ΔTartan: how far apart are two tartans?

Every cloth page in the Dictionary carries a nearest tartans table: the most similar cloths in the whole corpus, each with a number in the ΔTartan column. This post explains what that number measures — why a faded, two-hundred-year-old kilt sits a whisker from its own younger self while two tartans that share most of their colours sit far apart.

Judged the way a weaver would

A tartan is a small palette deployed in a strict order, and that is how ΔTartan reads it. Each sett becomes a colour ladder: its colours are grouped into hue families (red, green, blue…), ranked dark-to-light within each family, and split by prominence into the ground — the broad foundation colours that set the cloth's character — and the decoration — the thin overstripes and dark accents laid over it.

To compare two setts, the Dictionary first re-reads the second cloth's stripes in the first cloth's ladder — a deep charcoal-navy takes the role of the other's bright navy if it holds the same rung of the blue family — and then counts the jogs: the fewest stripe edits (a recolour, an added band, a removed band) that turn one sequence into the other, tried at every alignment, so a sett recorded from the other end or from a different starting stripe costs nothing. Before any of that, both setts are folded to one canonical form, so a source that wrote out the full mirrored sett and one that wrote the half sett are read as the same recording; and the measure is scale-free, so the same proportions woven at twice the thread count read identical.

The edit costs are a weaver's judgement, not a statistician's:

• an edit on a ground colour costs 1 — change the foundation and you have made a new tartan;
• an edit on a decoration stripe costs about 0.3 — overstripes come and go within one tartan's life, added by a weaver's flourish or lost in a coarse recording;
• stripe-width drift and the ground ratio — the balance between the two main colours, say red against green — add smaller fractional terms, so that geometry alone stays below 1;
• shade travel — how far each dye has drifted within its rung, measured perceptually — adds the smallest term of all, scaled so it can never carry a pair across the same-tartan line: it grades within the band, it never decides it.

The same-tartan band: Δ < 1

Those costs are calibrated so that one line can be read off the number: ΔTartan below 1 means the same tartan. Decoration changes, width drift, re-shading, fading, scale — everything a single tartan does across two centuries of weavings — lands inside [0, 1). Only a change of ground clears 1.

The worked example is the Drummond of Megginch collection: eight recorded weavings of one design — a plaid at roughly twice the kilt's scale, three kilts, a Victorian child's kilt, even a carpet, in dye lots from bright Victorian reds to the deep, almost-charcoal 1997 reweave. Every pair in the family reads at 0.83 or less — one tartan, as the family has always known. The starkest single reading: the 1849 kilt against its own faded outer face is 0.14. Fading moved every dye in the cloth but not a single stripe, and the number says exactly that — a little bit different, nowhere near a different tartan. (Only a bit-identical recording reads 0.00; the deliberate deep re-dye of 1997 reads 0.30, twice the fade, still family.)

A distance built on colour statistics gets this badly wrong: it puts the kilt 1.4 from its own faded face, and scatters the Megginch family out to 4.5 — further apart than many pairs of genuinely different tartans. That is the failure ΔTartan was calibrated against.

The boundary behaves too. Wilson's 1819 New Bruce — the sett sold as Grant or Drummond, the same layout in a different colourway — reads 2.6 to 4.7 from the Megginch weavings: visibly kin, never inside the band.

Grading the far field

Past the calibrated range the edit count goes coarse. Once nearly every stripe differs, "fourteen edits" against "twelve" is no longer a judgement a weaver would stand behind — both just mean not this tartan. So in the far field the Dictionary changes instrument: distance is graded by the cloths' overall colour and complexity statistics — ground shade, colourfulness, stripe complexity, the same statistics that place each cloth on the neighbour map — proportioned onto the jog scale so the two readings meet without a seam. The exact rule:

ΔTartan = w·J + (1 − w)·max(2.13·L, 1)

J = the ladder distance (the weighted jogs, width, ratio and shade terms)
L = the colour/complexity statistics distance
w = 1 up to J = 2, 0 from J = 4, blending smoothly between

Up to ΔTartan 2 the number is the weaver's reading, untouched. From 4 upward it is the statistical grading, scaled to match. The floor of 1 inside the far term is deliberate: the statistics may order the strangers, but only the ladder is ever allowed to pronounce two cloths the same tartan.

Where you meet the number

The nearest-tartans table on each cloth page lists neighbours out to ΔTartan 4 — the near ones at full strength, anything past 2 progressively muted — with each cloth's sett drawn to the same width under the page's own, so the comparison is made by eye as well as by number. The ΔTartan figure itself is a link: it opens that neighbour in the tartan designer pinned against the cloth you came from, so you can see exactly where the distance lives.

A short reading guide:

The calibration study — the Megginch ground truth, the Grant/Drummond distance triangles, and the fitted constants — lives in the engine's research notes (research/delta-blend.md in the source repository) and is reproducible with the deltablend tool.