Summary
Highlights
Sound is vibration of air. The simplicity of the ratio between two frequencies determines their consonance. The harmonic series demonstrates how consonant sounds are related by simple integer ratios (2:1 for octave, 3:1 for fifth). Dissonant sounds produce 'beating' due to waves going in and out of sync. This principle is used to build musical scales, starting with a pentatonic scale (5 notes per octave) and then expanding to a seven-note major scale. The most consonant notes relative to the tonic in a major scale are the fifth and fourth.
Natural sounds are composed of a fundamental frequency and overtones. Harmonic overtones are multiples of the fundamental frequency and contribute to the musical note we hear. Inharmonic overtones are not part of the harmonic series and influence the 'tone' or timbre of a sound. Instruments like guitars produce harmonics that are naturally present in the sound. The combination of harmonic and inharmonic overtones is why different instruments sound distinct even when playing the same note. Sine, square, and sawtooth waves are examples of purely harmonic sounds used in synthesizers, while white noise is purely inharmonic.
Instruments were historically tuned using the fifth, an interval found in the harmonic series. The Pythagorean temperament, based on tuning in perfect fifths (3:2 ratio), was used until the 15th century. This method led to two main issues: the 'wolf interval' at the end of the circle of fifths, which was highly dissonant, and impure major thirds. This became problematic as music began to favor thirds and sixths.
Mean-tone temperaments attempted to pure the thirds by slightly lowering each fifth, but this exacerbated the wolf interval. Various irregular temperaments followed, each giving unique character to different tonalities but limiting modulation. The 19th century saw the rise of equal temperament, which is still used today. This system divides the octave into 12 equal semitones, making all intervals slightly impure but uniformly so. This allows for seamless modulation between all keys and transposition of music without re-tuning, offering greater musical freedom.
An octave is divided into 12 intervals, with most common scales having 7 notes. Scales are defined by their fundamental note (tonic) and the sequence of intervals between notes. The C major scale follows a specific tone-semitone pattern (T-T-ST-T-T-T-ST). Related to each major scale is a relative minor scale, using the same notes but starting on the sixth note of the major scale. Modes are derived from a major scale by starting on different notes, each creating a distinct intervallic sequence and emotional character. Examples include Ionian (major), Dorian, Phrygian, Lydian, Mixolydian, Aeolian (natural minor), and Locrian modes.
A chord is a set of notes played together. Triads, made of three notes (root, third, fifth), are the foundation of all chords. A major chord (root + major third + perfect fifth) sounds 'happy'. A minor chord (root + minor third + perfect fifth) sounds 'sad'. Altering the fifth to augmented or diminished creates augmented or diminished chords, respectively, which have distinct 'bright' or 'dark' qualities. Chords can be seen as stacks of thirds (e.g., major chord = major third + minor third). Chords are built from a scale by stacking notes every other step, and their nature (major, minor, diminished) depends on the intervals within the scale.
Extended chords are formed by adding notes beyond the triad, typically stacking thirds. Adding a seventh creates a seventh chord, a ninth creates a ninth chord, and so on. Intervals are defined by their distance from the root note. Perfect intervals (fourth, fifth) are augmented or diminished by a semitone. Major/minor intervals (second, third, sixth, seventh) change quality when altered. The C major scale provides a reference for identifying these intervals. Different types of seventh chords (dominant, major, minor, half-diminished, fully diminished, augmented) are formed by combining major/minor triads with major/minor/diminished sevenths, each with distinct sonic characteristics.
Chord progressions are sequences of chords that create a musical narrative. Specific degrees in a scale evoke different feelings; the fifth degree (dominant) creates tension, while the first degree (tonic) brings resolution. These tension-resolution dynamics are crucial in music. Cadences are recognizable chord progressions that act as musical punctuation. The authentic cadence (V-I) is conclusive, often preceded by a II chord (II-V-I). The half cadence (ending on V) creates an unresolved feeling. The deceptive cadence (V-VI) provides an unexpected turn. The plagal cadence (IV-I) offers a gentler resolution.
The ability to play a major chord on the fifth degree of a minor scale introduces an altered note, creating the harmonic minor scale. This scale improves authentic cadences in minor tonalities by providing a leading tone. Melodically, the interval jump in the harmonic minor led to the melodic minor scale, which raises both the sixth and seventh notes by a semitone. Composers often use all three minor scales (natural, harmonic, melodic) interchangeably within a minor tonality, depending on harmonic or melodic context, enriching the musical possibilities.
The circle of fifths is a powerful tool illustrating the relationship between notes and tonalities. Moving clockwise shows perfect fifths; counter-clockwise shows perfect fourths. It maps out all major and relative minor tonalities and indicates how many sharps or flats are in each. This chart helps identify adjacent scales, which share many notes and allow for smooth modulations (transitions between tonalities). Modulating to an adjacent scale is easier than to a distant one, as it involves fewer altered notes.
Dominant seventh chords (V7) are highly effective for modulations due to their inherent tension and strong pull to the tonic. Adding a V7 chord before a target tonic makes the transition smoother, even between major and minor keys. Harmonic sequences involve repeating a chord pattern at different intervals, often following the circle of fifths for smooth transitions (e.g., descending fifths creating a series of authentic cadences). Chromatic modulation, moving a semitone up or down, is more drastic and often signals a definitive key change.
Accidental notes (outside the current tonality) can add tension and flavor. Borrowing a dominant chord from an anticipated key in an authentic cadence (e.g., V of II, V of IV) introduces accidentals by momentarily treating another chord as a tonic. Borrowing chords from parallel major/minor tonalities (e.g., C major borrowing from C minor) allows for greater harmonic variety. Altering the fifth degree chord (e.g., diminished seventh, augmented, or tritone substitution) can also create unique harmonic effects and add surprise, especially in cadences.
The Neapolitan Sixth is a major chord on the flattened second degree (♭II), often in first inversion, used as a subdominant substitution to lead to the dominant. The Picardy Third involves ending a minor piece with a major tonic chord, creating a 'happy ending'. The Fauré Cadence is a specific half-cadence (IV7/5 - V), often combining a borrowed note from the melodic minor scale. The Andalusian Cadence (Am-G-F-E in A minor) is a common flamenco progression that functions as a loop or a cadence in Phrygian mode. Sus4 and Sus2 chords omit the third and add a fourth or second, creating a 'suspended' or ambiguous quality often used for transition or to blur tonality.
Chord inversions occur when notes other than the root are in the bass (e.g., first inversion has the third in the bass, second inversion has the fifth). The order of notes above the bass determines the voicing. Key voicing tips include: maintaining common notes at the same pitch between successive chords, grouping notes to define melody and harmony, avoiding parallel fifths between root and fifth in classical contexts, being intentional when doubling notes (avoiding doubling the fifth unless for specific effects), and placing strong tonal notes in the bass. Prioritizing ear judgment over rigid rules is essential.
Embellishing tones are non-chord tones that add variety and interest to melodies, often creating mild dissonance before resolving. Neighbor tones (or auxiliary tones) move up/down a step from a chord tone and return. Passing tones connect two chord tones by stepwise motion in the same direction. Escape tones move stepwise from a chord tone and then skip in the opposite direction to another chord tone. Anticipations play a note from the next chord early, while suspensions delay a note from the previous chord. Appoggiaturas are unprepared, often accented non-chord tones that resolve stepwise to a chord tone.
Rhythm organizes notes in time, distinct from tempo (overall speed). Note lengths include whole notes (4 beats), half notes (2 beats), quarter notes (1 beat), eighth notes (1/2 beat), and sixteenth notes (1/4 beat), divisible by two. Corresponding rests denote silence. Modifiers like ties (linking notes to add durations) and dots (extending a note's length by half) further vary durations. Triplets and other tuplets (quintuplets, sextuplets, septuplets) divide notes into irregular groupings (e.g., 3, 5, 6, 7 equal parts), creating complex rhythmic patterns that don't fit binary divisions.
Bars are containers for notes, with length defined by a time signature. The top number indicates beats per bar; the bottom indicates the note value of one beat. For example, 4/4 (common time) means four quarter notes per bar. This notation evolved from medieval practices of 'tempus' (binary/ternary division) and 'prolatio' (subdivision). Simple time signatures (e.g., 4/4, 3/4) have beats divisible by two. Compound time signatures (e.g., 6/8, 9/8) have dotted beats divisible by three. Asymmetric meters (e.g., 5/4, 7/8) combine groupings of two and three notes, creating irregular accents.
Polyrhythm is the superposition of two rhythms with different numbers of beats but the same overall duration (e.g., 3 beats played over 4 beats within the same bar). Polymeter is the superposition of two different time signatures, meaning their bars have different lengths, causing their patterns to offset and resynchronize over larger cycles. The ratio between the rhythmic layers is key to their perceived consonance or dissonance; simpler ratios like 2:3 (hemiola) are more easily internalized. Hemiola, for example, is both a polyrhythm and a polymeter, often blending 3/4 and 6/8 patterns.
A comprehensive cheat sheet for music theory is introduced, based primarily on the key of C for ease of visualization. It includes a table of intervals, major and minor scale structures (natural, harmonic, melodic), and lists of chords (triads, seventh chords) for each degree of these scales. It also features a list of various cadences (authentic, plagal, half, deceptive, tritone substitution, Neapolitan sixth, Fauré, Andalusian), a circle of fifths for understanding key relationships and modulations, and modes (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian) with their intervallic structures. Additionally, it details secondary dominants and chord substitutions for the fifth degree. The document includes pages for all 12 root notes, eliminating the need for manual transposition.
Pentatonic scales, consisting of five notes, are common globally. In Western music, they are often derived from major or minor scales by removing semitones (e.g., C major pentatonic removes F and B from C major). They are useful for improvisation and modular synthesis due to their inherent consonance. However, pentatonic scales have diverse origins and structures in other cultures, often not aligning with Western equal temperament, highlighting that Western music theory is just one framework among many. Indian ragas, for instance, combine scales with specific musical structures, motifs, emotional, and symbolic associations, often utilizing intervals not present in Western scales.
The blues scale, essential in rock, blues, and jazz, is typically a minor pentatonic scale with an added diminished fifth (a 'blue note'). For example, A minor blues adds an E♭ to the A minor pentatonic. The major blues scale adds a minor third to the major pentatonic, creating a unique dissonance between major and minor thirds within the scale. This 'blue note' (minor third over a major chord) creates tension that typically resolves to the major third. This characteristic dissonance is central to the blues sound, even found in chords like the dominant 7#9 (Hendrix chord).
Ninth chords extend seventh chords by adding a ninth (an octave higher than the second, to reduce friction with the tonic and third). They introduce more color and tension, requiring resolution. Common types include dominant ninth (major triad + minor seventh + major ninth), often used as a V chord resolving to I. Altered dominant ninths (flat nine, sharp nine) increase tension further. Major ninth chords (major triad + major seventh + major ninth) are used on I and IV in major keys, or III and VI in minor keys. Minor ninth chords (minor triad + minor seventh + major ninth) are found on II and VI in major, or I and IV in minor. Specialized ninth chords also exist for harmonic minor contexts.
Harmonizing a melody begins by identifying its tonality (e.g., D harmonic minor based on notes present). Next, build available chords for each degree of that scale. Assign chords to sections of the melody, ensuring melody notes align with chord tones (root, third, fifth) and considering rhythmic placement (strong vs. weak beats). Chord changes contribute to harmonic rhythm, influencing the song's feel. Using extended chords (sevenths, ninths) adds richness. Non-diatonic chords, like secondary dominants, can be borrowed from other tonalities to create dynamic transitions and introduce accidentals, expanding harmonic possibilities beyond the base scale. This showcases how various music theory concepts combine in practical composition.
Diminished chords are dissonant and tense, requiring resolution. They are versatile tools for modulation because they can function in multiple ways. The seven flat five chord (root + major third + diminished fifth + minor seventh) can lead to a target tonic and acts as a pivot point for modulation due to its dual interpretation (e.g., B7♭5 and F7♭5 sharing notes). The diminished seventh chord (root + minor third + diminished fifth + diminished seventh) is highly dissonant, with a symmetrical structure where any note can be interpreted as the root. This allows it to resolve to many different chords (semitone above, same root note, major or minor), offering extensive flexibility for transitions and creating intriguing harmonic ambiguity.
To effectively use an unfamiliar scale, first identify the triads that can be built from its notes. Exotic scales may yield multiple options for certain chord qualities or unusual intervals that don't fit standard chord types. Prioritize chords that sound good and inspire you. When soloing, holding a stable chord (like the tonic) allows for extensive improvisation over the entire scale. You can also mix scales by temporarily switching to a new scale if its notes can form the current chord, then returning to the original scale or continuing with the new one. This flexibility allows for dynamic and expressive musical exploration. An updated cheat sheet with chords for all modes is available as a resource.
To compose in a specific mode (e.g., Dorian, Phrygian, Lydian, Mixolydian), the goal is to make that mode's tonic feel like home, using its unique characteristic chords. Compare each mode's chord degrees to the major and natural minor scales to identify these unique chords. For Dorian, emphasize the minor first degree and major fourth degree. For Phrygian, use the minor first degree with the major flat-two chord. For Lydian, combine the major first degree with the major second degree. For Mixolydian, highlight the major first degree with the major flat-seven degree. Using these distinct chord combinations helps to establish the unique 'flavor' of each mode, moving beyond just major or minor tonalities.