📚 AS CAIE Music: Formula & Theorem Quick Reference | AS CAIE 音乐：公式定理速查手册
In the AS CAIE Music syllabus, understanding the underlying ‘formulas’ of music theory is as crucial as mastering equations in the sciences. This handbook compiles the essential rules and patterns – from interval counting to cadence resolutions – that will sharpen your analysis, composition, and listening skills for the exam. Use it as your rapid recall companion.
在 AS CAIE 音乐大纲中，掌握音乐理论的底层’公式’就像掌握科学中的方程式一样关键。本手册汇编了从音程计算到终止式解决的基本规则与模式，这些都将提升你在考试中的分析、作曲与聆听能力。把它当作你的快速记忆伴侣。
1. Interval Formula | 音程公式
Every interval quality and size is defined by the number of semitones between two notes. An interval name has two parts: a number (2nd, 3rd, 4th, etc.) and a quality (perfect, major, minor, augmented, diminished). The ‘formula’ is to count letter names for the number, then count semitones to determine the quality. For example, C to E spans three letter names (C-D-E) and 4 semitones, so it is a major 3rd. Turning that into a minor 3rd requires lowering the top note by a semitone (3 semitones).
每个音程的属性和大小由两个音符之间的半音数决定。音程名称有两部分：数字（二度、三度、四度等）和属性（纯、大、小、增、减）。’公式’是先数字母名称来确定度数，再数半音来确定属性。例如，C 到 E 跨越三个字母名 (C-D-E) 并且有 4 个半音，所以是大三度。变成小三度则需要把上方音降低一个半音（3 个半音）。
| Interval | Semitones | Common Example | 音程 | 半音数 |
|---|---|---|---|---|
| Perfect Unison | 0 | C–C | 纯一度 | 0 |
| Minor 2nd | 1 | E–F | 小二度 | 1 |
| Major 2nd | 2 | C–D | 大二度 | 2 |
| Minor 3rd | 3 | A–C | 小三度 | 3 |
| Major 3rd | 4 | C–E | 大三度 | 4 |
| Perfect 4th | 5 | C–F | 纯四度 | 5 |
| Perfect 5th | 7 | C–G | 纯五度 | 7 |
| Minor 6th | 8 | E–C | 小六度 | 8 |
| Major 6th | 9 | C–A | 大六度 | 9 |
| Minor 7th | 10 | G–F | 小七度 | 10 |
| Major 7th | 11 | C–B | 大七度 | 11 |
| Perfect Octave | 12 | C–C | 纯八度 | 12 |
2. Major and Minor Scale Formulas | 大小调音阶公式
All major scales follow the same pattern of tones (T) and semitones (S): T-T-S-T-T-T-S. Starting from any note, applying this sequence generates the exact major scale. The natural minor scale formula is T-S-T-T-S-T-T. The harmonic minor raises the 7th degree by a semitone, creating a characteristic augmented 2nd between the 6th and 7th degrees. The melodic minor raises both the 6th and 7th degrees when ascending, and reverts to the natural minor when descending.
所有大调音阶都遵循相同的大全音（T）与半音（S）模式：全-全-半-全-全-全-半。从任何音开始，应用此序列就能生成准确的大调音阶。自然小调音阶公式为全-半-全-全-半-全-全。和声小调将第七级音升高半音，在第六与第七级之间产生特征性的增二度。旋律小调在上行时升高第六和第七级，下行时恢复为自然小调。
Major: T – T – S – T – T – T – S
大调：全 全 半 全 全 全 半
Natural minor: T – S – T – T – S – T – T
3. Triad Construction Formula | 三和弦构造公式
A triad is built by stacking two 3rds on top of a root. The quality depends on the combination of intervals: a major third plus a perfect fifth creates a major triad; a minor third plus a perfect fifth creates a minor triad. The exact semitone stacks are: major triad = root + 4 semitones + 3 semitones (from the previous note). Minor triad = root + 3 semitones + 4 semitones. Diminished triad = root + 3 + 3; Augmented triad = root + 4 + 4.
三和弦是在根音上叠加两个三度构成。其属性取决于音程的组合：大三度加纯五度构成大三和弦；小三度加纯五度构成小三和弦。具体的半音堆叠为：大三和弦 = 根音 + 4 个半音 + 3 个半音（从前一个音算起）。小三和弦 = 根音 + 3 个半音 + 4 个半音。减三和弦 = 根音 + 3 + 3；增三和弦 = 根音 + 4 + 4。
Major: R + 4st + 3st Minor: R + 3st + 4st
大三：根 + 4半 + 3半 小三：根 + 3半 + 4半
4. Seventh Chord Formulas | 七和弦公式
Seventh chords add another 3rd on top of a triad. The most common are: dominant 7th = major triad + minor 7th (R+4+3+3); major 7th = major triad + major 7th (R+4+3+4); minor 7th = minor triad + minor 7th (R+3+4+3); half-diminished 7th = diminished triad + major 3rd (R+3+3+4); fully diminished 7th = diminished triad + minor 3rd (R+3+3+3). Knowing these stacks helps you spell any chord rapidly.
七和弦是在三和弦上再叠加一个三度。最常见的有：属七和弦 = 大三和弦 + 小七度 (R+4+3+3)；大七和弦 = 大三和弦 + 大七度 (R+4+3+4)；小七和弦 = 小三和弦 + 小七度 (R+3+4+3)；半减七和弦 = 减三和弦 + 大三度 (R+3+3+4)；全减七和弦 = 减三和弦 + 小三度 (R+3+3+3)。熟悉这些堆叠方法可助你快速拼出任何和弦。
Dominant 7th: R + 4 + 3 + 3 属七：根 + 4 + 3 + 3
Major 7th: R + 4 + 3 + 4 大七：根 + 4 + 3 + 4
5. Circle of Fifths Progression | 五度循环进行
The circle of fifths is a fundamental formula for key relationships and harmonic motion. Moving clockwise, each key is a perfect 5th higher, adding one sharp (or removing one flat). Moving anticlockwise, each key is a perfect 5th lower, adding one flat. In functional harmony, root movement by descending 5th (or ascending 4th) forms the strongest progression: iii → vi → ii → V → I. This pattern underpins countless Baroque, Classical, and Romantic works.
五度循环是调性关系与和声运动的基本公式。顺时针方向，每个调比前一个高纯五度，增加一个升号（或减少一个降号）。逆时针方向，每个调降低纯五度，增加一个降号。在功能和声中，根音下行五度（或上行四度）的进行最强有力：iii → vi → ii → V → I。无数巴洛克、古典及浪漫主义作品都以此模式为基础。
Common progression: iii → vi → ii → V → I
常见进行：iii → vi → ii → V → I
6. Cadence Formulas | 终止式公式
Cadences are musical punctuation marks, each with a fixed chord formula. The perfect (authentic) cadence is V → I (or V⁷ → I), both chords in root position and the final I on a strong beat. The plagal cadence (Amen) is IV → I. The imperfect cadence ends on V, usually preceded by I, ii, or IV. The interrupted (deceptive) cadence typically goes V → vi. In a minor key, the dominant chord is usually major (with raised leading tone).
终止式是音乐的标点符号，各有固定的和弦公式。正格终止（完全终止）是 V → I（或 V⁷ → I），两者根音位置且最后的 I 在强拍上。变格终止（阿们终止）是 IV → I。不完全终止以 V 结尾，前面常接 I、ii 或 IV。阻碍终止通常为 V → vi。在小调中，属和弦一般是大三和弦（含升高导音）。
Perfect: V – I Plagal: IV – I Imperfect: any – V Interrupted: V – vi
正格：V – I 变格：IV – I 不完全：任何 – V 阻碍：V – vi
7. Time Signature and Rhythm Formulas | 拍号与节奏公式
A time signature can be thought of as a formula: top number = number of beats per bar, bottom number = beat unit value. Simple time (2/4, 3/4, 4/4) divides each beat into two equal parts. Compound time (6/8, 9/8, 12/8) divides each beat into three equal parts, where the dotted crotchet is the beat unit. For fast analysis, remember that compound duple (6/8) has two dotted beats; compound triple (9/8) has three; compound quadruple (12/8) has four.
拍号可以看作一个公式：上方数字 = 每小节的拍数，下方数字 = 拍的时值单位。单拍子（2/4、3/4、4/4）将每拍分为两个相等部分。复拍子（6/8、9/8、12/8）将每拍分为三个相等部分，其中附点四分音符是拍单位。为快速分析，记住复二拍 (6/8) 有两拍附点；复三拍 (9/8) 有三拍；复四拍 (12/8) 有四拍。
Simple time: beat = undotted note, subdivides into 2
单拍子：拍 = 非附点音符，每拍分为两部分
Compound time: beat = dotted note, subdivides into 3
复拍子：拍 = 附点音符，每拍分为三部分
8. Melodic and Harmonic Intervals | 旋律与和声音程规则
A melodic interval is sounded successively; a harmonic interval sounds simultaneously. The same semitone-counting formula applies to both. In addition, you must be aware of interval inversion: a 3rd inverts to a 6th, a 4th to a 5th, etc. The sum of the numbers always equals 9. The quality flips major to minor, perfect stays perfect, augmented to diminished. This is a rapid way to identify large intervals—simply invert!
旋律音程是先后奏响；和声音程是同时奏响。相同的半音计数公式对两者都适用。此外，你还需要了解音程转位：三度转位成六度，四度转位成五度，等等。转位前后度数之和总是 9。属性翻转：大变小，小变大，纯保持不变，增变减。这是识别大跨度音程的快捷方法——只需转位！
Inversion rule: 9 – original number = inversion number
转位规则：9 – 原度数 = 转位度数
9. Key Signature Formula | 调号公式
Key signatures follow a strict order of sharps and flats. Sharps: F# C# G# D# A# E# B#. Flats: Bb Eb Ab Db Gb Cb Fb. You can recall them with mnemonics, but the formula is: each new sharp is a perfect 5th above the last; each new flat is a perfect 5th below. For a given key signature, the major key is one semitone above the last sharp, or the second-to-last flat from the right. The relative minor is a minor 3rd below the major tonic.
调号的升降记号有严格的顺序。升号：F# C# G# D# A# E# B#。降号：Bb Eb Ab Db Gb Cb Fb。你可以用口诀记忆，但公式是：每个新升号在前一个基础上升高纯五度；每个新降号则是降低纯五度。对于给定的调号，大调主音是最后一个升号上方半音，或者倒数第二个降号所在的音。关系小调的主音比大调主音低小三度。
Major key from sharps: last sharp + semitone 从升号找大调：最后升号 + 半音
Major key from flats: penultimate flat 从降号找大调：倒数第二个降号
10. Modulation Pivot Chords | 转调枢纽和弦公式
A pivot chord is one that is diatonic in both the old and the new key. The formula for spotting a pivot in a score is to label the chord with roman numerals in the original key, then test whether it has a plausible function in the new key. Common-chord modulation typically uses ii, IV, vi, or I in the original key that can act as a different function in the target key. For example, a vi chord in G major (E minor) could become a ii chord in D major, bridging a move to the dominant.
枢纽和弦是在原调与新调中都自然存在的和弦。在乐谱中识别枢纽的公式是：先在原调中标记罗马数字和弦，然后检验它在目标调中是否具有合理的功能。共同和弦转调通常使用原调中的 ii、IV、vi 或 I，它们可以在新调中充当不同功能。例如，G 大调的 vi 和弦（E 小三和弦）可成为 D 大调的 ii 和弦，搭建通往属调的桥梁。
Example: G major vi (Em) = D major ii (Em)
例子：G大调 vi (Em) = D大调 ii (Em)
11. Roman Numeral Analysis | 罗马数字分析规则
Roman numerals describe chords’ scale-degree function within a key. Uppercase denotes major/augmented triads; lowercase denotes minor/diminished. The formula for diatonic triads in a major key is: I ii iii IV V vi vii°. For harmonic minor, typical roman numerals are i ii° III⁺ iv V VI vii°. Inversions are indicated by figured bass shorthand: ‘b’ for 1st inversion (6), ‘c’ for 2nd inversion (6/4). Applying these rules consistently builds a precise harmonic map of any test excerpt.
罗马数字描述和弦在调内的音级功能。大写表示大三/增三和弦，小写表示小三/减三和弦。大调中的自然三和弦公式是：I ii iii IV V vi vii°。和声小调的典型罗马数字为 i ii° III⁺ iv V VI vii°。转位通过数字低音简写表示：’b’ 表示第一转位 (6)，’c’ 表示第二转位 (6/4)。持续应用这些规则可为任何考试片段构建精确的和声地图。
Major triads: I – ii – iii – IV – V – vi – vii°
大调三和弦：I – ii – iii – IV – V – vi – vii°
12. Musical Form and Structure Patterns | 曲式结构模式
Forms can be expressed as formulas of letter schemes. Binary form: AB, where A ends in a related key and B returns. Ternary form: ABA, with a contrasting middle section. Rondo form: ABACA (or ABACABA). Theme and variations uses A A’ A” A”’… Each variation retains the phrase structure and harmonic plan of the theme. Sonata form has a more complex formula: Exposition (Theme 1 – transition – Theme 2 in dominant/relative major) – Development – Recapitulation (Theme 1 – transition – Theme 2 in tonic) – coda.
曲式可以表达为字母图式的公式。二部曲式：AB，其中 A 结束在关系调，B 再返回。三部曲式：ABA，有一个对比的中段。回旋曲式：ABACA（或 ABACABA）。主题与变奏使用 A A’ A” A”’… 每个变奏保留主题的乐句结构与和声计划。奏鸣曲式有更复杂的公式：呈示部（主题 1 – 连接部 – 主题 2 在属调或关系大调）– 展开部 – 再现部（主题 1 – 连接部 – 主题 2 在主调）– 尾声。
Ternary: A B A Rondo: A B A C A Sonata: Expo – Dev – Recap
三部曲式：A B A 回旋曲式：A B A C A 奏鸣曲式：呈示部 – 展开部 – 再现部
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