📚 Year 12 CIE Music: Quick-Reference Formula & Theorem Handbook | Year 12 CIE 音乐：公式定理速查手册
This quick-reference handbook is designed for Year 12 CIE Music students to consolidate essential theoretical ‘formulae’ and ‘theorems’—the core building blocks of Western tonal music. From interval structures and chord construction to cadential progressions and formal patterning, the rules collected here will support both score analysis and practical musicianship throughout the AS course.
这本速查手册专为 Year 12 CIE 音乐学生编写，旨在梳理最重要的理论“公式”和“定理”——也就是西方调性音乐的基本构建模块。从音程结构、和弦构成到终止式进行和曲式规律，这里汇集的所有规则将助力你完成 AS 课程中的乐谱分析与音乐实践。
1. Interval Formulae | 音程公式
Intervals are calculated by counting the letter names (generic size) and then measuring in semitones. A perfect unison (P1) is 0 semitones; a perfect fourth (P4) is 5 semitones; a perfect fifth (P5) is 7 semitones; an octave (P8) is 12 semitones. Major seconds (M2) span 2 semitones, major thirds (M3) 4 semitones, major sixths (M6) 9 semitones, major sevenths (M7) 11 semitones. To obtain a minor interval, reduce a major interval by one semitone; to obtain an augmented interval, expand a major or perfect interval by one semitone; to obtain a diminished interval, shrink a minor or perfect interval by one semitone.
音程计算需先数音名（距离度数），再用半音测量。纯一度 (P1) 为 0 半音，纯四度 (P4) 5 半音，纯五度 (P5) 7 半音，纯八度 (P8) 12 半音。大二度 (M2) 为 2 半音，大三度 (M3) 4 半音，大六度 (M6) 9 半音，大七度 (M7) 11 半音。将大音程减少半音即得到小音程；将大或纯音程扩大半音得到增音程；将小或纯音程缩小半音得到减音程。
2. Chord Construction Formulae | 和弦构建公式
Triads are built by stacking thirds above a root. A major triad: root + M3 + P5 (e.g. C–E–G). A minor triad: root + m3 + P5 (e.g. C–E♭–G). A diminished triad: root + m3 + dim5 (e.g. C–E♭–G♭). An augmented triad: root + M3 + aug5 (e.g. C–E–G♯). Seventh chords add another third above the fifth: a dominant seventh (V⁷) is a major triad plus a minor seventh (root + M3 + P5 + m7); a major seventh (e.g. I⁷) is a major triad plus a major seventh (root + M3 + P5 + M7); a minor seventh is a minor triad plus a minor seventh (root + m3 + P5 + m7). In Roman numeral analysis, capital numerals denote major/minor quality, and figured bass symbols such as 6/4 or 5/3 indicate inversions.
三和弦是在根音上叠置三度构成。大三和弦：根音 + 大三度 + 纯五度（如 C–E–G）。小三和弦：根音 + 小三度 + 纯五度（如 C–E♭–G）。减三和弦：根音 + 小三度 + 减五度（如 C–E♭–G♭）。增三和弦：根音 + 大三度 + 增五度（如 C–E–G♯）。七和弦则在五度之上再加一个三度：属七和弦 (V⁷) 为大三和弦加小七度（根音 + 大三度 + 纯五度 + 小七度）；大七和弦（如 I⁷）为大三和弦加大七度（根音 + 大三度 + 纯五度 + 大七度）；小七和弦为小三和弦加小七度（根音 + 小三度 + 纯五度 + 小七度）。罗马数字分析中，大写数字表示大/小调性，数字低音记号如 6/4 或 5/3 表示转位。
3. Scale Patterns | 音阶公式
The major scale follows the tone–tone–semitone–tone–tone–tone–semitone pattern (TTSTTTS). The natural minor scale is TSTTSTT. The harmonic minor raises the seventh degree, creating an augmented second between the sixth and seventh degrees: TSTTS+2S (where +2 indicates an interval of three semitones). The melodic minor (ascending) raises the sixth and seventh degrees: TSTTTTS; descending it reverts to natural minor. The chromatic scale fills all semitone steps within an octave. Modes can be derived from the major scale by rotating the starting note: Dorian (tone–semitone–tone–tone–tone–semitone–tone), Phrygian, Lydian, Mixolydian etc., each with its own semitone placement formula.
大调音阶遵循全音–全音–半音–全音–全音–全音–半音 (TTSTTTS)。自然小调为 TSTTSTT。和声小调升高第七级，导致第六级与第七级之间出现增二度：TSTTS+2S（其中 +2 表示三个半音的距离）。旋律小调上行升高第六、七级：TSTTTTS，下行恢复自然小调。半音音阶填充八度内所有半音。调式可通过旋转大调音阶的起始音获得：多利亚调式（全–半–全–全–全–半–全），弗里几亚、利底亚、混合利底亚等，各有其半音位置公式。
4. Time Signature and Rhythm Relations | 拍号与节奏关系
Time signatures encode a beat hierarchy. The top number indicates beats per bar; the bottom number indicates the note value that equals one beat (2=minim, 4=crotchet, 8=quaver). Simple time (2/4, 3/4, 4/4) divides each beat into two equal parts; compound time (6/8, 9/8, 12/8) divides each beat into three equal parts. The dotted rhythm formula: a dotted note equals the original note plus half its value. Triplets may temporarily convert duple division into triple division; the triplet bracket indicates that three notes fill the time normally taken by two. Syncopation shifts accent onto a normally weak beat or between beats, often expressed by ties or rests.
拍号编码了拍子的强弱层次。上方数字表示每小节拍数，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（2/4、3/4、4/4）每一拍均分为二部；复拍子（6/8、9/8、12/8）每一拍均分为三部。附点节奏公式：附点音符等于原音符加其一半时值。三连音可将二均分临时转为三均分，三连音标记表示三个音在两拍的正常时值内奏出。切分音将重音移至通常的弱拍或拍子之间，常通过连线或休止符表现。
5. Cadences and Their Progressions | 终止式及其进行
A perfect cadence (V–I) gives the strongest sense of closure. An imperfect cadence (any chord landing on V, usually I–V, II–V, IV–V) sounds incomplete. A plagal cadence (IV–I) is the ‘Amen’ effect. An interrupted cadence (V–vi) creates a surprise deflection. In major keys, V⁷–I requires the leading note to rise and the seventh to fall. In minor keys, V usually uses the raised seventh from harmonic minor. A chord progression formula such as I–IV–V–I underpins countless classical and popular pieces; functional harmony often follows the cycle of fifths: iii → vi → ii → V → I.
完全终止 (V–I) 形成最强的收束感。不完全终止（任何和弦进行到 V，常见有 I–V、II–V、IV–V）听感未结束。变格终止 (IV–I) 即“阿们”效果。阻碍终止 (V–vi) 带来意外的折转。大调中的 V⁷–I 要求导音上行、七音下行。小调中的 V 通常使用和声小调的升高七级音。和弦进行公式如 I–IV–V–I 支撑着无数古典与流行作品；功能性和声常遵循五度圈：iii → vi → ii → V → I。
6. Ornamentation and Symbols | 装饰音与符号
Trill (tr) alternates rapidly between the written note and the upper neighbour. Mordent (∼) goes to the upper or lower neighbour and returns immediately. Turn (𝄐) wraps around the main note: upper note – main – lower – main. Appoggiatura is a stressed non‑harmonic note taking half the value of the following note; acciaccatura is a crushed grace note played as quickly as possible. The fermata (𝄐) indicates a pause of undefined length. Slurs indicate legato phrasing; staccato dots indicate detached articulation. All these symbols must be interpreted stylistically according to the period.
颤音 (tr) 在本音与上方邻音之间快速交替。涟音 (∼) 向上或下方邻音快速折返。回音 (𝄐) 环绕本音：上方音–本音–下方音–本音。倚音是占后方音符一半时值的重音非和弦音；短倚音是尽可能快速奏出的碎音。延音记号 (𝄐) 表示任意长度的停延。连线表示连奏分句，断音点表示分离奏法。所有这些符号必须根据风格时期加以诠释。
7. Texture Types | 织体类型
Monophony: a single unaccompanied melodic line. Homophony: melody with chordal accompaniment moving in the same rhythm (often called melody‑dominated homophony). Polyphony (contrapuntal texture): two or more independent melodic lines woven together; imitative polyphony uses entries of the same melodic idea in different voices. Heterophony: simultaneous variation of a single melody by different performers. The formula for identifying texture: ask whether there is a clear melody + accompaniment (homophonic) or whether each line is equally important and rhythmically independent (polyphonic). In CIE set works, texture often shifts between these types, so describe exactly where and how the change occurs.
单声部织体：单一无伴奏的旋律线条。主调织体：旋律加和弦伴奏且节奏基本一致（常称旋律主导的主调织体）。复调织体（对位织体）：两个或多个独立的旋律线条交织在一起；模仿复调运用同一旋律动机在不同声部进入。支声复调：不同演奏者对同一旋律进行即时的变奏。识别织体的公式：问自己是否有明显的旋律+伴奏（主调）还是各声部同样重要且节奏独立（复调）。在 CIE 指定作品中，织体常在这些类型间转换，因此务要准确描述变化的位置与方式。
8. Modulation and Key Relations | 转调与调性关系
Modulation to the dominant (V) is the most common in major‑key music; to the relative major (III) is typical in minor keys. Other closely related keys are iv, ii, vi in major; v, iv, VI in minor. The pivot chord formula: a chord that belongs diatonically to both the old key and the new key serves as the pivot. Chromatic modulation involves an alteration of one note to create a dominant seventh that resolves into the new tonic. The circle of fifths provides a map of key distances: each step clockwise adds a sharp; anticlockwise adds a flat. Enharmonic modulation re‑spells a chord to reach a remote key. In analysis, always identify the pivot chord and confirm the new key with a cadence.
大调作品中最常见的转调是转到属调 (V)；小调则常转到关系大调 (III)。其他近关系调在大调中是 iv、ii、vi；在小调中是 v、iv、VI。共同和弦公式：一个同时属于原调和目标调自然音体系的和弦充当转调枢纽。半音转调通过改变某一音，构成一个解决至新主音的属七和弦。五度圈提供了调性距离地图：顺时针每步增加一个升号，逆时针增加一个降号。等音转调通过重新解释某和弦的音高抵达远关系调。在分析中，务必指出共同和弦，并用终止式确证新调。
9. Harmonic Progressions and Figured Bass | 和声进行与数字低音
Figured bass uses numbers and accidentals beneath a bass line to indicate chord inversion and chromatic alteration. 5/3 (unfigured) indicates root position; 6 (or 6/3) first inversion; 6/4 second inversion. A slash through a number raises that interval by a semitone (e.g., 4+ means augmented fourth). The cadential 6/4 formula: a I6/4 chord placed on the dominant, resolving to V(⁷)–I. Sequential progressions, such as descending fifths (circle‑of‑fifths sequence: I–IV–vii°–iii–vi–ii–V–I), provide a strong motor for tonal music. In Year 12 analysis, be able to realise a short figured bass excerpt in four‑part vocal style, observing voice leading rules: no parallel fifths or octaves, leading note resolves upwards, and avoid augmented intervals melodically.
数字低音在低音线条下方使用数字和临时记号来标示和弦转位与半音变化。5/3（无数字）表示原位；6（或6/3）为第一转位；6/4 为第二转位。数字上打斜线表示将该音程升高半音（如 4+ 表示增四度）。终止四六和弦公式：在属音上配置 I6/4 和弦，解决至 V(⁷)–I。模进进行，例如下行五度模进（五度圈模进：I–IV–vii°–iii–vi–ii–V–I），为调性音乐提供了强大动力。在 Year 12 分析中，要能根据简短的数字低音编写四声部声乐曲式的实现，遵守声部进行规则：避免平行五、八度，导音上行解决，避免旋律增音程。
10. Melodic Writing Rules | 旋律写作规则
Melody in the Common Practice style is shaped by stepwise motion (conjunct) with occasional leaps (disjunct), which should normally be resolved by step in the opposite direction. The melodic contour formula: an arch shape (rise‑peak‑fall), a single peak note, and a sense of finality towards the tonic. Avoid augmented intervals and leaps of a seventh or larger. Chromatic notes follow the rule of ‘chromatic passing note’ (between two notes a whole tone apart) or ‘chromatic auxiliary’ (a semitone away from a main note). Sequences that repeat a pattern at a higher or lower pitch give coherence. In CIE composition exercises, always consider phrase balance (typically 4‑bar or 8‑bar phrases) and a clear sense of cadence points.
共晓时期风格旋律多以级进（连贯进行）为主，偶有跳进（分离进行），跳进后通常反向级进解决。旋律轮廓公式：拱形（上行–高点–下行），一个高点音，并朝向主音具有收束感。避免增音程以及七度或更大的跳进。半音可遵循“半音经过音”（两个全音之间的经过音）或“半音助音”（距主音半音的辅助音）的规则。在较高或较低音高上重复模式的模进能赋予连贯性。在 CIE 作曲练习中，始终考虑乐句的平衡（通常为 4 小节或 8 小节乐句）和清晰的终止感。
11. Dynamics, Articulation and Instrumental Techniques | 力度、演奏法与乐器技巧
Dynamics range from pp (pianissimo) to ff (fortissimo) with gradual changes marked by crescendo (cresc.) and diminuendo (dim.). The standard articulation signs—legato (slur), staccato (dot), tenuto (line), accent (>), marcato (^)—carry specific formulae for duration and attack. String instruments: arco (bowed), pizzicato (plucked), con sordino (muted), sul ponticello (near the bridge). Woodwinds: flutter tongue, harmonics, key clicks. Brass: mute types (straight, cup, harmon), glissando. Piano: use of pedal (una corda, sustain), una corda softens tone; sustain pedal allows legato across wide leaps. In CIE extracts, recognise these timbral variants and describe their effect.
力度从 pp（很弱）到 ff（很强），渐变用 cresc.（渐强）和 dim.（渐弱）标记。常见的演奏法记号——连音线、断音点、保持音线、重音 (>)、特重音 (^)——携带着关于时值和起音的特定公式。弦乐器：arco（运弓）、pizz.（拨奏）、con sordino（加弱音器）、sul ponticello（近马奏）。木管：花舌、泛音、键击声。铜管：弱音器类型（直、杯、瓦答）、滑奏。钢琴：踏板使用（弱音踏板、延音踏板），弱音踏板柔化音色，延音踏板实现大跳间的连奏。在 CIE 选段中，识别这些音色变化并描述其效果。
12. Musical Form Structures | 曲式结构
Binary form (AB) typically repeats each section; a rounded binary (A BA’) returns the opening material before closing. Ternary form (ABA) has a contrasting middle section, often in a related key. Sonata form consists of Exposition (two subjects in tonic and dominant, joined by a transition), Development (motivic fragmentation and modulation), and Recapitulation (both subjects now in the tonic) with optional Coda. Minuet and Trio is a compound ternary with a da capo repeat. Theme and variations follows the formula AA₁A₂A₃… Strophic form repeats the same music for each stanza. Through‑composed sets new music throughout. The ‘formula’ for recognising form: map the material using letters, identify cadence keys, and note any repeat signs or da capo markings.
二部曲式 (AB) 通常每段重复；再现二部曲式 (A BA’) 结束前回顾开篇材料。三部曲式 (ABA) 有对比中段，常位于近关系调。奏鸣曲式包含呈示部（主部与副部，通过连接部从主调转到属调）、展开部（动机分裂与转调）、再现部（两个主题均回到主调），可附尾声。小步舞曲与中段为带反复的复三部曲式。主题与变奏遵循 AA₁A₂A₃… 公式。分节歌曲式为每节歌词重复相同音乐。通谱歌曲式完全谱以新乐。识别曲式的“公式”：用字母标记材料，判断终止调性，留意反复记号和 Da Capo 标记。
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