📚 Year 10 CAIE Music: Quick Reference Handbook of Formulas and Theorems | Year 10 CAIE 音乐：公式定理速查手册
Music may not be built from mathematical formulas in the traditional sense, but its structure relies on precise rules and patterns that function just like theorems. This quick reference handbook gathers the essential ‘formulas’ every Year 10 CAIE Music student needs to memorise — from interval patterns and scale constructions to cadences and rhythmic groupings. Keep this guide close when practising analysis, composition or aural skills, and let these musical building blocks become second nature.
音乐或许不是由传统意义上的数学公式构建的，但它的结构依赖于如同定理般精确的规则与模式。这本速查手册汇集了每一位 Year 10 CAIE 音乐学生需要牢记的基本“公式”——从音程模式、音阶构建到终止式与节奏组合。在练习分析、作曲或听力技能时，请将这本指南放在手边，让这些音乐构成的基石成为你的第二天性。
1. Pitch Names and Keyboard Layout | 音名与键盘布局
The musical alphabet consists of seven letters: A, B, C, D, E, F, G. On a piano keyboard, these repeat in octaves. The black keys are sharps (#) or flats (b) of the white keys and are named as accidentals. The pattern of black keys — groups of two and three — helps identify the white notes: C is always immediately to the left of a group of two black keys.
音乐字母表由七个字母组成：A、B、C、D、E、F、G。在钢琴键盘上，它们以八度叠加重复。黑键是白键的升号(#)或降号(b)形式，被称为变化音。黑键的排列模式——两个一组和三个一组——有助于识别白键：C 始终紧邻两个黑键组的左侧。
The distance from any note to its next occurrence is an octave. The octave can be divided into 12 equal semitones (half steps) in equal temperament. A semitone is the smallest interval on the keyboard; two semitones make a whole tone (whole step).
从一个音符到它再次出现的距离就是一个八度。在平均律中，八度可以被均分为12个等大的半音。半音是键盘上最小的音程；两个半音构成一个全音。
2. Scale Formulas | 音阶公式
A scale is a sequence of notes ordered by pitch. The two foundational scales in Western music are the major scale and the natural minor scale, each defined by a fixed pattern of tones (T) and semitones (S).
音阶是按音高顺序排列的一串音符序列。西方音乐中两个基础音阶是大调音阶和自然小调音阶，每个都由固定的全音(T)与半音(S)模式定义。
Major Scale Formula: T – T – S – T – T – T – S
Natural Minor Scale Formula: T – S – T – T – S – T – T
大调音阶公式：T – T – S – T – T – T – S
自然小调音阶公式：T – S – T – T – S – T – T
Using these formulas, you can build any major or minor scale starting from any tonic note. For example, C major: C (T) D (T) E (S) F (T) G (T) A (T) B (S) C.
利用这些公式，你可以从任何一个主音开始构建任何大调或小调音阶。例如，C大调：C (T) D (T) E (S) F (T) G (T) A (T) B (S) C。
The harmonic minor scale raises the seventh degree of the natural minor by one semitone, creating an exotic step-and-a-half interval between the sixth and seventh degrees. The melodic minor scale raises both the sixth and seventh degrees when ascending but reverts to the natural minor when descending.
和声小调音阶将自然小调的第七级音升高半音，在第六级和第七级之间制造了一个特有的增二度音程。旋律小调音阶在上行时升高第六和第七级音，但在下行时回到自然小调的形式。
3. Intervals: Counting and Quality | 音程：计算与性质
An interval is the distance in pitch between two notes. It is named by two components: the numerical distance (2nd, 3rd, 4th, etc.) and the quality (perfect, major, minor, augmented, diminished).
音程是两个音符之间的音高距离。它的名称由两部分组成：数字距离（二度、三度、四度等）和性质（纯、大、小、增、减）。
To find the number, count the letter names from the lower note to the higher note, inclusive. For quality in major scales, all intervals from the tonic are either perfect (unison, 4th, 5th, octave) or major (2nd, 3rd, 6th, 7th).
要确定音数，请从较低音符开始数字母名称直到较高音符，包括两端。在大调音阶中，从主音开始的所有音程要么是纯音程（同度、四度、五度、八度），要么是大音程（二度、三度、六度、七度）。
If a major interval is reduced by a semitone, it becomes minor. If a perfect interval is reduced by a semitone, it becomes diminished; if enlarged by a semitone, it becomes augmented. For example, C to E is a major 3rd; C to Eb is a minor 3rd; C to G is a perfect 5th; C to G# is an augmented 5th.
如果一个大音程被减小一个半音，它就变成小音程。如果一个纯音程被减小一个半音，它变成减音程；如果被增大一个半音，它变为增音程。例如，C到E是大三度；C到Eb是小三度；C到G是纯五度；C到G#是增五度。
Memorise the number of semitones for each interval in its simple form:
Minor 2nd = 1 semitone, Major 2nd = 2, Minor 3rd = 3, Major 3rd = 4, Perfect 4th = 5, Augmented 4th/Diminished 5th (Tritone) = 6, Perfect 5th = 7, Minor 6th = 8, Major 6th = 9, Minor 7th = 10, Major 7th = 11, Octave = 12.
记住每个单音程的半音数：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=11，八度=12。
4. Chord Construction: Triads to Sevenths | 和弦构建：从三和弦到七和弦
A triad is a three-note chord consisting of a root, a third and a fifth. There are four types of triads, each defined by the intervals stacked above the root.
三和弦是由根音、三音和五音组成的三个音符的和弦。三和弦有四种类型，每种都由叠加在根音之上的音程定义。
Major triad: root + Major 3rd + Perfect 5th (e.g. C-E-G)
Minor triad: root + Minor 3rd + Perfect 5th (e.g. C-Eb-G)
Diminished triad: root + Minor 3rd + Diminished 5th (e.g. C-Eb-Gb)
Augmented triad: root + Major 3rd + Augmented 5th (e.g. C-E-G#)
大三和弦：根音+大三度+纯五度（例如C-E-G）
小三和弦：根音+小三度+纯五度（例如C-Eb-G）
减三和弦：根音+小三度+减五度（例如C-Eb-Gb）
增三和弦：根音+大三度+增五度（例如C-E-G#）
Triads built on the degrees of a major scale follow a fixed pattern of chord qualities: I (major), ii (minor), iii (minor), IV (major), V (major), vi (minor), vii° (diminished). This pattern is crucial for harmonic analysis and chord progressions.
在大调音阶各音级上构建的三和弦，遵循固定的和弦性质模式：I（大）、ii（小）、iii（小）、IV（大）、V（大）、vi（小）、vii°（减）。这个模式对于和声分析及和弦进行至关重要。
Seventh chords add a fourth note, a seventh above the root. The dominant seventh chord (V⁷) is built on the fifth degree: major triad plus a minor 7th above the root (e.g. G-B-D-F in C major). The diminished seventh chord (vii°⁷) contains three stacked minor 3rds (e.g. B-D-F-Ab).
七和弦增加第四个音符，即根音上方的七度音。属七和弦（V⁷）建立在第五级上：大三和弦加上根音上方的小七度（例如，C大调中的G-B-D-F）。减七和弦（vii°⁷）包含三个叠加的小三度（例如，B-D-F-Ab）。
5. Key Signatures and the Circle of Fifths | 调号与五度圈
Key signatures are written at the beginning of a piece, indicating which notes should consistently be played sharp or flat. The order of sharps is F#, C#, G#, D#, A#, E#, B#. The order of flats is Bb, Eb, Ab, Db, Gb, Cb, Fb.
调号写在乐曲开头，指示哪些音符应始终演奏为升号或降号。升号的顺序是F#、C#、G#、D#、A#、E#、B#。降号的顺序是Bb、Eb、Ab、Db、Gb、Cb、Fb。
To determine the major key from a sharp key signature, go up one semitone from the last sharp. For flat key signatures, the second-to-last flat gives the major key name. The key of C major has no sharps or flats; A minor shares the same key signature (its relative minor), which is the sixth degree of C major.
要从升调号判断大调，请从最后一个升号向上移动一个半音。对于降调号，倒数第二个降号即为大调名称。C大调没有升降号；A小调共享相同的调号（其关系小调），即C大调的第六级音。
The circle of fifths organises all 12 major and relative minor keys in a diagram. Moving clockwise, each new key adds one sharp; moving anticlockwise, each new key adds one flat. Adjacent keys on the circle are closely related and share many common chords, making it an essential tool for modulation and composition.
五度圈将全部12个大调及其关系小调组织在一个图表中。顺时针移动，每个新调增加一个升号；逆时针移动，每个新调增加一个降号。五度圈上相邻的调是近关系调，共享许多共同和弦，使其成为转调和作曲的重要工具。
6. Rhythm and Time Signatures | 节奏与拍号
A time signature is a symbol placed after the clef and key signature that specifies how many beats are in each bar and which note value represents one beat. The top number indicates the number of beats per bar; the bottom number indicates the beat unit (4 = crotchet, 8 = quaver, 2 = minim).
拍号是放在谱号和调号之后的符号，指明每小节有多少拍以及什么音符值代表一拍。上方数字表示每小节的拍数；下方数字表示拍单位（4=四分音符，8=八分音符，2=二分音符）。
Simple time signatures have a top number of 2, 3 or 4 and each beat divides naturally into two equal parts. Compound time signatures have a top number of 6, 9 or 12 and each beat divides into three equal parts (dotted beat). For example, 6/8 has two dotted-crotchet beats per bar, each beat subdivided into three quavers.
简单拍号的上方数字为2、3或4，每拍自然地分为两个等份。复合拍号的上方数字为6、9或12，每拍分为三个等份（附点拍）。例如，6/8拍每小节有两个附点四分音符的拍子，每拍再细分为三个八分音符。
Common rhythmic patterns include syncopation (emphasis on weak beats or off beats), hemiola (a shift from 6/8 grouping in three twos to two threes, or vice versa), and anacrusis (an upbeat that starts the melody before the first full bar).
常见节奏模式包括切分音（强调弱拍或拍中错位）、赫米奥拉（在6/8拍中由三个二音组切换为两个三音组，反之亦然）以及弱起（在第一个完整小节之前开始旋律的上拍）。
Note value tree (UK terminology):
Semibreve (4 beats) → Minim (2 beats) → Crotchet (1 beat) → Quaver (1/2 beat) → Semiquaver (1/4 beat). Dotted notes increase the duration by half of the original value.
音符时值树（英式术语）：全音符（4拍）→二分音符（2拍）→四分音符（1拍）→八分音符（1/2拍）→十六分音符（1/4拍）。附点音符将原有时值增加一半。
7. Cadences: The Punctuation of Music | 终止式：音乐的标点
A cadence is a progression of two or more chords that concludes a phrase, section or piece of music. There are four primary types of cadences in Common Practice harmony, each with a characteristic sound and function.
终止式是由两个或更多和弦组成的进行，用来结束一个乐句、乐段或整首乐曲。在共性实践时期和声中，有四种主要类型的终止式，每种都有其特有的音响效果和功能。
Perfect Cadence: V – I (or V⁷ – I). It sounds final and conclusive, like a full stop. Both chords are typically in root position, and the tonic note is in the melody of the I chord.
完全终止：V – I（或V⁷ – I）。听上去具有结束感和终结性，就像一个句号。两个和弦通常都是根音位置，且I和弦的旋律音为主音。
Plagal Cadence: IV – I. Often called the ‘Amen’ cadence, it sounds gentle and resolved, frequently used at the end of hymns.
变格终止：IV – I。常被称为“阿们”终止，听上去柔和且带有解决感，常用于赞美诗的结尾。
Imperfect Cadence: I, ii, IV or vi – V. It sounds unfinished, like a comma, and demands continuation. Any chord moving to V creates an imperfect cadence.
不完全终止：I、ii、IV或vi – V。听上去未完成，像一个逗号，需要继续发展。任何和弦进行到V都构成不完全终止。
Interrupted Cadence: V – vi (or sometimes V – IV/VIb). The listener expects V to go to I, but the progression ‘surprises’ by moving to the submediant, creating an unexpected break or interruption.
阻碍终止：V – vi（有时为V – IV/VIb）。听众期待V解决到I，但和声却“意外地”进行到下中音和弦，制造出一种意想不到的停顿或阻断感。
8. Chord Progressions and Harmonisation Rules | 和弦进行与和声化规则
Functional harmony assigns specific roles to chords within a key. The three primary chords, I (tonic), IV (subdominant) and V (dominant), form the backbone of tonal music. The tonic is the home chord, the dominant creates tension that strongly pulls back to the tonic, and the subdominant provides a complementary departure.
功能和声为调内的和弦赋予了特定的角色。三个主要和弦，I（主和弦）、IV（下属和弦）和V（属和弦），构成了调性音乐的骨架。主和弦是家的感觉，属和弦制造紧张感并强烈地倾向回主和弦，下属和弦则提供一种补充性的离调。
Common chord progression patterns include:
- I – IV – V – I (basic primary triad progression)
- I – vi – IV – V (the 1950s doo-wop progression)
- ii – V – I (essential jazz and Classical turnaround)
- I – V – vi – IV (modern pop four-chord song pattern)
常见的和弦进行模式包括：
- I – IV – V – I（基础主要三和弦进行）
- I – vi – IV – V（20世纪50年代的doo-wop进行）
- ii – V – I（爵士乐和古典音乐中必不可少的回转进行）
- I – V – vi – IV（现代流行音乐的四和弦歌曲模式）
When harmonising a melody, select chords that contain the melody note, aiming for smooth voice leading. Avoid parallel fifths and octaves between any two voices; move the bass and upper voices in contrary or oblique motion where possible. Leading notes (7th degree) should generally rise to the tonic when in the melody or an outer voice.
在为旋律配置和声时，选择包含该旋律音的和弦，力求平稳的声部进行。避免任何两个声部之间出现平行五度和平行八度；尽可能使低音和上方声部以反向或斜向运动。导音（第七级音）在旋律或外声部时，通常应上行解决到主音。
9. Melody Writing: Structure and Shape | 旋律写作：结构与形态
A well-crafted melody typically exhibits balance, a clear phrase structure and motivic development. Phrases are often 4 or 8 bars long and arranged in ‘question and answer’ (antecedent and consequent) pairs, where the first phrase ends on an imperfect cadence and the second on a perfect cadence.
一段精心雕琢的旋律通常展现出平衡、清晰的乐句结构以及动机的展开。乐句通常是4或8个小节长，并以“问答句”（前句和后句）成对排列，其中前句以不完全终止结束，后句以完全终止结束。
Use stepwise (conjunct) motion as the foundation, and introduce leaps (disjunct) sparingly to create interest. A leap of a 4th, 5th or octave should ideally be followed by a step in the opposite direction to maintain singability. The overall contour should have a single high point (climax) and a smooth descent towards the cadence.
应将级进(conjunct)作为旋律的基础，并谨慎地引入跳进(disjunct)以增添趣味。四度、五度或八度的跳进之后，最好反向级进，以保持可唱性。旋律的整体轮廓应具有一个高潮点(climax)，并向终止式平滑下行。
Motivic development techniques include repetition (exact or at a different pitch), sequence (repeating a pattern at a higher or lower pitch level), inversion (flipping the interval direction), augmentation (lengthening note values) and diminution (shortening note values).
动机发展手法包括重复（精确重复或在不同音高上重复）、模进（在更高或更低的音高层面重复一个模式）、倒影（翻转音程方向）、增值（加长音符时值）和减值（缩短音符时值）。
10. Musical Texture and Ensemble Types | 音乐织体与合奏类型
Texture describes how many layers of sound are heard at once and how they relate. Monophonic texture consists of a single unaccompanied melodic line. Homophonic texture features a melody with chordal accompaniment (melody-dominated homophony) or block chords moving together (chorale style). Polyphonic (contrapuntal) texture combines two or more independent melodic lines simultaneously.
织体描述了一时间能听到多少声音层次以及它们之间的关系。单音织体由一条无伴奏的单旋律线组成。主调织体以一条旋律加和弦伴奏为特征（旋律支配型主调），或者以块状和弦一起移动为特征（众赞歌风格）。复调（对位）织体则同时结合了两条或更多独立的旋律线条。
Common chamber ensembles in the CAIE syllabus include the string quartet (2 violins, viola, cello), wind quintet (flute, oboe, clarinet, bassoon, horn), piano trio (piano, violin, cello) and jazz trio/quartet. Orchestras are larger ensembles divided into strings, woodwinds, brass and percussion sections.
CAIE大纲中常见的室内乐合奏包括弦乐四重奏（两把小提琴、中提琴、大提琴）、管乐五重奏（长笛、双簧管、单簧管、巴松、圆号）、钢琴三重奏（钢琴、小提琴、大提琴）以及爵士三重奏/四重奏。管弦乐队是更大的合奏，分为弦乐、木管、铜管和打击乐四个乐器组。
Recognising texture changes within a piece — for example, a shift from homophonic to polyphonic texture during a development section, or a sudden monophonic unison passage — is an important analytical skill for the listening component of the exam.
识别乐曲内部织体的变化——例如，在展开部中从主调织体转为复调织体，或是突然出现的单音齐奏段落——是考试听力部分重要的分析技能。
11. Musical Forms and Structures | 音乐曲式与结构
Musical form gives a composition its framework. Binary form (AB) consists of two sections, each usually repeated. Ternary form (ABA) presents a contrast in the B section before returning to the A section, which may be varied. Rondo form follows the pattern ABACA (or sometimes ABACADA), where the principal theme (A) keeps returning.
音乐曲式赋予作品以框架。二部曲式(AB)由两个段落组成，每段通常各自反复。三部曲式(ABA)在B段呈现对比，然后回到A段，A段可能会有变化。回旋曲式遵循ABACA（有时为ABACADA）的模式，其中主要主题(A)不断回归。
Theme and variations takes a basic theme and alters it in successive iterations through changes in melody, harmony, rhythm, metre, tonality, instrumentation or texture. Sonata form (often found in first movements of Classical symphonies) has three main sections: exposition (first and second subjects in contrasting keys), development (thematic material is developed and modulated) and recapitulation (both subjects return in the tonic key).
主题与变奏以一个基本主题为基础，通过改变旋律、和声、节奏、节拍、调性、乐器法或织体，在随后的反复中进行变化。奏鸣曲式（常见于古典主义交响曲的第一乐章）有三个主要部分：呈示部（呈示在对比调性上的第一主题和第二主题）、展开部（主题材料被发展并频繁转调）和再现部（两个主题都回到主调再现）。
For Year 10, you should be able to identify these structures in scores and audio excerpts, and describe how the composer creates contrast and unity within the chosen form.
对于Year 10，你应该能够在乐谱和音频选段中识别这些结构，并描述作曲家在所选曲式中如何创造对比与统一。
12. Stylistic Periods and Key Features | 风格时期与关键特征
The three main stylistic periods you will encounter in the CAIE IGCSE course are Baroque, Classical and Romantic. Each has characteristic features that help identify the era of a piece.
你在CAIE IGCSE课程中会遇到的三个主要风格时期是巴洛克、古典主义和浪漫主义。每个时期都有帮助识别作品时代的特征性标志。
Baroque (c.1600-1750): Use of harpsichord and basso continuo (continuous bass line with figured bass), terraced dynamics, polyphonic textures (fugue, canon), ornate ornamentation (trills, mordents, turns), and dance suites. Key composers: Bach, Handel, Vivaldi.
巴洛克时期（约1600-1750）：使用羽管键琴和持续低音（带数字低音的连续低音线条）、阶梯式力度、复调织体（赋格、卡农）、华丽装饰音（颤音、波音、回音）以及舞蹈组曲。代表作曲家：巴赫、亨德尔、维瓦尔第。
Classical (c.1750-1820): Emphasis on balance, clarity and singable melodies; homophonic texture with Alberti bass and other accompaniment patterns; use of crescendo and diminuendo; clear phrasing with periodic structures; development of the symphony, string quartet and sonata form. Key composers: Haydn, Mozart, early Beethoven.
古典主义时期（约1750-1820）：强调平衡、清晰与可唱性的旋律；以阿尔贝蒂低音及其他伴奏音型为特征的主调织体；使用渐强和渐弱；清晰的分句与周期性的结构；交响曲、弦乐四重奏和奏鸣曲式的发展。代表作曲家：海顿、莫扎特、早期贝多芬。
Romantic (c.1820-1900): Expressive, emotional melodies with wide leaps and chromaticism; expanded orchestra with richer harmonies and varied tone colours; programme music and character pieces; rubato tempo and wide dynamic range; nationalism and exoticism. Key composers: Chopin, Schumann, Tchaikovsky, Brahms.
浪漫主义时期（约1820-1900）：富有表现力、情感化的旋律，包含大跳和半音阶；编制扩大、和声更丰富、音色更多变的管弦乐队；标题音乐和特性小品；弹性速度和大动态范围；民族主义和异国情调。代表作曲家：肖邦、舒曼、柴可夫斯基、勃拉姆斯。
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