📚 Pre-U AQA Music: Formula & Theorem Quick Reference Handbook | 公式定理速查手册
Music is often described as a language of emotion, but at its core lies a structure governed by precise mathematical relationships, interval ratios, and syntactic rules. This quick reference handbook compiles the essential 'formulas' and 'theorems' of Western tonal music as required by the AQA Pre-U Music specification. From interval construction to harmonic progression, each entry provides a concise definition and practical application. Whether you are analysing a Bach chorale or composing a sonata-form movement, these formulas will serve as your reliable toolkit.
音乐常被誉为情感的语言，但其核心建立在由精确的数学关系、音程比例和句法规则支配的结构之上。本速查手册汇编了AQA Pre-U音乐大纲所要求的西方调性音乐中的基本“公式”与“定理”。从音程构建到和声进行，每一条均提供简明定义和实际应用。无论你在分析巴赫众赞歌还是创作奏鸣曲乐章，这些公式都将是你可靠的实用工具箱。
1. The Interval Formula | 音程公式
An interval is the distance between two pitches, measured in semitones and named by its ordinal number (size) and quality. The formula for naming any interval is: Quality + Ordinal Number the actual semitones; compare with the perfect/major versions from the diatonic series: Perfect (unison, 4th, 5th, octave) and Major (2nd, 3rd, 6th, 7th). If semitone count matches major/perfect, it is major/perfect; if one semitone wider, augmented; if one semitone narrower, minor (for 2nd, 3rd, 6th, 7th) or diminished (for perfect intervals).
音程是两个音高之间的距离，以半音数计量，并通过序数（度数）和性质命名。命名任何音程的公式为：性质 + 序数。首先通过字母音名包含两端来获取序数（例如C到E为三度）。然后测量实际半音数，与自然音阶中的纯音程（同度、四度、五度、八度）和大音程（二度、三度、六度、七度）比较。若半音数与大/纯音程匹配，即为大/纯音程；若增宽一个半音，为增音程；若缩减一个半音，则为小音程（针对二、三、六、七度）或减音程（针对纯音程）。
2. Major Scale Formula | 大调音阶公式
The major scale is built from a fixed pattern of tones (T) and semitones (S): T T S T T T S this pattern to generate the seven diatonic notes. For example, D major: D (T) → E (T) → F♯ (S) → G (T) → A (T) → B (T) → C♯ , 6 (submediant), 7 (leading note). The semitone steps always fall between scale degrees 3–4 and 7–8.
大调音阶由固定的全音（T）和半音（S）模式构建：T T S T T T S。从任一主音出发，应用此模式生成七个自然音级。例如D大调：D (T)→ E (T)→ F♯ (S)→ G (T)→ A (T)→ B (T)→ C♯ (S)→ D。音级依次标记为1级（主音）、2级（上主音）、3级（中音）、4级（下属音）、5级（属音）、6级（下中音）、7级（导音）。半音步总是落在第3–4级与第7–8级之间。
3. Minor Scale Formulas | 小调音阶公式
The natural minor scale follows the pattern: T S T T S T T ented 2nd between degrees 6 and 7; its pattern is T S T T S T+S S (where T+S = 3 semitones). The melodic minor (ascending) raises both 6th and 7th degrees by a semitone to smooth the leap; its pattern is T S T T T T S. Descending melodic minor usually reverts to natural minor. Each formula is essential for recognising key signatures and constructing chords in minor keys.
自然小调遵循模式：T S T T S T T，等同于爱奥利亚调式。和声小调将第7级升高半音以形成导音，从而在第6与第7级间产生特色的增二度；其模式为T S T T S T+S S（T+S = 3个半音）。旋律小调（上行）同时升高第6和第7级以平滑跳跃；其模式为T S T T T T S。下行旋律小调通常回归自然小调。每条公式对于识别调号与构建小调内的和弦都至关重要。
4. Chord Construction Formula – Triads | 三和弦构建公式
A triad is formed by stacking two thirds above a root. The interval formula for a major triad: root + major 3rd (4 semitones) + perfect 5th (7 semitones). Minor triad: root + minor 3rd (3 semitones) + perfect 5th. Diminished triad: root + minor 3rd + diminished 5th (6 semitones). Augmented triad: root + major 3rd + augmented 5th (8 semitones). In Roman numeral analysis, uppercase indicates major (I), lowercase minor (ii), lowercase with ° diminished (vii°), uppercase with + augmented (V+). The diatonic triads in major keys follow the pattern: I, ii, iii, IV, V, vi, vii°.
三和弦由在根音上方叠加两个三度构成。大三和弦的音程公式：根音 + 大三度（4个半音）+ 纯五度（7个半音）。小三和弦：根音 + 小三度（3个半音）+ 纯五度。减三和弦：根音 + 小三度 + 减五度（6个半音）。增三和弦：根音 + 大三度 + 增五度（8个半音）。在罗马数字分析中，大写表示大三（I），小写表示小三（ii），小写加°表示减三（vii°），大写加+表示增三（V+）。大调中的自然三和弦遵循模式：I, ii, iii, IV, V, vi, vii°。
5. Seventh Chord Formulas | 七和弦公式
Seventh chords add a 7th interval above the root to a triad. The five common qualities are defined by their constituent intervals: Major seventh (maj7): root + M3 + P5 + M7. Dominant seventh (7): root + M3 + P5 + m7. Minor seventh (m7): root + m3 + P5 + m7. Half-diminished (m7♭5): root + m3 + d5 + m7. Diminished seventh (dim7): root + m3 + d5 + d7 (equivalent to d7 = 9 semitones). The dominant 7th chord is the most crucial functional chord, containing a tritone between its 3rd and 7th that demands resolution.
七和弦在三和弦基础上叠加一个七度音程。五种常见性质通过其内部音程定义：大七和弦 (maj7)：根音 + 大三度 + 纯五度 + 大七度。属七和弦 (7)：根音 + 大三度 + 纯五度 + 小七度。小七和弦 (m7)：根音 + 小三度 + 纯五度 + 小七度。半减七和弦 (m7♭5)：根音 + 小三度 + 减五度 + 小七度。减七和弦 (dim7)：根音 + 小三度 + 减五度 + 减七度（相当于减七度 = 9个半音）。属七和弦是最关键的功能和弦，其三音与七音间包含一个三全音，具有强烈的解决倾向。
6. Chord Inversion Formula | 和弦转位公式
Inversions are determined by the lowest-sounding note (the bass). The formula for naming inversions uses figured bass: Root position: 5/3 (often omitted); First inversion: 6/3 (or simply 6), the 3rd is in the bass; Second inversion: 6/4, the 5th is in the bass. For seventh chords: root position (7), first inversion (6/5), second inversion (4/3), third inversion (4/2 or simply 2). The intervals in the figures indicate the notes above the bass (e.g. a 6/4 chord has a 6th and a 4th above its bass note). Knowing this formula allows precise harmonic analysis.
转位由最低音（低音）决定。命名转位的公式使用数字低音标记：原位：5/3（常省略）；第一转位：6/3（或简写6），三音在低音部；第二转位：6/4，五音在低音部。对于七和弦：原位（7），第一转位（6/5），第二转位（4/3），第三转位（4/2或简写2）。数字中的音程表示低音上方的音高（例如6/4和弦表示低音上方有六度和四度音）。掌握这条公式能进行精确的和声分析。
7. Cadence Formulas | 终止式公式
Cadences are two-chord progressions that close a musical phrase. The four archetypal cadence formulas are: Perfect cadence (V–I): both chords in root position, with the tonic in the soprano of the I chord, creates a strong, conclusive close. Plagal cadence (IV–I): often the 'Amen' cadence. Imperfect cadence (I/ii/IV/vi – V): ends on V, leaving an open, expectant feeling. Interrupted cadence (V–vi): a deceptive resolution, substituting vi for the expected I. The soprano movement and bass line determine the effect: perfect closes require the leading note to rise to tonic, reinforcing the tonal centre.
终止式是收束乐句的两个和弦进行。四种原型终止式公式为：正格终止 (V–I)：两个和弦均处原位，I 和弦的高音部为主音，营造强烈收束感。变格终止 (IV–I)：常为“阿们”终止。半终止 (I/ii/IV/vi – V)：结束于V，留下开放、期待的感觉。阻碍终止 (V–vi)：虚假解决，以vi代替预期的I。高音进行与低音线条决定效果：完全收束要求导音上行至主音，巩固调性中心。
8. Harmonic Progression Formula | 和声进行公式
Tonal harmony generally follows a strong directional pull from tonic through pre-dominant to dominant and back to tonic. The circle-of-fifths progression formula is one of the most fundamental: I – IV – vii° – iii – vi – ii – V – I. Root movement by descending 5th (or ascending 4th) is the strongest harmonic progression. Common patterns include: I – vi – IV – V (50s progression), ii – V – I (jazz turnaround), I – V – vi – IV. In minor keys, the formulas adapt: i – iv – V – i, or i – VI – iv – V. Cadential 6/4 (I6/4–V–I) is a standard expansion of the cadential dominant.
调性和声通常遵循从主和弦经下属功能到属功能再回到主和弦的强烈方向性拉力。五度循环进行公式是最基础的之一：I – IV – vii° – iii – vi – ii – V – I。根音作下行五度（或上行四度）运动是最强的和声进行。常见模式包括：I – vi – IV – V（1950年代进行），ii – V – I（爵士终止式），I – V – vi – IV。在小调中，公式相应调整：i – iv – V – i，或 i – VI – iv – V。终止四六和弦 (I6/4–V–I) 是属终止的标准扩展形式。
9. Modulation Formula – Pivot Chords | 转调公式 – 共同和弦
y keys that differ by no more than one sharp or flat in the key signature (e.g. C major to G major, or A minor to D minor). Select a chord that functions as a pre-dominant or dominant in the new key and resolve accordingly. Common pivot chords: I in the old key = IV in the new key (when modulating to the dominant); vi in major key = i in relative minor. The pivot chord must be clearly established in the old key before reinterpretation, and the new key must be confirmed by a perfect cadence.
在近关系调之间转调时，使用一个在旧调与新调中均为自然音级的共同和弦。平滑转调的公式是：选择调号相差不超过一个升号或降号的调性（如C大调转G大调，或a小调转d小调）。选一个在新调中充当下属或属功能的和弦并相应解决。常见共同和弦：旧调的I = 新调的IV（转调至属调时）；大调中的vi = 关系小调的i。共同和弦必须在旧调明确建立后方可重新诠释，且新调须通过正格终止予以确认。
10. Rhythm Division Formula | 节奏分割公式
Meter is a regular grouping of beats, and the formula for subdividing beats depends on time signature. In simple time (e.g. 2/4, 3/4, 4/4), the beat is divided into two equal parts: 1 beat = 2 eighth notes. In compound time (e.g. 6/8, 9/8, 12/8), the beat is a dotted note divided into three equal parts: 1 beat = 3 eighth notes. Dotted rhythms, triplets, and duplets alter the standard division. A triplet in simple time fits three notes into the time of two; a duplet in compound time fits two notes into the time of three. Syncopation displaces accents to weak beats or off-beats, governed by the formula: anticipation + accent on a normally unaccented subdivision.
节拍是有规律的拍子分组，节拍分割公式取决于拍号。在单拍子中（如2/4、3/4、4/4），一拍被均分为两部分：1拍 = 2个八分音符。在复拍子中（如6/8、9/8、12/8），一拍是一个附点音符，均分为三部分：1拍 = 3个八分音符。附点节奏、三连音和二连音改变标准划分。单拍子中的三连音在原本两拍的时间内包含三个音符；复拍子中的二连音在原本三拍的时间内包含两个音符。切分音将重音移至弱拍或弱位，其公式为：预现 + 在通常不加重音的位置出现重音。
11. Species Counterpoint Rules – The First Species Formula | 对位法规则 – 第一类对位公式
In first species (note-against-note) counterpoint, every note in the added voice (contrapuntal line) is a perfect consonance (unison, 8ve, 5th) or imperfect consonance (3rd, 6th) against the cantus firmus. The formula for a correct succession: begin and end on perfect consonances (tonic or octave); prefer contrary and oblique motion; avoid parallel perfect 5ths and 8ves; unisons may only be used at the beginning and end. The rule of ‘no similar motion to a perfect interval’ (hidden 5ths/8ves) except between inner and an upper voice, applies rigorously. Stepwise motion is preferred, with leaps compensated by opposite stepwise movement.
在第一类对位（音对音）中，附加声部（对位旋律）的每个音相对定旋律必须是纯协和音程（同度、八度、五度）或不完全协和音程（三度、六度）。正确接续的公式为：开始与结束必须用纯协和音程（主音或八度）；优先采用反向和斜向进行；避免平行纯五度与平行八度；同度仅可用于开头与结尾。除内声部与上声部外，严格禁止同向进行进入纯音程（隐伏五度/八度）。级进优先，跳进需用反向级进予以平衡。
12. Sonata Form Blueprint | 奏鸣曲式结构公式
Sonata form, the most important large-scale structure from the Classical period, consists of three main sections. The formula: Exposition: (First subject in tonic) – transition (modulating) – Second subject (in dominant or relative major) – codetta. Development: thematic fragments from exposition, moving through distant keys, building harmonic tension. Recapitulation: First subject (tonic) – transition (altered to stay in tonic) – Second subject (tonic) – coda. The essential theorem is that the tonal conflict set up in the exposition (two contrasting themes in different keys) is resolved in the recapitulation, where both themes appear in the tonic. An introduction and coda may frame the movement.
奏鸣曲式是古典时期最重要的大型结构，包含三个主要部分。其公式为：呈示部：（主调第一主题）– 连接段（转调）– 第二主题（在属调或平行大调）– 小结尾。展开部：来自呈示部的主题片段，游离于远关系调，累积和声张力。再现部：第一主题（主调）– 连接段（修改以保持在主调）– 第二主题（主调）– 尾声。其中的核心定理是，呈示部所建立的两个不同调性中的对比主题之间的调性冲突，在再现部中通过将两个主题均置于主调而得以解决。引子和尾声可框定整个乐章。
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