📚 Year 8 CAIE Music: Quick Reference Handbook of Formulas and Theorems | Year 8 CAIE 音乐公式定理速查手册
Music, like mathematics, is built on patterns, relationships, and rules. This handbook collects the essential ‘formulas’ and ‘theorems’ you need for Year 8 CAIE Music – from scale constructions and interval inversions to chord building and rhythmic grouping. Each rule is presented as a clear statement, followed by its practical application.
音乐如同数学，建立于模式、关系和规则之上。本手册汇集了 Year 8 CAIE 音乐所需的核心“公式”与“定理”——从音阶构成、音程转位，到和弦构建和节奏组合。每条规则都以清晰陈述的形式呈现，并附上实际应用说明。
1. Scale Formulas | 音阶公式
Major Scale: W – W – H – W – W – W – H (where W = whole tone, H = semitone).
大调音阶：全 – 全 – 半 – 全 – 全 – 全 – 半（全=全音，半=半音）。
Natural Minor Scale: W – H – W – W – H – W – W.
自然小调音阶：全 – 半 – 全 – 全 – 半 – 全 – 全。
Harmonic Minor Scale: W – H – W – W – H – Augmented 2nd – H (raises the 7th degree).
和声小调音阶：全 – 半 – 全 – 全 – 半 – 增二度 – 半（升高第VII级）。
Melodic Minor Scale (ascending): W – H – W – W – W – W – H (raised 6th and 7th).
旋律小调音阶（上行）：全 – 半 – 全 – 全 – 全 – 全 – 半（升高第VI、VII级）。
To find the relative minor, count down three semitones from the major tonic.
寻找关系小调的方法：从大调主音向下数三个半音。
2. Interval Theorems | 音程定理
Inversion Formula: New interval number = 9 – original interval number. A 3rd inverts to a 6th, a 4th to a 5th.
转位公式：新音程度数 = 9 – 原音程度数。三度转成六度，四度转成五度。
Quality Change on Inversion: Major ↔ minor, Augmented ↔ diminished, Perfect remains Perfect.
音程性质转位规律：大音程 ↔ 小音程，增音程 ↔ 减音程，纯音程保持纯音程。
Counting Semitones:
半音数量计算：
- Minor 2nd = 1 semitone | 小二度 = 1 个半音
- Major 2nd = 2 semitones | 大二度 = 2 个半音
- Minor 3rd = 3 semitones | 小三度 = 3 个半音
- Major 3rd = 4 semitones | 大三度 = 4 个半音
- Perfect 4th = 5 semitones | 纯四度 = 5 个半音
- Augmented 4th / Diminished 5th = 6 semitones | 增四度/减五度 = 6 个半音
- Perfect 5th = 7 semitones | 纯五度 = 7 个半音
- Minor 6th = 8 semitones | 小六度 = 8 个半音
- Major 6th = 9 semitones | 大六度 = 9 个半音
- Minor 7th = 10 semitones | 小七度 = 10 个半音
- Major 7th = 11 semitones | 大七度 = 11 个半音
- Octave = 12 semitones | 纯八度 = 12 个半音
3. Triad Construction Formulas | 三和弦构成公式
Major Triad: Root + Major 3rd (4 semitones) + Perfect 5th (7 semitones).
大三和弦：根音 + 大三度（4个半音）+ 纯五度（7个半音）。
Minor Triad: Root + Minor 3rd (3 semitones) + Perfect 5th (7 semitones).
小三和弦：根音 + 小三度（3个半音）+ 纯五度（7个半音）。
Diminished Triad: Root + Minor 3rd (3 semitones) + Diminished 5th (6 semitones).
减三和弦：根音 + 小三度（3个半音）+ 减五度（6个半音）。
Augmented Triad: Root + Major 3rd (4 semitones) + Augmented 5th (8 semitones).
增三和弦：根音 + 大三度（4个半音）+ 增五度（8个半音）。
Triad positions: Root position (1 – 3 – 5), 1st inversion (3 – 5 – 1), 2nd inversion (5 – 1 – 3).
三和弦位置：原位（1-3-5），第一转位（3-5-1），第二转位（5-1-3）。
4. Key Signature Theorems | 调号定理
Sharp Keys Rule: The last sharp in the key signature is the leading note. The major key is one semitone above it. For G major, F♯ is the last sharp → tonic G.
升号调规则：调号中最后一个升号是导音，大调主音在其上方半音。例如 G 大调，最后一个升号为 F♯ → 主音 G。
Flat Keys Rule: The second-to-last flat is the major key tonic. For F major (B♭ only), the tonic is F.
降号调规则：倒数第二个降号即为大调主音。对于只有一个降号的 F 大调（B♭），主音是 F。
Order of Sharps: F♯ C♯ G♯ D♯ A♯ E♯ B♯. (Father Charles Goes Down And Ends Battle)
升号顺序：F♯ C♯ G♯ D♯ A♯ E♯ B♯。
Order of Flats: B♭ E♭ A♭ D♭ G♭ C♭ F♭. (Battle Ends And Down Goes Charles’ Father)
降号顺序：B♭ E♭ A♭ D♭ G♭ C♭ F♭。
Relative keys share the same key signature. To find the relative minor, take the major tonic and go down a minor 3rd (3 semitones).
关系大小调共享同一调号。寻找关系小调：从大调主音向下小三度（3个半音）。
5. Rhythm & Time Signature Formulas | 节奏与拍号公式
Simple Time: Upper number = beats per bar, lower number = beat unit (4 = quarter note, 2 = half note, 8 = eighth note).
单拍子：上方数字 = 每小节拍数，下方数字 = 单位拍时值（4 = 四分音符，2 = 二分音符，8 = 八分音符）。
Compound Time: Beats are grouped in threes. Upper number = 6, 9, 12; each beat is a dotted note. In 6/8, there are 2 dotted-quarter beats.
复拍子：单位拍由附点音符构成，每小节拍数通常为 6、9、12。6/8 拍表示每小节两个附点四分音符。
Dotted Note Formula: Dotted note = note value + half of its value. A dotted crotchet (quarter) = 1 + ½ = 1½ beats in simple time.
附点音符公式：附点音符时值 = 原音符时值 + 其一半时值。附点四分音符 = 1 + ½ = 1½ 拍（单拍子）。
Duple, Triple, Quadruple: 2/4, 2/2 = duple; 3/4, 3/8 = triple; 4/4, 4/2 = quadruple.
拍子类型：二拍子（2/4、2/2），三拍子（3/4、3/8），四拍子（4/4、4/2）。
6. Note Grouping Rules | 音值组合规则
Beaming Rule: Beam notes together to show whole beats. In 4/4, four eighth notes can be beamed per half-note beat, or two per quarter beat.
符杠组合原则：按单位拍将音符用符杠连接，以清晰显示拍子。4/4 拍中，八分音符可按四分音符单位拍两两组合，或按二分音符单位拍四四组合。
Rest Rule: In simple time, a rest that covers a whole beat should not be broken into smaller rests unless necessary.
休止符规则：单拍子中，覆盖整个单位拍的休止符不应拆分为更小的休止符，除非不可避免。
Tie vs Slur: A tie joins two notes of the SAME pitch and adds their durations. A slur joins notes of DIFFERENT pitches for legato playing.
连音线与圆滑线：连音线连接音高相同的两个音，将时值相加；圆滑线连接不同音高的音，指示圆润连奏。
Syncopation Formula: A syncopated rhythm places accent on a weak beat or between beats, often using ties or rests on strong beats.
切分音公式：切分节奏将重音置于弱拍或拍子之间，常通过强拍上的连音线或休止符实现。
7. Transposition Formulas | 移调公式
Chromatic Transposition: Move every note by the same number of semitones up or down. (e.g., C up a major 3rd becomes E).
半音移调：将每个音向上或向下移动相同的半音数（如 C 向上大三度移至 E）。
Diatonic Transposition: Move every note the same number of scale steps within the new key signature. For example, transposing from C major to D major means every note moves up a major 2nd, adjusting accidentals to fit the key.
自然音移调：在新调号的音阶内，将每个音向上或向下移动相同的音级数。例如从 C 大调移到 D 大调，每个音上行大二度，并调整临时记号以符合新调。
Interval Key Change: If transposing up a perfect 5th, add one sharp or remove one flat from the key signature (and vice versa).
音程移调与调号变化：向上移纯五度，调号增加一个升号或减少一个降号；反之亦然。
8. Musical Structure Formulas | 音乐结构公式
Binary Form: A–B, each section often repeated: ||: A :||: B :||. Common in Baroque dances.
二段体：A–B 结构，各段通常反复：||: A :||: B :||。常见于巴洛克舞曲。
Ternary Form: A–B–A, where B is contrasting in key, mood or melody. The return of A may be exact or varied.
三段体：A–B–A 结构，B 段在调性、情绪或旋律上形成对比，A 段原样或变化再现。
Rondo Form: A–B–A–C–A–D–A…, where the A theme always returns between contrasting episodes.
回旋曲式：A–B–A–C–A–D–A…，主题 A 不断在对比插部之间重现。
Theme and Variations: A (theme) – A’ – A” – A”’ …, where the theme is altered melodically, rhythmically, harmonically or in texture.
主题与变奏：A – A’ – A” – A”’…，主题在旋律、节奏、和声或织体上进行变化。
9. Ornament Formulas | 装饰音公式
Trill (tr): Rapid alternation between the written note and the note a step above. Performed for the entire written duration.
颤音 (tr)：本音与上方二度音快速交替，持续整个音符时值。
Turn (𝄐): Four-note figure: note above → written note → note below → written note. If placed between two notes, the rhythm is adapted to the surrounding notes.
回音 (𝄐)：四音组：上方邻音 → 本音 → 下方邻音 → 本音。若记于两音之间，节奏需根据前后音符调整。
Mordent (𝄞 upper mordent, 𝄟 lower mordent): Upper mordent: written note – upper note – written note, played rapidly. Lower mordent: written note – lower note – written note.
波音（上波音 𝄞，下波音 𝄟）：上波音为本音-上方邻音-本音快速演奏；下波音为本音-下方邻音-本音快速演奏。
Appoggiatura: A small grace note that takes half the time of the main note (or two-thirds if main note is dotted) and is played ON the beat.
倚音：小号装饰音，通常占主要音符时值的一半（若主音为附点音符则占三分之二），并在拍点上奏出。
Acciaccatura: A very short grace note with a stroke through the stem, played just before the beat, ‘crushed’ into the main note.
短倚音：带斜线的小装饰音，在拍点前极短奏出，迅速滑入主音。
10. Dynamics & Tempo Abbreviations | 力度与速度术语速查
Dynamics (soft to loud): pp (pianissimo, very soft), p (piano, soft), mp (mezzo-piano, moderately soft), mf (mezzo-forte, moderately loud), f (forte, loud), ff (fortissimo, very loud). Crescendo (cresc.) = gradually louder, diminuendo (dim.) = gradually softer.
力度（弱到强）：pp 很弱，p 弱，mp 中弱，mf 中强，f 强，ff 很强。cresc. 渐强，dim. 渐弱。
Tempo Marks (slow to fast): Largo (very slow and broad), Adagio (slow and stately), Andante (walking pace), Moderato (moderate speed), Allegro (fast, cheerful), Presto (very fast). Accelerando (accel.) = gradually faster, Ritardando (rit.) = gradually slower.
速度术语（慢到快）：Largo 广板（很慢），Adagio 柔板（缓慢庄严），Andante 行板（步行速度），Moderato 中板，Allegro 快板（快速欢快），Presto 急板（非常快）。accel. 渐快，rit. 渐慢。
Italian Terms for Expression: legato (smoothly tied), staccato (short and detached), marcato (marked, emphasised), tenuto (held, full length), cantabile (in a singing style), dolce (sweetly).
表情术语：legato 连奏，staccato 断奏，marcato 强调、突出，tenuto 保持足拍，cantabile 如歌地，dolce 甜美地。
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