Formula & Theorem Quick Reference Handbook | 公式定理速查手册

📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This quick reference guide compiles the essential formulas, rules, and theoretical frameworks encountered in the Year 13 AQA Music specification. From interval calculations to chord construction, cadential patterns to frequency ratios, these ‘formulae’ serve as the building blocks of Western tonal harmony and musical analysis. Mastery of these fundamentals will support both your listening and appraising, as well as your composing tasks. Use this handbook to reinforce your theoretical knowledge and to develop fluency in recognising and applying musical structures.

本速查手册汇编了 Year 13 AQA 音乐课程中涉及的核心公式、规则与理论框架。从音程计算到和弦构造，从终止式模式到频率比，这些“公式”构成了西方调性和声与音乐分析的基本单元。掌握这些基础，将帮助你更好地完成听力评析与作曲任务。利用本手册巩固你的理论知识，提升对音乐结构的识别与运用能力。

1. Intervals and Inversions | 音程与转位

An interval is the distance between two pitches. The number of semitones determines the interval quality: minor 2nd = 1 semitone, 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.

音程是两音之间的距离。半音数量决定音程性质：小二度=1个半音，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。

Inversion of intervals: a major interval inverted becomes minor, a minor becomes major, a perfect stays perfect. The sum of the two interval numbers always equals 9 (e.g., 3rd inverted becomes 6th, 3+6=9). Augmented and diminished intervals invert to each other.

音程转位规则：大音程转位后变小音程，小变大，纯保持纯。两音程度数之和恒为9（如三度转位成六度，3+6=9）。增音程与减音程互为转位。

2. Triad Construction Formulas | 三和弦构造公式

A triad is formed by stacking two thirds above a root. Major triad = root + major 3rd (4 semitones) + perfect 5th (7 semitones). Minor triad = root + minor 3rd (3 semitones) + perfect 5th (7 semitones).

三和弦由根音上方叠加两个三度构成。大三和弦 = 根音 + 大三度（4个半音）+ 纯五度（7个半音）。小三和弦 = 根音 + 小三度（3个半音）+ 纯五度（7个半音）。

Diminished triad = root + minor 3rd (3) + diminished 5th (6). Augmented triad = root + major 3rd (4) + augmented 5th (8).

减三和弦 = 根音 + 小三度（3）+ 减五度（6）。增三和弦 = 根音 + 大三度（4）+ 增五度（8）。

Formula in semitones: Major: 0-4-7. Minor: 0-3-7. Diminished: 0-3-6. Augmented: 0-4-8.

半音数公式：大三：0-4-7。小三：0-3-7。减三：0-3-6。增三：0-4-8。

3. Seventh Chord Construction | 七和弦构造

Seventh chords add a third above the fifth. Major 7th: major triad + major 7th (11 semitones from root). Dominant 7th: major triad + minor 7th (10 semitones). Minor 7th: minor triad + minor 7th (10). Half-diminished: diminished triad + minor 7th (10). Fully-diminished: diminished triad + diminished 7th (9).

七和弦在五音上方再加一个三度。大七和弦：大三和弦 + 大七度（距根音11个半音）。属七和弦：大三和弦 + 小七度（10个半音）。小七和弦：小三和弦 + 小七度（10）。半减七和弦：减三和弦 + 小七度（10）。全减七和弦：减三和弦 + 减七度（9）。

Semitone formulas: Maj7: 0-4-7-11. Dom7: 0-4-7-10. Min7: 0-3-7-10. ø7: 0-3-6-10. o7: 0-3-6-9.

半音数公式：大七：0-4-7-11。属七：0-4-7-10。小七：0-3-7-10。半减七：0-3-6-10。全减七：0-3-6-9。

4. Inversions and Figured Bass | 转位与数字低音

Root position: 5/3 (intervals of a 3rd and 5th above the bass). 1st inversion: 6/3 (3rd and 6th above the bass, usually abbreviated as 6). 2nd inversion: 6/4 (4th and 6th above the bass). Seventh chords add 7th intervals: root position 7/5/3 (abbreviated 7), 1st inv. 6/5/3 (6/5), 2nd inv. 6/4/3 (4/3), 3rd inv. 6/4/2 (4/2 or simply 2).

原位：5/3（低音上方有三度和五度）。第一转位：6/3（低音上方有三度和六度，常简写为6）。第二转位：6/4（低音上方有四度和六度）。七和弦增加七度音程：原位7/5/3（简写7），第一转位6/5/3（6/5），第二转位6/4/3（4/3），第三转位6/4/2（4/2或2）。

To identify inversion: find the chord’s root, then analyse the bass note. If bass is root → root position; if 3rd → 1st inversion; if 5th → 2nd inversion; if 7th → 3rd inversion (for seventh chords).

判断转位：找出和弦根音，再看低音。低音为根音→原位；低音为三音→第一转位；低音为五音→第二转位；低音为七音→第三转位（七和弦）。

5. Cadence Types | 终止式类型

Perfect cadence: V–I (both in root position, soprano ends on tonic). Imperfect cadence: any chord–V (often I–V, ii–V, IV–V). Plagal cadence: IV–I. Interrupted (deceptive) cadence: V–vi (V–VI in minor).

正格终止：V–I（均为原位，高音部结束在主音）。不完全终止：任何和弦–V（常见 I–V、ii–V、IV–V）。变格终止：IV–I。阻碍终止：V–vi（小调中 V–VI）。

Phrygian cadence: iv⁶–V in minor (first inversion subdominant to dominant). Tierce de Picardie: a minor-key piece ends with a major tonic chord (I instead of i).

弗里几亚终止：小调中 iv⁶–V（下属和弦第一转位到属和弦）。辟卡迪三度：小调作品以大三和弦主和弦结束（I 代替 i）。

6. Chord Progressions and Functions | 和弦进行与功能

Functional harmony revolves around tonic (I, i), subdominant (IV, ii, vi in major; iv, ii°, VI in minor), and dominant (V, vii°). The typical functional flow: Tonic → Subdominant → Dominant → Tonic. This can be abbreviated as I–IV–V–I.

功能和声围绕主功能（I, i）、下属功能（IV, ii, 大调中 vi；小调中 iv, ii°, VI）和属功能（V, vii°）展开。典型功能流向：主→下属→属→主，可简写为 I–IV–V–I。

Common diatonic progressions: circle of fifths progression (vi–ii–V–I or iii–vi–ii–V–I); descending thirds (I–vi–IV–ii–V–I); ascending seconds (I–ii–iii–IV–V). The cadential ⁶/₄ (I⁶/₄) functions as a dominant preparation, moving to V then I.

常见自然音进行：五度圈进行（vi–ii–V–I 或 iii–vi–ii–V–I）；下行三度（I–vi–IV–ii–V–I）；上行二度（I–ii–iii–IV–V）。终止六四和弦（I⁶/₄）作为属准备，进行到 V 再到 I。

Roman numeral analysis: uppercase for major chords (I, IV, V), lowercase for minor (ii, vi). Diminished chords use lowercase with ° (vii°), augmented uses uppercase with + (III+). Inversions indicated by figured bass numbers.

罗马数字分析：大写表示大三和弦（I, IV, V），小写表示小三和弦（ii, vi）。减三和弦用小写加 °（vii°），增三和弦用大写加 +（III+）。转位用数字低音标记。

7. Non-Chord Tones | 非和弦音

Passing note (P): stepwise movement between two consonant tones, approached and left by step in the same direction. Neighbour note (N): step away from a consonant tone and return to it (upper or lower). Suspension (S): a note tied over from the previous chord, creating a dissonance, then resolving down by step. Appoggiatura (App): approached by leap and resolved by step, often on a strong accent. Escape tone (E): approached by step, left by leap in the opposite direction. Anticipation (Ant): a chord tone of the next harmony sounded early. Pedal: a sustained note (usually tonic or dominant) while harmonies change above it.

经过音（P）：在两个协和音之间以级进同方向进出。辅助音（N）：从协和音级进离开再返回（上辅助或下辅助）。延留音（S）：从前和弦延留的音形成不协和，然后下行级进解决。倚音（App）：跳进进入，级进解决，常出现在强拍。逃逸音（E）：级进进入，反向跳进离开。先现音（Ant）：后一和弦的音提前出现。持续音：上方和声变化时，一个音（常为主音或属音）持续保持。

Formulae for handling suspensions: preparation – suspension – resolution. 4-3 suspension is typical above a bass (the 4th resolving to 3rd); 7-6 and 9-8 also common.

延留音处理公式：准备 – 延留 – 解决。低音上方的4-3延留音（四度解决到三度）最为典型；7-6和9-8延留也很常见。

8. Voice Leading Rules | 声部进行规则

Voice leading governs the smooth movement of individual parts. Adjacent voices (SATB) should stay within an octave of each other (except tenor-bass, which can be up to a 12th). Avoid parallel 5ths and parallel 8ves – if two voices move in parallel perfect intervals, it weakens independence.

声部进行规则控制各声部的平稳进行。相邻声部（S-A-T-B）间距离一般不超过八度（男高-男低可到十二度）。避免平行五度与平行八度——两声部以平行纯音程进行会削弱独立性。

Other rules: avoid voice crossing, avoid large leaps (more than a 6th) in upper voices, leading tone (7th degree) must resolve upwards to tonic when in an outer voice, diminished intervals should be resolved by step inward.

其他规则：避免声部交错，避免上方声部大跳（超过六度），导音（七级音）在外声部时必须上行级进解决到主音，减音程应向内级进解决。

Spacing rule: the three upper voices (S, A, T) should lie within the same octave, while the bass can be widely separated. Double the root of a chord whenever possible; never double the leading tone or chromatic notes.

排列规则：三个上方声部（S、A、T）应处于同一八度内，低音则可远离。尽可能重复根音；切勿重复导音或变化半音。

9. Frequency and Pitch (Equal Temperament) | 频率与音高（十二平均律）

In equal temperament, each semitone step multiplies the frequency by the twelfth root of 2, approximately 1.05946. The formula: f = f₀ × 2^(n/12), where f₀ is a reference pitch (e.g., A₄ = 440 Hz) and n is the number of semitones above (positive) or below (negative).

十二平均律中每个半音将频率乘以2的十二次方根，约1.05946。公式：f = f₀ × 2^(n/12)，其中 f₀ 为参考音高（如 A₄=440 Hz），n 为高于（正值）或低于（负值）的伴音数。

Octave ratio is exactly 2:1. A perfect 5th in equal temperament is 2^(7/12) ≈ 1.4983 (close to pure 3:2 ratio). A major 3rd is 2^(4/12) ≈ 1.2599 (pure 5:4 = 1.25). Beats occur when two pitches are slightly mistuned.

八度频率比严格为2:1。平均律纯五度为 2^(7/12) ≈ 1.4983（纯律3:2接近）。大三度为 2^(4/12) ≈ 1.2599（纯律5:4=1.25）。两音轻微失谐时会产生拍频。

10. Circle of Fifths | 五度圈

The circle of fifths arranges keys by ascending perfect fifths (clockwise) or descending perfect fifths/ascending fourths (anti-clockwise). Each step clockwise adds one sharp (or removes one flat); each step anti-clockwise adds one flat (or removes one sharp).

五度圈以纯五度上行（顺时针）或纯五度下行/四度上行（逆时针）排列调性。顺时针每步增加一个升号（或减少一个降号）；逆时针每步增加一个降号（或减少一个升号）。

Relative major and minor keys share the same key signature; the minor tonic is a minor 3rd below the major tonic (e.g., C major – A minor). To find the number of sharps in a major key: for sharp keys, the last sharp is the leading tone (a semitone below tonic); for flat keys, the penultimate flat is the tonic (except F major).

关系大小调共享调号；小调主音比大调主音低小三度（如C大调-a小调）。大调升号数目规律：升号调中最后一个升号为导音（主音下方半音）；降号调中倒数第二个降号即为主音（F大调除外）。

Apply the circle for modulation: closely related keys are those adjacent on the circle (difference of one sharp/flat). The dominant (V) is one step clockwise, subdominant (IV) one step anti-clockwise from the tonic.

应用五度圈进行转调：近关系调是五度圈上相邻的调（相差一个升降号）。属调（V）为顺时针一步，下属调（IV）为逆时针一步。

11. Ornaments and Articulations | 装饰音与演奏法

Trill: rapid alternation between the written note and the note above, indicated by ‘tr’ or a wavy line. Turn: a four-note figure: upper note, main, lower, main. Mordent: main – lower – main (lower mordent); main – upper – main (upper mordent/inverted mordent). Appoggiatura: small note that takes half the value of the following note (if undotted) or two-thirds (if dotted), approached by leap. Acciaccatura: crushed grace note, usually a semitone or tone, played very quickly on the beat.

颤音：在本音与上方二度音之间快速交替，记号为“tr”或波状线。回音：四音组合：上邻音-本音-下邻音-本音。波音：本音-下邻音-本音（下波音）；本音-上邻音-本音（上波音/逆波音）。倚音：小音符，占后音符时值的一半（无附点）或三分之二（有附点），跳进进入。碎音：短小装饰音，通常为半音或全音，在拍子上极快奏出。

Articulation marks affect attack and duration: staccato (short, detached), legato (smooth, connected), tenuto (held for full value, slightly stressed), accent (>), marcato (^ or wedge), sforzando (sfz, sudden strong accent).

演奏法标记影响触键与时值：跳音（短而断开），连音（平滑连接），保持音（奏满时值并稍加重），重音（>），特重音（^），突强（sfz，突加强调）。

12. Common Scale Formulas | 常见音阶公式

Major scale: W-W-H-W-W-W-H (W = whole tone, H = semitone). Natural minor: W-H-W-W-H-W-W. Harmonic minor: raise 7th degree of natural minor, creating an augmented 2nd between 6th and 7th. Melodic minor ascending: raise 6th and 7th; descending: revert to natural minor.

大调音阶：全-全-半-全-全-全-半。自然小调：全-半-全-全-半-全-全。和声小调：升高自然小调第7级，在6-7级间产生增二度。旋律小调上行：升高6、7级；下行：还原为自然小调。

Chromatic scale: all twelve semitones in sequence. Whole-tone scale: entirely whole tones (e.g., C-D-E-F#-G#-A#), only two transpositions. Pentatonic major: 1-2-3-5-6 of major scale. Pentatonic minor: 1-♭3-4-5-♭7. Blues scale: minor pentatonic plus ♭5.

半音阶：顺序排列十二个半音。全音阶：完全由全音组成（如C-D-E-升F-升G-升A），只有两种移位。大调五声音阶：大调音阶的1-2-3-5-6级。小调五声音阶：1-降3-4-5-降7。蓝调音阶：小调五声音阶加降5级。

Modes of the major scale: Ionian (W-W-H-W-W-W-H), Dorian (W-H-W-W-W-H-W), Phrygian (H-W-W-W-H-W-W), Lydian (W-W-W-H-W-W-H), Mixolydian (W-W-H-W-W-H-W), Aeolian (W-H-W-W-H-W-W), Locrian (H-W-W-H-W-W-W). Modes are often used in composition and analysis.

大调调式：伊奥尼亚（全-全-半-全-全-全-半），多利亚（全-半-全-全-全-半-全），弗里几亚（半-全-全-全-半-全-全），利底亚（全-全-全-半-全-全-半），混合利底亚（全-全-半-全-全-半-全），爱奥利亚（全-半-全-全-半-全-全），洛克里亚（半-全-全-半-全-全-全）。调式常用于作曲与分析。

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