor 7th (10 semitones). Major 7th: major triad + major 7th (11 semitones). Minor 7th: minor triad + minor 7th (10 semitones). Diminished 7th: diminished triad + diminished 7th (9 semitones). Half-diminished 7th: diminished triad + minor 7th (10 semitones).

四种主要七和弦是在三和弦基础上加根音上方的七度音。属七和弦：大三和弦 + 小七度（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（10个半音）。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（10个半音）。

Chord 和弦	Intervals from root 根音上方音程
Dominant 7th	M3, P5, m7 (4+7+10)
Major 7th	M3, P5, M7 (4+7+11)
Minor 7th	m3, P5, m7 (3+7+10)
Diminished 7th	m3, d5, d7 (3+6+9)
Half-diminished 7th	m3, d5, m7 (3+6+10)

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4. Cadence Recipes | 终止式配方

A cadence is a two-chord formula that closes a phrase. Perfect cadence: V → I (or V⁷ → I). Plagal cadence: IV → I. Imperfect cadence: I → V, ii → V, or IV → V. Interrupted (deceptive) cadence: V → vi (or V → VI in minor). Always check the leading note’s resolution in the perfect cadence: 7→8 and 2→3 in major keys.

终止式是结束乐句的两和弦公式。完全终止：V → I（或 V⁷ → I）。变格终止：IV → I。不完全终止：任何和弦到 V，如 I → V、ii → V、IV → V。阻碍终止：V → vi（小调中 V → VI）。完全终止中务必检查导音的解-决：大调中 7→8，2→3。

Perfect: V–I | Plagal: IV–I | Imperfect: →V | Interrupted: V–vi

完全终止: V–I | 变格终止: IV–I | 不完全终止: →V | 阻碍终止: V–vi

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5. Circle of Fifths & Key Signatures | 五度圈与调号

Sharps are added in the order F–C–G–D–A–E–B; flats in the reverse order B–E–A–D–G–C–F. The number of sharps or flats in a major key can be found by moving clockwise (sharps) or anticlockwise (flats) around the circle of fifths from C major. The relative minor lies a minor 3rd below the major tonic.

升号调按照 F–C–G–D–A–E–B 的顺序添加；降号调按反向顺序 B–E–A–D–G–C–F。大调调号中升降号的数量可通过从 C 大调出发，顺时针（升号）或逆时针（降号）绕五度圈得出。关系小调的主音位于大调主音下方小三度。

Major Key	Sharps/Flats	Relative Minor
G	1♯ (F♯)	E minor
D	2♯ (F♯ C♯)	B minor
A	3♯	F♯ minor
E	4♯	C♯ minor
F	1♭ (B♭)	D minor
B♭	2♭ (B♭ E♭)	G minor

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6. Harmonic Series & Chord Functions | 和声序列与和弦功能

In a major key, the diatonic triads are: I (tonic), ii (supertonic, minor), iii (mediant, minor), IV (subdominant), V (dominant), vi (submediant, minor), vii° (leading note, diminished). Primary triads (I, IV, V) establish the key; secondary triads (ii, iii, vi, vii°) provide colour and substitute functions. The chord progression formula I–IV–V–I is the most fundamental tonal closure.

在大调中，自然音三和弦为：I（主和弦）、ii（上主和弦，小三）、iii（中和弦，小三）、IV（下属和弦）、V（属和弦）、vi（下中和弦，小三）、vii°（导和弦，减三）。正三和弦（I、IV、V）确立调性；副三和弦（ii、iii、vi、vii°）提供色彩与替代功能。和弦进行公式 I–IV–V–I 是最基本的调性闭合。

Diatonic chords in C major: C, Dm, Em, F, G, Am, B°

C大调自然音和弦：C，Dm，Em，F，G，Am，B°

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7. Modulation Shortcuts | 转调快捷方式

Modulation to a closely related key uses the pivot chord—a chord common to both the old and the new key. The most common modulations are to the dominant (V), the relative minor (vi), and the subdominant (IV). In a major key, the V of the new key (applied dominant) can be used to strengthen the modulation. Example: C major to G major, pivot chord is C major (I in C = IV in G) or Am (vi in C = ii in G).

向近关系调转调需使用中介和弦——旧调与新调共有的和弦。最常见的转调方向是到属调（V）、关系小调（vi）和下属调（IV）。在大调中，可使用新调的属和弦（副属和弦）加强转调。例：C大调转G大调，中介和弦可以是C大三和弦（C调I = G调IV）或Am（C调vi = G调ii）。

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8. Voice Leading Rules | 声部进行规则

In four-part harmony, avoid parallel octaves and parallel fifths between any pair of voices. Unequal fifths (d5 → P5) should also be avoided. The leading note should rise to the tonic when in an outer voice. Leaps of a diminished or augmented interval are to be avoided. Each voice should remain within its typical range and move as smoothly as possible (stepwise preferred).

在四部和声中，任何两个声部之间应避免平行八度和平行五度。不等五度（d5 → P5）同样应避免。导音在外声部时应上行解决到主音。应避免增减音程的跳进。每个声部应保持在典型音域内，并尽可能平稳进行（级进优先）。

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9. Figured Bass Decoder | 数字低音解码

Figured bass numbers indicate intervals above the bass note. A ‘5’ alone is understood as a root-position triad (no figure often means 5 3). ‘6’ alone means first inversion (a 6th and a 3rd above bass). ‘6 4’ means second inversion. For seventh chords: ‘7’ implies root-position; ‘6 5’ first inversion; ‘4 3’ second inversion; ‘4 2’ (or ‘2’) third inversion. Accidentals in the figuring alter the indicated interval.

数字低音中的数字表示低音上方的音程。单独一个“5”代表原位三和弦（通常没有数字也意味着 5 3）。“6”单独表示第一转位（低音上方六度和三度）。“6 4”表示第二转位。七和弦的数字：“7”代表原位；“6 5”第一转位；“4 3”第二转位；“4 2”（或“2”）第三转位。数字中的临时升降号改变所指的音程。

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10. Ornament Interpretation | 装饰音演奏公式

Ornament	Sign	Execution
Trill	tr	Rapid alternation between main note and upper auxiliary note, usually starting on the main note in Classical style.
Turn	~	Upper auxiliary, main note, lower auxiliary, main note.
Mordent (upper)	short squiggle	Main note, upper auxiliary, main note.
Appoggiatura	Small note	Takes half the value of the main note (or two thirds if dotted), on the beat.
Acciaccatura	Small note with slash	Crushed, played very quickly before the main note.

装饰音演奏公式：颤音（tr）在主音与上方辅助音之间快速交替，古典时期通常从主音开始。回音（~）演奏为：上方辅助音-主音-下方辅助音-主音。波音（上波音）演奏为主-上辅-主。倚音（小音符）占用主音一半时值（附点音符占三分之二），在拍上演奏。碎音（带斜线的小音符）极快地挤在主音前演奏。

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11. Transposing Instrument Calculation | 移调乐器换算式

A transposing instrument sounds at a different pitch from its written note. The formula: sounding pitch = written pitch + transposition interval. For B♭ instruments (clarinet, trumpet): written C sounds B♭ (major 2nd lower). For F instruments (horn): written C sounds F (perfect 5th lower). For E♭ instruments (alto sax): written C sounds E♭ (major 6th lower). To write for a B♭ instrument in concert pitch, write the note a major 2nd higher.

移调乐器的实际音高与记谱音高不同。公式：实际音高 = 记谱音高 + 移调音程。B♭调乐器（单簧管、小号）：记谱C实际为B♭（低大二度）。F调乐器（圆号）：记谱C实际为F（低纯五度）。E♭调乐器（中音萨克斯）：记谱C实际为E♭（低大六度）。为B♭调乐器写总谱时，要把音乐会音高移高大二度记谱。

Written C = Sounding: B♭ (Trumpet/Clar.) | F (Horn) | E♭ (Alto Sax)

记谱C = 实际音高: B♭ (小号/单簧管) | F (圆号) | E♭ (中音萨克斯)

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12. Sonata Form Blueprint | 奏鸣曲式蓝图

Sonata form is a large-scale structure with three main sections. Exposition: First subject in tonic key, transition (modulating), second subject in a contrasting key (usually dominant in major, relative major in minor), closing theme. Development: motivic development of exposition material, exploring distant keys. Recapitulation: restatement of both subjects in the tonic key, often with a coda. The ‘formula’ helps shape analytical essays and composition tasks.

奏鸣曲式是一种包含三个主要部分的大型结构。呈示部：主调中的第一主题、连接部（转调）、建立在对比调性上的第二主题（大调中通常为属调，小调中为关系大调），以及结束主题。展开部：对呈示部材料进行动机发展，探索远关系调。再现部：两个主题在主调中重现，常带有尾声。这个“公式”有助于组织分析论文和作曲任务。

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