📚 SQA Music Formula & Theorem Quick Reference Handbook | SQA 音乐公式定理速查手册
This quick reference handbook distils the essential 'formulas' and 'theorems' of music theory for Year 10 SQA learners. Think of each concept not as a vague artistic rule but as a precise equation — semitone counts for intervals, step patterns for scales, stacked ratios for chords, and logic tricks for key signatures. Internalising these patterns will transform your sight‑reading, aural skills and composing, much like memorising mathematical identities accelerates problem‑solving. Use this collection as your rapid revision toolkit, whether you are preparing for Understanding Music listening papers or consolidating your practical work.
本速查手册为 Year 10 SQA 学习者提炼了音乐理论中最核心的“公式”与“定理”。请将每一个概念看作一个精确的方程式，而非模糊的艺术法则——音程的半音计数、音阶的音程排列、和弦的堆叠比例以及调号的逻辑窍门。内化这些模式将大大提升你的视奏、听觉和作曲能力，正如熟记数学恒等式能加速解题一般。无论是备战 Understanding Music 听力试卷，还是巩固实践作品，均可将这本手册用作快速复习的工具箱。
1. Interval Formulas | 音程公式
An interval is the pitch distance between two notes, measured in semitones (half steps). The quality and number can be derived from a simple lookup table. For the perfect intervals (unison, 4th, 5th, octave), the pure form is 'perfect'. For seconds, thirds, sixths and sevenths, we distinguish between major and minor. A major interval becomes diminished if lowered by one semitone from perfect/minor, or augmented if raised by one semitone from perfect/major.
音程是两个音之间的音高距离，以半音（半音阶）为单位。性质和度数可通过一个简明的速查表得出。纯音程（同度、四度、五度、八度）的原始状态为“纯”；二度、三度、六度、七度则区分大、小。若比纯/小音程降低一个半音，则变为减音程；若比纯/大音程升高一个半音，则变为增音程。
| Interval | Semitones | Common example |
|---|---|---|
| Perfect Unison (P1) | 0 | C – C |
| Minor 2nd (m2) | 1 | E – F |
| Major 2nd (M2) | 2 | C – D |
| Minor 3rd (m3) | 3 | D – F |
| Major 3rd (M3) | 4 | C – E |
| Perfect 4th (P4) | 5 | C – F |
| Tritone (A4 / d5) | 6 | F – B |
| Perfect 5th (P5) | 7 | C – G |
| Minor 6th (m6) | 8 | E – C |
| Major 6th (M6) | 9 | C – A |
| Minor 7th (m7) | 10 | D – C |
| Major 7th (M7) | 11 | C – B |
| Perfect Octave (P8) | 12 | C – C |
2. Key Signature Formulas | 调号公式
Key signatures indicate which notes are consistently sharp or flat. The order of sharps is F♯, C♯, G♯, D♯, A♯, E♯, B♯, and the order of flats is B♭, E♭, A♭, D♭, G♭, C♭, F♭ — simply the reverse. You can find the tonic major key with two quick rules: for sharps, the last sharp raised by a semitone gives the major key name. For flats, the second‑last flat is the major key (except F major, which has one flat).
调号标示乐曲中固定升高或降低的音。升号出现的顺序为 F♯, C♯, G♯, D♯, A♯, E♯, B♯，降号顺序为 B♭, E♭, A♭, D♭, G♭, C♭, F♭，两者恰好互为逆行。可通过两条速查规则推断大调主音：升号调中，最后一个升号升高半音即为主音；降号调中，倒数第二个降号即为主音（一个降号的 F 大调除外）。
Sharp key rule: Major key = last sharp + 1 semitone. E.g., if the key signature contains F♯, C♯, G♯, the last sharp is G♯; G♯ + 1 semitone = A → A major.
升号调规则：大调 = 最后一个升号音 + 1 半音。例如，若调号含 F♯, C♯, G♯，最后一个升号为 G♯；G♯ + 半音 = A → A 大调。
Flat key rule: Major key = second‑last flat. For B♭, E♭, A♭, the second‑last flat is E♭ → E♭ major. For one flat (B♭), the key is F major by memory.
降号调规则：大调 = 倒数第二个降号。如 B♭, E♭, A♭，倒数第二个为 E♭ → 降 E 大调。一个降号(B♭)时直接记忆为 F 大调。
3. Scale Formulas | 音阶公式
Every scale can be expressed as a sequence of tones (T, whole step) and semitones (S, half step), or by a numeric semitone pattern from the tonic. These formulas are the same for every key, making them powerful transposition tools.
任何一种音阶都可以用全音（T）和半音（S）的序列，或从主音出发的半音数字模式来表示。这些公式在所有调中保持不变，因此是强有力的移调工具。
Major scale formula: T T S T T T S (semitone sequence: 0, 2, 4, 5, 7, 9, 11, 12).
大调音阶公式：全 全 半 全 全 全 半（半音序列：0, 2, 4, 5, 7, 9, 11, 12）。
Natural minor formula: T S T T S T T (semitone sequence: 0, 2, 3, 5, 7, 8, 10, 12).
自然小调公式：全 半 全 全 半 全 全（半音序列：0, 2, 3, 5, 7, 8, 10, 12）。
Harmonic minor: Raise the 7th degree by a semitone, creating an augmented 2nd between 6 and 7. (e.g., A harmonic minor: A B C D E F G♯ A).
和声小调：升高第七级音一个半音，使第六与第七级间形成增二度。（如 A 和声小调：A B C D E F G♯ A）。
Melodic minor (ascending): Raise the 6th and 7th degrees; descending uses the natural minor.
旋律小调（上行）：升高第六、七级音；下行时恢复为自然小调。
4. Chord Building Formulas | 和弦构建公式
A chord formula states the exact number of semitones between consecutive notes above the root. Triads are built by stacking two thirds, and seventh chords add another third on top. The quality of each third determines the chord type.
和弦公式明确指出根音上方各相邻音之间的半音数。三和弦由两个三度叠加而成，七和弦再追加一个三度。每个三度的性质决定了和弦类型。
| Chord type | Formula (semitones from root) | Example on C |
|---|---|---|
| Major triad | 0 – 4 – 3 | C – E – G |
| Minor triad | 0 – 3 – 4 | C – E♭ – G |
| Diminished triad | 0 – 3 – 3 | C – E♭ – G♭ |
| Augmented triad | 0 – 4 – 4 | C – E – G♯ |
| Dominant 7th | 0 – 4 – 3 – 3 | C – E – G – B♭ |
| Major 7th | 0 – 4 – 3 – 4 | C – E – G – B |
| Minor 7th | 0 – 3 – 4 – 3 | C – E♭ – G – B♭ |
To build a chord in any key, simply take the root note and count the semitones upwards according to the formula.
在任何调中构建和弦，只需取根音并按公式向上数出相应的半音数即可。
5. Note and Rest Values | 音符与休止符时值
chet (quarter note) 1 beat, and a quaver (eighth note) ½ beat. Each shorter value halves the beat. Adding a dot increases the note's duration by 50% ormula is dotted note = original value × 1.5. A tie adds the values of two notes together.
节奏时值遵循简洁的算术关系。在 4/4 拍中，全音符为 4 拍，二分音符为 2 拍，四分音符为 1 拍，八分音符为 ½ 拍。每缩短一级，时值减半。附点将原时值增加一半：公式为附点音符 = 原时值 × 1.5。延音线则把两个音符的时值相加。
Dotted crotchet value: 1 + 0.5 = 1.5 crotchet beats.
附点四分音符时值：1 + 0.5 = 1.5 个四分音符拍。
Triplet rule: A triplet bracket tells you to play three notes evenly in the time normally taken by two of the same note value. For example, a quaver triplet in 4/4 fits three quavers into one crotchet beat.
三连音规则：三连音括线表示在通常演奏两个相同音符的时值内，均匀地奏出三个音。例如 4/4 拍中的八分音符三连音，将三个八分音符填入一个四分音符的拍中。
6. Time Signatures | 拍号公式
A time signature is not a fraction, but it can be read with a formula: the top number tells how many beats per bar; the bottom number tells the type of note that gets one beat (4 stands for a crotchet, 8 stands for a quaver). In simple time, each beat divides naturally into two; in compound time, each beat divides into three.
拍号并非分数，但可按公式理解：上方数字表示每小节的拍数，下方数字表示以何种音符为一拍（4 代表四分音符，8 代表八分音符）。在单拍子中，每一拍自然二均分；在复拍子中，每一拍三均分。
Simple duple/triple/quadruple: beats are 2, 3, or 4 crotchets.
单二拍/单三拍/单四拍：每小节 2、3 或 4 个四分音符拍。
Compound duple: 6/8 has two dotted‑crotchet beats, each dividing into three quavers.
复二拍：6/8 拍有两个附点四分音符拍，每拍再分为三个八分音符。
Compound triple: 9/8 = three dotted‑crotchet beats.
复三拍：9/8 = 三个附点四分音符拍。
Compound quadruple: 12/8 = four dotted‑crotchet beats.
复四拍：12/8 = 四个附点四分音符拍。
7. Tempo and Dynamics Quick Reference | 速度与力度术语速查
Italian terms are the standard musical 'commands'. Treat them as living vocabulary to be recognised instantly. Tempo markings give a beats‑per‑minute (bpm) range, while dynamics indicate volume and its changes.
意大利术语是标准音乐“指令”，应将其视为须立即识别的活词汇。速度标记给出近似 bpm 范围，力度标记则指示音量及其变化。
| Term | Meaning | Term | Meaning |
|---|---|---|---|
| Largo | Very slow, broad | pp (pianissimo) | Very soft |
| Adagio | Slow and stately | p (piano) | Soft |
| Andante | Walking pace | mp (mezzo‑piano) | Moderately soft |
| Moderato | Moderate | mf (mezzo‑forte) | Moderately loud |
| Allegro | Fast, lively | f (forte) | Loud |
| Presto | Very fast | ff (fortissimo) | Very loud |
Crescendo (cresc.): gradually louder. Diminuendo (dim.): gradually softer.
Crescendo（渐强）：音量逐渐增强。Diminuendo（渐弱）：音量逐渐减弱。
8. Cadence Formulas | 终止式公式
A cadence is a two‑chord progression that ends a phrase, acting like musical punctuation. The Roman numeral formula reveals the harmonic function. Use scale degrees as I, IV, V, vi (lower‑case for minor).
终止式是乐句结尾的两个和弦进行，起到音乐标点的作用。用罗马数字标示的公式揭示了和声功能。音阶级数记作 I、IV、V、vi（小写表示小调）。
Perfect cadence (V – I): Dominant to tonic; sounds final and complete. In C major: G – C.
正格终止（V – I）：属到主；听感完满终结。C 大调中：G – C。
Plagal cadence (IV – I): Subdominant to tonic; known as the 'Amen' cadence. In C major: F – C.
变格终止（IV – I）：下属到主；常称“阿门终止”。C 大调中：F – C。
Imperfect cadence (I – V, ii – V, IV – V): Ends on the dominant, sounding unfinished.
不完全终止（I – V、ii – V、IV – V）：停于属和弦，听感未完结。
Interrupted / Deceptive cadence (V – vi): The dominant resolves to the submediant instead of the tonic, creating surprise. In C major: G – Am.
阻碍终止（V – vi）：属和弦进行至下中音和弦，而非主和弦，产生意外效果。C 大调中：G – Am。
9. Interval Inversion Theorem | 音程转位定理
When you invert an interval — moving the lower note up an octave so that the upper note becomes the lower — a simple theorem applies: the sum of the original interval number and the inverted interval number always equals 9. Perfect intervals remain perfect; major becomes minor and vice versa; augmented becomes diminished and vice versa.
当转位音程时——将较低音移高八度，使原较高音变为低音——可运用一条简约定理：原音程度数与转位音程度数之和恒为 9。纯音程保持不变；大音程变为小音程，反之亦然；增音程变为减音程，反之亦然。
Inversion formula: Inverted number = 9 – original number. Example: 3rd inverts to 6th (9 – 3 = 6). The quality flips: major 3rd becomes minor 6th.
转位公式：转位度数 = 9 – 原度数。例：三度转位为六度（9 – 3 = 6）。性质翻转：大三度转为小六度。
Quick check: Perfect 5th inverts to Perfect 4th (9 – 5 = 4). Augmented 4th (tritone) inverts to diminished 5th (still 6 semitones).
快速验证：纯五度转位为纯四度（9 – 5 = 4）。增四度（三全音）转位为减五度（仍为 6 个半音）。
10. Chord Inversion Markings | 和弦转位标记
When a chord's lowest note is not the root, it is in inversion. Figured bass numbers, originally used in Baroque music, still provide the clearest formula for identifying inversions. Triads have two inversions; seventh chords have three.
当和弦的最低音不是根音时，即为转位和弦。数字低音标记虽源于巴洛克音乐，但至今仍是识别转位和弦最清晰的公式。三和弦有两种转位；七和弦有三种转位。
| Chord type | Inversion | Figured bass | Intervals above bass |
|---|---|---|---|
| Triad | Root position | 5/3 | 3rd and 5th |
| 1st inversion | 6/3
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