📚 Year 11 CIE Music Formula & Theorem Quick Reference Handbook | CIE 音乐公式定理速查手册
Music theory, like mathematics, is built upon a set of fundamental rules, patterns, and relationships that govern how melodies, harmonies, and rhythms are constructed. This quick reference handbook distills the essential 'formulas' and 'theorems' of the CIE IGCSE Music (0410) syllabus into a concise, easy-to-revise format. Whether you are memorizing interval qualities, decoding chord progressions, or checking the structure of a minuet and trio, these are the core theoretical building blocks you need at your fingertips for the listening paper, the prepared listening extract, and your composition coursework.
音乐理论如同数学，建立在一套控制旋律、和声与节奏如何构建的基本规则、模式和关系之上。这本速查手册将 CIE IGCSE 音乐（0410）课程大纲中的基本"公式"和"定理"提炼成简洁、易于复习的格式。无论你是在记忆音程性质、解读和弦进行，还是在核对小步舞曲与三声中部结构，这些都是你在听力试卷、指定听力作品分析以及作曲课程作业中需要随时掌握的核心理论构件。
1. Interval Formula | 音程公式
An interval is the distance in pitch between two notes, measured in semitones and described by both a number (size) and a quality. The quality of a perfect, major, or minor interval follows fixed semitone counts measured from the lower note upwards.
音程是两个音符之间的音高距离，以半音为单位测量，并用数字（度数）和性质来描述。纯音程、大音程或小音程的性质遵循从下方音向上测量的固定半音数。
Perfect intervals (within one octave): ones; Perfect 5th = 7 semitones; Perfect octave = 12 semitones. These intervals are called 'perfect' y because they appear in both major and minor scales unchanged and produce a hollow, stable consonance.
纯音程（一个八度内）：纯一度 = 0 半音；纯四度 = 5 半音；纯五度 = 7 半音；纯八度 = 12 半音。这些音程被称为"纯"是因为它们在大调和小调音阶中都保持不变，并产生空洞、稳定的协和感。
Major intervals (within one octave): 1 semitones. Each major interval is exactly one semitone wider than its minor counterpart.
大音程（一个八度内）：大二度 = 2 半音；大三度 = 4 半音；大六度 = 9 半音；大七度 = 11 半音。每个大音程恰好比对应的小音程宽一个半音。
Interval inversion formula: l number, and the quality flips: major becomes minor, minor becomes major, perfect remains perfect, augmented becomes diminished, and diminished becomes augmented.
音程转位公式：当一个音程被转位时（下方音移高一个八度），新的度数 = 9 − 原度数，性质翻转：大变小，小变大，纯保持纯，增变减，减变增。
| Interval 音程 | Semitones 半音 | Example from C 以C为例 |
|---|---|---|
| Minor 2nd (m2) 小二度 | 1 | C – D♭ |
| Major 2nd (M2) 大二度 | 2 | C – D |
| Minor 3rd (m3) 小三度 | 3 | C – E♭ |
| Major 3rd (M3) 大三度 | 4 | C – E |
| Perfect 4th (P4) 纯四度 | 5 | C – F |
| Perfect 5th (P5) 纯五度 | 7 | C – G |
| Minor 6th (m6) 小六度 | 8 | C – A♭ |
| Major 6th (M6) 大六度 | 9 | C – A |
| Minor 7th (m7) 小七度 | 10 | C – B♭ |
| Major 7th (M7) 大七度 | 11 | C – B |
| Perfect Octave (P8) 纯八度 | 12 | C – C (higher) |
2. Chord Construction Formula | 和弦构建公式
Every type of triad and seventh chord is built from a root note stacked with specific intervals measured in semitones. The formula remains constant regardless of the starting pitch, which is the foundational 'chord construction theorem' of Western harmony.
每种三和弦和七和弦都是由根音加上以半音计量的特定音程叠置而成。无论从哪个音开始，公式保持不变，这是西方和声的基本"和弦构建定理"。
Triad formulas from root: s) + augmented 5th (8 semitones). Diminished triad = root + m3 (3 semitones) + diminished 5th (6 semitones).
以根音为基础的三和弦公式：大三和弦 = 根音 + 大三度（4 半音）+ 纯五度（7 半音）。小三和弦 = 根音 + 小三度（3 半音）+ 纯五度（7 半音）。增三和弦 = 根音 + 大三度（4 半音）+ 增五度（8 半音）。减三和弦 = 根音 + 小三度（3 半音）+ 减五度（6 半音）。
Seventh chord formulas (all intervals measured from root): major 7th). Minor 7th = m3 + P5 + m7 (minor triad plus minor 7th). Half-diminished 7th (m7♭ ished 7th = m3 + dim5 + dim7 (all intervals stacked as minor thirds, a symmetrical chord).
七和弦公式（所有音程从根音测量）：属七和弦 = 大三度 + 纯五度 + 小七度（大三和弦加根音上方小七度）。大七和弦 = 大三度 + 纯五度 + 大七度（大三和弦加大七度）。小七和弦 = 小三度 + 纯五度 + 小七度（小三和弦加小七度）。半减七和弦（m7♭5）= 小三度 + 减五度 + 小七度（减三和弦加小七度）。减七和弦 = 小三度 + 减五度 + 减七度（所有音程均为小三度叠置，是一个对称和弦）。
3. Scale Formula | 音阶公式
Scales are ordered sequences of notes that follow fixed patterns of tones (whole steps, T) and semitones (half steps, S). The three most essential scale formulas for CIE IGCSE are the major scale, the natural minor scale, and the harmonic minor scale. The chromatic scale, built entirely of semitones, underlies all twelve-note resources.
音阶是按照固定的全音（T）和半音（S）模式排列的音符序列。CIE IGCSE 课程中最重要的三种音阶公式是大调音阶、自然小调音阶与和声小调音阶。完全由半音构成的半音阶则是一切十二音素材的基础。
Major scale formula (ascending): T – T – S – T – T – T – S. This produces the familiar pattern of whole and half steps: between degrees 3–4 and 7–8 are semitones; all other adjacent pairs are whole tones. This formula applies to all twelve major keys.
大调音阶公式（上行）：T – T – S – T – T – T – S。这产生了熟悉的全音半音排列：第3–4级和第7–8级之间为半音；其余相邻音级之间均为全音。此公式适用于所有十二个大调。
Natural minor scale formula (ascending): T – S – T – T – S – T – T. Semitones fall between degrees 2–3 and 5–6. This scale shares the same key signature as its relative major (the major scale starting on the third degree of the minor scale).
自然小调音阶公式（上行）：T – S – T – T – S – T – T。半音位于第2–3级和第5–6级之间。该音阶与其关系大调（从该小调第三级开始的大调音阶）共用同一调号。
Harmonic minor scale formula (ascending): T – S – T – T – S – Augmented 2nd (3 semitones) – S. This is the natural minor scale with a raised (sharpened) 7th degree from the key signature, creating the characteristic augmented 2nd between degrees 6 and 7.
和声小调音阶公式（上行）：T – S – T – T – S – 增二度（3 半音）– S。这是将调号中第7级升高（升半音）的自然小调音阶，在第6级和第7级之间产生了特有的增二度。
Melodic minor scale formula: Ascending: T – S – T – T – T – T – S (raised 6th and 7th degrees). Descending: identical to the natural minor (T – T – S – T – T – S – T). The ascending form smooths the augmented 2nd of the harmonic minor for melodic ease.
旋律小调音阶公式：上行：T – S – T – T – T – T – S（升高第6级和第7级）。下行：与自然小调相同（T – T – S – T – T – S – T）。上行形式为了旋律流畅性而消解了和声小调的增二度。
4. Key Signature Theorem | 调号定理
Key signatures are determined by two immutable patterns: the cycle of sharps (Father Charles Goes Down And Ends Battle) and the cycle of flats (Battle Ends And Down Goes Charles' Father). The number of sharps or flats uniquely identifies the major key, and the relative minor is always the sixth degree (la) of that major scale.
调号由两个不变的循环决定：升号循环（Father Charles Goes Down And Ends Battle，即 FCGDAEB）和降号循环（Battle Ends And Down Goes Charles' Father，即 BEADGCF）。升号或降号的数量唯一地标识大调，而关系小调始终是该大调音阶的第六级（la）。
Sharp key identification theorem: To find the major key from a sharp key signature, take the last sharp in the sequence and go up one semitone (a minor 2nd). For example, if the last sharp is D♯, the major key is E major. The relative minor is the note a minor 3rd below the major tonic.
升号调识别定理：根据升号调号找大调，取升号序列中的最后一个升号，再上升一个半音（小二度）。例如，若最后一个升号是 D♯，则大调为 E 大调。关系小调位于大调主音下方小三度处。
Flat key identification theorem: For flat key signatures, the second-to-last flat in the sequence names the major key (for two or more flats). If there is only one flat (B♭), the key is F major. The relative minor is a minor 3rd below.
降号调识别定理：对于降号调号，序列中倒数第二个降号即为大调名称（适用于两个及以上降号）。若只有一个降号（B♭），则为 F 大调。关系小调在下方小三度处。
Relative minor theorem: Relative major and minor keys share the same key signature. To find the relative minor from a major key, count down three semitones from the major tonic. To find the relative major from a minor key, count up three semitones from the minor tonic.
关系小调定理：关系大小调共用同一调号。从大调主音向下数三个半音即可找到关系小调；从小调主音向上数三个半音即可找到关系大调。
5. Time Signature Decoder | 拍号解码器
eceives one beat (2 = minim half note, 4 = crotchet quarter note, 8 = quaver eighth note). Simple, compound, and irregular time categories each follow distinct internal grouping rules.
拍号由两个数字组成。上方数字表示每小节有多少拍；下方数字表示以何种音符为一拍（2 = 二分音符，4 = 四分音符，8 = 八分音符）。单拍子、复拍子和不规则拍子各自遵循不同的内部分组规则。
Simple time theorem: ivides into two equal subdivisions. The beat is an undotted note. Examples: 2/4 (two crotchet beats per bar), 3/4 (three crotchet beats), 4/4 (four crotchet beats), 2/2 (two minim beats), 3/8 at a slow tempo (three quaver beats).
单拍子定理：在单拍子中（上方数字 = 2、3 或 4），每一拍自然地二等分。拍子单位是不附点的音符。例如：2/4（每小节两个四分音符拍）、3/4（三个四分音符拍）、4/4（四个四分音符拍）、2/2（两个二分音符拍）、慢速的 3/8（三个八分音符拍）。
Compound time theorem: umber of beats, divide the upper number by 3. Examples: 6/8 = 2 dotted-crotchet beats per bar; 9/8 = 3 dotted-crotchet beats; 12/8 = 4 dotted-crotchet beats.
复拍子定理：在复拍子中（上方数字 = 6、9 或 12），每一拍是一个附点音符，自然地三等分。拍子数量 = 上方数字 ÷ 3。例如：6/8 = 每小节2个附点四分音符拍；9/8 = 3个附点四分音符拍；12/8 = 4个附点四分音符拍。
Irregular time signatures: lly grouped in alternating patterns of 2s and 3s (e.g., 5/4 = 2+3 or 3+2 crotchet groupings).
不规则拍号：上方数字不是 2、3、4、6、9 或 12 的拍号（如 5/4、7/8、11/8）为不规则或不对称拍号。小节内的拍子通常以 2 拍和 3 拍的交替模式分组（例如 5/4 = 2+3 或 3+2 的四分音符分组）。
6. Dynamics Scale | 力度等级表
Italian dynamic markings are the universal language of volume in Western classical music. They range from the barely audible to the overwhelmingly loud and are often modified by gradual-change instructions. Memorising the standard eight-level scale plus the two gradual-change terms covers all CIE IGCSE requirements.
意大利语力度标记是西方古典音乐中通用的音量语言。它们从几乎听不见到势不可挡地响亮，并且通常由渐变指示来修饰。记住标准的八级标尺加上两个渐变术语即可覆盖 CIE IGCSE 的所有要求。
| Abbreviation 缩写 | Italian Term 意大利语 | Meaning 含义 | Approximate Level 大致级别 |
|---|---|---|---|
| ppp | pianississimo | Very, very soft 极极弱 | Barely audible |
| pp | pianissimo | Very soft 很弱 | Whisper-soft |
| p | piano | Soft 弱 | Gentle |
| mp | mezzo-piano | Moderately soft 中弱 | Half-soft |
| mf | mezzo-forte | Moderately loud 中强 | Half-loud |
| f | forte | Loud 强 | Strong |
| ff | fortissimo | Very loud 很强 | Forceful |
| fff | fortississimo | Very, very loud 极极强 | Overwhelming |
Gradual dynamic changes: crescendo (cresc.) = gradually becoming louder. diminuendo (dim.) or decrescendo (decresc.) = gradually becoming softer. These are often notated with hairpins (< for crescendo, > for diminuendo) that open or close over the affected passage.
渐变的力度变化：crescendo（cresc.）= 逐渐变强。diminuendo（dim.）或 decrescendo（decresc.）= 逐渐变弱。这些通常用渐强渐弱符号表示（< 表示渐强，> 表示渐弱），在受影响的段落上张开或闭合。
7. Tempo Marking Reference | 速度标记参考
d expressive character of a piece. They fall into slow, moderate, fast, and very fast categories, with numerous modifiers that fine-tune the intended character. The metronome mark (e.g., ♩ xact beats-per-minute value, but expressive terms convey the overall mood alongside speed.
速度标记指示乐曲的速度和表现特征。它们分为慢速、中速、快速和极快速类别，并有大量修饰词来微调所意图的表现特征。节拍器标记（如 ♩ = 120）规定精确的每分钟拍数，但表现性术语在速度之外还传达整体的情绪氛围。
Slow tempi (approximately 40–60 BPM): Largo — broad, very slow, and dignified. Lento — slow. Adagio — slow and stately, 'at ease'. Grave — very slow, solemn, and heavy.
慢速（大约 40–60 BPM）：Largo — 宽广的，非常缓慢且庄严。Lento — 缓慢的。Adagio — 缓慢而庄重，"从容的"。Grave — 极为缓慢、肃穆、沉重的。
Moderate tempi (approximately 66–108 BPM): Andante — at a walking pace, flowing. Andantino — slightly faster than Andante (or sometimes slightly slower, context-dependent). Moderato — moderate speed. Allegretto — moderately fast, lighter than Allegro.
中速（大约 66–108 BPM）：Andante — 行板，步行速度，流动的。Andantino — 比行板略快（或有时略慢，视语境而定）。Moderato — 中速。Allegretto — 中等偏快，比快板轻盈。
Fast and very fast tempi (approximately 120–200+ BPM): Allegro — fast, lively, and cheerful. Vivace — lively and brisk, faster than Allegro. Presto — very fast. Prestissimo — as fast as possible, the extreme of speed.
快速和极快速（大约 120–200+ BPM）：Allegro — 快板，快速、活泼、欢快的。Vivace — 生气勃勃、轻快的，比快板更快。Presto — 急板，非常快。Prestissimo — 最急板，尽可能快，速度的极致。
Common tempo modifiers: molto — very (e.g., Allegro molto — very fast). ma non troppo — but not too much (e.g., Allegro ma non troppo — fast but not too much). poco — a little (e.g., poco più mosso — a little more movement). ritardando (rit.) — gradually slowing down. accelerando (accel.) — gradually speeding up. a tempo — return to the original tempo after a deviation. rubato — expressive, flexible tempo with slight speeding up and slowing down within a phrase.
常见速度修饰词：molto — 非常（如 Allegro molto — 非常快）。ma non troppo — 但不过分（如 Allegro ma non troppo — 快但不过分）。poco — 一点点（如 poco più mosso — 稍微更多动感）。ritardando（rit.）— 逐渐减速。accelerando（accel.）— 逐渐加速。a tempo — 回到原速。rubato — 弹性速度，在乐句内稍快稍慢的自由表现性速度。
8. Ornamentation Decoder | 装饰音解码器
Ornaments are melodic decorations that add expressive detail to a melody. In Baroque and Classical repertoire — central to the CIE IGCSE set works and score-reading — they are indicated by specific symbols or small notes. Each ornament has a fixed execution pattern relative to the principal written note.
装饰音是增加旋律表现性细节的旋律装饰。在巴洛克和古典曲目中——这是 CIE IGCSE 指定作品和乐谱阅读的核心——它们由特定的符号或小音符标示。每个装饰音相对于主要书面音符有固定的演奏模式。
Trill (tr): A rapid alternation between the written principal note and the note above (the upper auxiliary, usually diatonic to the key). The trill starts on the upper auxiliary in Baroque and early Classical practice, and on the principal note from the late Classical period onward. In CIE exams, context determines the starting note; the trill typically lasts for the full written duration.
颤音（tr）：在所写的主要音符与其上方音符（上方辅助音，通常属于本调的自然音）之间快速交替。在巴洛克和古典早期实践中，颤音从上方辅助音开始；自古典晚期起从主要音符开始。在 CIE 考试中，上下文决定起始音；颤音通常持续整个所写时值。
Mordent (upper mordent, written as a short wavy line with a vertical stroke through it): The principal note, the note above, and the principal note again, played rapidly. The lower mordent (wavy line without vertical stroke) alternates principal note, note below, principal note. Both are compact, three-note patterns.
波音（上波音，写作短波浪线加纵向短线）：主要音符、上方音符、再回到主要音符，快速演奏。下波音（无纵向短线的波浪线）交替演奏主要音符、下方音符、主要音符。两者都是紧凑的三音模式。
Turn (written as a sideways S-shape symbol ∼ above the note): A four-note figure consisting of the upper auxiliary note, the principal note, the lower auxiliary note, and the principal note again. When placed between two notes, the turn occurs after the principal note has sounded. If the symbol includes an accidental above or below, apply it to the corresponding auxiliary.
回音（写在音符上方的横置 S 形符号 ∼）：四音音型，依次为上方辅助音、主要音符、下方辅助音、再回到主要音符。当符号置于两个音符之间时，回音在主要音符发出后进行。若符号上方或下方包含临时变音记号，则应用于对应的辅助音。
Appoggiatura (small note before a principal note, usually printed in smaller type without a stroke through the stem): A dissonant 'leaning' note that takes half the value of the following principal note (or two-thirds if the principal note is dotted). The appoggiatura is played on the beat, creating an expressive dissonance that resolves onto the principal note. This is different from the acciaccatura (small note with
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