📚 Year 11 CIE Music: Formula & Theorem Quick Reference Handbook | Year 11 CIE 音乐：公式定理速查手册
Music theory can be thought of as a mathematical system of sound. This handbook compiles the essential formulas, theorems, and rules that govern intervals, scales, chords, cadences, and compositional techniques tested in the CIE IGCSE Music syllabus. Treat each entry as a proven theorem that you can apply directly to your listening, composing, and performing tasks.
音乐理论可以被看作是声音的数学系统。本手册汇编了控制音程、音阶、和弦、终止式和作曲技法的核心公式、定理和法则，这些全部是 CIE IGCSE 音乐大纲的考察重点。请将每一个条目视为一条经过验证的定理，直接运用于你的听力、作曲和演奏任务中。
1. Interval Formulas | 音程公式
An interval is measured by the number of semitones between two pitches. The interval number (2nd, 3rd, 4th, etc.) is determined by the letter names. The quality (major, minor, perfect, augmented, diminished) follows a strict semitone count formula.
音程由两个音高之间的半音数量来测量。音程的度数（二度、三度、四度等）由字母音名决定。音程的性质（大、小、纯、增、减）严格遵循半音计数公式。
s; Augmented 4th / Diminished 5th = 6 semitones; Perfect 5th = 7 semitones; Minor 6th = 8 semitones; Major 6th = 9 semitones; Minor 7th = 10 semitones; Major 7th = 11 semitones; Perfect Octave = 12 semitones.
纯一度 = 0 半音；小二度 = 1 半音；大二度 = 2 半音；小三度 = 3 半音；大三度 = 4 半音；纯四度 = 5 半音；增四度 / 减五度 = 6 半音；纯五度 = 7 半音；小六度 = 8 半音；大六度 = 9 半音；小七度 = 10 半音；大七度 = 11 半音；纯八度 = 12 半音。
| Interval Name | Semitone Formula |
|---|---|
| Perfect Unison (P1) | 0 |
| Minor 2nd (m2) | 1 |
| Major 2nd (M2) | 2 |
| Minor 3rd (m3) | 3 |
| Major 3rd (M3) | 4 |
| Perfect 4th (P4) | 5 |
| Augmented 4th / Diminished 5th (A4/d5) | 6 |
| Perfect 5th (P5) | 7 |
| Minor 6th (m6) | 8 |
| Major 6th (M6) | 9 |
| Minor 7th (m7) | 10 |
| Major 7th (M7) | 11 |
| Perfect Octave (P8) | 12 |
Inverted interval theorem: The sum of the interval numbers in an inversion is always 9. A major interval inverts to a minor interval; a perfect interval inverts to a perfect interval; an augmented interval inverts to a diminished interval.
音程转位定理：转位前后的音程度数之和恒为 9。大音程转位变为小音程；纯音程转位仍为纯音程；增音程转位变为减音程。
Example: M3 (C–E) inverts to m6 (E–C). Check: 3+6=9, major becomes minor.
示例：大三度（C–E）转位成为小六度（E–C）。验证：3+6=9，大变小。
2. Major Scale Construction Theorem | 大调音阶构建定理
Every major scale is built using the exact pattern of tones (T) and semitones (S): T–T–S–T–T–T–S. Starting from any tonic, apply this formula and you generate a major scale.
每个大调音阶均按照全音（T）与半音（S）的固定模式构建：T–T–S–T–T–T–S。从任何一个主音出发，套用此公式就能生成大三阶。
For C major: C–D (T), D–E (T), E–F (S), F–G (T), G–A (T), A–B (T), B–C (S). The semitones fall between steps 3-4 and 7-8.
C 大调：C–D (T)，D–E (T)，E–F (S)，F–G (T)，G–A (T)，A–B (T)，B–C (S)。半音出现在第 3-4 级和第 7-8 级之间。
The key signature theorem: The number of sharps or flats in a major key follows the circle of fifths. Order of sharps: F♯, C♯, G♯, D♯, A♯, E♯, B♯. Order of flats: B♭, E♭, A♭, D♭, G♭, C♭, F♭.
调号定理：大调中的升号或降号数量遵循五度循环律。升号出现顺序：F♯, C♯, G♯, D♯, A♯, E♯, B♯。降号出现顺序：B♭, E♭, A♭, D♭, G♭, C♭, F♭。
To find the major key from a sharp signature, take the last sharp and go up a semitone. For flat keys, the second-to-last flat names the key.
由升号调号判断大调：取最后一个升号再升高半音即为调名。由降号调号判断大调：倒数第二个降号即为调名。
3. Minor Scale Formulas | 小调音阶公式
Three minor scale forms exist, all derived from the natural minor formula: T–S–T–T–S–T–T. The harmonic minor raises the 7th degree by a semitone, creating a distinctive augmented 2nd between degrees 6 and 7. The melodic minor raises the 6th and 7th degrees when ascending, and reverts to natural minor when descending.
小调音阶有三种形式，均源于自然小调公式：T–S–T–T–S–T–T。和声小调将第 7 级升高半音，从而在第 6 级与第 7 级之间产生一个独特的增二度。旋律小调在上行时升高第 6 级和第 7 级，下行时恢复为自然小调。
e minor, which is found a minor 3rd below the major tonic. For example, C major → A minor.
关系小调定理：每个大调与它的关系小调共享同一调号，该关系小调位于大调主音下方小三度处。例如 C 大调 → A 小调。
Parallel minor formula: Use the same tonic but alter the 3rd, 6th, and 7th degrees down by a semitone to get natural minor.
同主音小调公式：使用相同主音，但将第 3、6、7 级降低半音即可得到自然小调。
4. Chord Construction Formulas | 和弦构建公式
nor triad = Root + 3 semitones + 4 semitones. Diminished triad = Root + 3 semitones + 3 semitones. Augmented triad = Root + 4 semitones + 4 semitones.
三和弦由根音上方叠置两个三度构成。半音跨度决定和弦性质。大三和弦 = 根音 + 4 半音 + 3 半音。小三和弦 = 根音 + 3 半音 + 4 半音。减三和弦 = 根音 + 3 半音 + 3 半音。增三和弦 = 根音 + 4 半音 + 4 半音。
triad + minor 3rd (root + 4 + 3 + 3 semitones). The formula for a major 7th chord is root + 4 + 3 + 4 semitones.
七和弦定理：在三和弦上方再叠置一个三度即得七和弦。例如属七和弦 = 大三和弦 + 小三度（根音 + 4 + 3 + 3 半音）。大七和弦的公式为根音 + 4 + 3 + 4 半音。
on = 6/3 (or 6); 2nd inversion = 6/4. For 7th chords, the additional 3rd inversion is 4/2.
转位公式：原位 = 5/3；第一转位 = 6/3（或记作 6）；第二转位 = 6/4。对七和弦而言，还存在第三转位 4/2。
5. Cadence Theorems | 终止式定理
root position, with the tonic in the soprano of the I chord). Plagal cadence: IV → I. Imperfect cadence: any chord (often I, ii, IV) → V. Interrupted cadence: V → vi (in major) or V → VI (in minor).
终止式是结束一个乐句的两个和弦的进行。四种常规终止式遵循固定的和弦进行逻辑。完全终止：V → I（两者均为原位，且 I 和弦的高音部为根音）。变格终止：IV → I。不完全终止：任何和弦（常为 I、ii、IV）→ V。阻碍终止：V → vi（大调）或 V → VI（小调）。
The strongest ‘theorem’ of tonal music: V–I defines the key. The leading note (7th degree) resolves up by a semitone to the tonic, and the 7th of V7 falls by step.
调性音乐中最重要的“定理”：V–I 确立了调性。导音（第 7 级）向上半音解决至主音，属七和弦的七音则级进下行。
Perfect cadence voice-leading formula: In V–I, if the leading note is in the soprano, it must move to the tonic. If the soprano is scale degree 2 (supertonic), it may move to the tonic or mediant.
完全终止的声部进行公式：在 V–I 中，若导音位于高音部，它必须进行到主音。若高音部为音阶第 2 级（上主音），它可以进行到主音或中音。
6. Circle of Fifths Rule | 五度圈法则
The circle of fifths is a visual theorem that arranges the 12 keys so that each step clockwise adds a sharp (or removes a flat), and each step anticlockwise adds a flat (or removes a sharp). It demonstrates that keys a perfect 5th apart are closely related.
五度圈是一种视觉定理，将 12 个调按顺时针每次升高纯五度则增加一个升号（或去掉一个降号），逆时针每次则增加一个降号（或去掉一个升号）。它显示相距纯五度的调是近关系调。
Relative keys sit opposite across the circle: C major / A minor share the same point. The dominant key (V) is one step clockwise; the subdominant key (IV) is one step anticlockwise.
关系大小调在圈上位于同一位置：C 大调 / A 小调共享同一点。属调 (V) 为顺时针一步；下属调 (IV) 为逆时针一步。
| Major Key | Sharps/Flats | Relative Minor |
|---|---|---|
| C | 0 | A |
| G | 1♯ (F♯) | E |
| D | 2♯ (F♯, C♯) | B |
| A | 3♯ (F♯, C♯, G♯) | F♯ |
| F | 1♭ (B♭) | D |
| B♭ | 2♭ (B♭, E♭) | G |
ubmediant; vii° = leading tone (dominant function). This triad quality pattern is identical for all major keys: I ii iii IV V vi vii°.
音阶各级和弦功能定理：I = 主；ii = 上主（属前）；iii = 中；IV = 下属（属前）；V = 属；vi = 下中；vii° = 导（属功能）。该三和弦性质模式适用于所有大调：I ii iii IV V vi vii°。
7. Transposition Formula | 移调公式
oves every pitch up or down by the same interval. The formula: new pitch = old pitch + (or – nterval. You must adjust accidentals to preserve the exact pattern of tones and semitones.
移调是将每个音向上或向下移动相同的音程。公式：新音高 = 旧音高 +（或 –）移调音程。必须调整临时记号，以保留完全相同的全音和半音模式。
Concert pitch to B♭ instrument: Write the part a major 2nd higher than sounding pitch. For E♭ instruments, write a minor 3rd lower (or major 6th higher, depending on register). These are standard transposition theorems tested in the CIE paper.
实音至 B♭ 乐器：将分谱写为比实际音高高一个大二度。对 E♭ 乐器，记谱比实音低一个小三度（或视音区而定亦可高大六度）。这些是 CIE 考试中涉及的标准移调定理。
Octave transposing theorem: change the clef or apply an 8va sign to shift all notes by a perfect octave. The key signature and accidentals remain unchanged.
八度移调定理：更换谱号或标注 8va 记号即可将全部音符移动一个纯八度。调号与临时记号保持不变。
8. Chord Progression Rules | 和弦进行法则
5th: I → IV → vii° → iii → vi → ii → V → I. This is called the circle progression theorem.
调性和声遵循强有力的根音运动原则。最常见的进行模式是五度下行：I → IV → vii° → iii → vi → ii → V → I。此即所谓的循环进行定理。
Substitution rule: iii can replace I; vi can replace I; ii can replace IV. These substitutions preserve harmonic function while adding variety.
替代法则：iii 可替代 I；vi 可替代 I；ii 可替代 IV。这些替代能在保持和声功能的同时增添变化。
Perfect authentic cadence rule: V must be in root position, and the tonic chord must have the root in the melody. Imperfect authentic cadence has V–I but the tonic chord is not in root position or the soprano does not land on the tonic.
完全正格终止法则：V 必须为原位，主和弦的高音部必须为根音。不完全正格终止虽然也是 V–I，但主和弦并非原位或高音部未落在主音上。
9. Melodic Writing Theorems | 旋律写作定理
Conjunct motion (stepwise) should dominate; disjunct motion (leaps) must be balanced by stepwise motion in the opposite direction. This is the theorem of melodic smoothness.
级进（音阶式进行）应占主导；跳进必须由反方向的级进加以平衡。此为旋律平滑性定理。
Leap resolution theorem: A leap of a major 6th or larger, or a diminished/augmented interval, must be followed by a stepwise movement in the opposite direction. Dissonant intervals resolve inward.
跳进解决定理：大六度或更大的跳进，以及减音程/增音程之后，必须跟随一个反方向的级进。不协和音程向内解决。
Melodic climax rule: The highest note should appear once, about two-thirds into the phrase, forming a single peak. Avoid multiple identical climaxes.
旋律高潮法则：最高音应唯一出现，大约在乐句的三分之二处，构成单一峰值。避免出现多个平齐的高潮点。
Sequence formula: A melodic fragment repeated at a higher or lower pitch, often following the pattern of the scale or chord tones. Each repetition typically shifts by the same interval.
模进公式：一个旋律片段在更高或更低的音高上重复，常按音阶或和弦音的模式进行。每次重复通常移动相同音程。
10. Time Signature Laws | 拍号定律
Simple time: each beat divides naturally into two equal parts. The upper number is 2, 3, or 4. Compound time: each beat divides into three equal parts. The upper number is 6, 9, or 12.
单拍子：每一拍自然分为两个均等部分，上方数字为 2、3 或 4。复拍子：每一拍分为三个均等部分，上方数字为 6、9 或 12。
Compound beat identification theorem: Divide the top number by 3 to get the number of beats per bar. For example, 6/8 has 2 dotted-crotchet beats; 9/8 has 3; 12/8 has 4.
复拍子拍数识别定理：将上方数字除以 3 即得每小节的拍数。例如 6/8 有 2 个附点四分音符拍；9/8 有 3 个；12/8 有 4 个。
Irregular time signatures (5/4, 7/8) are grouped as 2+3, 3+2, or 2+2+3 etc. Syncopation rule: accent falls on a normally weak beat or weak part of the beat, often tied across the barline.
不规则拍号（5/4, 7/8）常被分组为 2+3、3+2 或 2+2+3 等。切分音法则：重音落在正常的弱拍或弱部分之上，并经常通过连线跨越小节线。
11. Ornamentation & Articulation Theorems | 装饰音与运音法定理
Ornaments are formulas for decorating a principal note. Trill: rapid alternation between the written note and the note above, usually starting on the upper note in Baroque practice. Turn: the written note, note above, written note, note below, written note. Mordent: the written note, the note below, and back.
装饰音是修饰主要音的一些固定模式。颤音：在本音与上方音之间快速交替，巴洛克时期通常从上方音开始。回音：本音—上助音—本音—下助音—本音。波音：本音—下助音—本音。
Acciaccatura theorem: a crushed grace note played as quickly as possible before the main note, taking minimal value from the main note.
倚音定理：一个被“碾碎”的装饰音，在主音之前尽可能快地奏出，并从主音时值中获取极短的长度。
Appoggiatura formula: a grace note that takes half the value of the main note (or two-thirds if the main note is dotted), creating a dissonance on the beat that resolves.
长倚音公式：一个装饰音，占主音时值的一半（若主音为附点音符，则占三分之二），在强拍上构成不协和音并予以解决。
Staccato law: the note is shortened to about half its written value. Legato: smooth, connected, with no perceptible silence between notes. These marks directly affect phrase shaping.
断奏定律：音符被缩短至原时值的大约一半。连奏：平滑、连贯，音符之间没有可察觉的间隙。这些标记直接影响乐句塑造。
12. Form and Structure Theorems | 曲式与结构定理
Binary form theorem: ||: A :||: B :||, where A modulates to a related key (usually the dominant), and B returns to the tonic. Both sections are often repeated.
二段体定理：||: A :||: B :||，A 段转调至一个近关系调（通常为属调），B 段返回主调。两段通常均反复。
Ternary form formula: A B A, with a contrasting B section in a different key or texture, and a return of A exactly or with variation.
三段体公式：A B A，具有对比的 B 段，通常采用不同的调性或织体，最后 A 段完整或变化再现。
Rondo structure: A B A C A (or more episodes). The A (refrain) returns in the tonic while episodes present contrasting material in other keys.
回旋曲结构：A B A C A（或更多插部）。A（主部）每次在主调上回归，插部则在其他调上呈现对比材料。
Theme and variations theorem: a theme is stated and then altered through techniques like melodic decoration, change of mode (major↔minor), rhythmic variation, and change of accompaniment texture, but the phrase structure remains largely intact.
主题与变奏定理：先呈示主题，随后通过旋律装饰、调式变换（大调↔小调）、节奏变化和伴奏织体变化等手段加以改造，但乐句结构基本保持不变。
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