📚 Pre-U CIE Music: Formula & Theorem Quick Reference Handbook | Pre-U CIE 音乐：公式定理速查手册
This quick-reference handbook distils the essential ‘formulae’ and structural principles required for Pre-U CIE Music. From scale construction and interval arithmetic to harmonic syntax, voice-leading rules, and formal architecture, each section presents a concise rule or model that can be applied analytically or creatively. Mastery of these patterns will strengthen your score-reading, aural perception, and composition skills.
这本速查手册提炼了 Pre-U CIE 音乐课程所必需的核心“公式”与结构原理。从音阶构建、音程计算到和声语法、声部进行规则以及曲式结构，每一节都以简明的方式呈现一条可以用于分析或创作的规则或模型。掌握这些模式将增强你在读谱、听觉辨识和作曲方面的能力。
1. Major Scale Formula | 大调音阶公式
A major scale follows the fixed pattern of whole tones (T) and semitones (S): T–T–S–T–T–T–S. Starting on any tonic note, this sequence generates all eight degrees with the characteristic leading-tone semitone between degrees 7 and 8.
大调音阶遵循固定的全音（T）与半音（S）模式：T–T–S–T–T–T–S。从任一主音开始，这个序列可以生成全部八个音级，并在第7级与第8级之间形成特有的导音半音。
- Example: C major — C (T) D (T) E (S) F (T) G (T) A (T) B (S) C
- 示例：C大调 — C（全） D（全） E（半） F（全） G（全） A（全） B（半） C
For any tonic note, apply the interval sequence of major 2nd, major 2nd, minor 2nd, major 2nd, major 2nd, major 2nd, minor 2nd. The semitone placement is always between 3–4 and 7–8.
对任何主音，依次使用大二度、大二度、小二度、大二度、大二度、大二度、小二度。半音位置永远在3–4级与7–8级之间。
2. Minor Scale Variants | 小调音阶变体公式
The natural minor scale applies the pattern T–S–T–T–S–T–T. Compared with the major, the 3rd, 6th, and 7th degrees are lowered by a semitone. The harmonic minor raises the 7th degree to create a leading tone, producing an augmented 2nd between 6th and 7th. The melodic minor raises both 6th and 7th when ascending and reverts to natural minor when descending.
自然小调采用 T–S–T–T–S–T–T 模式。与大调相比，第3级、第6级和第7级降低半音。和声小调升高第7级以形成导音，在第6级与第7级之间产生增二度。旋律小调上行时升高第6级和第7级，下行时还原为自然小调。
- A natural minor: A–B–C–D–E–F–G–A
- A和声小调: A–B–C–D–E–F–G♯–A
- A旋律小调（上行）: A–B–C–D–E–F♯–G♯–A; 下行同自然小调
3. Interval Number & Quality Rule | 音程度数与性质公式
Interval size is determined by counting letter names from the lower note to the upper, inclusive. Quality (perfect, major, minor, augmented, diminished) is then calculated by checking the number of semitones against the standard values for that numeric interval in a major scale built on the lower note.
音程的大小由从低音到高音包含的音名字母个数决定（含两端）。性质（纯、大、小、增、减）则通过检查实际半音数与以低音为主音的大调音阶中该度数对应的标准半音数来确定。
| Interval / 音程 | Semitones in major / 大调半音数 | Quality rule / 性质规则 |
|---|---|---|
| Unison, 4th, 5th, 8ve | 0, 5, 7, 12 | Perfect; one semitone smaller = diminished; one larger = augmented |
| 2nd, 3rd, 6th, 7th | 2, 4, 9, 11 | Major; one semitone smaller = minor; two smaller = diminished; one larger = augmented |
Example: C to A♭ is a 6th (C–D–E–F–G–A = 6 letters). In C major, A is natural, giving a major 6th of 9 semitones. A♭ reduces that to 8 semitones, so the interval is a minor 6th.
示例：C到A♭是六度（C-D-E-F-G-A = 6个字母）。在C大调中A为还原A，形成9个半音的大六度。A♭使半音减为8个，因此该音程是小六度。
4. Triad Construction Formula | 三和弦构建公式
A triad consists of a root, a 3rd, and a 5th. The quality is defined by the stacked thirds: major triad = major 3rd + minor 3rd (root to 5th = perfect 5th); minor triad = minor 3rd + major 3rd; diminished triad = minor 3rd + minor 3rd; augmented triad = major 3rd + major 3rd.
三和弦由根音、三音和五音构成。性质由叠置的三度决定：大三和弦 = 大三度 + 小三度（根音到五音为纯五度）；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。
Major triad: root → +4 semitones (M3) → 3rd → +3 semitones (m3) → 5th
大三和弦：根音 → +4半音（大三度）→ 三音 → +3半音（小三度）→ 五音
In Roman numeral analysis, triads built on scale degrees follow this pattern in major keys: I, ii, iii, IV, V, vi, vii°.
在罗马数字分析中，大调音阶各级上构建的三和弦模式为：I, ii, iii, IV, V, vi, vii°。
5. Seventh Chord Recipe | 七和弦配方
Adding a 7th above the root produces four types of seventh chords: major–major 7th (MM7) = major triad + major 7th; major–minor 7th (Mm7, ‘dominant 7th’) = major triad + minor 7th; minor–minor 7th (mm7) = minor triad + minor 7th; half–diminished 7th = diminished triad + minor 7th; fully diminished 7th = diminished triad + diminished 7th.
在根音上方增加七音可生成四种七和弦：大大七和弦（MM7）= 大三和弦 + 大七度；大小七和弦（Mm7，即属七和弦）= 大三和弦 + 小七度；小小七和弦（mm7）= 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。
| Chord type / 和弦类型 | Intervallic formula / 音程公式 | Example on C / C上的示例 |
|---|---|---|
| Dominant 7th | M3 + m3 + m3 | C–E–G–B♭ |
| Major 7th | M3 + m3 + M3 | C–E–G–B |
| Minor 7th | m3 + M3 + m3 | C–E♭–G–B♭ |
| Half-diminished 7th | m3 + m3 + M3 | C–E♭–G♭–B♭ |
| Diminished 7th | m3 + m3 + m3 | C–E♭–G♭–B♭♭ |
6. Cadence Formulas | 终止式公式
Cadences function as musical punctuation. The four principal cadences and their harmonic formulae are: perfect (V–I), imperfect (I–V, ii–V, IV–V, ending on V), plagal (IV–I), and interrupted/deceptive (V–vi). In a perfect cadence, both chords must be in root position and the tonic must be in the soprano of the final chord for a ‘perfect authentic’ effect.
终止式充当音乐的标点符号。四种主要终止式及其和声公式为：正格终止（V–I）、半终止（I–V、ii–V、IV–V，结束于V）、变格终止（IV–I）和阻碍终止（V–vi）。在正格终止中，两个和弦必须为原位，且结束和弦的高音声部须为主音，方可形成“完满正格终止”。
Perfect cadence: V → I | Plagal cadence: IV → I | Interrupted: V → vi
正格终止：V → I | 变格终止：IV → I | 阻碍终止：V → vi
The leading note (7th degree) in chord V must rise to the tonic; in root position V–I, the bass leaps a 5th down or a 4th up.
V和弦中的导音（第7级）必须上行到主音；在原位V–I中，低音部下行五度或上行四度跳进。
7. Fundamental Voice-Leading Rules | 基本声部进行规则
Smooth part-writing requires observing several key prohibitions and tendencies: avoid parallel perfect 5ths and octaves between any pair of voices; avoid direct 5ths and octaves by contrary motion unless the upper voice moves by step; leading note resolves up to the tonic; the 7th of any chord resolves down by step; double the root of root-position triads unless forced otherwise.
平稳的声部写作需要遵守以下核心禁令和倾向：避免任何两声部之间的平行纯五度和平行八度；避免反向进行的隐伏五度与八度，除非上声部级进；导音应上行解决到主音；任何和弦的七音应下行级进解决；原位三和弦优先重复根音，除非不得已。
- Parallel 5ths — same two voices move by the same interval to another perfect 5th: strictly forbidden in Common Practice style.
- 平行五度 — 相同的两个声部以相同音程进行到另一个纯五度：在共性写作时期严格禁止。
- Voice spacing — keep upper three voices within an octave of each other; bass may be up to a 12th from tenor.
- 声部间距 — 保持上方三声部彼此在一个八度之内；低音与男高音可相距最多十二度。
8. Chord Progression Grammar | 和弦进行语法
Tonal harmony tends to follow a functional cycle: tonic → predominant → dominant → tonic. In major keys, the typical progression map is: I can go to any chord; ii and IV move to V or vii°; V and vii° move to I (or vi for deception); vi can move to ii or IV; iii is the weakest and usually moves to vi or IV. The descending circle‑of‑5ths progression (iii→vi→ii→V→I) is a fundamental sequential formula.
调性和声通常遵循功能循环：主功能 → 下属功能 → 属功能 → 主功能。在大调中，典型进行图为：I 可进行到任何和弦；ii 与 IV 进行到 V 或 vii°；V 和 vii° 进行到 I（或通过阻碍终止到 vi）；vi 可进行到 ii 或 IV；iii 功能最弱，通常进行到 vi 或 IV。下行五度圈进行（iii→vi→ii→V→I）是一个基本的模进公式。
Circle progression: iii → vi → ii → V → I
五度圈进行：iii → vi → ii → V → I
The strongest root movement is by descending perfect 5th (or ascending perfect 4th). Stepwise root movement (e.g. IV–V) is also strong; movements by 3rd are weaker but common.
最强的根音进行是下行纯五度（或上行纯四度）。级进根音进行（例如IV–V）同样强有力；三度进行较弱，但仍常见。
9. Modulation Procedures | 转调操作
Modulation to the dominant (V) or relative major (III in minor) is achieved via a pivot chord — a chord diatonic to both keys — followed by a cadential progression in the new key. The pivot chord is typically a pre‑dominant chord in the new key. For distant modulations, chromatic pivot chords or enharmonic reinterpretation of diminished 7th chords can provide smooth transitions.
转向属调（大调V）或关系大调（小调III）需通过中介和弦——一个同属于两个调的自然音和弦——随后在新调上完成终止式进行。中介和弦通常是新调的下属功能类和弦。对于远关系转调，可使用半音中介和弦或通过对减七和弦的等音重新解释来实现平滑过渡。
Close-related keys: keys whose key signatures differ by no more than one accidental — relative major/minor, dominant, subdominant, and their relative minors.
近关系调：调号相差不超过一个升号或降号的调——关系大/小调、属调、下属调以及它们的关系小调。
Formula for a pivot‑chord modulation: establish original key → introduce pivot chord → confirm new key with its V⁷–I cadence. Avoid modulating to the leading-note key in major without careful preparation; use mode mixture for colour.
中介和弦转调公式：建立原调 → 引入中介和弦 → 用新调的V⁷–I终止式确认新调。在大调中需谨慎转至导音调，可使用调式交替增添色彩。
10. Rhythm and Metre Formulae | 节奏与节拍公式
Metre is organised hierarchically: simple time has beats divisible into two equal parts; compound time has beats divisible into three. Duple, triple, and quadruple refer to the number of beats per bar. The formula for relating simple and compound time signatures: a simple metre in 2/4, 3/4, 4/4 becomes compound by dotting the beat (6/8, 9/8, 12/8) where the lower number indicates the subdivision (8 = quaver).
节拍是分级组织的：简单拍子的每拍可分为两个等份；复拍子的每拍可分为三个等份。二拍子、三拍子、四拍子指每小节的拍数。简单拍号与复拍号的关系公式：简单的2/4、3/4、4/4转换为复拍子时，拍子加附点（6/8、9/8、12/8），下方数字表示时值细分（8 = 八分音符）。
Simple triple: 3/4 (beat = crotchet, divided into 2 quavers) → Compound duple: 6/8 (beat = dotted crotchet, divided into 3 quavers)
简单三拍子：3/4（拍子=四分音符，细分为2个八分音符）→ 复二拍子：6/8（拍子=附点四分音符，细分为3个八分音符）
Syncopation formula: displace the natural accent by tying a weak beat or weak part of a beat to a strong beat. Hemiola: in triple metre, articulating two-beat groupings to create a temporary 3:2 cross-rhythm (e.g., three duplets over two bars of 3/4).
切分音公式：将弱拍或拍中弱部分用延音线与强拍相连，从而移位置自然重音。赫米奥拉：在三拍子中，通过划分出两拍一组的节奏，临时形成3:2的交叉节奏（例如在两小节3/4上演奏三个二连音）。
11. Formal Architecture Templates | 曲式结构模板
Sonata form (first-movement form) follows a tripartite plan: Exposition (first subject in tonic, bridge modulating to dominant or relative major, second subject in new key, codetta) — Development (thematic fragmentation, sequential treatment, harmonic instability) — Recapitulation (first subject in tonic, bridge altered to stay in tonic, second subject in tonic, coda). The structural formula is fundamentally a tonal journey: I → V (or III) → development (various keys) → I.
奏鸣曲式（第一乐章曲式）遵循三部结构：呈示部（主部主题在主调，连接部转至属调或关系大调，副部主题在新调，结束部）——发展部（主题碎片化、模进处理、和声不稳定）——再现部（主部主题回到主调，连接部改动以保持在主调，副部主题在主调，尾声）。其结构公式本质上是一段调性旅程：I → V（或III）→ 发展部（多调）→ I。
Rondo formula: A B A C A (or A B A C A B A), where A is the refrain in the tonic, contrasting episodes occur in related keys, and the return of A provides formal closure.
回旋曲公式：A B A C A（或A B A C A B A），其中A为在主调上的叠句，对比插部出现在关系调上，A的回归提供曲式收束。
Ternary form: A (tonic, closed) — B (contrasting key and material) — A’ (return of A, often abbreviated). Binary: (A) ||: (B) A’ :|| where the first section modulates to V, the second returns to I.
三部曲式：A（主调，闭合终止）— B（对比调性与材料）— A’（A的回归，常缩短）。二部曲式：(A) ||: (B) A’ :||，第一部分转至V，第二部分返回I。
12. Non-Harmonic Tones & Embellishment Rules | 和弦外音与装饰音规则
Non‑harmonic tones (NHTs) are classified by their approach and departure. Passing tone: approached by step, left by step in the same direction. Neighbour tone: step away from a chord tone, step back to the same tone. Suspension: prepared (same pitch as previous chord), suspended (held over while harmony changes), resolved down by step. Appoggiatura: leapt to, resolved by step, usually opposite direction. Escape tone: stepped to, leapt away in opposite direction. Anticipation: approached by step or leap, becomes a chord tone when the next chord arrives.
和弦外音按其进入和离开的方式分类。经过音：级进进入，同向级进离开。辅助音：从和弦音级进离开，再级进返回同一音。延留音：预备（与前和弦同音高），延留（和声变化时保持），下行级进解决。倚音：跳进进入，级进解决，通常反向。规避音：级进进入，反向跳进离开。先现音：级进或跳进进入，当下一个和弦到达时成为和弦音。
Suspension formula: preparation (consonant) — suspension (dissonant, tied or repeated) — resolution (down by step). The 4–3 suspension is the most common (suspended 4th resolving to 3rd above bass).
延留音公式：预备（协和）——延留（不协和，用延音线或重复）——解决（下行级进）。4–3延留最为常见（低音上方的四度延留到三度）。
All NHTs create metrically weak dissonances, except the suspension and appoggiatura which fall on the strong part of the beat and demand expressive resolution.
所有和弦外音均产生在节拍弱位上的不协和音，但延留音和倚音落在拍子的强部分，需要进行富有表现力的解决。
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