What's the difference between \tolerance and \badness ?

Question

I'm having some difficulty understanding this sentence on page 29, Chapter 6 of Knuth's TeXbook:

Run TeX again, and begin this time by saying

\hsize=2in \tolerance=1600 \input story

so that lines with badness up to 1600 will be tolerated. Hurray! There are no overfull boxes this time. (But you do get a message about an underfull box, since TeX reports all boxes whose badness exceeds a certain threshold called \hbadness; plain TeX sets \hbadness=1000.)

Answer 1

The \tolerance setting influences the paragraph breaking routine itself: changes to \tolerance (and \pretolerance) actually affects which line breaks are chosen. Higher values allow worse lines (usually meaning: with stretched inter-word spaces) to be accepted, with the value 10000 indicating a 'panic mode' where anything at all is acceptable. Normally the lower the value, the better the paragraph will look, but you run the risk of reducing the list of possible breaks so much that you end up with overfull lines.

The \hbadness setting only influences the user report (the messages you see on screen and in the log) about the actually chosen lines, it has no effect on the breaking routine itself.

Answer 2

You can think of three groups of parameters that are used by TeX when typesetting a paragraph, those that are set by the typesetter and the font designer, and those calculated by TeX internally. The \badness is calculated by TeX internally, whereas \tolerance is set by the typesetter to tell TeX what can be tolerated in terms of line and overall paragraph badness.

Tweaking these parameters is a bit of a black art.

Run the following minimal with pdfTeX.

\overfullrule=0.1pt
\hsize150pt

\def\astory{Fast pace and mechanically intensive 
  facilities such as data-centers, hospitals, and 
  laboratories typically require the most intense 
  coordination efforts. Architectural and structural 
  systems are often designed first with allowances 
  for MEP systems. Tensions between the size of 
  these MEP spaces, usable floor space, 
  and ceiling height exist. }

\astory

\the\tolerance, 
\the\pretolerance, 
\the\hbadness, 
\the\hfuzz, 
\the\emergencystretch, 
% Second test    
\pretolerance=150
\tolerance=753
\hbadness=752
\hfuzz0pt
\emergencystretch=0em

\astory
\bye

In the minimal the paragraph is typeset, firstly with default values and second without. The first one produces a lot of overfull and underfull boxes. We get rid of them by suitably adjusting the \tolerance. In this particular example I set the tolerance to one higher than the hbadness value reported by TeX. This eliminated all the underfull warnings.

There is a nice surprise when you set the \hsize from 150pt to 160pt and this is where the "black art" part comes along!