One of these eight-bar phrases accelerates

Two eight-bar phrases. Both start with a two-bar idea. Both end with a cadence. Both are, by the arithmetic of the previous rung, the same span of seconds and therefore the same kind of perceptual object.

Musicians distinguish them without effort and without counting anything, and the names for them are old: the period, in which a first phrase asks and a second answers, and the sentence, in which an idea is stated, repeated, and then broken up and driven to a close. William Caplin’s Classical Form is the modern statement of the distinction and the source of most current teaching; the terms themselves go back through Schoenberg to the eighteenth-century theorists.

The distinction is normally taught by example — this opening is a sentence, that one is a period, listen to the difference. There is a number in it.

The number #

Write out the phrase as the lengths of its constituent units, in bars, in order.

A period is a four-bar antecedent and a four-bar consequent, each built from a two-bar basic idea and a two-bar continuation: 2, 2, 2, 2.

A sentence is a two-bar basic idea, the same idea repeated or transposed, and then a four-bar continuation which does not stay in two-bar units — it breaks into one-bar fragments before the cadence pulls back to two: 2, 2, 1, 1, 2.

Now take the mean unit length in the second half of the phrase and divide it by the mean in the first half.

The period gives $2/2 = 1.00$. The sentence gives $(1+1+2)/3$ over $2$, which is $1.33/2 = 0.67$.

That is the whole of it, and the hero figure is that arithmetic drawn. One phrase’s units are the same size throughout; the other’s shrink by a third and then recover.

Fragmentation is an acceleration #

The right word for what the sentence does is the one Caplin uses: fragmentation. The unit gets shorter, so events arrive more often, so the music speeds up without the tempo changing.

This is worth separating from tempo carefully, because they are easy to conflate and are not the same. The beat is unchanged. The bar is unchanged. What changes is the rate at which ideas arrive — the harmonic rhythm typically doubles with the fragmentation, the melodic figure gets shorter, and the sense of pressure toward the cadence is manufactured out of that and nothing else.

The compound version is the check that the measure is measuring the right thing. If the ratio were an artefact of the number four or of eight bars it would move when the whole plan doubled. It does not. The quantity is a proportion, so it survives a change of scale, which is what a description of a shape ought to do.

What does move with the scale is the measure’s resolution. At eight bars it can take eleven values; at sixteen, with units of one to eight, it can take many more, and the intermediate scores an eight-bar phrase cannot produce become available. So the measure is scale-free in its answer for a given shape and not in how finely it can distinguish shapes — which is the ordinary situation for a ratio of means over a small number of parts, and is worth knowing before reading a score of 0.67 as a precise one.

Why this is not a taxonomy #

There is a version of this material that is a list of shapes to recognise, and it is what the field of musical form usually turns into. The reason to run it through a number instead is that the number keeps being useful after the labels stop applying.

Real phrases are not always one shape or the other. A great many are hybrids — an antecedent that behaves like a presentation, a consequent that fragments, a period whose second half accelerates and whose first does not. Asking “is this a sentence or a period” of such a phrase has no answer; asking what its ratio is always has one.

And the ratio grades rather than sorting. A phrase that fragments slightly scores more than one that fragments hard, which is a description of a degree, and degree is what most of the interesting cases are about.

Continuous would be too strong, and enumerating the eight-bar plans says how much too strong. Filling eight bars with units of one, two and four gives 55 arrangements, of which 36 split cleanly at bar four and the other nineteen have a unit straddling the midpoint — so the measure has no value at all for a third of the possible shapes, including every plan with a four-bar unit across bars three to six.

Of the 36 that do split, the ratio takes eleven distinct values: 0.25, 0.33, 0.50, 0.67, 0.75, 1.00, 1.33, 1.50, 2.00, 3.00 and 4.00. It is a quotient of two small means and it is as coarse as they are, so the 0.9 a mildly fragmenting phrase might be expected to score is not a number this measure produces at this scale. The nearest available below one are 0.75 and 0.67.

Three other things fall out of the census and each of them bounds the measure.

The sentence’s score is shared by exactly three plans — 2+2+1+1+2, 2+2+1+2+1 and 2+2+2+1+1 — and all three open with the presentation and fragment afterwards. So 0.67 picks out a coherent family rather than a coincidence, which is the best that could be hoped for.

The period’s score is shared by twelve, and they are not a family. 1.00 covers 2+2+2+2 and also 1+1+1+1+1+1+1+1, which is a phrase in one-bar units throughout — no fragmentation because there was never a longer unit to halve. A measure that gives the same answer to nothing changed and nothing to change is reporting an absence rather than a shape, and the period’s 1.00 is the less informative of the essay’s two numbers for that reason.

And half the range is above one, which nothing in this essay has said. A plan that begins in one-bar units and closes with a four-bar one scores 4.00, and that is a real device — the broadening into a cadence that a fragmenting phrase is the mirror of. The measure has always been able to see it and the essay has only ever looked downward.

What the two shapes do to a listener #

The two plans are not interchangeable, and the difference is about what each one licenses next.

A period is a closed argument. It states, it answers, and the answer’s cadence is the strong one — the ending that supplies every signal there is — while the antecedent’s is deliberately weaker. Nothing about a period demands a continuation. It is a complete musical sentence in the ordinary sense of the word, which is why the term for it is so unhelpfully close to the term for the other thing.

A sentence is an open one. Its second half is an acceleration into a cadence, and an acceleration establishes a rate that the cadence then interrupts. That is a machine for producing expectation: a listener who has been counting has just been given evidence that events are speeding up, and the arrival stops the process rather than completing it.

So the two shapes fill the same eight bars and leave a listener in different states. The period has settled something; the sentence has spent something.

The repetition that starts a sentence #

There is a second thing the sentence does that the ratio does not capture, and it is worth putting beside the ratio so that the ratio’s coverage is clear.

The sentence’s first two units are the same idea twice. Sometimes literally the same; far more often the same shape a step or a fifth away, over a different chord. Schoenberg’s term for the pair is the presentation, and the repetition is not decoration — it is what establishes the two-bar unit as the unit, so that halving it later is an event rather than an accident.

That is a claim about what a listener has to have been given before an acceleration can be perceived. A unit length cannot be halved unless it has first been established, and one statement does not establish anything. The presentation is the measurement stage of the phrase.

A shape that keeps its identity when it is moved is the thread that runs through the whole of this subject — a scale, a chord, a rhythm and now a two-bar idea — and the sentence is the place where the moved repeat has a job rather than just a name. It is there to make the fragmentation legible.

The offset is the diagnostic. A stripe at an offset of two says an idea has been immediately restated, which is a presentation; a stripe at an offset of four or eight says a phrase has been answered, which is a period. Both are visible in the matrix and neither needs a label.

Both of them are the same number of seconds #

Nothing in this essay has mentioned tempo, and that is deliberate: the ratio is dimensionless and survives any tempo at all. But the two shapes are being compared as alternatives available to a composer, and that comparison is only fair if both are perceptible objects.

At an ordinary minuet or allegro tempo an eight-bar phrase lasts between twelve and sixteen seconds, which is past the two-to-eight-second window inside which a series of events is heard as one thing. So neither shape is a perceptual unit; both are held in memory rather than in the present, and the acceleration a sentence performs is an acceleration of something a listener is counting rather than of something they are inside. The two cadences the period is built from are the site’s own five-component vector at two strengths: the antecedent’s arrival on the dominant supplies the metre and the duration and withholds the goal, and the consequent’s supplies everything.

The figure makes a point the ratio cannot. The eight-bar phrase is not itself a perceptual object at these tempos. Its two-bar units are, and its four-bar halves are near the edge. So what a listener holds is a small number of units and a memory of how long they were, which is precisely the information the ratio is computed from — and it is why fragmentation is audible while a description in terms of the phrase as a whole would not be.

Where the unit lengths come from #

The measurement takes the unit lengths as given, and it is fair to ask where they come from, because if they come from an analyst’s judgement then the number is a judgement wearing a number’s clothes.

For the plans above they come from the definitions: a sentence is a presentation of two two-bar units followed by a continuation that fragments, so writing 2, 2, 1, 1, 2 is restating the definition rather than measuring anything. The figures are therefore a demonstration of what the quantity does, not a measurement of a piece.

Getting the unit lengths out of actual music is a different job and a harder one, and it is exactly what a novelty operator does at a smaller scale. Run a narrow kernel along a phrase and the peaks are the internal group boundaries; the gaps between the peaks are the unit lengths; the ratio follows. That chain works, and every link in it is one this site has already built.

What it needs, and what these encodings do not have, is a melody. Fragmentation is overwhelmingly a melodic and rhythmic device — the harmony often does change with it, but a bar-level chord scheme is far too coarse to see a two-bar figure become a one-bar figure. So the honest position is that the quantity is well defined and computable and that the input for computing it on real music is not present here.

The classic examples, and what is actually being claimed about them #

The two standard illustrations are the opening of Mozart’s Piano Sonata K331, which is a period, and the opening of Beethoven’s Piano Sonata Op. 2 No. 1, which is a sentence. Those attributions are not this site’s; they are the textbook cases, used by Caplin and by nearly everybody since.

What is being claimed here is not that those two openings are correctly labelled — that is somebody else’s analysis and it is not being checked. The claim is narrower and it is about the measure: if a phrase is described as a sentence, the description entails a fragmentation, a fragmentation entails a shorter mean unit in the second half, and that is a number somebody can disagree with. A label cannot be wrong in an interesting way. A ratio can.

It is also worth noticing which repertoire this belongs to. The sentence and the period as described here are Viennese classical constructions, roughly 1770 to 1830, and the theory was built to explain that repertoire. The eight-bar norm behind both of them is a property of that music and of the styles descended from it. It is not a fact about music, and a raga’s exposition, a piece of organum and a west African bell cycle each fail to be either shape in a different way.

Fed the phrase beginnings of a sixteen-bar plan, the site’s own metre induction returns an eight-bar period — so a phrase is one unit of a hypermetre, and the fragmentation this essay measures happens inside a single hypermetric unit rather than across two.

The device is not confined to its repertoire #

The Viennese pedigree above makes the sentence sound like a museum object. The device is not.

Fragmentation is what a drum fill is. Four bars of a two-bar pattern, then a bar of half-bar figures, then a bar of quarter-bar figures arriving at the downbeat of the next section: that is the sentence’s proportion, played on a kit, and its ratio comes out at 0.33 — the same direction as the sentence’s 0.67 and further along it. The same shape runs a build in dance music, where the unit that shortens is the length of the loop rather than the length of a melodic idea, and the arrival it accelerates into is a drop rather than a cadence.

It runs a raga’s development too, though the arithmetic there is doing something else — the acceleration in a jor or a jhala is continuous and over a much longer span, and it is a change of rate rather than a halving of a unit. That is a real difference and it is the kind the measure is good for: run it and the ratio comes out near 1 with a long slow drift, which is not the sentence’s signature at all.

So the claim worth making is the general one. An acceleration of the unit is a way of manufacturing arrival, it is available in any music that has a unit, and the ratio measures it wherever it occurs. What belongs to eighteenth-century Vienna is the particular pairing of that device with an eight-bar span and a cadence, which is a convention. The device underneath is not.

What this cannot show #

The ratio is computed from unit lengths and knows nothing else. Two phrases with identical unit lengths and completely different content score identically, which is correct and is also most of what a phrase is.

It cannot distinguish a repetition from a transposition — the sentence’s second unit is usually the first unit moved, and that “moved” is doing a great deal of work that a length cannot capture. And it says nothing about which cadence ends the phrase, which is the other half of what makes a period a period and is measured separately.

It is also entirely blind to what is being repeated. A presentation whose two statements are over the tonic and then over the subdominant is doing something a presentation over one chord is not, and both write 2, 2 in the unit list. That is a case where the harmonic rhythm is the variable and the unit lengths are the constant, which is the exact inverse of what this measure is sensitive to — so the two quantities have to be read together or neither is worth much.

Where the ladder goes #

Both shapes end at a cadence, and both depend on one of their cadences being weaker than the other. That weakness is not a defect and it is not vagueness: it is a specific, computable shortfall on specific components, and an ending that exists so that a bigger one can is the rung that measures it.

The other direction goes upward. A phrase is a unit; units group into larger units; and the machinery that groups beats into bars turns out to group bars into hypermeasures without being modified at all.