📚 Year 13 OCR Music: Formula & Theorem Quick Reference Handbook | Year 13 OCR 音乐：公式定理速查手册
As a Year 13 OCR Music student, you need rapid recall of the rules, patterns, and constructs that form the backbone of analysis, composition and performance. This compact handbook draws together the essential ‘formulas’ and ‘theorems’ encountered across the specification, from scale construction to serialism, so you can check them at a glance.
作为 Year 13 OCR 音乐学生，你需要快速调用构成分析、作曲与表演骨干的规则、模式与结构。这本小手册汇集了从音阶构建到序列主义在内、考纲涉及的必备“公式”与“定理”，方便你随时查阅。
1. Major Scale Formula | 大调音阶公式
A major scale is built from a fixed pattern of tones (T) and semitones (S): T-T-S-T-T-T-S. In terms of semitone steps, the pattern is 2-2-1-2-2-2-1.
大调音阶由固定的全音（T）与半音（S）模式构成：全-全-半-全-全-全-半。用半音步数表示即为 2-2-1-2-2-2-1。
The scale degrees have functional names: tonic (1st), supertonic (2nd), mediant (3rd), subdominant (4th), dominant (5th), submediant (6th), leading note (7th), and the octave returns to tonic.
音阶级数有功能性名称：主音（Ⅰ）、上主音（Ⅱ）、中音（Ⅲ）、下属音（Ⅳ）、属音（Ⅴ）、下中音（Ⅵ）、导音（Ⅶ），八度回到主音。
C Major: C-D-EF-G-A-BC (T-T-S-T-T-T-S)
C 大调：C-D-EF-G-A-BC（全-全-半-全-全-全-半）
2. Minor Scale Types | 小调音阶类型
There are three main minor scale forms in common-practice harmony, each with its own intervallic formula using semitone steps between successive degrees.
在共性实践和声中主要有三种小调音阶，每一种都有其相邻音级间的半音步数公式。
| Scale | Ascending pattern (semitones) | Characteristic |
|---|---|---|
| Natural minor | 2-1-2-2-1-2-2 | No raised leading note |
| Harmonic minor | 2-1-2-2-1-3-1 | Raised 7th, augmented 2nd between 6th and 7th |
| Melodic minor | 2-1-2-2-2-2-1 (up) / 2-2-1-2-2-1-2 (down) | Raised 6th and 7th ascending, natural minor descending |
Natural minor shares the same key signature as its relative major. Harmonic minor introduces a raised 7th to create a leading tone, while melodic minor raises both the 6th and 7th when ascending to avoid the augmented 2nd leap.
自然小调与其关系大调共享相同的调号。和声小调通过升高Ⅶ级产生导音，旋律小调在上行时同时升高Ⅵ级和Ⅶ级以避免增二度跳进。
A harmonic minor: A-B-C-D-E-F-G♯-A (2-1-2-2-1-3-1)
A 和声小调：A-B-C-D-E-F-G♯-A（2-1-2-2-1-3-1）
3. Chord Construction Formulas | 和弦构成公式
Triads and seventh chords are built by stacking intervals. The formula defines the number of semitones between consecutive chord members: root–3rd, 3rd–5th, and for seventh chords, 5th–7th.
三和弦与七和弦通过叠置音程构成。公式定义了各和弦音之间的半音数：根音–三音、三音–五音，对七和弦还包括五音–七音。
| Chord type | Intervals (semitones) | Example on C |
|---|---|---|
| Major triad | 4 + 3 | C-E-G |
| Minor triad | 3 + 4 | C-E♭-G |
| Diminished triad | 3 + 3 | C-E♭-G♭ |
| Augmented triad | 4 + 4 | C-E-G♯ |
| Dominant 7th | 4 + 3 + 3 | C-E-G-B♭ |
| Major 7th | 4 + 3 + 4 | C-E-G-B |
| Minor 7th | 3 + 4 + 3 | C-E♭-G-B♭ |
| Half-diminished 7th | 3 + 3 + 4 | C-E♭-G♭-B♭ |
| Diminished 7th | 3 + 3 + 3 | C-E♭-G♭-B♭♭ |
You can memorise these as stacking major and minor thirds. A diminished seventh chord is entirely symmetrical, dividing the octave into four equal minor-third steps.
你可以将这些公式记忆为大三度与小三度的叠置。减七和弦完全对称，将八度均分为四个小三度。
4. Circle of Fifths | 五度圈
The circle of fifths organises keys by ascending perfect fifths. Each step clockwise adds a sharp or removes a flat; each step anticlockwise adds a flat or removes a sharp. The pattern reveals the key signature accumulation.
五度圈以纯五度上行来组织调性。顺时针每走一步增加一个升号或减少一个降号；逆时针每走一步增加一个降号或减少一个升号。该模式揭示了调号的累积方式。
Sharp keys in order: G (1♯), D (2♯), A (3♯), E (4♯), B (5♯), F♯ (6♯), C♯ (7♯). Flat keys in order: F (1♭), B♭ (2♭), E♭ (3♭), A♭ (4♭), D♭ (5♭), G♭ (6♭), C♭ (7♭).
升号调顺序：G（1♯）、D（2♯）、A（3♯）、E（4♯）、B（5♯）、F♯（6♯）、C♯（7♯）。降号调顺序：F（1♭）、B♭（2♭）、E♭（3♭）、A♭（4♭）、D♭（5♭）、G♭（6♭）、C♭（7♭）。
The relative minor key lies a minor third below each major. For example, the relative minor of G major is E minor; of F major is D minor. Every key signature thus serves a major and its relative minor.
关系小调位于每个大调下方小三度处。例如，G 大调的关系小调是 E 小调；F 大调的关系小调是 D 小调。因此每个调号服务于一个大调及其关系小调。
5. Cadence Patterns | 终止式模式
A cadence is a pair of chords that punctuate a phrase. The four primary cadence types follow predictable chord progressions and voice-leading norms.
终止式是一对用来为乐句做出收束的和弦。四种主要终止式有着可预见的和弦进行与声部进行规范。
Perfect cadence: V – I (in major or minor). The leading note resolves to the tonic, the bass leaps 4th/5th. Plagal cadence: IV – I, often heard as the ‘Amen’ cadence. Imperfect cadence: any chord to V, leaving a sense of expectation (e.g. I–V, II–V, IV–V). Interrupted (deceptive) cadence: V – vi (or VI in minor), where the expected I is replaced by the submediant.
正格终止：V – I（大调或小调）。导音解决到主音，低音作四度/五度跳进。变格终止：IV – I，常被称作“阿们终止”。不完全终止：任何和弦进行到 V，留下期待感（如 I–V、II–V、IV–V）。阻碍终止：V – vi（小调中为 VI），期待的 I 被下中音和弦取代。
Perfect cadence in C major: G – C; Interrupted: G – Am
C 大调中的正格终止：G – C；阻碍终止：G – Am
6. Intervals and Semitone Count | 音程与半音数
An interval is named by its number (2nd, 3rd, etc.) and quality (major, minor, perfect, augmented, diminished). Counting semitones helps determine the quality quickly.
音程由其度数（二度、三度等）和性质（大、小、纯、增、减）命名。通过数半音可以快速判断音程性质。
| Interval | Semitones | Example from C |
|---|---|---|
| Unison | 0 | C-C |
| Minor 2nd | 1 | C-D♭ |
| Major 2nd | 2 | C-D |
| Minor 3rd | 3 | C-E♭ |
| Major 3rd | 4 | C-E |
| Perfect 4th | 5 | C-F |
| Augmented 4th / Diminished 5th | 6 | C-F♯ / C-G♭
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