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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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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

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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 3, 2018 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 3, 2018 at 11:32
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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 3, 2018 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 3, 2018 at 11:44
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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 3, 2018 at 21:25
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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

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