Part B - Translate, combine and improve notations

NOT.5 - Translate between Notations while Tracking Lost Distinctions

Type: Method Status: Stable Normativity: Normative

NOT.5:1 - Problem frame

Use this when work must continue in another notation and you need to construct or repair the translation. A diagram may need a textual form, an ordered performance may need a compact score, or an expression may need a representation accepted by another reader or tool.

Start with the operation the receiving notation must support. Construct its expression from the source’s interpreted parts and connections, then use it to obtain the needed answer. Find a pair of source expressions that the translation would make indistinguishable. If the work needs their difference, retain it, enrich the target or restrict the translation’s use.

The first result is a usable translation with an answer-recovery procedure, or a demonstrated loss and a concrete way to repair it. Recovering a needed answer, reproducing the source expression and carrying a target edit back are different demands; choose the ones the work actually has.

The reader needs the two notations’ formation and interpretation rules, supplied by NOT.2 and NOT.3 where they are missing. Subject knowledge establishes which consequences matter. The examples need directed connections, ordered sequences and elementary arithmetic.

Use an established translation directly when its conditions meet the need. NOT.4 handles transformations within one notation. NOT.6 handles continuing coordination of several representations and their changes. CMP.12 develops an effective translator and its execution-preservation argument when software must perform the translation. A.6.3.RT governs the underlying representation change and the claims it can preserve.

NOT.5:2 - Problem

How can a change of notation make the next operation possible while keeping its answer connected to the source and exposing distinctions that can no longer be recovered?

Two source expressions can produce the same target expression. The target may still answer a useful question, but a reverse converter cannot infer which source was used from that target alone. A familiar default can conceal this loss. Conversely, requiring complete reversibility can exclude a cheap summary that is sufficient for the actual question.

A target edit adds another difficulty. There may be several source changes consistent with it, each preserving different features of the earlier source. Successful conversion in one direction does not choose among them.

NOT.5:3 - Forces

NOT.5:4 - Solution

Choose the receiving operation → construct the translation → obtain and return an answer → expose collapsed distinctions → repair the needed loss → establish the required return.

NOT.5:4.1 - Choose what must remain possible

Name the source notation, target notation and intended reader operation. Specify the source expressions admitted for this use and the context required to interpret them. A drawing may use position as meaning or only as layout; a spoken sequence may rely on a separately maintained beat. Carry the context that the question uses.

Decide what the recipient must return: a value, a construction, an explanation, a reconstructed expression or a proposed change. These demands can require different information. An answer expressed in target units or names needs a way back to the source question.

If a source choice is unresolved, preserve the alternatives or the unresolved status that matters to the question. A default introduced by the translation is an added decision. Accept it only when making that decision is part of the intended work and its basis is available.

NOT.5:4.2 - Construct target expressions from interpreted source structure

For every kind of source part used by the operation, give its target construction. Then give a rule for translating their composition. Preserve operand order, connections, scope and reference where the interpretation uses them. Copying labels does not supply those rules.

Translate an expression by following its construction. For a diagram, this can mean declaring its nodes and translating each directed connection; for a sequence, translating each event while retaining its position or duration. Derive the target composite from those contributions. NOT.2 supplies the formation and reference rules; NOT.3 supplies the reading operations.

When a source ingredient has no target counterpart, choose an explicit treatment. Extend the target, carry an annotation, leave a visible unresolved part, or limit the admitted source expressions. Make this choice from the receiving operation. Silently deleting the ingredient or guessing its counterpart leaves the recipient unable to locate the loss.

For a mathematical interpretation, MATH.18 develops preservation and reflection of the required constructions and assertions. For a bijective change of mathematical representation, MATH.7 transports the operations through the inverse. Use those arguments when their conditions hold; a correspondence between printed signs alone does not establish them.

NOT.5:4.3 - Perform the receiving operation and recover its answer

Translate a small source expression, perform the intended operation in the target and interpret the result as an answer to the original question. Show the return explicitly. If the target gives a list of numbered nodes, say which source nodes those numbers refer to. If it supplies only a bound or a set of possible answers, retain that limitation on return.

Compare this result with what follows from the source under its declared interpretation. A mismatch locates a failed translation rule, a missing premise or a target operation that answers a different question. Repair the affected correspondence before relying on that answer.

A target-side answer can be spurious for the source if the target admits additional possibilities. Retain the source restriction or construct the needed reflection argument. For example, allowing arbitrary real values does not preserve a source question that admits only whole counts.

One successful expression establishes that case. A reusable translation needs an argument for its admitted family, such as a rule-by-rule construction over expressions. A separating example can refute a proposed general claim. Additional checking is warranted when its result can change the use or repair of the translation.

NOT.5:4.4 - Expose distinctions the target collapses

Look for different admitted source expressions with the same target expression. Ask whether the required operation gives different answers on them. If it does, the target alone cannot determine that answer: the translation has removed something the work needs. Section :5.2 makes this failure visible through cue order.

If every collapsed pair gives the same needed answer, the loss need not obstruct that question. For a claim about the whole admitted family, justify this independence over that family rather than infer it from a few pairs. MATH.2 supplies the mathematical construction through equivalence classes when that form is useful.

Include differences in assumptions, unfinished choices, references and reading context when they can change the answer. Equal visible strings can have different interpretations under different contexts. Conversely, two differently laid out diagrams can express the same directed structure when position has no role in their interpretation.

For several translation stages, follow the information needed by the final operation through each stage. A later, richer format cannot recover a distinction that an earlier stage removed unless another input supplies it.

NOT.5:4.5 - Repair the loss needed by the work

Choose the least burdensome repair that actually restores the operation. Retain the original expression when an occasional return is enough. Add the missing distinction to the target when the recipient must work independently. Otherwise, carry supplementary information with a defined recovery rule.

Call that supplementary information a complement when the target and the supplement together permit the required source recovery. State what it contains and how the return uses it. In :5.2 the position of a cue complements its count; in :5.3 a retained phase duration selects one backward update. A statement that the conversion is reversible cannot replace this construction.

Keep the target and its supplement associated with the same source. If the target is edited, check whether the old supplement still permits the intended reconstruction. A stored position can become invalid after deleting an event. Locate that conflict instead of returning an invented original.

The repair may instead be an explicit narrower use. A cue count remains useful for inventory even when it cannot recover order. Retain that useful result and identify the additional contribution needed by an order-sensitive question.

NOT.5:4.6 - Establish the return that the work requires

For an unchanged round trip, translate and reconstruct the source at the required level: the same expression, the same interpreted structure or the same needed consequence. State which level is obtained. Recovering a directed graph up to layout does not recover where its boxes were drawn.

For a target edit, construct a backward update using the edited target and retained source information. Specify what remains fixed and what may change. In the ordinary single-view setting, check two properties: returning an unchanged view leaves the source unchanged; translating the updated source yields the requested view. These properties still leave a choice of update policy, worked in :5.3. They do not mean that all possible edits must be accepted.

When several views constrain a shared source, distinguish the user’s changed requirement from values merely copied from the earlier view. NOT.6 handles their propagation and possible conflict. An unchanged value in a submitted view is not necessarily an instruction to freeze it.

If conversion performs effects such as asking a reader for missing information or changing external state, include the relevant behavior in the return claim. Recovering the same output value need not undo those effects. CMP.12 supplies the computational correspondence when an executable implementation is required.

Stop once the chosen operation and required return work within their stated conditions. Retain the translation rule, any supplement and the loss that changes later use where the recipient can find them. Reopen the affected construction when the receiving question, source interpretation or admitted edits change.

NOT.5:5 - Archetypal Grounding

NOT.5:5.1 - Translate a dependency drawing without inventing geometric meaning

A drawing has named nodes A, B, C and D. An arrow X→Y means that Y takes X’s result as its direct input. Box positions carry no meaning. In one drawing the arrows are A→B and B→C; D is isolated. The receiver must determine C’s direct input using ordinary text.

Construct this textual notation:

nodes: A, B, C, D
direct-input pairs (supplier, receiver): (A, B), (B, C)

The reading rule selects the pair whose receiver is C and returns its supplier, B. Translating back draws one node per declared name and one arrow per pair. This recovers the named directed structure, including isolated D. It need not recover the original layout.

A list of node names alone fails. The second drawing with A→B and A→C would give the same list but a different answer for C. Retaining the pairs repairs that loss. Retaining only pairs would create a different loss: isolated D would disappear from a reconstructed drawing.

Now suppose horizontal position additionally records a planned start time. The earlier translation still answers the direct-input question, but it cannot reconstruct that timing. Add a start-time value for each node if timing is now needed. That extension follows a changed interpretation or question; it is not evidence that the original drawing already specified times.

NOT.5:5.2 - Keep a useful count without pretending it preserves a rhythm

A toy score is a sequence of four equally spaced cue positions. D and T name two distinct sounds; there are no rests. Translate a score into the counts of D and T. Both D T D D and D D T D become D:3, T:1.

The target answers how many sounds are required: four. It cannot answer which sound occurs second. In the first score it is T; in the second it is D. No reverse choice made from the counts alone can recover both originals.

For this family with one T, retain its position as a supplement. Counts plus T-position:2 reconstruct D T D D: place T at position 2 and fill the remaining positions with D. This permits both the count and the second-cue question. If the score may contain several T cues, retain their positions, check that they are distinct and within the score, and check that their number agrees with the count. If rests or unequal intervals become relevant, the chosen description needs further distinctions.

The counts remain a sufficient result for inventory. Reconstructing performance order requires the supplement; reconstructing tempo would require a time scale that this toy score never supplied.

NOT.5:5.3 - Choose what a backward update preserves

A plan describes two successive phases by [A:3; B:5], in whole minutes. A summary notation shows only their total, 8. The recipient changes that total to 10. Several source plans fit: [A:3; B:7] and [A:5; B:5] both do. The new total does not choose between them.

Suppose the work requires A to remain fixed. Retain A’s duration, 3, and reconstruct B as new total - 3. The requested total 10 gives [A:3; B:7]. Returning the unchanged total 8 recovers [A:3; B:5]; summing the updated plan returns 10. If instead B must remain fixed, the other update policy gives [A:5; B:5].

Assume phase durations must be nonnegative. A requested total of 2 is incompatible with retaining A at 3. Report that conflict and obtain a changed requirement if the work allows one. Substituting B = -1 would preserve the sum while violating the admitted source plan.

NOT.5:6 - Bias-Annotation

A familiar target can look more definite than its source. Preserve the source’s unresolved choices and restrictions where they affect the receiving use. A translator’s chosen default is additional content.

Conversely, fear of information loss can make every conversion retain a complete original. Start with the actual question. The cue inventory needs counts; an editable score needs more. Retain the extra information when it opens a needed operation.

NOT.5:7 - Conformance Checklist

When the translation needs checking, use the questions relevant to its declared use.

• Can the recipient construct the target from the source parts, connections and required context?
• Can a target result be returned as an answer to the source question at the stated strength?
• Do two collapsed source cases require different answers for any promised operation?
• Does a necessary supplement have a usable recovery rule and remain associated with the right source?
• Is the round-trip claim about expressions, interpreted structures or selected consequences?
• For an edited target, is the backward policy explicit, including incompatible edits?
• Does a changed question or translation stage require a distinction already discarded?

NOT.5:8 - Common Anti-Patterns and How to Avoid Them

NOT.5:9 - Consequences

The recipient gains a usable expression and can tell which answers, reconstructions and changes it supports. A loss becomes a reason to choose a supplement, a richer notation or a narrower use instead of an unexplained conversion failure.

Retained information has an access and maintenance cost. A more restrictive reverse policy can protect a source feature while excluding useful edits. These trade-offs remain choices about the work; a round-trip law alone does not settle them.

NOT.5:10 - Architectural Rationale

Translation changes the means through which an operation is performed. Its success therefore depends on both interpretation and the receiving operation. Two expressions may be interchangeable for counting and different for performing an ordered sequence.

The constructive loss test gives a local reason for a limit: when one target stands for two sources with different required answers, the target cannot by itself select the answer. Supplementary information repairs that failure only through a defined recovery operation.

Backward change is a further construction. Keeping one source feature fixed and changing another can realize the same target edit as a different policy. This is why two-way translation needs its own use conditions, and why continuing coordination of shared views belongs with NOT.6.

NOT.5:11 - SoTA-Echoing

Choosing the translation method. A bijection with transported operations is a strong choice when both notations express the same required information and a suitable inverse exists. It unnecessarily restricts a useful summary that deliberately omits detail. For that case, construct the receiving answer and retain only the information needed by the required return. A lens-style backward policy becomes useful when edits must propagate; it adds conditions and design choices that a one-way question need not carry. The pure view-and-complement account remains adequate for pure single-view conversions. Effects or interacting views select the stronger lines above when those difficulties actually occur.

This synthesis adapts those computational constructions to notation design by making the reader’s operation determine the needed recovery. It does not claim that every diagram, performance or human interpretation already has an effective bidirectional implementation. Reopen the chosen method when a previously harmless loss obstructs a new question, an edit violates its preserved feature, or conversion effects change the relied-on result.

NOT.5:12 - Relations

NOT.5:End

NOT.6 - Coordinate Complementary Representations through Shared References

Type: Method Status: Stable Normativity: Normative

NOT.6:1 - Problem frame

Use this when different representations support different parts of the work, but a reader cannot reliably combine their results or carry a change between them. A formula can support derivation, a table direct lookup and a diagram recognition of a relationship. Their usefulness together depends on knowing what corresponds and which claims each supplies.

Start with one operation that needs contributions from two representations. Identify the objects, quantities or relations those contributions concern. Construct the correspondence, perform the combined operation, then follow one relevant change through it.

The result is a usable combination: readers can obtain the joined answer, locate a disagreement and update the affected contributions. A discovered incompatibility can instead identify the subject decision that must be made before combining them.

The reader needs the interpretations of the participating representations and the subject relations used to connect them. NOT.2 and NOT.3 supply missing formation and reading rules; NOT.5 supplies a needed translation. A name or explicit correspondence beside the two representations can suffice for a small use.

Keep a single adequate representation when another adds no useful operation. Use NOT.5 for a one-time translation whose target alone supports the remaining work. This method is for continuing joint use, including representations that cannot be fully translated into each other. When different uses of a working method need complementary descriptions, C.37.MP relates the claims those uses require; this construction supplies their notational connections.

NOT.6:2 - Problem

How can several representations jointly support reading, reasoning and change without requiring every reader to reconstruct their correspondence or silently reconcile their disagreements?

Side-by-side placement leaves that work unresolved. Equal names can denote different things; one thing can have different names. A table cell may be calculated from a model or measured independently. A displayed value may be old context or a constraint that a proposed edit must preserve. These differences determine which results can be combined and what should change next.

Forcing all representations into one form can remove the different operations that made them useful. The task is to construct the necessary connections while retaining those useful differences.

NOT.6:3 - Forces

NOT.6:4 - Solution

Choose complementary operations → recover referents → construct correspondences → perform a joined reading → propagate the intended change → retain the useful combination.

NOT.6:4.1 - Select representations for the operations they support

Name the working question and the contributions needed to answer it. Use each representation for an operation it actually supports. A table may expose a recorded value, a formula generate values outside the table, and a graph reveal a shape under a declared scale. State what the reader obtains from each.

Keep a representation because it adds a needed distinction, constrains a plausible misreading or enables a useful operation. Do not infer a gain merely from using different media. Try the intended combination before expanding it.

The representations can overlap partially. They need not carry every fact or support identical operations. Identify the overlap needed for this use and the information that remains available only in one of them. Use NOT.5’s loss and recovery construction when movement between them needs a translation.

NOT.6:4.2 - Recover what each sign refers to

For the joined operation, follow the relevant labels, positions and references to their referents. Establish whether two occurrences concern the same thing, different parts or versions of it, or things connected by a model or measurement relation. The same spelling does not decide this.

Give the correspondence a form the reader can follow: shared labels with a declared meaning, a small pairing table, a drawn connection or an explicit mapping rule. Use stable identifiers when names or positions can change independently. For a temporary two-item comparison, a direct explanation may be enough.

Include the context that makes the reference determinate. A column named time needs its unit and origin when compared with an axis; a position in a repeated score needs its cycle when several cycles are shown. Distinguish actual, planned and modeled values where they are different claims.

One source item can correspond to several target items, or several items to a summary. Show that multiplicity where it affects the operation. NOT.5 identifies what such a projection loses; a repeated label alone does not recover the missing distinctions.

NOT.6:4.3 - Construct the relations that let results be combined

Give the reading or construction rule connecting the matched contributions. A formula-to-table relation specifies which inputs generate which cells. A table-to-plot relation specifies axes, units and which cells determine a point. A procedure-to-flow-diagram relation specifies which described actions and conditions each node and connection expresses.

Distinguish a derived representation from an independent contribution. A graph generated from the same table can check rendering but does not supply an independent observation supporting the table’s values. An observed trajectory and a predicted trajectory require the subject correspondence and comparison that relate them; C.29 supplies that modeling use.

Decide how a change will be handled. When a value is derived, give its source and transformation. When either side can be edited, give the backward policy or the condition under which a human choice is needed. Preserve information that the other representation cannot reconstruct.

For a chain of correspondences, follow the connection through the intermediate representations. If two routes reach what is supposed to be the same result, determine the agreement the work requires. A numerical tolerance can be appropriate for a plot, while an identity-sensitive reference needs the same referent. MATH.18 supplies the mathematical argument when the correspondence compares formal structures.

NOT.6:4.4 - Perform the joined operation

Take one input or object and follow it across the representations. Obtain each contribution, combine them using the stated relation and return the answer to the working question. Show where a reader switches representations and what is carried across that switch.

If the connection exists only in the author’s memory, add the missing reference or operation. If the correspondence is clear but the subject inference is missing, obtain that contribution from the subject method or a collaborator. Merely adding another display will not perform the inference.

Use one consequential contrast to challenge the connection. A row with the same label but a different unit, a node in another version or a modeled value beside an observation may require different treatment.

For an immediate use, a worked joined reading may settle the question. A broader claim about how readily users can combine the representations requires evidence appropriate to their preparation and work; C.2.8 characterizes that recovery. Do not infer learning or transfer from the existence of the correspondence.

NOT.6:4.5 - Propagate the intended change and expose conflicts

Identify what changed and what the change asks to preserve. Separate an explicit new requirement from values merely retained in an old display. A change to one coefficient can require several table cells to change even if those cells were not directly edited.

Follow the affected references and derivations. Recompute dependent contributions or update their correspondence; leave independent information unchanged unless the subject work calls for its revision. If a representation no longer agrees and cannot yet be updated, make that limitation visible before its next joined use.

When edits arrive through several representations, combine their requirements and check whether a compatible result exists. The result can be several admissible revisions or a conflict. Do not manufacture a unique update when the supplied constraints leave a choice; use the declared policy or obtain the missing decision.

Distinguish changes with the same visible outcome but different meanings. Removing an item from an unfinished-work list can mean completion, cancellation or deletion. The intended action determines which other representations change. A suitable change description can be more informative than comparing two finished displays.

If a model and an observation disagree, return to their governing subject account. Updating the observation to match the model would destroy the independent contribution; changing the model may require a modeling decision. Coordination makes the disagreement usable rather than hiding it.

NOT.6:4.6 - Retain the combination at the needed effort

Keep the participating interpretations, references and change rules available where the joint work happens. For a small construction, shared labels, one mapping rule and a worked change can suffice. Repeated edits or many interdependent elements can justify automated propagation, but the software needs the same declared correspondence and conflict behavior.

Stop when the intended joined operation works and the selected change can be handled or its unresolved decision is located. Remove a representation that adds upkeep without enabling a needed operation. NOT.7 helps redesign a connection that is correct but difficult to use.

Reopen the affected correspondence when the subject identity, version, unit, interpretation, reader operation or permitted edit changes. Unaffected representations may remain useful for their own questions even when a particular combined answer is temporarily unavailable.

NOT.6:5 - Archetypal Grounding

NOT.6:5.1 - Combine a formula, sample table and plot without freezing old values

A mathematical account defines f(x) = 2x + b for real x. A table stores its values at x = 0, 1 and 2, and a plot shows those same sample points with x horizontally and f(x) vertically. With b = 1, the table values are 1, 3 and 5.

The formula supports a value outside the table: f(3) = 7. The table supplies direct sample lookup; the plot exposes the relative positions of the samples. Connect each row’s x to the formula input and each plotted point to that row’s pair. The plot is generated from the table. Its agreement is not additional evidence that the formula describes some physical phenomenon.

Now the user requests f(1) = 4 while keeping the coefficient 2 fixed. The relation 2*1 + b = 4 gives b = 2. Recompute the table as 2, 4, 6 and move the plotted points accordingly; the formula now gives f(3) = 8. The old values at 0 and 2 were earlier derived values, not constraints freezing those points.

If the user instead requires f(0) = 1 and f(2) = 5 to remain, the three requested values cannot belong to a line of this form. f(0) fixes b = 1 and hence f(1) = 3. Expose the conflict. Choosing a different model or dropping a requirement changes the mathematical task; silently moving one representation would only hide it.

Finally, the three samples alone do not establish the formula for other inputs. Their generating formula is supplied here. If the table instead contains measurements, recover that different status before treating interpolation as a derivation.

NOT.6:5.2 - Link a component label, an adopted dimension and a mesh

A drawing identifies a beam as B9. A dimension note gives its adopted modeling length as 2.00 m. A one-dimensional model uses the variable L for that length, and a computational representation divides it into twenty equal segments. A correspondence states that B9’s adopted length supplies L and that segment length is L/20.

The reader can now identify which beam a segment calculation concerns and obtain 0.10 m per segment. B9, the variable L and a segment-array entry are different things connected by the stated construction. Replacing their names by one common name would not express that relation.

An accepted model revision changes B9’s adopted length to 2.02 m while keeping twenty equal segments. Update L to 2.02 and recompute the segment length as 0.101 m. The component reference remains B9. A copied old segment length of 0.10 would contradict the new construction because twenty such segments total only 2.00 m.

Suppose instead that 2.02 m is a new observation while the dimension note still describes a nominal 2.00 m design. Those values can coexist. Keep their different meanings and let the modeling work decide whether its adopted L changes. The correspondence does not itself turn an observation into a revised design.

This example coordinates quantities and references. It establishes no claim that twenty segments are sufficient for a physical simulation; that is a separate modeling and computational question.

NOT.6:6 - Bias-Annotation

Visual agreement can be mistaken for independent support. Two displays generated from the same source share its error unless one checks a different contribution. Use the correspondence to distinguish repeated presentation from independent evidence.

The opposite tendency treats every difference as inconsistency. A nominal dimension and a measured dimension, or a prediction and an observation, can legitimately differ. Recover their claim and subject before choosing an update.

NOT.6:7 - Conformance Checklist

When the combination needs checking, ask the questions that can change its use.

• Does each retained representation support a named operation or needed distinction?
• Can the reader recover the referents and the relation between their occurrences?
• Are derived values distinguished from independent contributions and different claim kinds?
• Can one joined answer be obtained by following the stated correspondences?
• Does a change preserve its actual requirements rather than every value copied from an earlier display?
• Are incompatible edits or interpretations exposed at the affected contribution?
• Can the work stop with the usable combination, without unnecessary synchronized copies?

NOT.6:8 - Common Anti-Patterns and How to Avoid Them

NOT.6:9 - Consequences

Different representations can contribute to one answer without pretending that they contain the same information or that their referred things are identical. Readers can move between their useful operations and find where a changed assumption affects the combination.

The connections cost interpretation and upkeep. More automatic propagation can reduce repetitive work but makes the update policy consequential. Some disagreements locate a genuine subject decision; notation design cannot settle that decision by synchronizing appearances.

NOT.6:10 - Architectural Rationale

Complementarity concerns operations, not media count. Two textual forms can support different work, while a text and picture can duplicate the same limited information. The useful whole consists of contributions and the relations through which they can be combined.

Reference and derivation are distinct relations. A variable may represent a quantity of a component; a table value may result from evaluating that variable’s formula. Retaining those relations allows a change to reach the right contribution without making the representations or their subjects identical.

Update intention is also content. A finished state can conceal whether an unchanged value is fixed by the user or merely carried forward. Making that intention recoverable permits local propagation and reveals conflicting demands before one silently replaces another.

NOT.6:11 - SoTA-Echoing

Choice of coordination. A single sufficient representation avoids the effort of maintaining correspondence. Several become useful when their operations contribute to the actual answer. Manual labels and a direct rule can then be enough. Repeated edits across shared information can justify a maintained transformation; the 2026 partial-state approach addresses an important failure of replacing whole views indiscriminately. It also requires a model of update intentions and admissible merging. Choose that added construction when propagation and conflicts matter, rather than adopting it merely because several displays exist.

The synthesis retains cognitive and computational contributions at their own scopes: a formally consistent update need not be easy to understand, and an easy comparison need not preserve a user’s edit. Reopen the chosen arrangement when a useful operation remains inaccessible, navigation or upkeep outweighs its gain, or a changed edit exposes an inadequate correspondence.

NOT.6:12 - Relations

NOT.6:End

NOT.7 - Redesign a Notation around the Reader’s Difficult Operations

Type: Method Status: Stable Normativity: Normative

NOT.7:1 - Problem frame

Use this when a notation can express the required content, yet reading or changing that content is unnecessarily difficult. A reader repeatedly loses a dependency, has to reconstruct a grouping, or must change the same assumption in several places. Another symbol or a shorter inscription may leave the difficult operation unchanged.

Begin with one such operation on an actual expression. Find what the reader has to locate, infer, remember or alter. Change the notation or its editing arrangement at that point, then compare the operation and another operation that the change could burden.

The result is a repaired notation with a reason to prefer it for the intended work, or a decision to retain the existing form. A useful diagnosis may instead identify missing subject knowledge, an unavailable operation or inaccessible content that a notation change cannot supply.

The reader needs the subject criterion for a correct result and enough knowledge of the existing notation to inspect its interpretation. NOT.1 supplies a missing use requirement; NOT.2–.3 supply formation and interpretation. A sketch and a worked comparison can be enough for a local repair.

Keep a sufficient familiar form when no consequential reading or change difficulty is present. If the underlying account is wrong, repair that account. Use EXD.1–.3 when the needed work is to construct an explanation; use the instructional method appropriate to the missing capability when the reader cannot yet perform the required subject operation.

NOT.7:2 - Problem

How can a notation be made easier to use without removing the distinctions, dependencies or operations on which its usefulness depends?

Shortness, visual neatness and familiarity can each help particular work, but none identifies the difficult operation. A repeated value may be easy to read and costly to change. A shared name may make the change local while sending readers elsewhere to recover the value. A new arrangement must be judged through those operations.

The repair also has to separate expression from assistance. An author can explain the missing connection during a trial, making an unchanged notation appear successful. A trained reader can reconstruct a relation that the expression never supplies. Those contributions may be useful, but they change what the notation itself has achieved.

NOT.7:3 - Forces

NOT.7:4 - Solution

Locate the difficult operation → explain its burden → construct a targeted alternative → preserve the required meaning → compare use and displaced effort → retain or revise.

NOT.7:4.1 - Locate the operation and its failure

Select a concrete question and the expression on which the reader must work. Ask for the answer or modification, not only an opinion of the notation. Recover the route taken: where the reader searches, which reference is followed, what has to be held in memory and where the result goes wrong or the work becomes costly.

Use an available response or observed difficulty when one exists. Otherwise, walk through the operation and treat the diagnosis as a design estimate. An actual trial is useful when plausible reader responses would change the repair decision; it is not a prerequisite for every obvious correction.

State the relevant preparation, permitted help, display or access conditions and effort limit. A reader who knows the local grouping convention faces a different question from one learning it. A page that hides half a diagram can create a different difficulty from the diagram’s structure.

Keep the intended result in view. If the reader has correctly recovered the notation but lacks the subject inference, obtain that inference or capability. If an interpretation is ambiguous, NOT.2 or NOT.3 repairs the rule before comparisons of ease can settle its use.

NOT.7:4.2 - Explain which work the current form imposes

Describe the burden through operations: repeated searching, uncertain grouping, reconstruction of an unstated dependency, simultaneous retention of distant values, or many edits for one intended change. Choose the description that explains the observed case.

Locate the source of the burden. The expression, the construction rule, the editing environment and the reader’s preparation can make different contributions. An editor that cannot show two needed passages together may be changed without introducing a new notation. A missing dependency needs expression even when the display is large enough.

Use Cognitive Dimensions as a vocabulary for such trade-offs when it helps the design conversation. Its activity-relative descriptions suggest where to look; a list of dimension names does not establish that a reader can perform the operation.

Retain successful uses of the current form. They identify what the repair must preserve and may reveal why an apparently awkward convention exists.

NOT.7:4.3 - Construct a change at the source of the burden

Choose a change that alters the difficult operation. The following constructions supply starting points; their costs determine whether to keep them.

A local label, grouping mark or rearrangement can suffice. Use NOT.4 when an expression transformation needs a preservation argument, NOT.5 when the alternative omits distinctions, and NOT.6 when both forms will remain in joint use.

Keep the alternative small enough to inspect. If construction reveals that the original question was wrong or incomplete, revise the requirement through NOT.1. Do not keep optimizing the old operation merely to preserve the initial design brief.

NOT.7:4.4 - Preserve what the work needs from the expression

Identify the meaning and operations that must survive the change. Work through one consequential contrast: reversed dependency, local exception, changed grouping or unresolved input, as appropriate to this notation. The new form must make the required difference recoverable.

If the repair adds a claim that was absent before, treat it as a content addition. If it supplies a new procedure or teaches a missing convention, include that contribution in the comparison. A useful combined repair is allowed; attribute its effect to the actual combination.

Preserve the path from a compressed form to the detail needed for its use. A shared definition needs a resolvable reference; an intermediate name needs its construction; an omitted branch needs a rule showing when it can be omitted. Compression is useful when it reduces the relevant work, not merely the displayed size.

NOT.7:4.5 - Compare the same operation and the burden it moves

Perform the selected operation with the old and new forms under comparable conditions. Compare correct and missing results first, then the effort relevant to the use: searches, unresolved references, edits, elapsed time or another suitably defined measure. C.2.8 distinguishes structure recovered from the effort of recovering it.

Also perform an operation plausibly made harder by the repair. A shared parameter may make a global change easy and a local exception harder. An outgoing-dependency index may improve downstream search while an incoming-dependency list remains better for finding prerequisites. Keep that trade-off visible in the choice.

For a human or agent trial, retain the first response before explanation or correction. If the same reader sees both forms, practice can contribute to the second result. For a local decision, that limitation may be acceptable. A general superiority or learning claim requires a comparison that supports that stronger conclusion; EXD.6 and, for instructional use, HCD.19 supply the corresponding evaluation methods.

A family of alternatives can justify matched examples and computed measures. Use those measures to locate an expression or change that needs inspection. Shorter code, fewer symbols or a smaller edit count does not alone establish better understanding. Do not build a comparison gallery when one bounded contrast can settle the local repair.

NOT.7:4.6 - Choose the repair at the needed scale

Retain the change when its demonstrated or reasonably expected contribution is worth the new burden for the actual work. Keep the incumbent when it remains sufficient. If different forms serve different important operations, preserve the useful alternatives and their correspondence rather than force one universal winner.

Use the existing contribution and choice methods in C.11.CRC/C.11 when cost or competing improvements affect the decision. When the work needs maintained comparison and selection rules, use A.19.CPM for comparison and A.19.SelectorMechanism for selection. This method supplies notation changes and their use differences, not a separate scoring system.

Leave the needed convention, reference or operation available with the retained expression. A repair that works only while its author explains it has not supplied a self-sufficient notation for that reading condition.

Stop when the present use is adequately supported and further comparison would not change the choice. Reopen when the operation, preparation, access, scale of use or meaning changes, or an actual response contradicts the claimed benefit.

NOT.7:5 - Archetypal Grounding

NOT.7:5.1 - Turn prerequisite lookup into downstream search

A small dependency account gives the immediate inputs of four tasks:

This form answers “what does D need?” directly: B and C. The new working question is “after A changes, which tasks may need reconsideration?” The reader must repeatedly search the input column for dependents, then do so again for each newly found task.

Construct the reverse index: A → {B, C}, B → {D}, C → {D}, D → {}. Here an arrow goes from a supplied input to a task using it. Starting at A, follow arrows and retain newly reached tasks until none remains. The resulting set is {B, C, D}; D is retained once even though two paths reach it.

The repair changes the lookup arrangement and supplies a reading procedure. It does not add a new dependency. It makes the required direction directly available, but finding the immediate inputs of D is now less direct. Keep the original table as a complementary view if that operation remains frequent, using NOT.6 to maintain the connection.

Now remove C from D’s input list. Remove the arrow from C to D as well. A still reaches D through B. A reader who merely drops D from the downstream set has confused removal of one path with removal of every path. The changed case tests the reading operation, not the prettiness of the index.

This construction demonstrates the dependencies and operations preserved. It supplies no measured claim that every reader finds the index faster; the local display and reading conditions can change that comparison.

NOT.7:5.2 - Share a parameter while retaining a local exception

Three quoted amounts use the same tariff, 3 units per item, for quantities 2, 5 and 7. The expressions are 3*2, 3*5 and 3*7. Their intended common assumption is known, but the written expressions repeat it independently.

To make a common tariff change local, introduce p = 3 and write p*2, p*5 and p*7. A change to p = 4 then gives 8, 20 and 28. There is one tariff definition to change, while a reader of an individual expression now follows p to that definition. The definition must remain accessible.

The next request keeps the second quote at the old tariff and changes the others. A global change of p alone cannot express that request. Write the second expression as 3*5, with its fixed-tariff meaning stated, while the others remain p*2 and p*7. The amounts are now 8, 15 and 28. If later exceptions acquire their own shared rule, give that rule its own name.

For work dominated by common tariff changes, the shared parameter can be useful. For independently negotiated quotes, the assumption of a common tariff may be wrong. The notation repair therefore needs the subject’s sharing relation; identical printed numbers alone do not establish it.

NOT.7:6 - Bias-Annotation

A designer’s fluency can hide the work a new reader performs. Recover an actual route or state the preparation assumed by a design estimate. When the author supplies missing information during use, include it among the enabling contributions.

Optimizing a visible measure can also remove the method’s value. A shorter inscription can hide the dependency needed for a change. Compare the operation and its result before treating size or edit count as an improvement.

NOT.7:7 - Conformance Checklist

When the repair needs checking, ask the questions that can change its adoption.

• Is a concrete difficult operation identified, with its intended result?
• Does the repair change the cause of that difficulty?
• Can the needed distinctions and references still be recovered?
• Are content, notation, preparation, help and access changes distinguished where they affect attribution?
• Has an operation plausibly burdened by the repair been considered?
• Does the comparison support the claimed scope of benefit?
• Can the work stop with a sufficient local repair or a justified choice to retain the current form?

NOT.7:8 - Common Anti-Patterns and How to Avoid Them

NOT.7:9 - Consequences

A repair can make a needed relation or operation available with less reconstruction, or make an intended change less error-prone. It also makes the reason for keeping a familiar convention explicit when a proposed alternative would burden important work.

New conventions require learning and migration. Complementary views add maintenance. A local improvement remains local until broader use supports a wider claim. Some failures return to subject knowledge or explanation rather than notation.

NOT.7:10 - Architectural Rationale

The unit of redesign is the reader’s operation on meaningful content. Appearance matters through what it makes available, distinguishable or changeable in that operation. This lets the method apply to formulae, diagrams, programs and other notations without prescribing one visual style.

Meaning preservation and ease of use answer different questions. A transformation can preserve the result while making its construction harder to inspect. Conversely, a helpful explanation may add missing content. Keeping those contributions separate permits a fair comparison without forbidding useful combined repairs.

An activity-relative choice also avoids inventing a universal quality scale. The existing characterization and contribution methods compare the relevant gains and burdens. Notational engineering supplies the constructions and the operation-level differences those methods need.

NOT.7:11 - SoTA-Echoing

Choice of repair method. A local walkthrough can settle an explicit lost dependency or burdensome edit. Activity-based diagnosis helps discover the trade-off when the trouble is less obvious. Matched galleries become useful for recurring comparisons across a family, but add construction and interpretation cost. Reader trials answer a remaining recoverability question when their outcomes can alter the choice. None of these methods replaces subject correctness.

The synthesis connects those notation-design methods with C.2.8’s qualified structural comparison and the existing marginal-choice methods. Keep design estimates, measured responses and formal preservation arguments at their own scopes. Reopen when observed use defeats the proposed mechanism, a newly important operation changes the trade-off, or stronger evidence changes a claimed benefit.

NOT.7:12 - Relations

NOT.7:End

NOT.8 - Construct Temporal or Embodied Notation with a Reading Procedure

Type: Method Status: Stable Normativity: Normative

NOT.8:1 - Problem frame

Use this when a notation carries important distinctions through timing, sound, gesture or bodily position, and recognizing its individual signs does not yet let a reader follow or perform the intended sequence. The reader may lose the cycle, confuse a gesture’s reference frame, or recognize a command without knowing how to enact it.

Start with the distinction the work needs: an onset within a beat, two simultaneous events, a body-relative direction or another stated relation. Choose how the carrier expresses it, provide its reference and segmentation, then construct the procedure by which a reader recovers and uses it.

The result is a temporal or embodied expression with a usable interpretation and reading procedure. It states what a prepared reader can obtain, what remains for performance or control, and how a consequential change affects the reading.

This is notational engineering. Rhythm, movement, laboratory signals and interaction traces supply different subjects; their subject methods determine which timing or motion relations matter. The notation designer needs that subject meaning and the relevant capabilities of the intended reader. NOT.1–.3 supply the general design, formation and interpretation methods.

Reuse a familiar adequate convention when it already supports the operation. An ordinary static expression may suffice when no temporal or embodied distinction matters. Knowing the notation is not by itself a claim that a person can perform a movement or a machine can realize it; obtain the missing performance or control method when the work requires enactment.

NOT.8:2 - Problem

How can a reader recover and manipulate relations that unfold in time or action, including those no longer immediately present when the next sign arrives?

A sequence of recognized sounds can lack a known starting point or grouping. A pause can mean a specified rest, an unobserved interval or a wait for an event. A gesture to the right can refer to the performer’s body, a room or a drawing. These interpretations permit different continuations.

The carrier’s behavior also matters. Playing a recording faster changes its audible duration, but need not change the duration represented by a coded timestamp. A score can specify a desired action while leaving its bodily or algorithmic realization to the performer. The reading procedure must make the relevant convention available.

NOT.8:3 - Forces

NOT.8:4 - Solution

Select the temporal or embodied distinction → establish references → segment and encode → construct the reading or enactment route → vary a consequential condition → retain a usable expression.

NOT.8:4.1 - Select what is expressed and what the reader should do

Identify the subject relation and the needed operation. The work may be to recognize an order, reproduce a rhythm, compare two movements, locate a missing event or construct instructions for another performer. Choose the distinctions that operation consumes.

Say whether the expression describes an observed occurrence, proposes a future performance or presents a construction for analysis. The same cue sequence can serve these different uses, but a plan does not establish that its events happened.

Separate the sign’s physical properties from the represented properties. A sustained tone might represent duration by its length, or merely identify an event kind. A hand position might be an action being performed, a sign for another movement, or a reference against which a different action is timed. Use the convention that supports the selected work.

State the reader’s available perception and operations. Distinguishing two timbres, maintaining a pulse, recognizing a body part and following a formal event code are different prerequisites. A computational reader needs a suitable input representation and interpretation, not an assumed human sensory skill.

NOT.8:4.2 - Establish the references that locate an occurrence

Give the origin, unit, frame or recurring reference needed to interpret the signs. For cyclic time, identify the cycle and position within it when occurrences across cycles must be distinguished. For movement, identify the participant, body part, spatial frame and starting condition that affect the interpretation.

Choose whether timing is absolute, relative to another event or relative to a maintained pulse. If a tempo change should retain relative positions, preserve that relation when changing elapsed durations. If a deadline is in seconds, a mere proportional rescaling may change the required result.

Give different participants a way to align their references when joint action or comparison needs it. A shared visible pulse, a named event or a common clock can supply alignment under its own conditions. Separately timestamped records do not establish event order unless their clock relation supports that inference.

Keep the reference identifiable during use. A repeated syllable can denote the same position in successive beats while being a different occurrence each time. A rightward direction can change in room coordinates when its body frame turns.

NOT.8:4.3 - Segment and encode the needed relations

Choose recognizable units and a rule for joining them: separate events, intervals, phases, gestures or other parts appropriate to the subject. Mark a beginning, boundary or return to a cycle where losing it would change the reading.

Assign signs and their temporal or spatial arrangement to the selected distinctions. Make order, overlap and duration recoverable when the work needs them. Writing two signs successively does not express simultaneity unless a grouping or interpretation rule supplies it.

Provide a way to distinguish a specified absence from a missing observation. A rest within a known beat and a recording gap can both be silent, but only the former supplies the intended absence. Use a marker, an accompanying record or another recoverable convention where the difference affects the next operation.

Inspect discriminability under the intended conditions: speed, overlap, background sound, viewing direction or available sensors. A symbol family that is distinct on a page can become confusable when sounded or gestured. Change the encoding or access arrangement if the needed contrast disappears.

Choose detail for the receiving use. A movement score can constrain endpoints while leaving the route open; a different use can require the route, timing or body-part coordination. Keep an omitted choice with the performer or construction method that is to supply it, rather than silently giving the notation credit for it.

NOT.8:4.4 - Construct the reader’s route through the expression

State how the reader establishes the reference, identifies the current part, combines it with retained information and obtains the next result. Make the necessary repetition, count, comparison, transformation or enactment available. A legend naming sounds or gestures may need this additional procedure.

For a transient carrier, choose how earlier parts remain usable: replay, a persistent companion record, a repeated reference, trained retention or another available means. Use only what the receiving conditions permit. NOT.6 connects a score, sound and event table when several forms remain necessary.

If the task is performance, connect the decoded instruction to the performer’s available method. A person may already know how to realize a gesture or syllable. A machine may need a parser, state representation, control algorithm and physical realization. CMP.12 constructs an effective interpreter; B.5.MPC connects the abstract construction to preparation, execution and interpreted observation. Notation alone does not supply those methods.

A demonstration can be part of the supplied interpretation. Identify what it teaches and keep it available if the text relies on it. If another prepared reader can use the expression without that demonstration, the reading condition is different; assess the contribution at the condition actually offered.

NOT.8:4.5 - Challenge the relation with a meaningful change

Carry out or inspect a small use and a contrast that can change its result. Shift a cue while retaining its identity, change a body’s orientation, remove a timing reference or ask for simultaneity rather than order. Choose a contrast grounded in the intended operation.

Distinguish reading from successful physical performance. A correctly decoded sequence can still be infeasible for a particular body or controller. Locate that missing realization contribution rather than changing the meaning until the observed performance appears correct.

When using replay, decide which property it can preserve. Slowing a recording can aid segmentation. It cannot by itself demonstrate that a performer meets the original speed, that two live processes stay synchronized or that a paused physical system stops evolving.

A worked interpretation supports its stated use. Claims about human fluency, learning or transfer require their own basis; C.2.8 and the applicable capability-development method characterize the relevant recovery or performance. Use the current sufficient result, and seek an additional trial only when its outcome can change the next choice.

NOT.8:4.6 - Retain the expression with its operative convention

Keep the reference, segmentation, interpretation and needed reading procedure available with the expression. A short local convention may suffice; reusable performance material may also need a demonstration, an interpreter or an accessible alternative carrier.

State an unresolved performer choice or lost distinction where the next use needs it. A score specifying endpoints can be useful before a motion path is chosen. A recording with an unobserved interval can support the observed segments without settling the missing order.

Stop when the selected operation is supported or the missing contribution is located. Use NOT.7 if the interpretation is determinate but difficult to follow. Reopen the affected convention when the reference frame, timing, medium, required observation or reader capability changes.

NOT.8:5 - Archetypal Grounding

NOT.8:5.1 - Recover a cue’s place in a beat

Choose a beat divided into four equal positions. In the Takadimi convention these positions are voiced as ta, ka, di and mi. For this example, a dot in the written grid means no new onset at that position; it does not specify how long an earlier sound continues.

Establish a steady beat, divide it into four and keep the beat’s start identifiable. The first pattern has onsets at 0 and 1/2. A shift of one quarter-beat, within this one-beat example, yields onsets at 1/4 and 3/4. The new syllables identify the new positions. Repeating the same two spoken syllables without their reference would not express this change of placement.

At a beat duration of one second, the first onsets occur at 0 and 0.5 seconds; the shifted ones at 0.25 and 0.75 seconds. At a beat duration of half a second, the corresponding times halve while the relative positions remain. A requirement for an onset at an absolute 0.5 seconds is therefore a different requirement.

If the sound between two known positions is unrecorded, do not replace that gap by a written dot claiming no onset. Preserve the missing-observation status. If the next operation asks how long notes are held, add the needed duration convention: the onset grid alone does not answer it.

This is a notation construction, not a claim that reading the table teaches fluent rhythmic performance. Maintaining the beat and coordinating speech are available skills or contributions to be learned.

NOT.8:5.2 - Read a movement relative to a changing body frame

A tabletop movement score uses two signs. R means move one grid unit to the performer’s right without turning; F(2) means move two units forward without turning. The signs describe successive completed displacements. The starting position is (0, 0), with east as positive x and north as positive y.

A performer initially facing north reads R; F(2) as an eastward unit displacement followed by two northward units. The endpoint is (1, 2). The reading procedure first recovers the starting orientation, translates each body-relative direction into the grid frame, updates the position and continues from there.

Now the performer starts facing east. The same score yields one unit south and two units east, ending at (2, -1). Treating R as page-right would give a different route. If the intended task is to preserve the original grid-space path regardless of orientation, choose a grid-relative notation or construct the appropriate body-relative instructions; keeping the marks unchanged does not meet that new task.

The score leaves the speed and path within each displacement unspecified. It can guide a prepared person moving a token. Making a physical robot perform it additionally requires an applicable motion method and an account of its realization. Reading the endpoint establishes neither that method nor a performed movement.

NOT.8:6 - Bias-Annotation

Expert fluency can conceal the reading procedure. A musician or dancer may supply pulse, segmentation and bodily realization without noticing those contributions. Recover them before expecting a newcomer or computational agent to use the same expression.

A familiar carrier can also invite the wrong relation. Spoken order can be mistaken for event order, silence for a rest and a facing gesture for a room-fixed direction. Use a consequential contrast to determine which reference the notation supplies.

NOT.8:7 - Conformance Checklist

When the expression needs checking, ask the questions that can change its use.

• Is the represented relation and intended reading or performance operation recoverable?
• Are the needed origin, unit, cycle, participant and frame available?
• Can the reader segment occurrences and distinguish the relevant order, overlap or duration?
• Is a specified absence distinguishable from missing information where it matters?
• Does the reading procedure identify the retained state or replay that it needs?
• Are perception, interpretation and physical performance contributions supplied at their stated scopes?
• Does a meaningful change preserve the intended relation or reveal the required repair?

NOT.8:8 - Common Anti-Patterns and How to Avoid Them

NOT.8:9 - Consequences

Temporal and embodied expression can become a manipulable part of reasoning: readers can locate occurrences, vary a relation, compare performances or hand an interpretable instruction to another agent. A score can retain useful constraints while allowing more than one realization.

The method exposes preparation and access costs that a fluent practitioner might leave tacit. Additional references can burden performance; a persistent companion representation can help analysis while requiring coordination. Some expressions remain useful only with a trained performer or an available interpreter.

NOT.8:10 - Architectural Rationale

Temporal or embodied expression adds reference and availability problems to ordinary formation and interpretation. What a sign means can depend on phase, orientation or the reader’s retained state. These conditions belong in the operative convention rather than an assumed resemblance between mark and action.

Segmentation serves the receiving operation. An action can be described by its intended result, selected intermediate constraints or a detailed movement account. Those descriptions require different contributions from a performer. Keeping that distinction permits transfer between people and machines while exposing what must be reconstructed.

Reading and enactment are connected methods. Their separation makes the connection constructive: decode what the score requires, determine what the performer must supply, obtain that method and interpret the resulting action. This retains the relation between mathematical structure, algorithms and physical realization.

NOT.8:11 - SoTA-Echoing

Choosing a form for the same operation. In :5.1 the reader must locate and shift two onsets within a beat. Compare a static position grid with a sounded score, for a reader who knows the syllables and can count the four positions. The grid already answers the question and remains available for inspection. Keep it when that is the whole task; adding voiced practice would cost effort without improving this answer. A familiar convention also needs no extra account when its references and reading operations are already available to the reader.

For following the onsets during a performance, both a retained grid and a sounded score still need a way to keep place in the beat. Here we adapt Nelson’s recurring reference and Takadimi’s positional reading into :4.2 and :4.4: maintain the reference, locate each occurrence and retain or replay the parts needed for the next operation. This repairs the case in which a sign legend identifies syllables but leaves their positions unavailable. It does not require a new symbol vocabulary. A prepared reader may maintain the reference while reading the grid; a sounded or embodied form is preferable only when its coordination benefit warrants the added perception or practice. Those costs are the accepted trade-off, not a demonstrated general learning advantage.

The beat calculation in :5.1 shows the information gained from the reference; it does not compare human performance speeds. The source table supplies the limits of transferring this reasoning to movement and audible interfaces: a recovered instruction still needs its performer method, and preferred sounds need not improve interpretation. Reopen this choice if a simpler available convention supports the same operation and timing with less burden, or if the chosen reference or reading procedure fails under the receiving conditions.

The synthesis uses these different practices to expose recurring design operations; it does not merge their symbol systems. A position syllable, a stroke-associated syllable and a motion sign can carry different relations. Reopen the construction when a relevant contrast is not perceptible, a reader lacks an assumed operation, a new realization cannot supply the omitted work or a receiving use needs a previously unexpressed relation.

NOT.8:12 - Relations

NOT.8:End