Simple and Compound Meter and Major Keys Analysis

Simple and Compound Meter

• Meter provides the underlying metric framework and rhythmic organization in music, grouping beats into recurring patterns of accented (strong) and unaccented (weak) pulses.

• Meter classification relies on two core criteria:

  • The division of individual beats (Simple vs. Compound).

  • The number of beats per measure (Duple, Triple, or Quadruple).

Beat Division: Simple vs. Compound

• Simple Meter:

  • A meter in which the primary beat naturally divides into two equal sub-beats (duple division).

  • The main beat unit is an undotted note value (such as a quarter note, half note, or eighth note).

  • Subdivisions are counted using standard fractional patterns such as $1\,\&,\,2\,\&,\,3\,\&,\,4\,\&$.

• Compound Meter:

  • A meter in which the primary beat naturally divides into three equal sub-beats (triple division).

  • The main beat unit is always a dotted note value (such as a dotted quarter note, dotted half note, or dotted eighth note).

  • Subdivisions group pulses into sets of three, typically counted as $1\,\text{la}\,\text{li},\,2\,\text{la}\,\text{li}$ or $1\,-2\,-3,\,4\,-5\,-6$.

Classification by Beats per Measure

• Duple Meter: Contains $2$ main beats per measure.

• Triple Meter: Contains $3$ main beats per measure.

• Quadruple Meter: Contains $4$ main beats per measure.

Simple Time Signatures

In simple time signatures, the top number indicates the exact number of beats per measure, and the bottom number indicates which note duration receives one beat ($2$ = half note, $4$ = quarter note, $8$ = eighth note).

• Simple Duple Meters ($2$ beats per measure):

  • $\frac{2}{4}$: Two beats per measure; the quarter note gets one beat. Each beat divides into two eighth notes.

  • $\frac{2}{2}$ (also known as Cut Time or Alla Breve): Two beats per measure; the half note gets one beat. Each beat divides into two quarter notes.

  • $\frac{2}{8}$: Two beats per measure; the eighth note gets one beat. Each beat divides into two sixteenth notes.

• Simple Triple Meters ($3$ beats per measure):

  • $\frac{3}{4}$: Three beats per measure; the quarter note gets one beat. Each beat divides into two eighth notes.

  • $\frac{3}{2}$: Three beats per measure; the half note gets one beat. Each beat divides into two quarter notes.

  • $\frac{3}{8}$: Three beats per measure; the eighth note gets one beat. Each beat divides into two sixteenth notes.

• Simple Quadruple Meters ($4$ beats per measure):

  • $\frac{4}{4}$ (also known as Common Time, represented by $\mathbf{C}$): Four beats per measure; the quarter note gets one beat. Each beat divides into two eighth notes.

  • $\frac{4}{2}$: Four beats per measure; the half note gets one beat. Each beat divides into two quarter notes.

  • $\frac{4}{8}$: Four beats per measure; the eighth note gets one beat. Each beat divides into two sixteenth notes.

Compound Time Signatures

In compound time signatures, the top number represents the total number of sub-beat divisions (not main beats). Dividing the top number by $3$ yields the total number of main beats per measure. The bottom number indicates the note value of each sub-beat division.

• Compound Duple Meters ($2$ main dotted beats per measure, top number $6$):

  • $\frac{6}{8}$: Six eighth notes per measure, organized into $2$ main beats. Each main beat is a dotted quarter note ($\text{dotted quarter} = 3\,\text{eighth notes}$).

  • $\frac{6}{4}$: Six quarter notes per measure, organized into $2$ main beats. Each main beat is a dotted half note.

  • $\frac{6}{16}$: Six sixteenth notes per measure, organized into $2$ main beats. Each main beat is a dotted eighth note.

• Compound Triple Meters ($3$ main dotted beats per measure, top number $9$):

  • $\frac{9}{8}$: Nine eighth notes per measure, organized into $3$ main beats (dotted quarter notes).

  • $\frac{9}{4}$: Nine quarter notes per measure, organized into $3$ main beats (dotted half notes).

  • $\frac{9}{16}$: Nine sixteenth notes per measure, organized into $3$ main beats (dotted eighth notes).

• Compound Quadruple Meters ($4$ main dotted beats per measure, top number $12$):

  • $\frac{12}{8}$: Twelve eighth notes per measure, organized into $4$ main beats (dotted quarter notes).

  • $\frac{12}{4}$: Twelve quarter notes per measure, organized into $4$ main beats (dotted half notes).

  • $\frac{12}{16}$: Twelve sixteenth notes per measure, organized into $4$ main beats (dotted eighth notes).

Relationships Between Meter Types

• Top Number Summary:

  • Simple Duple: $2$ | Compound Duple: $6$

  • Simple Triple: $3$ | Compound Triple: $9$

  • Simple Quadruple: $4$ | Compound Quadruple: $12$

• Mathematical conversion between corresponding simple and compound beat divisions:

  • Multiply the top number of a simple time signature by $3$ and the bottom number by $2$ to derive its equivalent compound time signature with the same number of main beats.

Key of C, A, F, and G Major

Structural Foundations of Major Scales

• A major scale is a diatonic scale constructed using a specific sequence of whole steps ($W$) and half steps ($H$):   $W - W - H - W - W - W - H$

• Standard Scale Degrees and Functional Terminology:

  • Scale Degree $\hat{1}$: Tonic (the foundational tonality center)

  • Scale Degree $\hat{2}$: Supertonic (positioned one whole step above tonic)

  • Scale Degree $\hat{3}$: Mediant (midway between tonic and dominant)

  • Scale Degree $\hat{4}$: Subdominant (a perfect fifth below tonic / perfect fourth above)

  • Scale Degree $\hat{5}$: Dominant (a perfect fifth above tonic)

  • Scale Degree $\hat{6}$: Submediant (midway between tonic and upper subdominant)

  • Scale Degree $\hat{7}$: Leading Tone (positioned one half step below tonic, resolving upward)

Detailed Analysis of Specific Major Keys

Key of C Major

• Key Signature: $0$ sharps and $0$ flats (no accidentals in key signature).

• Pitches of the C Major Scale: C - D - E - F - G - A - B - C

• Interval Breakdown:

  • C to D: Whole step ($W$)

  • D to E: Whole step ($W$)

  • E to F: Half step ($H$)

  • F to G: Whole step ($W$)

  • G to A: Whole step ($W$)

  • A to B: Whole step ($W$)

  • B to C: Half step ($H$)

• Relative Minor: A minor ($A\text{m}$), sharing the same key signature of zero accidentals.

• Primary Triads:

  • Tonic ($I$): C Major (C - E - G)

  • Subdominant ($IV$): F Major (F - A - C)

  • Dominant ($V$): G Major (G - B - D)

Key of G Major

• Key Signature: $1$ sharp ($F\sharp$).

• Pitches of the G Major Scale: G - A - B - C - D - E - F$\sharp$ - G

• Interval Breakdown:

  • G to A: Whole step ($W$)

  • A to B: Whole step ($W$)

  • B to C: Half step ($H$)

  • C to D: Whole step ($W$)

  • D to E: Whole step ($W$)

  • E to F$\sharp$: Whole step ($W$)

  • F$\sharp$ to G: Half step ($H$)

• Structural Relationship: G Major serves as the dominant key (5th scale degree) of C Major.

• Relative Minor: E minor ($E\text{m}$).

• Primary Triads:

  • Tonic ($I$): G Major (G - B - D)

  • Subdominant ($IV$): C Major (C - E - G)

  • Dominant ($V$): D Major (D - F$\sharp$ - A)

Key of F Major

• Key Signature: $1$ flat ($B\flat$).

• Pitches of the F Major Scale: F - G - A - B$\flat$ - C - D - E - F

• Interval Breakdown:

  • F to G: Whole step ($W$)

  • G to A: Whole step ($W$)

  • A to B$\flat$: Half step ($H$)

  • B$\flat$ to C: Whole step ($W$)

  • C to D: Whole step ($W$)

  • D to E: Whole step ($W$)

  • E to F: Half step ($H$)

• Structural Relationship: F Major serves as the subdominant key (4th scale degree) of C Major.

• Relative Minor: D minor ($D\text{m}$).

• Primary Triads:

  • Tonic ($I$): F Major (F - A - C)

  • Subdominant ($IV$): B$\flat$ Major (B$\flat$ - D - F)

  • Dominant ($V$): C Major (C - E - G)

Key of A Major

• Key Signature: $3$ sharps ($F\sharp$, $C\sharp$, $G\sharp$).

• Pitches of the A Major Scale: A - B - C$\sharp$ - D - E - F$\sharp$ - G$\sharp$ - A

• Interval Breakdown:

  • A to B: Whole step ($W$)

  • B to C$\sharp$: Whole step ($W$)

  • C$\sharp$ to D: Half step ($H$)

  • D to E: Whole step ($W$)

  • E to F$\sharp$: Whole step ($W$)

  • F$\sharp$ to G$\sharp$: Whole step ($W$)

  • G$\sharp$ to A: Half step ($H$)

• Relative Minor: F$\sharp$ minor ($F\sharp\text{m}$).

• Primary Triads:

  • Tonic ($I$): A Major (A - C$\sharp$ - E)

  • Subdominant ($IV$): D Major (D - F$\sharp$ - A)

  • Dominant ($V$): E Major (E - G$\sharp$ - B)