How beats and chords work

Music is something you enjoy with your ears, but underneath it there are rules made of numbers: how beats are counted, how long notes last, how notes are stacked. This page looks at the ideas behind the site’s eight works: Meter Tap, Note Fill, Chord Ear, Polyrhythm, Rhythm Echo, Melody Memory, Interval Steps and Tempo Feel.

Meter: strong beats and weak beats

The numbers at the start of a score, like 4/4 or 3/4, are the time signature. The top number says how many beats are in a bar; the bottom number says which note counts as one beat (4 for a quarter note, 8 for an eighth note). Beats are not all equally strong. The first beat of the bar is the strongest and the others are weaker. In 3/4 you feel strong–weak–weak; in 4/4 it is strong–weak–medium–weak.

When the top number is 2, 3 or 4, each beat splits into two: this is “simple” meter. When it is 6, 9 or 12, each beat splits into three: “compound” meter. For example, 6/8 has six eighth notes, but they form two groups of three, and most people feel it as two beats. Some books treat 6/8 differently, so the classification varies a little from one textbook to another.

This site’s program checked that each of the five meters (2, 3, 4, 6 and 5) puts its strong accent only on the first beat of each bar, and that the 6-pulse bar alone has a secondary accent on the fourth pulse (the start of the second group of three).

Note lengths halve step by step

Starting from the whole note, a half note is half as long, a quarter note half again, then the eighth and the sixteenth. Rests (silent time) are counted in the same system.

A dot after a note adds half of its own length. A dotted quarter note is a quarter plus an eighth, 1.5 times the original. If the shortest note, the sixteenth, counts as 1, then a whole note is 16, a half 8, a quarter 4, an eighth 2 and a dotted quarter 6. Everything becomes whole-number addition. This site’s program checked this counting (16 down to 1, the dotted values 12, 6 and 3, and rests of 4 and 2) and that all five Note Fill challenges can be filled exactly to the end of the bar.

Triads: three notes stacked

The basic chord is the triad, three notes stacked in thirds. Distances are counted in semitones. With the root as 0, there are four kinds.

• Major (bright): 0, 4, 7
• Minor (calmer, darker): 0, 3, 7
• Augmented (floating): 0, 4, 8
• Diminished (tense): 0, 3, 6

This site’s program checked how the four are built and that every root stays within the keyboard range.

Neat ratios and a small difference

Write the frequencies (vibrations per second) of a major triad as the simplest whole-number ratio and you get 4:5:6. This is the “just intonation” major triad, and it blends very smoothly.

Modern pianos, however, use “twelve-tone equal temperament”, which divides the octave (a doubling of frequency) into twelve equal steps. One semitone is a factor of the twelfth root of 2 (2^(1/12)), about 1.0595. This lets music move into any key and sound the same, but the chords drift slightly away from 4:5:6. The equal-tempered major third is about 14 cents (roughly one seventh of a semitone) higher than the just one, and the fifth is about 2 cents lower. Scaled so the root is 4, an equal-tempered major triad is 4 : 5.04 : 5.993. This site’s program checked these values.

Neither is “correct”. Just intonation sounds clear and pure; equal temperament lets you change key freely.

There is also a standard for the pitch itself. The international standard ISO 16 sets the A in the treble staff at 440 hertz (440 vibrations per second). It dates from 1975 and was reviewed and confirmed in 2022. It is an agreement, though, and not every performance uses exactly this pitch.

Two rhythms at once

“Three against two” means spacing three beats evenly in one part and two beats evenly in the other, over the same stretch of time. Three notes taking the time of two is called a hemiola, and repeating the layering is called a polyrhythm or cross-rhythm. Scholars use these words slightly differently, so here we simply say that 3:2 is a classic example.

Put on a grid of six steps (the least common multiple of 3 and 2), both rhythms land exactly on grid points. For 4 against 3, the grid has twelve steps. The two parts sound together only at the start of a cycle and are out of step in between. This site’s program checked that 3:2 uses six steps, 4:3 uses twelve, and that the two parts coincide only at the start of a cycle.

Rhythm Echo: copying the gaps

You hear a rhythm and tap it back. What counts is not the exact time of each tap but the gaps between taps. Starting a little late is fine as long as the gaps match. This site’s program checked that it compares the spacing from the first tap and accepts a difference of up to 0.16 seconds (160 milliseconds), and that too few or too many taps both fail. There are four levels, from even spacing to eighth notes, dotted spacing and two-bar patterns with off-beats (notes that fall between the beats).

Melody Memory: a sequence that grows by one

Four pads light up and sound in order. If you repeat the order correctly, the sequence gets one step longer. This site’s program checked that a longer sequence keeps all earlier notes unchanged and adds exactly one at the end, that every note is one of the four pads, and that your answer is checked from the start in order, so repeated pads cause no mistakes.

Interval Steps: the distance between two notes

The difference in pitch between two notes is an “interval”. C to D is a second, C to E is a third, and so on: you count which note it is from the lower one. In the major scale, the seconds up to the octave (the 8th) are 2, 4, 5, 7, 9, 11 and 12 semitones from the starting note. An octave is exactly a doubling of frequency. The frequency ratio of one semitone is the twelfth root of 2 (about 1.0595), and twelve of them multiply to 2. This site’s program checked that questions on seconds through octaves, both going up and going down, match those semitone counts, and that the frequency ratio over 12 semitones is exactly 2 (or one half going down).

Tempo Feel: beats per minute

Tempo is how many beats there are in a minute, written BPM. If the gap between beats is 0.5 seconds, that is 120 beats a minute, or 120 BPM. Here the BPM comes from the average of your tap gaps, and the unevenness of the gaps is shown in milliseconds. The program checked that four taps 0.5 seconds apart give 120 BPM with zero unevenness, and that gaps mixing 0.4 and 0.6 seconds give 100 milliseconds. With too few taps, times going backwards, or values that are not numbers, it gives no result. The demonstration has eight beats, and you pass if you are within 5 percent of the target (at 120 BPM, 126 passes and 127 does not).

Spot the differences

• Meter: the same number of beats can feel different depending on which are strong.
• Notes: different shapes can still add up to the same bar length.
• Copying and remembering: rhythm is copied by its gaps, melody by its order.
• Chords: “neat whole-number ratios” and “works in every key” are a little different things.

Find the strong beat in Meter Tap, fill the bar in Note Fill, test your ears in Chord Ear, and hear the drift and meeting points in Polyrhythm.