📚 Year 13 SQA Music: Formula & Theorem Quick Reference Handbook | SQA Year 13 音乐公式定理速查手册
This quick reference handbook brings together the essential formulas, rules, and structural principles that underpin the Year 13 SQA Music course. From interval calculation and chord construction to key signatures and transposition, these musical ‘theorems’ provide a reliable toolkit for analysis, composition, and listening. Use it to consolidate your understanding and to sharpen your accuracy under exam conditions.
本速查手册汇集了 SQA Year 13 音乐课程所依赖的核心公式、规则和结构原理。从音程计算、和弦构建，到调号与移调，这些音乐‘定理’为分析、创作与听力理解提供了可靠的工具包。利用它来巩固你的理解，并在考试条件下提升准确性。
1. Interval Formulas | 音程公式
An interval is defined by the number of semitones (half steps) between two notes and their staff-degree separation. The quality—perfect, major, minor, augmented, diminished—follows a precise semitone count. In a major scale from the tonic, all intervals are either major or perfect: M2 (2 semitones), M3 (4), P4 (5), P5 (7), M6 (9), M7 (11), P8 (12).
音程由两个音之间的半音（半音级）数量及其谱表度数距离来定义。音程性质——纯、大、小、增、减——遵循精确的半音数量。从主音出发的自然大调音阶中，所有音程要么是大音程，要么是纯音程：大二度为2个半音，大三度为4个半音，纯四度5半音，纯五度7半音，大六度9半音，大七度11半音，纯八度12半音。
To identify an interval outside the major scale, compare its semitone count to the major/perfect reference. A minor interval is one semitone smaller; a diminished interval is one semitone smaller than minor or perfect; an augmented interval is one semitone larger than major or perfect. The formula for any interval can be written as: Interval = reference quality ± semitone alteration.
要识别大调音阶以外的音程，请将其半音数与对应的纯/大音程基准进行比较。小音程比大音程少一个半音；减音程比小音程或纯音程少一个半音；增音程比大音程或纯音程多一个半音。任何音程的公式均可表示为：音程 = 基准性质 ± 半音变化量。
| Interval | Semitones | Example from C |
| Perfect unison (P1) | 0 | C |
| Minor 2nd (m2) | 1 | D♭ |
| Major 2nd (M2) | 2 | D |
| Minor 3rd (m3) | 3 | E♭ |
| Major 3rd (M3) | 4 | E |
| Perfect 4th (P4) | 5 | F |
| Tritone (A4/d5) | 6 | F♯ / G♭ |
| Perfect 5th (P5) | 7 | G |
| Minor 6th (m6) | 8 | A♭ |
| Major 6th (M6) | 9 | A |
| Minor 7th (m7) | 10 | B♭ |
| Major 7th (M7) | 11 | B |
| Perfect octave (P8) | 12 | C |
上表罗列了从C出发的基本音程及其半音总数。当音程包含升降号时，只需调整半音数以及对应的谱表度数，并据此修正性质名称。
2. Interval Inversion Rules | 音程转位规则
When an interval is inverted, the lower note is raised by an octave or the upper note is lowered by an octave. The inversion follows this numeric formula: original quantity + inverted quantity = 9. For example, a 3rd inverts to a 6th (3+6=9), and a 7th inverts to a 2nd.
当音程转位时，将下方音升高八度或将上方音降低八度。转位遵循以下数字公式：原度数 + 转位度数 = 9。例如，三度转位变成六度（3+6=9），七度转位变成二度。
Quality also transforms predictably: major becomes minor, minor becomes major, augmented becomes diminished, diminished becomes augmented. Perfect intervals remain perfect. Thus a M6 inverts to a m3, a P4 inverts to a P5, and a d5 inverts to an A4.
性质也有规律地转换：大音程变成小音程，小音程变成大音程，增音程变成减音程，减音程变成增音程。纯音程保持纯。因此，大六度转位为小三度，纯四度转位为纯五度，减五度转位为增四度。
3. Scale Formulas | 音阶公式
Scales are built using a pattern of tones (T, whole step) and semitones (S, half step). The major scale formula is: T – T – S – T – T – T – S. Starting on C, this yields C D E F G A B C, using only white notes.
音阶通过全音（T，全音级）和半音（S，半音级）的特定模式构建。大调音阶公式为：T – T – S – T – T – T – S。从C开始，得到C D E F G A B C，只使用白键。
The natural minor scale uses the pattern: T – S – T – T – S – T – T, equivalent to a major scale starting on its 6th degree. The harmonic minor scale raises the 7th degree, creating a distinctive augmented 2nd (A2, 3 semitones) between the 6th and 7th degrees: T – S – T – T – S – A2 – S.
自然小调音阶使用模式：T – S – T – T – S – T – T，等同于从大调第六级音开始的音阶。和声小调升高第七级音，在第六级与第七级之间形成一个特色的增二度（A2，3个半音）：T – S – T – T – S – A2 – S。
The melodic minor ascending pattern is: T – S – T – T – T – T – S (raised 6th and 7th), while descending it reverts to the natural minor. Whole-tone scales consist entirely of tones: T – T – T – T – T, creating a 6-note symmetrical scale.
旋律小调上行公式为：T – S – T – T – T – T – S（升高第六、七级），下行则还原为自然小调。全音音阶完全由全音构成：T – T – T – T – T，形成一个六音对称音阶。
4. Triad Construction Formulas | 三和弦构建公式
A triad is formed by stacking two thirds on a root. The combination of major and minor thirds defines the triad quality. The semitone formulas measured from the root are:
三和弦是在根音上方叠加两个三度音构成。大三度与小三度的组合决定了三和弦的性质。从根音测量的半音公式如下：
- Major triad: root + M3 (4 semitones) + P5 (7 semitones) → stacked M3 + m3. Example: C–E–G.
- 大三和弦：根音 + 大三度（4半音）+ 纯五度（7半音）→ 叠加结构为 M3 + m3。示例：C–E–G。
- Minor triad: root + m3 (3 semitones) + P5 (7 semitones) → stacked m3 + M3. Example: A–C–E.
- 小三和弦：根音 + 小三度（3半音）+ 纯五度（7半音）→ 叠加结构为 m3 + M3。示例：A–C–E。
- Diminished triad: root + m3 (3) + d5 (6 semitones) → stacked m3 + m3. Example: B–D–F.
- 减三和弦：根音 + 小三度（3）+ 减五度（6半音）→ 叠加结构为 m3 + m3。示例：B–D–F。
- Augmented triad: root + M3 (4) + A5 (8 semitones) → stacked M3 + M3. Example: C–E–G♯.
- 增三和弦：根音 + 大三度（4）+ 增五度（8半音）→ 叠加结构为 M3 + M3。示例：C–E–G♯。
In Roman numeral analysis within a major key, triads on scale degrees are: I (major), ii (minor), iii (minor), IV (major), V (major), vi (minor), vii° (diminished). This pattern itself is a useful formula for harmonic analysis.
在大调的罗马数字分析中，各音级上的三和弦为：I（大三），ii（小三），iii（小三），IV（大三），V（大三），vi（小三），vii°（减三）。这一模式本身就是一个有用的和声分析公式。
5. Seventh Chord Construction Formulas | 七和弦构建公式
Adding a fourth note (a seventh above the root) to a triad creates a seventh chord. The quality depends on the triad type and the seventh interval. Semitone distances from the root are given below:
在三和弦基础上添加第四个音（根音上方七度音）即构成七和弦。其性质取决于三和弦类型以及七度音程。从根音开始的半音距离如下：
| Chord | Formula from root | Semitones (root-M3-P5-7th) | Example (on C) |
| Major 7th | Major triad + M7 | 0-4-7-11 | C–E–G–B |
| Dominant 7th | Major triad + m7 | 0-4-7-10 | C–E–G–B♭ |
| Minor 7th | Minor triad + m7 | 0-3-7-10 | C–E♭–G–B♭ |
| Half-diminished 7th (m7♭5) | Diminished triad + m7 | 0-3-6-10 | C–E♭–G♭–B♭ |
| Diminished 7th | Diminished triad + d7 | 0-3-6-9 | C–E♭–G♭–B𝄫 (A) |
牢记这些半音公式可以让你在任何一个根音上快速构建七和弦，无需依赖键盘记忆。属七和弦因其三全音（根音与七音之间6个半音）而具有强烈的解决倾向。
6. Chord Inversion Symbols | 和弦转位标记
Inversions are indicated by figured bass symbols that show the intervals above the lowest sounding note. For triads: root position uses ⁵₃ (often omitted), first inversion uses ⁶ or ⁶₃, second inversion uses ⁶₄. In a C major triad, first inversion is E–G–C (third in bass), and second inversion is G–C–E (fifth in bass).
转位通过数字低音标记来表示低音上方构成的音程。三和弦：原位使用⁵₃（常省略），第一转位用⁶ 或 ⁶₃，第二转位用⁶₄。以C大三和弦为例，第一转位为E–G–C（三音在低音），第二转位为G–C–E（五音在低音）。
For seventh chords: root position uses ⁷, first inversion ⁶₅, second inversion ⁴₃, third inversion ⁴₂ (or simply ²). These numbers reflect the pitch classes above the bass note: e.g., a ⁴₂ inversion has a second and a fourth above the bass. The formula is fixed regardless of the chord quality.
七和弦：原位使用⁷，第一转位⁶₅，第二转位⁴₃，第三转位⁴₂（或简写为²）。这些数字反映了低音上方的音级结构：例如，⁴₂转位表示低音上方有二度和四度。无论和弦性质如何，这一公式均固定不变。
7. Key Signatures and the Circle of Fifths | 调号与五度圈
The circle of fifths is a visual theorem for key relationships. Progessing clockwise by ascending perfect fifths adds one sharp each step: G (1♯: F♯), D (2♯: F♯ C♯), A (3♯), E (4♯), B (5♯), F♯ (6♯), C♯ (7♯). The order of sharps follows the mnemonic: Father Charles Goes Down And Ends Battle (F♯ C♯ G♯ D♯ A♯ E♯ B♯).
五度圈是调性关系的视觉化定理。沿顺时针方向每升高一个纯五度便增加一个升号：G（1♯：F♯），D（2♯：F♯ C♯），A（3♯），E（4♯），B（
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