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Inspired by a comment by Mico in a totally unrelated question, I want to know if someone of you know why font sizes are based on a linear progression of 1.2. The author of the comment suggest that probably it has to do with the fact that 1.2 ≈ ⁴√2̅ . Someone of you know why and, maybe, has some reference to it?

Note This question is on typography in general more than on LaTeX but I think that probably this is the best place to ask, otherwise feel free to migrate my question to a more proper place.

Edit Mico pointed out that the linear progression of 1.2 is only for larger font sizes of \normalsize while for smaller font sizes the linear progression of 0.7 is used (0.7 ≠ 1/1.2 = 0.833). So the main question still remain and a new one arise: why for smaller font sizes of \normalsize the linear progression of 0.7 is used and why it's different from the one used for larger font sizes of \normalsize ?

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gvgramazio
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2 Answers2

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The LaTeX size names are related to the fonts available in the earliest releases and they in turn are related to this comment in the TeXBook:

\danger At many computer centers it has proved convenient to supply fonts at magnifications that grow in geometric ratios---something like equal-tempered tuning on a ^{piano}. The idea is to have all fonts available at their true size as well as at magnifications 1.2 and~1.44 (which is $1.2\times1.2$); perhaps also at magnification~1.728 ($=1.2\times1.2\times1.2$) and even higher. Then you can magnify an entire document by 1.2 or~1.44 and still stay within the set of available fonts. Plain \TeX\ provides the abbreviations ^|\magstep||0| for a scale factor of 1000, |\magstep1| for a scaled factor of 1200, |\magstep2| for 1440, and so on up to |\magstep5|.

To answer the extra question in the edit above

Note that smaller sizes like 7pt and 5pt are not (in computer modern) made by scaling down the 10pt font but are generated at that design size, so (more or less) the available fonts were base fonts at sizes 5pt, 7pt and 10pt, scaled up by magsteps of 0.5,1,2,3,4,5

David Carlisle
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  • This answer to my question on linear progression. But there is a reasons for the value 1.2? Also, the comment is only partially true since if you magnify an entire document by 1.2, 1.44, etc. you don't stay within the set of available fonts using font sizes smaller than \normalsize – gvgramazio May 03 '18 at 11:28
  • @giusva you do, as the base fonts were all provided at all magsteps, 1.2 is convenient as it gets between the common sizes of 10pt and 12pt, I guess:-) – David Carlisle May 03 '18 at 11:29
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    Thanks. Has Knuth provided similar comments on the \textstyle-\scriptstyle-\scriptscriptstyle geometric progression, which involves magnification steps of 0.7?Interestingly (and, I suspect, not coincidentally), 0.7 is rather close to 1/\sqrt{2}. – Mico May 03 '18 at 11:30
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    @Mico not as far as I can see but that is rather different cm uses design size 5pt and 7pt fonts, so they are not scaled in the same sense. and if you scale a plain tex document by magstep 1 then you stay within the available fonts as the 5pt , 7pt and 10pt fonts were all provided at magstep 1, you don't need the 7pt scaled by 1.2 to equal 10 (which is fortunate as it doesn't:-) – David Carlisle May 03 '18 at 11:32
  • @David I think I see your point. So while for font sizes bigger than \normalsize we could speak of magnification while for font sizes smaller than \normalsize we cannot since they use a different design. Right? – gvgramazio May 03 '18 at 11:34
  • yes so for larger sizes you could save disk space so that normalsize at 12pt and large at 10pt could use the same scaled font you would never want to use the 5pt cm font scaled to 10pt as a normalsize font (try it, it looks quite weird:-) @giusva – David Carlisle May 03 '18 at 11:36
  • Thanks for answer to my curiosity. Could you just edit your answer adding the links to TeXBook and to the comment for future readers. I know that one can simply google it and use ctrl+F as I did but readers are lazy. If you want I could do it myself. – gvgramazio May 03 '18 at 11:41
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    @DavidCarlisle - The 5pt-7pt-10pt math mode progression (with a scaling factor of ca 1.4) still begs the question, "why 1.4 and not either 1.35 or 1.45, say"? I have a hunch that 0.7\approx 1/\sqrt{2} factor is not entirely a coincidence. However, I've never come a formal reference on this either. – Mico May 03 '18 at 11:42
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    @Mico the texbook just states as fact that the script and scriptscript fonts are 7pt and 5pt, so unless there is a transcript of an interview somewhere I think "why" is not directly answerable may be just traditional sizing going back to Gutenberg, I can't say. – David Carlisle May 03 '18 at 11:44
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    @giusva the texbook is a book on paper, there are no (legal) online links to a typeset version. – David Carlisle May 03 '18 at 11:45
  • @David I'm quite sure about the first link, even if it specify that The source has ‘pro­tec­tion’ against use to pro­duce a doc­u­ment: such use is only al­lowed with the per­mis­sion of the Copy­right holder and of the pub­lisher (Ad­di­son-Wes­ley). But i'm not quite sure about the latter. – gvgramazio May 03 '18 at 11:53
  • The A series in ISO 216 also uses the square root of 2 in defining paper sizes. Using the same factor to scale fonts used on that paper would make sense. – chepner May 03 '18 at 18:42
  • Minor correction to: "the available fonts were base fonts at sizes 5pt, 7pt and 10pt". These are not the only ones. The complete list is: 5pt, 6pt, 7pt, 8pt, 9pt, 10pt, 12pt, and 17.28pt. Thus, the following common sizes are achieved by scaling: 10.95pt (geometric mean of 10 and 12), and also 14.4pt, 20.736pt, and 24.8832pt; they're normally rounded to 2 decimal places though (not that anyone could tell the difference without a microscope). EDIT: forgot link; Complete list of fonts is here: https://ctan.org/tex-archive/fonts/cm/mf – Mark May 03 '18 at 20:26
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    @Mico I suppose that there was a need for two approximately equal size ratios, with all sizes in whole points, and the smallest size not illegibly small. When starting form 10pt one is then almost forced to what we have: 4pt might already be too small for the purpose, 6pt would make either 10pt:8pt < 8pt:6pt or 10pt:7pt>7pt:6pt, whereas 10pt:7pt = 7pt:5pt, approximately – Hagen von Eitzen May 03 '18 at 21:25
  • @chepner Also other series use the square root of 2, even if in a different way. In the case of the A series is for practical reasons, i.e. you can obtain an a5 paper by cut in half an a4 paper. Anyway I agree that it could be related. – gvgramazio May 04 '18 at 08:05
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The TeXbook describes this choice making a reference to the equal-tempering in musical instruments. Don't forget that Knuth is a musician himself and plays the organ.

The octave is divided in semitones having respective ratio the 12th root of 2. This gives slightly “untuned” notes, because, for instance, the dominant should have a ratio 3/2 with respect to the tonic, whereas

27/12 = 1.498...

(there are seven semitones to go to the dominant). The difference is very small, but noticeable for people with “absolute pitch”.

By choosing an “equal-tempered” scale based on 1.2, we have that the square root of 1.2 is 1.095 (not so different from 1.1) and scaling a 10pt font with these ratios we get

10pt 10.95pt 12pt 14.4pt 17.28pt 20.736pt 24.88pt

which are remarkably near to the point sizes actually used in metal typography:

10pt 11pt 12pt 14pt 18pt 20pt 24pt

see https://en.wikipedia.org/wiki/Traditional_point-size_names

egreg
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