📚 Year 9 CAIE Music: Formula & Theorem Quick Reference | 九年级 CAIE 音乐：公式定理速查手册
This handbook compiles the essential 'formulas' and 'theorems' for Year 9 CAIE Music – the core rules, patterns, and structures that underpin scales, intervals, chords, rhythm, and harmony. Use it as a quick reference to reinforce your understanding and speed up your analysis and composition work.
本手册汇总了九年级 CAIE 音乐课程中的核心‘公式’与‘定理’——支撑音阶、音程、和弦、节奏与和声的关键规则、模式与架构。以此作为速查指南，帮助巩固理解，提升分析与创作效率。
1. The Major Scale Formula | 大调音阶公式
A major scale is built using the step pattern: Whole, Whole, Half, Whole, Whole, Whole, Half. This can be expressed as a simple formula using tones (T) and semitones (S): T T S T T T S.
大调音阶由全音半音模式构成：全、全、半、全、全、全、半。可以使用全音 (T) 和半音 (S) 表示为公式：T T S T T T S。
Major Scale Formula: W W H W W W H (or T T S T T T S)
For example, C major: C (W) D (W) E (H) F (W) G (W) A (W) B (H) C. No sharps or flats are needed; the formula generates the correct pitches for any starting note.
例如 C 大调：C (全) D (全) E (半) F (全) G (全) A (全) B (半) C。不需要任何升降记号；从任意起始音开始，该公式都能生成正确的音高。
2. Minor Scale Formulas | 小调音阶公式
There are three common forms of the minor scale, each with its own interval formula. The natural minor follows: T S T T S T T. The harmonic minor raises the 7th degree: T S T T S T+S S. The melodic minor raises both the 6th and 7th degrees ascending, but reverts to natural minor descending.
小调音阶有三种常见形式，每种都有各自的音程序列公式。自然小调：T S T T S T T。和声小调升高第 VII 级：T S T T S 增二度 S。旋律小调上行升高第 VI 和第 VII 级，下行还原为自然小调。
Natural Minor: 1 2 ♭3 4 5 ♭6 ♭7 8 (T S T T S T T)
Harmonic Minor: 1 2 ♭3 4 5 ♭6 7 8 (T S T T S A2 S)
Melodic Minor (ascending): 1 2 ♭3 4 5 6 7 8 (T S T T T T S)
Understanding these formulas allows you to construct any minor scale on a given tonic, such as A harmonic minor: A B C D E F G♯ A.
掌握这些公式便能在给定主音上构建任意小调音阶，例如 A 和声小调：A B C D E F G♯ A。
3. Key Signatures and the Circle of Fifths | 调号与五度圈
The order of sharps in key signatures follows a fixed sequence: F♯ C♯ G♯ D♯ A♯ E♯ B♯. Flats appear in the reverse order: B♭ E♭ A♭ D♭ G♭ C♭ F♭. These sequences are derived from the circle of fifths.
调号中升记号的顺序固定为：F♯ C♯ G♯ D♯ A♯ E♯ B♯。降记号则反向排列：B♭ E♭ A♭ D♭ G♭ C♭ F♭。这些序列源自五度圈。
Every major key has a relative minor key sharing the same key signature, located a minor third below the major tonic. For C major, the relative minor is A minor; for G major, it is E minor.
每个大调都有一个关系小调，共享相同的调号，且位于大调主音下方小三度。例如 C 大调的关系小调为 a 小调，G 大调的为 e 小调。
| Major Key | Relative Minor | Sharps/Flats |
|---|---|---|
| C major | A minor | none |
| G major | E minor | F♯ |
| D major | B minor | F♯ C♯ |
| F major | D minor | B♭ |
| B♭ major | G minor | B♭ E♭ |
To identify a major key with sharps, go up a semitone from the last sharp. For flats, the second-last flat names the major key. These rules help you decode any key signature quickly.
识别升号调时，最后一个升号上行半音即为主音。降号调则看倒数第二个降号，其音名即大调主音。这些规则能帮你快速破译任何调号。
4. Interval Formulas | 音程公式
An interval measures the distance between two pitches in semitones. The quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.) together describe the interval exactly. Unisons, fourths, fifths, and octaves can be perfect; seconds, thirds, sixths, and sevenths are major/minor.
音程衡量两个音高之间以半音计的距离。性质（纯、大、小、增、减）与度数（一度、二度、三度等）共同准确描述该音程。同度、四度、五度、八度可以为纯；二度、三度、六度、七度则为大/小。
| Interval | 中文名称 | Semitones |
|---|---|---|
| Perfect Unison | 纯一度 | 0 |
| Minor 2nd | 小二度 | 1 |
| Major 2nd | 大二度 | 2 |
| Minor 3rd | 小三度 | 3 |
| Major 3rd | 大三度 | 4 |
| Perfect 4th | 纯四度 | 5 |
| Augmented 4th / Diminished 5th | 增四度 / 减五度 | 6 |
| Perfect 5th | 纯五度 | 7 |
| Minor 6th | 小六度 | 8 |
| Major 6th | 大六度 | 9 |
| Minor 7th | 小七度 | 10 |
| Major 7th | 大七度 | 11 |
| Perfect Octave | 纯八度 | 12 |
To invert an interval, subtract the number from 9 and flip the quality: major becomes minor, perfect stays perfect, augmented becomes diminished. A major 3rd inverts to a minor 6th.
转位音程时，用 9 减去度数并反转性质：大变小，纯不变，增减互换。大三度转位后成为小六度。
5. Triad Chord Formulas | 三和弦公式
A triad is a three-note chord built on a root by stacking thirds. The formula defines the intervals above the root in semitones. Major triad: root + 4 (major 3rd) + 7 (perfect 5th). Minor triad: root + 3 (minor 3rd) + 7 (perfect 5th).
三和弦是由根音上方叠加三度构成的三个音组成的和弦。公式用根音上方的半音数定义各音。大三和弦：根音 + 4（大三度）+ 7（纯五度）。小三和弦：根音 + 3（小三度）+ 7（纯五度）。
Major: 1 3 5 → semitones: 0 – 4 – 7
Minor: 1 ♭3 5 → semitones: 0 – 3 – 7
Diminished: 1 ♭3 ♭5 → semitones: 0 – 3 – 6
Augmented: 1 3 ♯5 → semitones: 0 – 4 – 8
The quality of triads built on each degree of a major scale follows a fixed pattern: I major, ii minor, iii minor, IV major, V major, vi minor, vii° diminished. This chord-scale rule is fundamental in tonal harmony.
大调音阶每一级上构建的三和弦性质有固定规律：I 大，ii 小，iii 小，IV 大，V 大，vi 小，vii° 减。这一和弦-音阶规律是调性和声的基础。
6. Cadence Rules | 终止式规则
A cadence is a pair of chords that concludes a phrase. The four principal cadences are defined by their chord progression formulas. A perfect cadence moves from V to I (or V7 to I); it sounds resolved and final.
终止式是结束乐句的一对和弦。四种主要终止式由其和弦进行公式定义。正格终止从 V 进行到 I（或 V7 至 I），听感完满、终结。
Plagal cadence: IV to I (the 'Amen' cadence). Imperfect cadence: any chord to V, commonly I-V, ii-V, or IV-V; it sounds unresolved and expects continuation. Interrupted cadence: V to vi, a deceptive resolution that surprises the listener.
变格终止：IV 到 I（‘阿门’终止）。不完全终止：任意和弦到 V，常见 I-V、ii-V 或 IV-V，听感未解决，期待继续。阻碍终止：V 到 vi，属欺骗性解决，令听众意外。
Perfect: V – I | Plagal: IV – I | Imperfect: I/ii/IV – V | Interrupted: V – vi
In a minor key, the perfect cadence often uses V with a raised leading note (harmonic minor), so V – i. The same formulas apply, reinforcing the pull back to the tonic.
在小调中，正格终止通常使用升高导音的 V（和声小调），即 V 到 i。同样的公式适用，强化回归主音的倾向。
7. Note and Rest Duration Formulas | 音符与休止符时值公式
Note durations are relative; the formula for a dotted note adds half of the original value. A dotted crotchet (quarter note) equals 1½ beats. A double-dotted note adds another half of that half (total added ¾ of the original).
音符时值是相对的；附点音符公式是在原有时值上增加一半。附点四分音符等于 1½ 拍。复附点再增加那一半的一半（总计增加原时值的¾）。
Dotted note = original + ½ original
Double-dotted = original + ½ original + ¼ original
The hierarchy of note values in simple time: semibreve (4 beats), minim (2), crotchet (1), quaver (½), semiquaver (¼). Equivalent rest symbols follow the same relationship. Beaming rules group beats together to show metric structure.
简单节拍中的音符时值层级：全音符（4拍），二分音符（2拍），四分音符（1拍），八分音符（½拍），十六分音符（¼拍）。相应的休止符也遵循相同关系。符杠连接规则按拍分组以体现节拍结构。
8. Time Signature Formulas | 拍号公式
A time signature is a fraction-like symbol where the top number indicates the number of beats per bar, and the bottom number indicates the note value that counts as one beat. For example, 4/4 means 4 crotchet beats per bar.
拍号是一个类似分数的符号，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。例如 4/4 表示以四分音符为一拍，每小节四拍。
Simple time: beat divides into two equal parts (e.g., 2/4, 3/4, 4/4). Compound time: beat divides into three equal parts; top numbers are 6, 9, or 12. In 6/8, there are two dotted-crotchet beats, each subdividing into three quavers.
单拍子：每一拍均分为二（如 2/4、3/4、4/4）。复拍子：每一拍均分为三；上方数字为 6、9 或 12。在 6/8 拍中，有两个附点四分音符拍，每拍再分为三个八分音符。
Simple Duple: 2/4, 2/2 | Simple Triple: 3/4, 3/2 | Simple Quadruple: 4/4, 4/2
Compound Duple: 6/8 | Compound Triple: 9/8 | Compound Quadruple: 12/8
one compound dotted beat. Thus a crotchet in 2/4 corresponds to a dotted crotchet in 6/8.
单复拍子转换原则：一个单拍子的一拍等于复拍子的一个附点拍。因此 2/4 中的四分音符相当于 6/8 中的附点四分音符。
9. Harmonic Progression Formulas | 和声进行公式
Common chord progressions follow strong root movements of descending fifths, descending thirds, and stepwise motion. The most archetypal progression is I – IV – V – I, which establishes the tonic, subdominant, and dominant functions.
常见和弦进行遵循下五度、下三度和级进的强根音进行。最典型的进行式 I – IV – V – I，建立起主、下属与属功能。
Functional harmony assigns each chord a role: tonic (I, vi), predominant (IV, ii), and dominant (V, vii°). The formula for a perfect authentic cadence is V – I, but progressions such as ii – V – I and I – vi – IV – V are staples in classical and popular music.
功能和声为每个和弦指派角色：主功能（I, vi），下属功能（IV, ii），属功能（V, vii°）。完美正格终止式为 V – I，而诸如 ii – V – I 和 I – vi – IV – V 的进行是古典与流行音乐的基石。
Strong progressions: I – IV – V – I ; ii – V – I ; I – V – vi – IV
When writing or analysing, always check that the leading note (7th degree) resolves upward to the tonic in a V – I cadence. These formulas ensure smooth voice leading and functional clarity.
写作或分析时，始终检查导音（第 VII 级）在 V – I 终止中是否上行解决到主音。这些公式确保平稳的声部进行和清晰的功能性。
10. Transposition Formulas | 移调公式
to the new key signature. The formula is simple: new pitch = old pitch + interval in semitones.
移调将每个音符移动固定的音程。向上移高大二度，即将每个音高增加 2 个半音，并根据新调号调整临时记号。公式简化为：新音高 = 原音高 + 半音数。
When transposing instruments are involved (e.g., clarinet in B♭, horn in F), the written note must be adjusted so that the sounding pitch matches concert pitch. For a B♭ clarinet, the written note sounds a major 2nd lower, so to sound C, write D.
当涉及移调乐器时（如 降 B 调单簧管、F 调圆号），需调整记谱音高使实际音高与音乐会音高一致。降 B 调单簧管实际音高比记谱低大二度，因此若想奏出 C 音，需记谱为 D。
A reliable method: determine the interval from the instrument's key to C. If the instrument is in B♭, the interval is a major 2nd down to C. Apply the opposite interval in writing: notated part is a major 2nd above concert pitch. This formula prevents many ensemble mishaps.
可靠方法：确定乐器调号与 C 的音程关系。若乐器为 B♭ 调，到 C 音程为下行大二度。写作时则使用反向音程：记谱比音乐会音高高大二度。这则公式可避免许多合奏差错。
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