It's not the umlaut which is sitting lower, but the i-dot which is sitting higher. No pun. In most typefaces the umlaut is in line with other accents.
The i-dot may sit higher for aesthetic reasons, but I am not sure if this is a must. (I think Mr Gaultney raises the issue in his essay on diacritic design, and following it I have tried to drop the i-dot a bit but found the result unpleasing. At least in German, the combination of i and umlaut-letter doesn't occur often -- if at all.)

Karsten

I can think of two possible reasons:
1) Since the tittle spans further left, it collides with the "f" more [often], so bringing it down helps.
2) Unless one makes the dots much smaller (which would cause a disturbance similar to the one caused by the height mismatch you mention/imply) they combine to have too much more presence than the single dot on the "i", so bringing them down makes them stand out less (see below).

> The i-dot may sit higher for aesthetic reasons

Or a functional one: since the dot is the main thing the "i"
can contribute to readability, making it more prominent helps.
I've done that (as well as make it slightly "too big") in Patria.

hhp

I am of the school that says the diaeresis should always sit at the same height as the dot on the i, along with the dot on ċ ė ġ ż ȯ ṁ ṅ ṗ ṡ ṩ

Of course, this presumes that the dot on the i isn't in the stratosphere. For a language like English, of course, the dot on the i can be virtually any height, since there are almost no accent marks to be concerned with. But as soon as you have a language with a rich diacritical set you want the height of the dot on the i to be constrained to a reasonable height. In a typically proportioned text face, I usually align the top of the dot somewhere close to the cap height.

If I recall correctly, Adobe revised the diaeresis height in all their fonts based on user feedback from several European countries. In all recent Adobe fonts, the dots align as I have shown above.

I concur with Mr Hudson, and I also align my dot and diaeresis/dieresis/umlaut accents. One of the oddest dot location things I have ever seen is in the text face used in The New Yorker, where the dots on the i and j are at slightly different heights. Looks very odd in a word like "Nijinsky".

> user feedback

Ah, the solution to all our problems...

> the dots on the i and j are at slightly different heights.

Yes, another useful trick. :-)

hhp

I usually make my dieresis dots a bit smaller than the i dot, for two reasons:

1. So that, compared with other accents, it (the dieresis) is not too heavy -- it has to have "single accent" weight, and not look like two 'i' dots.

2. Because of the f, as Hrant says, but also because of ascenders that may follow, such as b, h, k and l.

I line them up along the bottom, with the i dot.
I sometimes vary the width betwen the dots, depending on the width of the character.
A lot of little "cheats".

Seems that my i-dot placement is a bit narrow minded, and some design decisions turn out to be national peculiarities which get evened out as character sets grow. Approaching diacritics a more systematic way is one consequence. Also the peculiar Czech caron comes to mind, or Ä Ö Ü with dropped umlaut (in German typefaces) which are just out of place in larger character sets.
As regards your design samples, it is interesting that versals and ascenders are comparably small which doesn't leave much choice for placing accents.

"The doubling of the ї is not unusual either (e.g., її, ‘her/hers’)."

This is good to know!

...national peculiarities which get evened out as character sets grow.

Not necessarily even -- it seems that the more demanding languages are paid more attention in the mega-language font; for instance, when some languages have proponents who are quite adamant about having steep acute accents, and others traditionally have a flatter angle, and no strong voice arguing for that in the type design community, then the steeper accent will predominate.

Nick, I don't usually make my diaeresis dots as much smaller than the i-dot as in your examples, but I do make them smaller and, like you, align them to the bottom of the i-dot.

> align them to the bottom

Optically or mathematically?

BTW, when you wrote "the diaeresis should always sit at the same height as the dot on the i" I assumed you meant their entire bodies should look to be the same height (which would mean aligning the centers, not the bottoms).

hhp

Optically or mathematically?

Given that they are similarly shaped forms of similar size, I don't see a difference between optical and mathematical in this instance, at least given the coarseness of the typical UPM grid.

Aligning the diaeresis dots to the bottom of the i-dot works better than aligning the centres of the dots. If you align the centres, the diaresis tends to sit too high relative to other accent marks.

> The doubling of the ї is not unusual either

I'd missed this...
Well, well, Mana's kerning pair for її wasn't that anal after all! :-)

> I don’t see a difference between optical
> and mathematical in this instance

It's probably small, but probably more than 1/1000 em.
Not that I adjust stuff below 2 em myself generally,
except to combat grid coarseness (like in an italic).

Anyway, I really don't know, since I've only
made the diaresis visibly lower (and smaller,
dotwise). Probably with the Baskerville I'll
have to be more Modernist though.

> If you align the centres, the diaresis tends
> to sit too high relative to other accent marks.

Even the ones with a lot of vertical span?

hhp

Dear Maxim G. Zhukov, You remembered about such typical ukrainian costruction as /quoteright+idieresis :)))