A note on a staff says which frequency and nothing about when, and a page of pitches with no durations is not yet music. This lesson supplies the other half of a notehead's meaning.
Duration is a fraction, not a time
The first thing to accept is that written durations are relative. A symbol on the page says how long a note lasts compared with the other notes around it, and says nothing about seconds until a tempo mark is added at the end of this lesson. That separation is deliberate and useful: it lets the same rhythm be performed fast or slow without rewriting a single notehead.
The unit at the top of the system is the whole note, and every other value is reached by halving it. Halve a whole note and you have a half note, halve again for a quarter note, again for an eighth, a sixteenth, a thirty-second, a sixty-fourth. British names describe the same objects: semibreve, minim, crotchet, quaver, semiquaver, demisemiquaver, hemidemisemiquaver. Both naming systems appear in print, and the American one is used here because it says the arithmetic out loud.
The symbols are built to make the halving visible. A whole note is a hollow notehead with no stem. Add a stem and it is a half note. Fill the head in and it is a quarter note. From there, each further halving adds a flag to the stem: one flag for an eighth, two for a sixteenth, three for a thirty-second. So a filled head with two flags is a sixteenth note, and you can read the value off the drawing without recognising the shape as a whole. Above the whole note sits one survivor of older notation, the double whole note or breve, worth two whole notes and now rare outside choral music and final chords.
Rests are the same tree, silent
Every note value has a matching rest, drawn differently but worth exactly the same. A whole rest is a small filled rectangle hanging below the fourth line of the staff, a half rest is the same rectangle sitting on the third line, a quarter rest is a zigzag, and eighth and shorter rests carry flags in the same pattern the notes do: one flag for an eighth rest, two for a sixteenth.
The two rectangles are worth learning properly, because hanging and sitting are the whole difference between two beats and four. There is also one deliberate inconsistency: a whole rest, in most contexts, means a whole bar of silence rather than four quarters of it, so in a bar of three quarters it is worth three, not four. That is a convention about bars, which the next lesson defines, and it is the one place where the halving tree is quietly overruled by context.
Dots, and what halving cannot reach
Halving alone can only produce durations that are powers of two: one whole, one half, one quarter. It can never produce a duration of three quarters, because 3 is not a power of 2 times anything in the tree. Since music in three is at least as old as music in two, notation needs a repair, and the repair is the dot.
A dot to the right of a notehead adds half of that note's value to it. A dotted half note is worth quarter notes. A dotted quarter is worth one and a half quarters, which is three eighths. A dotted whole note is worth six quarters.
A second dot adds half of the first dot, that is, a quarter of the original. A double-dotted half note is therefore worth quarter notes. The pattern is a geometric series: with dots the total is
times the undotted value. One dot gives times, two dots times, three dots times. Notice what the formula says about the limit: no number of dots ever reaches twice the value, which is why a dotted note is always shorter than the next note up the tree and never equal to it. Three dots is the practical maximum and is already unusual.
Example. How many quarter notes is a double-dotted whole note worth?
A whole note is 4 quarters, and two dots multiply by , so quarter notes. The same result by addition: .
Now you. How many eighth notes is a double-dotted quarter note worth?
Answer
A quarter note is 2 eighths, and eighths. By addition, , so it equals seven sixteenth notes.
Ties, and durations no symbol can spell
Dots close many of the gaps but not all of them. A duration of five sixteenths, for instance, is out of reach: a quarter note is four sixteenths, a dotted quarter is six, and nothing in between exists. The general repair is the tie, a curved line joining two noteheads of the same pitch, which instructs the player to sound the first and hold it through the second without rearticulating. Five sixteenths is a quarter note tied to a sixteenth.
Because any duration can be written as a sum of tree values, ties plus dots can express anything. They are also the only way to write a note that continues across a barline, since a notehead cannot be printed in two bars at once, and that is their commonest use by far.
A tie looks exactly like a slur, which is also a curved line, and means something entirely different: a slur joins notes of different pitches and asks for them to be played smoothly and connectedly, each still sounded. The pitches tell you which is which. Two identical noteheads joined by a curve is a tie and one sound; a curve over a rising phrase is a slur and as many sounds as there are notes.
Example. Write a duration of seven sixteenth notes.
Check the dots first. A quarter is 4 sixteenths, so a double-dotted quarter is sixteenths exactly. No tie is needed at all, which is worth checking for before reaching for one.
Now you. Write a duration of nine sixteenth notes.
Answer
Dots cannot do it: a half note is 8 sixteenths, a dotted half is 12, and a double-dotted half is 14. So tie a half note to a sixteenth, . A dotted quarter tied to a dotted eighth also gives , and both are correct arithmetic, though the first is far easier to read.
A tempo mark makes it seconds
Now the fractions become time. A metronome mark at the head of a piece names a note value and a number of them per minute: quarter note = 120 means 120 quarter notes pass in a minute, so each quarter note lasts s. In general, a note worth quarter notes at a mark of quarter notes per minute lasts
The mark does not have to be in quarters. Half note = 138, the marking on the first movement of Beethoven's Hammerklavier sonata, means 138 half notes per minute, so a half note is s and a quarter note is 0.217 s. Converting to quarters first, then applying the formula, keeps this from going wrong.
The device that made such marks possible is recent. Johann Maelzel patented a workable metronome in 1815, and Beethoven published metronome marks for his symphonies in 1817, the first major composer to do so. Before that a tempo instruction was a word like allegro, which is a character more than a speed, and the difference between the two kinds of instruction has never been fully settled: the Hammerklavier marking is still argued about, since at 138 the movement is close to unplayable.
Example. At a tempo of quarter note = 92, how long does a dotted half note last?
A dotted half is 3 quarter notes, and each quarter is s, so s.
Now you. At quarter note = 132, how long does a passage of five dotted quarter notes last?
Answer
A dotted quarter is 1.5 quarters, so it lasts s, and five of them last s.
The same arithmetic gives a whole movement a running time. Thirty-two bars of four quarter notes each, at quarter note = 120, contain quarter notes at 0.5 s apiece, so 64 s. That is why a metronome mark and a bar count are enough to know whether a piece will fit on one side of a record, and why publishers have cared about them since long before anyone could measure a performance.
What the written durations do not say
Notated rhythm is a good approximation of performed rhythm and not a transcription of it, and three gaps are worth naming.
The first is that performers stretch time on purpose. Rubato means exactly that, taking from one note and giving to another, and in most repertoire a phrase slows slightly at its end whether or not anything is written. A performance that matched the arithmetic exactly would sound mechanical, which is the usual complaint about a first pass at engraving software playback.
The second is that some traditions systematically play what is written unequally. Swung eighth notes in jazz are written as plain pairs and played closer to long-short in a ratio near 2:1, so at quarter note = 120 the pair occupies 0.5 s split roughly 0.33 s and 0.17 s rather than 0.25 s each. French baroque notes inégales did the same thing two centuries earlier. The convention lives outside the notation entirely, and a reader who does not know the style will read the page correctly and play it wrongly.
The third is that the tree assumes division by two, patched by dots for three. Genuine division of a beat into three, five or seven equal parts needs a further device, the tuplet, which belongs with the next lesson because it depends on what a beat is.
And that is the gap this lesson leaves open. Every duration on the page is now measurable, and nothing yet says which notes are strong. A row of eight identical eighth notes could be four groups of two or two groups of four, and those are different pieces of music. Supplying the grouping is what metre does.