Comprehensive Music Theory Guide: Harmony, Keys, and the Circle of Fifths

Rhythm, Pulse, and Harmony Fundamentals

• Beat:

  • Represents a regular underlying pulse.

  • Is not always explicitly audible in performance, but is continuously felt or imagined.

• Meter:

  • Refers to the grouping of beats.

  • Establishes repeating patterns of strong and weak beats.

• Harmony:

  • Occurs whenever two or more tones sound simultaneously.

Common-Practice Tonality

• Definition:

  • The system of organizing pitch and harmony that serves as the intuitive baseline in Western musical culture.

• Historical Development:

  • Originated in Europe during the Middle Ages.

  • Formally codified by approximately $1750$.

  • Subsequent developments introduced increased layers of complexity, direct challenges by modern composers, and integration of non-Western musical traditions.

  • Governs nearly all musical works produced or consumed within the Western world.

Chords and Triads

• Chord Definition:

  • A collection of three or more pitches sounding simultaneously.

  • Pressing a forearm or a book down onto a piano keyboard creates a chord.

  • Functional and common chords avoid using immediately adjacent pitches.

• Triad Definition:

  • A three-note chord consisting of two stacked intervals of a third.

• The Four Triad Qualities:

  • Major Triad ($M$): Contains a major third between its lower two pitches and a minor third between its upper two pitches.

  • Minor Triad ($m$): Contains a minor third on the bottom and a major third on top.

  • Diminished Triad ($d$): Consists of two stacked minor thirds.

  • Augmented Triad ($A$): Consists of two stacked major thirds.

• Basic Scale Triads:

  • The core chords of any musical composition consist of triads constructed above each note of the scale.

• Anatomy of a Triad:

  • Root: The lowest note of the triad when stacked in thirds.

  • Third: The middle note of the triad.

  • Fifth: The highest note of the triad.

  • Root Position: The state of a triad when its root is located in the lowest voice.

• Examples of Triads:

  • Major Triad ($M$): $C\text{--}E\text{--}G$ or $B\text{--}D\text{\#}\text{--}F\text{\#}$

  • Minor Triad ($m$): $C\text{--}E\flat\text{--}G$

  • Augmented Triad ($A$): $A\text{--}C\text{\#}\text{--}E\text{\#}$

  • Diminished Triad ($d$): $B\text{--}D\text{--}F$

Triad Inversions and Voicing Spacing

• Pitch Displacement:

  • Any pitch within a triad can be octave-transposed up or down without altering the fundamental chord classification.

• Inversion Classifications:

  • First Inversion:

  • Occurs when the third of the triad is positioned as the lowest note.

  • Notated by placing a figure six ($6$) after the chord symbol (e.g., $C^6$ or $C_6$).

  • Second Inversion:

  • Occurs when the fifth of the triad is positioned as the lowest note.

  • Notated using a vertically aligned six and four ($_4^6$) after the chord symbol (e.g., $C_4^6$).

• Structural Properties of Inversions:

  • The lowest pitch strictly determines the inversion state.

  • The upper remaining pitches may appear in any vertical arrangement.

  • Any of the triad's three member pitches can be duplicated (doubled) across identical or different octaves without changing its structural identity as a triad.

• Spacing Variations (e.g., $C$ Triad Formations):

  • Root Position ($C$): $C\text{--}E\text{--}G$, $C\text{--}G\text{--}E$, $C\text{--}G\text{--}E\text{--}C$, $C\text{--}G\text{--}C\text{--}E$

  • First Inversion ($C^6$): $E\text{--}G\text{--}C$, $E\text{--}C\text{--}G$, $E\text{--}C\text{--}G\text{--}C$

  • Second Inversion ($C_4^6$): Variations with $G$ as the lowest pitch.

Musical Keys and Key Signatures

• Definition of Key:

  • The complete environment of pitch relationships within which a composition or major musical section exists.

  • Must be distinguished from a physical key on a piano keybed.

  • Represents a set of seven notes (scale) selected as the pitch foundation for a piece.

• Tonic Pitch:

  • The central gravitational pitch of a key.

  • Imparts its pitch name directly to the overall key (e.g., key of $D$ has a tonic of $D$; key of $A$ major uses the seven pitches of the $A$ scale).

• Major vs. Minor Distinction:

  • Differentiated by specific scale degrees: scale degree three ($\hat{3}$), scale degree six ($\hat{6}$), and scale degree seven ($\hat{7}$).

  • Unqualified references such as "key of $C$" default to $C$ major.

  • Additional common-practice tonality variants include $C$ natural minor, $C$ harmonic minor, and $C$ melodic minor.

• Structural vs. Chromatic Pitches:

  • Music in $C$ major primarily utilizes the seven main scale pitches of $C$ major and their octave transpositions.

  • Pitches foreign to the scale are termed chromatic pitches, serving decorative or expressive purposes rather than structural ones.

• Key Signatures:

  • A collection of sharps or flats placed at the start of every musical staff.

  • Dictates which seven pitches form the underlying scale by systematically raising or lowering specific letter names.

  • An $F\sharp$ key signature entry automatically sharpens all $F$ pitches throughout the entire work unless overridden by a natural sign ($\natural$).

  • Only two scales require exactly one sharp ($F\sharp$):

  • $G$ major (requires the $7\text{th}$ scale degree raised to fit the major scale whole/half step pattern).

  • $E$ minor (requires the $2\text{nd}$ scale degree raised to fit the natural minor pattern).

• Notation of Minor Keys:

  • All three forms of minor scales (natural, harmonic, melodic) utilize the key signature of the natural minor scale.

  • Harmonic and melodic variants incorporate explicit accidentals on individual notes in the score.

• Key Signature Efficiency Example ($E$ Major):

  • Notated without a key signature: Requires individual sharp accidentals placed before every $F\sharp$, $C\sharp$, $G\sharp$, and $D\sharp$.

  • Notated with a key signature: Includes four sharps ($F\sharp$, $C\sharp$, $G\sharp$, $D\sharp$) at the start of the staff, automatically modifying all corresponding pitches.

Hierarchy of Keys and the Circle of Fifths

• System Overview:

  • Comprises $15$ major scales and $15$ minor scales, yielding $30$ theoretical key options.

  • Shared key signatures reduce the required distinct signatures to $15$.

  • Relative keys share identical pitch collections and key signatures (e.g., $C$ minor is the relative minor of $E\flat$ major; $E\flat$ major is the relative major of $C$ minor).

• Interval of a Perfect Fifth:

  • Defined as an interval spanning $7$ half steps subsumed within $5$ scale letter names (e.g., $C\text{--}G$ or $B\flat\text{--}F$).

• Clockwise Progression (Sharps):

  • Begins at the top with $C$ major ($0$ sharps/flats, all natural pitches).

  • Each clockwise step ascends by a perfect fifth and adds one sharp:

  • $0$ sharps: $C$ major

  • $1$ sharp ($F\sharp$): $G$ major

  • $2$ sharps ($F\sharp, C\sharp$): $D$ major

  • $3$ sharps ($F\sharp, C\sharp, G\sharp$): $A$ major

  • $4$ sharps ($F\sharp, C\sharp, G\sharp, D\sharp$): $E$ major

  • $5$ sharps ($F\sharp, C\sharp, G\sharp, D\sharp, A\sharp$): $B$ major

  • $6$ sharps ($F\sharp, C\sharp, G\sharp, D\sharp, A\sharp, E\sharp$): $F\sharp$ major

  • $7$ sharps ($F\sharp, C\sharp, G\sharp, D\sharp, A\sharp, E\sharp, B\sharp$): $C\sharp$ major (raises every pitch of $C$ major by one half step)

  • Standard order of sharps: $F\text{--}C\text{--}G\text{--}D\text{--}A\text{--}E\text{--}B$

• Counterclockwise Progression (Flats):

  • Each counterclockwise step descends by a perfect fifth and adds one flat:

  • $1$ flat ($B\flat$): $F$ major

  • $2$ flats ($B\flat, E\flat$): $B\flat$ major

  • $3$ flats ($B\flat, E\flat, A\flat$): $E\flat$ major

  • $4$ flats ($B\flat, E\flat, A\flat, D\flat$): $A\flat$ major

  • $5$ flats ($B\flat, E\flat, A\flat, D\flat, G\flat$): $D\flat$ major

  • $6$ flats ($B\flat, E\flat, A\flat, D\flat, G\flat, C\flat$): $G\flat$ major

  • $7$ flats ($B\flat, E\flat, A\flat, D\flat, G\flat, C\flat, F\flat$): $C\flat$ major

  • Standard order of flats: $B\text{--}E\text{--}A\text{--}D\text{--}G\text{--}C\text{--}F$

• Minor Keys Layout:

  • Top ($0$ sharps/flats): $a$ minor

  • Clockwise: $e$, $b$, $f\sharp$, $c\sharp$, $g\sharp$ / $a\flat$, $d\sharp$ / $e\flat$

  • Counterclockwise: $d$, $g$, $c$, $f$, $b\flat$, $e\flat$

Consonance, Dissonance, and Tritone Resolution

• Consonance vs. Dissonance:

  • Dissonant chords generate acoustic instability and harmonic tension.

  • Consonant chords sound stable, relaxed, and melodically balanced.

• Tritone Resolution Dynamics:

  • The tritone interval formed between $F$ and $B$ contains high harmonic tension.

  • Resolves stepwise into consonant pitch pairs:

  • Inward motion: $F$ steps down to $E$ while $B$ steps up to $C$ (resolving to a third).

  • Outward motion: Voices expand outward to reach stable consonant intervals.