📚 Cambridge Year 10 Music: Formula & Theorem Quick Reference Handbook | 剑桥十年级音乐：公式定理速查手册
Music theory may appear to be an art of feelings and expressions, yet behind every scale, chord and interval lies a precise mathematical structure. This quick reference handbook gathers all essential formulas, patterns and theorems you need for the Cambridge Year 10 curriculum – from the construction of major and minor scales to triad recipes, key signature rules and cadence types. Keep these formulas close while practising, and theory will soon become second nature.
音乐理论看似是关于感觉与表达的艺术，但在每一种音阶、和弦与音程背后都隐藏着精确的数学结构。这本速查手册汇集了剑桥十年级课程所需的全部核心公式、模式与定理——从大小调音阶的构建到和弦配方、调号法则和终止式类型。练习时将这些公式放在手边，乐理很快就会成为你的第二本能。
1. The Major Scale Formula | 大调音阶公式
A major scale is built by following a fixed sequence of whole tones (T) and semitones (S). Starting from any tonic, apply the pattern T-T-S-T-T-T-S to produce the familiar bright sound. In terms of semitones counted from the root, the major scale degrees are: 0, 2, 4, 5, 7, 9, 11, 12.
大调音阶通过一个固定的全音（T）与半音（S）序列来构建。从任意主音出发，运用模式全-全-半-全-全-全-半，即可得到那熟悉明亮的音响。若从根音按半音计数，大调音阶各音级的位置为：0、2、4、5、7、9、11、12。
T T S T T T S (e.g. C – D – E – F – G – A – B – C)
全 全 半 全 全 全 半 例如 C – D – E – F – G – A – B – C
Always check that the notes follow the letter order with exactly one version of each letter name. For example, in G major, the F becomes F♯ to preserve the tone–semitone pattern; we never mix F and F♯ within the same scale spelling.
始终确保音名按字母顺序出现且每个字母仅使用一次。例如，在 G 大调中，F 必须升高为 F♯ 以保持全音–半音模式；同一个音阶内绝不能同时出现 F 与 F♯。
2. Natural Minor Scale Formula | 自然小调音阶公式
The natural minor scale is essentially the major scale pattern shifted to start on the sixth degree (the relative minor). Its tone–semitone sequence is T-S-T-T-S-T-T. In semitone steps from the tonic: 0, 2, 3, 5, 7, 8, 10, 12.
自然小调实质上就是将大调模式从第六级音开始演奏（即关系小调）。其全音–半音序列为全-半-全-全-半-全-全。从主音按半音计数为：0、2、3、5、7、8、10、12。
T S T T S T T (e.g. A – B – C – D – E – F – G – A)
全 半 全 全 半 全 全 例如 A – B – C – D – E – F – G – A
Remember that the natural minor shares the same key signature as its relative major. A minor is the relative of C major, so it uses no sharps or flats in its key signature.
请记住，自然小调与其关系大调共用相同的调号。A 小调是 C 大调的关系小调，因此调号中不含任何升号或降号。
3. Harmonic Minor Scale Formula | 和声小调音阶公式
The harmonic minor scale introduces a strong leading tone by raising the seventh degree by a semitone. This creates an augmented second (A2) interval between the sixth and seventh degrees. The tone–semitone pattern is T-S-T-T-S-A2-S. As semitone steps from the tonic: 0, 2, 3, 5, 7, 8, 11, 12.
和声小调通过将第七级音升高半音而引入一个强烈的导音，并在第六级与第七级之间形成增二度（A2）音程。全音–半音模式为全-半-全-全-半-增二度-半。半音级数序列为：0、2、3、5、7、8、11、12。
T S T T S A2 S (e.g. A – B – C – D – E – F – G♯ – A)
全 半 全 全 半 增二 半 例如 A – B – C – D – E – F – G♯ – A
The distinctive exotic flavour of this scale comes from the gap of three semitones between scale degrees ♭6 and 7. Use this formula to construct harmonic minor on any tonic; for instance, D harmonic minor gives D–E–F–G–A–B♭–C♯–D.
该音阶独特的异域色彩源自 ♭6 级与 7 级之间三个半音的跳跃。使用此公式可在任何主音上构建和声小调；例如 D 和声小调为 D–E–F–G–A–B♭–C♯–D。
4. Melodic Minor Scale Formula | 旋律小调音阶公式
The melodic minor scale has two forms: ascending and descending. When ascending, both the sixth and seventh degrees are raised by a semitone to smooth the melodic line, producing the pattern T-S-T-T-T-T-S. When descending, it reverts to the natural minor form (T-T-S-T-T-S-T). In semitones ascending: 0, 2, 3, 5, 7, 9, 11, 12; descending: 0, 2, 3, 5, 7, 8, 10, 12.
旋律小调具有上行和下行两种形式。上行时，第六级和第七级均升高半音以使旋律线条更平滑，形成全-半-全-全-全-全-半模式。下行时回归自然小调形态（全-全-半-全-全-半-全）。半音级数上行为：0、2、3、5、7、9、11、12；下行为：0、2、3、5、7、8、10、12。
Ascending 上行: T S T T T T S
Descending 下行: T T S T T S T
For A melodic minor ascending: A–B–C–D–E–F♯–G♯–A; descending: A–G–F–E–D–C–B–A. This formula helps to avoid the awkward augmented second of the harmonic minor in melody writing.
A 旋律小调上行：A–B–C–D–E–F♯–G♯–A；下行：A–G–F–E–D–C–B–A。运用这一公式可以避免旋律写作中和声小调增二度的别扭感。
5. Major Triad Formula | 大三和弦公式
A major triad consists of a root, a major third above the root (4 semitones) and a perfect fifth above the root (7 semitones). This can also be thought of as stacking a major third (4 st) and then a minor third (3 st) on top. The semitone formula from the root is: 0, 4, 7.
大三和弦由根音、其上大三度（4 个半音）和纯五度（7 个半音）组成。也可以理解为堆叠一个大三度（4 半音）再叠加一个小三度（3 半音）。从根音起的半音公式为：0、4、7。
Root + 4 semitones + 3 semitones → {0, 4, 7}
根音 + 4 半音 + 3 半音 → {0, 4, 7}
Example: C major triad = C (0) – E (4) – G (7). Any note that fits the 0-4-7 semitone pattern above a root forms a major triad.
示例：C 大三和弦 = C(0) – E(4) – G(7)。任何满足根音上方 0-4-7 半音模式的音便构成一个大三和弦。
6. Minor Triad Formula | 小三和弦公式
The minor triad is built from a root, a minor third (3 semitones) and a perfect fifth (7 semitones). The stacking order is a minor third plus a major third. Semitone pattern: 0, 3, 7.
小三和弦由根音、小三度（3 个半音）和纯五度（7 个半音）构成。堆叠顺序为小三度加大三度。半音模式：0、3、7。
Root + 3 semitones + 4 semitones → {0, 3, 7}
根音 + 3 半音 + 4 半音 → {0, 3, 7}
A minor triad: A (0) – C (3) – E (7). Dimming the major third of a major triad by one semitone always produces the parallel minor triad.
A 小三和弦：A(0) – C(3) – E(7)。将大三和弦的大三度降低半音，总会得到同名小三和弦。
7. Augmented and Diminished Triad Formulas | 增三与减三和弦公式
An augmented triad is formed by two stacked major thirds: root + 4 st + 4 st, giving the semitone set {0, 4, 8}. A diminished triad stacks two minor thirds: root + 3 st + 3 st, resulting in {0, 3, 6}.
增三和弦由两个大三度堆叠而成：根音 + 4 半音 + 4 半音，半音集合为 {0, 4, 8}。减三和弦则由两个小三度堆叠：根音 + 3 半音 + 3 半音，得出 {0, 3, 6}。
Augmented 增三: {0, 4, 8} (C – E – G♯)
Diminished 减三: {0, 3, 6} (B – D – F)
These symmetrical triads create tension: the augmented triad wants to resolve outward, while the diminished triad feels unstable and typically resolves inward to a major or minor third.
这两种对称结构和弦都会制造紧张感：增三和弦倾向于向外解决，而减三和弦听感不稳定，通常向内解决到大三度或小三度。
8. Interval Quick Reference | 音程快速参考表
Use the table below to identify any interval by counting the semitones between two notes. Remember that the number part of the interval name depends on the letter distance (e.g., C to E is a third because it spans three letter names), while the quality (perfect, major, minor, augmented, diminished) is determined by the exact semitone count.
通过计算两个音之间的半音数，便可借助下表识别任何音程。请记住，音程名称中的“度数”取决于字母跨度（如 C 到 E 跨三个字母，故为三度），而音程的性质（纯、大、小、增、减）则由精确的半音数决定。
| Semitones 半音数 | Interval Name 音程名称 |
|---|---|
| 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 | Augmented 4th / Diminished 5th (Tritone) 增四度/减五度(三全音) |
| 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 |
For quick mental calculation: perfect intervals (unison, 4th, 5th, octave) invert to remain perfect; major intervals invert to minor intervals and vice versa. The numbers always add to 9 (e.g., a 3rd inverts to a 6th).
快速心算口诀：纯音程（一度、四度、五度、八度）转位后仍为纯音程；大音程转位变为小音程，小音程转位变为大音程。度数之和永远为 9（如三度转位为六度）。
9. Circle of Fifths & Key Signature Formulas | 五度圈与调号公式
The order of sharps in key signatures follows the ascending circle of fifths: F♯ – C♯ – G♯ – D♯ – A♯ – E♯ – B♯. A common mnemonic is Father Charles Goes Down And Ends Battle. For major keys, the tonic is a semitone above the last sharp. For example, if the last sharp is D♯, the major key is E major.
升号调的升号出现顺序遵循上行五度圈：F♯ – C♯ – G♯ – D♯ – A♯ – E♯ – B♯。常用的记忆口诀为 Father Charles Goes Down And Ends Battle。对于大调而言，主音位于最后一个升号的上方半音。例如，若最后一个升号为 D♯，则大调为 E 大调。
The order of flats is the reverse of the sharps order: B♭ – E♭ – A♭ – D♭ – G♭ – C♭ – F♭ (Battle Ends And Down Goes Charles' Father). For all flat major keys except F major, the tonic is the second-to-last flat in the signature. Thus, a key signature with three flats (B♭, E♭, A♭) is E♭ major.
降号调的降号顺序与升号相反：B♭ – E♭ – A♭ – D♭ – G♭ – C♭ – F♭（Battle Ends And Down Goes Charles’ Father）。除 F 大调外，所有降号大调的主音都是调号中倒数第二个降号。因此，三个降号（B♭、E♭、A♭）的调号为 E♭ 大调。
Sharps rule: Last ♯ + 1 semitone = Major tonic
