Ugly things indeed, Mojca. And even more so given my limited skills.  
Here is what I have for  “H” with a breve accent below.
\def\Hbrevebelow{H\smash{\lower1.9ex\hbox{\kern-3pt\llap{\char'010}}} 
\kern3pt}
This only “works” in an italic environment by the way.

With apologies to the wizards on this list,

Alan

On May 14, 2008, at 13;17,49 , Mojca Miklavec wrote:

> On Wed, May 14, 2008 at 4:41 PM, Alan Bowen wrote:
>> Many thanks, Idris and Mojca.
>>
>> If all goes well, I will move to luatex this summer---that really  
>> does
>> seem to be the way to go---and then investigate XeTeX, which I gather
>> is now functioning well with luatex. For now, however, I will have to
>> cobble a macro using hints that Mojca provided.
>
> My hint(s) only work in XeTeX and LuaTeX.
>
>> By the way I am using Latin Modern (ec encoding).
>
> With ec encoding you really need to do ugly things. You need to
> compose a character by more or less manually lowering the breve accent
> and take care for italic correction yourself. Not even accent
> primitive can be used for it. And you cannot use the combining accent
> after the character or the proper Unicode glyph (though the latter
> might be possible by patching some definition in ConTeXt core), you
> always need to use a macro.
>
> With XeTeX and LuaTeX all that comes (almost) for free.
>
> Mojca
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