📚 Year 8 CIE Music: Quick-Reference Formula & Theorem Handbook | 音乐公式定理速查手册
Welcome to your Year 8 CIE Music quick-reference handbook. While music rarely uses mathematical formulas, it is full of patterns, rules and 'recipes' that help you read, write and analyse music accurately. Think of these as your musical formulas: from note durations and key signatures to interval calculations and chord building. This guide brings together all the essential facts in one place, with every explanation paired in English and Chinese so you can revise confidently.
欢迎使用这本 Year 8 CIE 音乐速查手册。音乐虽不像数学那样充满公式，但它有着丰富的规律、法则和"配方"，帮助你准确地读谱、记谱和分析音乐。把这些知识当作你的音乐公式吧：从音符时值、调号到音程计算与和弦构建，应有尽有。本手册将核心知识点汇集一处，并以英中对照逐一说明，助你自信备考。
1. Note Values & Time Signatures | 音符时值与拍号
In CIE Year 8 Music, you must instantly recall note shapes and their durations. A semibreve (whole note) ♩ is equal to 2 minims, 4 crotchets, 8 quavers or 16 semiquavers. In simple time, such as 4/4, the semibreve gets 4 beats, the minim gets 2, the crotchet gets 1, the quaver gets ½, and the semiquaver gets ¼.
在 CIE Year 8 音乐中，你必须快速辨认音符形状及其时值。全音符 ♩ 等于 2 个二分音符、4 个四分音符、8 个八分音符或 16 个十六分音符。在单拍子（如 4/4 拍）中，全音符 4 拍，二分音符 2 拍，四分音符 1 拍，八分音符 ½ 拍，十六分音符 ¼ 拍。
ats per bar, bottom number = beat unit. For example, 3/4 means 3 crotchet beats per bar. A 'C' (semibreve), 2 (minim), 4 (crotchet), 8 (quaver), 16 (semiquaver) or 32 (demisemiquaver).
拍号公式：上方数字 = 每小节拍数，下方数字 = 拍子单位。例如 3/4 拍表示以四分音符为一拍，每小节 3 拍。符号 "C"（普通拍）即 4/4 拍的另一种写法。下方数字只能是 1（全音符）、2（二分音符）、4（四分音符）、8（八分音符）、16（十六分音符）或 32（三十二分音符）。
2. Dotted Notes & Ties | 附点音符与连线
rotchet (crotchet + quaver) lasts 1½ beats in 4/4. A second dot adds half of the first dot's value, making a double-dotted minim equal to 2 + 1 + ½ = 3½ beats.
音符后的附点将其时值延长原有时值的一半。公式：附点音符时值 = 基本音符时值 × 1.5。因此，在 4/4 拍中，附点四分音符（四分音符 + 八分音符）为 1½ 拍。第二个附点再增加第一个附点时值的一半，因此双附点二分音符 = 2 + 1 + ½ = 3½ 拍。
r line. The total tied duration = sum of the tied notes. Triplets are written with a small '3' into the time of two. The triplet formula: three triplet quavers equal one crotchet beat.
连线（延音线）连接两个相同音高的音符，将它们的时值相加。与附点不同，连线可以跨小节延长音符。总时值 = 被连接音符时值之和。三连音以小型数字 "3" 标记，将三个音符的时值压缩为两个音符的时间。三连音公式：三个三连音八分音符 = 一个四分音符的拍子。
3. Simple & Compound Time | 单拍子与复拍子
s 6, 9 or 12, where the beat divides into three. The rule of thumb: compound beat = dotted note. So 6/8 means 2 dotted-crotchet beats per bar (each beat worth 3 quavers).
单拍子的拍号上方数字为 2、3 或 4，每拍自然地一分为二。复拍子的拍号上方数字为 6、9 或 12，每拍一分为三。其核心规律：复拍子每拍 = 一个附点音符。因此 6/8 拍表示每小节有 2 个附点四分音符拍子（每拍含 3 个八分音符）。
. The bottom number tells you which note value equals one-third of the beat, so 8 means the quaver is the division unit.
计算复拍子拍数的公式：将上方数字除以 3，即得每小节的拍数。6/8 → 6 ÷ 3 = 2 拍；9/8 → 3 拍；12/8 → 4 拍。下方数字指示了每一等分的音符时值，因此 8 表示以八分音符作为拍子的三等分单元。
4. Key Signatures & The Circle of Fifths | 调号与五度圈
Key signatures collect the sharps or flats belonging to a key. The order of sharps is always F♯ C♯ G♯ D♯ A♯ E♯ B♯ (Father Charles Goes Down And Ends Battle). To find the major key from a sharp key signature, look at the last sharp and go up one semitone. For example, if the last sharp is D♯, the key is E major.
调号汇集了某个调所有需要升高或降低的音。升号调的排列顺序固定为：F♯ C♯ G♯ D♯ A♯ E♯ B♯。从升号调号推断大调名称的公式：找到最后一个升号，再往上一个半音。例如，最后一个升号为 D♯，则调名是 E 大调。
The order of flats is the reverse: B♭ E♭ A♭ D♭ G♭ C♭ F♭ (Battle Ends And Down Goes Charles Father). For flat keys, the penultimate flat in the signature gives the major key name. For example, in a key signature with B♭ E♭ A♭, the second-to-last flat is E♭, so the key is E♭ major.
降号调的顺序正好相反：B♭ E♭ A♭ D♭ G♭ C♭ F♭。从降号调号推大调名称的公式：看倒数第二个降号，其所在音名即调名。如调号含 B♭ E♭ A♭，倒数第二个降号为 E♭，故为 E♭ 大调。
The Circle of Fifths organises keys by perfect-fifth steps. Moving clockwise adds one sharp (or removes one flat), and moving anticlockwise adds one flat. C major (0 sharps/flats) rises to G major (1 sharp), D major (2 sharps), etc. Relative minors sit a minor third below their major; for instance, A minor is the relative of C major.
五度圈按纯五度关系排列各调。顺时针方向每步增加一个升号（或抵消一个降号），逆时针方向每步增加一个降号。C 大调（无升降号）→ G 大调（1 升号）→ D 大调（2 升号），依此类推。关系小调位于其大调下方小三度；例如 a 小调是 C 大调的关系小调。
5. Major Scale Formula | 大调音阶公式
Every major scale follows the same interval pattern using tones (T) and semitones (S): T T S T T T S. In terms of half steps: 2 – 2 – 1 – 2 – 2 – 2 – 1. Starting from any note, apply this sequence to find the correct notes and accidentals.
所有大调音阶遵循相同的全音 / 半音模式：T T S T T T S。用半音数表示就是：2 – 2 – 1 – 2 – 2 – 2 – 1。从任意起始音出发，按此顺序即可得到正确的音名及临时升降号。
rpen the 7th degree to keep the formula: G (T) → A (T) → B (S) → C (T) → D (T) → E (T) → F♯ → G. Accidentals are never mixed in a standard major scale; use only sharps or only flats.
以 C 大调为例：C (T) → D (T) → E (S) → F (T) → G (T) → A (T) → B (S) → C。构建 G 大调时，需要升高第 VII 级音来保持模式：G (T) → A (T) → B (S) → C (T) → D (T) → E (T) → F♯ (S) → G。标准大调音阶中升降号从不混用，要么全用升号，要么全用降号。
6. Minor Scales (Natural, Harmonic, Melodic) | 小调音阶（自然、和声、旋律）
T) → E (S) → F (T) → G (T) → A. This scale uses the same notes as its relative major but starts on the 6th degree.
Year 8 学生需要掌握三种小调音阶形式。自然小调的模式为：T S T T S T T。以 a 小调为例：A (T) → B (S) → C (T) → D (T) → E (S) → F (T) → G (T) → A。自然小调使用的音符与其关系大调完全一致，只是从第 VI 级开始。
The harmonic minor raises the 7th degree by a semitone, creating an exotic-sounding interval of an augmented 2nd between the 6th and 7th degrees. Pattern: T S T T S 1½ S. A harmonic minor: A B C D E F G♯ A. The melodic minor raises both the 6th and 7th degrees when ascending, but reverts to the natural minor descending. Ascending: T S T T T T S (A B C D E F♯ G♯ A); descending: natural minor.
和声小调将第 VII 级音升高半音，在 VI 级与 VII 级之间产生一个增二度的异域音响。其模式：T S T T S 1½ S。a 和声小调上行：A B C D E F G♯ A。旋律小调上行时升高 VI、VII 两级，下行时还原为自然小调。上行模式：T S T T T T S（A B C D E F♯ G♯ A），下行即自然小调。
7. Intervals: Quality & Number | 音程：性质与度数
An interval describes the distance between two notes. First find the number by counting letter names inclusive (C to E is a 3rd, because C-D-E counts three letters). Then determine the quality: perfect, major, minor, augmented or diminished. Use the semitone count as a formula in the table below.
音程描述两个音之间的距离。先确定度数：按音名依次包含的数量计算（如 C 至 E 为三度，因为 C-D-E 共三个音）。再判断性质：纯音程、大/小音程、增/减音程。可借助下表的半音数公式加以确认。
| Interval 音程 | Semitones 半音数 | Example 示例 |
|---|---|---|
| Minor 2nd 小二度 | 1 | C – D♭ |
| Major 2nd 大二度 | 2 | C – D |
| Minor 3rd 小三度 | 3 | C – E♭ |
| Major 3rd 大三度 | 4 | C – E |
| Perfect 4th 纯四度 | 5 | C – F |
| Augmented 4th/Diminished 5th 增四度/减五度 | 6 | C – F♯ |
| Perfect 5th 纯五度 | 7 | C – G |
| Minor 6th 小六度 | 8 | C – A♭ |
| Major 6th 大六度 | 9 | C – A |
| Minor 7th 小七度 | 10 | C – B♭ |
| Major 7th 大七度 | 11 | C – B |
| Perfect 8ve 纯八度 | 12 | C – C |
If an interval is one semitone larger than perfect or major, it becomes augmented; one semitone smaller becomes diminished. For example, C to F♯ is an augmented 4th, while C to G♭ is a diminished 5th, even though they sound the same on a piano.
若某音程比纯音程或大音程大一个半音，即为增音程；小一个半音则为减音程。例如 C 至 F♯ 是增四度，C 至 G♭ 是减五度，尽管在钢琴上两者声音相同。
8. Triads & Chord Construction | 三和弦及和弦构建
A triad is a three-note chord built by stacking two 3rds on top of a root. The formula for a major triad: root + major 3rd (4 semitones) + perfect 5th (7 semitones from root). C major: C (root) – E (4 semitones up) – G (7 semitones up). The outer frame is a perfect 5th.
三和弦是由根音上方叠加两个三度构成的三音和弦。大三和弦公式：根音 + 大三度（距根音 4 个半音） + 纯五度（距根音 7 个半音）。C 大三和弦：C（根音）– E（升高 4 半音）– G（升高 7 半音）。外框架永远是纯五度。
A minor triad formula: root + minor 3rd (3 semitones) + perfect 5th (7 semitones). A minor: A – C (3 semitones) – E (7 semitones). A diminished triad: root + minor 3rd + diminished 5th (6 semitones). B diminished: B – D – F (6 semitones from B). An augmented triad: root + major 3rd + augmented 5th (8 semitones).
小三和弦公式：根音 + 小三度（3 半音） + 纯五度（7 半音）。a 小三和弦：A – C（3 半音）– E（7 半音）。减三和弦：根音 + 小三度 + 减五度（6 半音）。B 减三和弦：B – D – F（距 B 6 半音）。增三和弦：根音 + 大三度 + 增五度（8 半音）。
When writing chords in root position, the three notes are stacked as evenly as possible. Inversions change the lowest note: first inversion uses the 3rd as the bass, second inversion uses the 5th. The chord symbol system uses Roman numerals based on scale degrees (I, ii, iii, IV, V, vi, vii°).
原位和弦尽可能均匀叠置三音。转位会改变最低音：第一转位以三音为低音，第二转位以五音为低音。和弦标记系统采用基于音级的功能标记，如 I、ii、iii、IV、V、vi、vii°。
9. Primary Chords (I, IV, V) & Cadences | 正三和弦（I, IV, V）与终止式
ontain every note of the scale and form the basis of functional harmony. In C major: I = C–E–G, IV = F–A–C, V = G–B–D.
在任何大调中，正三和弦分别是 I（主和弦）、IV（下属和弦）和 V（属和弦）。这三个和弦涵盖了音阶所有的音，构成了功能和声的基础。C 大调中：I = C–E–G，IV = F–A–C，V = G–B–D。
A cadence is a two-chord formula that rounds off a phrase. The perfect cadence V–I sounds final and strong; the plagal cadence IV–I is the 'Amen' cadence; the imperfect cadence ends on V (e.g. I–V, ii–V, IV–V), leaving the music unfinished; the interrupted cadence goes V–vi, surprising the listener.
终止式是收束乐句的两个和弦组合。正格终止 V–I 听感完满有力；变格终止 IV–I 即 "阿门终止"；半终止以 V 结束（如 I–V、ii–V、IV–V），让音乐有意犹未尽之感；阻碍终止 V–vi 出其不意地连接到 VI 级和弦。
To identify a cadence, look at the last two chords of a phrase and name the Roman numerals. The formula for a perfect cadence is always V (or V⁷) moving to I. In C major, G–dominant to C–tonic. Recognising these patterns by sound and by score is a key skill.
识别终止式时，关注乐句最后两个和弦，并用罗马数字标出。正格终止必定为 V（或 V⁷）进行到 I。C 大调中就是 G 属和弦接 C 主和弦。通过听觉和乐谱辨认这些进行模式，是重要的能力。
10. Transposition | 移调
al note ± interval. When transposing up a major 2nd, a melody starting C D E becomes D E F♯ s and key signatures must be adjusted to preserve the same pattern of tones and semitones.
移调是指将旋律的所有音符以相同的音程度数整体上移或下移。公式：新音高 = 原音高 ± 指定音程。若整体上移大二度，旋律 C D E 便成为 D E F♯。过程中所有临时升降号与调号都须调整，以保持原有的全音半音关系。
For transposing instruments, such as the clarinet in B♭, Year 8 students need to know that the written note and sounding pitch are different. A B♭ clarinet sounds a major 2nd lower than written, so a written C sounds as B♭. The transposition formula for B♭ instruments: sounding pitch = written pitch – major 2nd.
针对移调乐器（如降 B 调单簧管），Year 8 学生要明白记谱音高与实际发音不同。降 B 调单簧管发音比记谱低大二度，因此记谱 C 实际发音为 B♭。降 B 调乐器的移调公式：实际音高 = 记谱音高 – 大二度。
11. Dynamics, Articulation & Tempo Terms | 力度、演奏法与速度术语
Dynamics indicate volume. The standard Italian terms form a spectrum: 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 include crescendo (cresc., getting louder) and diminuendo (dim., getting softer).
力度标记表示音量。标准意大利术语形成一个序列：pp（极弱）、p（弱）、mp（中弱）、mf（中强）、f（强）、ff（极强）。渐变力度包括 cresc.（渐强）和 dim./decresc.（渐弱）。
Articulation shapes how notes are played. Staccato (dot) means detached and short; legato (slur) means smoothly connected; accent ( > ) means stressed; tenuto (line) means held for full value. Tempo marks define speed: Largo (broadly, very slow), Adagio (slow), Andante (walking pace), Moderato (moderate), Allegro (fast, cheerful), Presto (very fast).
演奏法决定音符的触键方式。断奏（staccato, 圆点）短促分离；连奏（legato, 连线）圆滑连贯；重音（accent, > 符号）加力强调；保持音（tenuto, 横线）奏满时值。速度标记定义快慢：Largo（广板，很慢）、Adagio（柔板）、Andante（行板）、Moderato（中板）、Allegro（快板）、Presto（急板）。
Metronome marks give precise beats per minute (bpm). ♩ 120 crotchet beats per minute. A dotted crotchet in compound time often carries the beat: ♩. = 60. Learn to pair Italian terms with their bpm ranges: Andante ≈ 76–108, Moderato ≈ 108–120, Allegro ≈ 120–168.
节拍器标记给出精确的每分钟拍数 (bpm)。♩ = 120 表示每分钟 120 个四分音符拍。复拍子中常用附点四分音符打拍，如 ♩. = 60。要熟记意大利术语与大致 bpm 范围：行板约 76–108，中板约 108–120，快板约 120–168。
12. Ornaments & Symbols | 装饰音与符号
Ornaments are decorative notes that add expression. The trill (tr) alternates rapidly between the written note and the upper diatonic neighbour. The turn ( 𝄐 ) goes: upper note – main note – lower note – main note. The mordent (𝄝) is a rapid main – upper – main; the inverted mordent (𝄞) is main – lower – main. An appoggiatura is a leaning note that takes half the value of the following note.
装饰音是增添表现力的附加音符。颤音（tr）在主音与其上方的自然邻音之间
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