📚 A-Level CAIE Music: Quick Reference to Formulas and Theorems | A-Level CAIE 音乐：公式定理速查手册
Music may seem all about emotion, but behind every soaring melody and gripping harmony lies a set of structural 'formulas' and 'theorems' that A-Level musicians must master. This quick-reference guide condenses the essential rules and patterns of Western tonal music into bite‑sized principles, helping you navigate scales, chords, part‑writing, form, and more with mathematical clarity.
音乐看似全凭情感，但每段飞扬的旋律和扣人心弦的和声背后，都有一套结构的“公式”与“定理”，这是A-Level音乐学子必须掌握的。这本速查手册将西方调性音乐的核心规则与模式浓缩成精要条目，以数学般的清晰度帮助你掌握音阶、和弦、声部写作、曲式等内容。
1. Major and Minor Scale Formulas | 大调与小调音阶公式
A major scale always follows the intervallic pattern: Tone, Tone, Semitone, Tone, Tone, Tone, Semitone (T-T-S-T-T-T-S). Starting on any note and applying this sequence yields the correct diatonic major scale.
大调音阶始终遵循全音、全音、半音、全音、全音、全音、半音（T-T-S-T-T-T-S）的音程模式。以任何音为起点应用此序列，即可得到正确的自然大调音阶。
The natural minor scale uses the pattern: Tone, Semitone, Tone, Tone, Semitone, Tone, Tone (T-S-T-T-S-T-T). The harmonic minor raises the 7th degree by a semitone; the melodic minor raises both the 6th and 7th degrees when ascending, reverting to natural minor when descending.
自然小调使用的模式为：全音，半音，全音，全音，半音，全音，全音（T-S-T-T-S-T-T）。和声小调将第7级音升高半音；旋律小调上行时升高第6、7级，下行时还原为自然小调。
- Major: C D E F G A B C – steps: 2,2,1,2,2,2,1
- 中文：大调：C D E F G A B C – 步数：2,2,1,2,2,2,1
- Natural minor: A B C D E F G A – steps: 2,1,2,2,1,2,2
- 中文：自然小调：A B C D E F G A – 步数：2,1,2,2,1,2,2
2. Interval Quality and Size Theorems | 音程性质与度数定理
An interval is measured by the number of letter names (size) and its quality (perfect, major, minor, augmented, diminished). The 'rule of nine' states that the inversion of a 2nd is a 7th, 3rd ↔ 6th, 4th ↔ 5th, and perfect remains perfect, major becomes minor, augmented becomes diminished.
音程由音名字母的数目（度数）与性质（纯、大、小、增、减）来度量。 “九规则”指出，二度的转位是七度，三度↔六度，四度↔五度；纯音程转位后仍为纯，大音程转位后变为小，增音程转位后变为减。
nd = 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, octave = 12.
在大调音阶中，从主音出发的所有音程要么是大音程，要么是纯音程。要辨认音程，可数半音数：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=11，八度=12。
3. Triad and Seventh-Chord Construction Formulas | 三和弦与七和弦构建公式
ajor triad = major 3rd + minor 3rd (e.g., C-E-G); minor triad = minor 3rd + major 3rd (C-E♭-G); diminished triad = minor 3rd + minor 3rd (C-E♭-G♭); augmented triad = major 3rd + major 3rd (C-E-G♯).
三和弦由两个三度叠置而成。原位和弦中：大三和弦=大三度+小三度（如C-E-G）；小三和弦=小三度+大三度（C-E♭-G）；减三和弦=小三度+小三度（C-E♭-G♭）；增三和弦=大三度+大三度（C-E-G♯）。
dominant seventh = major triad + minor 3rd; minor seventh = minor triad + minor 3rd; half‑ seventh = diminished triad + major 3rd; diminished seventh = diminished triad + minor 3rd.
七和弦再叠加一个三度：大七和弦=大三和弦+大三度；属七和弦=大三和弦+小三度；小七和弦=小三和弦+小三度；半减七和弦=减三和弦+大三度；减七和弦=减三和弦+小三度。
4. Circle of Fifths and Key Relationships | 五度圈与调性关系定理
Moving clockwise around the circle of fifths adds one sharp or removes one flat; each step raises the tonic by a perfect fifth. The order of sharps: F♯ C♯ G♯ D♯ A♯ E♯ B♯; order of flats: B♭ E♭ A♭ D♭ G♭ C♭ F♭. Relative minors share the same key signature; the tonic minor lies three semitones below the major tonic.
顺时针移动五度圈，每步增加一个升号或减少一个降号，主音升高纯五度。升号出现的顺序为：F♯ C♯ G♯ D♯ A♯ E♯ B♯；降号顺序：B♭ E♭ A♭ D♭ G♭ C♭ F♭。关系小调与大调共享同一调号，其主音比大调主音低小三度。
The circle also encodes closely related keys: any key's dominant, subdominant, and relative minor are only one accidental away. Modulation to these keys sounds smooth and is governed by common‑chord pivots.
五度圈还编码了近关系调：任何调的属调、下属调及关系小调都只相差一个调号。向这些调转调听起来最为平滑，并受共同和弦枢轴支配。
| C major / A minor | 0 ♯/♭ |
| G major / E minor | 1 ♯ (F♯) |
| D major / B minor | 2 ♯ (F♯, C♯) |
| F major / D minor | 1 ♭ (B♭) |
| B♭ major / G minor | 2 ♭ (B♭, E♭) |
5. Cadence Formulas | 终止式公式
A perfect cadence (V–I) creates a full close, with both chords in root position and the tonic in the soprano of the final chord. An imperfect cadence ends on V, often preceded by I, ii or IV. A plagal cadence (IV–I) is the 'Amen' formula. An interrupted cadence moves V to vi (or occasionally IV⁶ to V), denying closure.
正格终止（V–I）产生完全收束，两和弦均用原位，且最终和弦的高音部为主音。不完全终止结束在V级上，前面常用I、ii或IV。变格终止（IV–I）即“阿们”公式。阻碍终止则由V进行到vi（有时IV⁶至V），不提供终止感。
In chorale‑style harmony, the leading note (7th degree) must resolve upwards to the tonic in a perfect cadence. The bass typically leaps a 5th down or 4th up.
在众赞歌风格的和声中，导音（第7级）在正格终止中必须向上解决到主音。低音部通常下行五度或上行四度跳进。
6. Four‑Part Harmony (Chorale‑Style) Rules | 四部和声（众赞歌风格）规则
Voice ranges must not be exceeded: Soprano C₄–G₅, Alto G₃–D₅, Tenor C₃–G₄, Bass E₂–C₄. The spacing between adjacent upper voices (S‑A, A‑T) should not exceed an octave; between T‑B it may be wider.
声部音域不得超出：女高音C₄–G₅，女低音G₃–D₅，男高音C₃–G₄，男低音E₂–C₄。相邻上三声部（S‑A，A‑T）之间的间距不得超过八度；T‑B之间可稍宽。
Parallel perfect fifths and octaves are forbidden between any two voices. Exposed (hidden) fifths and octaves—approached by similar motion and resulting in a perfect interval—should be avoided unless the higher voice moves by step. Consecutive fifths by contrary motion are also prohibited.
任何两声部之间禁止平行纯五度与平行八度。显露（隐伏）五度与八度——即同向进行并造成纯音程——应避免，除非较高声部作级进。反向进行的连续五度同样被禁止。
Leading note resolution: the 7th degree must rise to the tonic when it occurs in a dominant‑function chord. Chordal sevenths must resolve downwards by step. Avoid augmented and diminished melodic intervals; use oblique or contrary motion to maintain independence.
导音解决法则：导音在属功能和弦中出现时，必须上行至主音。和弦七音必须下行级进解决。避免旋律增、减音程；使用斜向或反向进行来保持声部独立。
7. Species Counterpoint Rules | 分类对位法规则
First species (note against note) permits only consonant intervals: perfect unisons, octaves, fifths, and major/minor thirds and sixths. The perfect fourth is considered a dissonance in two‑voice texture. Begin and end on perfect consonances; the final bar must have the leading note rising to tonic in one voice, with the other voice moving by step.
第一类（全音符对全音符）只允许协和音程：纯一度、纯八度、纯五度，以及大小三度、大小六度。在二声部织体中，纯四度被视为不协和音程。开始与结束必须使用完全协和音程；最后一小节须有一个声部导音上行至主音，另一声部级进。
Second species (two notes against one) introduces the passing tone on unaccented beats; the downbeat must be consonant. Third species (four notes against one) adds the neighbour note and double neighbour figures. Dissonances may occur only on weak beats as passing or neighbour tones, and must be approached and left by step.
第二类（二音对一音）在弱拍上引入经过音；强拍必须协和。第三类（四音对一音）加入助音和双助音形态。不协和音只能出现在弱拍上，作为经过音或助音，并且必须级进进入和级进离开。
8. Sonata Form Theorem | 奏鸣曲式定理
Sonata form comprises Exposition, Development, and Recapitulation. Exposition: first subject (tonic), transition (modulating), second subject (dominant or relative major), closing section, often repeated. Development: motivic fragmentation, sequential treatment, modulation through remote keys. Recapitulation: both subjects now in the tonic key.
奏鸣曲式由呈示部、展开部和再现部组成。呈示部：第一主题（主调）、连接部（转调）、第二主题（属调或关系大调）、结束部，通常反复。展开部：动机分裂、模进处理、远关系转调。再现部：两个主题均回到主调。
The 'double return' marks the recapitulation—return of the tonic key together with the first theme. A coda may reinforce the tonic after the closing material has been restated.
“双重回归”标志着再现部的开始——主调与第一主题一同回归。结束段材料再次陈述后，尾声可进一步加强主调。
9. Melodic Writing Formulas | 旋律写作公式
A well‑formed melody balances conjunct (stepwise) and disjunct (leap) motion. After a leap of a 4th or larger, the melody should reverse direction by step. Leaps of augmented intervals, 7ths, and intervals greater than an octave are avoided in classical style.
一条优美的旋律应平衡好级进与跳进。四度及以上的跳进后，旋律应反向级进。古典风格中应避免增音程、七度以及超过八度的大跳。
The structural notes often outline chords implied by the harmony, while non‑harmonic tones (passing notes, auxiliary notes, appoggiaturas) provide motion. Sequences—repetition of a pattern at different pitch levels—create unity and direction.
结构音常常勾勒出和声所暗含的和弦轮廓，而非和弦音（经过音、辅助音、倚音）则提供运动感。模进——在不同音高上重复同一模式——可创造统一性与方向感。
10. Figured Bass and Harmonic Shorthand | 数字低音与和声速记公式
Figured bass symbols denote intervals above the bass note. A root‑position triad has no figures (or 5/3); first inversion is ⁶ (or 6/3); second inversion is ⁶/₄. Seventh chords: root position ⁷, first inversion ⁶/₅, second inversion ⁴/₃, third inversion ⁴/₂ (or ²).
数字低音记号表示低音上方应奏出的音程。原位三和弦无数字（或记5/3）；第一转位为⁶（或6/3）；第二转位为⁶/₄。七和弦：原位⁷，第一转位⁶/₅，第二转位⁴/₃，第三转位⁴/₂（或²）。
Accidentals in figures apply to the third above the bass unless a specific number is modified. A slash through a number indicates raising that note by a semitone.
数字中的临时记号作用于低音上方的三度音，除非在特定数字上修改。数字上的一条斜线表示将该音升高半音。
11. Orchestration and Instrumental Range Formulas | 配器法与乐器音域公式
Each orchestral instrument has a practical written range and characteristic registers. For precise writing, know transpositions: Clarinet in B♭ sounds a major 2nd lower; Horn in F sounds a perfect 5th lower; Trumpet in B♭ sounds a major 2nd lower; Double bass sounds an octave lower than written.
每件管弦乐器都有实际的记谱音域和特性音区。写作须明确移调：B♭调单簧管实际音高比记谱低大二度；F调圆号低纯五度；B♭调小号低大二度；低音提琴实际音高比记谱低八度。
| Flute | C₄–C₇ (written) |
| Oboe | B♭₃–F₆ |
| Violin | G₃–C₈ |
| Cello | C₂–A₅ |
12. Performance Direction 'Equations' | 演奏指示 “方程式”
Italian terms act like operators: crescendo (cresc.) = gradually louder; diminuendo (dim.) = gradually softer; ritardando (rit.) = gradually slower; accelerando (accel.) = gradually faster. Tempo markings give the speed equation: Allegro ≈ 120–168 bpm, Andante ≈ 76–108 bpm, Adagio ≈ 66–76 bpm.
意大利术语类似算子：crescendo（渐强）=逐渐响亮；diminuendo（渐弱）=逐渐柔和；ritardando（渐慢）=逐渐减速；accelerando（渐快）=逐渐加速。速度标记给出速度方程：快板约120–168 bpm，行板约76–108 bpm，柔板约66–76 bpm。
Articulation: staccato reduces a note's duration by about half; legato indicates smooth connection; accent (> ) stresses the attack. Dynamics span from ppp to fff, creating a loudness function over time.
运音法：断奏将音符时值缩短大约一半；连奏表示圆滑的连接；重音记号（>）强调音头。力度范围从极弱到极强，形成随时间变化的响度函数。
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