The Immutable Past · What Is
What Is a Compromise
A compromise is pictured as a point on a line between two positions, reached by each side giving up a proportion of what it wanted. The foundation I work from says there is no such line. Two conditions are two complete forms, not two ends of one scale — so a settlement is a single mark serving one of them, and the shortfall against the other is placed rather than paid off.
The number that is settled before the tuner arrives
23.46 cents
Not a limit of craftsmanship, and not a manufacturing tolerance. A consequence of 312 ≠ 219.
Twelve pure fifths stacked up overshoot seven pure octaves by the ratio 531441:524288, which is 23.46 cents — the Pythagorean comma. No tuning of twelve notes has removed it, because removing it would require a whole number of pure fifths to equal a whole number of pure octaves, and unique factorisation forbids that. What a temperament decides is not how much shortfall there will be. It is where to put it.
A compromise is rarely announced with pride. It is the word we reach for when an outcome has to be defended rather than reported — we met in the middle, we split the difference, half a loaf, both sides gave something up. Underneath that vocabulary is a picture so ordinary it is seldom stated: two positions at either end of a line, two sides walking toward each other along it, and a settlement somewhere in between. The measure of the result is how far each side had to travel. The virtue of the result is that the distances were roughly equal.
One neighbour named up front so the two do not get merged. I wrote a piece yesterday on a tie, and it also concerns two things and a rule. That piece runs on the compass — on whether a symmetry broke and whether an ordering is present — and its subject is what gets recorded when a contest hands back a level result. This one runs on exemplarity, and its subject is what happens to two complete forms when one mark has to answer to both. The foundation is the same and almost none of the machinery carries over.
What the ordinary picture commits you to
State it strongly. The weak version is not what anybody acts on.
Three commitments. First, that the two positions are degrees of one quantity — otherwise there is no line for a midpoint to sit on. Second, that the settlement is a diluted version of both, holding some fraction of each. Third, that the measure of a compromise is therefore distance travelled: how much each side conceded, and whether the concessions were balanced.
Three cases do not fit inside those commitments. The first comes from arithmetic, the second from a room full of people with no interest in metaphysics, and the third from a management theorist who reached this boundary from the other side a century ago.
Case one · the shortfall that is fixed before anyone negotiates
Two intervals in music are defined by exact whole-number ratios. The octave is 2:1. The perfect fifth is 3:2. Neither is an approximation of the other and neither is a degree of the other; they are two exact things.
Now stack twelve fifths and seven octaves from the same starting note. Twelve fifths multiply out to 312/212; seven octaves to 27. For those to land on the same note you would need 312 = 219 — 531,441 = 524,288. They differ by about one part in seventy-three, which is 23.46 cents, a little under a quarter of a semitone. This is the Pythagorean comma, and no improvement in instrument-making touches it, because the obstruction is that a power of three is not a power of two.
So a keyboard has to be tempered, and the interesting part is what the choices look like. Equal temperament keeps the octave exactly pure and narrows each of the twelve fifths by 1.955 cents — a twelfth of the comma each — so that the cycle closes. The octave is left exactly where it was, and the whole discrepancy is loaded onto the fifths. Quarter-comma meantone, the tuning of much of the sixteenth and seventeenth centuries, makes the opposite election: it keeps the major third exactly pure at 5:4, narrows eleven fifths by about 5.38 cents each to do it, and lets the remainder collect in the twelfth. That last interval comes out at about 737.6 cents against a pure 702 — roughly 35.7 cents wide, which is audible from across a room. It is called the wolf.
Both are compromises in the literal sense and neither is a midpoint. Each one keeps a form exactly and places the shortfall somewhere. They differ in which form is kept whole and where the residue is banked — spread thinly across twelve intervals, or concentrated in one and then avoided by composers for two centuries. The first commitment says the positions are degrees of one quantity. Nothing here is a degree of anything. The second says the settlement dilutes both. Equal temperament dilutes the octave not at all.
Case two · the famous compromise that refused to average
In the summer of 1787 the Constitutional Convention nearly came apart over representation. The Virginia Plan, drafted by Madison and introduced on 29 May, proposed a two-house legislature in which the weight of each state in both houses was proportional to its size. The small-state delegates would not have it. William Paterson’s New Jersey Plan, presented on 15 June, proposed a single house in which each state had one vote. On 2 July a vote on equal representation in the upper house came back tied.
The arithmetic middle was available and obvious. Give every state a floor and a population component in one chamber. Weight the votes. Any number of formulas would have produced a single house in which a large state counted for more than a small one but less than its population, and each side could have reported how far it had moved.
That is not what was adopted on 16 July, by a single vote. What was adopted gave the Virginia Plan’s principle an entire chamber, undiluted — seats in the House apportioned by population — and gave the small-state principle an entire chamber, undiluted: two senators per state, Wyoming equal to California. No state received one and a half senators. Neither principle was reduced by a percentage. Each was instantiated completely, in its own house, and the settlement was the construction of a second place to put one.
The second commitment says a compromise holds a fraction of each position. The most consequential compromise in American constitutional history holds all of one and all of the other, and the shortfall shows up somewhere else entirely — in a two-century argument about whether a Senate majority represents a population majority. It did not vanish. It moved.
Case three · the manager who noticed the word was doing two jobs
Mary Parker Follett, lecturing to personnel administrators in 1925, separated three ways of ending a conflict: domination, compromise, and what she called integration. Her own example is a room in a library. Somebody wanted the window open; she wanted it shut. They opened a window in the next room instead, where nobody was sitting. Her gloss is the part that matters: this was not a compromise, she says, because there was “no curtailing of desire” — both parties got what they actually wanted.
Read that carefully. The two stated positions — open, shut — do look like two ends of one line, and on that line there is a midpoint: a window slightly ajar, which would have suited neither of them. What Follett noticed is that the positions were standing in for two different requirements, air and no draught, and that once those were named the line disappeared. She was not proposing a better bargaining technique. She was pointing out that the line was an artefact of how the disagreement had been described.
The third commitment says the measure of a compromise is distance travelled. In Follett’s case the distance travelled was zero and the outcome was better than any midpoint. Whatever is being measured, it is not that.
So the two sides are not reliably degrees of one thing, the settlement is not reliably a fraction of each, and the distances people moved are not the measure of anything. Something else is going on, and the rest of this is about what.
What the foundation says instead
I work from a short list of numbered primitives and the results derived from them. The one doing the work today is the tenth, and it is the thinnest element in the whole document — cited by four published pieces, and in two of those it was the subject rather than the tool.
P10, exemplarity: every condition has an exemplar — a complete and exact form against which any artifact can be measured. One clause, and the clause I need is the one that gets skipped: the exemplar belongs to a condition, one each. The primitive is distributive. It hands out a complete form per condition and says nothing about arranging them.
Two supports. P6: a condition is a prerequisite for something to happen or exist. R10: every artifact is a miss — the mark falls short of the form, by construction and not by misfortune. And the locked definition of an artifact, which is a mark carrying a magnitude and a phase: a magnitude for how far the actual departed from what was predicted, and a phase for which condition it serves.
Now put P10 and P6 together on a case with two conditions in it. Two conditions, two exemplars. Two complete and exact forms, each entire, neither one a diminished version of the other and neither one a position on a scale that runs to the other. There is nothing between them, because between is a relation that needs a line, and a line is what a scale supplies. The primitive supplies no scale. It supplies a form apiece.
Figure 01
What the ordinary picture puts on a line, the primitive hands out one apiece
- Condition A The octave: 2:1, exactly. A complete and exact form. Not 90 % of anything and not an approximation of a fifth. A mark either matches it or falls short of it, by a readable amount.
- Condition B The fifth: 3:2, exactly. Also a complete and exact form. Not a degree of the octave; the two are independent. A mark either matches it or falls short of it, by a readable amount.
Why there is no midpoint to find
The ordinary picture needs the two positions to be commensurable — comparable in degree, so that a proportion of one can be traded against a proportion of the other. A scale would deliver that, and so would a single quantity that both sides were asking for in different amounts. Sometimes there is one: two people arguing about a price are arguing about one quantity, and the midpoint is there to be found.
But when two conditions are in play, the scale is not there to be found, and looking harder for it is not a method. That is the whole of the first result, and the ordinary vocabulary obscures it because a percentage can be computed between any two numbers whether or not the numbers are on a common scale. Three-fifths of a population is an arithmetically impeccable operation. Arithmetic will produce a number between two things that have no between.
First marked construction
Two conditions are two complete forms, not two ends of one scale. A person standing between them is standing between two complete forms rather than at a degree of one thing — so there is no midpoint to find, and a settlement is not a fraction of either.
Consequence of P10 with P6. Mine: the primitive says every condition has an exemplar and says nothing about what it is to stand among several. I am reading the distributive clause as load-bearing, and that reading is the construction. Note the boundary the foundation puts on this and I am keeping: it records as open whether conditions are abstract objects, and nothing here is an answer to that. The exemplar is what an artifact can be measured against. What it is made of is not settled and I am not improvising it.
Arithmetic will produce a number between two things that have no between.
A settlement is one mark, and one mark has one phase
Here is where the framework does something the ordinary picture cannot. An artifact carries a single complex value: one magnitude and one phase. The phase is which condition the mark serves, and it is relational — fixed by which idea has the actualizer, not by the mark itself and not by the person who made it.
A signed settlement is an artifact. It is one mark. So it has one phase, which means it serves one condition, and its fidelity is read against that condition’s form. It is not two half-marks stapled together, and there is no operation in the algebra that makes a mark half-face one form and half-face another.
This sounds abstract until you notice what it rules out. It rules out the comfortable thought that a compromise is partly faithful to both sides — the thought that licenses the phrase we each got half of what we wanted. What the machinery gives instead is a mark aimed at one condition, at some fidelity, plus a shortfall against the other condition that is not reduced by the mark existing.
Equal temperament is the clean demonstration. It is not slightly faithful to the pure fifth. It serves a different condition entirely — a cycle of twelve notes that closes and can be played in any key — and it serves that one exactly, with no residue at all. The fifth’s shortfall of 1.955 cents is not a partial success at being a fifth. It is the price of a mark that was aimed at something else.
Second marked construction
A compromise is one mark with one phase, not two half-marks. It serves one condition and is read against that condition’s form; the shortfall against the other condition is placed somewhere, and it is not reduced by the agreement.
Consequence of the artifact algebra — a mark carries one magnitude and one phase, and the phase is fixed by which idea has the actualizer — with R10 and R11. Mine: the application to a negotiated settlement, and the claim that the condition a settlement serves is characteristically a third condition rather than either party’s. The foundation gives the algebra and does not name which ordinary objects are single marks; that a treaty or a constitution is one artifact rather than many is my reading and a reader could contest it.
Figure 02
Four settlements, what each kept exact, and where the shortfall was placed
| The settlement | Forms in play | Kept exact | Where the shortfall went |
|---|---|---|---|
| Equal temperament | The octave at 2:1; the fifth at 3:2; a cycle of twelve that closes | The octave, and the closing cycle | Spread thinly. Each fifth narrowed by 1.955 cents, one twelfth of the comma. Major thirds land 13.69 cents sharp of 5:4, a residue that falls out rather than being elected |
| Quarter-comma meantone | The same three, with the major third at 5:4 added to the list | The octave, and the major third | Concentrated. Eleven fifths narrowed by about 5.38 cents; the twelfth left about 35.7 cents wide — the wolf — and then avoided |
| The Connecticut Compromise, 16 July 1787 | Representation proportional to population; equal standing of states | Both, each in a chamber of its own | Displaced into the structure. Not into either principle but into their combination, and argued over ever since |
| Follett’s library window | Air in the room; no draught on the person sitting there | Both, by opening a window elsewhere | Not applicable — and that is the finding. There were two conditions and one description that had merged them; renaming them removed the apparent conflict |
What can be conceded, and what cannot
The second result comes from two more primitives, and it is the one I would keep if I had to keep one.
P7: an idea is a named condition, and naming does not create it — naming makes the relationship conscious. P8: a condition is prior to what it conditions, and where a person’s naming is itself conditioned, that person is not the condition’s origin. Ideas have people; people do not have ideas.
Put those against a negotiating table. What sits on the table is a set of stated positions, and a position is a naming. What the naming points at is a condition, which was not authored by the person naming it and is not in that person’s gift. So a party can concede a position. A party cannot concede a condition, because it did not hold one to begin with.
This is why the shortfall survives the handshake. When two sides agree to load the discrepancy onto one of them, what has changed is which position is being asserted. The condition is where it was. The octave does not become 1.998:1 because a tuner elected to bend the fifths, and a principle does not stop conditioning what it conditions because a committee voted it down. The agreement is a mark with genuine standing — it settles what will be done — and it is not a finding about the form that was missed.
And because the circumference of nameable conditions is not closed by any enumeration, these two options is itself a naming rather than an inventory. That is what Follett walked into. The two positions on her table were a finite naming; the thing that ended the argument was naming a third condition, which had been available and unnamed the whole time. I am carrying that principle over from earlier pieces rather than deriving it again: what a host can work with is what has become readable, and the named conditions at any moment are finite while the nameable ones are not.
Which gives the practical form of the whole argument. Before splitting a difference, it is worth asking whether the difference is one quantity or two conditions wearing one description — because the answer determines whether there is a midpoint to walk to or a third form to go and name.
Compromise, corruption, and settling are three different things
The English word does two jobs and the framework separates them cleanly, so it is worth doing explicitly. A compromise is a settlement between conditions. To compromise oneself is something else: it is phase drift, an artifact that has rotated away from the condition it was made for and toward the maker’s own interest, while the maker continues to describe it in the old vocabulary. The locked term for that is corruption, and its distinguishing feature is not the size of the concession but whose interest fixed the phase.
And a third thing again is settling, which the framework treats as the one sin available: ceasing to correct. Not low fidelity — low fidelity under continuing correction is called approximation and is the ordinary condition of everybody working at anything. The difference between approximation and settling is direction of travel, not distance from the form.
So three questions, and only the last is about the size of what was given up:
- Whose interest fixed the phase? If the mark was aimed at a condition, it is a compromise, however much was conceded. If it was aimed at the maker and described as the condition, it is corruption, however little was conceded.
- Which direction is it travelling? A settlement you expect to revise is approximation. A settlement you have stopped reading against the form is settling.
- How far is the mark from the form it serves? Third, and the only one the ordinary conversation ever asks.
A person can concede a great deal and be doing fine work, and a person can concede almost nothing and have quietly changed what the work is for. The ordinary conversation about selling out gets this backwards roughly as often as it gets it right, because it measures the concession.
What it costs
Three things follow that are less comfortable than the argument that produced them.
The reassurance is gone. If a settlement is one mark with one phase, then we each got half of what we wanted is not a description of anything, and the feeling that a fair compromise has discharged both claims has nothing behind it. What there is instead is a mark serving one condition and a shortfall against the other, sitting wherever it was placed. Agreement does not retire it. That is why the observation that a good compromise leaves everybody unhappy is accurate rather than cynical — and the framework says why without anybody having to be unreasonable. R14: every idea is biased toward itself, orientation and prejudice arriving together. Each party reads the mark against the form its own orientation brought to the room, and each of those readings is correct about its own condition and blind in the same motion to the other. Two accurate readings, both of them partial, and no view from nowhere that would adjudicate them.
The settlement did not find the answer. R11: the evaluative question is fidelity, not success. R10: every artifact is a miss. A compromise is an artifact like any other — and so, for that matter, is the procedure that produced it. R12 keeps the people who made it accountable for the mark they made, which is the correct location for the accountability and not a comfortable one, because it does not permit the process decided. A process is a mark somebody made too.
And the thing I most want to claim, I cannot. The music case gives an exact figure for the shortfall in each direction, in cents, and the temptation is to generalise: every compromise has a measurable residue, and the design freedom is over placement rather than magnitude. The first half of that is not available to me. Cents are a counter in the present — a logarithmic scale somebody built, applied to ratios — and the foundation does not supply a magnitude for the distance between a mark and its exemplar. It gives an artifact a magnitude for surprisal, which is a different quantity measuring a different thing. Whether the shortfall against an exemplar carries a magnitude or only an order is an open question in the document and I am leaving it open.
The open question, and what closing it would take
- The question
- Does the shortfall of a mark against its exemplar carry a magnitude, or only an order? The tenth primitive gives a form to measure against. It does not give a metric on the measuring.
- Why this piece runs into it
- The tuning case has exact numbers — 1.955 cents against the pure fifth, 13.69 against the pure major third — and those numbers are doing work in the argument about placement. But they come from a counter, not from the primitives: cents are a present instrument applied to present ratios, and importing them into the general claim would be borrowing a magnitude the foundation has not issued.
- What closing it would take
- An expression in the primitives that yields a magnitude for the distance between an artifact and its exemplar, without routing through a counter supplied by the present and without collapsing into surprisal, which measures the departure of an actual from a prediction and not the departure of a mark from a form. I do not have one.
- Status
- Open, and the argument above is written so that it survives either answer. The claim that a shortfall is placed needs only that there is a shortfall in each direction and that a settlement can concentrate or spread it. It does not need the shortfalls to be commensurable, and I have not assumed they are.
Third marked construction
The freedom in a compromise is over placement rather than over whether there is a shortfall. Which forms are kept exact, and which absorbs the residue, is a decision somebody makes; that a mark answering to two forms misses at least one of them is not.
Provisional, and the provisionality is the honest part. The second half follows from R10 with P10 and is safe. The first half is stated in the tuning case with exact numbers and is generalised here without a metric behind it, per the open question above. Read it as a claim about structure — there is a choice about where — and not as a claim that the amounts in different directions can be added, compared, or traded.
The compromises that were not negotiations
Most of what people call a compromise in their own lives had nobody on the other side of the table. A career shaped around an illness, a city chosen for somebody else, the version of the work that fitted the hours that were actually available. The argument above says something narrow about those and I want to be exact about how narrow. It says the shortfall is there and is not reduced by having agreed to it, which is a harder sentence than the usual reassurance. It also says the thing you settled for is a mark serving a condition, at whatever fidelity, and is not a fraction of a life you failed to have — because there was no scale running between them for a fraction to be taken on.
That is not offered as consolation, and I am not reaching for the balance in this framework to soften it: that ledger is not invoked here, and nothing outside the present is keeping accounts on anyone’s behalf. Understanding the structure does not make the shortfall smaller and is on no schedule. If this is a loop — running many times an hour, not stopping when you are busy, arriving in the small hours — that is work for a person and the person should be a clinician. A metaphysics can take a false weight off a reading. It cannot treat a mind that is grinding, and it must not be offered as though it could.
On a Tuesday
- Before splitting the difference, write down what each side’s complete version would be. If you can state both forms and they are not degrees of one thing, there is no middle and you are choosing which one to serve. If you cannot state them, you are haggling over a number, which is a fine thing to do and a different thing.
- Say out loud where the shortfall is going. Equal temperament and meantone are the same decision made two ways: which form stays exact, and which interval absorbs the rest. If you do not name the interval you are loading, you have loaded one anyway, and somebody finds it later.
- Spend one pass looking for the third condition before you start bargaining. Follett’s window took a sentence. The two positions on the table are a naming, not an inventory, and the question is whether the description has merged two requirements that could both be met.
- Judge a compromise by direction of travel, not by distance. The useful question about a concession is whether the mark is still being read against the form it serves. A large concession under continuing correction is approximation. A small one you have stopped reading is the other thing.
None of which is fatalism, and the foundation is careful about the difference. P5: the host can occupy a position from which its response to a readable condition is revisable. R13: agency is relocated, not abolished, and the host is the site of conscious responsibility. Which form a settlement keeps exact, and where it puts the rest, is a decision people make in rooms — and it stays a decision right up until the mark is made. Sherman and Ellsworth had the averaging move available and did not take it.
A compromise is not a point between two positions, because two conditions do not lie on a line. It is one mark, serving one form, with a shortfall against the other that somebody decided where to put. Say where you put it.