📚 Music Formula & Theorem Quick Reference Guide | 音乐公式定理速查手册
This quick reference guide compiles the essential formulas and theoretical ‘theorems’ encountered in the Edexcel A Level Music course (Year 13). From interval construction to 12-tone serialism, these patterns form the backbone of analysis, composition, and aural perception. Mastering these formulas will equip you to decode any score and tackle appraising questions with confidence.
本速查手册汇编了 Edexcel A Level 音乐（13 年级）课程中遇到的基本公式与理论“定理”。从音程构建到十二音序列主义，这些模式构成了分析、作曲和听觉辨识的骨干。掌握这些公式将使你能够解读任何乐谱，并自信应对评鉴题目。
1. Interval Construction Formula | 音程构成公式
Every interval is defined by the number of semitones between two pitches. Use this key to identify or build any interval instantly.
每个音程由两个音高之间的半音数定义。使用以下关键数据可立即识别或构建任何音程。
Perfect Intervals: P1 (0) | P4 (5) | P5 (7) | P8 (12)
完全音程：纯一度 (0) | 纯四度 (5) | 纯五度 (7) | 纯八度 (12)
Major Intervals: M2 (2) | M3 (4) | M6 (9) | M7 (11)
大音程：大二度 (2) | 大三度 (4) | 大六度 (9) | 大七度 (11)
To derive other qualities: lower a Major interval by a semitone to get a Minor interval; lower a perfect or minor interval by a semitone to get a Diminished interval; raise a perfect or major interval by a semitone to get an Augmented interval.
推导其他性质：将大音程降低半音得到小音程；将完全或小音程降低半音得到减音程；将完全或大音程升高半音得到增音程。
The Inversion Theorem: The sum of an interval and its inversion always equals 9 (e.g., M3 inverts to m6; 3 + 6 = 9). Perfect intervals invert to perfect; major inverts to minor; augmented inverts to diminished.
转位定理：一个音程与其转位的度数之和始终为 9（例如大三度转为小六度，3 + 6 = 9）。完全音程转位后仍为完全；大音程转位变小；增音程转位变减。
2. Major & Minor Scale Formulas | 大小调音阶公式
Scales are built on a fixed sequence of tones (T) and semitones (S). The formulas below generate the required key signatures and accidentals.
音阶建立在固定的全音 (T) 与半音 (S) 序列上。以下公式可用于生成所需的调号与临时记号。
Major Scale: T – T – S – T – T – T – S
大调音阶：全 – 全 – 半 – 全 – 全 – 全 – 半
Natural Minor: T – S – T – T – S – T – T
自然小调：全 – 半 – 全 – 全 – 半 – 全 – 全
Harmonic Minor: T – S – T – T – S – A2 – S (A2 = augmented 2nd = 3 semitones)
和声小调：全 – 半 – 全 – 全 – 半 – 增二度 – 半（增二度 = 3 个半音）
Melodic Minor Theorem (Ascending): T – S – T – T – T – T – S (raises 6th and 7th). Descending follows natural minor.
旋律小调定理（上行）：全 – 半 – 全 – 全 – 全 – 全 – 半（升高 VI 级与 VII 级）。下行使用自然小调。
Relative minor is found a minor 3rd below the major tonic. Parallel minor shares the same tonic with altered 3rd, 6th, 7th degrees.
关系小调位于大调主音下方小三度。同主音小调共享相同主音，但改变 III、VI、VII 级音。
3. Chord Construction Formulas | 和弦构成公式
All triads are stacks of thirds. Knowing the semitone distances from the root makes chord identification and spelling automatic.
所有三和弦都是三度叠置。掌握根音上方的半音距离可使和弦辨识与拼写自动化。
Major Triad: Root + 4 semitones (M3) + 7 semitones (P5). Minor Triad: Root + 3 (m3) + 7 (P5).
大三和弦：根音 + 4 半音（大三度）+ 7 半音（纯五度）。小三和弦：根音 + 3（小三度）+ 7（纯五度）。
Diminished Triad: Root + 3 (m3) + 6 (diminished 5th). Augmented Triad: Root + 4 (M3) + 8 (augmented 5th).
减三和弦：根音 + 3 + 6（减五度）。增三和弦：根音 + 4 + 8（增五度）。
Stacking Theorem: Close position triads are built by stacking two 3rds. The quality of the 3rds determines the chord: Major = M3 + m3; Minor = m3 + M3; Diminished = m3 + m3; Augmented = M3 + M3.
叠置定理：密集位置的三和弦由两个三度叠置构成。三度音程的性质决定和弦性质：大三和弦 = 大三度 + 小三度；小三和弦 = 小三度 + 大三度；减三和弦 = 小三度 + 小三度；增三和弦 = 大三度 + 大三度。
4. Seventh Chord Formulas | 七和弦公式
Seventh chords add a 3rd above the triad. Master these four archetypes for all harmonic analysis.
七和弦在三和弦上方再叠加一个三度。掌握以下四种原型可应对所有和声分析。
Dominant 7th: Major triad + minor 7th (10 semitones). Structure: M3 + m3 + m3.
属七和弦：大三和弦 + 小七度（10 半音）。结构：大三度 + 小三度 + 小三度。
Major 7th: Major triad + major 7th (11 semitones). Structure: M3 + m3 + M3.
大七和弦：大三和弦 + 大七度（11 半音）。结构：大三度 + 小三度 + 大三度。
Minor 7th: Minor triad + minor 7th (10 semitones). Structure: m3 + M3 + m3.
小七和弦：小三和弦 + 小七度。结构：小三度 + 大三度 + 小三度。
Half-diminished 7th: Diminished triad + minor 7th. Fully diminished 7th adds a diminished 7th (9 semitones): m3 + m3 + m3.
半减七和弦：减三和弦 + 小七度。完全减七和弦：减三和弦 + 减七度（9 半音）：小三度 + 小三度 + 小三度。
5. Cadence Formulas | 终止式公式
Cadences are the punctuation marks of tonal music. Their chord pairs are governed by strict harmonic formulas.
终止式是调性音乐的标点符号。其和弦连接由严格的和声公式支配。
Perfect Cadence: V – I (both chords in root position; soprano finishes on tonic). Creates the strongest sense of closure.
正格终止：V – I（两和弦均位于根音位置；女高音结束于主音）。产生最强的收束感。
Plagal Cadence: IV – I (often called the ‘Amen’ cadence).
变格终止：IV – I（通常称为“阿门”终止）。
Imperfect Cadence: Any chord moving to V, e.g., I – V, ii – V, IV – V. Sounds unfinished.
半终止：任何和弦进行到 V，如 I – V、ii – V、IV – V。听起来未完成。
Interrupted (Deceptive) Cadence: V – vi (in a major key) or V – VI (in a minor key). Surprises the ear by avoiding the tonic.
阻碍终止（诈伪终止）：V – vi（大调中）或 V – VI（小调中）。避免主音，给听觉带来意外感。
Cadential 6-4 Theorem: A I 6/4 chord acts as a double suspension over a V bass, resolving to a V(7) before the final I. Formula: I 6/4 – V(7) – I.
终止四六和弦定理：I 6/4 和弦作为属音低音上的双重留音，解决到 V(7) 再至 I。公式：I 6/4 – V(7) – I。
6. Functional Harmony Progression Axioms | 功能和声进行公理
Chord functions (tonic, pre-dominant, dominant) follow predictable paths. These axioms help you decode both Classical and popular harmony.
和弦功能（主、下属、属）遵循可预测的路径。这些公理有助于解读古典与流行和声。
Tonic Expansion: I – vi – IV – V – I is a standard cycle. I vi ii V I is the ’50s progression.
主功能扩展：I – vi – IV – V – I 是标准循环。I vi ii V I 是“50 年代进行”。
Circle of Fifths Progression: iii – vi – ii – V – I (in major) follows descending perfect 5ths (or ascending 4ths). V – I motion is fundamental.
五度圈进行：iii – vi – ii – V – I（大调中）按照下行纯五度（或上行纯四度）移动。V – I 的进行是根本。
Voice Leading Theorems: Avoid parallel 5ths and octaves. Leading note (7th) resolves up to tonic. Chordal 7ths resolve down by step.
声部进行定理：避免平行五度与平行八度。导音（VII 级）上行解决到主音。和弦七音下行级进解决。
Functional Substitutions: ii substitutes for IV (both pre-dominant). viio (diminished) substitutes for V (dominant function). In jazz, tritone substitution replaces V7 with bII7.
功能替代：ii 替代 IV（均属下属功能）。viio（减和弦）替代 V（属功能）。爵士中，三全音替代用 bII7 替换 V7。
7. Key Signature Theorem (Circle of Fifths) | 调号定理（五度圈）
Key signatures follow a strict pattern. Add a sharp for each perfect 5th up; add a flat for each perfect 4th up (or perfect 5th down).
调号遵循严格的模式。每上行纯五度增加一个升号；每上行纯四度（或下行纯五度）增加一个降号。
Sharp Key Mnemonic: Father Charles Goes Down And Ends Battle (F♯, C♯, G♯, D♯, A♯, E♯, B♯).
升号调口诀：Father Charles Goes Down And Ends Battle（F♯、C♯、G♯、D♯、A♯、E♯、B♯）。
Flat Key Mnemonic: Battle Ends And Down Goes Charles’ Father (B♭, E♭, A♭, D♭, G♭, C♭, F♭).
降号调口诀：Battle Ends And Down Goes Charles’ Father（B♭、E♭、A♭、D♭、G♭、C♭、F♭）。
Relative Minor Formula: To find the relative minor, go down a minor 3rd from the major tonic. Example: G major → E minor.
关系小调公式：从大调主音下行小三度即得到关系小调。例：G 大调 → e 小调。
Enharmonic Theorem: Keys like F♯ major (6 sharps) sound identical to G♭ major (6 flats). This is enharmonic equivalence.
等音调定理：如 F♯ 大调（6 个升号）与 G♭ 大调（6 个降号）音响相同。此为等音等同。
8. Serialism: 12-Tone Matrix Formula | 序列主义：十二音矩阵公式
In serial composition, a prime row (P0) generates three other forms: Retrograde (R), Inversion (I), and Retrograde Inversion (RI). Use a matrix to find all 48 row transpositions.
在序列音乐中，一个原型音列（P0）可生成另外三种形式：逆行（R）、倒影（I）和倒影逆行（RI）。使用矩阵可找出全部 48 个移调形式。
Matrix Construction Theorem: Write P0 horizontally. The first note of P0 is the pitch class 0. For each row I_n, use the formula: I_n’s first note = the note that makes the interval from P0[0] to the leftmost note equal to the negative of the interval to the original P0’s corresponding note. Practically, the inversion starting on the same pitch as the first note of P0 forms the first column. Then transpose each I row to start on the successive notes of P0.
矩阵构建定理：将原型横写。P0 的首音为音级 0。对于每一倒影行 I_n，公式为：使之与 P0 首音的音程等于原排中对应音程的相反数。实践上，以与 P0 首音相同音高开始的倒影构成首列，然后依次移调，以 P0 的各个音作为各行倒影的起始音。
Pitch Class Integer Notation: C = 0, C♯/D♭ = 1, D = 2, …, B = 11. To compute RI of a given P0, reverse the I row that starts on the last note of P0.
音级整数记谱：C = 0, C♯/D♭ = 1, D = 2, …, B = 11。计算既定原型的逆行倒影 RI：找到以 P0 末音开始的倒影行，将其逆行即可。
The Invariance Theorem: Some rows contain invariant segments (e.g., a trichord that maps onto itself under inversion or retrograde). These are used for structural coherence.
不变性定理：某些音列包含不变片段（如一个三音组在倒影或逆行下映射到自身）。用于构建结构连贯性。
9. Blues & Jazz Scale Formulas | 蓝调与爵士音阶公式
Blues and jazz rely on specific scale patterns and chord-scale relationships. Memorise these formulas for improvisation and analysis of set works.
蓝调与爵士依赖特定的音阶模式及和弦-音阶关系。熟记这些公式有助即兴演奏与分析作品集。
Minor Blues Scale: 1 – ♭3 – 4 – ♯4/♭5 – 5 – ♭7. (Added ‘blue note’ between 4 and
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