📚 Formula & Theorem Quick Reference for Year 12 CAIE Music | Year 12 CAIE 音乐：公式定理速查手册
This quick-reference handbook distils essential music theory concepts tested in the CAIE AS Music syllabus into clear formulas, rules, and patterns. Use it to rapidly revise intervals, scales, chords, cadences, and other building blocks of Western tonal music, as well as to reinforce aural and analytical skills.
本速查手册将 CAIE AS 音乐课程中的核心乐理概念提炼为清晰的公式、定理与规则。快速复习音程、音阶、和弦、终止式等西方调性音乐的构成要素，同时巩固听觉分析与笔试所需的关键知识。
1. Interval Quick Formulas | 音程快速公式
An interval is the distance between two pitches. Count the letter names from the lower note to the higher note (including both) to determine the interval number. Then adjust the quality using semitone counts measured against the major scale built on the lower note.
音程是两个音高之间的距离。从较低音到较高音数音名字母（包含两端）得到度数，然后以较低音的大调音阶为基准，用半音数判断音程性质。
- Perfect intervals (unison, 4th, 5th, octave): if the upper note matches the major scale, it is perfect. One semitone smaller = diminished; one semitone larger = augmented.
纯音程（同度、四度、五度、八度）：若上方音符合大调音阶则为纯。减少半音为减，增加半音为增。 - Major/minor intervals (2nd, 3rd, 6th, 7th): if the upper note matches the major scale, it is major. One semitone smaller = minor; two semitones smaller = diminished; one semitone larger than major = augmented.
大/小音程（二度、三度、六度、七度）：符合大调音阶为大音程，减少半音为小，再减半音为减，增加半音为增。
| Interval name | Semitones from tonic | Example from C |
|---|---|---|
| Minor 2nd | 1 | C–D♭ |
| Major 2nd | 2 | C–D |
| Minor 3rd | 3 | C–E♭ |
| Major 3rd | 4 | C–E |
| Perfect 4th | 5 | C–F |
| Augmented 4th / Diminished 5th | 6 | C–F♯ / C–G♭ |
| Perfect 5th | 7 | C–G |
| Minor 6th | 8 | C–A♭ |
| Major 6th | 9 | C–A |
| Minor 7th | 10 | C–B♭ |
| Major 7th | 11 | C–B |
| Perfect Octave | 12 | C–C |
2. Major Scale Formula | 大调音阶公式
The major scale follows a fixed pattern of whole tones (T) and semitones (S): T T S T T T S. Starting on any tonic, apply this sequence to generate all seven scale degrees.
大调音阶遵循全音（T）与半音（S）的固定排列：T T S T T T S。从任意主音出发，套用此序列即可生成全部七个音级。
T T S T T T S
For example, C major: C–D (T), D–E (T), E–F (S), F–G (T), G–A (T), A–B (T), B–C (S). This pattern works identically for all major keys; only the starting note changes.
例如 C 大调：C–D（T），D–E（T），E–F（S），F–G（T），G–A（T），A–B（T），B–C（S）。这一模式对所有大调均适用，仅起始音不同。
3. Minor Scale Formulas | 小调音阶公式
There are three forms of the minor scale, each built by altering the natural minor. The natural minor itself uses the pattern T S T T S T T, which is the Aeolian mode and can be derived from the 6th degree of the relative major.
小调有三种形式，均通过改变自然小调获得。自然小调使用 T S T T S T T，即伊奥利安调式，可从关系大调的第六级音推得。
Natural minor: T S T T S T T
自然小调：T S T T S T T
Harmonic minor: raise the 7th degree by a semitone from the natural minor. Pattern: T S T T S TS S (augmented 2nd between 6th and 7th).
和声小调：将自然小调的第七级音升高半音。模式：T S T T S TS S（第六与第七级间为增二度）。
Melodic minor (ascending): raise both the 6th and 7th degrees by a semitone from the natural minor. Pattern: T S T T T T S. Descending is usually the natural minor.
旋律小调（上行）：将自然小调的第六与第七级音升高半音。模式：T S T T T T S。下行通常恢复为自然小调。
| Type | Formula from tonic | A minor example |
|---|---|---|
| Natural minor | T S T T S T T | A B C D E F G A |
| Harmonic minor | T S T T S TS S | A B C D E F G♯ A |
| Melodic minor asc. | T S T T T T S | A B C D E F♯ G♯ A |
4. Chord Construction Formulas | 和弦构成公式
Chords are built by stacking intervals above a root note. The most fundamental are triads (three notes) and seventh chords (four notes). Each type has a fixed semitone pattern above the root.
和弦通过根音上方叠加音程构成。最基本的是三和弦（三个音）和七和弦（四个音）。每种类型都有一个以根音为基准的固定半音距离模式。
| Chord type | Formula (semitones above root) | Example from C |
|---|---|---|
| Major triad | 0 + 4 + 7 | C–E–G |
| Minor triad | 0 + 3 + 7 | C–E♭–G |
| Diminished triad | 0 + 3 + 6 | C–E♭–G♭ |
| Augmented triad | 0 + 4 + 8 | C–E–G♯ |
| Dominant 7th | 0 + 4 + 7 + 10 | C–E–G–B♭ |
| Major 7th | 0 + 4 + 7 + 11 | C–E–G–B |
| Minor 7th | 0 + 3 + 7 + 10 | C–E♭–G–B♭ |
| Half-diminished 7th | 0 + 3 + 6 + 10 | C–E♭–G♭–B♭ |
| Diminished 7th | 0 + 3 + 6 + 9 | C–E♭–G♭–B♭♭ (A) |
In Roman numeral analysis, chord quality is indicated by case and symbols: I (major tonic), ii (minor supertonic), iii (minor mediant), IV (major subdominant), V (major dominant), vi (minor submediant), vii° (diminished leading-tone). In a minor key using harmonic minor, the V becomes major and the vii° becomes diminished.
在罗马数字分析中，和弦性质通过大小写和符号表示：I（大主和弦）、ii（小上主和弦）、iii（小中音和弦）、IV（大下属和弦）、V（大属和弦）、vi（小下中音和弦）、vii°（减导音和弦）。在使用和声小调的小调中，V 变为大三和弦，vii° 为减三和弦。
5. Key Signature Quick Reference | 调号速查
For sharps, the order is F♯, C♯, G♯, D♯, A♯, E♯, B♯. To find the major key from sharps, go one semitone above the last sharp. For flats, the order is B♭, E♭, A♭, D♭, G♭, C♭, F♭; the major key is the second-to-last flat, except one flat (F major).
升号调的顺序是 F♯、C♯、G♯、D♯、A♯、E♯、B♯。由升号找大调：最后一个升号上方半音即为大调主音。降号调顺序为 B♭、E♭、A♭、D♭、G♭、C♭、F♭；大调是倒数第二个降号，只有一个降号时是 F 大调。
| Accidentals | Major key | Minor key (same signature) |
|---|---|---|
| 0 | C major | A minor |
| 1♯ (F♯) | G major | E minor |
| 2♯ (F♯ C♯) | D major | B minor |
| 3♯ (F♯ C♯ G♯) | A major | F♯ minor |
| 4♯ (F♯ C♯ G♯ D♯) | E major | C♯ minor |
| 1♭ (B♭) | F major | D minor |
| 2♭ (B♭ E♭) | B♭ major | G minor |
| 3♭ (B♭ E♭ A♭) | E♭ major | C minor |
6. Scale Degree Names and Chord Functions | 音级名称与和弦功能
Each note of the diatonic scale has a technical name and the chord built upon it fulfils a specific harmonic role. These relationships form the grammar of tonal music.
自然音阶的每个音级都有专业名称，其上构成的和弦承担特定的和声功能。这些关系构成了调性音乐的语法。
| Degree | Name | Chord function |
|---|---|---|
| I | Tonic | Home, stability; point of resolution |
| II | Supertonic | Pre-dominant; often moves to V |
| III | Mediant | Variable; can substitute for I or V |
| IV | Subdominant | Pre-dominant; sets up the dominant |
| V | Dominant | Maximum tension; wants to resolve to I |
| VI | Submediant | Variable; often after V as deceptive cadence |
| VII | Leading tone (raised, in major) or Subtonic | Dominant function (vii° resolves to I) |
In a major key, the primary triads I, IV, V are all major; in a minor key (harmonic), i and iv are minor, V is major. This asymmetry is essential for the characteristic pull of minor-mode harmony.
大调中，正三和弦 I、IV、V 均为大三和弦；在小调（和声小调）中，i 和 iv 是小三和弦，V 是大三和弦。这种不对称性对于小调和声的独特倾向至关重要。
7. Cadence Formulas | 终止式公式
A cadence is a two-chord formula that closes a musical phrase. The four principal cadences are defined by the chords involved and the melodic movement in the upper voice.
终止式是结束乐句的双和弦公式。四种主要终止式由所涉和弦及旋律声部的进行决定。
| Cadence | Chords | Effect | Typical melody note |
|---|---|---|---|
| Perfect (authentic) | V → I (both in root position; upper voice moves to tonic) | Full closure, sense of finality | Leading tone (7) → tonic (1) |
| Imperfect | I → V, ii → V, IV → V (ending on V) | Half-closure; expectation continues | Any note → 2 or 5 often |
| Plagal | IV → I | Amen cadence, softer finality | Subdominant (4) → mediant (3) often |
| Interrupted (deceptive) | V → vi (or sometimes vi⁷, IV⁶) | Surprise; avoids expected tonic | Leading tone (7) → submediant (6) in vi |
In all cadences the bass movement is critical: perfect cadence leaps a fifth down or fourth up; plagal cadence leaps a fourth down or fifth up; imperfect often moves by step or a leap onto the dominant. Recognise these in both keyboard and full-score textures.
所有终止式中低声部进行至关重要：正格终止向下五度或向上四度跳进；变格终止向下四度或向上五度跳进；不完全终止常作级进或跳进至属音。在键盘织体和总谱中均需能识别这些公式。
8. Common Harmonic Progression Patterns | 常见和声进行模式
Tonality relies on functional progressions that create motion towards the tonic. The circle of fifths progression and the tonic–subdominant–dominant axis provide the skeleton of most classical works.
调性依赖于创造向主音运动的功能进行。五度圈进行与主–下属–属轴构成了大多数古典作品的骨架。
- Circle-of-fifths progression: iii → vi → ii → V → I (in major). For example, in C major: Em → Am → Dm → G → C.
五度圈进行：iii → vi → ii → V → I（大调）。例如 C 大调：Em → Am → Dm → G → C。 - Strong cadential progressions: ii⁷ – V⁷ – I or IV – V – I. The leading tone in V must resolve upwards to the tonic.
强终止进行：ii⁷ – V⁷ – I 或 IV – V – I。V 中的导音须上行解决至主音。 - Plagal extensions: I – IV – I – IV – I is common in hymns and popular music.
变格扩展：I – IV – I – IV – I 常见于赞美诗与流行音乐。 - Deceptive extension: V – vi delays the final resolution and allows phrase expansion.
阻碍延长：V – vi 延迟最终解决，便于扩展乐句。
9. Time Signatures and Rhythmic Grouping | 拍号与节奏分组
Time signatures encode both the metre and the beat subdivision. The top number indicates how many beats per bar; the bottom number indicates the note value that counts as one beat. Simple metres have beats that divide naturally into two; compound metres split beats into three.
拍号既表示节拍类型又表示拍子细分方式。上方数字为每小节拍数；下方数字表示以几分音符为一拍。单拍子每拍自然二分；复拍子每拍三分。
| Category | Examples | Subdivision |
|---|---|---|
| Simple duple | 2/4, 2/2 | Beat divides into 2 |
| Simple triple | 3/4, 3/8 | Beat divides into 2 |
| Simple quadruple | 4/4 (common time) | Beat divides into 2 |
| Compound duple | 6/8 (two dotted-crotchet beats) | Beat divides into 3 |
| Compound triple | 9/8 | Beat divides into 3 |
| Compound quadruple | 12/8 | Beat divides into 3 |
Beaming must show the beat or half-bar grouping. In simple time, beam together notes that belong to the same beat; in compound time, beam a full beat of three quavers together. Syncopation, hemiola, and cross-rhythms are additional rhythmic devices that CAIE expects you to identify aurally and in notation.
符尾连接必须显示拍子或半小节分组。单拍子中将同一拍的音符连在一起；复拍子中将一拍三个八分音符连在一起。切分、赫米奥拉比例和交错节奏是 CAIE 要求能在听觉和记谱中识别的额外节奏手法。
10. Transposing Instruments Formula | 移调乐器公式
Many orchestral instruments sound at a different pitch than written. To find the sounding pitch, apply the transposition interval above or below the written note. The most common transposing instruments and their rules are listed below.
许多管弦乐器发出的音高与记谱不同。要找出实际音高，需在记谱音的上方或下方增加相应的音程。以下是常见移调乐器及其规则。
| Instrument | Key | Sounds | Transposition from written C |
|---|---|---|---|
| Clarinet in B♭ | B♭Published by TutorHao | Year 12 音乐 Revision Series | aleveler.com
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