📚 Music Formulas & Theorems Quick Reference | 音乐公式定理速查手册
Music is built on patterns, and these patterns can be expressed as simple 'formulas'—rules that help you read, write, and understand music quickly. This handbook gathers the essential formulas and theorems you need for Year 8 CCEA Music, from note values and time signatures to scales, intervals, and chords. Keep it handy as you practise, and you will soon spot the logic behind every piece of music you play or hear.
音乐由规律构成，这些规律可以表达为简单的“公式”——帮助你快速读谱、写谱和理解音乐的规则。这本手册汇集了八年级CCEA音乐所需的核心公式与定理，涵盖音符时值、拍号、音阶、音程以及和弦等内容。练习时随时查阅，你会逐渐发现每首乐曲背后都有清晰的逻辑。
1. Note Value Formula | 音符时值公式
A whole note (semibreve) is the longest standard note. Each shorter note is half the length of the previous one.
全音符（semibreve）是最长的标准音符，每个更短的音符都是前一个音符时值的一半。
Formula:
公式：
- 1 semibreve = 2 minims (½ notes)
- 1 minim = 2 crotchets (¼ notes)
- 1 crotchet = 2 quavers (⅛ notes)
- 1 quaver = 2 semiquavers (1/16 notes)
In symbols: 𝅝 = 𝅗𝅥 𝅗𝅥 ; 𝅗𝅥 = 𝅘𝅥 𝅘𝅥 ; 𝅘𝅥 = 𝅘𝅥𝅮 𝅘𝅥𝅮 ; 𝅘𝅥𝅮 = 𝅘𝅥𝅯 𝅘𝅥𝅯
用符号表示：𝅝 = 𝅗𝅥 𝅗𝅥 ; 𝅗𝅥 = 𝅘𝅥 𝅘𝅥 ; 𝅘𝅥 = 𝅘𝅥𝅮 𝅘𝅥𝅮 ; 𝅘𝅥𝅮 = 𝅘𝅥𝅯 𝅘𝅥𝅯
2. Dotted Note Rule | 附点音符规则
A dot after a note increases its duration by half of its original value.
音符后面的附点将该音符的时值增加其原有时值的一半。
Formula: Dotted note value = original note value + (original note value × ½)
公式： 附点音符时值 = 原音符时值 + (原音符时值 × ½)
Example: 𝅘𝅥 . = 𝅘𝅥 + 𝅘𝅥𝅮 (1 crotchet + 1 quaver = 1½ beats in 4/4 time)
例子：𝅘𝅥 . = 𝅘𝅥 + 𝅘𝅥𝅮 （在4/4拍中，1个四分音符加1个八分音符，共1½拍）
Double dot: the second dot adds half of the first added value. Rule: original + ½ + ¼. Useful in advanced rhythms.
双附点：第二个附点增加第一个增加值的一半。规则：原值 + ½ + ¼。在复杂节奏中有用。
3. Time Signature Theorem | 拍号定理
A time signature is like a fraction that tells you how many beats are in each bar (top number) and what kind of note gets one beat (bottom number).
拍号就像一个分数，上方数字告诉你每小节有多少拍，下方数字告诉你哪种音符算一拍。
Common time signatures:
常见拍号：
| Top number (beats) | Bottom number (note value) | Meaning |
|---|---|---|
| 4 | 4 | 4 crotchet beats per bar |
| 3 | 4 | 3 crotchet beats per bar |
| 2 | 4 | 2 crotchet beats per bar |
| 6 | 8 | 6 quaver beats (grouped in 2s: compound duple) |
rotchet, 8 = quaver, 2 = minim. This theorem helps you decode any time signature at sight.
下方数字规则：4代表四分音符，8代表八分音符，2代表二分音符。这个定理帮助你看一眼就知道任何拍号的含义。
4. Beat Grouping Formula | 拍子组合公式
Notes are beamed together to show the beat structure. In simple time (2/4, 3/4, 4/4), each beat is a crotchet beat group. In compound time (6/8, 9/8, 12/8), each beat is a dotted crotchet group made of three quavers.
音符用符杠连接起来以显示节拍结构。在单拍子（2/4、3/4、4/4）中，每一拍是一个四分音符组合；在复拍子（6/8、9/8、12/8）中，每一拍是一个附点四分音符组合，由三个八分音符组成。
Simple time beaming rule: Never beam across the middle of a bar in 4/4; beat 3 must show a new beam group.
单拍子符杠规则：4/4拍中不要跨过小节中线连接符杠；第三拍必须开始一个新的符杠组。
Compound time beaming rule: Six quavers in 6/8 should be beamed as two groups of three: (𝅘𝅥𝅮𝅘𝅥𝅮𝅘𝅥𝅮) (𝅘𝅥𝅮𝅘𝅥𝅮𝅘𝅥𝅮). This shows the two main beats clearly.
复拍子符杠规则：6/8拍的六个八分音符应连接成两组三个：(𝅘𝅥𝅮𝅘𝅥𝅮𝅘𝅥𝅮) (𝅘𝅥𝅮𝅘𝅥𝅮𝅘𝅥𝅮)，以清楚显示两个主要拍点。
5. Rest Duration Equivalents | 休止符时值等价公式
Every note value has a matching rest of identical duration. Use the same hierarchy: semibreve rest, minim rest, crotchet rest, etc.
每个音符时值都有一个完全相等时值的休止符。遵循相同的等级：全休止符、二分休止符、四分休止符等。
Formula: semibreve rest (hanging from the 4th line). Only in 4/2 time use a breve rest for a full bar.
公式： 任何拍号中整个小节的休止都使用全休止符（挂在第四线上）。仅在4/2拍中用二全休止符表示整个小节。
Remember: In 3/4, a whole bar rest is still a semibreve rest, not three crotchet rests.
记住：在3/4拍中，整个小节的休止仍然使用全休止符，而不是三个四分休止符。
6. Major Scale Formula | 大调音阶公式
A major scale follows a fixed pattern of tones (T) and semitones (S): T-T-S-T-T-T-S.
大调音阶遵循固定的全音（T）和半音（S）模式：T-T-S-T-T-T-S。
Starting on any note, apply this formula to find the correct notes. Example: C major: C (T) D (T) E (S) F (T) G (T) A (T) B (S) C. No sharps or flats needed.
从任意音开始，使用这个公式找到正确的音符。例如：C大调：C (T) D (T) E (S) F (T) G (T) A (T) B (S) C。不需要升降号。
G major: G (T) A (T) B (S) C (T) D (T) E (T) F♯ (S) G. F must be sharpened to keep the semitone at the end.
G大调：G (T) A (T) B (S) C (T) D (T) E (T) F♯ (S) G。F必须升半音以保持最后一个半音关系。
7. Natural Minor Scale Formula | 自然小调音阶公式
The natural minor scale pattern: T-S-T-T-S-T-T.
自然小调音阶模式：T-S-T-T-S-T-T。
Example: A natural minor: A (T) B (S) C (T) D (T) E (S) F (T) G (T) A. This uses the same key signature as C major (no sharps or flats).
例子：a自然小调：A (T) B (S) C (T) D (T) E (S) F (T) G (T) A。其调号与C大调相同（无升降号）。
or scale is the tonic of its relative natural minor. C major → A minor; G major → E minor.
关系小调定理：大调音阶的第六级音是其关系自然小调的主音。C大调 → a小调；G大调 → e小调。
8. Key Signature Formula (Major Keys) | 调号公式（大调）
The order of sharps: F♯ C♯ G♯ D♯ A♯ E♯ B♯. Remember: Father Charles Goes Down And Ends Battle.
升号顺序：F♯ C♯ G♯ D♯ A♯ E♯ B♯。记忆口诀：Father Charles Goes Down And Ends Battle。
The order of flats: B♭ E♭ A♭ D♭ G♭ C♭ F♭. Reverse of sharps: Battle Ends And Down Goes Charles Father.
降号顺序：B♭ E♭ A♭ D♭ G♭ C♭ F♭。是升号顺序的反向：Battle Ends And Down Goes Charles Father。
Formula to find major key from sharps: Go up a semitone from the last sharp. Last sharp = G♯? key = A major.
由升号找大调调性公式： 最后一个升号升高半音。最后一个升号是G♯？调性 = A大调。
Formula for flats: The second-last flat is the major key. Key signature B♭, E♭, A♭? Second-last = E♭, so E♭ major.
降号公式： 倒数第二个降号即为大调调性。调号含B♭、E♭、A♭？倒数第二个是E♭，所以E♭大调。
9. Interval Number & Quality Theorem | 音程度数与性质定理
An interval is the distance between two notes. Count the letter names (inclusive) to get the interval number. Then determine quality: perfect, major, minor, augmented, or diminished.
音程是两个音之间的距离。计算字母名（含首尾）得到音程度数，然后判断性质：纯、大、小、增或减。
Formula for perfect intervals: Unison, 4th, 5th, octave are perfect if they match the major scale of the lower note. A perfect 4th widened by a semitone becomes augmented; narrowed becomes diminished.
纯音程公式： 同度、四度、五度、八度，若与低音的大调音阶相符则为纯。纯四度扩大半音变为增四度，缩小半音变为减四度。
Formula for major/minor intervals: 2nd, 3rd, 6th, 7th are major if the upper note is in the major scale of the lower note. If a semitone smaller, they become minor. Further semitone alterations create augmented or diminished.
大小音程公式： 二度、三度、六度、七度，如果上方音在低音的大调音阶中则为大音程；小半音则变为小音程。进一步半音变化产生增或减音程。
Example: C to E is a major 3rd (E is in C major). C to E♭ is a minor 3rd (one semitone smaller).
例子：C到E是大三度（E在C大调中）。C到E♭是小三度（小半音）。
10. Triad Chord Construction Formula | 三和弦构成公式
A triad is a three-note chord built by stacking two thirds. The formula depends on the quality.
三和弦是由两个三度叠置而成的三个音的和弦，其构成公式取决于和弦性质。
Major triad: perfect 5th (e.g., C-E-G). Distance: root to 3rd = 4 semitones, 3rd to 5th = 3 semitones.
大三和弦： 根音 + 大三度 + 纯五度（如C-E-G）。距离：根音到三音 = 4个半音，三音到五音 = 3个半音。
Minor triad: perfect 5th (e.g., A-C-E). Distance: root to 3rd = 3 semitones, 3rd to 5th = 4 semitones.
小三和弦： 根音 + 小三度 + 纯五度（如A-C-E）。距离：根音到三音 = 3个半音，三音到五音 = 4个半音。
Diminished triad: root + minor 3rd + diminished 5th (e.g., B-D-F). Both thirds are minor: 3 + 3 semitones.
减三和弦： 根音 + 小三度 + 减五度（如B-D-F）。两个三度均是小三度：3+3个半音。
Augmented triad: root + major 3rd + augmented 5th (e.g., C-E-G♯). Two major thirds: 4 + 4 semitones.
增三和弦： 根音 + 大三度 + 增五度（如C-E-G♯）。两个大三度：4+4个半音。
11. Primary Triads Theorem | 正三和弦定理
In any major key, the triads built on the 1st (tonic), 4th (subdominant), and 5th (dominant) degrees are primary chords. They define the key strongly.
在任何大调中，建立在第I级（主音）、IV级（下属音）和V级（属音）上的三和弦是正三和弦，它们强烈地定义了调性。
Formula for major keys: I = major, IV = major, V = major. Example C major: C (I), F (IV), G (V).
大调公式： I = 大三和弦，IV = 大三和弦，V = 大三和弦。例如C大调：C (I)，F (IV)，G (V)。
Formula for minor keys (using harmonic minor): = major (because the leading note is raised). A minor: Am (i), Dm (iv), E major (V, with G♯).
小调公式（和声小调）： i = 小三和弦，iv = 小三和弦，V = 大三和弦（因为导音被升高）。a小调：Am (i)，Dm (iv)，E大调（V，含G♯）。
This theorem helps with composing cadences and harmonising melodies.
这一定理有助于创作终止式和为旋律配和声。
12. Cadence Recognition Formula | 终止式识别公式
A cadence is a two-chord progression that ends a musical phrase. There are four main types, each with a distinct sound and chord pattern.
终止式是结束乐句的两个和弦进行。共有四种主要类型，每种都有独特的声音与和弦模式。
| Cadence | Chords | Feel |
|---|---|---|
| Perfect (V-I) | Dominant → Tonic | Strong, finished |
| Plagal (IV-I) | Subdominant → Tonic | Amen, softer |
| Imperfect (I-V, II-V, IV-V) | Any → Dominant | Unfinished, pause |
| Interrupted (V-vi) | Dominant → Submediant | Surprise, diversion |
To identify, look at the last two chords of a phrase. In a perfect cadence, both chords are in root position and the tonic note is in the top voice of the final chord for a complete finish.
识别时，看乐句最后两个和弦。在完全终止中，两个和弦都用原位，且终止和弦的高音部是主音，以获得完满的结束感。
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