📚 Year 12 SQA Music: Formula & Theorem Quick Reference Handbook | Year 12 SQA 音乐：公式定理速查手册
This handbook distills the core ‘formulae’ of music theory that every Year 12 SQA Music student needs at their fingertips. From scale construction and chord building to cadence patterns and frequency ratios, these rules and templates will sharpen your analysis, composition, and aural skills. Use it as a quick-reference revision tool alongside your practical work.
本手册提炼了每位 Year 12 SQA 音乐学生需要烂熟于心的核心音乐理论”公式”。从音阶构建、和弦构成到终止式模式以及频率比，这些规则和模板将提升你的分析、创作与听觉训练能力。请将其作为实践练习的快速参考复习工具。
1. Interval Formula: Semitone Map | 音程公式：半音映射
Intervals are measured in semitones. The quality and number combine to name the interval. Below is the semitone count for intervals up to an octave: Unison (0), minor 2nd (1), Major 2nd (2), minor 3rd (3), Major 3rd (4), Perfect 4th (5), Tritone (6), Perfect 5th (7), minor 6th (8), Major 6th (9), minor 7th (10), Major 7th (11), Octave (12). Augmented intervals add a semitone, diminished intervals lose one.
音程按半音数测量。性质与度数结合构成音程名称。以下是八度以内音程的半音数：同度（0）、小二度（1）、大二度（2）、小三度（3）、大三度（4）、纯四度（5）、三全音（6）、纯五度（7）、小六度（8）、大六度（9）、小七度（10）、大七度（11）、八度（12）。增音程增加一个半音，减音程减少一个半音。
- Perfect unison: P1, 0 semitones | 纯一度：P1, 0 半音
- Major 2nd: M2, 2 semitones | 大二度：M2, 2 半音
- Minor 3rd: m3, 3 semitones | 小三度：m3, 3 半音
- Perfect 5th: P5, 7 semitones | 纯五度：P5, 7 半音
- Major 7th: M7, 11 semitones | 大七度：M7, 11 半音
2. Major Scale Formula: Whole and Half Steps | 大调音阶公式：全音与半音
The major scale is defined by the step pattern: Whole, Whole, Half, Whole, Whole, Whole, Half (W‑W‑H‑W‑W‑W‑H). Starting on any tonic, this sequence yields the major diatonic collection. In C major: C–D–E–F–G–A–B–C, showing semitones between E–F and B–C.
大调音阶由以下全半音模式定义：全、全、半、全、全、全、半（W‑W‑H‑W‑W‑W‑H）。从任意主音出发按此序列即得大调自然音集合。C大调：C–D–E–F–G–A–B–C，半音出现在E–F与B–C之间。
Tonic → W → W → H → W → W → W → H → Octave
| Scale Degree | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Interval above tonic | P1 | M2 | M3 | P4 | P5 | M6 | M7 | P8 |
3. Minor Scale Formulae: Natural, Harmonic, Melodic | 小调音阶公式：自然、和声、旋律
The natural minor scale follows the pattern W‑H‑W‑W‑H‑W‑W (Aeolian mode). The harmonic minor raises the 7th degree by a semitone, creating an augmented 2nd between the 6th and 7th degrees (W‑H‑W‑W‑H‑WH‑H). The melodic minor ascends with a raised 6th and 7th (W‑H‑W‑W‑W‑W‑H) and descends as natural minor.
自然小调音阶遵循全-半-全-全-半-全-全的模式（爱奥里亚调式）。和声小调升高第七级音半音，在第六与第七级之间形成增二度（全-半-全-全-半-增二度-半）。旋律小调上行升高第六和第七级（全-半-全-全-全-全-半），下行恢复为自然小调。
A natural minor: A–B–C–D–E–F–G–A (W‑H‑W‑W‑H‑W‑W)
A harmonic minor: A–B–C–D–E–F–G♯–A (W‑H‑W‑W‑H–3 semitones–H)
A melodic minor ascending: A–B–C–D–E–F♯–G♯–A (W‑H‑W‑W‑W‑W‑H)
4. Triad Formulae: Major, Minor, Augmented, Diminished | 三和弦公式：大、小、增、减
Triads are built by stacking two thirds on a root. The quality depends on the interval makeup: Major triad = M3 + m3 (root, M3, P5). Minor triad = m3 + M3 (root, m3, P5). Diminished triad = m3 + m3 (root, m3, d5). Augmented triad = M3 + M3 (root, M3, A5). In semitones above root: major 0‑4‑7, minor 0‑3‑7, diminished 0‑3‑6, augmented 0‑4‑8.
三和弦由根音上方叠加两个三度构成。性质取决于音程组合：大三和弦 = 大三度+小三度（根音、大三度、纯五度）；小三和弦 = 小三度+大三度（根音、小三度、纯五度）；减三和弦 = 小三度+小三度（根音、小三度、减五度）；增三和弦 = 大三度+大三度（根音、大三度、增五度）。以根音上方半音数表示：大 0‑4‑7，小 0‑3‑7，减 0‑3‑6，增 0‑4‑8。
| Triad type | Semitone pattern (root=0) | Example (C root) |
|---|---|---|
| Major | 0, 4, 7 | C–E–G |
| Minor | 0, 3, 7 | C–Eb–G |
| Diminished | 0, 3, 6 | C–Eb–Gb |
| Augmented | 0, 4, 8 | C–E–G♯ |
5. Seventh Chord Formulae | 七和弦公式
A seventh chord adds a fourth note a seventh above the root. The common types: Major 7th (Maj7) = M3 + m3 + M3 (0‑4‑7‑11). Dominant 7th = M3 + m3 + m3 (0‑4‑7‑10). Minor 7th (min7) = m3 + M3 + m3 (0‑3‑7‑10). Half‑diminished 7th (ø7) = m3 + m3 + M3 (0‑3‑6‑10). Fully diminished 7th (°7) = m3 + m3 + m3 (0‑3‑6‑9). The structure can also be expressed as triad + interval above root: e.g., dominant 7th = major triad + minor 7th.
七和弦在根音上方增加一个七度音。常见类型：大七和弦（Maj7）= 大三度+小三度+大三度（0‑4‑7‑11）；属七和弦 = 大三度+小三度+小三度（0‑4‑7‑10）；小七和弦（min7）= 小三度+大三度+小三度（0‑3‑7‑10）；半减七和弦（ø7）= 小三度+小三度+大三度（0‑3‑6‑10）；减七和弦（°7）= 小三度+小三度+小三度（0‑3‑6‑9）。结构也可表示为三和弦加七度音程：如属七和弦 = 大三和弦 + 小七度。
Cmaj7: C–E–G–B (0‑4‑7‑11) C7: C–E–G–Bb (0‑4‑7‑10)
6. Circle of Fifths and Key Signature Formula | 五度循环与调号公式
The circle of fifths arranges keys by the number of sharps or flats. Starting from C (0 sharps/flats), ascending by perfect 5ths adds a sharp at each step (G 1♯, D 2♯, A 3♯, E 4♯, B 5♯, F♯ 6♯, C♯ 7♯). Descending by perfect 5ths (or ascending by perfect 4ths) adds a flat (F 1b, Bb 2b, Eb 3b, Ab 4b, Db 5b, Gb 6b, Cb 7b). The order of sharps: F♯ C♯ G♯ D♯ A♯ E♯ B♯ (Father Charles Goes Down And Ends Battle). Order of flats: Bb Eb Ab Db Gb Cb Fb (Battle Ends And Down Goes Charles’s Father). The relative minor is a minor 3rd below the major tonic.
五度循环按升降号数目排列调性。从C大调（0个升降号）开始，向上纯五度每步增加一个升号（G 1♯, D 2♯, A 3♯, E 4♯, B 5♯, F♯ 6♯, C♯ 7♯）。向下纯五度（或向上纯四度）每步增加一个降号（F 1b, Bb 2b, Eb 3b, Ab 4b, Db 5b, Gb 6b, Cb 7b）。升号次序：F♯ C♯ G♯ D♯ A♯ E♯ B♯。降号次序：Bb Eb Ab Db Gb Cb Fb。关系小调在主音下方小三度。
Relative minor = Major tonic – 3 semitones (e.g., C major → A minor)
7. Cadence Formulae: Perfect, Plagal, Imperfect, Interrupted | 终止式公式：正格、变格、不完全、阻碍
Cadences are chord progressions that end a phrase. Perfect cadence: V → I (or V7 → I) giving a strong sense of resolution. Plagal cadence: IV → I (the ‘Amen’ cadence). Imperfect cadence: any chord → V, ending on the dominant, sounding incomplete. Interrupted (deceptive) cadence: V → vi (or V → VI in minor), surprising the ear. In Roman numerals, the bass and soprano movement often follows voice‑leading conventions.
终止式是结束乐句的和弦进行。正格终止：V → I（或V7 → I），具有强烈的解决感。变格终止：IV → I（“阿门”终止）。不完全终止：任意和弦 → V，结束在属和弦上，听感未完。阻碍终止：V → vi（或小调V → VI），给人以意外。罗马数字标记中，低音与高声部进行常遵循声部写作规则。
- Perfect: G major (V) → C major (I) | 正格：G大和弦(V) → C大和弦(I)
- Plagal: F major (IV) → C major (I) | 变格：F大和弦(IV) → C大和弦(I)
- Imperfect: C major (I) → G major (V) | 不完全：C大和弦(I) → G大和弦(V)
- Interrupted: G major (V) → A minor (vi) | 阻碍：G大和弦(V) → A小和弦(vi)
8. Non‑Chord Tone Formulae: Passing, Neighbour, Suspension | 和弦外音公式：经过音、辅助音、延留音
Non‑chord tones are melodic notes not part of the underlying harmony, approached and resolved by step. Passing tone: stepwise motion connecting two chord tones in the same direction (e.g., chord tone – passing tone – chord tone). Neighbour tone (auxiliary): step away from a chord tone and return to the same tone (upper or lower). Suspension: a note held over from the previous chord (preparation) that creates a dissonance on the strong beat and resolves down by step (e.g., 4‑3 suspension). Appoggiatura: approached by leap and resolved by step. Escape tone: approached by step and resolved by leap in opposite direction.
和弦外音是不属于当前和声的旋律音，以级进引入并解决。经过音：同向级进连接两个和弦音（和弦音-经过音-和弦音）。辅助音（邻音）：从和弦音级进出去再回到原音（上辅助或下辅助）。延留音：前和弦的音延续并停留（预备），在强拍形成不协和，下行级进解决（如4‑3延留）。倚音：跳进引入、级进解决。跳进辅助音：级进引入、反向跳进解决。
Suspension formula: preparation (consonant) – suspension (dissonant) – resolution (step down)
9. Inversions and Figured Bass Shorthand | 转位与数字低音速记
Chords can be inverted by placing a note other than the root in the bass. Triads: root position (5/3), first inversion (6 or 6/3), second inversion (6/4). Seventh chords: root position (7/5/3, often abbreviated 7), first inversion (6/5), second inversion (4/3), third inversion (4/2 or simply 2). The numbers represent the intervals above the bass note. For example, a first‑inversion major triad has a 3rd and a 6th above the bass, hence 6.
和弦可通过将非根音放在低音部进行转位。三和弦：原转位（5/3）、第一转位（6或6/3）、第二转位（6/4）。七和弦：原转位（7/5/3，常简写为7）、第一转位（6/5）、第二转位（4/3）、第三转位（4/2或简写为2）。数字表示低音上方音程。例如，三和弦第一转位低音上方有3度和6度，故标记为6。
- C major root: C‑E‑G (5/3) | C大调原位：C‑E‑G (5/3)
- C major 1st inversion: E‑G‑C (6) | C大调第一转位：E‑G‑C (6)
- C major 2nd inversion: G‑C‑E (6/4) | C大调第二转位：G‑C‑E (6/4)
- Dominant 7th 1st inversion (V6/5 in C): B‑D‑F‑G | 属七第一转位（C大调V6/5）：B‑D‑F‑G
10. Modes of the Major Scale: Interval Constructions | 大调调式：音程构造
The modes are seven scales sharing the same notes as a major scale but starting on a different degree. Ionian (1st): W‑W‑H‑W‑W‑W‑H (major). Dorian (2nd): W‑H‑W‑W‑W‑H‑W (minor with raised 6th). Phrygian (3rd): H‑W‑W‑W‑H‑W‑W (minor with lowered 2nd). Lydian (4th): W‑W‑W‑H‑W‑W‑H (major with raised 4th). Mixolydian (5th): W‑W‑H‑W‑W‑H‑W (major with lowered 7th). Aeolian (6th): W‑H‑W‑W‑H‑W‑W (natural minor). Locrian (7th): H‑W‑W‑H‑W‑W‑W (diminished, lowered 2nd and 5th).
调式是共享大调音阶各音但以不同音级为主音的七种音阶。伊奥尼亚（第1级）：全‑全‑半‑全‑全‑全‑半（大调）。多利亚（第2级）：全‑半‑全‑全‑全‑半‑全（升高六度的小调）。弗里几亚（第3级）：半‑全‑全‑全‑半‑全‑全（降低二度的小调）。利底亚（第4级）：全‑全‑全‑半‑全‑全‑半（升高四度的大调）。混合利底亚（第5级）：全‑全‑半‑全‑全‑半‑全（降低七度的大调）。爱奥里亚（第6级）：全‑半‑全‑全‑半‑全‑全（自然小调）。洛克里亚（第7级）：半‑全‑全‑半‑全‑全‑全（减音阶，降2级和降5级）。
D Dorian (D‑E‑F‑G‑A‑B‑C‑D): W‑H‑W‑W‑W‑H‑W
11. Rhythm and Metre Formulae: Dots, Ties, Tuplets | 节奏与节拍公式：附点、连线、连音
Rhythmic formulae govern note durations. A dot adds half the value of the note it follows: dotted crotchet = crotchet + quaver (1.5 beats in simple time). A tie connects two notes of the same pitch, summing their durations. A triplet brackets three notes in the time of two: in simple time, three quavers in the space of two. Similarly, duplets occur in compound time: two notes in the time of three. The time signature top number indicates beats per bar; the bottom number the beat unit (4 = crotchet, 8 = quaver). Simple time has beats divisible by 2; compound time beats divide into 3.
节奏公式管控音符时值。附点增加原音符时值的一半：附点四分音符 = 四分音符 + 八分音符（单拍子中1.5拍）。连线连接同音高音符，总和时值。三连音在通常两音的时间内演奏三个音：单拍子中，三个八分音符代替两个八分音符。同样，复拍子中出现二连音：二音代替三音。拍号上方数字表示每小节拍数；下方数字表示拍的单位（4为四分音符，8为八分音符）。单拍子每拍可二等分；复拍子每拍可三等分。
Dotted minim = minim + crotchet. Triplet quaver group = 3 quavers in time of 2.
12. Frequency and Intonation Formula (12‑Tone Equal Temperament) | 频率与音准公式（十二平均律）
In equal temperament, the octave is divided into 12 equal semitones. The frequency of a note n semitones above a reference pitch f₀ is given by f = f₀ × 2^(n/12). For example, A4 = 440 Hz, then A♯4 = 440 × 2^(1/12) ≈ 466.16 Hz. The frequency ratio of a perfect 5th is 2^(7/12) ≈ 1.498, close to the just ratio 3/2. The cent is a logarithmic unit where one octave = 1200 cents, one semitone = 100 cents. This formula underpins tuning and harmonic understanding.
十二平均律将八度分为12个相等的半音。高于基准音f₀ n个半音的音高频率为 f = f₀ × 2^(n/12)。例如A4=440 Hz，则A♯4 = 440 × 2^(1/12) ≈ 466.16 Hz。纯五度的频率比为2^(7/12) ≈ 1.498，接近纯律比3/2。音分是对数单位，一个八度为1200音分，一个半音为100音分。此公式是调律与和声感的基础。
f = f₀ × 2^(n/12) (n = number of semitones from reference)
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