📚 Year 9 Cambridge Music: Formula & Theorem Quick Reference | 剑桥 Year 9 音乐：公式定理速查手册
Welcome to your essential pocket guide for Year 9 Cambridge Music. Although music is an art, its building blocks follow clear patterns and rules. This handbook distils the most important 'formulas' and 'theorems' behind scales, chords, intervals, rhythm, cadences and key signatures. Use it to check your understanding, speed up your theory work, and apply these patterns in listening and composing tasks.
欢迎使用这本 Year 9 剑桥音乐核心速查手册。音乐虽然是艺术，但其基本要素都遵循着清晰的模式与规则。这本小手册提炼了音阶、和弦、音程、节奏、终止式与调号背后最重要的"公式"与"定理"。用它来查漏补缺、提高乐理作业速度，并将这些模式运用到听辨与作曲任务中。
1. The Major Scale Formula | 大调音阶公式
All major scales are built from a fixed sequence of whole steps (W) and half steps (H): W W H W W W H. Starting on any note, applying this formula gives the correct sharps or flats automatically. For C major (C D E F G A B C), the half steps fall between E–F and B–C.
所有大调音阶都由一个固定的全音(W)与半音(H)序列构成：全全半全全全半。从任何一个音出发，套用这个公式就能自动得到正确的升降号。以 C 大调 (C D E F G A B C) 为例，半音位置落在 E–F 与 B–C 之间。
The formula works because the major scale pattern is symmetric in terms of intervallic structure. This is the foundation of all key signatures, chord progressions and melody writing in tonal music.
该公式之所以成立，是因为大调音阶在音程结构上具有对称性。它是调号、和弦进行和调性音乐旋律写作的基础。
2. The Minor Scale Formulas | 小调音阶公式
The natural minor scale uses the formula W H W W H W W (a descending pattern from its relative major). For A natural minor: A B C D E F G A. The harmonic minor raises the 7th degree by a half step, creating an exotic step of three semitones between the 6th and 7th: W H W W H WH H. The melodic minor raises the 6th and 7th degrees when ascending (W H W W W W H) but reverts to the natural minor descending.
自然小调音阶公式为：全半全全半全全（即从关系大调下行的模式）。A 自然小调：A B C D E F G A。和声小调将第七级音升高半音，在第六级与第七级之间产生一个增二度：全半全全半增半。旋律小调在上行时升高第六级与第七级（全半全全全全半），下行时还原为自然小调。
Memorising these three variants is vital for understanding minor-key pieces and for dictation. Always check whether the 6th and 7th degrees are natural, sharp or flat in the context.
牢记这三种变体对于理解小调作品和听写至关重要。在实际作品中，务必观察第六、七级音是还原、升还是降。
3. Chord Construction Formulas | 和弦构成公式
A triad is a three-note chord built in thirds. The quality of the triad depends on the number of semitones between the notes. The formulas (in semitones from the root) are:
- Major triad: root + 4 semitones + 3 semitones (e.g., C–E–G)
- Minor triad: root + 3 semitones + 4 semitones (e.g., C–E♭–G)
- Augmented triad: root + 4 semitones + 4 semitones (e.g., C–E–G♯)
- Diminished triad: root + 3 semitones + 3 semitones (e.g., C–E♭–G♭)
三和弦是按三度叠置的三音和弦。和弦的性质取决于各音之间的半音数。以根音为起点的半音公式如下：
- 大三和弦：根音 + 4 个半音 + 3 个半音（例如 C–E–G）
- 小三和弦：根音 + 3 个半音 + 4 个半音（例如 C–E♭–G）
- 增三和弦：根音 + 4 个半音 + 4 个半音（例如 C–E–G♯）
- 减三和弦：根音 + 3 个半音 + 3 个半音（例如 C–E♭–G♭）
These construction formulas allow you to build any triad on the keyboard or staff without guessing. They also underpin chord recognition in aural tests.
这些构成公式能让你在键盘或五线谱上准确构建任何三和弦，无需猜测。它们也是听觉测试中和弦辨识的基础。
4. The Circle of Fifths and Key-Signature Theorems | 五度圈与调号定理
The circle of fifths arranges keys so that each step clockwise adds one sharp, and each step counter-clockwise adds one flat. The order of sharps: F♯ C♯ G♯ D♯ A♯ E♯ B♯. The order of flats: B♭ E♭ A♭ D♭ G♭ C♭ F♭.
五度圈将各调按上行五度排列，顺时针方向每步增加一个升号，逆时针方向每步增加一个降号。升号出现的顺序为：F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号出现的顺序为：B♭ E♭ A♭ D♭ G♭ C♭ F♭。
Key theorem for sharps: the major key name is a minor second above the last sharp. Key theorem for flats: the major key name is the penultimate flat in the signature.
升号调定理：大调主音是最后一个升号上方的小二度音。降号调定理：大调主音是调号中倒数第二个降号。
Without these quick rules, identifying keys from signatures would be slow trial and error. Apply them until they become automatic.
没有这些快捷规则，从调号推断主调就会变成缓慢的试错。反复运用直到变成本能反应。
5. Interval Calculation Formula | 音程计算公式
An interval's numerical size (2nd, 3rd, 4th, etc.) is counted by letter names alone. Its quality (major, minor, perfect, augmented, diminished) is determined by the number of semitones. The following table gives the most common conversions in a major-key context:
音程的度数（二度、三度、四度等）只由音名字母的个数决定。它的性质（大、小、纯、增、减）由所含半音数决定。下表列出了大调背景下最常见的转换：
| Semitones | Interval | 半音数 | 音程 |
|---|---|---|---|
| 0 | Perfect Unison / Diminished 2nd | 0 | 纯一度 / 减二度 |
| 1 | Minor 2nd | 1 | 小二度 |
| 2 | Major 2nd | 2 | 大二度 |
| 3 | Minor 3rd | 3 | 小三度 |
| 4 | Major 3rd | 4 | 大三度 |
| 5 | Perfect 4th | 5 | 纯四度 |
| 6 | Augmented 4th / Diminished 5th | 6 | 增四度 / 减五度 |
| 7 | Perfect 5th | 7 | 纯五度 |
| 8 | Minor 6th | 8 | 小六度 |
| 9 | Major 6th | 9 | 大六度 |
| 10 | Minor 7th | 10 | 小七度 |
| 11 | Major 7th | 11 | 大七度 |
| 12 | Perfect Octave | 12 | 纯八度 |
Always count letter names first to get the number, then count semitones to fix the quality. For example, C to G♯ is a 5th (C D E F G) with 8 semitones, making it an augmented 5th.
先数音名个数得出度数，再数半音数确定性质。例如，C 到 G♯ 是五度（C D E F G），含 8 个半音，因此是增五度。
6. Time Signature and Rhythm Division Formulas | 拍号与节奏划分公式
Time signatures are not random; they follow a division logic. In simple time (2/4, 3/4, 4/4), each beat divides into two equal parts. In compound time (6/8, 9/8, 12/8), each beat divides into three. The top number tells you how many beats per bar in compound time when divided by three.
拍号并非随意设定，它们遵循划分逻辑。在简单拍子 (2/4, 3/4, 4/4) 中，每一拍均分成两个等份。在复合拍子 (6/8, 9/8, 12/8) 中，每一拍分成三个等份。复合拍子的上方数字除以三，就是每小节的拍数。
wo dotted-crotchet beats), 9/8 compound triple, 12/8 compound quadruple. This grouping theorem is often tested when adding bar lines or beaming quavers.
速记公式：附点四分音符拍 = 3 个八分音符。6/8 是复二拍子（两个附点四分音符拍），9/8 是复三拍子，12/8 是复四拍子。这种分组定理在标写小节线和符尾连接考题中经常出现。
7. Cadence Formulas | 终止式公式
Cadences are the punctuation marks of music. The four essential formulas using Roman numerals in a major key are:
- Perfect cadence: V → I (authentic, sounds finished)
- Plagal cadence: IV → I (Amen cadence)
- Imperfect cadence: I → V (or any chord → V, sounds unfinished)
- Interrupted cadence: V → vi (surprise, the 'deceptive' cadence)
终止式是音乐的标点符号。大调中四个核心公式（用罗马数字表示）为：
- 正格终止：V → I（稳定结束感）
- 变格终止：IV → I（阿门终止）
- 半终止：I → V（或任何和弦到 V，未完成感）
- 阻碍终止：V → vi（意外感，伪终止）
ust contain the raised leading note (harmonic minor). Always check the final chords of a piece to identify the home key and the cadence type.
在小调中，正格终止同样用 V → i，但属和弦必须含有升高导音（即和声小调）。分析作品时，永远先看最后几个和弦来判定主调与终止式类型。
8. Triad Inversion Formulas | 三和弦转位公式
Any triad can appear in root position, first inversion or second inversion. The bass note determines the inversion, and the intervals above the bass give each inversion its figured-bass name:
任何三和弦均可以原位、第一转位或第二转位出现。低音决定了转位，低音上方的音程赋予了各转位的数字低音名称：
| Inversion | Bass note | Figured bass | Intervals |
|---|---|---|---|
| Root position | Root | ⁵₃ | 3rd and 5th above bass |
| First inversion | 3rd of the chord | ⁶₃ | 3rd and 6th above bass |
| Second inversion | 5th of the chord | ⁶₄ | 4th and 6th above bass |
中文对应：原位（低音为根音，上方三度五度，标记 ⁵₃）；第一转位（低音为三音，上方三度六度，标记 ⁶₃）；第二转位（低音为五音，上方四度六度，标记 ⁶₄）。
The same formulas apply to minor, augmented and diminished triads. Inversions provide smooth bass lines and are essential for chorale-style harmony.
相同的转位公式适用于小三和弦、增三和弦与减三和弦。转位能提供平滑的低音线条，是众赞歌风格和声的基础。
9. Relative and Parallel Keys Theorem | 关系大小调与平行调定理
The relative minor of any major key starts on the 6th degree (a minor third below the tonic). For example, C major's relative minor is A minor; G major's is E minor. They share the same key signature. The parallel key theorem states that a major key and its parallel minor have the same tonic but different key signatures, with the minor having three more flats (or three fewer sharps).
任何大调的关系小调从其第六级音开始（主音下方小三度）。例如，C 大调的关系小调是 a 小调，G 大调的关系小调是 e 小调。两者共用同一个调号。平行调定理指出：大调与其平行小调共有相同的主音，但调号不同，小调多三个降号（或少三个升号）。
A fast check: C major (0 sharps/flats) parallel C minor (3 flats: B♭, E♭, A♭). These relationships are used constantly in modulation and composition tasks.
快速检验：C 大调（无升降号）的平行小调 c 小调（3 个降号：B♭, E♭, A♭）。这些关系在转调和作曲任务中被反复运用。
10. The Blues Scale Formula | 布鲁斯音阶公式
The blues scale is derived from the minor pentatonic with an added 'blue note'. Its formula in scale degrees: 1, ♭3, 4, ♭5, 5, ♭7. In C, this gives: C, E♭, F, G♭, G, B♭. The chromatic clash between the flat fifth and perfect fifth creates the characteristic blues sound.
布鲁斯音阶由小调五声音阶加入一个"蓝调音"派生而成。其音级公式为：1, ♭3, 4, ♭5, 5, ♭7。C 布鲁斯音阶为：C, E♭, F, G♭, G, B♭。降五级与纯五级之间的半音冲突营造出典型的布鲁斯色彩。
This formula is a staple in improvisation and composition exercises in Year 9. Knowing the pattern means you can instantly play a convincing blues line over a 12-bar progression in any key.
这个公式是 Year 9 即兴与作曲练习中的基本素材。掌握该模式，你就能在任何调的 12 小节布鲁斯进行上即刻弹出地道的布鲁斯线条。
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