📚 Pre-U CCEA Music: Essential Formulas and Theorems Quick Reference | Pre-U CCEA 音乐：公式定理速查手册
This handbook distils the core formulas, theorems and structural rules required for the CCEA Pre-U Music course. It covers acoustics, pitch organisation, harmonic syntax, formal archetypes and voice-leading conventions, serving as a rapid revision aid for both the written and aural components of the examination.
本手册浓缩了 CCEA Pre-U 音乐课程的核心公式、定理与结构规则，涵盖声学基础、音高组织、和声句法、曲式原型与声部进行惯例，可作为笔试与听觉考核的快速复习工具。
1. Sound and Frequency Equations | 声音与频率方程
Sound is a longitudinal pressure wave. The frequency f (in Hz) is the number of complete cycles per second and is the reciprocal of the period T: f = 1/T. The wavelength λ is related to frequency and the speed of sound v (approximately 340 m s⁻¹ in air at 20 °C) by the formula λ = v / f. Higher frequency yields shorter wavelength, and the human ear typically perceives frequencies from 20 Hz to 20 000 Hz.
声音是一种纵波。频率 f（单位 Hz）为每秒完整循环的次数，是周期 T 的倒数：f = 1/T。波长 λ 与频率及声速 v（20 °C 空气中约为 340 m s⁻¹）的关系为 λ = v / f。频率越高波长越短，人耳通常可感知 20 Hz 到 20 000 Hz 的频率。
2. Equal Temperament and Pitch Formula | 十二平均律与音高公式
In the modern equal‑tempered system the octave is divided into twelve semitones of identical frequency ratio. A note n semitones above a reference pitch f₀ has the frequency fₙ = f₀ × 2^(n/12). For example, if A₄ = 440 Hz, then the A an octave higher (n = +12) is 440 × 2^(12/12) = 880 Hz. A semitone above A₄ is B♭₄ ≈ 440 × 2^(1/12) ≈ 466.16 Hz. The same multiplier applies for descending intervals by using negative values of n.
在现代十二平均律中，一个八度被均分为十二个频率比相同的半音。比基准音高 f₀ 高 n 个半音的音符，其频率 fₙ = f₀ × 2^(n/12)。例如，设 A₄ = 440 Hz，则高八度的 A（n = +12）为 440 × 2^(12/12) = 880 Hz。高出 A₄ 一个半音的 B♭₄ ≈ 440 × 2^(1/12) ≈ 466.16 Hz。下行音程用负的 n 值即可。
3. Interval Calculation | 音程计算公式
Intervals are measured in semitones. The table below lists the number of semitones for standard simple intervals (within an octave). Quality is determined by the semitone count relative to the diatonic size.
音程以半音数度量。下表列出了标准单音程（八度以内）的半音数。音程的性质由其相对于自然音级大小的半音数决定。
| Interval Name / 音程名称 | Semitones / 半音数 | Example from C / 从 C 开始的例子 |
|---|---|---|
| Perfect unison / 纯一度 | 0 | C – C |
| Minor second / 小二度 | 1 | C – D♭ |
| Major second / 大二度 | 2 | C – D |
| Minor third / 小三度 | 3 | C – E♭ |
| Major third / 大三度 | 4 | C – E |
| Perfect fourth / 纯四度 | 5 | C – F |
| Augmented fourth / Diminished fifth (tritone) / 增四度 / 减五度 | 6 | C – F♯ / C – G♭ |
| Perfect fifth / 纯五度 | 7 | C – G |
| Minor sixth / 小六度 | 8 | C – A♭ |
| Major sixth / 大六度 | 9 | C – A |
| Minor seventh / 小七度 | 10 | C – B♭ |
| Major seventh / 大七度 | 11 | C – B |
| Perfect octave / 纯八度 | 12 | C – C |
Compound intervals (wider than an octave) are obtained by adding 12 semitones for each additional octave. Inversion of an interval follows the rule: the sum of the two interval numbers equals 9, and qualities invert as major ↔ minor, augmented ↔ diminished, while perfect remains perfect.
复音程（超过八度）只需每增加一个八度加上 12 个半音。音程转位规律：两个音程度数之和为 9；性质互变：大 ↔ 小，增 ↔ 减，纯音程保持为纯。
4. Scale Formulas | 音阶构式
Diatonic scales are built from a pattern of tones (T, whole step) and semitones (S, half step). The table gives the step formulas for the most commonly examined scales in Pre‑U Music.
自然音阶由全音（T）与半音（S）的模式构建。下表为 Pre‑U 音乐中最常见的音阶模式。
| Scale / 音阶 | Formula (ascending) / 上行公式 | Notes from C / 以 C 为主音 |
|---|---|---|
| Major / 大调 | T T S T T T S | C D E F G A B C |
| Natural minor / 自然小调 | T S T T S T T | C D E♭ F G A♭ B♭ C |
| Harmonic minor / 和声小调 | T S T T S 1½ S | C D E♭ F G A♭ B C |
| Melodic minor (ascending) / 旋律小调上行 | T S T T T T S | C D E♭ F G A B C |
| Melodic minor (descending) / 旋律小调下行 | T T S T T S T | C B♭ A♭ G F E♭ D C |
Other modes (e.g. Dorian = T S T T T S T, Mixolydian = T T S T T S T) follow similar tone‑semitone patterns derived from the white‑note series.
其他调式（如多利亚 T S T T T S T、米克索利底亚 T T S T T S T）同样遵循从自然音列衍生的全半音模式。
5. Chord Construction Formulas | 和弦构建公式
Triads and seventh chords are built by stacking intervals of a third. The quality of a chord is determined by the exact semitone distances between the root, third, fifth and, where applicable, seventh.
三和弦与七和弦由三度音程叠置而成，和弦的性质取决于根音、三音、五音及七音（若存在）之间的精确半音距离。
- Major triad / 大三和弦: root + 4 semitones + 3 semitones (e.g. C–E–G, 0–4–7).
- Minor triad / 小三和弦: root + 3 + 4 (e.g. C–E♭–G, 0–3–7).
- Augmented triad / 增三和弦: root + 4 + 4 (e.g. C–E–G♯, 0–4–8).
- Diminished triad / 减三和弦: root + 3 + 3 (e.g. C–E♭–G♭, 0–3–6).
For seventh chords:
七和弦：
- Major seventh / 大七和弦: major triad + major seventh (11 semitones from root) → 0–4–7–11.
- Dominant seventh / 属七和弦: major triad + minor seventh (10 semitones) → 0–4–7–10.
- Minor seventh / 小七和弦: minor triad + minor seventh → 0–3–7–10.
- Half‑diminished seventh / 半减七和弦: diminished triad + minor seventh → 0–3–6–10.
- Diminished seventh / 减七和弦: diminished triad + diminished seventh (9 semitones) → 0–3–6–9.
Roman numeral analysis assigns chords to scale degrees: in a major key, I ii iii IV V vi vii°; in a harmonic minor, i ii° III⁺ iv V VI vii°⁷.
罗马数字分析将和弦对应音级：大调中为 I ii iii IV V vi vii°；和声小调为 i ii° III⁺ iv V VI vii°⁷。
6. Harmonic Progressions and Cadences | 和声进行与终止式定理
Functional harmony organises chords around tonic (T, I/vi), subdominant (S, IV/ii) and dominant (D, V/vii°) categories. Standard progressions move T → S → D → T. The four principal cadences are defined by the chords that articulate phrase endings:
功能和声将和弦组织为主功能（T，I/vi）、下属功能（S，IV/ii）和属功能（D，V/vii°）。标准进行为 T → S → D → T。四种主要终止式由乐句结尾的和弦进行定义：
- Perfect cadence / 正格终止: V – I (or V⁷ – I), gives a strong sense of closure.
- Plagal cadence / 变格终止: IV – I, often referred to as the ‘Amen’ cadence.
- Imperfect cadence / 半终止: any chord moving to V (e.g. I–V, ii–V, IV–V), creates an expectation of continuation.
- Interrupted (deceptive) cadence / 阻碍终止: V – vi (or V – ♭VI), frustrates the expected tonic resolution.
In tonal composition voice‑leading conventions further constrain how chords connect; these are addressed in Section 10.
在调性写作中，声部进行惯例进一步约束了和弦的连接方式。详见第 10 节。
7. Rhythm and Metre Formulas | 节奏与节拍计算
Time signatures are classified as simple or compound. In simple time (e.g. 2/4, 3/4, 4/4) the beat divides into two equal parts; in compound time (e.g. 6/8, 9/8, 12/8) the beat is a dotted note that divides into three equal parts. Duple, triple and quadruple refer to the number of beats per bar.
拍号分为单拍子与复拍子。单拍子（如 2/4、3/4、4/4）每拍可均分为两个部分；复拍子（如 6/8、9/8、12/8）以附点音符为一拍，每拍均分为三个部分。二拍子、三拍子与四拍子指每小节的拍数。
Dotted notes and ties alter duration: a single dot adds half the original value; a double dot adds half of that half (i.e. three‑quarters of the original). For example, a dotted crotchet equals a crotchet + a quaver; a double‑dotted crotchet equals a crotchet + a quaver + a semiquaver. Syncopation shifts accent to a normally weak beat or off‑beat, often notated by ties or rests.
附点与连线改变时值：单附点增加原时值的一半；双附点再增加其一半的一半（即原时值的四分之三）。例如，附点四分音符 = 四分音符 + 八分音符；双附点四分音符 = 四分音符 + 八分音符 + 十六分音符。切分音将重音移至弱拍或弱位，常通过连线或休止符记号实现。
8. Circle of Fifths and Key Signatures | 五度圈与调号定理
The circle of fifths organises the twelve major and twelve minor keys by ascending (clockwise) or descending (anticlockwise) perfect fifths. Each step clockwise adds one sharp or removes one flat; each step anticlockwise adds one flat or removes one sharp. The order of sharps is F♯ C♯ G♯ D♯ A♯ E♯ B♯; the order of flats is the reverse: B♭ E♭ A♭ D♭ G♭ C♭ F♭.
五度圈按纯五度上行（顺时针）或下行（逆时针）排列十二大调与十二小调。顺时针每步增加一个升号或减少一个降号；逆时针每步增加一个降号或减少一个升号。升号出现的顺序为 F♯ C♯ G♯ D♯ A♯ E♯ B♯；降号顺序与之相反：B♭ E♭ A♭ D♭ G♭ C♭ F♭。
Relative minors share the same key signature and lie a minor third below their relative major. For a given major key, the tonic of its relative minor is the sixth degree; for a minor key, its relative major is the third degree. For example, C major (no sharps/flats) ↔ A minor; G major (1 sharp) ↔ E minor; F major (1 flat) ↔ D minor.
关系小调与关系大调共用同一调号，且位于其关系大调下方小三度。已知大调，其关系小调主音为六级音；已知小调，其关系大调主音为三级音。例如，C 大调（无升降号）↔ a 小调；G 大调（1 升）↔ e 小调；F 大调（1 降）↔ d 小调。
9. Musical Form and Structure Theorems | 曲式与结构定理
Form refers to the large‑scale architecture of a work. The most common frameworks examined at Pre‑U level include:
曲式指作品的宏观结构。Pre‑U 阶段考核最常见的框架包括：
- Binary form / 二部曲式: AB, each section usually repeated; the B section moves to a related key and then returns to the tonic.
- Ternary form / 三部曲式: ABA, with a contrasting middle section and a return of the opening material, often varied.
- Rondo form / 回旋曲式: ABACA (or ABACABA), where a principal theme (A) alternates with contrasting episodes.
- Sonata form / 奏鸣曲式: Exposition – Development – Recapitulation. The exposition presents a first subject in the tonic and a second subject in a related key (dominant for major keys, relative major or dominant for minor keys). The development explores and fragments thematic material through remote keys. The recapitulation restates both subjects in the tonic, sometimes followed by a coda.
- Theme and variations / 主题与变奏: a theme is stated then repeatedly transformed through changes in melody, harmony, rhythm, texture or orchestration.
In vocal music, strophic, through‑composed and modified strophic forms are frequent, governed by the relationship between identical or changing music and successive stanzas of text.
声乐作品中常见分节歌、通谱歌及变化分节歌等形式，由相同或
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