📚 Year 9 WJEC Music: Formula & Theorem Quick Reference Handbook | Year 9 WJEC 音乐：公式定理速查手册
This article is a compact revision guide for Year 9 WJEC Music, gathering all essential musical 'formulas' and 'theorems' – the rules that govern rhythm, pitch, and structure. Think of it as your musical maths cheat sheet: from note length equations to scale construction recipes, interval arithmetic, chord building blocks, and cadence patterns. Every concept is presented as a clear statement or formula, followed by a brief explanation, so you can swiftly recall the facts needed for listening, appraising, and composing tasks.
本文是 Year 9 WJEC 音乐的精简复习手册，汇集了所有基本的音乐“公式”与“定理”——即支配节奏、音高和结构的规则。把它看作你的音乐数学速查表：从音符时长的计算方程到音阶构造的配方，从音程的算术到和弦的积木块，再到终止式的模式。每个概念都以一条清晰的陈述或公式呈现，并附有简短说明，让你能快速调取在听觉分析、评估和作曲任务中所需的事实。
1. Note Value Mathematics | 音符时值的数学
inim = 2 crotchets; 1 crotchet = 2 quavers; 1 quaver = 2 semiquavers. Rests follow the same relationships.
全音符是基本单位，所有其他音符时值都是它的分数。基本等式：1 个全音符 = 2 个二分音符；1 个二分音符 = 2 个四分音符；1 个四分音符 = 2 个八分音符；1 个八分音符 = 2 个十六分音符。休止符遵循同样的关系。
ot adds half of the half: double dotted minim = minim + crotchet + quaver. Tied notes sum their durations: tie formula: Total duration = value₁ + value₂.
附点音符公式：附点增加原音符时值的一半。因此，附点二分音符 = 二分音符 + 四分音符（在单拍子里为 3 个四分音符拍）。复附点再加一半的一半：复附点二分音符 = 二分音符 + 四分音符 + 八分音符。连音线将时值相加：连音线公式：总时长 = 时值₁ + 时值₂。
Triplet rule: three notes played in the time of two of the same kind. The triplet sign '3' of a normal quaver. Group of 3 triplet quavers occupies one crotchet beat in simple time.
三连音规则：用三个音符演奏通常两个同类音符的时间。三连音标记“3”压缩了时值：每个三连音八分音符 = 正常八分音符的 (2/3)。三个三连音八分音符的组合在单拍子中占一个四分音符拍。
2. Time Signature Theorem | 拍号定理
nim). Simple time: beats divide naturally into two equal parts. Compound time: beats divide into three equal parts (dotted beat).
拍号以类似分数的符号表示：上方数字 = 每小节的拍数；下方数字 = 拍子类型（4 = 四分音符，8 = 八分音符，2 = 二分音符）。单拍子：拍子自然地分成两个相等部分。复拍子：拍子分成三个相等部分（附点节拍）。
otchet beats (each beat = 3 quavers); 9/8 = three dotted crotchet beats; 12/8 = four dotted crotchet beats.
单拍子举例：2/4 = 两个四分音符拍；3/4 = 三个四分音符拍；4/4（C）= 四个四分音符拍。复拍子举例：6/8 = 两个附点四分音符拍（每拍 = 3 个八分音符）；9/8 = 三个附点四分音符拍；12/8 = 四个附点四分音符拍。
Irregular time: 5/4, 7/8 – ts appear at the end. Summation rule: upbeat duration + final bar duration = one full bar.
不规则拍子：5/4、7/8——拍子分组不均匀（例如 5/4 可拆为 3+2 或 2+3）。弱起小节（上拍）：开头的不完全小节，缺失的拍子出现在结尾。求和规则：弱起时值 + 最后一小节时值 = 一个完整小节。
3. Major Scale Construction Formula | 大调音阶构建公式
any tonic, apply the pattern ascending to obtain the eight-note scale. Using scale degree numbers: 1-2-3-4-5-6-7-8.
每个大调音阶都遵循以下音程模式（全音和半音）：全-全-半-全-全-全-半。以任一主音开始，上行应用该模式便可得到八声音阶。用音级数字表示：1-2-3-4-5-6-7-8。
Key signature formula for sharps: Father Charles Goes Down And Ends Battle (F♯, C♯, G♯, D♯, A♯, E♯, B♯). Order of flats is the reverse: Battle Ends And Down Goes Charles' Father (B♭, E♭, A♭, D♭, G♭, C♭, F♭ or key with sharps = last sharp + one semitone; for flats = second last flat in signature.
升号调的调号公式：Father Charles Goes Down And Ends Battle（升 F、升 C、升 G、升 D、升 A、升 E、升 B）。降号调的顺序相反：Battle Ends And Down Goes Charles' Father（降 B、降 E、降 A、降 D、降 G、降 C、降 F）。含升号的大调主音 = 最后一个升号 + 一个半音；含降号的大调主音 = 调号中倒数第二个降号。
tive minor, count down three semitones from the major tonic (e.g., C major → A minor). Harmonic minor formula: 1-2-♭3-4-5-♭6-7-8 (raised 7th). Melodic minor ascending: 1-2-♭3-4-5-6-7-8 (raised 6th and 7th); descending is natural minor (♭7, ♭6, ♭3).
关系小调定理：每个大调都与其关系小调共享调号，关系小调是大调音阶的第 6 级音（下中音）。找出关系小调的方法：从大调主音向下数三个半音（如 C 大调 → a 小调）。和声小调公式：1-2-♭3-4-5-♭6-7-8（升高第 7 级）。旋律小调上行：1-2-♭3-4-5-6-7-8（升高第 6、7 级）；下行还原为自然小调（♭7、♭6、♭3）。
4. Pentatonic, Blues and Mode Formulas | 五声音阶、蓝调音阶与调式公式
Major pentatonic: use scale degrees 1, 2, 3, 5, 6 of the major scale. Interval pattern: T-T-1½-T-1½ (or W-W-m3-W-m3). Minor pentatonic: 1, ♭3, 4, 5, ♭7 of the natural minor scale. Pattern: 1½-T-T-1½-T.
大调五声音阶：使用大调音阶的第 1、2、3、5、6 级。音程模式：全-全-小三度-全-小三度。小调五声音阶：自然小调的第 1、♭3、4、5、♭7 级。模式：小三度-全-全-小三度-全。
Blues scale formula: minor pentatonic + ♭5 (the 'blue note'). Thus: 1, ♭3, 4, ♭5, 5, ♭7. This added ♭5 creates characteristic tension, often used in jazz, rock and blues.
蓝调音阶公式：小调五声音阶 + ♭5（“蓝调音”）。即：1、♭3、4、♭5、5、♭7。这个添加的 ♭5 创造出特有的紧张度，常用于爵士、摇滚和蓝调音乐。
ferent starting notes): Ionian = major (T-T-S-T-T-T-S); Dorian = T-S-T-T-T-S-T (minor with ♮6); Phrygian = S-T-T-T-S-T-T (minor with ♭2); Lydian = T-T-T-S-T-T-S (major with ♯4); Mixolydian = T-T-S-T-T-S-T (major with ♭7); Aeolian = natural minor (T-S-T-T-S-T-T); Locrian = S-T-T-S-T-T-T (diminished sound, ♭2, ♭3, ♭5, ♭6, ♭7).
调式公式（均基于白键 C 大调式样，移到不同起始音）：伊奥尼亚 = 大调（全-全-半-全-全-全-半）；多利亚 = 全-半-全-全-全-半-全（小调带 ♮6）；弗里几亚 = 半-全-全-全-半-全-全（小调带 ♭2）；利底亚 = 全-全-全-半-全-全-半（大调带 ♯4）；混合利底亚 = 全-全-半-全-全-半-全（大调带 ♭7）；爱奥利亚 = 自然小调（全-半-全-全-半-全-全）；洛克里亚 = 半-全-全-半-全-全-全（减音效，带 ♭2、♭3、♭5、♭6、♭7）。
5. Interval Arithmetic | 音程算术
to E = 3rd), quality is determined by exact semitone count. In a major scale, intervals from tonic to scale degrees are: unison (P1), Major 2nd, Major 3rd, Perfect 4th, Perfect 5th, Major 6th, Major 7th, Perfect 8ve.
音程是两个音之间的音高距离，以音阶级数计算。性质 + 度数规则：度数是音名距离（如 C 到 E = 三度），性质由确切的半音数决定。在大调音阶中，从主音到各音级的音程为：纯一度（P1）、大二度、大三度、纯四度、纯五度、大六度、大七度、纯八度。
Interval alteration shortcuts: a Major interval reduced by a semitone becomes Minor; reduced by two semitones becomes Diminished. A Perfect interval reduced by a semitone becomes Diminished; augmented by a semitone becomes Augmented. Augmented intervals: Major or Perfect interval widened by one semitone.
音程变化捷径：大音程减去一个半音变为小音程；减去两个半音变为减音程。纯音程减去一个半音变为减音程；增加一个半音变为增音程。增音程：大音程或纯音程扩大一个半音。
Semitone count reference (ascending) for quick identification:
| Interval | Semitones | Example |
|---|---|---|
| Minor 2nd | 1 | E-F |
| Major 2nd | 2 | C-D |
| Minor 3rd | 3 | A-C |
| Major 3rd | 4 | C-E |
| Perfect 4th | 5 | C-F |
| Perfect 5th | 7 | C-G |
| Minor 6th | 8 | E-C |
| Major 6th | 9 | C-A |
| Minor 7th | 10 | C-B♭ |
| Major 7th | 11 | C-B |
| Octave (P8) | 12 | C-C |
快速识别的半音数参考表（上行）：
nterval and its inversion always equals 9 (e.g., P5 inverts to P4; 5+4=9). Major becomes minor upon inversion and vice versa; perfect remains perfect.
复音程：大于八度的音程；将单音程度数加 7（如九度 = 复二度）。转位规则：音程与其转位的度数之和恒为 9（如纯五度转位成纯四度；5+4=9）。转位时大音程变小音程，反之亦然；纯音程仍为纯。
6. Triad Building Blocks | 三和弦积木
A triad is a three-note chord built in thirds. Formula for a major triad (root position): root + major 3rd + perfect 5th. Semitone pattern: 4 semitones from root to 3rd, then 3 semitones from 3rd to 5th (total 7). Minor triad: root + minor 3rd + perfect 5th (3 + 4 semitones). Diminished: root + minor 3rd + diminished 5th (3 + 3 semitones). Augmented: root + major 3rd + augmented 5th (4 + 4 semitones).
三和弦是由三度叠置而成的三个音的和弦。大三和弦（原位）公式：根音 + 大三度 + 纯五度。半音模式：根音到三音 4 个半音，三音到五音 3 个半音（共 7 个）。小三和弦：根音 + 小三度 + 纯五度（3 + 4 个半音）。减三和弦：根音 + 小三度 + 减五度（3 + 3 个半音）。增三和弦：根音 + 大三度 + 增五度（4 + 4 个半音）。
Triad inversion labels: root position (5/3), first inversion (6/3 or simply '6' ond inversion. Inversion changes the bass note: first inversion has 3rd in bass, second inversion has 5th in bass.
三和弦转位标记：原位（5/3）、第一转位（6/3 或简写“6”）、第二转位（6/4）。数字低音简写：无数字 = 原位；6 = 第一转位；6/4 = 第二转位。转位改变低音：第一转位三音在低音部，第二转位五音在低音部。
Diatonic triads in a major key (I-ii-iii-IV-V-vi-vii°): I, IV, V are major; ii, iii, vi are minor; vii° is diminished. The primary triads (I, IV, V) form the backbone of tonal harmonic progressions. V often includes a 7th (V⁷) for stronger pull to I.
大调中的自然音三和弦（I-ii-iii-IV-V-vi-vii°）：I、IV、V 是大三和弦；ii、iii、vi 是小三和弦；vii° 是减三和弦。正三和弦（I、IV、V）构成调性和声进行的骨干。V 常添加七音（V⁷）以增强向 I 的倾向。
7. Seventh Chord Formulae | 七和弦公式
A seventh chord adds a 7th above the root to a triad. Dominant 7th (V⁷): major triad + minor 7th (in a major key, built on 5th degree). Formula in semitones from root: 4 (M3) + 3 (P5) + 3 (m7) – total 10 semitones to 7th. Major 7th: major triad + major 7th (4+3+4). Minor 7th: minor triad + minor 7th (3+4+3). Half-diminished 7th (ø7): diminished triad + minor 7th (3+3+4). Diminished 7th (dim7): diminished triad + diminished 7th (3+3+3).
七和弦在三和弦之上再叠置一个根音上方的七度音。属七和弦（V⁷）：大三和弦 + 小七度（在大调中建立于第 5 级）。从根音算起的半音公式：4（大三度）+ 3（纯五度）+ 3（小七度）——到七音共 10 个半音。大七和弦：大三和弦 + 大七度（4+3+4）。小七和弦：小三和弦 + 小七度（3+4+3）。半减七和弦（ø7）：减三和弦 + 小七度（3+3+4）。减七和弦（dim7）：减三和弦 + 减七度（3+3+3）。
The dominant 7th contains a tritone (augmented 4th/diminished 5th) between the 3rd and 7th, which resolves inwards for major keys or outwards for minor. In root position, V⁷ often resolves to I (or i) with the 7th falling by step.
属七和弦包含一个三全音（增四度/减五度），位于三音与七音之间；在大调中向内解决，在小调中向外解决。原位 V⁷ 通常解决到 I（或 i），七音级进下行。
8. Chord Progression and Cadence Laws | 和弦进行与终止式法则
Cadences are the punctuation of music. Perfect cadence (V-I): full stop, sounds complete. Formula: V (or V⁷) moving to I. Plagal cadence (IV-I): 'Amen' cadence, sounds softer. Interrupted/Deceptive cadence (V-vi): breaks expectation; V moves to vi (minor submediant). Imperfect cadence (any chord - V): feels unfinished, like a comma; ends on V.
终止式是音乐的标点。完全终止（V-I）：句号，音响完满。公式：V（或 V⁷）进行到 I。变格终止（IV-I）：“阿们”终止，听感柔和。阻碍终止（V-vi）：打破期待；V 进行到 vi（小下中音）。不完全终止（任何和弦 - V）：感觉未完成，如逗号；结束于 V。
Common chord progression formulas in Year 9 context: I-IV-V-I (the most fundamental progression); I-vi-IV-V (doo-wop / 1950s progression); ii-V-I (jazz turn-around, strong root movement by 5ths); 12-bar blues pattern: I-I-I-I / IV-IV-I-I / V-IV-I-V (may use dominant 7ths). These formulas can be transposed to any key.
Year 9 语境中常见的和弦进行公式：I-IV-V-I（最基本的进行）；I-vi-IV-V（嘟哇/1950 年代进行）；ii-V-I（爵士回转，强五度根音运动）；12 小节蓝调模式：I-I-I-I / IV-IV-I-I / V-IV-I-V（可使用属七和弦）。这些公式可以移调到任何调。
Circle of 5ths progression: chords often move by descending perfect 5ths (or ascending 4ths). The sequence iii-vi-ii-V-I moves through the circle. This root movement is the strongest harmonic progression in tonal music.
五度圈进行：和弦常以下行纯五度（或上行纯四度）运动。序列 iii-vi-ii-V-I 沿五度圈移动。这种根音运动是调性音乐中最强的和声进行。
9. Form and Structure Blueprints | 曲式与结构蓝图
often with ornamentation). Binary form (AB): two sections, each usually repeated. Rounded binary adds a brief return of A at the end of B (A :||: B A' :||).
三部曲式（ABA）：呈示 - 对比 - 再现。公式：A（主题在主调）→ B（对比调，常为属调或关系小调）→ A（再现，常加装饰）。二部曲式（AB）：两个段落，通常各自反复。再现二部曲式在 B 段末尾简短再现 A（A :||: B A' :||）。
Theme and variations: A (theme) → A' → A'' → A''' ... . Each variation alters melody, harmony, rhythm, texture, or mode while the theme' , common in folk songs. Through-composed: continuous new music, no large-scale repetition.
主题与变奏：A（主题）→ A' → A'' → A''' ……。每次变奏改变旋律、和声、节奏、织体或调式，但主题的框架仍可辨认。分节歌形式：每段歌词使用相同音乐（AAA……），常见于民歌。通谱体：连续的新音乐，无大规模重复。
Pop song structure template: Intro - Verse - Chorus - Verse - Chorus - Bridge - Chorus - Outro. Verses tell the story (often same music, different lyrics), chorus is the hook (repeated), bridge provides contrast. 12-bar blues: three 4-bar phrases following the I-IV-V formula.
流行歌曲结构模板：前奏 - 主歌 - 副歌 - 主歌 - 副歌 - 桥段 - 副歌 - 尾奏。主歌讲故事（常同音乐，不同歌词），副歌是记忆点（重复），桥段提供对比。12 小节蓝调：三个 4 小节乐句，遵循 I-IV-V 公式。
10. Dynamics, Tempo and Articulation Lexicon | 力度、速度与演奏法术语表
oft; mf (mezzo-forte) = moderately loud; f (forte) = loud; ff (fortissimo) = very loud. Gradual changes: crescendo (cresc.) = get louder; diminuendo (dim.) or decrescendo = get softer. Sforzando (sfz) = sudden strong accent.
力度公式：音量层次用意大利语缩写。pp（很弱）；p（弱）；mp（中弱）；mf（中强）；f（强）；ff（很强）。渐变：crescendo（渐强）；diminuendo 或 decrescendo（渐弱）。sforzando（sfz）= 突强重音。
Tempo markings: Largo (very slow, 40–60 bpm), Adagio (slow, 66–76), Andante (walking pace, 76–108), Moderato (moderate, 108–120), Allegro (fast, 120–168), Vivace (lively, 168–176), Presto (very fast, 168–200). Modifiers: 'issimo' intensifies (e.g., prestissimo); '-etto' reduces (e.g., allegretto = moderately fast). Metronome mark formulas: ♩ = 120 means 120 crotchet beats per minute.
速度标记：Largo（极慢，40–60 bpm），Adagio（慢，66–76），Andante（步行速度，76–108），Moderato（中板，108–120），Allegro（快板，120–168），Vivace（活泼的快板，168–176），Presto（急板，168–200）。修饰词：'-issimo' 强化（如 prestissimo）；'-etto' 减弱（如 allegretto = 稍快）。节拍器标记公式：♩ = 120 表示每分钟 120 个四分音符拍。
ck; tenuto (line) = hold full value, slightly stressed; marcato (^) = marked, heavily accented. Combining dynamics and articulation shapes phrasing and expression.
演奏法：staccato（小圆点）= 断开、缩短（约原时值的一半）；legato（连线）= 平滑连接；重音（>）= 起音加强；tenuto（横线）= 保持足值，稍加强调；marcato（^）= 突出地，强烈重音。力度与演奏法的结合塑造乐句与表情。
11. Texture and Ensemble Density Equations | 织体与合奏密度方程
oice). Homophonic (melody + accompaniment): a clearly dominant melody supported by chords; often described by 'melody-dominated homophony'. Chordal homophony: all parts move in same rhythm (block chords).
织体描述听到多少声音层次以及它们如何关联。单音织体：单一旋律线条，无伴奏（密度 = 1 个声部）。主调织体（旋律 + 伴奏）：清晰的旋律由和弦支撑；常表述为“旋律主导的主调织体”。和弦式主调织体：所有声部以相同节奏运动（块状和弦）。
yphony: entries copy each other (canon, fugue); non-imitative: different independent lines. Heterophonic texture: simultaneous variation of a single melody (rare in Western music, common in some world traditions).
复调/对位织体：两条或以上独立旋律交织（密度 ≥ 2）。模仿式复调：声部依次进入相互模仿（卡农、赋格）；非模仿式：不同的独立线条。异音织体：同一旋律同时做不同变奏（西方音乐罕见，常见于某些世界传统音乐）。
Texture change methods: adding or subtracting instruments (orchestration), layering, octave doublings, and using pedal points or drones. Antiphonal texture: alternating groups of performers (stereo effect). Thick vs. thin: a full orchestra represents high density; a solo flute represents low density.
织体变化手段：增删乐器（配器）、层次叠加、八度重复、持续音或低音长音。对唱/应答织体：两组表演者交替（立体声效果）。厚与薄：完整管弦乐队代表高密度；长笛独奏代表低密度。
12. Ornamentation and Melodic Decoration Rules | 装饰音与旋律润色规则
Trill (tr): rapid alternation between the written note and the note above (diatonic, often starting on upper note). Formula: note - upper - note - upper ... ending with turn. Upper mordent (short trill with line through): single alternation: note - upper - note. Lower mordent (no line): note - lower - note.
颤音（tr）：本音与上方音快速交替（自然音，常从上音开始）。公式：本音 - 上音 - 本音 - 上音……以回音结束。上波音（带竖线的短颤音）：单次交替：本音 - 上音 - 本音。下波音（无竖线）：本音 - 下音 - 本音。
Turn (sideways S): four notes ornamenting the main note: upper - main - lower - main. Inverted turn: lower - main - upper - main. Appoggiatura (small note with slash): takes half the value of the main note (or two-thirds if dotted), creating a dissonant leaning. Acciaccatura (small note slashed, joined to main): 'crushed' very short before the beat.
回音（横 S）：四个音装饰本音：上音 - 本音 - 下音 - 本音。逆回音：下音 - 本音 - 上音 - 本音。倚音（带斜线的小音符）：占本音时值的一半（或三分之二，若本音带附点），形成不协和倚靠。短倚音（带斜线小音符，连至本音）：在节拍前极短地“挤”入。
Grace notes are not counted in the measure's rhythmic arithmetic; they steal time from the main note or preceding beat. Ornamentation symbols are realised differently across Baroque, Classical, and Romantic styles, but the core rhythmic formula remains as given.
装饰音不计入小节节拍算术；它们从本音或前一拍中偷取时值。装饰音符号在巴洛克、古典、浪漫时期的实际演奏方式有差异，但核心节奏公式如上所述。
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