📚 Year 10 CAIE Music: Quick Reference Handbook of Formulas & Theorems | Year 10 CAIE 音乐：公式定理速查手册
This handbook compiles the essential 'formulas' and rules in music theory that every Year 10 CAIE Music student needs to know. From interval counting and scale construction to chord building and rhythmic mathematics, these are the building blocks for analysis, composition, and performance. Think of them as the equations behind the sound.
本手册汇集了每位 Year 10 CAIE 音乐学生必须掌握的音乐理论核心“公式”与规则。从音程计算、音阶构成到和弦构建和节奏数学，这些是分析、作曲与表演的基石。不妨把它们看作声音背后的方程式。
1. Intervals: Semitone Counting Formula | 音程：半音计数公式
An interval is the distance between two pitches, measured in semitones (half steps). The quality (major, minor, perfect, augmented, diminished) is determined by the number of semitones and the letter names involved. You can always count up from the lower note using the chromatic scale.
音程是两音高之间的距离，以半音为单位。性质（大、小、纯、增、减）由半音数和音名字母共同决定。你可以从较低音开始，利用半音音阶向上计数。
Standard interval semitone counts in one octave (excluding compound intervals):
一个八度内标准音程的半音数（不含复音程）：
| Interval | Semitones | Example from C |
|---|---|---|
| Minor 2nd | 1 | C – D♭ |
| Major 2nd | 2 | C – D |
| Minor 3rd | 3 | C – E♭ |
| Major 3rd | 4 | C – E |
| Perfect 4th | 5 | C – F |
| Augmented 4th / Diminished 5th (Tritone) | 6 | C – F♯ / C – G♭ |
| Perfect 5th | 7 | C – G |
| Minor 6th | 8 | C – A♭ |
| Major 6th | 9 | C – A |
| Minor 7th | 10 | C – B♭ |
| Major 7th | 11 | C – B |
| Perfect Octave | 12 | C – C |
Remember: the number in the interval name (e.g., '2nd', '3rd') always relates to the letter-name distance, not the semitone count. The semitone count confirms the quality.
记住：音程名称中的数字（如“二度”、“三度”）始终对应音名字母的距离，而非半音数。半音数确定了音程的性质。
2. Major Scale Formula | 大调音阶公式
All major scales follow an identical pattern of tones (T) and semitones (S), or whole steps (W) and half steps (H):
所有大调音阶都遵循完全相同的全音（T）与半音（S）排列模式（或全音步W、半音步H）：
T – T – S – T – T – T – S
W – W – H – W – W – W – H
Starting on any note, simply climb up using this sequence: whole step, whole step, half step, whole step, whole step, whole step, half step. For example, from D: D (W) E (W) F♯ (H) G (W) A (W) B (W) C♯ (H) D.
从任意音开始，只需按该序列上行：全音、全音、半音、全音、全音、全音、半音。例如从D开始：D (W) E (W) F♯ (H) G (W) A (W) B (W) C♯ (H) D。
You can also generate the scale using the scale-degree pattern: I (tonic), II (supertonic), III (mediant), IV (subdominant), V (dominant), VI (submediant), VII (leading note), VIII (tonic). The semitones always fall between III–IV and VII–VIII.
也可以用音级模式生成音阶：I（主音）、II（上主音）、III（中音）、IV（下属音）、V（属音）、VI（下中音）、VII（导音）、VIII（主音）。半音总是出现在III–IV和VII–VIII之间。
3. Minor Scales: Natural, Harmonic, Melodic | 小调音阶：自然、和声、旋律
There are three common minor scale forms, each with its own interval formula. They all share a lowered 3rd degree (♭3) relative to the major.
有三种常见的小调音阶形式，各有其音程公式。它们都有相对于大调的降低的第三级音（♭3）。
Natural Minor: follows the key signature exactly. Formula (semitones between degrees):
自然小调：严格遵循调号。音级之间的半音公式：
T – S – T – T – S – T – T
W – H – W – W – H – W – W
Harmonic Minor: raises the 7th degree by a semitone, creating an augmented 2nd step between 6th and 7th. Formula:
和声小调：升高第VII级半音，在VI–VII之间形成增二度。公式：
T – S – T – T – S – 3S – S
W – H – W – W – H – WH – H (the step from VI to VII is 3 semitones)
Melodic Minor (ascending): raises both the 6th and 7th degrees. Formula:
旋律小调（上行）：升高第VI和VII级。公式：
T – S – T – T – T – T – S
W – H – W – W – W – W – H
Melodic minor descending often reverts to natural minor. This three-form system is essential for CAIE listening and composing.
旋律小调下行常回归自然小调。这三种形式对CAIE听辨和作曲至关重要。
4. Key Signatures: Circle of Fifths | 调号：五度圈
Key signatures tell you which notes are consistently sharpened or flattened. The circle of fifths arranges keys so that each clockwise step adds a sharp (or removes a flat), moving up a perfect fifth. Each anticlockwise step adds a flat (or removes a sharp), moving up a perfect fourth.
调号告诉你哪些音被持续升高或降低。五度圈将调性如此排列：顺时针每步增加一个升号（或减少一个降号），向上移动纯五度；逆时针每步增加一个降号（或减少一个升号），向上移动纯四度。
Order of sharps: F♯, C♯, G♯, D♯, A♯, E♯, B♯. Order of flats: B♭, E♭, A♭, D♭, G♭, C♭, F♭. You can memorise them with mnemonics, but the mathematical rule is: new sharp is the leading note (7th degree) of the new major key; new flat is the subdominant (4th degree) of the new major key.
升号顺序：F♯、C♯、G♯、D♯、A♯、E♯、B♯。降号顺序：B♭、E♭、A♭、D♭、G♭、C♭、F♭。记忆口诀虽好，但数学规则是：新升号是新大调的导音（VII级）；新降号是新大调的下属音（IV级）。
ound a minor 3rd below the major tonic. For example, C major → A minor (3 semitones down).
关系小调位于大调主音下方小三度。例如，C大调 → A小调（下行3个半音）。
5. Triad Chord Construction | 三和弦构成
A triad is a three-note chord built from two superimposed thirds. The formula depends on the intervals above the root.
三和弦是由两个叠加的三度构成的三音和弦。公式取决于根音上方的音程。
- Major triad: Root + Major 3rd (4 semitones) + Perfect 5th (7 semitones). Structure: M3 + m3.
- Major triad: 根音 + 大三度（4半音） + 纯五度（7半音）。结构：大三度+小三度。
- Minor triad: Root + Minor 3rd (3 semitones) + Perfect 5th (7 semitones). Structure: m3 + M3.
- Minor triad: 根音 + 小三度（3半音） + 纯五度（7半音）。结构：小三度+大三度。
- Diminished triad: Root + Minor 3rd (3 semitones) + Diminished 5th (6 semitones). Structure: m3 + m3.
- Diminished triad: 根音 + 小三度（3半音） + 减五度（6半音）。结构：小三度+小三度。
- Augmented triad: Root + Major 3rd (4 semitones) + Augmented 5th (8 semitones). Structure: M3 + M3.
- Augmented triad: 根音 + 大三度（4半音） + 增五度（8半音）。结构：大三度+大三度。
You can also derive triads from a scale: stacking every alternate note (1–3–5, 2–4–6, etc.) on each degree. The quality of each triad follows the scale's pattern.
也可以从音阶中导出三和弦：在每个音级上每隔一音叠加（1–3–5、2–4–6等）。各音级上三和弦的性质遵循音阶的模式。
6. Seventh Chords | 七和弦
A seventh chord adds another third on top of the triad, forming a 4-note chord. The quality is defined by the triad base plus the seventh interval from the root.
七和弦在三和弦上方再叠加一个三度，构成四音和弦。性质取决于三和弦基础加上从根音到七音的音程。
| Chord Type | Triad | 7th Interval | Example (on C) |
|---|---|---|---|
| Major 7th | Major | Major 7th (11 semitones) | C–E–G–B |
| Dominant 7th | Major | Minor 7th (10 semitones) | C–E–G–B♭ |
| Minor 7th | Minor | Minor 7th (10 semitones) | C–E♭–G–B♭ |
| Half-diminished 7th | Diminished | Minor 7th (10 semitones) | C–E♭–G♭–B♭ |
| Diminished 7th | Diminished | Diminished 7th (9 semitones) | C–E♭–G♭–B♭♭ (A) |
The dominant 7th (V7) is structurally the most tension-filled and resolves to the tonic. Diminished 7ths are symmetrical: they divide the octave into four equal minor thirds.
属七和弦（V7）结构紧张感最强，倾向解决到主和弦。减七和弦是对称的：将八度均分为四个小三度。
7. Time Signatures: Beat Hierarchy | 拍号：节拍层级
ue gets one beat (4 = crotchet/quarter note, 8 = quaver/eighth note, 2 = minim/half note).
拍号是音乐中的数学分数。上方数字表明每小节拍数；下方数字表明以何种音符为一拍（4=四分音符，8=八分音符，2=二分音符）。
Simple time: each beat divides into two equal parts (e.g., 2/4, 3/4, 4/4, 2/2, 3/2). Compound time: each beat divides into three equal parts (e.g., 6/8, 9/8, 12/8). In compound time, the upper number is a multiple of 3; the beat unit is a dotted note.
单拍子：每一拍均分为两个等分（如2/4、3/4、4/4、2/2、3/2）。复拍子：每一拍均分为三个等分（如6/8、9/8、12/8）。复拍子中，上方数字为3的倍数；拍子单位是附点音符。
Hierarchy formula: within each bar, the first beat is the strongest (downbeat). In 4/4, stress pattern is strong – weak – medium – weak. In 3/4, strong – weak – weak. In 6/8, the dotted crotchet beat is felt as two main pulses: strong – weak.
层级公式：每小节内，第一拍最强（强拍）。4/4中，重音模式为强–弱–次强–弱。3/4中，强–弱–弱。6/8中，以附点四分音符为拍感知为两个主脉冲：强–弱。
8. Rhythmic Mathematics: Note and Rest Values | 节奏数学：音符与休止符时值
Each rhythmic symbol has a fractional relationship to the semibreve (whole note). The formulas are based on powers of 2.
每个节奏符号与全音符存在分数关系。公式基于2的乘方。
- Semibreve (whole note) = 1 = 2 minims = 4 crotchets = 8 quavers = 16 semiquavers.
- 全音符 = 1 = 2个二分音符 = 4个四分音符 = 8个八分音符 = 16个十六分音符。
- Dotted note: iginal value × 1.5. For example, dotted crotchet = 1 + ½ beat = 1½ beats (in simple time).
- 附点音符：增加自身时值的一半。公式：附点时值 = 原值 × 1.5。例如附点四分音符 = 1 + ½ 拍 = 1½ 拍（单拍子）。
- Double dotted note: original + half + quarter. Formula: value × 1.75.
- 复附点音符：原值 + 一半 + 四分之一。公式：原值 × 1.75。
- Triplet: three notes in the time of two of the same kind. Each triplet eighth, for instance, equals 1/3 of a beat in simple time. No exact fraction can be expressed as a single dotted value, hence the irregular division.
- 三连音：三个音替代两个同类音的时值。例如，每个三连音八分音符在单拍子中等于1/3拍。无法表示为单一附点值，因此为不规则划分。
Rests follow identical duration mathematics. Tie joins two notes of the same pitch, adding their values.
休止符遵循相同的时值数学。延音线连接两个同音高音符，相加其时值。
9. Transposition: Interval Shift Formula | 移调：音程移动公式
Transposition means moving a passage of music to a different pitch level while keeping all intervals exactly the same. The formula requires you to shift every note by a constant interval (e.g., up a major 2nd, down a perfect 4th).
移调意味着将一段音乐移动到不同的音高位置，同时保持所有音程完全相同。公式要求你将每个音符移动一个恒定的音程（例如上行大二度、下行纯四度）。
When transposing instruments are written at a different pitch from concert pitch, use the interval between the instrument's key and concert pitch. For example, trumpet in B♭ sounds a major 2nd lower than written. So to get sounding pitch, transpose written notes down a major 2nd; to notate for trumpet from concert pitch, write them up a major 2nd.
当移调乐器的记谱与实际音高不同时，利用乐器调性与实际音高之间的音程。例如，降B调小号实际发音比记谱低大二度。要得到实际音高，将记谱下行大二度；若从实际音高为小号记谱，则上行大二度记写。
Accidentals must be recalculated: an interval shift preserves enharmonic correctness, but you must check key signatures and accidental spelling (e.g., up a minor 3rd from C major gives E♭ major, not D♯ major).
临时记号必须重新计算：音程移动保持等音正确性，但需检查调号与临时记号拼法（例如从C大调上行小三度得E♭大调，而非D♯大调）。
10. Cadences: Harmonic Formulas | 终止式：和声公式
A cadence is a two-chord progression that ends a phrase. The four principal cadences are defined by their chord formula and final effect.
终止式是结束一个乐句的两个和弦的进行。四种主要终止式由其和弦公式和最终效果定义。
- Perfect cadence (authentic): I). Sounds final, complete. Both chords in root position with tonic in the top voice of I.
- 完全终止（正格终止）：V → I（或V7 → I）。听感结束、完满。两个和弦均为原位，且I和弦最高声部为根音。
- Plagal cadence: IV → I. Often called 'Amen' cadence. Sounds softer but still conclusive.
- 变格终止：IV → I。常称“阿门”终止。听感柔和但仍具收束感。
- Imperfect cadence: : any chord → V. It ends on the dominant, creating expectation (I → V, ii → V, IV → V, vi → V).
- 不完全终止：任意和弦→V。终止在属和弦上，产生期待感（I → V、ii → V、IV → V、vi → V）。
- Interrupted (deceptive) cadence: V → vi (sometimes V → IV or other). The expected tonic is replaced, creating surprise.
- 阻碍终止（虚假终止）：V → vi（有时V → IV等）。预期的主和弦被替代，制造意外感。
The formulas rely on scale-degree relationships, not absolute chords. In a minor key, use the raised leading note in V to create a dominant major chord for perfect and imperfect cadences.
公式依赖于音级关系，而非绝对和弦。在小调中，完全终止和不完全终止的V使用升高导音构成大属和弦。
11. Musical Terms and Symbols Quick Reference | 音乐术语与符号速查
Essential Italian terms and symbols with their precise meanings function as performance 'formulas' in CAIE music. Here is a compact reference.
核心意大利术语与符号及其精确含义在CAIE音乐中扮演着表演“公式”的角色。以下为精简参考。
| Symbol / Term | Meaning | 中文 |
|---|---|---|
| p, piano | Soft | 弱 |
| f, forte | Loud | 强 |
| mp, mezzo piano | Moderately soft | 中弱 |
| mf, mezzo forte | Moderately loud | 中强 |
| crescendo (cresc.) | Gradually getting louder | 渐强 |
| diminuendo (dim.) | Gradually getting softer | 渐弱 |
| staccato (·) | Short, detached | 断奏 |
| legato (slur) | Smoothly connected | 连奏 |
| accent (> ) | Emphasised attack | 重音 |
| tenuto (–) | Hold full value, slightly stressed | 保持音 |
| fermata (𝄐) | Pause, hold note longer | 延长记号 |
| 8va | Play an octave higher (if above) / lower (if below) | 高八度/低八度演奏 |
Tempo markings act as BPM (beats per minute) ranges: Largo (40–60), Adagio (66–76), Andante (76–108), Moderato (108–120), Allegro (120–168), Presto (168–200). These metronome formulas help interpret style.
速度标记对应BPM范围：广板(40–60)、柔板(66–76)、行板(76–108)、中板(108–120)、快板(120–168)、急板(168–200)。这些节拍器公式有助于诠释风格。
12. Dynamics and Articulation Formulas | 力度与演奏法公式
Dynamics are not just random; they form a measurable scale from very quiet to very loud, often notated with Italian terms and corresponding symbols. The standard sequence formula is:
力度并非随意，它们构成从极弱到极强的可测量等级，常用意大利术语和对应符号标记。标准序列公式如下：
ppp – pp – p – mp – mf – f – ff – fff
Each step represents a distinct dynamic level. Subito dynamics (sudden changes) like sfz (sforzando) or fp follow a different rule: a strong accent immediately dropping to piano. Another mathematical aspect is the dynamic envelope: crescendo and diminuendo can be notated with hairpins ( < and > ) and may span any length, but their execution is governed by the music's expressive shape.
每一级代表一个明确的力度层次。突强（sfz）或强后即弱（fp）之类的突变动量遵循不同规则：强重音后立刻降至弱。另一个数学方面是力度包络：渐强和渐弱可用开口三角号（ < 和 > ）表示，可跨任意长度，但其执行受乐句表情形态支配。
Articulation formulas like legato (slur, indicating smooth connection), staccato (dots, indicating separated), and accent (>, >) can be combined mathematically: staccato shortens a note by about half its written value, while tenuto insists on full length. Understanding these proportional relationships is key to accurate interpretation in CAIE performance and listening exams.
演奏法公式如连奏（连线，表示圆滑连接）、断奏（圆点，表示分离）和重音
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