Pre-U Cambridge Music: Quick Reference Handbook of Formulas and Theorems | Pre-U 剑桥音乐：公式定理速查手册

📚 Pre-U Cambridge Music: Quick Reference Handbook of Formulas and Theorems | Pre-U 剑桥音乐：公式定理速查手册

Music theory often operates like a precise language of sonic mathematics – intervals measured in semitones, chord structures built from stacked thirds, and harmonic progressions governed by voice‑leading axioms. This quick reference handbook collects the essential ‘formulas’ and ‘theorems’ you will need for the Pre‑U Cambridge Music examination. Use it to reinforce aural skills, written harmony, and analytical precision across all set works and unfamiliar scores.

音乐理论常常像一门精确的声音数学——以半音为单位的音程计算、由三度叠置构成的和弦结构、以及受声部进行公理支配的和声进行。这份速查手册汇集了 Pre‑U 剑桥音乐考试所需的核心“公式”与“定理”。用它来加强听力、书面和声技巧，以及对所有指定作品和新谱例分析中的准确性。

1. Interval Formulas | 音程公式

An interval is defined by its quality (perfect, major, minor, augmented, diminished) and its number (unison through octave). The number of semitones provides the absolute ‘formula’. Memorise the following semitone‑count reference table for all simple intervals within one octave.

音程由性质（纯、大、小、增、减）和度数（同度到八度）定义。半音数量提供了绝对的“公式”。记住以下一个八度内所有简单音程的半音数参考表。

Interval Name	Abbreviation	Semitones	中文名称
Perfect Unison	P1	0	纯一度
Minor Second	m2	1	小二度
Major Second	M2	2	大二度
Minor Third	m3	3	小三度
Major Third	M3	4	大三度
Perfect Fourth	P4	5	纯四度
Augmented Fourth / Diminished Fifth	A4 / d5	6	增四度 / 减五度
Perfect Fifth	P5	7	纯五度
Minor Sixth	m6	8	小六度
Major Sixth	M6	9	大六度
Minor Seventh	m7	10	小七度
Major Seventh	M7	11	大七度
Perfect Octave	P8	12	纯八度

Theorem of interval inversion: The sum of the interval numbers always equals 9, and qualities invert as follows – major ↔ minor, augmented ↔ diminished, perfect remains perfect. For example, a M6 inverts to a m3.

音程转位定理：音程度数之和恒为 9，性质转位规则如下：大↔小，增↔减，纯保持纯。例如大三度转位为小六度，大六度转位为小三度。

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2. Scale Construction Formulas | 音阶构建公式

Every scale is a sequence of tones (T) and semitones (S). The major scale pattern acts as the template from which all other diatonic modes and minor scales are derived. The formula for a major scale ascending is: T – T – S – T – T – T – S.

每种音阶都是全音 (T) 和半音 (S) 的序列。大调音阶模式是所有其他自然调式和大小调音阶的模板。大调音阶上行公式为：全 – 全 – 半 – 全 – 全 – 全 – 半。

Harmonic minor scale formula: T – S – T – T – S – A2 – S, where A2 is an augmented second (3 semitones) between scale degrees ♭6 and ♮7.

和声小调公式：全 – 半 – 全 – 全 – 半 – 增二度 – 半，其中增二度（3个半音）位于 ♭6 级与 ♮7 级之间。

Melodic minor scale ascending: T – S – T – T – T – T – S (raised 6th and 7th); descending follows natural minor: T – T – S – T – T – S – T.

旋律小调上行：全 – 半 – 全 – 全 – 全 – 全 – 半（升六级和七级）；下行同自然小调：全 – 全 – 半 – 全 – 全 – 半 – 全。

Whole‑tone scale formula: T – T – T – T – T – T. Chromatic scale is simply all 12 semitones. The octatonic scale alternates T‑S or S‑T.

全音音阶公式：全 – 全 – 全 – 全 – 全 – 全。半音音阶即所有12个半音。八声音阶以全‑半或半‑全交替排列。

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3. Chord Construction Theorems | 和弦构成定理

A triad is built by stacking two thirds above a root. The quality depends on the combination of those thirds. Major triad = M3 + m3; minor triad = m3 + M3; diminished triad = m3 + m3; augmented triad = M3 + M3.

三和弦由根音上方叠置两个三度构成。性质取决于这些三度的组合。大三和弦 = 大三度 + 小三度；小三和弦 = 小三度 + 大三度；减三和弦 = 小三度 + 小三度；增三和弦 = 大三度 + 大三度。

Seventh chords add another third on top. Dominant 7th = major triad + m7 (M3 + m3 + m3). Major 7th = major triad + M7. Minor 7th = minor triad + m7. Half‑diminished 7th = diminished triad + m7 (m3 + m3 + M3). Fully diminished 7th = diminished triad + diminished 7th (m3 + m3 + m3). See the semitone stack formulas.

七和弦再叠加一个三度。属七和弦 = 大三和弦 + 小七度 (大三+小三+小三)。大七和弦 = 大三和弦 + 大七度。小七和弦 = 小三和弦 + 小七度。半减七和弦 = 减三和弦 + 小七度 (小三+小三+大三)。全减七和弦 = 减三和弦 + 减七度 (小三+小三+小三)。留意半音叠置公式。

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4. Chord Inversion and Figured Bass Formulas | 和弦转位与数字低音公式

Inversion changes the bass note. Root position triads use bass note = root. First inversion = 3rd in bass; second inversion = 5th in bass. Figured bass shorthand: root position ⁵₃ (often omitted), first inversion ⁶₃ or simply ⁶, second inversion ⁶₄.

转位改变低音。原位三和弦低音为根音。第一转位 = 三音在低音；第二转位 = 五音在低音。数字低音简写：原位 ⁵₃（常省略），第一转位 ⁶₃ 或简写 ⁶，第二转位 ⁶₄。

For seventh chords: root position ⁷, first inversion ⁶₅, second inversion ⁴₃, third inversion ⁴₂ or simply ². These numbers represent the intervals above the bass note.

七和弦：原位 ⁷，第一转位 ⁶₅，第二转位 ⁴₃，第三转位 ⁴₂ 或简写 ²。这些数字代表低音上方的音程。

Realising figured bass: the bass is given; you supply upper voices according to the intervals indicated. For example, a ⁶ under a bass note means you must add a 6th and a 3rd above that bass (first inversion triad). When no figures appear, a root‑position ⁵₃ triad is assumed.

数字低音的实现：低音已给；你根据标示的音程填充上方声部。例如低音下方的 ⁶ 表示你需在低音上方加上六度和三度（第一转位三和弦）。无数字标记时默认原位 ⁵₃ 三和弦。

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5. Diatonic Triad and Seventh Chord Tables | 自然音阶三和弦与七和弦表

Harmonising the major scale yields a fixed pattern of chord qualities. The theorem: chords built on each degree of a major scale follow the sequence I (major), ii (minor), iii (minor), IV (major), V (major), vi (minor), vii° (diminished). This is often written as M, m, m, M, M, m, dim.

大调音阶的和声化产生固定的和弦性质模式。定理：大调音阶每个音级上构成的和弦按顺序为 I (大), ii (小), iii (小), IV (大), V (大), vi (小), vii° (减)。常记作 大, 小, 小, 大, 大, 小, 减。

Seventh chords on a major scale: Imaj7, ii7, iii7, IVmaj7, V7, vi7, viiø7 (half‑diminished). The harmonic minor scale raises the 7th degree, altering dominant and leading‑tone chords. V becomes major (or V7) and vii° becomes diminished (or vii°7).

大调音阶上的七和弦：Imaj7, ii7, iii7, IVmaj7, V7, vi7, viiø7 (半减七)。和声小调升高第VII级，改变属和弦与导音和弦。V 变成大三和弦（或 V7），vii° 变成减三和弦（或 vii°7）。

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6. Cadence Formulas | 终止式公式

A cadence is a harmonic punctuation formula. Perfect cadence: V – I (or V⁷ – I) in root position, both chords in root position, soprano ends on tonic. Imperfect cadence: any chord (often I, ii, IV) proceeding to V. Plagal cadence: IV – I. Interrupted (deceptive) cadence: V – vi (in minor, V – VI).

终止式是和声的标点公式。正格终止：V – I（或 V⁷ – I），两和弦均为原位，女高音结束在主音。不完全终止：任何和弦（常为 I, ii, IV）进行到 V。变格终止：IV – I。阻碍终止：V – vi（小调中为 V – VI）。

The Phrygian half cadence is a special formula in minor: iv⁶ – V. The bass moves by descending semitone from ♭6 to 5. This creates a distinctive darkened cadence favoured in Baroque and Romantic styles.

弗里几亚半终止是小调中的特殊公式：iv⁶ – V。低音从 ♭6 级下行半音到 5 级。这在巴洛克和浪漫主义风格中营造出独特的阴暗终止。

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7. Harmonic Progression Theorems | 和声进行定理

The Circle of Fifths progression is a fundamental theorem of tonal music: chords move down by perfect fifths (or up by perfect fourths). A typical cycle in major: iii – vi – ii – V – I – IV – vii° – iii… Each step strengthens tonic direction.

五度循环进行是调性音乐的基本定理：和弦以纯五度下行（或纯四度上行）移动。大调中典型循环：iii – vi – ii – V – I – IV – vii° – iii… 每一步都增强主音方向。

Functional harmony theorem: chords are grouped into tonic (I, vi, iii), subdominant (IV, ii, vi) and dominant (V, vii°) families. Progressions typically move T – S – D – T, avoiding retrogression (D – S) unless stylistically justified.

功能和声定理：和弦分为主功能组 (I, vi, iii)、下属功能组 (IV, ii, vi)、属功能组 (V, vii°)。进行通常为 T – S – D – T，避免逆行 (D – S) 除非风格所需。

Voice‑leading axioms: common tones should be retained when possible; avoid parallel fifths and octaves; leading tone (7th degree) must resolve up to tonic; seventh of a chord resolves down by step.

声部进行公理：尽可能保持共同音；避免平行五度和八度；导音（第VII级）必须上行解决到主音；和弦七音级进下行解决。

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8. Time Signature & Rhythmic Formulas | 拍号与节奏公式

Time signatures express meter as a fraction theorem: top number = beats per bar, bottom number = beat unit (e.g., 4 = crotchet, 8 = quaver). Simple time beats divide into two; compound time beats divide into three. Compound signatures: beats are dotted notes, so 6/8 = two dotted crotchet beats.

拍号以分数定理表示节拍：上方数字 = 每小节拍数，下方数字 = 拍子单位（如 4 代表四分音符，8 代表八分音符）。单拍子每拍一分为二；复拍子每拍一分为三。复拍号：拍子是附点音符，故 6/8 拍为两个附点四分音符拍。

Rhythmic formula for swing or shuffle: written as even quavers but performed with triplet feel, first quaver gets 2/3 of the beat, second gets 1/3. Can be notated as pairs of quavers within a triplet bracket with the first two tied, or with an instruction ‘swing’.

摇摆或曳步节奏公式：记谱为均匀八分音符，但演奏时使用三连音感觉，第一个八分音符占拍子的 2/3，第二个占 1/3。可记谱为三连音括号内前两个音用连线连接，或标注“swing”。

For rhythmic dictation, subdivide beats into the smallest note value required; use the metronome beat theorem: tempo marking ♩=120 means 120 crotchets per minute. Semiquavers at this tempo pass at 480 notes per minute; apply the formula beat duration = 60 ÷ bpm.

节奏听写时，将每拍细分为所需的最小时值；运用节拍器节拍定理：速度标记 ♩=120 表示每分钟120个四分音符。此时十六分音符速度为480个/分钟；应用公式 拍长 = 60 ÷ bpm。

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9. Transposition Formulas for Transposing Instruments | 移调乐器移调公式

Many orchestral instruments are ‘in’ a key: the written pitch differs from sounding pitch. The theorem: the interval of transposition equals the difference between the instrument’s key and C. For a B♭ clarinet: written C sounds as B♭ a major second lower. To find written pitch from concert pitch, transpose up a major second.

许多管弦乐器是“移调乐器”：记谱音高与实际音高不同。定理：移调音程等于乐器调性与 C 的差值。对于 B♭ 单簧管：记谱 C 实际发音为下方大二度的 B♭。若从实际音高求记谱音高，则向上移高大二度。

Formula summary by instrument: B♭ instruments (clarinet, trumpet) – sound a M2 lower than written; F instruments (French horn, cor anglais) – sound a P5 lower than written; E♭ instruments (alto saxophone, Eb clarinet) – sound a M6 lower (alto sax) or m3 higher (Eb clarinet). For octave‑transposing instruments (piccolo, double bass, guitar) the sounding pitch is shifted by one or two octaves.

乐器移调公式总结：B♭ 乐器（单簧管、小号）– 发音比记谱低大二度；F 乐器（法国号、英国管）– 发音比记谱低纯五度；E♭ 乐器（中音萨克斯、降E单簧管）– 中音萨克斯低大六度，降E单簧管高小三度。八度移调乐器（短笛、低音提琴、吉他）发音移动一个或两个八度。

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10. Thematic & Form Analysis Theorems | 主题与曲式分析定理

Sonata form theorem: Exposition presents first subject (tonic), transition, second subject (dominant or relative major), closing section; Development explores and fragments themes; Recapitulation returns both subjects in tonic. Essential key relationships form the structural formula.

奏鸣曲式定理：呈示部呈现第一主题（主调）、过渡、第二主题（属调或关系大调）、结束段；展开部探索并分裂主题；再现部将两个主题都回到主调。核心的调性关系构成了结构公式。

Binary form formula: ||: A :||: B :|| where A modulates to a related key and B returns to tonic. Rounded binary: ||: A :||: B A’ :||. Ternary form: A – B – A with contrasting middle and exact or varied return. Rondo formulas: ABACA, ABACABA, etc.

二段式公式：||: A :||: B :||，其中 A 转至关系调，B 返回主调。再现二段式：||: A :||: B A’ :||。三段式：A – B – A，中部对比，再现相同或变化。回旋曲公式：ABACA, ABACABA 等。

Fugue theorem: Exposition states subject in each voice (tonic‑dominant‑tonic‑dominant in a 4‑voice fugue), followed by episodes (modulatory passages) and middle entries in related keys. Countersubject accompanies the subject in invertible counterpoint. Final entry returns firmly to tonic.

赋格定理：呈示部在每个声部分别陈述主题（四声部赋格中为 主‑属‑主‑属 顺序），接着是间插段（转调段落）和关系调上的中间进入。对题以可逆对位伴随主题。最后进入牢固地回归主调。

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11. Ornament Realisation Formulas | 装饰音实现公式

Baroque and Classical ornaments follow defined rhythm‑melodic models. The trill (tr) starts on the upper auxiliary note and alternates rapidly before finishing with a turn or anticipation. The number of alternations depends on tempo; a trill formula: start on the beat, upper note, then as many alternations as the note value allows, often ending with a lower auxiliary suffix.

巴洛克和古典装饰音遵循确定的节奏‑旋律模型。颤音 (tr) 从上助音开始快速交替，最后以下波音或先现音结束。交替次数取决于速度；颤音公式：从拍点上开始，上助音，然后在音符时值允许内尽可能多交替，常以下二度助音后缀收束。

The turn (𝄐) places the note itself, upper auxiliary, note, lower auxiliary, note in quick succession. Formula for a turn between two notes: the turn is played within the value of the principal note, starting with the upper auxiliary. The upper mordent (𝄒) plays note, upper auxiliary, note; lower mordent (𝄞) plays note, lower auxiliary, note.

回音 (𝄐) 快速演奏本音‑上助音‑本音‑下助音‑本音。两音之间的回音公式：回音在主音时值内演奏，以上助音开始。上波音 (𝄒) 演奏本音‑上助音‑本音；下波音 (𝄞) 演奏本音‑下助音‑本音。

Appoggiatura formula: the dissonant auxiliary takes half the value of a duple note, or two‑thirds of a triple note, and resolves by step onto the consonant harmony note. The acciaccatura is a crushed, very short dissonance played simultaneously with the main note.

倚音公式：不协和助音占二拍子音符时值的一半，或三拍子音符的三分之二，然后级进解决到协和和弦音。碎音是与主音同时奏出的极短的不协和音。

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12. Chromatic Harmony Quick‑Theorems | 半音化和声速成定理

Secondary dominants formula: any diatonic chord (except vii°) can be preceded by its own dominant. V/ii (major triad on scale degree 6 in major) resolves to ii. V/IV, V/V, V/vi are common. In minor keys, V/III and V/VI create rich expressive colours.

副属和弦公式：任何自然音和弦（vii° 除外）前可加其自身的属和弦。V/ii（大调中音阶第六级上的大三和弦）解决到 ii。V/IV, V/V, V/vi 常见。小调中，V/III 和 V/VI 创造丰富的表情色彩。

The diminished seventh chord theorem: a fully diminished seventh chord can resolve in four different keys, moving any of its notes up a semitone to become the root of a new chord. This equal division of the octave (four m3s) makes it the ultimate enharmonic pivot.

减七和弦定理：全减七和弦可以解决到四个不同的调，将其任一音上行半音即成为新和弦的根音。这种八度的等分（四个小三度）使其成为最终的等音转调枢纽。

Neapolitan sixth formula: a major triad built on the flattened supertonic (♭II) in first inversion, used as a pre‑dominant chord, usually progressing to V. In C major: D♭ major triad in first inversion (F–D♭–A♭). Augmented sixth chords (Italian, French, German) resolve outwards to the dominant note or tonic ⁶₄; each contains an augmented sixth interval expanding to an octave.

那波里六和弦公式：降二级音上构成的大三和弦第一转位 (♭II⁶)，用作前属和弦，通常进行到 V。C大调中：降D大三和弦第一转位 (F–D♭–A♭)。增六和弦（意大利、法国、德国）向外解决到属音或主四六；每个包含增六度音程扩展为八度。

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