The room chooses the harmonic rhythm

A room’s reverberation time is computable from its volume, its surface and how absorbent that surface is, and the figure that computes it is one of the site’s oldest. What has not been asked is what the number does to the music.

It does something specific and rather severe, and the arithmetic takes one line.

Decay is linear in decibels #

Reverberation time is defined as the time for a sound to fall by sixty decibels. The decay itself is exponential in amplitude, which means it is a straight line in decibels — a sound is $-60t/T_{60}$ decibels down after $t$ seconds, and that is exact rather than approximate for the diffuse part of the decay.

So if chords arrive every $1/r$ seconds in a room with reverberation time $T_{60}$, the chord $k$ changes ago is

$$\frac{60k}{r\,T_{60}} \ \text{decibels}$$

below the one just struck. Every chord in the sequence sits on a ladder with rungs of that size.

The hero figure draws six chords in a cathedral at one a second. The rungs are 7.5 decibels apart, so two earlier chords are still within twenty decibels of the new arrival — which is to say a listener hears three harmonies at once, continuously, for as long as the music goes on.

Why twenty decibels #

The floor is a choice and it deserves a defence, because the whole result scales with it — exactly, in fact: the ceiling is 60 over the floor times the decay time, so halving the floor doubles every number in every table below and reorders nothing. That is worth knowing before the defence rather than after, because it means the choice cannot be wrong in a way that changes any comparison, only in a way that changes every absolute figure together.

Sixty decibels is the wrong number. That is the definition of the reverberation time and it means inaudible-in-a-quiet-room; a chord sixty decibels below a new one is not contributing anything to the harmony. The question here is different: at what level does an earlier chord stop being heard as harmony underneath a new one?

That is a masking question, and the site has measured masking. A tone twenty decibels below a simultaneous masker of similar spectrum is at or below threshold across much of the spectrum, and one ten decibels down is plainly audible. Twenty is a defensible middle, it is on the conservative side, and moving it changes the numbers proportionally rather than changing the shape of the result.

At twenty decibels the calculation is arithmetic. The rate at which exactly one earlier chord remains above the floor works out at $60/(20\,T_{60})$, which is $3/T_{60}$ chords a second.

Six rooms, six ceilings #

Put six rooms through $3/T_{60}$:

A factor of twenty-two between the extremes, produced entirely by the buildings.

It is worth noticing that the two ends of that range are not equally binding. Eight and a half chords a second in a studio is above anything any repertoire does — the fastest style in the figure below is under two — so the studio’s ceiling is not a ceiling at all, and neither is the jazz club’s. Only the three reverberant rooms have a number that any music approaches. So the table is really three constraints and three non-constraints, and the interesting comparison is between the cathedral and the hall rather than across the whole range.

The slider on those figures is the composer’s variable. The slope belongs to the room and cannot be argued with; how many chords fall inside the shaded band is a decision, and it is the only decision available.

And the music does sit under it #

Now put the ceilings beside the rates.

Two things in that figure are worth stating carefully, because one is a fact and one is a hypothesis.

The fact. The earliest notated polyphony was written for stone rooms and it changes sonority very slowly — the organum of the Notre-Dame school holds a single note of chant underneath many notes of the upper voice, and the held note can last many seconds. Whatever else it is doing, it is well inside the ceiling its room imposes.

The ceiling is not one number, and the second number is the interesting one. Every figure above uses a single reverberation time, and a room does not decay evenly: absorption is a strong function of frequency, so the ceiling is too. Running 3/T₆₀ band by band:

In a cathedral the bass may change once every three seconds and the treble once a second — a factor of three inside one building, and the same factor in each of the other two. The single ceiling this essay has been quoting is the bass ceiling, because the bass is where the room holds on longest.

Which changes what the constraint permits. It does not say the music must be slow; it says the music must have a slow bass and may have a fast top, and those are different instructions. A texture with a held low voice and rapid upper parts is inside the constraint at every frequency; a texture that moves every part together at one rate is bound by the slowest band.

That is a description of organum. A held tenor under many notes of an upper voice is exactly the texture the band-by-band ceiling allows and the single-number ceiling does not distinguish from a slow chorale — and it is the texture the rooms in question produced. The same reading covers the faburden and fauxbourdon that follow, and it fails for the chorale, whose whole point is four voices moving together.

The hypothesis. That the room caused the style. This site cannot demonstrate that, and the honest reasons are worth listing. Music of that period is elaborating a plainchant whose notes are already long, so the slow tenor may be inherited rather than chosen; the notation available did not measure duration in the modern way; and liturgy, not acoustics, decided a great deal about the shape of the music. The most that can be said with the arithmetic in hand is that the style is consistent with the constraint and would not have been possible against it.

The negative version of the claim is stronger and is safe, and the band-by-band ceiling makes it stronger still. A chorale at chorale tempo cannot be performed intelligibly in an eight-second room — the arithmetic gives three chords sounding at once, permanently — and the reason is not that the music is fast but that all four of its voices move at once, so the whole texture is bound by the slowest band. There is no register a chorale can put its harmonic change in that the room does not hold. A texture with a static bass has one; that is the difference, and it is why the constraint sorts textures rather than tempos.

No repertoire of the chorale kind was written for such rooms. That is a real constraint doing real work, whatever caused what.

The same constraint, four ways #

The arithmetic above is one instance of a shape that runs through this site’s instrument field, and putting it beside the others makes it easier to see what kind of constraint it is.

An error that is a time cannot be fixed by anything that is a pitch, and vice versa. A tube’s end correction is a length against a sounding length that halves every octave, so one physical fact is 13 cents at the bottom of a register and 52 at the top; a guitar’s saddle is a length against errors that are frequencies, and moving it improves the worst case by nothing.

A reverberation time is the same kind of object. It is a fixed number of seconds, and the thing it is measured against — the interval between chords — is also a number of seconds, which is why the two divide cleanly and give an integer count of overlapping harmonies. That is what makes this constraint so much sharper than most acoustic ones: both sides of the ratio are in the same units, so there is no register dependence and no correction to argue about.

The consequence is that the ceiling is genuinely a ceiling rather than a trade. A maker can undercut a tone hole; a luthier can slant a saddle. Nobody can make a chord decay faster in a stone building, and the only variable on the composer’s side is how often to change.

The Renaissance objection, which is a good one #

An obvious counter-example is the polyphony of the fifteenth and sixteenth centuries, which was sung in exactly those stone rooms and whose sonorities plainly change faster than 0.38 a second.

The objection is right about the sonorities and it points at a distinction this essay has been sliding over. There are two rates here, not one.

The rate of sonority change — how often the set of pitches sounding is different — is fast in that music, because four or five independent lines are moving and every crossing produces a new vertical. The rate of harmonic motion — how often the music is somewhere else, in the sense that a bass and a modal centre have moved — is much slower. In much of that repertoire a phrase’s harmonic content is describable as one or two sonorities with a great deal of passing motion between them.

So the constraint bites on the second and not the first. A wash of overlapping passing notes in a resonant room is not a problem, because the notes are all drawn from one harmony and the reverberation blends them into it. Overlapping harmonies is the problem, because the reverberation then blends two things that were meant to be different.

That is a real distinction and it also explains a compositional habit. Music written for resonant spaces tends to keep its harmonic vocabulary within one collection for long stretches and to move by step within it, which produces overlap that is consonant. Music written for dry rooms can afford harmonies that clash if they overlap, because they do not.

The number a composer would want #

There is a more useful form of the same arithmetic, and it is worth stating because it inverts the question.

A composer does not usually get to choose the room. Given the room, the useful quantity is the longest a harmony can be allowed to change: $20\,T_{60}/60$ seconds, which is $T_{60}/3$.

For the six rooms that is 2.7 seconds in a cathedral, 1.3 in a large church, 0.67 in a concert hall, 0.47 in an opera house, 0.27 in a jazz club and 0.12 in a studio.

Read that way, the constraint stops being about styles and becomes a working number. A conductor deciding a tempo for a piece in a resonant church is choosing, whether or not it is put this way, how many chords will be sounding at once — and the usual response, which is to take it slower than the marking, is exactly the right one and is a direct consequence of the line above.

The instrument the room becomes #

The room is part of the instrument is the site’s older statement of this, and the harmonic version sharpens it into something a composer can act on.

The room adds a low-pass copy of everything, delayed and decaying. In a very reverberant space that copy is loud enough to be a second ensemble, playing what was played a second or two ago, and the composer’s real orchestration is the sum of the two. Writing for such a room is writing a canon at a delay one cannot control against an accompaniment one cannot silence.

Which is why the composers who worked in them wrote what they wrote. Slow harmonic motion, a limited vocabulary, stepwise voice leading, and a preference for sonorities that survive being smeared — that is a description of a style and it is also a list of the sensible responses to the arithmetic above.

What a listener in the room actually gets #

There is a second effect of the same overlap that the count of audible chords does not capture, and it works in the composer’s favour rather than against.

An overlapping chord is not only a competing harmony. It is also a sustain, and it is the reason a single line sung in a cathedral sounds like a texture. Reverberation supplies, for free, the thing an organ supplies with stops and an orchestra with a string section: continuity between notes, a bed under the melody, and a sense that the sound is coming from everywhere.

So the same number is a constraint at one rate and an instrument at another. Below the clear rate, the tail fills the gaps and makes a thin ensemble sound large; above it, the tail is another chord and makes a clear ensemble sound muddy. The transition between the two is not gradual in the way it looks — it is the difference between one earlier harmony sounding and two.

That is why the ceiling is worth computing rather than judging by ear. A room that flatters a slow piece will ruin a fast one, and the boundary between the two is at a rate that can be worked out in advance from a single published number about the building.

What this cannot show #

The single number $T_{60}$ is a coarse description of a room. Real decay is frequency-dependent: stone rooms are far more reverberant at low frequencies than at high, so a bass note lingers while the treble clears, and the overlap this essay computes is much worse in the register where harmony is decided. That makes the ceiling lower than the figures say, which is the safe direction, and it also means a single ceiling per room is a simplification.

The early part of a decay is not diffuse either. The first reflections arrive as discrete echoes rather than as a smooth tail, and for the first tenth of a second or so the straight line in the figures is not the right model at all. For a question about chords a second apart that does not matter; for one about consecutive notes it would.

Nor does any of this account for the listener’s position, the size of the ensemble, or the fact that a performer in a resonant room plays differently — shorter, more separated, and slower — which is itself a response to the constraint and partly conceals it from measurement.

The performer absorbs part of it #

One reason this constraint is easy to overlook is that it is partly hidden by the people playing under it.

An organist in a resonant church plays shorter than the notation says, detaching notes that would be joined in a dry room, so that each chord has some silence before the next arrives. A choir in the same building takes a slower tempo than the same music would get elsewhere. A brass ensemble reduces the number of moving parts. All three are reducing the overlap, and none of them is likely to describe what they are doing in decibels.

That has a consequence for anybody trying to check the arithmetic against recordings. A measurement of harmonic rhythm taken from performances in resonant spaces is already a measurement of the adapted rate rather than of the notated one, so it will agree with the ceiling rather better than the score alone would — which makes the agreement weaker evidence than it looks.

The clean test would compare the notated harmonic rhythm of repertoires written for rooms of known reverberation time, which is a musicological question with a real answer and is not one a figure on this site can settle.

Where the ladder goes #

The ceiling and the floor of harmonic rhythm are now both computed, and between them they define the band in which harmony can be the agent of musical change. Below the floor lies a whole class of music that does not use harmony that way at all: cycles, loops, drones, and forms that cannot cadence and close by density instead.