The unison is the coarsest thing in the room

The last two rungs price a tuning ritual: one match is good to two cents when a beat is available, and the graph multiplies that by the square root of its depth to give an ensemble a band of about eleven.

Both are about a unison. The oboe sounds an A and everybody puts an A on it, and that is the only interval the ritual contains. The music that follows contains almost no unisons at all.

An interval is a better test than a unison, by an exact factor, and the factor is the interval’s own upper term.

Where the coincidence is #

Two notes in the ratio p:q — the lower at f, the upper at f p/q — have their lowest coincidence at partial p of the lower and partial q of the upper, and both of them sit at p f.

For a unison that is f itself. For an octave it is 2f, for a fifth 3f, for a fourth 4f, for a major third 5f, for a minor third 6f.

A departure of ε cents on the upper note moves its partial by the same proportion, so the coincidence separates at p f times that proportion — which is to say the beat runs p times faster than a unison’s would for the same number of cents. Divide the detectable beat rate by that and the criterion improves by the same factor.

At A on a two-second note, taking one full beat cycle as what a player needs, the numbers are 1.97 cents for a unison, 0.98 for an octave, 0.66 for a fifth, 0.49 for a fourth, 0.39 for a third and 0.33 for a minor third.

This is the beating ladder’s own arithmetic read for a different purpose. That ladder establishes that any coincidence between two spectra generates a family of beats indexed by the multiplier, harmonic on an ideal string; here only the family’s slowest member matters, because it is the one that takes longest to show a cycle and therefore sets the criterion.

Which reverses what the ritual is for #

If a fifth is three times sharper a test than a unison, the tuning ritual is not the ensemble’s most accurate measurement of its own intonation. It is the least accurate one it makes.

That is not a criticism of the ritual, and stating why is the point of the rung. A unison is the only interval that can be checked against a single given reference — it is the interval that transfers a standard, and no other interval transfers anything, because a fifth from the oboe’s A puts a player on E and says nothing about where their A is.

So the ritual is doing a different job from the one the arithmetic evaluates. Its job is to move a number from one instrument to sixty, and the coarse criterion is the price of the transfer. Accuracy is bought back afterwards, in the playing, on every interval that is not a unison — which is nearly all of them.

Why string players check with fifths and nobody explained it #

There is a practice this predicts and it is universal enough that it is usually not remarked on.

A violinist does not tune four strings against a reference. They tune the A to the oboe and then set the other three by double-stopped fifths, bowing two strings at once and adjusting until the beat stops. Every teacher describes it, nobody offers a reason beyond convenience, and the convenience explanation is weak — playing two strings together is harder than playing one.

The arithmetic says the fifth is three times finer than the unison that produced the A it started from. So the instrument’s own internal tuning is set to about 0.7 cents while its relationship to the orchestra is set to about 2, and a violin is three times better in tune with itself than with the room.

The same picture says the E string is set more finely than the G. The E’s fifth sits on a coincidence at 1320 hertz and the G’s at 440, so the top of the instrument is tuned to about a third of a cent and the bottom to a whole one. That ordering matches the practice too — the E is the string a player is fussiest about — and it is usually attributed to the E being the audible one.

What the register does to a whole orchestra #

A coincidence is a frequency, so every criterion on the axis halves per octave, and an ensemble’s intonation is therefore held to a different standard in different registers.

A minor third at the top of the drawn range betrays a twentieth of a cent; a unison at the bottom betrays four. That is a factor of eighty between the easiest and hardest place in an orchestra to be out of tune, and it is entirely a property of where the coincidences fall rather than of anything about the players.

That is a strong claim and it has an obvious objection, which the next section takes up. Whatever it survives, its direction is the direction the practice already goes: an out-of-tune second violin in a high close-position chord is the standard example of an audible intonation failure, and a bass line’s octave is the standard example of one nobody hears.

The objection, which is depth #

Every criterion above is a rate, and the beating ladder spent five rungs establishing that a beat has a depth as well as a rate, and that a beat too shallow is not heard however slow it is.

Depth is where this argument is weakest, and it goes against it. A unison’s coincidence is between two fundamentals, which are the strongest components either note has, so the beat swings between near-total cancellation and near-doubling. A minor third’s coincidence is between partial six of the lower note and partial five of the upper, and on a string spectrum falling as one over n, those are twenty to thirty decibels down. A beat between two weak partials in the presence of two strong fundamentals is a small ripple on a large sound.

So the factor of p in the rate is bought with a loss in depth that also grows with p, and the two run against each other. The rate improves as p and the amplitude product falls roughly as one over p squared for a string-like spectrum — which would make the higher coincidences worse rather than better if audibility depended on depth alone.

What saves the argument is that the two quantities are compared against different thresholds and only one of them is a hard limit. A beat too slow to complete a cycle inside the note cannot be detected; a beat that is shallow can be, given enough level and attention, and the ladder’s own threshold curve puts a mistuned third’s family comfortably above detection in the middle of the keyboard.

The honest statement is therefore weaker than the drawing. The rate advantage is exact and the depth cost is real, and this rung has computed only the first. What the practice reports — that a fifth is easier to tune accurately than a unison, that a third is harder to tune than a fifth despite being sharper still — is consistent with the two effects crossing somewhere around the fifth, and computing where would need the depth threshold run over the coincidence family rather than over the fundamentals.

The interval a wind player has and a string player does not #

The register argument has a counterpart in the ensemble rather than in the compass, and it decides which players can use the fine criterion at all.

A coincidence needs both notes to have the partial the coincidence sits on. A minor third’s is at partial six of the lower note, and not every instrument has a sixth partial worth beating against.

A clarinet’s even partials are some twenty-five decibels below its odd ones, so a clarinet playing the lower note of a major third has almost nothing at partial four and its beat against the upper note’s partial five is between one strong component and one weak one — and the beating ladder’s own measurement of what unequal components do to a beat’s depth says such a pair barely modulates at all. A flute is three partials and stops. A stopped organ pipe has no even partials at all.

So the fine criterion is not distributed evenly across an orchestra, and it is distributed in the same direction the previous rung found the coarse one to be: the strings, whose spectra run to twenty partials or more, have every coincidence available, and the instruments with sparse spectra have only the low ones.

That is a second reason for a thing the practice already does. String sections are asked to tune more precisely than wind sections and are able to; and a wind player checking their intonation against a string player is using a coincidence the string supplies, which is why the check works in one direction better than the other.

The one interval that gets no help #

There is an exception in the table and it is the interval an ensemble most often has to place.

An octave is 2:1, so its coincidence is at partial two of the lower note — only twice as sharp a test as a unison, and the least improvement of anything on the axis. Every other interval does better.

That matters because an octave is what a bass line and a melody most often stand in, what a section doubling itself is playing, and what an orchestra’s outer parts do at almost every cadence. So the interval carrying the most structural weight in the texture is the second-worst measurement available, one step above the unison nobody plays.

It is also the interval whose just value is least certain. Listeners set the octave wide, by fifteen to twenty cents with pure tones, so a player nulling an octave’s beat is arriving at a 2:1 that a listener would call narrow — and the criterion says nothing about which of the two an ensemble should want. A coincidence has a zero; a preference does not have to be at it.

The two facts point opposite ways and the interval is common, which is why octaves in an orchestra are argued about and fifths are not.

What the pictures cannot show #

The one-cycle criterion is a convention with nothing measured behind it, and every number on this page is proportional to it. It is the same convention the previous two rungs used, so the ratios between intervals are safe whatever it is and the absolute cents are not.

Nothing here has a temperament in it. Every interval is drawn at its just ratio, which is where the beat actually goes to zero — and an orchestra playing tempered fifths has a beat at every fifth by construction, of about 0.6 hertz at A. A player nulling that beat is playing a pure fifth and is not playing in tune with a keyboard, which is the whole of the comma ladder’s subject and is why an ensemble with a piano tunes differently from one without.

There is also nothing here about which of the two players moves. A beat says that two notes disagree and says nothing whatever about which one is wrong — it is a symmetric measurement, and every correction it prompts is a negotiation between two people with no information about who should give way. The topology of the last rung decides that by convention, and inside a performance there is no convention at all. That is the whole subject of the rung after this one.

And a real interval is not two steady tones. Vibrato modulates both notes at five or six hertz, which is faster than every beat rate this page treats as detectable, and what a vibrato does to a beat is to smear it into something a listener may or may not be able to separate. A string section playing with vibrato has no clean beat anywhere in it, which bounds the whole argument to the instant of tuning and to the traditions that tune without it.

What this does to the eleven cents #

The two previous rungs left an ensemble at a band of about eleven cents and treated that as the answer to how well an orchestra is in tune. This rung says it is the answer to a different question.

Eleven cents is the width of the distribution of where each player’s A sits, measured at the moment the tuning stops. It is the accuracy with which a standard has been transferred, and nothing in it is about how the ensemble sounds.

What a listener hears is chords, and every chord is a set of coincidences each of which is a finer test than the transfer was — a whole family of them per interval, of which only the slowest sets the criterion. Two players eleven cents apart at A, playing a fifth, produce a beat at three times the lower note — 3.9 hertz on a fifth from A3 — which is a fluctuation nobody in the room could miss and which both of them will remove within a bar.

So the transfer’s band is not the performance’s band, and the performance’s is narrower. That is the honest reading of three rungs’ arithmetic put together, and it says the tuning ritual is a starting condition rather than a specification: it puts sixty players close enough that the fine criteria can take over, and then the fine criteria do.

The reverse reading is the one to be careful about. A band of eleven cents would be inaudible in unisons — nobody plays alone with the oboe — and is very audible in thirds, because a third’s coincidence multiplies it by five. An ensemble that never corrected after tuning would be worst wherever its harmony was richest, which is not where an account based on the transfer would predict.

Whose practice this describes #

The claim is about ensembles that tune by ear on sounding notes: orchestras, string quartets, wind bands, and any tradition in which two players sound together and adjust. It excludes anything tuned to a fixed instrument in advance, and it excludes singing without accompaniment, which has the notes but not the reference.

The prediction with the most content in it is the register one. If intonation errors are held to a criterion that halves per octave, then an ensemble’s measured pitch scatter should be smaller in the treble than in the bass, in cents, by about a factor of two per octave — which is the reverse of what a naive account predicts, since a bass instrument’s own pitch is the hardest thing to control.

That is a measurement anybody with a recording and an analyser could make. It has not been made here and it is not, as far as this collection knows, in the literature; what is there is the observation that bass intonation is forgiving, which is usually explained by the ear’s poorer frequency resolution at low pitches and would be explained equally well by this. The two accounts are separable, and cheaply: the frequency-resolution account predicts the forgiveness follows the critical band, which widens by a factor of twelve from the top of the compass to the bottom, and this one predicts it follows the coincidence frequency, which changes by a factor of eighty over the same range. A measurement of intonation scatter against register would tell them apart on the exponent alone.

The ladder from here #

Thirteen rungs. The last three take a standard out of a document and put it in a room: it reaches an ensemble through a limen, through a graph, and then through intervals that are finer tests than the one it arrived on.

All three describe the ritual and the minutes after it. What none of them touches is the hour that follows, and the hour is where a performance’s pitch actually lives. Once the oboe has stopped, no reference exists: each player corrects toward what they hear around them, which is other players correcting toward them. That is a consensus with no anchor in it, it has a fixed point at every common value, and what it does to an ensemble is not what anybody assumes.