📚 Year 12 CCEA Music: Formula & Theorem Quick Reference Handbook | Year 12 CCEA 音乐：公式定理速查手册
Music is often seen as an art, but behind its expressive surface lie precise patterns, intervallic ‘formulas’ and harmonic ‘theorems’ that bring order to sound. This handbook gathers the essential theoretical rules you must internalise for Year 12 CCEA Music — from scale construction and chord building to cadential progressions and rhythmic proportions. Treat each rule as a formula: memorise it, apply it in your compositions, and use it to decode the set works in your listening paper.
音乐常被视为艺术，但在其表现力之下隐藏着精确的模式、音程“公式”与和声“定理”，它们赋予声音秩序。这本速查手册汇集了 Year 12 CCEA 音乐必需的理论规则——从音阶构成、和弦构建到终止式进行和节奏比例。把每一条规则当作公式来对待：记住它，在作曲中运用它，并用它来解读听力试卷中的规定作品。
1. Pitch & Keyboard Formulas | 音高与键盘公式
A semitone is the smallest step on the keyboard; a whole tone equals two semitones. These two distances form the building blocks of all scales and intervals.
半音是键盘上最小的步幅；一个全音等于两个半音。这两种距离构成了所有音阶与音程的基本积木。
1 semitone (half step) = adjacent key, 2 semitones = 1 whole tone
1 个半音 = 相邻琴键，2 个半音 = 1 个全音
Accidentals raise or lower pitch by semitone: ♯ (sharp) raises by one semitone; ♭ (flat) lowers by one semitone; ♮ (natural) cancels previous accidentals.
变音记号以半音为单位升降音高：♯（升号）升高半音；♭（降号）降低半音；♮（还原号）取消先前的变音记号。
2. Major Scale Formula | 大调音阶公式
The major scale follows a fixed sequence of whole and half steps, regardless of the starting note. This is the most fundamental formula in tonal music.
大调音阶遵循固定的全音与半音序列，与起始音无关。这是调性音乐中最基本的公式。
W – W – H – W – W – W – H (W = whole tone, H = semitone)
全 – 全 – 半 – 全 – 全 – 全 – 半 (全 = 全音，半 = 半音)
Apply this formula to any tonic to generate the scale. For example, D major: D – E – F♯ – G – A – B – C♯ – D. The pattern of tones and semitones remains constant; only the specific pitches change.
将此公式应用于任何主音即可生成音阶。例如 D 大调：D – E – F♯ – G – A – B – C♯ – D。全音与半音的模式保持不变；变化的只是具体音高。
3. Minor Scale Formulas | 小调音阶公式
Three distinct minor scales exist, each governed by its own formula. The natural minor uses the key signature alone and follows the pattern: W – H – W – W – H – W – W.
存在三种不同的自然小调音阶，每种都由各自的公式支配。自然小调仅使用调号，并遵循模式：全 – 半 – 全 – 全 – 半 – 全 – 全。
The harmonic minor raises the 7th degree by a semitone (creating a leading note) while keeping the key signature. This produces the characteristic augmented 2nd between the 6th and 7th degrees.
和声小调在保持调号的同时将第Ⅶ级音升高半音（创造导音），从而在Ⅵ级与Ⅶ级之间产生特征性的增二度。
The melodic minor raises both the 6th and 7th degrees when ascending, and reverts to the natural minor when descending. As a formula: ascending = natural minor with ♮6 and ♮7; descending = natural minor.
旋律小调上行时升高第Ⅵ和第Ⅶ级音，下行时回归自然小调。作为公式：上行 = 自然小调 + ♮6和♮7；下行 = 自然小调。
Natural: 1 2 ♭3 4 5 ♭6 ♭7 8 | Harmonic: 1 2 ♭3 4 5 ♭6 7 8 | Melodic ascending: 1 2 ♭3 4 5 6 7 8
自然：1 2 ♭3 4 5 ♭6 ♭7 8 | 和声：1 2 ♭3 4 5 ♭6 7 8 | 旋律上行：1 2 ♭3 4 5 6 7 8
4. Interval Calculation Theorems | 音程计算定理
An interval is measured by counting the letter names (size) and then determining the number of semitones (quality). Use the following table as a quick lookup for interval qualities within an octave.
音程通过计算字母名（度数）然后确定半音数（性质）来衡量。使用下表作为八度内音程性质的速查。
| Semitones | Interval Quality (English) | 中文音程性质 |
|---|---|---|
| 0 | Perfect Unison | 纯一度 |
| 1 | Minor 2nd | 小二度 |
| 2 | Major 2nd | 大二度 |
| 3 | Minor 3rd | 小三度 |
| 4 | Major 3rd | 大三度 |
| 5 | Perfect 4th | 纯四度 |
| 6 | Augmented 4th / Diminished 5th | 增四度/减五度 |
| 7 | Perfect 5th | 纯五度 |
| 8 | Minor 6th | 小六度 |
| 9 | Major 6th | 大六度 |
| 10 | Minor 7th | 小七度 |
| 11 | Major 7th | 大七度 |
| 12 | Perfect Octave | 纯八度 |
Compound intervals (wider than an octave) follow the same quality rules; simply add 7 to the simple interval number. For example, a major 2nd plus an octave becomes a major 9th.
复音程（宽于八度）遵循相同的性质规则；只需将单音程度数加7。例如大二度加一个八度成为大九度。
5. Triad Construction Formulas | 三和弦构建公式
All triads consist of three stacked notes: a root, a third and a fifth. The quality of the triad is determined by the precise pattern of semitones between these notes.
所有三和弦由三个叠置的音组成：根音、三音和五音。三和弦的性质由这些音之间精确的半音模式决定。
| Triad Type | Semitone Formula | 中文公式 |
|---|---|---|
| Major | root + 4st + 3st | 根音 +4半音 +3半音 |
| Minor | root + 3st + 4st | 根音 +3半音 +4半音 |
| Augmented | root + 4st + 4st | 根音 +4半音 +4半音 |
| Diminished | root + 3st + 3st | 根音 +3半音 +3半音 |
Using the C major scale, the primary triads (I, IV, V) are naturally major; the supertonic and mediant (ii, iii) are minor; the leading-note triad (vii°) is diminished. This is a direct consequence of the major scale step formula.
以 C 大调音阶为例，正三和弦（I、IV、V）自然为大三和弦；上主和弦和中音和弦（ii、iii）为小三和弦；导音和弦（vii°）为减三和弦。这是大调音阶全半音模式的直接结果。
6. Seventh Chord & Inversion Formulas | 七和弦与转位公式
A seventh chord adds a fourth note (the 7th) above the root. The most common types in Year 12 repertoire are the dominant 7th and the diminished 7th. Their formulas rely on stacking intervals over a triad.
七和弦在根音上方加入第四个音（七音）。Year 12 作品中最常见的类型是属七和弦和减七和弦。它们的公式基于在三和弦上叠置音程。
- Dominant 7th (V⁷): major triad + minor 7th. — 属七和弦：大三和弦 + 小七度。
- Major 7th (I⁷, IV⁷): major triad + major 7th. — 大七和弦：大三和弦 + 大七度。
- Minor 7th (ii⁷, vi⁷): minor triad + minor 7th. — 小七和弦：小三和弦 + 小七度。
- Half-diminished 7th (viiØ⁷): diminished triad + minor 7th. — 半减七和弦：减三和弦 + 小七度。
- Diminished 7th (vii°⁷): diminished triad + diminished 7th. — 减七和弦：减三和弦 + 减七度。
Inversions are determined by which chord member is in the bass: root position (root in bass), 1st inversion (3rd in bass, figured as ⁶₃), 2nd inversion (5th in bass, figured as ⁶₄), and for seventh chords, 3rd inversion (7th in bass, figured as ⁴₂).
转位取决于哪一个和弦音在低声部：原位（根音在低音），第一转位（三音在低音，数字低音标记为⁶₃），第二转位（五音在低音，标记⁶₄），七和弦还有第三转位（七音在低音，标记⁴₂）。
7. Cadence Theorems | 终止式定理
Cadences function like harmonic punctuation. Each type follows a fixed chord-pair formula that establishes or disrupts a sense of closure.
终止式如同和声的标点符号。每一种类型都遵循固定的和弦双对公式，确立或打破结束感。
| Cadence (English) | Chord Formula | 中文终止式与公式 |
|---|---|---|
| Perfect (V–I) | V → I (both in root position, I upper note tonic) | 正格终止：V → I（两者原位，I 最高音为主音） |
| Imperfect (I–V, IV–V, etc.) | any chord → V (ends on V) | 半终止：任何和弦 → V（结束在V上） |
| Plagal (IV–I) | IV → I | 变格终止：IV → I |
| Interrupted (V–vi) | V → vi (minor vi, avoids tonic) | 阻碍终止：V → vi（小调性vi，避开主音） |
In a perfect cadence, the leading note (7th degree of the scale) in V must rise to the tonic, reinforcing the formula’s finality.
在正格终止中，V 和弦中的导音（音阶第Ⅶ级）必须上行解决到主音，强化该公式的结束感。
8. Common Chord Progression Formulas | 常见和弦进行公式
Many harmonic progressions behave like templates. The circle-of-fifths progression and the I–IV–V–I loop reappear in countless set works.
许多和声进行如同模板。五度圈进行和I–IV–V–I循环在无数规定作品中反复出现。
Circle of 5ths progression (major): iii → vi → ii → V → I (each chord’s root a 5th higher than the next).
五度圈进行（大调）： iii → vi → ii → V → I（每个和弦的根音比下一个高五度）。
Blues / rock formula: I – IV – V – IV – I (often with 7ths).
蓝调/摇滚公式： I – IV – V – IV – I（常加入七音）。
In minor keys the harmonic minor scale strongly influences progressions, making V a major chord (with raised 7th) and often using the augmented 2nd melodic feature in the bass line.
在小调中，和声小调强烈地影响进行，使 V 成为大三和弦（升高第Ⅶ级音），并常常在低音线条中使用增二度的旋律特征。
9. Rhythm & Time Signature Formulas | 节奏与拍号公式
The time signature is a numeric formula: the top number tells how many beats are in each bar; the bottom number indicates the note value that receives one beat.
拍号是一个数字公式：上方数字表示每小节有多少拍；下方数字表示以何种音符为一拍。
In simple time the beat divides naturally into two equal parts. In compound time each beat divides into three equal parts. This distinction governs beaming and accent patterns.
在单拍子中，拍子自然分成两个均等部分；在复拍子中，每拍分成三个均等部分。这一区分支配着符杠连接和重音模式。
Simple duple: 2/4, 2/2 | Simple triple: 3/4, 3/2 | Simple quadruple: 4/4
二拍子单拍：2/4, 2/2 | 三拍子单拍：3/4, 3/2 | 四拍子单拍：4/4
Compound duple: 6/8 | Compound triple: 9/8 | Compound quadruple: 12/8
二拍子复拍：6/8 | 三拍子复拍：9/8 | 四拍子复拍：12/8
Dotted rhythms and syncopation can be understood as variations of the basic beat-division formula. The dot adds half the value of the note: dotted quarter = 3 eighth notes in simple time.
附点节奏和切分音可以理解为基本拍子分割公式的变体。附点增加音符时值的一半：单拍子中附点四分音符 = 3个八分音符。
10. Ornamentation & Articulation Symbols | 装饰音与演奏法符号
Ornaments are small melodic formulas written with special symbols. Knowing their precise execution rules is essential for score analysis and performance.
装饰音是用特殊符号记写的微型旋律公式。了解它们精确的执行规则对于乐谱分析和表演至关重要。
- Trill (tr): rapid alternation between the written note and the one above. — 颤音：本音与上方音快速交替。
- Mordent (upper mordent): note, upper neighbour, return note. — 波音（上波音）：本音-上方邻音-本音。
- Inverted mordent: note, lower neighbour, return note. — 下波音：本音-下方邻音-本音。
- Turn: upper note, main note, lower note, main note. — 回音：上方音-本音-下方音-本音。
- Appoggiatura: leans on the main note, taking half its value. — 倚音：靠向主音，占据其一半时值。
- Acciaccatura: crushed grace note played very quickly. — 碎音：极快速地演奏的短倚音。
Articulation marks, such as staccato (detached), legato (smooth), accent and tenuto, further define the attack and release formula for each note.
演奏法记号，如断奏（staccato）、连奏（legato）、重音和保持音，进一步定义了每个音符的起音和释放公式。
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