📚 Year 10 CIE Music: Formulas and Theorems Quick Reference | 10年级 CIE 音乐：公式定理速查手册
In the context of CIE IGCSE Music, 'formulas and theorems' are not mathematical equations but the fundamental rules and patterns that shape melody, harmony, rhythm and structure. This handbook compiles the essential concepts every Year 10 student must internalise — from note values and intervals to cadences and ornamentation. Treat these pages as your go-to revision map for listening, composing and performing with confidence.
在CIE IGCSE音乐课程中，“公式与定理”并不是数学公式，而是构建旋律、和声、节奏和曲式的基本规则与模式。本手册汇集了每位10年级学生必须内化的核心概念——从音符时值、音程到终止式与装饰音。将这几页当作你复习时的随身地图，帮助你在听力、作曲和演奏方面自信应对。
1. Note Values and Rests | 音符时值与休止符
Note durations in Western notation follow a simple binary logic: each symbol indicates a fixed proportional length. A whole note (semibreve) is divisible into two half notes (minims), four quarter notes (crotchets), eight eighth notes (quavers) and so on. Rests mirror these durations.
西方记谱法中的音符时值遵循简单的二进制逻辑：每个符号表示固定的比例时长。一个全音符可分成两个二分音符、四个四分音符、八个八分音符，依此类推。休止符与对应的音符时值完全相同。
Semibreve = 2 minims = 4 crotchets = 8 quavers = 16 semiquavers
全音符 = 2个二分音符 = 4个四分音符 = 8个八分音符 = 16个十六分音符
| English Name | 中文名称 | Note Symbol | Rest Symbol | Beats in 4/4 |
|---|---|---|---|---|
| Semibreve | 全音符 | 𝅝 | 𝄻 | 4 |
| Minim | 二分音符 | 𝅗𝅥 | 𝄼 | 2 |
| Crotchet | 四分音符 | 𝅘𝅥 | 𝄽 | 1 |
| Quaver | 八分音符 | 𝅘𝅥𝅮 | 𝄾 | ½ |
| Semiquaver | 十六分音符 | 𝅘𝅥𝅯 | 𝄿 | ¼ |
A dot placed after a note or rest increases its duration by half of its original value. A double dot adds a further quarter.
附点写在音符或休止符之后，将其时值延长原时值的一半。复附点则再增加原时值的四分之一。
Dotted crotchet = 1 + ½ = 1½ beats; Double-dotted minim = 2 + 1 + ½ = 3½ beats
附点四分音符 = 1 + ½ = 1½拍；复附点二分音符 = 2 + 1 + ½ = 3½拍
2. Time Signatures and Rhythm | 拍号与节奏
ives the number of beats per bar; the lower digit indicates the beat unit (4 = crotchet, 8 = quaver, 2 = minim).
拍号是用来规定每小节有几拍、以何种音符为一拍的记谱规则。上方数字表示每小节拍数；下方数字表示以几分音符为一拍（4=四分音符，8=八分音符，2=二分音符）。
Time Signature = (Beats per bar) / (Note value of one beat)
拍号 = (每小节拍数) / (一拍所对应的音符时值)
Simple time signatures have beats that divide naturally into two equal parts (e.g. 2/4, 3/4, 4/4). Compound time signatures have dotted-note beats that divide into three equal parts (e.g. 6/8, 9/8, 12/8).
单拍子的每一拍可自然划分为两个均等的部分（如2/4、3/4、4/4）。复拍子的每一拍是附点音符，可划分为三个均等的部分（如6/8、9/8、12/8）。
| Type | Time Signature | Beat Unit | Beats per Bar |
|---|---|---|---|
| Simple duple | 2/4, 2/2 | Crotchet, Minim | 2 |
| Simple triple | 3/4, 3/2 | Crotchet, Minim | 3 |
| Simple quadruple | 4/4 (C), 4/2 | Crotchet, Minim | 4 |
| Compound duple | 6/8 | Dotted crotchet | 2 |
| Compound triple | 9/8 | Dotted crotchet | 3 |
| Compound quadruple | 12/8 | Dotted crotchet | 4 |
Irregular time signatures (e.g. 5/4, 7/8) contain beats of unequal length groupings, requiring careful accent patterns.
混合拍子（如5/4、7/8）包含不均等分组的拍子，需要仔细安排重音模式。
3. Major and Minor Scales | 大调与小调音阶
A scale is an ascending or descending sequence of notes following a specific pattern of tones (T) and semitones (S). The major scale formula is T-T-S-T-T-T-S. When applied to any starting note, this pattern produces the diatonic major scale.
音阶是按特定全音（T）和半音（S）模式上行或下行排列的音列。大调音阶的公式为全-全-半-全-全-全-半。在任何起始音上套用该模式，即可生成自然大调音阶。
Major: T – T – S – T – T – T – S (e.g. C D E F G A B C)
大调：全 – 全 – 半 – 全 – 全 – 全 – 半（如 C D E F G A B C）
The natural minor scale follows T-S-T-T-S-T-T. The harmonic minor raises the 7th degree by a semitone, creating an augmented 2nd between the 6th and 7th degrees. The melodic minor raises both the 6th and 7th degrees when ascending and reverts to the natural minor when descending.
自然小调音阶公式为全-半-全-全-半-全-全。和声小调升高第七级音，在六级与七级之间形成一个增二度。旋律小调上行时升高六级和七级，下行时还原为自然小调。
Natural minor: T – S – T – T – S – T – T
自然小调：全 – 半 – 全 – 全 – 半 – 全 – 全
4. Key Signatures and Circle of Fifths | 调号与五度圈
A key signature collects all the sharps or flats that consistently appear in a key paradigm. The order of sharps is F♯, C♯, G♯, D♯, A♯, E♯, B♯. The order of flats is B♭, E♭, A♭, D♭, G♭, C♭, F♭.
调号是集中标记在谱号之后、表示该调中需要持续升高或降低的变音记号。升号调的顺序为：F♯、C♯、G♯、D♯、A♯、E♯、B♯。降号调的顺序为：B♭、E♭、A♭、D♭、G♭、C♭、F♭。
To identify a major key from sharps: take the last sharp and go up a semitone. For flats: the penultimate flat in the key signature is the tonic (except for F major, which has one flat). Relative minors share the same key signature and are a minor third below the major tonic.
从升号判断大调调名：找到最后一个升号，再上行半音。从降号判断：倒数第二个降号即为主音（F大调仅一个降号除外）。关系小调与该大调共享同一调号，主音位于大调主音下方小三度。
Key signature with 3 sharps (F♯, C♯, G♯) → A major / F♯ minor
三个升号调（F♯, C♯, G♯）→ A大调 / 升F小调
| Sharps/Flats | Major Key | Relative Minor |
|---|---|---|
| 0 | C major | A minor |
| 1♯ | G major | E minor |
| 1♭ | F major | D minor |
| 2♯ | D major | B minor |
| 2♭ | B♭ major | G minor |
5. Intervals | 音程
An interval measures the distance in pitch between two notes, counted in scale steps (size) and further qualified by semitone count (quality). The number of letter names spanned gives the numerical size: 2nd, 3rd, 4th, etc. Quality may be perfect, major, minor, augmented or diminished.
音程表示两个音之间的音高距离，首先按音级数确定度数（大小），再按半音数确定性质。跨越几个音名就是几度：二度、三度、四度等。性质可分为纯、大、小、增、减。
Interval quality formula: count letter names for the number, then check semitones against the diatonic pattern.
音程性质公式：先数音名求得度数，再对比自然音阶中的半音数以确认性质。
| Interval | Semitones | Common Name |
|---|---|---|
| Perfect unison (P1) | 0 | 同度 |
| Minor 2nd (m2) | 1 | 小二度 |
| Major 2nd (M2) | 2 | 大二度 |
| Minor 3rd (m3) | 3 | 小三度 |
| Major 3rd (M3) | 4 | 大三度 |
| Perfect 4th (P4) | 5 | 纯四度 |
| Augmented 4th / Diminished 5th (A4/d5) | 6 | 增四度/减五度 |
| Perfect 5th (P5) | 7 | 纯五度 |
| Minor 6th (m6) | 8 | 小六度 |
| Major 6th (M6) | 9 | 大六度 |
| Minor 7th (m7) | 10 | 小七度 |
| Major 7th (M7) | 11 | 大七度 |
| Perfect octave (P8) | 12 | 纯八度 |
Inverting an interval: sum of the size equals 9 (e.g. a 3rd inverts to a 6th). Qualities flip: major becomes minor, perfect remains perfect, augmented becomes diminished.
音程转位规则：原音程度数加转位后度数等于9（如三度转位后成六度）。性质发生互换：大变小，小变大，纯不变，增减互换。
6. Triads and Chord Symbols | 三和弦与和弦符号
A triad is a three-note chord consisting of a root, a third and a fifth. The quality of the third and fifth defines the chord type. The formula for a major triad is a root, a major third (4 semitones) and a perfect fifth (7 semitones). A minor triad uses a minor third (3 semitones) with a perfect fifth.
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