Formulas & Theorems Quick Reference for Year 13 Cambridge Music | A Level 音乐公式定理速查手册

📚 Formulas & Theorems Quick Reference for Year 13 Cambridge Music | A Level 音乐公式定理速查手册

Understanding the structural ‘formulas’ behind musical language is essential for A Level analysis and composition. This handbook distils core concepts of harmony, counterpoint, rhythm, and 20th-century techniques into concise, exam-ready references. Each section pairs a theoretical model with practical application, helping you navigate chromatic harmony, serial composition, and rhythmic structuring with confidence.

掌握音乐语言背后的结构性“公式”对于 A Level 分析与创作至关重要。本手册将和声、对位、节奏及二十世纪技法的核心概念提炼成简洁、适考的参考资料。每一节都将理论模型与实际应用配对，帮助你自信地驾驭半音化和声、序列作曲与节奏结构设计。

1. Interval Identification Formula | 音程识别公式

An interval is defined by both its numeric size (number of letter names) and its quality (perfect, major, minor, augmented, diminished). The formula: Count the lines and spaces inclusive of the bottom note to get the number, then count semitones to determine quality. A table of semitone counts for simple intervals provides quick identification.

音程由数字大小（字母名称数量）和性质（纯、大、小、增、减）共同定义。识别公式：从下方音开始逐行逐间数音名个数得到度数，再数半音数确定性质。常用自然音程的半音数表格可助快速判断：

Interval	Semitones	Example
Perfect Unison	0	C–C
Minor 2nd	1	C–D♭
Major 2nd	2	C–D
Minor 3rd	3	C–E♭
Major 3rd	4	C–E
Perfect 4th	5	C–F
Augmented 4th / Diminished 5th	6	C–F♯ / C–G♭
Perfect 5th	7	C–G
Minor 6th	8	C–A♭
Major 6th	9	C–A
Minor 7th	10	C–B♭
Major 7th	11	C–B
Perfect Octave	12	C–C’

To invert an interval, subtract its semitone count from 12; the quality inverts as well (major ↔ minor, augmented ↔ diminished, perfect ↔ perfect).

转位音程的半音数等于12减去原位半音数；性质也随之转换：大⇌小、增⇌减、纯⇌纯。

2. Triad Construction & Chord Symbols | 三和弦构建与和弦标记

All triads are built by stacking two thirds above a root. The quality is determined by the combination of intervals: major triad = major 3rd + minor 3rd (root–M3–P5); minor triad = minor 3rd + major 3rd (root–m3–P5); diminished triad = minor 3rd + minor 3rd (root–m3–d5); augmented triad = major 3rd + major 3rd (root–M3–A5).

所有三和弦由根音上叠置两个三度构成。性质取决于音程组合：大三和弦=大三度+小三度（根–大三–纯五）；小三和弦=小三度+大三度；减三和弦=小三度+小三度（根–小三–减五）；增三和弦=大三度+大三度（根–大三–增五）。

In Roman numeral analysis, uppercase indicates major (I), lowercase indicates minor (ii), lowercase with ° for diminished (vii°), and uppercase with + for augmented (V⁺). Figured bass symbols use 5/3 for root position, 6 for first inversion, and 6/4 for second inversion.

罗马数字分析中，大写表示大三和弦（I），小写表示小三和弦（ii），小写加“°”为减三和弦（vii°），大写加“⁺”为增三和弦。数字低音标记：原位用5/3，第一转位用6，第二转位用6/4。

3. Seventh Chord Formulas | 七和弦公式

A seventh chord adds a third above a triad, creating a four-note stack. The five common seventh chords in tonal music are defined by the quality of triad plus the interval of the seventh from the root: Major 7th = major triad + major 7th (Cmaj7: C–E–G–B); Dominant 7th = major triad + minor 7th (C7: C–E–G–B♭); Minor 7th = minor triad + minor 7th (Cm7: C–E♭–G–B♭); Half-diminished 7th = diminished triad + minor 7th (Cø7: C–E♭–G♭–B♭); Diminished 7th = diminished triad + diminished 7th (C°7: C–E♭–G♭–B♭♭).

七和弦在三和弦上方再叠加一个三度，形成四音结构和声。调性音乐中常见的五种七和弦由三和弦性质与根音上七度音程决定：大七和弦=大三和弦+大七度；属七和弦=大三和弦+小七度；小七和弦=小三和弦+小七度；半减七和弦=减三和弦+小七度；减七和弦=减三和弦+减七度。

The formula for resolution: The seventh of the chord resolves down by step, the leading tone (third of dominant seventh) resolves up, and the root moves up a fourth or down a fifth when progressing to the tonic. This voice-leading principle is paramount in harmonic dictation and part-writing.

解决公式：和弦的七音必须级进下行；导音（属七和弦的三音）上行解决；根音上行四度或下行五度进入主和弦。这一声部进行原则在和声听写与声部写作中至关重要。

4. Circle of Fifths Progression | 五度圈进行公式

The circle of fifths sequence moves roots by descending perfect fifths (or ascending perfect fourths), creating an archetypal tonal progression. The diatonic circle in major: I – IV – vii° – iii – vi – ii – V – I. Each step shifts one note of the scale, producing smooth voice leading and strong harmonic momentum.

五度圈进行以纯五度下行（或纯四度上行）的方式移动根音，形成典型的调性推进。大调自然音五度圈：I–IV–vii°–iii–vi–ii–V–I。每一步仅改变一个音级，带来平滑的声部进行和强劲的和声推动力。

Applied to chord progressions, the formula often appears as ii – V – I, or more extensively iii – vi – ii – V – I. Recognizing these patterns in score analysis reveals underlying structural frameworks in Baroque, Classical, and Romantic repertoire.

在和弦进行中，此公式常表现为 ii–V–I，或更扩展的 iii–vi–ii–V–I。在乐谱分析中识别这些模式，有助于揭示巴洛克、古典及浪漫主义曲目背后的结构性框架。

5. Cadence Formulas | 终止式公式

Cadences are formulaic punctuation points in tonal music. Perfect cadence (authentic): V – I, with both chords in root position and the tonic in the soprano for a perfect authentic cadence. Imperfect cadence: any chord – V (usually I–V, ii–V, IV–V). Plagal cadence: IV – I. Interrupted (deceptive) cadence: V – vi (or V – VI in minor).

终止式是调性音乐的公式化标点。正格终止（完全终止）：V–I，两和弦均需原位且旋律音落在主音上形成完全正格终止。半终止：任何和弦进行到V（常见 I–V、ii–V、IV–V）。变格终止：IV–I。阻碍终止：V–vi（或小调的 V–VI）。

In minor keys, the leading tone must be raised in the dominant chord, using the harmonic minor scale. The Phrygian half cadence (iv⁶ – V) in minor is a staple of late Romantic vocabulary.

小调中属和弦必须升高导音，即使用和声小调音阶。小调中的弗里几亚半终止（iv⁶–V）是晚期浪漫主义语汇的标志性公式。

6. Chromatic Harmony: Applied Dominants & Modal Mixture | 半音化和声：副属和弦与调式交替

Applied dominants (secondary dominants) are V or V⁷ chords that tonicize a diatonic chord other than the tonic. Formula: V/x (where x is any major or minor triad in the key, except the leading-tone triad). For example, in C major, V/vi is E major (E–G♯–B) resolving to Am.

副属和弦是将调内某一自然音和弦（主和弦除外）临时作为临时主和弦的属和弦或属七和弦。公式：V/x（x 为调内除导三和弦外的任何大三或小三和弦）。例如 C 大调中，V/vi 为 E 大三和弦（E–G♯–B）解决到 Am。

Modal mixture borrows chords from the parallel minor or major. Common borrowings in a major key: ♭VI, ♭III, ♭VII, and iv from the parallel minor. The formula: replace a diatonic chord with its parallel mode equivalent – e.g., in C major, using A♭ major (♭VI) instead of Am. This enriches harmonic colour without modulating.

调式交替从平行小调或大调借用和弦。大调中常见的借入和弦包括来自平行小调的 ♭VI、♭III、♭VII 和 iv。公式：用平行调式的同度数和弦替换原自然音和弦——例如 C 大调中用 A♭ 大和弦（♭VI）替代 Am。这一手法丰富和声色彩而无须转调。

7. Species Counterpoint Rules | 分类对位法则

First species (note against note) governs perfect consonances (unison, 5th, 8ve) at the start and end; motion must be contrary, oblique, or direct but avoid parallel perfect intervals. Perfect consonances should be approached by contrary or oblique motion.

第一类对位（音对音）规定开始和结束必须使用完全协和音程（同度、五度、八度）；声部进行须为反向、斜向或同向，但避免平行完全协和音程。进入完全协和音程须用反向或斜向进行。

Second species (two notes against one) introduces the passing tone and the neighbour tone on the weak beat. Formula for dissonance: the weak-beat note may be dissonant if it moves by step within the same direction (passing) or leaves and returns by step (neighbour).

第二类对位（二对一）在弱拍上引入经过音与助音。不协和音公式：弱拍音符若以级进同向进行（经过）或级进离开并返回（助音），则可为不协和音。

These rules distill Renaissance vocal polyphony into a pedagogical system that underpins common-practice counterpoint. In analysis, identifying suspensions (4-3, 7-6, 9-8, 2-3 in bass) as formulaic dissonance treatments is crucial.

这些法则将文艺复兴声乐复调浓缩为支撑共性实践时期对位的教学体系。在分析中，识别悬留音（4–3、7–6、9–8、低音声部的2–3）作为公式化的不协和音处理至关重要。

8. Fugue Structure Formula | 赋格结构公式

A fugue is built on a recurring subject with a formulaic exposition: subject (tonic key) – answer (dominant key, real or tonal) – countersubject – episodes modulating through related keys – entries in stretto. The exposition typically presents the subject in all voices, alternating tonic and dominant entries.

赋格建立在无数次回归的主题之上，具有公式化的呈示部：主题（主调）– 答题（属调，真答题或守调答题）– 对题 – 在不同关系调间转调的插部 – 密接和应中的主题进入。呈示部通常在所有声部中交替以主调和属调陈述主题。

Key elements: the countersubject is a recurring melodic line accompanying the subject; episodes develop motives from the subject or countersubject; stretto overlaps subject entries at a closer distance. The whole form often reprises the subject in the tonic near the end, sometimes with a pedal point.

关键要素：对题是与主题相伴反复出现的旋律线条；插部发展主题或对题的动机；密接和应以更短的间距叠合主题进入。全曲常在结尾附近于主调再现行，有时加上持续音。

9. Sonata Form Blueprint | 奏鸣曲式蓝图

Sonata form is a dramatic tonal scheme: Exposition (first subject in tonic – transition – second subject in dominant or relative major – codetta); Development (thematic manipulation through various keys); Recapitulation (first subject in tonic – transition altered to stay in tonic – second subject in tonic – coda).

奏鸣曲式是一种富有戏剧性的调性蓝图：呈示部（主调第一主题–连接部–属调或关系大调第二主题–小尾声）；展开部（主题素材在不同调上展开）；再现部（主调第一主题–改变后保持主调的连接部–主调第二主题–尾声）。

This formula underpins first movements, overtures, and many slow movements from Haydn to Brahms. In analysis, mark the thematic boundaries and key areas; note whether the development uses fragmentation, contrapuntal combination, or harmonic remoteness.

这一公式支撑了从海顿至勃拉姆斯的首乐章、序曲及许多慢乐章。在分析中，标记主题边界和调性区域；注意展开部是否运用了片段化、对位结合或远关系调性。

10. Serialism: The 12-Tone Row Matrix | 序列主义：十二音序列矩阵

A tone row is an ordered arrangement of all twelve chromatic pitches, used as the source material for a composition. The four basic permutations: Prime (P, original order), Retrograde (R, backwards), Inversion (I, mirroring intervals in opposite direction), Retrograde Inversion (RI). The matrix is constructed by transposing the prime row to start on each chromatic pitch.

序列是将十二个半音按特定顺序排列，作为作品的原素材。四种基本形态：原型（P，原序）、逆行（R，逆向）、倒影（I，反向音程镜像）、倒影逆行（RI）。矩阵通过将原序列逐次移调到各个半音起始而构建。

To generate the matrix: write the prime row in semitone numbers (0 = first pitch). The inversion starting on 0 is the inversion row’s pitch class numbers, calculated by subtracting each P integer from 12. Transpositions follow by adding the index number mod 12. Analysing serial works involves identifying row forms and their combinatorial properties.

生成矩阵：以半音数（0=首音）写出原序列。从0开始的倒影由每个原型整数减去12（模12）得出。移调则将指数数目加上模12。分析序列作品涉及识别音列形态及其组合特性。

A prime row example from Webern’s Op. 21: P0 = C, B, D, E♭, G, F♯, G♯, E, F, A, B♭, C♯ (intervals in semitones: -1, +2, +1, +4, -1, +2, -3, +1, +4, +1, +2). This interval pattern becomes the generative ‘formula’.

例如韦伯恩作品21的原序列：P0 = C, B, D, E♭, G, F♯, G♯, E, F, A, B♭, C♯（半音音程：–1, +2, +1, +4, –1, +2, –3, +1, +4, +1, +2）。这一音程模式成为生成性“公式”。

11. Rhythmic Formulas: Cross-Rhythm & Metric Modulation | 节奏公式：交错节奏与节拍转换

Cross-rhythm is the simultaneous use of contrasting rhythmic divisions, such as three equal notes against two (3:2) or four against three (4:3). The formula for a 3:2 cross-rhythm: the beat is divided into six equal sub-pulses; the triplet part attacks every two sub-pulses, the duplet part every three sub-pulses.

交错节奏是同时使用形成对比的节奏划分，如三对二（3:2）或四对三（4:3）。3:2交错节奏公式：一拍被均分为六等份微脉动；三连音部分每两个微脉动奏一个音，二连音部分每三个微脉动奏一个音。

Metric modulation (tempo modulation) uses a note value in the original tempo becoming equal to a note value in the new tempo. Elliott Carter pioneered this: a formula such as ‘♩ = ♩.’ or ‘♩ = ♪.’ must be precisely calculated with mathematical ratios relative to a common pulse.

节拍转换（速度转换）使用原速度中的某一时值等于新速度中的另一时值。艾略特·卡特对此进行了开创：类似“♩ = ♩.”或“♩ = ♪.”的公式必须基于共同脉动精确计算数学比值。

12. Extended Techniques & Notation Formulas | 扩展技巧与记谱公式

20th-century works demand formula-based approaches to notation. Harmonics: natural harmonics are indicated by a small circle above the note or a diamond-shaped notehead; artificial harmonics use a diamond notehead a perfect fourth above the stopped note, sounding two octaves above.

二十世纪作品要求基于公式的记谱方法。泛音：自然泛音用音符上方小圆圈或菱形符头表示；人工泛音用实按音上方纯四度的菱形符头，实际音高为两八度上方音。

The formula for ‘feathered beaming’ (accelerando/ritardando within a bar) shows a gradual increase or decrease in beam angle or notehead spacing. Graphic scores often provide performance formulas: a wavy line = trill, a thick line = cluster, a rising line = glissando. These formulas bridge notation and realisation.

“渐密符尾”（小节内渐快/渐慢）的公式通过符尾角度或符头间距的渐变来表示。图形谱常提供演奏公式：波浪线=颤音，粗线=音簇，上行线=滑奏。这些公式连接了记谱与实际演奏。

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Formulas & Theorems Quick Reference for Year 13 Cambridge Music | A Level 音乐公式定理速查手册