Music Harmony.

Chords

A chord is several notes played together, usually stacked in thirds on the root; sus chords and the sixth chord are exceptions. Pick a root on the wheel, then a type: the wheel draws its figure, the keyboard lights it up and plays it.

Switch to the circle of fifths: the same figure points to other neighbours. On the chromatic wheel, two chords are perfectly symmetric: the augmented, three major thirds (an equilateral triangle), and the diminished seventh, four minor thirds (a square). With no note in charge, they pivot towards many keys.

The simpler the frequency ratio, the more consonant the interval. On the circle of fifths, neighbours are consonant; the tritone sits diametrically opposite.

Scales

A scale picks, out of the twelve notes, those of a key. Choose one: the wheel shows it, the keyboard plays it upwards from the chosen note.

Between two neighbouring notes on the wheel lies a semitone, the smallest step; two semitones make a tone. The major scale doesn't move evenly: it follows the pattern tone, tone, semitone, tone, tone, tone, semitone. In C major the two semitones fall between E and F and between B and C, where the keyboard has no black key.

Seven modes, brightest to darkest

A mode is nothing but a reservoir of notes plus a reference note - the drone the bass holds, against which the ear compares everything else. Same reservoir, different reference: different mode. Major and minor are merely the two modes that became frequent enough to earn their own name. Here they are, ordered by brightness, play them from the chosen note.

Two routes to build a mode: by its characteristic alteration (Lydian is a major with ♯4), or by its degree (Lydian is the 4^th mode: if C is the 4^th degree, the parent scale is G - reimport its alterations). Special mention for the acoustic scale (Lydian dominant: ♯4 and ♭7 at once), the closest to the natural harmonic series. And the modal/tonal border is only a matter of duration: a Lydian moment lasts three seconds in Wagner, three quarters of an hour in a raga. - after Pur Pasteur's "periodic table of modes"

Well-known progressions

The chord sequences heard everywhere, played in the key of the chosen note. Roman numerals name the degrees of the scale: upper case for major chords, lower case for minor ones.

It's the tool for finding key signatures (sharps clockwise, flats the other way) and reading chord progressions, which often turn leftward - the ii-V-I cadence.

Counting the distances

The gap between two notes is counted in degrees from the tonic. Each distance has a name and a sonic colour.

Two families of intervals. The perfect ones - unison, fourth, fifth, octave - have no major or minor version: highly consonant, open, almost cold. The imperfect ones - second, third, sixth, seventh - come in two sizes, major or minor; in chord names they default to major… except the seventh: “C7” carries a minor seventh, because the seventh chord was born on the V^th degree, where the natural seventh is minor. In between, the tritone - six semitones, exactly between fourth and fifth, nicknamed diabolus in musica: the phrase dates from the 18th century, and the medieval ban attributed to it is a legend.

The counting trap: intervals are counted including the starting note (C→G = C-D-E-F-G = a fifth), semitones without it (a fifth = 7 semitones). Count note names first to find the interval type, then refine with semitones.

Intervals come in pairs: inverted (the bottom note moves on top), fourth ↔ fifth, third ↔ sixth, second ↔ seventh - and the imperfect ones swap quality when inverted: a major third becomes a minor sixth. The tritone inverts into a tritone. And the third remains the queen: major or minor, it is the composer's first emotional lever. - after Pur Pasteur's interval map

Harmonic series

The natural harmonics (×1, ×2, ×3…): what a single note already contains. The octave, then the fifth, then the major third emerge from it. Each harmonic is a vibration mode of the string: it vibrates in 2, 3, 4… equal parts at once.

The harmonic series almost folds onto itself: each new harmonic re-divides the octave slightly above the center, like a fractal. Almost - the major scale departs from the series at two precise points: the 7^th harmonic gives a B♭ (the scale has a B), the 11^th lands between F and F♯ (the scale has an F). And those two gaps are exactly the composers' two favorite alterations: the F♯ that releases accumulated tension toward the dominant (G major), and the B♭ that relaxes toward the subdominant - Bach's favorite ending marker. The scale's imperfection is the engine of its modulations. - after "Is the major scale a fractal?", Pur Pasteur

Equal temperament and just intonation

Two ways to define the twelve frequencies, depending on whether you want purity or the freedom to modulate.

The octave split into twelve strictly equal semitones. Every key sounds alike - you can modulate anywhere - at the cost of slightly 'off' intervals. In the 18th century keyboards were still tuned in unequal temperaments (Bach's Well-Tempered Clavier is not in equal temperament); it became the standard in the 19th.

Intervals as whole-number ratios: pure, with no beating. Sublime in one key, but off as soon as you modulate elsewhere - hence the unequal semitones.

Beats come from the harmonics: the 5^th harmonic of C and the 4^th of E land at almost the same pitch. In just intonation they coincide; in equal temperament they rub and give about ten beats per second (C at 262 Hz). The fifth beats only about once per second.

Three things to tell apart.
Octave 2:1 - the only interval identical in both systems.
Tempered fifth ≈ 700 cents vs just fifth 3:2 ≈ 702 cents: 2 hundredths of a semitone apart, almost imperceptible on their own.
Major third: this is where it shows most - the tempered third sits noticeably higher than the just 5:4.

Harmony is a very fast rhythm

Slow a fifth down a few hundred times: what remains is a 3-against-2 polyrhythm. Intervals are accelerated polyrhythms - exactly the same ratios, at another speed scale. Drag the slider: the same ratio moves from rhythm to pitch.

The 3 against 2 runs through all of music: it is the baroque hemiola, the generating cross-rhythm of sub-Saharan African traditions, the right hand in 3 over left hand in 2 of neoclassical piano. Chopin went further: his Fantaisie-Impromptu stacks 4 over 3 - literally a slowed-down fourth. - after "Are harmony and rhythm the same thing?", Pur Pasteur

Why these progressions work

A single note carries no meaning; a chord says a little more; but it is the key - a reservoir of notes around a tonic - that gives each one its role. The chords of the scale fall into three functions.

Tonic (I · vi · iii) : Rest and gravity: melodies tend to land there. The ambiguous iii mostly stands in for I.

Predominant (IV · ii) : The departure: it prepares the dominant.

Dominant (V · vii°) : Tension, strongly pointed back toward I. The vii° holds the tritone of V⁷.

I · Peace : Homeostasis: every need met, nothing threatens.

ii · Resignation : Acceptance - with a lightness that can slip toward major.

iii · Loneliness : The darkest of the minors; made major, it turns to anger.

IV · The chill : Novelty and comfort at once - "a big hug".

V · Joy : Optimism: the happy expectation of resolving to I.

vi · Sadness : The sad version of I: stable, but on the dark side.

vii° · The drama : The unforeseen event - the transition chord between the two worlds.

The circle of fifths reads as a narrative arc - the reason it is everywhere, from classical to pop: I (exposition, peace) → IV (the departure, warmth) → vii° (drama strikes) → iii (loneliness) → vi (stable sadness) → ii (resignation) → V (the joy of return) → I. Around the circle, the three minors follow each other, then the three majors: the dark side, then the light.

The first lever for shifting an emotion: the relative - the minor chord three semitones below each major, sharing two notes (I↔vi, IV↔ii, V↔iii). A IV too sweet? Try its relative ii, or simply slip its note into the harmony to shade it. - after "The secret meaning of the tonal system", Pur Pasteur. Probabilities, not prescriptions: tempo, rhythm and instrumentation weigh just as much.