📚 Year 11 SQA Music: Essential Formulas and Theorems Quick Reference | 音乐公式定理速查手册
Music is a language built on precise patterns, intervals, and predictable sonic relationships. In the SQA National 5 and Higher Music courses, a solid grasp of these 'formulas' – from scale construction to chord voicings – transforms listening, performing, and composing from guesswork into confident application. This quick reference collects the essential numerical and structural rules you will encounter in the exam, in your folio, and in the performance hall. Bookmark it, memorise it, and let the maths behind the music unlock every mark.
音乐是建立在精确模式、音程和可预测的声音关系之上的语言。在 SQA National 5 和 Higher 音乐课程中，牢牢掌握这些“公式”——从音阶构建到和弦排列——能将听力、表演和作曲从猜测变为胸有成竹的应用。这份速查手册汇集了你在考试、作品集和演奏厅中将遇到的所有关键数字与结构规则。把它加入书签，牢记于心，让音乐背后的数学为你拿下每一分。
1. Note Values and Time Signatures | 音符时值与拍号
In simple time, a semibreve (whole note) equals 4 crotchet beats. A dotted minim lasts 3 crotchet beats, a minim 2, a crotchet 1, a quaver ½, and a semiquaver ¼. The upper number of a time signature tells you how many beats are in a bar; the lower number tells you what type of note gets one beat. For example, in 4/4 there are 4 crotchet beats per bar; in 6/8 (compound duple time) there are 2 dotted crotchet beats, each dividing into 3 quavers.
在单拍子中，全音符等于 4 个四分音符的时值。附点二分音符持续 3 拍，二分音符 2 拍，四分音符 1 拍，八分音符半拍，十六分音符四分之一拍。拍号的分子表示每小节有多少拍；分母表示以什么音符为一拍。例如，4/4 拍表示每小节有 4 个四分音符拍；6/8 拍（复二拍）每小节有 2 个附点四分音符拍，每个细分为 3 个八分音符。
- Dotted note rule: adds half the value of the note it follows. Dotted crotchet = 1 + ½ = 1½ crotchet beats. / 附点规则：附点增加前面音符时值的一半。附点四分音符 = 1 + ½ = 1½ 拍。
- Compound time formula: mber represents the note value for each division. 9/8 = 3 dotted crotchet beats per bar. / 复拍子公式：分子 ÷ 3 = 主要拍数；分母表示细分的音符时值。9/8 = 每小节 3 个附点四分音符拍。
2. Key Signatures and Circle of Fifths | 调号与五度圈
Sharp keys follow the order Father Charles Goes Down And Ends Battle (F♯, C♯, G♯, D♯, A♯, E♯, B♯). For flat keys, the order reverses: Battle Ends And Down Goes Charles' Father (B♭, E♭, A♭, D♭, G♭, C♭, F♭ nd the major key from a sharp signature, take the last sharp and go up a semitone (e.g., F♯ → G major). For flat keys, the second-last flat is the major key (e.g., B♭, E♭, A♭ → E♭ major). The relative minor is always down a minor 3rd from the major tonic (3 semitones).
升号调的排列顺序是 Father Charles Goes Down And Ends Battle（F♯, C♯, G♯, D♯, A♯, E♯, B♯）。降号调则相反：Battle Ends And Down Goes Charles' Father（B♭, E♭, A♭, D♭, G♭, C♭, F♭）。根据升号调号找大调的方法是，取最后一个升号上行一个半音（例如 F♯ → G 大调）。降号调则看倒数第二个降号就是大调名（如 B♭, E♭, A♭ → E♭ 大调）。关系小调总是在大调主音下方小三度（3 个半音）。
- Sharp key formula: Tonic = last sharp + 1 semitone. / 升号调口诀：主音 = 最后一个升号 + 1 个半音。
- Flat key formula: Tonic = penultimate flat in key signature. / 降号调口诀：主音 = 调号中倒数第二个降号。
- Relative minor distance: Major tonic – 3 semitones = relative minor tonic. / 关系大小调音程：大调主音 – 3 个半音 = 关系小调主音。
3. Major and Minor Scales Formula | 大小调音阶公式
A major scale is built using the pattern Tone–Tone–Semitone–Tone–Tone–Tone–Semitone (T-T-S-T-T-T-S). Starting on C, this gives all natural notes. The natural minor scale follows T-S-T-T-S-T-T (e.g., A natural minor: A–B–C–D–E–F–G–A). Harmonic minor raises the 7th degree by a semitone (T-S-T-T-S-3S-S), creating the characteristic augmented 2nd between degrees 6 and 7. Melodic minor ascending raises both the 6th and 7th (T-S-T-T-T-T-S), and descending reverts to the natural minor pattern.
大调音阶的结构是全音-全音-半音-全音-全音-全音-半音（T-T-S-T-T-T-S）。从 C 开始，便得到所有白键音。自然小调的模式为全-半-全-全-半-全-全（T-S-T-T-S-T-T），例如 a 自然小调：A-B-C-D-E-F-G-A。和声小调升高第 7 级音半音（T-S-T-T-S-3S-S），在第 6 级与第 7 级之间形成一个增二度。旋律小调上行时升高第 6 和第 7 级（T-S-T-T-T-T-S），下行则还原为自然小调。
- Major scale formula (tones and semitones): T–T–S–T–T–T–S. / 大调公式：全-全-半-全-全-全-半。
- Harmonic minor formula: 1 2 ♭3 4 5 ♭6 7 8. Intervals (in semitones from tonic): 0-2-3-5-7-8-11-12. / 和声小调半音序列：0-2-3-5-7-8-11-12。
4. Intervals: Semitones and Qualities | 音程：半音数与性质
ajor 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, perfect octave = 12. The numerical size (2nd, 3rd, etc.) depends on the letter names, but the quality depends on the exact semitone count and context.
音程的性质由两音之间的半音数决定。纯一度 = 0 个半音，小二度 = 1，大二度 = 2，小三度 = 3，大三度 = 4，纯四度 = 5，增四度/减五度（三全音）= 6，纯五度 = 7，小六度 = 8，大六度 = 9，小七度 = 10，大七度 = 11，纯八度 = 12。音程的度数（如二度、三度）取决于字母名，但性质由精确的半音数与上下文决定。
| Interval / 音程 | Semitones / 半音 | Example from C / 从 C 举例 |
|---|---|---|
| Perfect unison / 纯一度 | 0 | C–C |
| Minor 2nd / 小二度 | 1 | C–D♭ |
| Major 2nd / 大二度 | 2 | C–D |
| Major 3rd / 大三度 | 4 | C–E |
| Perfect 5th / 纯五度 | 7 | C–G |
| Perfect octave / 纯八度 | 12 | C–C' |
5. Triad and Seventh Chord Construction | 三和弦与七和弦构建
triad flattens the 3rd: root + 3 + 4 (minor 3rd + major 3rd). A diminished triad has two minor 3rds: root + 3 + 3; an augmented triad has two major 3rds: root + 4 + 4. Seventh chords add a third above the triad. Dominant 7th = major triad + minor 7th (root + 4 + 3 + 3 semitones); major 7th = major triad + major 7th (root + 4 + 3 + 4); minor 7th = minor triad + minor 7th (root + 3 + 4 + 3).
三和弦通过叠置两个三度构成。大三和弦包含大三度（4 个半音）和纯五度（7 个半音）：根音 + 4 + 3。小三和弦将三音降低半音：根音 + 3 + 4（小三度 + 大三度）。减三和弦由两个小三度叠置：根音 + 3 + 3；增三和弦由两个大三度叠置：根音 + 4 + 4。七和弦则在三和弦上方再加一个三度。属七和弦 = 大三和弦 + 小七度（根音 + 4 + 3 + 3 个半音）；大七和弦 = 大三和弦 + 大七度（根音 + 4 + 3 + 4）；小七和弦 = 小三和弦 + 小七度（根音 + 3 + 4 + 3）。
- Major triad formula (semitones): 0-4-7. Minor: 0-3-7. Diminished: 0-3-6. Augmented: 0-4-8.
- Dominant 7th: 0-4-7-10. Major 7th: 0-4-7-11. Minor 7th: 0-3-7-10.
6. Chord Inversions and Figured Bass | 和弦转位与数字低音
In root position, the chord is stacked 1-3-5. First inversion uses the 3rd as the lowest note, giving intervals of a 3rd and a 6th above the bass (figured bass: 6/3, often shortened to 6). Second inversion places the 5th in the bass, forming a 4th and a 6th above (figured bass: 6/4). For seventh chords, first inversion is 6/5, second inversion is 4/3, third inversion (7th in bass) is 4/2. Identifying inversions by ear relies on the bass note and the 'floating' quality of a chord not grounded by its root.
原位和弦按照 1-3-5 排列。第一转位以三音为低音，低音与上方音构成三度和六度（数字低音标记为 6/3，常简写为 6）。第二转位以五音为低音，形成四度和六度（标记为 6/4）。七和弦的第一转位标记为 6/5，第二转位为 4/3，第三转位（七音在低音）为 4/2。通过听辨识别转位时，关键在于捕捉低音以及和弦不扎根于根音时那种“飘浮”的音响特质。
| Triad inversion / 三和弦转位 | Bass note / 低音 | Figured bass / 数字低音 |
|---|---|---|
| Root position / 原位 | Root / 根音 | 5/3 (often omitted) |
| 1st inversion / 第一转位 | 3rd / 三音 | 6 (or 6/3) |
| 2nd inversion / 第二转位 | 5th / 五音 | 6/4 |
7. Cadences: Perfect, Plagal, Imperfect, Interrupted | 终止式：正格、变格、半、阻碍
Cadences are the punctuation marks of music. A perfect cadence moves from chord V (dominant) to I (tonic); both chords must be in root position for a 'perfect' V–I, but the term is loosely applied. A plagal cadence is IV–I (amen cadence). An imperfect cadence ends on V, often approached by I, ii, or IV. An interrupted (deceptive) cadence goes V–vi, surprising the listener. Recognising these endings is crucial for structure analysis in SQA listening papers.
终止式是音乐的标点符号。正格终止从 V 级（属和弦）进行到 I 级（主和弦）；严格的正格终止要求两个和弦均为原位，但演奏中常宽泛使用。变格终止是 IV–I（阿门终止）。半终止结束在 V 级和弦上，常用 I、ii 或 IV 作为前导。阻碍终止（伪终止）为 V–vi，制造出听觉上的意外。辨识这些结束方式对于 SQA 听力试卷中的结构分析至关重要。
- Perfect cadence: V → I (major or minor). / 正格终止：V → I（大调或小调）。
- Plagal cadence: IV → I. / 变格终止：IV → I。
- Imperfect cadence: any chord → V. / 半终止：任一和弦进行到 V。
- Interrupted cadence: V → vi (or occasionally VI in minor). / 阻碍终止：V → vi（小调中偶尔 V → VI）。
8. Ornaments and Symbols | 装饰音与符号
Ornaments are musical flourishes with standardised notation. A trill (tr) means rapid alternation between the written note and the note above. A turn (𝄐) goes: upper note – main note – lower note – main note. An upper mordent (𝆯) is a single rapid alternation with the upper neighbour; a lower mordent (𝆰) uses the lower neighbour. An appoggiatura takes half the value of the main note and leans on the beat; an acciaccatura is a crushed grace note played very quickly before the beat.
装饰音是音乐中的华彩音型，拥有标准化的记谱方式。颤音（tr）表示在本音与上方音之间快速交替。回音（𝄐）的奏法为：上方音-本音-下方音-本音。上波音（𝆯）是与上方邻音进行一次快速交替；下波音（𝆰）则使用下方邻音。倚音占据本音时值的一半，并落在拍点上；碎音（短倚音）是在拍点前极其快速地奏出的装饰音。
- Trill: rapid note + upper note alteration. / 颤音：本音与上方邻音快速交替。
- Turn pattern: upper – main – lower – main. / 回音模式：上-主-下-主。
- Acciaccatura: squashed note before the beat, very short. / 碎音：拍前极短奏出的装饰音。
9. Dynamics and Tempo Indications | 力度与速度记号
ortissimo = very loud. Gradual changes: cresc. (crescendo) = getting louder; dim. (diminuendo) = getting softer. Common tempo marks: Largo (broadly, ~40–60 bpm), Adagio (slow, ~66–76), Andante (walking pace, ~76–108), Moderato (moderate, ~108–120), Allegro (fast, ~120–168), Presto (very fast, ~168–200). Metronome marks precisely link a note value to beats per minute.
意大利术语在 SQA 乐谱中支配着力度和速度。pp（极弱），p（弱），mp（中弱），mf（中强），f（强），ff（极强）。渐变术语：cresc.（渐强）；dim.（渐弱）。常见速度记号：Largo（宽广地，约 40-60 bpm），Adagio（柔板，约 66-76），Andante（行板，约 76-108），Moderato（中板，约 108-120），Allegro（快板，约 120-168），Presto（急板，约 168-200）。节拍器标记将特定音符时值与每分钟拍数精确关联。
- Dynamics scale (quiet to loud): pp – p – mp – mf – f – ff.
- Tempo formula: BPM = number of specified beats per minute. ♩ = 120 means 120 crotchets per minute.
10. Transposition Formulas | 移调公式
For transposing instruments, the written pitch differs from the sounding pitch. Clarinet in B♭ sounds a major 2nd lower than written (write up a tone). Alto saxophone in E♭ sounds a major 6th lower (write up a major 6th). Trumpet in B♭ sounds a major 2nd lower (same as clarinet). French horn in F sounds a perfect 5th lower. To find the concert pitch from a written part, subtract the transposition interval accordingly. For SQA, you must be able to transpose short melodies on paper or identify concert pitch.
移调乐器的记谱音高与实际音高不同。降 B 调单簧管实际发音比记谱低大二度（记谱需高大二度）。降 E 调中音萨克斯实际发音低大六度（记谱高大六度）。降 B 调小号实际发音低大二度（与单簧管相同）。F 调圆号实际发音低纯五度。从记谱推算实际音高时，相应减去移调音程。在 SQA 中，你必须能够在纸上移调简短旋律或识别实际音高。
- B♭ instruments: Concert pitch = written pitch – major 2nd (or add 2 sharps / remove 2 flats).
- E♭ instruments: Concert pitch = written pitch – major 6th.
- F instruments: Concert pitch = written pitch – perfect 5th.
11. Rhythm and Meter | 节奏与节拍
Rhythm involves the arrangement of durations. Dotted rhythms, syncopation (emphasis on weak beats), and triplets (three notes in the space of two) are common devices. A triplet bracket with '3' indicates that three equal notes fill the time normally taken by two notes of the same kind. Duplets (2 in the space of 3) occur in compound time. Cross-rhythms and hemiola (2 against 3) create tension. The SQA will test your ability to identify rhythmic devices by ear and notate basic rhythms from dictation.
节奏涉及时值的组织。附点节奏、切分音（弱拍上的重音）和三连音（以三个音代替两个音的时值）都是常见手法。带有数字“3”的三连音记号表示三个均等的音符占据了通常属于两个同种音符的时值。二连音（在复拍子中以二代三）则是相反情况。交叉节奏与赫米奥拉比例（2 对 3）产生张力。SQA 会测试你听辨节奏手法以及通过听写记谱基本节奏的能力。
- Triplet rule: 3 notes in the time of 2. E.g., 3 quavers in the time of 2 quavers.
- Dot rule: Dotted note = 1.5 × original duration. Double dot = 1.75 × original.
12. Texture and Form Structures | 织体与曲式结构
Texture terms describe the layers of sound: monophonic (single melody line), homophonic (melody with chordal accompaniment), polyphonic (independent interweaving lines), and heterophonic (simultaneous variation of one melody). Forms are structural blueprints. Binary form (AB) has two distinct sections, often repeated. Ternary form (ABA) returns to the opening material. Rondo form (ABACA) features a recurring A section. Theme and variations, strophic (same music each verse), and through-composed (new music throughout) shapes are equally testable. Memorise the definitions and listen for them in excerpts.
织体术语描述声音的层次：单声部（一条旋律）、主调（旋律加和弦伴奏）、复调（多条独立交织的线条）和支声复调（同一旋律同时进行变奏）。曲式是结构蓝图。二部曲式（AB）有两个对比段落，常反复。三部曲式（ABA）回归开头素材。回旋曲式（ABACA）的特征是 A 段反复出现。主题与变奏、分节歌（每段歌词用相同音乐）和通谱歌（音乐从头到尾不断变化）等形式同样会出现在考试中。记住定义，并在片段中听辨它们。
- Binary: A – B. Ternary: A – B – A.
- Rondo: A – B – A – C – A.
- Strophic: Same music for every stanza. / 分节歌：每段歌词重复相同音乐。
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