📚 Year 10 CIE Music: Formulae & Rules Handbook | CIE 音乐公式定理速查手册
This quick-reference handbook distils the essential 'formulae' and rules that underpin the CIE IGCSE Music syllabus. From scale construction and interval arithmetic to chord spelling and cadence patterns, every core concept is presented as a clear, memorisable equation or principle. Use these pages to drill theory fundamentals, analyse set works, and approach aural and written examinations with confidence.
这本速查手册提炼了 CIE IGCSE 音乐课程的核心“公式”与法则。从音阶构成、音程计算到和弦拼写与终止式规律，每一个基础概念都以清晰的公式或原理呈现。通过这些页面强化乐理基本功，你可以更自信地分析规定作品，应对听力与笔试。
1. The Major Scale Formula | 大调音阶公式
Every major scale follows the same pattern of whole tones (T) and semitones (S): T – T – S – T – T – T – S. Starting on any tonic, apply this sequence to obtain the correct set of seven notes. For example, C major: C (T) D (T) E (S) F (T) G (T) A (T) B (S) C. The formula also gives the key signature indirectly: the number of sharps or flats needed to preserve the pattern.
所有大调音阶都遵循相同的全音（T）与半音（S）模式：T – T – S – T – T – T – S。从任一主音出发，应用这一序列即可得到正确的七个音。例如 C 大调：C (T) D (T) E (S) F (T) G (T) A (T) B (S) C。该公式亦间接决定调号：需要多少个升号或降号来维持这一排列。
Major scale formula: T – T – S – T – T – T – S
2. The Minor Scale Formulae | 小调音阶公式
Three variants of the minor scale are examined. The natural minor follows a rotation of its relative major: T – S – T – T – S – T – T. The harmonic minor raises the 7th degree by a semitone, creating an augmented 2nd between degrees 6 and 7. Its formula is T – S – T – T – S – T+S – S (where T+S is 3 semitones). The melodic minor ascending raises both the 6th and 7th degrees, giving T – S – T – T – T – T – S; descending it reverts to the natural minor. Always specify which form a question refers to.
考试涉及三种小调音阶。自然小音阶是关系大调的轮转：T – S – T – T – S – T – T。和声小音阶将第 7 级音升高半音，在 6、7 级间产生增二度，公式为 T – S – T – T – S – T+S – S（T+S 为 3 个半音）。旋律小音阶上行时同时升高第 6、7 级，公式为 T – S – T – T – T – T – S；下行则还原为自然小音阶。务必留意题目要求的是哪一种形式。
| Natural minor | T – S – T – T – S – T – T |
| Harmonic minor | T – S – T – T – S – 3 semitones – S |
| Melodic minor (ascending) | T – S – T – T – T – T – S |
3. Interval Calculation | 音程计算
An interval is defined by its number (1st, 2nd, 3rd, etc.) and quality (perfect, major, minor, augmented, diminished). To label any interval, count the number of letter names from the lower note to the upper note, inclusive. Then count the semitones between them and consult the standard table. For unison to octave, the semitone values are: perfect unison (0), minor 2nd (1), major 2nd (2), minor 3rd (3), major 3rd (4), perfect 4th (5), augmented 4th/diminished 5th (6), perfect 5th (7), minor 6th (8), major 6th (9), minor 7th (10), major 7th (11), perfect octave (12). When a perfect interval is widened by a semitone it becomes augmented; when narrowed it becomes diminished. A major interval made one semitone larger becomes augmented; one semitone smaller becomes minor; two semitones smaller becomes diminished.
音程由其度数（1 度、2 度……）和性质（纯、大、小、增、减）界定。标记任何音程时，先将较低音到较高音的字母名（包含两端）数出度数，再数半音，并查表对照。从纯一度到纯八度的标准半音值：纯一度（0），小二度（1），大二度（2），小三度（3），大三度（4），纯四度（5），增四度/减五度（6），纯五度（7），小六度（8），大六度（9），小七度（10），大七度（11），纯八度（12）。纯音程扩大半音为增，缩小半音为减；大音程扩大半音为增，缩小半音为小，再缩小半音为减。
Interval number = (count of staff degrees from lower to upper) + 1
4. Triad Construction Rules | 三和弦构建规则
ve the root, then 3 semitones above the third, giving a total of 7 semitones from root to fifth). Minor triad: root + minor 3rd + perfect 5th (3 semitones + 4 semitones). Diminished triad: root + minor 3rd + diminished 5th (3 + 3 = 6 semitones). Augmented triad: root + major 3rd + augmented 5th (4 + 4 = 8 semitones). When writing triads on a stave, ensure the note names reflect the correct interval numbers, not enharmonic equivalents. In a key, triads built on degrees I, IV and V are major; on ii, iii and vi are minor; on vii° is diminished.
三和弦是由根音向上叠置两个三度构成的三音和弦。大三和弦公式：根音 + 大三度 + 纯五度（根音上方 4 半音得三音，再上 3 半音得五音，根至五共 7 半音）。小三和弦：根 + 小三度 + 纯五度（3 + 4 = 7 半音）。减三和弦：根 + 小三度 + 减五度（3 + 3 = 6 半音）。增三和弦：根 + 大三度 + 增五度（4 + 4 = 8 半音）。记谱时，音名必须体现正确的音程数，不可使用等音异名。在调内，I、IV、V 级上的三和弦为大三和弦；ii、iii、vi 级为小三和弦；vii° 为减三和弦。
Major triad = root + M3 (4 st) + P5 (7 st from root)
5. Chord Inversion Patterns | 和弦转位模式
Any triad can appear in root position, first inversion or second inversion. In root position the root is the lowest note. First inversion places the third in the bass; the notes above are the fifth and the root, giving an interval of a sixth above the bass. Figured bass notation for first inversion is ⁶⁄₃ (often abbreviated as ⁶). Second inversion places the fifth in the bass, with root and third stacked above, producing a sixth and a fourth above the bass, notated as ⁶⁄₄. For seventh chords, root position is ⁷, first inversion ⁶⁄₅, second inversion ⁴⁄₃, third inversion ⁴⁄₂ (or 2). Memorising these bass-note relationships helps you quickly identify chord inversions in score analysis.
三和弦可以以原位、第一转位或第二转位出现。原位时根音在低音部。第一转位将三音放在低音，上方依次为五音和根音，低音与上方两音分别形成六度和三度，数字低音记作 ⁶⁄₃（常简写为 ⁶）。第二转位将五音放在低音，上方为根音与三音，低音与上方音形成六度与四度，记为 ⁶⁄₄。七和弦的原位是 ⁷，第一转位 ⁶⁄₅，第二转位 ⁴⁄₃，第三转位 ⁴⁄₂（或 2）。牢记这些低音关系可帮助你在总谱分析时快速辨认和弦转位。
| Root position | Root in bass (5/3) |
| First inversion | 3rd in bass (6/3) |
| Second inversion | 5th in bass (6/4) |
6. Circle of Fifths Key Signatures | 五度圈调号规律
fifths adds one sharp each step: C (0 sharps) → G (1) → D (2) → A (3) → E (4) → B (5) → F♯ (6) → C♯ (7). The order of sharps is F♯, C♯, G♯, D♯, A♯, E♯, B♯. Moving anticlockwise by perfect fourths adds one flat each step: C (0) → F (1) → B♭ (2) → E♭ (3) → A♭ (4) → D♭ (5) → G♭ (6) → C♭ (7). Order of flats: B♭, E♭, A♭, D♭, G♭, C♭, F♭ ature but the minor tonic is a minor third below the major tonic (e.g. C major / A minor).
五度圈为调号提供直观公式。顺时针以纯五度推进，每步增加一个升号：C（0 升）→ G（1）→ D（2）→ A（3）→ E（4）→ B（5）→ F♯（6）→ C♯（7）。升号出现的顺序：F♯、C♯、G♯、D♯、A♯、E♯、B♯。逆时针以纯四度推进，每步增加一个降号：C（0 降）→ F（1）→ B♭（2）→ E♭（3）→ A♭（4）→ D♭（5）→ G♭（6）→ C♭（7）。降号顺序：B♭、E♭、A♭、D♭、G♭、C♭、F♭。关系大小调共享同一调号，但小调主音位于大调主音的下方小三度（如 C 大调 / a 小调）。
To find relative minor: go down 3 semitones from the major tonic.
7. Rhythm Value Equations | 节奏时值等式
hree notes into the time of two: a triplet quaver group (3:2) fits into one crotchet beat. Dotted notes add half the original value: a dotted minim is 3 crotchets; a dotted crotchet is 1½ crotchets (a crotchet + a quaver). Double dots add a further quarter: double-dotted minim = 3 + ¾ = 3¾ crotchets. The written equation is: duration = note value × (1 + 0.5 [if dotted] + 0.25 [if double-dotted]). All tie and beam groupings must respect the underlying beat structure of the time signature.
单拍子中的音符时值采用二分系统：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。三连音将三个音挤压到两个音的时长中，例如八分三连音（3:2）占一个四分音符时值。附点增加原音符一半的时值：附点二分音符为 3 个四分音符；附点四分音符为 1½ 个四分音符（一四分 + 一八分）。复附点再增加四分之一：复附点二分音符 = 3 + ¾ = 3¾ 四分音符。公式为：总时值 = 音符时值 × （1 + 0.5［有附点］+ 0.25［有复附点］）。所有延音线和符尾组合都须体现拍号内在的拍子结构。
Dotted note total = note value × 1.5; double-dotted = note value × 1.75
8. Time Signature Classification | 拍号分类法则
ivides into two equal quavers. In compound time the upper number is 6, 9 or 12, and each beat is dotted, subdividing into three quavers. The lower number indicates the note value that represents one subdivision: 4 = crotchet, 8 = quaver. Thus 6/8 has 2 dotted-crotchet beats per bar (each of 3 quavers), 9/8 has 3 beats, and 12/8 has 4 beats. Irregular time signatures (e.g. 5/4, 7/8) have unequal beat groupings; always annotate the beats (e.g. 5/4 as 3+2 or 2+3).
拍号按拍子的划分方式分为单拍子与复拍子。单拍子上方数字为 2、3 或 4，每拍均分为两个八分音符。复拍子上方数字为 6、9 或 12，每拍是附点音符，均分为三个八分音符。下方数字表示单位拍的时值：4 为四分音符，8 为八分音符。因此 6/8 每小节有 2 个附点四分音符拍（各含三个八分音符），9/8 有 3 拍，12/8 有 4 拍。混合拍子（如 5/4、7/8）的拍子分组不均等，须标明节拍分组（如 5/4 记作 3+2 或 2+3）。
| Simple duple | 2/4, 2/2 |
| Simple triple | 3/4, 3/2 |
| Simple quadruple | 4/4 |
| Compound duple | 6/8 |
| Compound triple | 9/8 |
| Compound quadruple | 12/8 |
9. Tempo Markings & Metronome Equivalents | 速度术语与节拍器换算
kings are standard in CIE exams. Common terms with approximate metronome ranges: Largo (40–60 bpm, very slow), Adagio (66–76), Andante (76–108, walking pace), Moderato (108–120), Allegro (120–168, fast and lively), Presto (168–200, very fast). A metronome mark such as ♩ 20 means 120 crotchet beats per minute. In compound time the beat value is usually dotted: ♩ minute. Changes of tempo are indicated by accelerando (gradually faster) and ritardando/ritenuto (gradually slower), with a tempo returning to the original speed. Always translate the mark to a precise beats-per-minute figure in your mind when preparing a performance or dictation question.
意大利速度术语是 CIE 考试的标准用语。常见术语及节拍器大致范围：Largo（40–60 bpm，极慢），Adagio（66–76），Andante（76–108，步行速度），Moderato（108–120），Allegro（120–168，快速活泼），Presto（168–200，极快）。节拍器标记如 ♩ = 120 表示每分钟 120 个四分音符拍。复拍子中拍的单位通常为附点音符：♩. = 60 指每分钟 60 个附点四分音符拍。速度变化用语：accelerando（渐快）和 ritardando/ritenuto（渐慢），a tempo 则回到原速。准备演奏题或听写题时，应在脑中将这些术语换算成精确的每分钟拍数。
BPM formula: Beats per minute = number of indicated note values counted in 60 seconds.
10. Ornament Symbols & Realisation | 装饰音符号与演奏公式
Ornaments must be identified by symbol and interpreted correctly. The trill (tr) alternates the principal note with the upper auxiliary note rapidly for the full value of the written note. The upper mordent (short squiggle) plays the principal note, the upper auxiliary, and the principal note again quickly, typically as three notes. The lower mordent (same symbol with a vertical line through it) uses the lower auxiliary. The turn (∼) encircles the note: a turn on a note goes upper auxiliary, principal, lower auxiliary, principal; a turn placed between notes occurs after the first note. Acciaccatura (small quaver with a stroke) is crushed as quickly as possible onto the main note. Appoggiatura (small note without a stroke) takes half the main note's value (or two-thirds if dotted). Follow the exact formula for each ornament based on the chronological context of the piece.
装饰音需辨识符号并正确演奏。颤音（tr）在主音与上方助音之间快速交替，持续整个写出的音符时值。上波音（短波浪线）顺序为主音–上方助音–主音，通常快速演奏为三个音。下波音（符号中间加一竖线）使用下方助音。回音（∼）环绕主音：记写在音符上方的回音奏上方助音–主音–下方助音–主音；记写在两音之间的回音则在第一音之后奏出。短倚音（带斜线的小八分音符）尽可能快地“粉碎”到主要音符上。长倚音（无斜线的小音符）占主要音符时值的一半（若主音符附点则占三分之二）。应根据作品所属时期，严格按照每种装饰音的公式处理。
Trill formula: principal ↔ upper auxiliary repeated for full note duration.
11. Cadence Formulas | 终止式公式
A cadence is a two-chord progression closing a phrase. Four types are tested. Perfect cadence (V – I) sounds complete and final; both chords are in root position, and the tonic note is in the soprano of the I chord. Plagal cadence (IV – I) is the 'Amen' cadence. Imperfect cadence ends on V, preceded by any chord (commonly I, ii or IV); it sounds unfinished. Interrupted cadence (V – vi) surprises the listener by substituting a minor chord for the expected tonic. In a minor key, the dominant chord must use the raised leading note (harmonic minor) to create a major V. Learn these formulas as harmonic templates: V–I, IV–I, ?–V, V–vi.
终止式是收束乐句的两个和弦进行，共考四种。正格终止（V – I）听感完满结束，两和弦均为原位，I 和弦的旋律音为调式主音。变格终止（IV – I）即“阿门”终止。半终止停在 V 级上，前方可以是 I、ii 或 IV 等和弦，听感未完成。阻碍终止（V – vi）以 vi 替代预期的 I，造成意外感。在小调中，属和弦必须使用升高了第七级音的导音（和声小调）构成大三和弦。将这些公式作为和声模板记牢：V–I、IV–I、?–V、V–vi。
Perfect = V–I; Plagal = IV–I; Imperfect = …–V; Interrupted = V–vi
12. Italian Terms Quick Reference | 意大利术语速查
ing style), dolce (sweetly), maestoso (majestically), leggiero (lightly), scherzando (playfully). Articulation: legato (smooth, indicated by a slur), staccato (detached, indicated by dots), tenuto (held, indicated by a dash). Dynamics arranged from softest to loudest: pp (pianissimo) → p (piano) → mp (mezzo-piano) → mf (mezzo-forte) → f (forte) → ff (fortissimo). Gradual changes: crescendo (cresc., growing louder), diminuendo (dim., growing softer), sforzando (sfz, sudden strong accent). Apply these terms as an interpretive formula to shape phrasing and articulation in both performance and score-based questions.
除速度与力度记号外，考纲还涵盖表情、奏法及技巧术语。常用表情术语：cantabile（如歌地）、dolce（甜美地）、maestoso（庄严地）、leggiero（轻巧地）、scherzando（诙谐地）。奏法：legato（连奏，以连线标示）、staccato（断奏，以圆点标示）、tenuto（保持，以短横线标示）。力度从最弱到最强排列：pp（很弱）→ p（弱）→ mp（中弱）→ mf（中强）→ f（强）→ ff（很强）。渐变用语：crescendo（cresc.，渐强）、diminuendo（dim.，渐弱）、sforzando（sfz，突强）。将这些术语作为演绎公式，在演奏题和读谱题中塑造乐句和奏法。
Dynamic gradient: pp < p < mp < mf < f < ff
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