Notes, octaves and the keyboard

Western music uses 12 pitch classes per octave: C, C♯/D♭, D, D♯/E♭, E, F, F♯/G♭, G, G♯/A♭, A, A♯/B♭, B. On the keyboard the white keys are the natural notes and the black keys the sharps/flats, grouped in twos and threes; C is always the white key just left of a group of two black keys. A full piano has 88 keys from A0 (27.5 Hz) to C8 (4186 Hz). Octave numbers (scientific pitch notation) change at C, so the note above B3 is C4 (“middle C”).

An octave is a doubling of frequency. The interval between adjacent keys is a semitone (half step); two semitones make a tone (whole step).

Equal temperament and A4 = 440 Hz

Modern pianos are tuned in 12-tone equal temperament (12-TET): each semitone multiplies frequency by the same ratio $2^{1/12}\approx1.059463$. Numbering the piano keys $n = 1$ (A0) to $n = 88$ (C8), with A4 as key 49:

Equivalently with MIDI note numbers $m$ (A4 = 69, middle C = 60, and $n = m-20$): $f = 440\cdot2^{(m-69)/12}$. Intervals are often measured in cents: 100 cents per equal-tempered semitone, $1200\log_2(f_2/f_1)$ cents in general. A just (pure) perfect fifth is 3:2 = 701.96 cents; the tempered fifth is 700 cents, about 2 cents narrow, which is why fifths on a piano beat slowly.

The tone of this piano is synthesised: each note is built from harmonics whose amplitudes follow a string struck at one-eighth of its length (partial $k$ has amplitude $\propto |\sin(k\pi/8)|/k$), with a brightness filter that closes as the note decays, slightly detuned unison strings, a hammer “thump” and longer decay for bass notes. As on a real piano, the top 1½ octaves have no dampers. Real piano strings are also slightly inharmonic, which this model ignores.

Intervals

Scales and keys

The major scale follows the step pattern W–W–H–W–W–W–H. Starting the same pattern on different notes of the major scale gives the modes: Ionian (major), Dorian, Phrygian, Lydian, Mixolydian, Aeolian (natural minor) and Locrian. The harmonic minor raises the 7th degree of natural minor (creating a leading note and an augmented 2nd), and the melodic minor (ascending) raises the 6th and 7th. Pentatonic scales use five notes and are central to many African, Asian and folk traditions; the blues scale adds the ♭5 “blue note” to the minor pentatonic.

A key signature lists the sharps or flats of a major key and its relative minor. Sharps are added in the order F C G D A E B, flats in the order B E A D G C F. Moving clockwise around the circle of fifths adds one sharp (or removes one flat). The relative minor starts on the 6th degree of the major scale (three semitones below the tonic) and shares its key signature: C major → A minor, F major → D minor.

Chords

A triad stacks two thirds: major (M3 + m3, e.g. C–E–G), minor (m3 + M3), diminished (m3 + m3) and augmented (M3 + M3). Adding another third gives seventh chords: dominant 7th (major triad + m7, e.g. G7 = G–B–D–F), major 7th, minor 7th, half-diminished (m7♭5) and diminished 7th. An inversion puts a note other than the root in the bass: 1st inversion has the 3rd in the bass, 2nd inversion the 5th, 3rd inversion the 7th.

In a major key the diatonic triads are I, ii, iii, IV, V, vi, vii° (upper case = major, lower = minor, ° = diminished). The dominant (V or V7) pulls strongly back to the tonic (I) because its 3rd is the leading note, a semitone below the tonic. Benga, the guitar-driven popular music of Kenya, and much Afro-pop often cycle quickly through I, IV and V with a bouncing bass and interlocking picked guitar lines, which you can hear in the Benga bounce rhythm (a simplified illustration).

Cadences

A cadence is the pair of chords that ends a phrase, like punctuation in a sentence.

Play I–IV–V–vi in C (C–F–G–Am) and hear the interrupted cadence; then play I–IV–V–I and hear the perfect cadence resolve.

1. Frequency of middle C

2. Which key is 1000 Hz?

3. Building E♭ major

4. Spelling and inverting a chord

5. Roman numerals: ii–V–I in B♭

6. Beats and tempo

7. Beating between unison strings

1. Find the octaves. Play A2, A3, A4, A5 and read the frequency each time: 110, 220, 440, 880 Hz. Each octave doubles the frequency.
2. Scale practice (auto-checked). In Scales, choose G major and press Practise, then play the scale upwards from any G. A toast confirms success; a wrong note restarts the attempt.
3. Hear the modes. Play D Dorian and D natural minor back to back: they differ only in the 6th degree (B vs B♭). Then compare C Lydian (F♯) with C major.
4. Inversions. Play C–E–G, then E–G–C, then G–C–E. The detector names all three “C major” and tells you the inversion.
5. Circle walk. Click C, G, D, A on the circle and watch the key signature on the staff gain a sharp each time.
6. Ear training challenge. Get a streak of 5 correct intervals at the Intermediate level (toast on success).
7. Play-along challenge. Play Ode to Joy in wait mode with 90 % accuracy or better, then turn wait mode off and try at 100 % tempo.
8. Record and listen. Start the metronome, record a I–IV–V–I progression in your own rhythm and play it back.

Computer keyboard

Two rows of your computer keyboard act as piano keys (positions shown for a QWERTY layout; any layout works by physical position):

Mouse and touch

Click or tap keys; play chords with several fingers (multi-touch). Slide across the keys for a glissando. Striking a key nearer its front edge plays louder. Drag the mini-map above the keys to move along the 88-key keyboard, or choose how many octaves to show.