📚 AS Cambridge Music: Quick Formula & Theorem Handbook | AS剑桥音乐：公式定理速查手册
Music theory can be thought of as a language of patterns and rules. This quick-reference handbook presents the essential 'formulas' and 'theorems' you need for the AS Cambridge Music examination. From interval construction to chord progressions, these building blocks will support your listening, analysis, and composition tasks.
音乐理论可以被视为一种由模式与规则构成的语言。本速查手册为你呈现 AS 剑桥音乐考试中必不可少的“公式”与“定理”。从音程构建到和弦进行，这些基础模块将助力你的听力、分析与作曲任务。
1. Interval Formulas | 音程公式
An interval is the distance between two pitches, measured in semitones. The following table shows the names, abbreviations, and semitone counts for simple intervals within an octave.
音程是两个音高之间的距离，以半音为单位。下表显示了八度以内简单音程的名称、缩写和半音数。
| Interval | Abbreviation | Semitones | Example from C |
|---|---|---|---|
| Perfect Unison | P1 | 0 | C – C |
| Minor Second | m2 | 1 | C – D♭ |
| Major Second | M2 | 2 | C – D |
| Minor Third | m3 | 3 | C – E♭ |
| Major Third | M3 | 4 | C – E |
| Perfect Fourth | P4 | 5 | C – F |
| Tritone (Augmented 4th / Diminished 5th) | TT | 6 | C – F♯ / G♭ |
| Perfect Fifth | P5 | 7 | C – G |
| Minor Sixth | m6 | 8 | C – A♭ |
| Major Sixth | M6 | 9 | C – A |
| Minor Seventh | m7 | 10 | C – B♭ |
| Major Seventh | M7 | 11 | C – B |
| Perfect Octave | P8 | 12 | C – C |
The formula for any interval is simply the number of semitones between the two notes. You must also consider the letter names to determine the correct interval quality (e.g., C–E♭ is a minor third because it spans three letters C-D-E, not a second).
任何音程的公式就是两个音符之间的半音数。你还必须考虑字母名称以确定正确的音程性质（例如，C–E♭是小三度，因为它跨越三个字母C-D-E，而不是二度）。
2. Major Scale Formula | 大调音阶公式
The major scale follows a fixed pattern of whole steps (W) and half steps (H). In British terminology, a whole step is a tone (T) and a half step is a semitone (S).
大调音阶遵循全音（W）和半音（H）的固定模式。在英国术语中，全音称为 tone (T)，半音称为 semitone (S)。
Major Scale: T – T – S – T – T – T – S
大调音阶公式：全 – 全 – 半 – 全 – 全 – 全 – 半
Using the key of C major as a model, this gives the notes C – D – E – F – G – A – B – C. The semitones occur between E–F and B–C, which are natural half steps.
以C大调为模型，得到音符 C – D – E – F – G – A – B – C。半音出现在E–F和B–C之间，它们是自然的半音关系。
To build any major scale, start on the tonic and apply the T-T-S-T-T-T-S sequence. For G major, this yields G – A – B – C – D – E – F♯ – G, where the F♯ is required to maintain the pattern.
要构建任何大调音阶，从主音开始套用 T-T-S-T-T-T-S 序列。对于G大调，得到 G – A – B – C – D – E – F♯ – G，其中F♯是为了保持模式所必需的。
3. Minor Scale Formulas | 小调音阶公式
There are three common forms of the minor scale, each with its own intervallic structure. The natural minor uses the same key signature as its relative major.
小调音阶有三种常见形式，每种都有其自身的音程结构。自然小调使用与其关系大调相同的调号。
Natural Minor: T – S – T – T – S – T – T
自然小调：全 – 半 – 全 – 全 – 半 – 全 – 全
The harmonic minor raises the 7th degree by a semitone, creating a characteristic augmented second (A2) between the 6th and 7th degrees.
和声小调将第VII级升高半音，在VI级与VII级之间形成一个特征性的增二度（A2）。
Harmonic Minor: T – S – T – T – S – A2 – S
和声小调：全 – 半 – 全 – 全 – 半 – 增二度 – 半
The melodic minor has two versions: ascending raises both the 6th and 7th degrees, while descending reverts to the natural minor form.
旋律小调有两种形式：上行时升高VI级和VII级，下行时恢复到自然小调的形式。
Melodic Minor (Ascending): T – S – T – T – T – T – S
旋律小调（上行）：全 – 半 – 全 – 全 – 全 – 全 – 半
In A harmonic minor, the notes are A – B – C – D – E – F – G♯ – A. The gap between F and G♯ is an augmented second (3 semitones). The melodic ascending form would be A – B – C – D – E – F♯ – G♯ – A.
在a和声小调中，音符是 A – B – C – D – E – F – G♯ – A。F与G♯之间的空隙是增二度（3个半音）。旋律上行形式则是 A – B – C – D – E – F♯ – G♯ – A。
4. Triad Chord Formulas | 三和弦公式
A triad is a three-note chord built by stacking two intervals of a third above a root. The quality of the triad is determined by the sequence of intervals in semitones.
三和弦是在根音上叠加两个三度构成的三音和弦。三和弦的性质由半音间隔序列决定。
Major Triad: Root + 4 semitones + 3 semitones (M3 + m3)
大三和弦：根音 + 4半音 + 3半音（大三度+小三度）
Minor Triad: Root + 3 + 4 (m3 + M3)
小三和弦：根音 + 3 + 4（小三度+大三度）
Diminished Triad: Root + 3 + 3 (m3 + m3)
减三和弦：根音 + 3 + 3（小三度+小三度）
Augmented Triad: Root + 4 + 4 (M3 + M3)
增三和弦：根音 + 4 + 4（大三度+大三度）
For example, C major is C – E – G (C to E = 4 semitones, E to G = 3 semitones). C minor is C – E♭ – G (3 + 4). C diminished is C – E♭ – G♭ (3 + 3). C augmented is C – E – G♯ (4 + 4).
例如，C大三和弦是 C – E – G（C到E = 4半音，E到G = 3半音）。C小三和弦是 C – E♭ – G（3+4）。C减三和弦是 C – E♭ – G♭（3+3）。C增三和弦是 C – E – G♯（4+4）。
5. Seventh Chord Formulas | 七和弦公式
A seventh chord adds a fourth note above the triad root, formed by stacking an additional third. The most common types are identified by their triad quality and the interval of the seventh.
七和弦在三和弦根音上方增加第四个音，由再叠加一个三度构成。最常见的类型由其三和弦性质和七度的音程来识别。
Dominant 7th (Major-minor 7th): Major triad + minor 7th (4+3+3 semitones)
属七和弦（大小七）：大三和弦 + 小七度（4+3+3半音）
Minor 7th: Minor triad + minor 7th (3+4+3)
小七和弦：小三和弦 + 小七度（3+4+3）
Major 7th: Major triad + major 7th (4+3+4)
大七和弦：大三和弦 + 大七度（4+3+4）
Half-diminished 7th: Diminished triad + minor 7th (3+3+4)
半减七和弦：减三和弦 + 小七度（3+3+4）
Diminished 7th: Diminished triad + diminished 7th (3+3+3)
减七和弦：减三和弦 + 减七度（3+3+3）
Thus, a C dominant 7th would be C – E – G – B♭ (10 semitones above root); a C minor 7th is C – E♭ – G – B♭; a C major 7th is C – E – G – B; a C half-diminished is C – E♭ – G♭ – B♭; and a C diminished 7th is C – E♭ – G♭ – B♭♭ (enharmonically A). In harmonic analysis, these chords often serve functional roles in cadences and modulations.
因此，C属七和弦是 C – E – G – B♭（根音上方10半音）；C小七是 C – E♭ – G – B♭；C大七是 C – E – G – B；C半减七是 C – E♭ – G♭ – B♭；C减七是 C – E♭ – G♭ – B♭♭（等音为A）。在和声分析中，这些和弦常常在终止式和转调中扮演功能性角色。
6. Cadence Formulas | 终止式公式
Cadences are chord progressions that punctuate the end of a phrase. They are defined by the harmonic movement and the sense of closure they provide.
终止式是标点乐句结束的和弦进行。它们由和声运动和所提供的收束感来定义。
Perfect (Authentic) Cadence: V – I (or V⁷ – I)
完全终止：V – I（或 V⁷ – I）
Plagal Cadence: IV – I
变格终止：IV – I
Imperfect Cadence: any chord – V (commonly I – V, ii – V, IV – V)
不完全终止：任何和弦到V（常见 I – V，ii – V，IV – V）
Interrupted (Deceptive) Cadence: V – vi (or V – VI in minor)
阻碍终止：V – vi（或小调中 V – VI）
The perfect cadence creates the strongest sense of resolution. In the key of C major, a perfect cadence would be G major (V) to C major (I). An interrupted cadence in the same key would be G major to A minor (vi). Knowing these formulas helps identify phrase endings in score analysis and informs your own compositional phrasing.
完全终止产生最强的解决感。在C大调中，完全终止会是G大三和弦（V）到C大三和弦（I）。同一调中的阻碍终止会是G大三和弦到a小三和弦（vi）。了解这些公式有助于在乐谱分析中识别乐句结尾，并指导你自己的作曲分句。
7. Chord Inversion Rules | 和弦转位规则
When a chord tone other than the root is placed in the bass, the chord is inverted. Each inversion has a specific figured bass symbol and intervallic structure.
当根音以外的和弦音被置于低音声部时，和弦就发生了转位。每种转位都有特定的数字低音标记和音程结构。
Triads: Root position (⁵₃) – bass is root; First inversion (⁶) – bass is 3rd; Second inversion (⁶₄) – bass is 5th
三和弦：原位（⁵₃）– 低音为根音；第一转位（⁶）– 低音为三音；第二转位（⁶₄）– 低音为五音
Seventh chords: Root position (⁷) ; First inversion (⁶₅) – 3rd in bass; Second inversion (⁴₃) – 5th in bass; Third inversion (⁴₂) – 7th in bass
七和弦：原位（⁷）；第一转位（⁶₅）– 低音为三音；第二转位（⁴₃）– 低音为五音；第三转位（⁴₂）– 低音为七音
For example, a C major triad in first inversion has E as the lowest note, spelled E – G – C. A G dominant 7th (G – B – D – F) in second inversion has D – F – G – B as its typical voicing. These inversions affect harmonic stability and voice-leading choices.
例如，C大三和弦的第一转位以E为最低音，记作 E – G – C。G属七和弦（G – B – D – F）的第二转位通常排列为 D – F – G – B。这些转位影响和声稳定性与声部进行的选择。
8. Time Signature and Rhythm Division | 拍号与节奏划分
A time signature's upper number indicates the number of beats per bar, while the lower number specifies the note value that represents one beat. This simple formula governs the metric structure.
拍号的上方数字表示每小节的拍数，下方数字则指定代表一拍的
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