📚 Year 10 Cambridge Music: Quick Reference Handbook of Formulas & Theorems | Year 10 剑桥音乐：公式定理速查手册
This quick-reference handbook gathers the most essential music theory 'formulas and theorems' you need for Year 10 Cambridge IGCSE Music. Think of it as your go-to revision sheet for scales, intervals, chords, cadences, key signatures, and other core concepts that underpin listening, composing, and performing. Each rule is presented as a clear, memorable statement so you can recall it instantly in lessons, homework, or exam settings.
本速查手册汇集了 Year 10 剑桥 IGCSE 音乐最关键的乐理“公式与定理”。它如同你的随身复习单，涵盖音阶、音程、和弦、终止式、调号等支撑听力、作曲和演奏的核心概念。每一条规则都以清晰、好记的方式呈现，方便你在课堂、作业或考试中即时调用。
1. Major Scale Formula | 大调音阶公式
All major scales follow the step pattern: whole – whole – half – whole – whole – whole – half (W‑W‑H‑W‑W‑W‑H). Apply this sequence from any starting note and you will always build a correct major scale.
所有大调音阶都遵循全音与半音的固定模式：全 – 全 – 半 – 全 – 全 – 全 – 半。从任意起始音套用这个顺序，你就能构筑出正确的大调音阶。
Example: C major uses C–D (W), D–E (W), E–F (H), F–G (W), G–A (W), A–B (W), B–C (H). The half‑step positions naturally occur between scale degrees 3‑4 and 7‑8.
示例：C 大调音阶为 C–D（全）、D–E（全）、E–F（半）、F–G（全）、G–A（全）、A–B（全）、B–C（半）。半音位置自然出现在第 3‑4 级和第 7‑8 级音之间。
2. Natural Minor Scale Formula | 自然小调音阶公式
The natural minor scale uses the pattern: whole – half – whole – whole – half – whole – whole (W‑H‑W‑W‑H‑W‑W). This creates a darker, more sombre sound compared with the major scale.
自然小调音阶遵循的模式为：全 – 半 – 全 – 全 – 半 – 全 – 全。这赋予音乐一种更沉郁、忧伤的色彩。
Example: A natural minor uses A–B (W), B‑C (H), C–D (W), D–E (W), E–F (H), F–G (W), G–A (W). Notice the lowered 3rd, 6th and 7th degrees when compared to its relative major, C major.
示例：A 自然小调为 A–B（全）、B‑C（半）、C–D（全）、D–E（全）、E–F（半）、F–G（全）、G–A（全）。请注意，与其关系大调 C 大调相比，第 3、6、7 级音降低了半音。
3. Harmonic and Melodic Minor Formulas | 和声与旋律小调公式
Harmonic minor: raise the 7th degree of the natural minor scale by a half step while descending and ascending. This creates a distinct augmented‑2nd gap between the 6th and 7th degrees (e.g. A B C D E F G♯ A).
和声小调：将自然小调的第 7 级音升高半音，上行与下行均保持变化。这使第 6‑7 级之间形成独特的增二度（例如 A B C D E F G♯ A）。
Melodic minor (ascending): raise both the 6th and 7th degrees of the natural minor scale by a half step. Descending, it returns to the natural minor form. So A melodic minor ascending is A B C D E F♯ G♯ A, and descending is A G F E D C B A.
旋律小调（上行）：将自然小调的第 6 和第 7 级音各升高半音。下行时恢复自然小调形态。因此，A 旋律小调上行为 A B C D E F♯ G♯ A，下行为 A G F E D C B A。
4. Interval Identification Quick-Guide | 音程识别速查
Intervals are measured by counting half steps between two notes. Commit the following half‑step numbers to memory – they work for any starting pitch.
音程通过计算两个音之间的半音数来判定。牢记下列半音数对应关系——它们适用于任何起始音高。
th = 7, minor 6th = 8, Major 6th = 9, minor 7th = 10, Major 7th = 11, Perfect octave = 12.
小二度 = 1 个半音，大二度 = 2，小三度 = 3，大三度 = 4，纯四度 = 5，增四度 / 减五度（三全音）= 6，纯五度 = 7，小六度 = 8，大六度 = 9，小七度 = 10，大七度 = 11，纯八度 = 12。
To name an interval quickly, first count the letter names (e.g. C to E is a 3rd), then check the half‑step count to determine if it is major, minor or perfect. Learning these numbers as formulas saves time in exams.
快速命名音程时，先数音名个数（如 C 到 E 为三度），然后核对半音数判断是大、小还是纯音程。将这些数字当作公式记住可以为考试节省大量时间。
5. Triad Construction Theorems | 三和弦构建定理
Every triad is built by stacking thirds. The quality depends on the exact half‑step distances between the root, 3rd and 5th.
所有三和弦均由三度叠加而成。和弦的性质取决于根音、三音与五音之间的具体半音距离。
+ 3 half steps to the minor 3rd, then + 4 half steps to the perfect 5th (e.g. D → F → A).
大三和弦：根音 + 4 个半音得到大三度音，再加 3 个半音得到纯五度音（如 C→E→G）。小三和弦：根音 + 3 个半音得到小三度音，再加 4 个半音得到纯五度音（如 D→F→A）。
Diminished triad: root + 3 + 3 (two stacked minor 3rds). Augmented triad: root + 4 + 4 (two stacked major 3rds). These formulas guarantee correct spelling every time.
减三和弦：根音 + 3 + 3（两个小三度叠置）。增三和弦：根音 + 4 + 4（两个大三度叠置）。掌握这些公式能保证你每次都写对和弦。
6. Primary Chords and Cadence Rules | 正和弦与终止式规则
In any major key, the primary triads are I (tonic), IV (subdominant) and V (dominant). These chords form the backbone of tonal music and appear in the most important cadences.
在任何大调中，正三和弦是 I（主和弦）、IV（下属和弦）和 V（属和弦）。它们是调性音乐的骨架，并且出现在最重要的终止式中。
se formulas are vital for completing cadence exercises and analysing harmony in set works.
完满终止：V → I。变格终止：IV → I。不完全终止：任意和弦 → V。阻碍终止：V → vi。这些公式对于完成终止式练习和分析指定作品的和声至关重要。
7. Time Signature Rules and Beat Grouping | 拍号规则与节拍分组
you which note value receives one beat (4 = quarter note, 8 = eighth note, 2 = half note).
拍号公式：上方数字表示每小节有几拍；下方数字表示以哪种音符为一拍（4 代表四分音符，8 代表八分音符，2 代表二分音符）。
d time: beats divide into three equal parts, and the beat is a dotted note (6/8 = 2 dotted‑quarter beats, 9/8 = 3, 12/8 = 4).
单拍子：拍子可二等分（2/4、3/4、4/4）。复拍子：拍子可三等分，且单位拍为附点音符（6/8 = 两拍附点四分音符，9/8 = 三拍，12/8 = 四拍）。
When beaming notes, always group to show the beat structure. In 4/4, join four eighth notes in two beamed pairs; in 6/8, beam six eighth notes into two groups of three. This rule makes your notation professional and easy to read.
符杠的连接须体现节拍结构。在 4/4 拍中，将四个八分音符连成两对使用符杠；在 6/8 拍中，将六个八分音符分成两组各三个进行连接。这条规则让你的记谱既专业又易读。
8. Key Signature Order – A Memorisation Theorem | 调号顺序记忆定理
Sharp key signatures always appear in the order: F♯ C♯ G♯ D♯ A♯ E♯ B♯. Flat key signatures follow: B♭ E♭ A♭ D♭ G♭ C♭ F♭. This fixed sequence never changes.
升号调的调号始终按以下顺序出现：F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号调的调号为：B♭ E♭ A♭ D♭ G♭ C♭ F♭。这一固定序列永不变更。
a major key from sharps: take the last sharp and go up a half step. (E.g. last sharp is D♯ → the key is E major.) For flats: the second‑to‑last flat is the major key. (E.g. key signature B♭, E♭, A♭ → A♭ major.) For F major with one flat, simply memorise it.
从升号识别大调：取最后一个升号再上行半音即为主音。（如最后升号为 D♯ → 调为 E 大调。）降号大调：倒数第二个降号即为主音。（如调号为 B♭, E♭, A♭ → A♭ 大调。）只有一个降号的 F 大调单独记忆即可。
9. Circle of Fifths – The Grand Unifying Model | 五度圈——大一统模型
Moving clockwise around the circle of fifths, each key adds one sharp (C–G–D–A–E–B–F♯–C♯…). Moving anticlockwise, each key adds one flat (C–F–B♭–E♭–A♭–D♭–G♭–C♭…).
沿五度圈顺时针移动，每个调增加一个升号（C–G–D–A–E–B–F♯–C♯…）。逆时针移动，每个调增加一个降号（C–F–B♭–E♭–A♭–D♭–G♭–C♭…）。
This geometric arrangement also reveals that every major key shares its key signature with a relative minor located a minor third below it. The circle of fifths is a powerful visual ‘formula’ for key relationships, chord progressions and modulation analysis.
这种几何排列还揭示出，每个大调与其下方小三度的关系小调共享同一调号。五度圈是分析调性关系、和弦进行和转调分析的强大视觉“公式”。
10. Relative Minor and Major Theorem | 关系大小调定理
To find the relative minor of any major key, count down a minor third (3 half steps). Example: C major down a minor third lands on A, so A minor is the relative minor of C major.
要找出任何大调的关系小调，向下数小三度（3 个半音）。例：C 大调向下小三度落于 A，因此 A 小调为 C 大调的关系小调。
Conversely, to find the relative major from a minor key, count up a minor third. Example: from E minor, up a minor third gives G major. This theorem instantly links pairs of keys that share the same key signature and is essential for score analysis and composition.
反之，由小调找关系大调，向上数小三度。例：E 小调上行小三度得 G 大调。该定理能瞬间关联共享同一调号的大小调对，对于乐谱分析和作曲不可或缺。
11. Transposing Instruments Rule | 移调乐器规则
Many wind and brass instruments are transposing instruments: the written note does not equal the sounding pitch. The most common formula is for B♭ instruments (clarinet, trumpet): when the player reads a C, the sounding pitch is B♭ – a major 2nd lower.
许多管乐器是移调乐器：谱面音符与实际音高并不相同。最常见的公式适用于 B♭ 调乐器（单簧管、小号）：演奏者读谱为 C 时，实际发出的音高是 B♭——低一个大二度。
To convert a written part to concert pitch for a B♭ instrument, transpose down a major 2nd. For E♭ instruments (alto sax), transpose down a major 6th or up a minor 3rd. For F instruments (French horn), transpose down a perfect 5th. Learn these intervals as fixed formulas so you can quickly read and write for any transposing instrument.
若要将 B♭ 乐器的记谱转为实际音高，向下移大二度。对于 E♭ 乐器（中音萨克斯），向下移大六度或向上移小三度。对于 F 乐器（圆号），向下移纯五度。将这些音程视为固定公式，便能快速为任何移调乐器读谱和写作。
12. Dynamics and Tempo Markings – Instant Decoder | 力度与速度标记速查
Although not algebraic formulas, standard Italian terms function like fixed codes that must be memorised. Dynamics: pp (pianissimo – very soft), p (piano – soft), mp (mezzo‑piano – moderately soft), mf (mezzo‑forte – moderately loud), f (forte – loud), ff (fortissimo – very loud). Gradual changes: cresc. (crescendo – gradually louder), dim. (diminuendo – gradually softer).
虽非数学公式，但标准意大利术语就像固定代码，必须牢记。力度：pp（pianissimo，很弱），p（piano，弱），mp（mezzo‑piano，中弱），mf（mezzo‑forte，中强），f（forte，强），ff（fortissimo，很强）。渐变：cresc.（渐强），dim.（渐弱）。
Tempo markings from slowest to fastest: Largo (very slow and broad), Adagio (slow and stately), Andante (walking pace), Moderato (moderate speed), Allegro (fast, lively), Presto (very fast). Many terms also have modifiers such as molto (very) and poco (a little). Keeping this vocabulary as a ready-made theorem helps you describe music accurately in listening papers and perform with correct style.
速度标记由慢到快：Largo（宽广的慢板），Adagio（从容的慢板），Andante（行板），Moderato（中板），Allegro（快板），Presto（急板）。许多术语还带有修饰词，如 molto（很）和 poco（稍）。将这组词汇视作现成的定理，有助于你在听力考试中准确描述音乐，并以正确的风格演奏。
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