📚 Year 10 CCEA Music: Quick Reference Handbook of Formulas & Theorems | 公式定理速查手册
Music theory can feel like a language of its own, but behind every melody and chord progression there are reliable patterns – what we might call 'formulas' and 'theorems'. This handbook collects the essential formulas you need for Year 10 CCEA Music, from scale construction to chord building, intervals, cadences and time signatures. Keep it close when you practise, compose or prepare for your exam.
音乐理论有时像一门独立的语言，但在每一段旋律与和弦进行背后，都有可靠的模式——我们可以称之为“公式”和“定理”。这份手册汇集了 Year 10 CCEA 音乐所需的核心公式：从音阶构成到和弦构建、音程、终止式以及拍号。无论是练习、作曲还是备考，请常备手边。
1. The Major Scale Formula | 大调音阶公式
A major scale follows the pattern of whole steps (W) and half steps (H): W – W – H – W – W – W – H. Starting on any tonic, apply this sequence to find the seven notes of the major key. For example, C major: C (W) D (W) E (H) F (W) G (W) A (W) B (H) C.
大调音阶遵循全音 (W) 与半音 (H) 的模式：全–全–半–全–全–全–半。从任意主音出发，按此序列即可得出该大调的七个音。例如 C 大调：C (全) D (全) E (半) F (全) G (全) A (全) B (半) C。
- The semitones always fall between degrees 3–4 and 7–8. | 半音总是出现在第 3–4 级和第 7–8 级之间。
- Use this formula to write any major scale without key signature memorisation. | 无需死记调号，用这个公式就能写出任何大调音阶。
W – W – H – W – W – W – H
2. Natural Minor Scale Formula | 自然小调音阶公式
The natural minor scale uses a different whole‑ and half‑step pattern: W – H – W – W – H – W – W. Starting on A, this gives A natural minor: A (W) B (H) C (W) D (W) E (H) F (W) G (W) A.
自然小调音阶采用不同的全音半音模式：全–半–全–全–半–全–全。从 A 开始，得到 A 自然小调：A (全) B (半) C (全) D (全) E (半) F (全) G (全) A。
- Semitones occur between degrees 2–3 and 5–6. | 半音位于第 2–3 级和第 5–6 级之间。
- Natural minor shares the same key signature as its relative major (three semitones higher). | 自然小调与其关系大调（高小三度）共用同一个调号。
W – H – W – W – H – W – W
3. Harmonic and Melodic Minor Adjustments | 和声小调与旋律小调的调整
Harmonic minor: raise the 7th degree of the natural minor by a semitone. This creates an augmented second between degrees 6 and 7. A harmonic minor becomes A B C D E F G♯ A.
和声小调：将自然小调的第 7 级升高半音。这会在第 6 级与第 7 级之间形成增二度。A 和声小调变为 A B C D E F G♯ A。
Melodic minor: ascending, raise the 6th and 7th degrees by a semitone; descending, follow the natural minor pattern. Ascending A melodic minor: A B C D E F♯ G♯ A; descending matches natural minor.
旋律小调：上行时升高第 6 级和第 7 级半音；下行时恢复自然小调模式。A 旋律小调上行：A B C D E F♯ G♯ A；下行等同于自然小调。
4. Interval 'Theorem' – Counting Semitones | 音程“定理” – 半音计数法
An interval is the distance between two notes, measured in semitones. Use this quick‑reference table for the simple intervals within an octave.
音程是两个音之间的距离，以半音为单位。使用下方的速查表，可快速对照八度以内的基本音程。
| Semitones | Interval Name | 中文名称 |
|---|---|---|
| 0 | Perfect Unison | 纯一度 |
| 1 | Minor 2nd | 小二度 |
| 2 | Major 2nd | 大二度 |
| 3 | Minor 3rd | 小三度 |
| 4 | Major 3rd | 大三度 |
| 5 | Perfect 4th | 纯四度 |
| 6 | Tritone (Aug 4th / Dim 5th) | 三全音（增四度 / 减五度） |
| 7 | Perfect 5th | 纯五度 |
| 8 | Minor 6th | 小六度 |
| 9 | Major 6th | 大六度 |
| 10 | Minor 7th | 小七度 |
| 11 | Major 7th | 大七度 |
| 12 | Perfect Octave | 纯八度 |
To work out any interval: count the letter names to get the numerical size (2nd, 3rd, etc.), then count the exact semitones to determine quality (major, minor, perfect, augmented, diminished).
计算任何音程：先数出字母名的数量得出度数（二度、三度等），再精确计算半音数来判定性质（大、小、纯、增、减）。
5. Chord Construction Formulas | 和弦构成公式
Chords are built by stacking thirds on top of a root. The formula is expressed in semitones above the root.
和弦通过在根音上方叠加三度构成。公式用根音上方的半音数表示。
- Major triad: root + 4 semitones (major 3rd) + 3 semitones (minor 3rd) → 0‑4‑7. | 大三和弦：根音 +4 半音（大三度）+3 半音（小三度）→ 0‑4‑7。
- Minor triad: root + 3 semitones (minor 3rd) + 4 semitones (major 3rd) → 0‑3‑7. | 小三和弦：根音 +3 半音（小三度）+4 半音（大三度）→ 0‑3‑7。
- Diminished triad: root + 3 semitones + 3 semitones → 0‑3‑6. | 减三和弦：根音 +3 半音 +3 半音 → 0‑3‑6。
- Augmented triad: root + 4 semitones + 4 semitones → 0‑4‑8. | 增三和弦：根音 +4 半音 +4 半音 → 0‑4‑8。
For seventh chords, add another third on top: dominant 7th = major triad + minor 3rd (0‑4‑7‑10); major 7th = major triad + major 3rd (0‑4‑7‑11); minor 7th = minor triad + minor 3rd (0‑3‑7‑10).
七和弦再叠加一个三度：属七和弦 = 大三和弦 + 小三度 (0‑4‑7‑10)；大七和弦 = 大三和弦 + 大三度 (0‑4‑7‑11)；小七和弦 = 小三和弦 + 小三度 (0‑3‑7‑10)。
6. Chord Inversion Formulas | 和弦转位公式
Inversions rearrange the same notes so a different chord member becomes the bass.
转位将相同音符重新排列，使不同的和弦音成为低音。
- Root position: root in the bass. | 原位：根音在低音部。
- 1st inversion: 3rd in the bass. The intervals above are a 3rd and a 6th (hence older name '6 chord'). | 第一转位：三音在低音部。上方音程为三度和六度（故旧称“六和弦”）。
- 2nd inversion: 5th in the bass. Intervals above are a 4th and a 6th ('6/4 chord'). | 第二转位：五音在低音部。上方音程为四度和六度（“四六和弦”）。
Seventh chords also have 3rd inversion (7th in the bass).
七和弦还有第三转位（七音在低音部）。
7. Primary Chords and the I‑IV‑V Theorem | 正三和弦与 I‑IV‑V 定理
mary triads. They contain all the notes of the scale and define the key. In C major: I = C‑E‑G, IV = F‑A‑C, V = G‑B‑D.
在任何大调中，建立在第 1、4、5 级上的和弦（I、IV、V）称为正三和弦。它们包含了该调音阶的全部音符，并确立调性。C 大调：I = C‑E‑G，IV = F‑A‑C，V = G‑B‑D。
back to I. The formula for the perfect cadence V → I is the strongest harmonic resolution.
V 级常加上七音 (V⁷)，增强返回 I 的倾向性。正格终止 V → I 是最强有力的和声解决。
8. Cadence Formulas | 终止式公式
Cadences are two‑chord progressions that end phrases. Learn them as formulas:
终止式是结束乐句的双和弦进行。将它们作为公式来记忆：
| Cadence | Chords (major key) | Sound | 中文名称 |
|---|---|---|---|
| Perfect | V → I | Final, complete | 正格终止 |
| Plagal | IV → I | 'Amen' | 变格终止 |
| Imperfect | I → V, II → V, etc. | Unfinished, pause | 半终止 |
| Interrupted | V → vi | Surprise, deflection | 阻碍终止 |
n minor keys, V still uses the raised 7th (harmonic minor), so V → i is a perfect cadence.
在小调中，V 级仍使用升高的第 7 级（和声小调），所以 V → i 是正格终止。
9. Time Signature Formulas | 拍号公式
p number = beats per bar; bottom number = beat unit (4 = crotchet, 8 = quaver, 2 = minim).
拍号看似分数，实为公式：上方数字表示每小节的拍数；下方数字表示以何种音符为一拍（4=四分音符，8=八分音符，2=二分音符）。
- Simple time: each beat divides into two equal parts. Top numbers usually 2, 3 or 4. | 单拍子：每拍可分为两个相等部分。上方数字通常为 2、3 或 4。
- Compound time: each beat divides into three equal parts. Top numbers usually 6, 9 or 12. | 复拍子：每拍可分为三个相等部分。上方数字通常为 6、9 或 12。
Dotted crotchet gets the beat in compound time: 6/8 = two dotted‑crotchet beats per bar.
复拍子中以附点四分音符为一拍：6/8 拍每小节有两拍。
10. Circle of Fifths as a Formula Wheel | 五度圈公式轮
The circle of fifths is a geometric formula for key relationships. Moving clockwise adds one sharp (or removes a flat); moving anticlockwise adds one flat (or removes a sharp). Each step is a perfect fifth.
五度圈是一种描述调性关系的几何公式。顺时针移动增加一个升号（或减少一个降号）；逆时针移动增加一个降号（或减少一个升号）。每一步为纯五度。
y find key signatures: C major (0 sharps) → G (1 sharp) → D (2 sharps) → A (3 sharps) etc.
关系小调位于其大调下方小三度。利用五度圈快速查找调号：C 大调 (0 升) → G (1 升) → D (2 升) → A (3 升) 等。
11. Transposition Formula | 移调公式
To transpose a melody, move every note by the same interval. Count the number of semitones (or scale degrees) between the original key and the new key. Apply that interval consistently. For transposing instruments (B♭ clarinet, trumpet), written C sounds as B♭; therefore you must write a major 2nd higher than the concert pitch.
移调时，将每个音符按相同的音程移动。计算原调与新调之间的半音数（或音阶级数），保持该音程不变。对于移调乐器（如 B♭ 单簧管、小号），记谱 C 实际发音为 B♭，因此记谱需比音乐会音高高一个大二度。
12. Exam Quick‑Check: The Formula Mindset | 考试快速检查：公式思维
When you see a scale, chord or cadence question, immediately recall the formula: W‑W‑H‑W‑W‑W‑H for major, 0‑4‑7 for major triad, V‑I for perfect cadence. Apply the formula step by step, checking each semitone or note spelling against the key signature.
遇到音阶、和弦或终止式题目时，立刻回想公式：大调全全半全全全半，大三和弦 0‑4‑7，正格终止 V‑I。逐步套用公式，将每个半音或音名与调号核对。
Keep this handbook with your practice diary. The more you use these patterns, the more they become instinctive musical thinking rather than abstract rules.
将这份手册与练习日记放在一起。使用这些模式越多，它们就越能成为本能的音乐思维，而非抽象的规则。
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