📚 Year 13 CCEA Music: Formulae & Theorems Quick-Reference Handbook | CCEA A2 音乐公式定理速查手册
This handbook distils the essential music theory ‘formulae’ and ‘theorems’ that underpin harmonic analysis, compositional technique, and aural perception in the CCEA Year 13 Music specification. It is designed as a rapid revision tool for A2 students, bringing together interval structures, chord-building rules, cadential patterns, voice-leading principles, and more in one accessible reference.
本手册浓缩了 CCEA Year 13 音乐课程中支撑和声分析、作曲技法与听觉感知的核心音乐理论“公式”与“定理”。它作为 A2 学生的快速复习工具，将音程结构、和弦构建法则、终止式模式、声部进行原则等内容汇于一处，便于查阅。
1. Interval Formulae | 音程公式
Intervals are the building blocks of all harmonic and melodic material. Each interval has a specific size in semitones and a quality (perfect, major, minor, augmented, diminished). The formula for a perfect unison is 0 semitones; a minor 2nd is 1 semitone (m2 = 1 st); a major 2nd is 2 st (M2 = 2 st); a minor 3rd is 3 st (m3 = 3 st); a major 3rd is 4 st (M3 = 4 st); a perfect 4th is 5 st (P4 = 5 st); an augmented 4th/diminished 5th is 6 st (A4/d5 = 6 st); a perfect 5th is 7 st (P5 = 7 st); a minor 6th is 8 st (m6 = 8 st); a major 6th is 9 st (M6 = 9 st); a minor 7th is 10 st (m7 = 10 st); a major 7th is 11 st (M7 = 11 st); and a perfect octave is 12 st (P8 = 12 st).
音程是所有和声与旋律素材的基石。每个音程都有特定的半音数与性质（纯、大、小、增、减）。纯一度为 0 个半音；小二度为 1 个半音（m2 = 1 st）；大二度为 2 个半音（M2 = 2 st）；小三度为 3 个半音（m3 = 3 st）；大三度为 4 个半音（M3 = 4 st）；纯四度为 5 个半音（P4 = 5 st）；增四度/减五度为 6 个半音（A4/d5 = 6 st）；纯五度为 7 个半音（P5 = 7 st）；小六度为 8 个半音（m6 = 8 st）；大六度为 9 个半音（M6 = 9 st）；小七度为 10 个半音（m7 = 10 st）；大七度为 11 个半音（M7 = 11 st）；纯八度为 12 个半音（P8 = 12 st）。
Inversions follow a simple complement rule: the sum of the interval and its inversion always equals 9 (e.g., P4 inverts to P5, M2 inverts to m7). A perfect interval remains perfect; a major interval becomes minor; an augmented intervals become diminished, and vice versa.
转位遵循简单的互补法则：原位音程与其转位音程的数字之和永远为 9（如纯四度转位成纯五度，大二度转位成小七度）。纯音程转位后仍为纯音程；大音程变为小音程；增音程变为减音程，反之亦然。
2. Triad Construction Formulae | 三和弦构建公式
A triad is a three-note chord stacked in thirds. The quality of a triad is determined by the combination of its two stacked thirds. The formula for a major triad is: root + major 3rd (4 st) + perfect 5th (7 st), which can be viewed as M3 + m3 stacked. A minor triad uses root + minor 3rd (3 st) + perfect 5th (7 st), i.e., m3 + M3. A diminished triad stacks two minor 3rds: root + m3 + diminished 5th (6 st), i.e., m3 + m3. An augmented triad stacks two major 3rds: root + M3 + augmented 5th (8 st), i.e., M3 + M3.
三和弦是由三度叠置构成的三音和弦。三和弦的性质取决于其内部两个三度的组合方式。大三和弦公式：根音 + 大三度（4 个半音）+ 纯五度（7 个半音），可看作 M3 + m3 叠置。小三和弦：根音 + 小三度（3 st）+ 纯五度（7 st），即 m3 + M3。减三和弦：根音 + 小三度 + 减五度（6 st），即 m3 + m3。增三和弦：根音 + 大三度 + 增五度（8 st），即 M3 + M3。
In Roman numeral notation within a major key, the quality of triads built on each scale degree follows this pattern: I (major), ii (minor), iii (minor), IV (major), V (major), vi (minor), vii° (diminished). In a harmonic minor key, the pattern is: i (minor), ii° (diminished), III+ (augmented, from the raised 7th’s influence on III), iv (minor), V (major), VI (major), vii° (diminished).
用罗马数字标记大调音级上的三和弦，其性质规律为：I（大）、ii（小）、iii（小）、IV（大）、V（大）、vi（小）、vii°（减）。在和声小调中，模式为：i（小）、ii°（减）、III+（增，因升 VII 级对 III 的影响）、iv（小）、V（大）、VI（大）、vii°（减）。
3. Seventh Chord Formulae | 七和弦构建公式
A seventh chord adds a fourth note, a seventh above the root, to a triad. The five most common seventh chords each have distinct interval stacks. The major seventh chord (Cmaj7) stacks M3 + m3 + M3: root, M3, P5, M7. The dominant seventh chord (C7) stacks M3 + m3 + m3: root, M3, P5, m7. The minor seventh chord (Cm7) stacks m3 + M3 + m3: root, m3, P5, m7. The half-diminished seventh chord (Cm7♭5) stacks m3 + m3 + M3: root, m3, d5, m7. The fully diminished seventh chord (Cdim7) stacks m3 + m3 + m3: root, m3, d5, d7 (equivalent to 9 st above root).
七和弦是在三和弦基础上加入根音上方的七度音。五种最常见的七和弦各有独特的音程叠置公式。大七和弦（Cmaj7）叠置 M3 + m3 + M3：根音、大三度、纯五度、大七度。属七和弦（C7）叠置 M3 + m3 + m3：根音、大三度、纯五度、小七度。小七和弦（Cm7）叠置 m3 + M3 + m3：根音、小三度、纯五度、小七度。半减七和弦（Cm7♭5）叠置 m3 + m3 + M3：根音、小三度、减五度、小七度。完全减七和弦（Cdim7）叠置 m3 + m3 + m3：根音、小三度、减五度、减七度（相当于根音上方 9 个半音）。
In a major key, diatonic seventh chords are: Imaj7, ii7 (minor seventh), iii7 (minor seventh), IVmaj7, V7 (dominant seventh), vi7 (minor seventh), viiø7 (half-diminished). In harmonic minor, the typical set includes: im(maj7) or i7, iiø7, III+maj7, iv7, V7, VImaj7, vii°7.
在大调中，自然音七和弦为：Imaj7、ii7（小七和弦）、iii7（小七和弦）、IVmaj7、V7（属七和弦）、vi7（小七和弦）、viiø7（半减七和弦）。在和声小调中，典型集合包括：im(maj7) 或 i7、iiø7、III+maj7、iv7、V7、VImaj7、vii°7。
4. Cadence Formulae | 终止式公式
Cadences are the punctuation marks of tonal music, defined by the progression of two chords at the end of a phrase. The perfect cadence moves from V (or V7) to I (or i), creating a strong sense of closure. Formula: V – I (V7 – I). The plagal cadence is IV – I (or iv – i), often heard as the ‘Amen’ cadence. The imperfect cadence ends on V, preceded by any chord, most commonly I–V, ii–V, or IV–V; it sounds incomplete. The interrupted (deceptive) cadence substitutes vi (or VI) for the expected I after V: V – vi (or V – VI).
终止式是调性音乐的标点，由乐句结尾处的两个和弦进行定义。正格终止从 V（或 V7）进行到 I（或 i），营造强烈的收束感。公式：V – I（V7 – I）。变格终止为 IV – I（或 iv – i），常被称为“阿门终止”。不完全终止结束在 V 级上，前面可以是任何和弦，常见 I–V、ii–V 或 IV–V；听感不完整。阻碍终止在 V 之后用 vi（或 VI）替代期待的 I：V – vi（或 V – VI）。
A perfect cadence is further strengthened by specific voice leading: the leading note rises by step to the tonic, and the bass leaps a 5th down or a 4th up. In interrupted cadences, the leading note still rises to the tonic, but the bass may move by step (e.g., V to vi). The tierce de Picardie is a variant where a minor-mode piece ends with a major tonic chord, i.e., v – I (major) instead of v – i.
正格终止通过特定声部进行进一步加强：导音级进上行至主音，低音下跳五度或上跳四度。阻碍终止中导音仍上行至主音，但低音常级进（如 V 到 vi）。辟卡迪三度是一种变体，指小调作品结尾用大三和弦代替小三和弦，即 v – I（大）而非 v – i。
5. Common Chord Progression Formulae | 常见和弦进行公式
Certain root-movement patterns recur so frequently in functional harmony that they function as progression ‘theorems’. The descending 5th (or ascending 4th) circle progression is the backbone of tonal music: iii – vi – ii – V – I (or vi – ii – V – I). The I – IV – V – I progression establishes a key firmly. The I – vi – IV – V (doo-wop progression) and its derivatives are staples in popular music. The ii – V – I progression is the quintessential jazz and Classical cadential pattern.
某些根音运动模式在功能和声中频繁出现，可被视作进行“定理”。下行五度（或上行四度）循环进行是调性音乐的支柱：iii – vi – ii – V – I（或 vi – ii – V – I）。I – IV – V – I 能够牢固地确立一个调。I – vi – IV – V（Doo-wop 进行）及其衍生是流行乐的常用语汇。ii – V – I 则是爵士乐与古典乐终止模式的精髓。
In minor keys, the progression i – iv – V – i is fundamental. The use of the subdominant before the dominant delays the sense of resolution. The progression I – V – vi – IV (or i – V – VI – iv) can cycle smoothly. The ‘Andalusian cadence’ i – VII – VI – V in minor is a descending tetrachord harmonisation. Always check that the leading note rises to tonic in V–I moves when the melody allows.
在小调中，i – iv – V – i 是基础进行。属和弦之前使用下属和弦会延迟解决感。I – V – vi – IV（或 i – V – VI – iv）可以平滑循环。小调的“安达卢西亚终止式”i – VII – VI – V 是下行四音列的和声化。当旋律允许时，务必确保 V–I 进行中导音上行解决到主音。
6. Modulation Pivot Chord Formulae | 转调枢纽和弦公式
A smooth modulation to a closely related key typically employs a pivot chord – a chord that belongs diatonically to both the old key and the new key. The ‘formula’ is to identify chords common to both keys. For modulation from C major to G major (dominant), chords common are: C: I = G: IV; C: iii = G: vi; C: V = G: I; C: vi = G: ii. The strongest pivot is often the one that functions as a pre-dominant in the new key (e.g., C: V becomes G: I, but that is the target; a good pivot is C: ii = G: vi, leading to V–I in G).
向近关系调平滑转调通常使用枢纽和弦——即在原调与新调中均属于自然音的和弦。其“公式”是找出两调共有的和弦。从 C 大调转至 G 大调（属调）时，共有和弦包括：C: I = G: IV；C: iii = G: vi；C: V = G: I；C: vi = G: ii。最强的枢纽和弦往往在新调中担任属前功能（例如 C: ii = G: vi，随后进行到 G 的 V–I）。
Modulation to the relative minor (e.g., C major to A minor) uses the pivot chord ii or IV, which are identical in both (C: ii = Am: iv; C: IV = Am: VI). The progress leading to V–i in the new minor confirms the modulation. Chromatic pivot chords, such as the Neapolitan 6th (♭II6) or augmented 6th chords, can initiate modulation to remote keys, but these are advanced devices used for dramatic effect in Romantic and 20th-century works.
转入关系小调（如 C 大调转 a 小调）时，使用两调中相同的 ii 或 IV 作为枢纽（C: ii = Am: iv；C: IV = Am: VI）。进行到新小调的 V–i 即可确认转调。半音枢纽和弦，如那波里六和弦（♭II6）或增六和弦，可以启动远关系转调，但它们是浪漫派和 20 世纪作品中为戏剧效果而使用的高级手法。
7. Voice Leading Rules | 声部进行规则
Voice leading ‘theorems’ govern the smooth movement of individual voices in part-writing. Consecutive perfect 5ths and octaves between any two voices are to be avoided in common-practice style because they destroy independence. The rule of thumb: if two voices move in similar motion into a perfect 5th or octave, check the interval: if it is approached by leap (hidden 5ths/octaves) it may be acceptable if the soprano moves by step. Direct octaves and 5ths with a leap in the soprano are generally forbidden.
声部进行“定理”控制着多声部写作中各个声部的平滑运动。在共性写作时期，两个声部之间的平行纯五度与纯八度应避免，因为它们会破坏声部独立性。经验法则：若两个声部以同向进行抵达纯五度或纯八度，需检查前一个音程；若高音部跳进进入，则可能形成隐伏五/八度；若高音部级进，则通常可接受。高音部跳进的直接八度与五度一般禁用。
In root-position triads, double the root most often, occasionally the fifth; avoid doubling the third unless it is a minor third, and never double the leading note. In first-inversion chords, double the soprano note or the third. The leading note in an outer voice should resolve upwards by step to the tonic, especially in V–I cadences. When the leading note is in an inner voice, it may fall to the fifth of the tonic chord if necessary. Avoid augmented melodic intervals; aim for contrary motion between bass and soprano where possible.
在根音位置三和弦中，最常重复根音，有时重复五音；避免重复三音，除非是小三度音，且永远不要重复导音。在第一转位和弦中，可重复高音声部的音或三音。外声部中的导音应级进上行解决到主音，尤其在 V–I 终止中。当导音处于内声部时，必要时可下行至主和弦的五音。避免增音程旋律进行；尽可能在低音与高音之间采用反向进行。
8. Figured Bass Shorthand | 数字低音速记
Figured bass notation is a shorthand that indicates intervals above a given bass note, functioning as a formula for harmonic realisation. A root-position triad is indicated by 5/3 (often omitted to just the note with no figures). A first-inversion triad is 6/3 (commonly written as 6), meaning a 6th and a 3rd above the bass. A second-inversion triad is 6/4 (fourth and sixth above bass). Seventh chords add the 7th to these patterns: root-position 7th is 7/5/3 (figures: 7); first inversion is 6/5/3 (abbreviated 6/5); second inversion is 6/4/3 (abbreviated 4/3); third inversion is 6/4/2 (abbreviated 4/2 or simply 2).
数字低音标记是一种简写，指示在给定低音上方应奏出的音程，是和声实现的一种公式。根音位置三和弦用 5/3 表示（常省略数字，仅写音名）。第一转位三和弦为 6/3（常简写为 6），表示低音上方奏出六度和三度。第二转位三和弦为 6/4（四度与六度）。七和弦在这些模式上添加七度：原位七和弦为 7/5/3（标记 7）；第一转位为 6/5/3（简记 6/5）；第二转位为 6/4/3（简记 4/3）；第三转位为 6/4/2（简记 4/2 或 2）。
Accidentals in figures apply to the interval above the bass. A sharp or flat next to a number raises or lowers that note chromatically. A slash through a number (often 6 or 5) indicates to raise the note a semitone. Knowing these formulae enables you to analyse the harmonic rhythm and cadential structure in CCEA score excerpts rapidly.
数字中的临时记号作用于低音上方的音程。数字旁的升号或降号将该音升高或降低半音。数字上划一斜线（通常为 6 或 5）表示将该音升高半音。掌握这些公式可使你快速分析 CCEA 乐谱选段中的和声节奏与终止结构。
9. Melodic Decoration Formulae | 旋律装饰音公式
Non-chord tones (melodic decorations) are classified by their approach and resolution. A passing note (P) connects two chord tones by step in the same direction; the formula is: chord tone – step to dissonance – step to next chord tone. A neighbour note (N) or auxiliary note moves stepwise away from a chord tone and returns to it: chord tone – step up/down – step back. An anticipation (Ant) sounds a note of the upcoming chord early, usually by step. An escape tone (E) is approached by step and resolved by leap in the opposite direction. An appoggiatura (App) is approached by leap and resolved by step, usually on a strong beat.
和弦外音（旋律装饰）按其进入与解决的方式分类。经过音（P）以同向级进连接两个和弦音；公式：和弦音 – 级进至不协和音 – 级进至下一和弦音。辅助音（N）或邻音从和弦音级进离开再级进返回：和弦音 – 上行/下行级进 – 级进返回。先现音（Ant）提前奏出下一和弦的某个音，通常为级进。规避音（E）级进进入后反向跳进解决。倚音（App）跳进进入后级进解决，通常落在强拍。
A suspension (S) consists of three parts: preparation (the dissonant note is consonant in the previous chord), suspension (the note is held over or repeated as a dissonance), and resolution (stepwise down to a consonance). The common suspensions are 4–3, 7–6, 9–8, and 2–3 (bass suspension). A retardation (R) is a suspension that resolves upward. Recognising these patterns is essential for aural analysis and completing a given texture in the CCEA exam.
延留音（S）由三部分组成：预备（该音在前一和弦中是协和的）、延留（作为不协和音保持或重复）与解决（级进下行至协和音）。常见的延留音包括 4–3、7–6、9–8 与 2–3（低音延留）。上行延留音（R）是向上解决的延留。识别这些模式对于 CCEA 考试中的听觉分析与完成所给织体至关重要。
10. Harmonic Rhythm and Phrasing Formulae | 和声节奏与乐句公式
Harmonic rhythm refers to the rate at which chords change. A common formula in Classical phrasing is that the harmonic rhythm accelerates towards a cadence. For example, in an 8-bar period, bars 1–4 might move from I to V with chords changing once per bar; bars 5–8 might see more frequent changes (e.g., ii – V – I within one bar) to drive towards the perfect cadence. In chorale-style writing, the pace of chord change often aligns with the strong beats of the metre.
和声节奏指和弦更换的速率。古典乐句的一个常见公式是：和声节奏在接近终止时加快。例如，在 8 小节的乐段中，前 4 小节可能从 I 进行到 V，每小节换一次和弦；后 4 小节则可能更频繁地变化（如一小节内完成 ii – V – I），以推动至正格终止。在众赞歌风格写作中，和弦更换的步速常与拍子的重音对齐。
Phrase structure can be described using formulae such as sentence (presentation + continuation) and period (antecedent + consequent). The antecedent ends with a half cadence (V) and the consequent with an authentic cadence (I); this call-and-response symmetry is a fundamental theorem of tonal form. The ‘rule of the octave’ in schema theory provides a scale harmonisation that was internalised by 18th-century composers: an ascending major scale could be harmonised I – V⁶ – I – IV – V – vi – V/V – V (or variants), reinforcing the basic pace of harmonic change.
乐句结构可用“句子”（呈示 + 延续）与“乐段”（前句 + 后句）等公式来描述。前句以半终止（V）结束，后句以完全终止（I）结束；这种呼应式对称是调性形式的一个基本定理。18 世纪作曲家内化的“八度规则”提供了音阶和声化公式：上行大调音阶可配和声 I – V⁶ – I – IV – V – vi – V/V – V（或变体），强化了基本的和声变化步速。
11. Atonal and Serial Formulae | 无调性与序列音乐公式
In CCEA Year 13, you may encounter post-tonal music that relies on interval-class content and set theory rather than functional harmony. The ‘Prime Form’ theorem: any pitch-class set can be reduced to its most compact and normalized representation, starting on 0, with intervals packed to the left. For example, the set {C, E, G} in pitch integers, transposed to start on 0, is [0,4,7], whose prime form is (037). The interval vector is a formula summarising the total interval-class content (e.g., for a major triad, vector is <001110> showing 1 semitone, 1 minor 3rd, etc., but actually it’s ic3, ic4, ic5 present).
在 CCEA Year 13 中，你可能会遇到依赖音级集合理论而非功能和声的后调性音乐。“原型”定理：任何音级集合都可以被缩减为最紧凑、以 0 为起点的标准形式，音程向左紧缩。例如，集合 {C, E, G} 以音高整数表示，转位至 0 起，即 [0,4,7]，其原型为 (037)。音程涵量是汇总全部音程级内容的公式（如大三和弦的向量为 <001110>，显示存在半音、小三度……实际上它包含 ic3、ic4、ic5）。
Twelve-tone rows (serialism) operate under the formulae of the four standard transformations: Prime (P), Retrograde (R), Inversion (I), and Retrograde-Inversion (RI). The matrix is a systematic way to generate all 48 forms. A row’s hexachordal combinatoriality can be a key structural theorem in works by Schoenberg and Webern. Analysis tasks in CCEA may ask you to identify row forms and note transformations.
十二音序列（序列主义）遵循四种标准转换公式：原型（P）、逆行（R）、倒影（I）与倒影逆行（RI）。矩阵是生成全部 48 种形式的系统方法。序列的六音组组合性是勋伯格与韦伯恩作品中的关键结构定理。CCEA 的分析任务可能要求你识别序列形式并记录转换。
12. Recapitulation of Essential Theorems | 核心定理总结
The essential theorems for CCEA Year 13 Music can be summarised as a mental checklist. All chords derive from stacked thirds; chord quality depends on the precise semitone pattern. Functional harmony rests on the dominant–tonic polarity; the V–I cadence is the strongest closure formula. Voice leading demands avoidance of parallel perfect intervals and proper resolution of tendency tones. Modulation exploits common chords and the circle of 5ths. Melodic decoration adds expressive tension within the rhythmic and harmonic framework. Post-tonal music replaces function with interval-class symmetry and ordered transformation. A firm grasp of these interrelated formulae transforms score analysis from guesswork into a precise, methodical process.
CCEA Year 13 音乐的核心定理可总结为一份脑内核查清单。所有和弦源于三度叠置；和弦性质取决于精确的半音模式。功能和声植根于属-主极性；V–I 终止式是最强的收束公式。声部进行要求避免平行纯音程并正确解决倾向音。转调利用共同和弦与五度循环。旋律装饰在和声节奏框架内增添表现张力。后调性音乐以音程对称性与有序转换取代功能。牢固掌握这些相互关联的公式，能将乐谱分析从猜测转变为精确、系统化的过程。
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