Music theory from scratch

A vibrating string or column of air pushes the air around it. Those pressure changes reach your ear and become sound. The wave's frequency (Hz) is heard as pitch; its amplitude is heard as loudness.

Real instruments are not pure sine waves. They mix a fundamental with harmonics, and that mixture is timbre - why a violin and a flute playing the same note sound different. The spectrum analyzer shows that mix live.

Phase says where a wave starts in its cycle. On its own it is mostly inaudible, but it decides how waves combine: two identical waves in phase add up, in opposite phase they cancel. That is the basis of interference, comb filtering and some effects.

Sound needs a medium. In a vacuum there is no sound, which is why space films are silent in reality. In air the speed is about 343 m/s at room temperature, and it rises with temperature. A wave has a period as well as a frequency: a 440 Hz tone repeats every 2.3 ms. The wavelength is speed divided by frequency, so a 440 Hz wave is about 78 cm long in air.

Pitch is how high or low a sound feels. A young ear hears roughly 20 Hz to 20 kHz; the range narrows with age. Frequency is physical, measured in cycles per second; pitch is perceptual. A4 = 440 Hz is a modern convention, not a law: orchestras have used 415, 430, 442 Hz and others. A4 = 440 was standardized in the 20th century.

Loudness is not amplitude. The ear is most sensitive around 2-5 kHz, and a sound can measure loud in decibels yet feel soft. Equal-loudness contours describe this; decibels are logarithmic. A mix can sound balanced at one level and dull at another.

Pitch is logarithmic too: an octave doubles the frequency, and a semitone multiplies it by 2^1/12 = 1.0594630943592953. Cents divide a semitone into 100 steps, which makes tunings comparable to a fraction of a cent.

Wavelength also decides how sound spreads: low notes wrap around obstacles, while high notes are more directional. That is why a bass is hard to locate and a cymbal is easy.

Double a frequency and you go up one octave: 220 Hz, 440 Hz and 880 Hz are all A. The higher wave completes exactly two cycles for every cycle of the lower one, so the ear groups them.

An octave is divided into 12 semitones in the standard scale. That is why note names repeat every 12 keys on a piano and C4 and C5 are both C.

The major scale has seven notes before the octave: C D E F G A B. It is built from the simplest ratios - the octave (2:1), the perfect fifth (3:2) and the perfect fourth (4:3). Stack fifths and the seven notes closest to C are F C G D A E B.

Seven is a practical compromise: enough steps for melody, more simple fifths than a pentatonic scale, less crowded than the chromatic twelve. Other traditions use five, seven, twelve or more. The major pattern is whole-whole-half-whole-whole-whole-half. See the scale on the board.

The hymn Ut queant laxis is addressed to St. John the Baptist. Each line begins one step higher; the first syllables of its lines gave the six notes of the hexachord. Singers call them Do, Re, Mi, Fa, Sol, La today; the original first syllable was Ut.

Guido d'Arezzo used them as a teaching device in the 11th century. Si arrived later, usually from Sancte Ioannes; the original Ut became Do in the 17th century, and English solfège uses Ti for the seventh. See the names on the board.

The letters A B C D E F G are simply the English alphabet laid over the diatonic scale from A to G: A B C D E F G is the natural minor scale of A. That is why the white-key major scale starts on C, the third letter, rather than on A.

Why five notes have no names

An accidental (from Latin accidens, "something added") changes a written note: a sharp raises it by a semitone, a flat lowers it, a natural cancels an earlier sign, and double signs move it by two semitones. The letter stays the same: F sharp is still an F, one step higher. The piano makes the asymmetry visible. Seven white keys per octave spell the C major scale and carry the letters A-G; five black keys sit between them. The black keys come in groups of two and three because the major scale has half steps only between E-F and B-C, and those pairs have no key between them. Even the colours are a convention: some early keyboards had dark natural keys and light accidentals. And the mapping is relational: relative to C major the five black keys are chromatic, but in F sharp major most of them are diatonic and E sharp is played on a white key.

The staff is a diatonic ladder: every line and space is one of the seven letters. Scales, intervals, chords and keys are built from those steps, so notation has to show which step is altered, not merely which of the twelve sounds is played. If the five chromatic notes had their own letters, a scale would stop looking like a scale: the visual pattern of steps would vanish, and the same pitch could not be spelled to fit the key. The five notes are therefore written as inflections of the seven - C sharp or D flat, depending on context. In equal temperament they sound identical; in older tunings they were different pitches, which is one reason the distinction survived.

Spell by letters, not by semitones. A seven-note scale uses each letter A-G once, in order, and accidentals adjust the pitch. D major is D E F sharp G A B C sharp, not D E G flat G A B D flat: every letter appears once, so the scale looks like a ladder on the staff. F sharp major needs E sharp rather than F, and C sharp major needs B sharp; extreme keys can require a double sharp or double flat. The same rule applies to intervals and chords: F sharp and G flat sound the same on a piano, but one is the seventh degree of G major and the other the fourth degree of D flat major. A mode follows the same order from its own tonic: D Dorian is D E F G A B C. Chromatic scales are the exception: ascending they usually use sharps, descending flats, so no letter is repeated more than necessary.

The first accidental was a flat, not a sharp. Medieval singers used hexachords, six-note patterns, and the hexachord on F needed B flat to avoid a tritone with F. The note B therefore had two forms: b molle (soft b, rounded) and b durum (hard b, square). The rounded b became the flat sign; the square b and its later variants are usually taken as the origin of the natural and sharp signs. German Dur (major) and Moll (minor) still preserve those names. Other accidentals spread as music became more chromatic.

Medieval and Renaissance singers also added accidentals that were not written, a practice called musica ficta: a raised leading tone at a cadence, a flattened note to avoid a tritone. As keys and modulations multiplied, composers wrote more of these signs, and key signatures collected them at the start of the staff. The order of sharps (F C G D A E B) and flats (B E A D G C F) is the circle of fifths in disguise. The natural sign cancels an earlier accidental within the bar, and double signs cover extreme modulations. None of this is a law of nature: it is a notation built around the seven-step scale.

From C to G is a perfect fifth (7 semitones); C to E is a major third (4). The name has a number and a quality (major, minor, perfect, augmented, diminished). Consonant intervals such as the octave and fifth sound stable; seconds and sevenths create tension. Try them on the board.

Interval names count letters, not semitones: a third spans three letter names (C-D-E), a fourth spans four (C-D-E-F), and a fifth spans five (C-D-E-F-G). That is why a fifth is seven semitones wide, not five. The quality - major, minor, perfect - fixes the exact size.

The same counting habit shows up in the calendar: September, October, November and December mean seventh, eighth, ninth and tenth. In the old Roman calendar, the year began in March; January and February were added later and eventually moved to the front, while the old names stayed. That is why months numbered 7-10 now sit at positions 9-12, and why a fifth is a "fifth" even though it spans seven semitones.

An octave is 1200 cents; each equal-tempered semitone is 100. Cents make tunings comparable: a pure fifth is about 702 cents, an equal-tempered fifth 700; a pure major third is about 386 cents, an equal-tempered major third 400. The logarithmic view in the demo spaces pitches by frequency, so each octave takes half the width of the one before.

The tritone divides the octave in half and later acquired the nickname diabolus in musica. It was not literally banned; the story is a colorful anecdote, not a rule.

Medieval theorists called the third an imperfect consonance: usable, but not as stable as a fifth or an octave. Its status rose with polyphony, and sweeter tunings made it a stable building block.

Intervals invert: a major third (4 semitones) plus a minor sixth (8) makes an octave, so the inversion of a major third is a minor sixth. The same note can be spelled differently: C sharp and D flat are enharmonic. The spelling matters in notation and harmony, not in sound.

The major scale is whole-whole-half-whole-whole-whole-half. The natural minor lowers the third, sixth and seventh. Modes use the same seven notes from a different starting point: Dorian is minor with a raised sixth, Mixolydian major with a lowered seventh. Hear them on the board.

Think of pitch as a pyramid of filters. The chromatic scale contains all twelve pitch classes. A key chooses a tonic and a mode and usually keeps seven of them; the other five become chromatic alterations. A chord keeps about three notes from those seven, and a melody picks one note at a time. Each step removes possibilities: the key filters out five notes, the chord four, and the melody leaves one. The pyramid is a hierarchy of stability, not a ban: a melody can pass through other notes, and a chord can borrow from outside the key, but the remaining notes define the key, the chord and the line. The widget below lets you filter the twelve notes down to a key, a chord and one melody note.

Other scales solve other problems. The pentatonic scale removes the two half steps and sounds at home almost anywhere; the blues scale adds a lowered fifth; harmonic minor raises the seventh for a stronger pull to the tonic; whole-tone and octatonic scales divide the octave evenly and blur the key. Locrian, built on the seventh degree, is the exception among modes: its fifth above the tonic is diminished, so it has no stable tonic triad and is rare in practice. Scale degrees are numbered 1-7, and the pattern of whole and half steps is what gives each scale its character.

Minor is not one scale. The natural minor uses the key signature; the harmonic minor raises the seventh, which creates a leading tone and an augmented second between the lowered sixth and raised seventh; the melodic minor raises the sixth and seventh on the way up and usually returns to the natural form on the way down. Harmonic minor is common in classical, metal and film music; melodic minor is common in jazz, where it is also the parent of several modes.

Scale degrees have names and jobs: tonic, supertonic, mediant, subdominant, dominant, submediant and leading tone. The third fixes major or minor, the seventh pulls to the tonic, and the sixth colors the mode. A mode is not just the same notes from a different start: the characteristic note matters. Dorian's raised sixth, Phrygian's lowered second, Lydian's raised fourth and Mixolydian's lowered seventh are the notes that make each mode recognizable. Raga and maqam go further: they prescribe ascending and descending lines, ornaments and phrasing, not only a set of pitches.

A triad is built from two thirds. C major is C-E-G; C minor is C-E flat-G - one note changes the mood. Diminished (C-E flat-G flat) and augmented (C-E-G sharp) triads are unstable. Inversions put another chord note in the bass; seventh chords add a fourth note and tension. Build them on the board.

Extensions continue the stack: add a ninth, eleventh or thirteenth for jazz and film colors. Suspended chords replace the third with a second or fourth, add9 keeps the third and adds the ninth, and a power chord is just root and fifth. The same notes can be voiced in many ways: close, open, spread across octaves.

In a key, chords are numbered with Roman numerals: I, ii, iii, IV, V, vi, vii. Uppercase is major, lowercase minor, the degree symbol diminished. The tonic (I) feels like home, the subdominant (IV) moves away, and the dominant (V) pulls back. A borrowed chord from the parallel minor (for example iv in a major key) adds a sudden shadow.

Lead sheets write chords as symbols: C, Cm, C7, Cmaj7, Cm7, Csus4, Cadd9, C/E for an inversion. The symbol says which notes are allowed, not how to voice them. The board shows the shapes; the synth playground lets you hear the voicing.

In tonal music the tonic feels like home, the subdominant moves away, and the dominant creates tension. Cadences end phrases: V-I (authentic), IV-I (plagal), ending on V (half), V-vi (deceptive). Progressions such as I-V-vi-IV balance motion and rest. Try them on the board.

Voice leading is the art of moving from chord to chord with small steps. Keep common tones, move the other voices by step when possible, and let the bass define the inversion. Good voice leading makes a simple progression sound smooth; bad spacing makes it muddy.

Functional harmony is a map of tension. Secondary dominants (V/V, V/ii) tonicize a chord that is not the tonic; modulation changes the home key for a while. Borrowed chords, the Neapolitan (flat II) and augmented sixth chords add color and pull. The circle of fifths explains why root motion by a fifth feels strong.

Modal harmony works differently: chords are built on the notes of a mode and do not necessarily resolve to a tonic. Jazz and film music often use static modal vamps, quartal voicings and extensions. Reharmonization keeps a melody and changes the chords under it.

Counterpoint adds independent lines. In species counterpoint, voices move mostly by step, avoid parallel fifths and octaves, and create consonance on strong beats. The result is polyphony: several melodies at once, not one melody with chords.

The Tonnetz (German for "tone network") is a map that turns pitch classes into a lattice. It was used by Leonhard Euler in the 18th century, developed by Arthur von Oettingen and Hugo Riemann, and revived by neo-Riemannian theory.

Horizontal steps are perfect fifths (+7 semitones), and the two diagonal directions are major thirds (+4) and minor thirds (+3). Each triangle is a triad: an upward triangle is major, a downward triangle is minor. Neighboring triangles share an edge, which means they share two notes.

That shared edge is the point. The transformations P (parallel), R (relative) and L (leading-tone) move to a neighboring triangle while changing only one note. For example, C major (C-E-G) becomes C minor (C-E flat-G) by lowering E. Chains of these moves produce the hexatonic and octatonic cycles.

In equal temperament the lattice wraps around after twelve fifths, so the map becomes a torus. In just intonation it spirals forever, because no number of pure fifths equals a number of octaves. Click a triangle below to hear its triad.

From C: G, D, A, E, B, F sharp, C sharp, G sharp, D sharp, A sharp, F, back to C. Neighboring keys share six of seven notes, so a step around the circle is a smooth modulation. It also explains key signatures: each step adds a sharp or removes a flat. See it on the board.

The inner circle pairs each major key with its relative minor: A minor under C major, E minor under G major, and so on. The parallel minor shares the tonic but not the key signature, so C minor sits three steps anticlockwise, under E flat major. The tritone is directly opposite: C and F sharp, the two poles of the circle.

The key signature is the circle written flat. Going clockwise from C major adds one sharp in the order F C G D A E B; going anticlockwise adds one flat in the reverse order B E A D G C F. The relative minor is three semitones below the major and shares its signature, so A minor has none, E minor one sharp, D minor one flat. The parallel minor keeps the tonic but adds three flats, which is why C minor is not next to C major on the circle but under E flat major.

Enharmonic keys meet at the bottom: B major and C flat major sound the same on a piano, as do F sharp major and G flat major, and C sharp major and D flat major. Composers choose the spelling that makes the voices and key relationships readable. Modulation often moves to a neighbor on the circle, to the relative minor, or through a pivot chord shared by both keys; a jump to a distant key needs a bridge. The circle is a map of distance, not a set of rules.

The circle is a map, not a rule. Music can move by seconds, thirds or chromatic steps, and modal music may never establish a tonic. Still, fifth motion is so common that it sounds like gravity. See it on the board and in the fifth circle above.

Twelve pure 3:2 fifths do not reach seven octaves; the gap is the Pythagorean comma, about 23.5 cents. Equal temperament divides the octave into twelve equal semitones, so every fifth is about two cents flat. Every key becomes usable, but only the octave is perfectly pure.

History is a series of compromises. Pythagorean tuning stacked pure fifths and produced pure fourths and fifths but harsh thirds. Just intonation tuned the major third to 5:4, which made chords sweet in one key. Meantone temperament narrowed the fifths to make the thirds pure, at the cost of a wolf fifth. Well temperaments spread the error unevenly so all keys worked but each had a different color. Equal temperament, known in China and described in Europe around 1600, became standard with the piano in the 19th century.

A piano is also stretched. Stiff strings do not vibrate in exact integer multiples: their partials are slightly sharp, and the effect grows in the treble. Tuners widen octaves to follow that inharmonicity, so the bass runs a little flat and the treble a little sharp. The result is the Railsback curve, not a straight line.

Tuning is audible. A pure fifth has no beating; a tempered fifth beats slowly. A pure major third is calm; an equal-tempered third beats audibly, about ten times a second around middle C. The spectrum analyzer shows the close peaks, and the temperament widget below lets you compare pure, just and equal fifths.

In practice, players tune to the ensemble, the room and the music. String quartets may use expressive intonation, singers adjust to harmony, and gamelan ensembles are deliberately tuned in pairs. There is no single correct tuning, only a choice of trade-offs.

Twelve equal semitones are one solution, not the only one. Indian classical music uses twelve swaras but treats the space between them as continuous; ornaments such as meend and gamak slide and shake around the written pitch. The theoretical 22 shrutis describe finer divisions, not a fixed keyboard.

Arabic maqam and Turkish makam use intervals that are not multiples of a semitone, including neutral thirds. The ear learns these intervals as scales with their own character. Indonesian gamelan uses slendro (five roughly equal steps) and pelog (seven unequal steps); paired instruments are tuned slightly apart, producing a shimmering beating.

Chinese, Japanese, Thai, West African and Andean traditions each have their own tuning and timbre. The blues uses bent notes and a lowered fifth that sit between the piano keys. None of these systems is a failed version of equal temperament; each answers a different musical problem.

Recorded music has spread equal temperament, but it has not erased the others. A single track can mix a retuned synth, a maqam melody and a gamelan sample. Listening to other systems is the fastest way to hear that tuning is a choice, not a law of nature.

The pulse is the beat; tempo is its speed in BPM. Meter groups beats into bars (4/4, 3/4, 6/8). Subdivisions, syncopation and swing create groove, and rhythm works without pitch. The curve editor has a beat grid and tempo you can play with.

Meter is a hierarchy, not just a count. In 4/4 the first beat is strongest and the third is next; in 3/4 the first beat leads. Syncopation accents weak beats or offbeats, and swing delays the second eighth of each pair. The brain also predicts the next beat, which is why a well-placed silence or a missed hit can feel as strong as a sound.

Tempo is measured in BPM, but perceived speed also depends on subdivision and density. A busy 90 BPM track can feel faster than a sparse 120 BPM one.

A grid is not a groove. Players push and pull against the beat: swing delays offbeats, funk accents the one and the offbeats, and a little timing and velocity variation keeps a loop from sounding mechanical. In a DAW, quantize is a starting point, not the finish.

Rhythm can layer different grids. A polyrhythm puts two conflicting subdivisions together, such as three against two; a hemiola shifts the accent so two bars of three feel like three bars of two. Metric modulation changes tempo by reinterpreting a note value as the new beat. These devices are common in West African drumming, jazz, progressive rock and drum and bass, and they work because the body can hold one pulse while the ear follows another.

A melody is not just a scale: contour, rhythm, repetition and phrasing make it memorable. Steps feel smooth, leaps feel dramatic, and a leap is usually balanced by a step in the opposite direction.

A motif is a short idea; a phrase is a musical sentence; a period pairs an antecedent phrase with a consequent one. Sequences repeat a pattern at a new pitch level. Inversion, retrograde and augmentation transform a motif without losing its identity.

Melodic tension comes from non-chord tones. A passing tone fills a step, a neighbor tone returns, an appoggiatura lands on a strong beat and resolves, and a suspension holds a note over a chord change. The listener hears the melody against the harmony, not in isolation.

Singability is a good test. If you can sing it, the contour and rhythm are clear. Transposition moves a melody to another key without changing its shape; the ear recognizes the intervals, not the absolute frequencies. Draw a melody as a curve in the vectoriano editor.

Form is how ideas repeat, contrast and return over time. The smallest units are the motif and the phrase; phrases group into periods and sections. A form is a map of tension and release.

Binary form is A-B; ternary form is A-B-A; rondo is A-B-A-C-A. Sonata form has an exposition, a development and a recapitulation, often with an introduction and a coda. Theme and variations keeps a theme and changes it. Popular music uses verse-chorus, strophic and 12-bar blues forms.

Repetition makes a piece understandable, contrast keeps it interesting, and return gives closure. The proportions matter: a long build needs a payoff, and a short loop needs small changes. Electronic music often works with layers entering and leaving rather than with chord progressions.

Form is not a container imposed from outside. It grows from the material: a motif suggests a sequence, a phrase suggests an answer, a climax suggests a place to stop. Listen for the moment when the music changes direction, and ask what changed.

A staff has five lines; the clef fixes which line is which note. Note heads give pitch, stems and beams give rhythm. A key signature collects sharps or flats; a time signature sets the meter. The piano roll is the same idea on a grid.

Clefs are reference points. The treble clef is a G clef: its curl circles the second line from the bottom, so that line is G4. The bass clef is an F clef: its two dots bracket the second line from the top, F3. The C clefs point to middle C: alto puts it on the third line, tenor on the fourth, and the rarer soprano, mezzo-soprano and baritone clefs move it to other lines. The lines from bottom to top are E G B D F in treble, G B D F A in bass, F A C E G in alto and D F A C E in tenor; the spaces fill the letters between them. Middle C (C4) is one ledger line below the treble staff and one above the bass staff, and ledger lines extend the ladder above and below. The widget below lets you change clef, note, accidental and duration. The board shows the same pitches on a keyboard and fretboard.

An accidental affects the rest of the bar on that line or space; a natural cancels it, and the key signature applies to every octave. Chords are written as stacked noteheads, usually in thirds, with one stem for the whole chord and accidentals in a column to the left. A triad in root position has its root at the bottom; an inversion puts the third or fifth below. The chord symbol above the staff names the harmony, not the exact voicing. Stem direction usually follows the note farthest from the middle line, and a chord uses one direction for all its notes.

The grand staff joins treble and bass clefs for piano. Ledger lines extend the staff, accidentals raise or lower a note for the bar, and ties connect durations across barlines. Dynamics (p, f, crescendo), articulation (staccato, legato, accent) and tempo marks tell the performer how the notes should feel.

Rhythm has its own arithmetic. A whole note lasts four beats in 4/4, a half two, a quarter one, an eighth half a beat. The shape of the notehead and stem encodes the value: a whole note has no stem, a half note is hollow, a quarter is filled, and an eighth adds a flag. A dot adds half the value, and a triplet fits three notes into the time of two. Rests are silence with the same values. Ties join notes across barlines; beams group eighths and sixteenths so the beat is visible. A score is also a map of the meter: the time signature says how many beats there are and which note value gets the beat, while the beaming shows the grouping.

Notation is written for transposing instruments. A B flat clarinet or trumpet reads a C but sounds a B flat; a horn in F reads a C but sounds an F; a piccolo sounds an octave above written, a double bass an octave below. The score transposes so the player can use familiar fingerings. Clefs do the same job for register: treble, bass, alto and tenor. Figured bass, used in the Baroque continuo, writes only the bass and figures for the intervals above it; the keyboard player realizes the chords.

Guitar and bass use tablature: lines for strings and numbers for frets. Drum notation places hits on a grid of lines. In a DAW the piano roll shows pitch on the vertical axis and time on the horizontal one, with velocity and automation in separate lanes. All of these are instructions, not the sound itself.

Timbre comes from the mix of harmonics, the noise component and the envelope. A flute has a strong fundamental and few harmonics; a violin has a dense series; a snare is mostly noise. The attack and decay matter as much as the spectrum. Shape it in the synth playground.

A vibrating string or air column supports a harmonic series: integer multiples of the fundamental. The relative strength of those partials is the spectral envelope. A clarinet, as a closed tube, suppresses even harmonics and sounds hollow; a violin bow excites a broad spectrum; a flute favors the low partials. The seventh harmonic is about 31 cents flatter than an equal-tempered minor seventh; barbershop quartets tune that chord by ear and get a noticeably different color.

Resonances shape the spectrum further. The vocal tract has formants, fixed frequency regions that make vowels sound like a, e, i, o, u. The body of a violin or guitar has its own resonances. Brass instruments brighten as the player tightens the lips and raises the amplitude.

Noise and transients carry information. The bow hiss, the breath, the pick attack and the snare rattle tell us what made the sound. A spectrogram shows noise as a broad band and a transient as a vertical stripe. Removing the attack can make a familiar instrument unrecognizable.

Inharmonicity adds character. Bells, gongs and stiff piano strings have partials that are not exact multiples. That is why a bell sounds like a bell and not like a flute. Additive synthesis can copy any spectrum; subtractive synthesis shapes it with filters; FM creates new sidebands.

Strings are bowed, plucked or struck; woodwinds move air through a tube or reed; brass players buzz their lips; percussion is struck or shaken. Keyboards, voice and electronic instruments complete the picture. The Sachs-Hornbostel system sorts instruments by the material that vibrates: idiophones, membranophones, chordophones, aerophones and electrophones. Play the instruments on the board.

The family name describes the source of the vibration, but the body does the shaping. A violin string alone is quiet; the bridge and the wooden body radiate the sound and add resonances. A flute's air column, a clarinet's reed and tube, a trumpet's lips and brass tubing, and a drum's membrane all couple a driver to a resonator. The player controls pitch with length, tension or fingering, and loudness with energy; the instrument's resonances decide the timbre.

Instruments are built for ranges and roles. A piccolo sounds an octave above its written part, a double bass an octave below, and a guitar an octave below the treble staff; transposing instruments are written in a comfortable key. Orchestration balances these ranges: strings carry the body and can swell or shimmer, woodwinds provide solos and color, brass adds weight and blaze, percussion marks rhythm and climax. Doubling two instruments in octaves or unison changes the color as much as the volume.

The voice is the original instrument: the lungs supply air, the vocal folds vibrate, and the throat, mouth and nose filter the result into vowels and consonants. Its range and tessitura differ by singer, and its formants make it cut through an orchestra. Electronic instruments replace the resonator with a circuit or an algorithm, but the same questions remain: what oscillates, what filters it, and how does the envelope shape it?

Hearing is not a microphone. The ear compresses loudness, masks quiet sounds and groups frequencies into critical bands. Two tones inside one critical band interfere and sound rough; tones in separate bands are heard as separate pitches. This is the sensory basis of roughness.

Masking hides sounds. A loud low tone can make a quiet high tone inaudible, and a transient can mask a sound that follows it. That is why a mix can sound clear in isolation and muddy in a busy arrangement. Equal-loudness contours explain why bass and treble seem weak at low volume.

The brain fills in missing information. The missing fundamental lets you hear a low pitch from harmonics even when the fundamental is absent. A Shepard tone seems to rise forever because its harmonics wrap around the spectrum. Octave equivalence groups frequencies by a 2:1 ratio. The auditory system also separates streams: in a crowded room you can follow one voice and tune out the rest, an effect known as the cocktail party problem.

Just noticeable differences are small but not zero: trained listeners hear a pitch change of about 5-10 cents, a level change of about 1 dB, and a timing difference of a few milliseconds. Perception also depends on context, expectation and training. Consonance is not one number: roughness, harmonicity, familiarity and culture all contribute.

Two ears, one scene

The brain compares the two ears. A sound on the left reaches the left ear first and is louder there; the right ear hears it later and quieter. The time difference is at most about 0.7 ms, and the level difference grows at high frequencies because the head casts a sound shadow. The duplex theory says the brain uses time differences mostly below about 1.5 kHz, where the phase of the wave is unambiguous, and level differences above, where the wavelength is short enough to be blocked by the head.

Left and right are only the beginning. The folds of the pinna act as a direction-dependent filter: a sound from above, below, in front or behind enters the ear canal with a different pattern of peaks and notches. This head-related transfer function (HRTF) supplies the cues for elevation and front/back. Time and level alone leave a cone of confusion, where front and back produce nearly the same cues; the pinna's spectral shape and small head movements break the tie. That is why we turn our heads when we are unsure, and why head-tracked spatial audio feels more solid than a fixed mix.

Distance is a separate calculation. A nearby source is louder, arrives with more high frequencies (air absorbs treble over distance), and has a higher ratio of direct sound to reverb. A distant source is quieter and wetter. The brain also uses familiarity: a known voice or instrument has an expected loudness, so a quiet recording of it can still feel close. Motion helps too: a moving source changes its angle faster when it is near.

Stereo is a trick

Stereo does not record a 3D scene; it records two channels and lets the brain build one. If the same signal comes from two speakers, you hear a phantom source in the middle. Pan a sound by making one speaker louder, and the image moves toward it; a delay of a few milliseconds does the same, though it also comb-filters the sum. The precedence effect helps: the first wavefront wins, so a delayed speaker can be loud without pulling the image away from the first one, as long as the delay stays within about 30 ms and the two signals fuse.

Stereo is not binaural. With speakers, each ear hears both channels and the room; with headphones, each ear hears one channel and the room disappears. A binaural recording is made with a dummy head or an HRTF and is meant for headphones; played on speakers, it often collapses. Phase is the other trap: out-of-phase content sounds wide but cancels in mono, and low frequencies are the first to suffer. That is why many mixes keep bass mono and check mono compatibility. Two channels mainly encode left and right; height and the rear are inferred from spectrum and reverb, so stereo is a 2D projection of a 3D scene.

The demo below lets you drag a source around the head and compare HRTF with level-only panning.

Non-obvious effects

Masking is asymmetric: a loud low tone hides a quiet high tone more effectively than the reverse, which is why bass can swallow a vocal. It also works backward and forward in time: a transient can mask a sound tens of milliseconds before it and a couple of hundred milliseconds after. The continuity illusion goes the other way: when a sound is interrupted by a louder noise, the brain fills the gap and you hear one continuous sound.

Vision captures hearing. In the ventriloquism effect, the voice follows the puppet's mouth; in the McGurk effect, the syllable you see changes the syllable you hear. Your own voice is another special case: you normally hear it partly through bone conduction, which adds low frequencies and internal resonance, so a recording, which reaches you only through air, sounds thinner and stranger. The brain also learns: with new headphones or a new HRTF, localization adapts over days, and spatial audio can be personalized by measuring your ears.

When a source moves toward you, each successive wavefront leaves from a point closer to you, so the wavelength shortens and the frequency you hear rises. When it moves away, the wavefronts are stretched and the pitch falls. For a stationary listener the shift is f' = f · c / (c - v), where f is the emitted frequency, c is the speed of sound and v is the source's velocity toward the listener. Only the component along the line between source and listener matters: a source crossing your view at high speed has no radial velocity at the closest point, so the pitch passes through its original value there.

At 343 m/s, a car moving at 30 m/s (108 km/h) with a 440 Hz horn sounds about 482 Hz as it approaches and about 405 Hz after it passes, a drop of roughly 78 Hz, almost three semitones. The faster the source, the larger the shift; as it approaches the speed of sound the formula grows steeply, and a source that exceeds it produces a shock wave, heard as a sonic boom.

The effect is used to measure motion. Doppler radar tracks rain and aircraft, speed cameras measure cars, medical ultrasound measures blood flow, and astronomers use the Doppler shift of starlight to find exoplanets. In astronomy, cosmological redshift is mostly the expansion of space, not a Doppler shift, though the motion of a star or galaxy adds a Doppler component on top.

For light the classical formula is not enough: the relativistic Doppler effect includes time dilation and gives a redshift when the source recedes and a blueshift when it approaches. At everyday speeds the difference is tiny. For sound, the effect is a pitch glide during a pass-by; the brain also uses the change as a cue for motion. Games and films add Doppler to passing cars and aircraft because it makes the movement feel real.

The demo below sends a 440 Hz tone past a listener. Change the speed and watch the outgoing wavefronts: at 343 m/s their edges meet in a line, and above it they form a shock cone.

Sensory dissonance is not the same as musical dissonance. Two pure tones sound rough when their frequencies are close enough to interfere but far enough to beat at an audible rate. The roughness is greatest when the separation is about a quarter of a critical band, and it falls to zero at unison and at wide separations.

Plomp and Levelt measured this curve in 1965. Sethares turned it into a model for complex tones: every pair of partials contributes roughness, weighted by amplitude, and the contributions are summed. The result explains why an octave or a fifth is smoother than a minor second, and why a bell can sound rough even at a simple interval.

The curve depends on register and timbre. At low frequencies the critical band is narrow in hertz, so small intervals are rougher; at high frequencies the same interval spans fewer critical bands and sounds smoother. A sound with many strong partials has more chances to create close pairs. That is why distortion can make a chord sound harsher.

The graph below plots roughness for two tones from unison to octave. With one partial you see the raw Plomp-Levelt curve; with more partials, local minima appear at simple ratios. Set the base frequency, change the partial count, and press and hold the curve to hear the interval.

This is a model, not a verdict: musical context can make a rough interval beautiful and a smooth one dull.

MIDI sends note on/off, velocity, pitch bend and controller changes; a synth turns them into sound. Sixteen channels carry sixteen parts. MPE gives each note its own channel, so pitch, pressure and timbre can move per note. Try it on drifter and vectoriano.

MIDI is coarse by design: note numbers, velocities and controller values are 7-bit, from 0 to 127. Note 69 is A4, and velocity is not volume: a synth may map it to loudness, brightness or both. A controller can carry a pedal, a knob or a breath sensor.

MIDI 1.0 arrived in 1983 as a serial protocol on five-pin DIN cables, and it is still everywhere. A cable carries messages at 31.25 kbit/s; channels 1-16 can address separate instruments, and a track in a DAW is a container for events, not a channel. Program change selects a patch, control change moves a knob or pedal, pitch bend is a 14-bit value, and clock or timecode keeps devices in sync. General MIDI fixed 128 program names and a drum map so a file would sound roughly the same on different gear.

The limits are part of the design. Seven-bit values are coarse for smooth fades; a single cable can choke on dense controller data; and note on/off messages can arrive late or jitter. MPE works around the channel limit by giving each note its own channel, with a master channel for the zone. MIDI 2.0, standardized in 2020, adds 16-bit resolution, per-note controllers, profiles that describe an instrument's capabilities, and property exchange; it can translate to MIDI 1.0 for older gear. Neither version carries audio: the sound comes from the synth, and latency comes from the interface, driver and buffer.

An envelope is a control signal that changes a parameter over time. On a synth it usually shapes amplitude, but it can also open a filter, bend pitch or modulate anything else. ADSR is the classic four-stage description. Some instruments add a hold stage (DAHDSR) or let you draw a free multi-stage envelope, but four stages cover most sounds.

Attack is the time from silence to the peak. Decay is the fall to the sustain level. Sustain is the level held while the key is down, not a time. Release is the fade after the key is released. A short attack and decay sound percussive; a long attack and release sound like a pad.

The same four numbers produce very different results. A plucked string has a fast attack, a fast decay and no real sustain; an organ has a fast attack, no decay and full sustain; a bowed pad has a slow attack and a long release. The envelope often matters more than the waveform.

Envelopes are not only for volume. A filter envelope makes a sound open and close; a pitch envelope makes a kick or a laser. In a DAW, envelope shapes are drawn as automation or as clip fades. The attack transient is a strong identity cue: remove it and a familiar instrument can become abstract.

Set the four stages in the widget below and play a note. The graph shows the shape and the sound follows it. Try a short attack with a long release, then the opposite. The synth playground has a full ADSR editor.

Additive: building from sines

Any periodic sound is a sum of a fundamental and its harmonics (Fourier series). Additive synthesis makes that sum explicit with one oscillator and level per partial; the drawbar organ is a familiar example. It is precise but expensive. A sawtooth is the extreme case with every harmonic at 1/n, which makes it the natural starting point for subtractive synthesis. Edit partials in the synth playground.

Subtractive: carving with filters

A saw or square wave has many harmonics. A low-pass filter cuts the highs; resonance boosts the cutoff. The classic path is oscillator, filter, amplifier, with envelopes and LFOs. It is still the basis of most virtual analog synths. Try the synth playground.

FM: the sound of the 1980s

One oscillator is the carrier, the other the modulator. The ratio sets the sideband spacing; the depth sets brightness. A single pair can make bells, brass or electric pianos. The DX7 made FM ubiquitous in the 1980s.

Wavetable and granular

Wavetable synthesis scans a table of single-cycle waveforms, changing the spectrum over time. Granular synthesis plays short grains at new speeds, densities and pitches, stretching time or freezing a moment. Both are made for evolving digital textures.

Reverb and delay add space and repeats; chorus and flanger thicken with modulated copies; distortion adds harmonics; compression controls dynamics. Dry sounds feel close, wet sounds feel far. Effects are also sound-design tools.

Reverb and delay are the two ways to put a sound in a space. Reverb is a dense cloud of reflections; its pre-delay separates the direct sound, its decay time says how long the tail lasts, and its early reflections give the size of the room. Delay repeats the signal at a set time; feedback sets how many repeats, and a tempo-synced dotted eighth is a pop staple. A plate or spring is a mechanical reverb with its own color; a convolution reverb uses a recording of a real space.

Modulation effects move a short delay. Chorus uses a slow, shallow modulation to thicken a sound; flanger uses a shorter delay and feedback to create a jet sweep; a phaser moves notches instead of a delay. Distortion clips the waveform and adds harmonics: overdrive is soft, distortion is harder, fuzz is extreme, and an amp or cabinet filter shapes the result. Compression reduces the dynamic range with a threshold, ratio, attack and release; it can glue a mix, make a vocal sit, or duck a bass under a kick with sidechain input.

EQ is the quietest effect and often the most important: a high-pass filter removes rumble, a narrow cut removes a resonance, a shelf adds air or weight. The order matters. A compressor before a reverb changes how the tail breathes; a reverb before a distortion turns the tail into a new sound. In a mix, effects create depth: drier and brighter sounds come forward, wetter and darker sounds recede. Use them in context, not solo, and check the mono sum.

A riser combines a rising pitch sweep, noise and an opening filter. An impact layers a sub hit, a mid crack and a reverb tail. A whoosh is filtered noise moving across the stereo field. Layers, pitch and volume envelopes do most of the work in sound design.

Most designed sounds are layers, not single recordings. A cinematic hit might combine a sub sine for weight, a mid transient for impact, a noisy crack for texture, and a reverb tail for size. Split the layers by frequency so they do not mask each other: sub below 80 Hz, body around 200-800 Hz, attack in the presence range, air above. Pitch, filter and amplitude envelopes make the layers move together or apart.

Noise is a material. Filtered white or pink noise becomes wind, rain, fire, a whoosh or a crowd; a short noise burst becomes a click, a snare or a footstep. Foley artists record ordinary objects and sync them to picture. Pitch shifting, reversing, stretching and granular processing turn a familiar recording into something else; a reversed reverb before a hit makes the hit feel inevitable.

A riser builds tension by rising in pitch, opening a filter and adding noise; an impact releases it with a low sub, a sharp transient and a tail. A whoosh moves filtered noise across the stereo field; a downlifter does the reverse. The story matters more than the plugin: what is approaching, how big is it, how far away, and what should the listener feel next? Automate the gesture, check it in mono and on small speakers, and leave space for dialogue or music.

The FFT shows which frequencies are present in a short block and how strong they are. A naive DFT compares the signal with every frequency and costs n-squared operations; the FFT reuses intermediate results and costs n log n, which is what makes real-time analyzers possible. Bin width is sample rate divided by block size: 1024 samples at 44.1 kHz gives bins about 43 Hz wide. Windowing reduces leakage; longer blocks give finer frequency detail but blur fast changes. The spectrum analyzer runs an FFT. The demo below builds a complex wave from draggable sine components: the left column shows the pure tones, the right column their cumulative sum.

The transform assumes the block repeats forever. If the block does not end where it begins, the jump creates broadband splatter called leakage. A window tapers the ends to zero, trading a wider main lobe for lower sidelobes. Hann is a common default; Hamming, Blackman and flat-top trade resolution against amplitude accuracy. Zero-padding interpolates the displayed spectrum but cannot add real resolution.

Frequency and time are a trade-off. A long block resolves close partials but smears a transient; a short block catches the attack but blurs pitch. A spectrogram stacks short FFTs over time. The FFT also returns phase, which most displays hide; phase matters for resynthesis, phase vocoders and cancellation. Real-time analyzers overlap blocks and average or peak-hold the magnitudes, which is why the display looks smooth even though each frame is discrete.

Do I need perfect pitch?

No. Relative pitch - hearing intervals and recognizing chords - is trainable and more useful for most musicians. Perfect pitch is rare and can even make transposed music feel wrong.

Is music theory a rulebook?

No. It describes what composers and performers do, and it lags behind practice. Learn the rules, then listen to the exceptions.

Why twelve notes?

Because twelve equal semitones divide the octave evenly and let every key work on one keyboard. Other numbers of steps are possible and are used in other traditions.

Is equal temperament out of tune?

Every interval except the octave is slightly mistuned. The error is small and spread across all keys. It is a compromise, not a mistake.

What is the difference between pitch and frequency?

Frequency is a physical measurement in hertz; pitch is what you hear. They usually track each other, but loudness, timbre and context can shift perception.

Can I learn theory without an instrument?

Yes, but an instrument makes it concrete. Use the board to see shapes and the synth playground to hear spectrum and partials.

Why does a piano sound stretched?

Stiff strings have inharmonic partials, so tuners widen octaves. The treble ends up slightly sharp and the bass slightly flat.

How do I train my ears?

Sing what you hear, learn intervals in songs you know, transcribe simple melodies, and compare pure and tempered intervals. Little and often beats a long session.

Why do minor keys sound sad?

The lowered third is the main cue, and slower tempos, darker timbres and cultural associations reinforce it. The effect is not universal: many joyful songs are in minor, and context matters more than the scale alone.

What is the difference between mixing and mastering?

Mixing balances the tracks: levels, panning, EQ, compression and effects. Mastering is the final pass on the whole record: tonal balance, loudness, spacing between songs and delivery formats. A good mix matters more; mastering cannot fix a broken balance.

Do I need to read music?

Not to produce in a DAW, but reading helps you learn repertoire, communicate with players and see harmony. Start with the staff, note values and the circle of fifths; you can add the rest as you need it.