📚 Year 12 CCEA Music: Quick Formula & Theorem Handbook | Year 12 CCEA 音乐公式定理速查手册
In CCEA AS Music, mastering the ‘formulae’ and ‘theorems’ of music theory is essential for analysis, composition, and performance. This handbook compiles the key theoretical rules, interval measurements, chord structures, and harmonic principles into a quick-reference guide.
在CCEA AS音乐课程中，掌握音乐理论的“公式”和“定理”对于分析、作曲和表演至关重要。本手册将关键的理论规则、音程度量、和弦结构和声原则汇编为速查指南。
1. Scale and Mode Formulas | 音阶与调式公式
The major scale formula is T-T-S-T-T-T-S, where T stands for Tone (whole step) and S stands for Semitone (half step).
大调音阶公式为全-全-半-全-全-全-半，其中“全”代表全音，“半”代表半音。
Major: T-T-S-T-T-T-S
大调：全-全-半-全-全-全-半
The natural minor scale follows the pattern T-S-T-T-S-T-T.
自然小调音阶遵循全-半-全-全-半-全-全的模式。
The harmonic minor scale raises the 7th degree, producing an augmented 2nd between the 6th and 7th degrees: T-S-T-T-S-1½-S.
和声小调升高第VII级音，在第VI级与第VII级之间形成增二度，公式为全-半-全-全-半-增二-半。
The melodic minor ascends with raised 6th and 7th degrees (T-S-T-T-T-T-S) and descends as natural minor.
旋律小调上行时升高第VI、VII级（全-半-全-全-全-全-半），下行时还原为自然小调。
The Dorian mode is a natural minor with a raised 6th (T-S-T-T-T-S-T). The Mixolydian mode is a major scale with a lowered 7th (T-T-S-T-T-S-T).
多利亚调式是升高第VI级的自然小调（全-半-全-全-全-半-全），混合利底亚调式是降低第VII级的大调（全-全-半-全-全-半-全）。
2. Interval Identification Rules | 音程识别规则
Interval quality and size are determined by counting semitones from the lower note. Use the major scale of the lower note as a reference.
音程的性质与大小由下方音向上的半音数决定。通常以下方音的大调为参照。
Unison = 0 semitones; Minor 2nd = 1; Major 2nd = 2; Minor 3rd = 3; Major 3rd = 4; Perfect 4th = 5; Augmented 4th/Diminished 5th (Tritone) = 6; Perfect 5th = 7; Minor 6th = 8; Major 6th = 9; Minor 7th = 10; Major 7th = 11; Octave = 12.
同度0个半音；小二度1个；大二度2个；小三度3个；大三度4个；纯四度5个；增四度/减五度（三全音）6个；纯五度7个；小六度8个；大六度9个；小七度10个；大七度11个；纯八度12个。
To identify a compound interval, subtract 7 from the number and find the equivalent simple interval.
识别复音程时，将数字减去7，转化为对应的单音程。
Inverted intervals always add up to 9. A 3rd inverts to a 6th; a 4th inverts to a 5th. Major becomes minor, perfect stays perfect.
音程转位的数字总和恒为9。三度转位为六度，四度转位为五度。大音程转位后变小，纯音程转位后仍为纯。
3. Triad and Seventh Chord Construction | 三和弦与七和弦构架
Triads are built by stacking two thirds on a root. The quality depends on the semitone pattern.
三和弦由根音上方叠加两个三度构成。其性质取决于半音间距模式。
Major triad: Root + 4 semitones (M3) + 3 semitones (m3) → R+4+3
大三和弦：根音 + 4半音（大三度） + 3半音（小三度） → R+4+3
Minor triad: Root + 3 semitones (m3) + 4 semitones (M3) → R+3+4.
小三和弦：根音 + 3半音（小三度） + 4半音（大三度） → R+3+4。
Augmented triad: R+4+4. Diminished triad: R+3+3.
增三和弦：R+4+4。减三和弦：R+3+3。
Seventh chords add a third above the fifth. Dominant 7th: major triad + minor 7th (10s) → 4+3+3. Major 7th: major triad + major 7th (11s) → 4+3+4. Minor 7th: minor triad + minor 7th → 3+4+3. Half-diminished 7th: diminished triad + minor 7th → 3+3+4. Diminished 7th: diminished triad + diminished 7th (9s) → 3+3+3.
七和弦在五音上方再加一个三度。属七：大三和弦+小七度（10半音）→4+3+3。大七：大三和弦+大七度（11半音）→4+3+4。小七：小三和弦+小七度→3+4+3。半减七：减三和弦+小七度→3+3+4。减七：减三和弦+减七度（9半音）→3+3+3。
4. Key Signatures and the Circle of Fifths | 调号与五度圈
Sharp key signatures follow the order F♯, C♯, G♯, D♯, A♯, E♯, B♯ (Father Charles Goes Down And Ends Battle). The last sharp is the leading tone; the tonic is a semitone above.
升号调顺序为 F♯, C♯, G♯, D♯, A♯, E♯, B♯。最后一个升号是导音，主音在其上方小二度。
Flat key signatures follow B♭, E♭, A♭, D♭, G♭, C♭, F♭ (Battle Ends And Down Goes Charles’ Father). The penultimate flat indicates the tonic in major keys.
降号调顺序为 B♭, E♭, A♭, D♭, G♭, C♭, F♭。倒数第二个降号即大调的主音。
The relative minor is found a minor third below the major tonic, sharing the same key signature. For example, A minor is the relative minor of C major.
关系小调在大调主音下方小三度，共享相同的调号。例如，A小调是C大调的关系小调。
The Circle of Fifths illustrates: clockwise adds a sharp (or removes a flat); counterclockwise adds a flat (or removes a sharp).
五度圈表明：顺时针方向增加升号（或减少降号），逆时针方向增加降号（或减少升号）。
5. Cadence Formulas and Classification | 终止式公式与分类
A cadence is a melodic/harmonic formula that ends a phrase. The four principal cadences are perfect, plagal, imperfect, and interrupted.
终止式是结束乐句的旋律/和声公式。主要有四种：完全终止、变格终止、不完全终止和阻碍终止。
Perfect cadence: V – I (Authentic) → feels final and complete.
完全终止：V–I（正格终止）→ 感觉完满结束。
Plagal cadence: IV – I (Amen cadence) → sounds softer and ‘Amen’-like. Imperfect cadence: any chord moving to V (e.g., I–V, ii–V, IV–V) → feels unfinished, expecting continuation.
变格终止：IV–I（阿门终止）→ 较为柔和，具有“阿门”感。不完全终止：任何和弦进行到V（如I–V, ii–V, IV–V）→ 感觉未完成，期待继续。
Interrupted cadence: V – vi (in major) or V – VI (in minor) → the expected I is replaced by a submediant chord, creating a surprise.
阻碍终止：V–vi（大调中）或V–VI（小调中）→ 预期的I被下中音和弦替代，产生意外感。
6. Chord Function and Harmonic Progressions | 和弦功能与和声进行
Chords in a key have specific functions: Tonic (I, vi), Subdominant / Predominant (IV, ii), and Dominant (V, vii°).
调内和弦具有特定功能：主功能（I, vi），下属/前属功能（IV, ii），以及属功能（V, vii°）。
The functional progression typically moves Tonic – Predominant – Dominant – Tonic. Avoid retrogression (moving back in functional order, e.g. V – IV).
功能性进行通常为：主功能 → 前属功能 → 属功能 → 主功能。应避免功能逆行（逆序进行，如V–IV）。
Common progressions: I-vi-IV-V (doo-wop), I-IV-V-I, ii-V-I (jazz cadence), I-V-vi-IV (pop progression).
常见和声进行：I-vi-IV-V（嘟·喔普进行），I-IV-V-I，ii-V-I（爵士终止），I-V-vi-IV（流行进行）。
In minor keys, the dominant chord is major (V) to provide a leading tone, often using harmonic minor. The chord iii is rarely used.
在小调中，属和弦通常为大大三和弦（V），以提供导音，常使用和声小调。iii级和弦很少使用。
7. Modulation Techniques | 转调技术
Modulation moves from one key to another. Common techniques include pivot chord, chromatic, phrase, and enharmonic modulation.
转调指从一个调转向另一个调。常用技术包括共同和弦转调、半音转调、乐句转调和等音转调。
Pivot chord modulation uses a chord common to both the old and new key as a hinge. For example, in C major to G major, the chord E minor (vi in C, ii in G) can serve as a pivot.
共同和弦转调使用原调与新调共有的和弦作为枢纽。例如，从C大调转到G大调，E小三和弦（C调vi级，G调ii级）可作为共同和弦。
Chromatic modulation alters one or more notes chromatically to approach the new key’s dominant, without a common chord. Phrase modulation simply begins a new phrase in the new key.
半音转调不通过共同和弦，而是通过变化音级半音化地引入新调的属和弦。乐句转调则直接在新乐句的开始更换调性。
Enharmonic modulation reinterprets a chord enharmonically, most commonly the diminished 7th or augmented 6th chords, to pivot seamlessly to distant keys.
等音转调通过等音重新解释某个和弦（最常见为减七或增六和弦），从而平滑地转向远关系调。
8. Melodic Shape and Voice Leading Principles | 旋律形态与声部进行原则
Good melody writing follows certain ‘theorems’. Leaps should be compensated by stepwise motion in the opposite direction. Avoid augmented and diminished intervals in melodic lines unless expressive.
好的旋律写作遵循一定的“定理”。跳进后应反向级进。旋律中避免增减音程，除非有意为之。
The leading tone (7th degree) must resolve up to the tonic in a melody. The 7th of a seventh chord should resolve downwards by step.
导音（第VII级）必须上行解决到主音。七和弦的七音应下行级进解决。
In SATB harmony, avoid parallel 5ths and octaves. Consecutive 5ths or octaves weaken independent part-writing. Keep common tones and move other voices by the smallest possible step.
在四部和声中，避免平行五度和平行八度。连续的纯五度或纯八度会削弱声部独立性。尽量保持共同音，其他声部作最小距离的平稳进行。
Contrary motion between outer parts is preferred. An ascending melody often fits a pedal note for contrast.
外声部反向进行更佳。上行旋律常可与持续音配合形成对比。
9. Rhythm, Metre and Time Signatures | 节奏、节拍与拍号
Simple time signatures (e.g. 2/4, 3/4, 4/4) subdivide the beat into two equal parts. Compound time (e.g. 6/8, 9/8, 12/8) subdivides each dotted beat into three quavers.
单拍子（如2/4, 3/4, 4/4）每拍可均分为两份。复拍子（如6/8, 9/8, 12/8）以附点音符为一拍，每拍均分为三个八分音符。
Syncopation accents a weak beat or offbeat, creating rhythmic tension. It frequently shifts the accent from strong to weak beats.
切分音强调弱拍或弱位，制造节奏张力，常用于将重音从强拍移向弱拍。
Dotted rhythms and triplets add variety. A dotted crotchet equals three quavers; a triplet squeezes three notes into the time of two.
附点节奏与三连音增添变化。附点四分音符等于三个八分音符；三连音将三个音压缩在通常两个音的时值中。
Hemiola occurs in triple metre when two beats are temporarily grouped as three duplets, creating a cross-rhythm.
希米奥拉节奏出现在三拍子中，临时将两拍重新组合为三组二拍，形成交错节奏。
10. Ornaments and Articulation Markings | 装饰音与演奏法记号
Ornaments embellish a melody with small, patterned notes. The appoggiatura is a stressed non-chord note, usually taking half the value of the main note and resolving by step.
装饰音以小型图案音修饰旋律。倚音是一个被强调的和弦外音，通常占主音一半时值，并级进解决。
The acciaccatura is a crushed, very brief grace note, played just before the beat. The mordent involves the main note, an upper or lower neighbour, and return (upper mordent: rapid alternation with upper note).
碎音是极短促的装饰音，在拍点前快速奏出。涟音由主音、上方或下方邻音再返回构成（上涟音：主音与上方音快速交替）。
A turn moves: upper note, main note, lower note, main note. A trill is rapid alternation between the written note and the note above.
回音按上方音-主音-下方音-主音的顺序演奏。颤音是本音与上方二度音快速交替。
Articulation marks indicate performance style: legato (smoothly, no break), staccato (short, detached), tenuto (held for full value, slightly stressed), accent (strong attack), marcato (marked, emphatic).
演奏法记号指示演奏风格：legato（连贯圆滑），staccato（短促断开），tenuto（保持充分时值并稍加强调），accent（有力重音），marcato（突出，加重）。
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