Dante Monotype typeface and fractions

Question

I am handling a text which is natively written in a proprietary LaTeX documentclass which implements Dante Monotype fontface. In this text there is a lot of maths and I would say that it works, overall, even if I would not advise in particular Dante font for maths applications.

Even exponents ($e^{x^2}$) are, say, well proportioned to my eye while what I do not like at all are fractions. As in the following picture:

the fraction circled in red, as well as the following ones, has numerator and denominator that are very far from the fraction bar.

How can I improve this in particular? Any package related to fractions may work currently with this font? Any command similar to \dfrac or \tfrac may help decreasing (rather than increasing) the total height of the fractions?

No further solutions come to my mind to fix such "length".

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Addendum:

I have MikTeX 2.9 on Windows 7 OS. The editors I use are TeXstudio and TeXnicCenter and I compile with pdfLaTeX. The behaviour of the fractions is the same when using TeXshop on Mac OS. Thank you.

The code for the picture above is:

$$y_0(t)=1$$
$$y_1(t)=1+\int^t_0 \mt\,\md\mt=1+\frac{t^2}{2}$$
$$y_2(t)=1+\int^t_0 \mt\left(1+\frac{\mt^2}{2}\right)\md\mt=1+\frac{t^2}{2}+\frac{t^4}{8}$$
\vspace{1pt}
$$\vdots$$
\vspace{1pt}
$$y_n(t)=1+\int^t_0 \mt\,y_{n-1}(\mt)\,\md\mt=1+\frac{t^2}{2}+\frac{1}{2!}\left(\frac{t^2}{2}\right)^2+\ldots+\frac{1}{n!}\left(\frac{t^2}{2}\right)^n$$

where \md and \mt were defined in my preamble. Thank you.

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Addendum (2):

I appreciated both the answers posted so far and I tried to define a newcommand as in lines 2 to 15 of the code by Steven B. (I preferred this approach because I never put my hands on LuaLaTeX before and this was meant to be "just fine tuning"). Furthermore I tried all of the following three:

\renewcommand{\frac}[2]{\displaystyle{\frac{\lower2pt\hbox{#1}}{\raise2pt\hbox{#2}}}}

and:

\renewcommand{\frac}[2]{\dfrac{\lower2pt\hbox{#1}}{\raise2pt\hbox{#2}}}

or:

\newcommand\newfrac[2]{\frac{\lower2pt\hbox{#1}}{\raise2pt\hbox{#2}}}

but all of these four candidate solutions generated several errors in my system (class). Do these three lines of me suggest something useful?

I'll post updates.

Answer 1

I don't have access to your font, but (as indicated in my comment), my answer at A special fraction macro with raised fraction bar can address your issue. Here, I create it as \newfrac.

I modified the dimensions from the cited answer to further squeeze the fractions together, for demonstration purposes. The optional arguments for \stackunder and \stackon are the gap distances to adjust to suit your needs. This MWE, of course, is using the CM fonts.

\documentclass{article}
\usepackage{stackengine}
\usepackage{ifthen}
\newlength\testwida
\newlength\testwidb
\newlength\mywidth
\newcommand\newfrac[2]{%
  \setlength\testwida{\widthof{$#1$}}%
  \setlength\testwidb{\widthof{$#2$}}%
  \ifthenelse{\lengthtest{\testwida>\testwidb}}%
    {\mywidth=\testwida}%
    {\mywidth=\testwidb}%
  \raisebox{.5ex}{%
    \stackunder[2pt]{\stackon[2pt]{\rule{\mywidth}{.4pt}}{$#1$}}{$#2$}}%
}
\def\mt{\mathbf{t}}
\def\md{\mathbf{d}}
%\let\newfrac\frac% UNCOMMENT TO SEE ORIGINAL \frac VERSION
\begin{document}
$$y_0(t)=1$$
$$y_1(t)=1+\int^t_0 \mt\,\md\mt=1+\newfrac{t^2}{2}$$
$$y_2(t)=1+\int^t_0 \mt\left(1+\newfrac{\mt^2}{2}\right)\md\mt=1+\newfrac{t^2}{2}+\newfrac{t^4}{8}$$
\vspace{1pt}
$$\vdots$$
\vspace{1pt}
$$y_n(t)=1+\int^t_0 \mt\,y_{n-1}(\mt)\,\md\mt=1+\newfrac{t^2}{2}+\newfrac{1}{2!}\left(\newfrac{t^2}{2}\right)^2+\ldots+\newfrac{1}{n!}\left(\newfrac{t^2}{2}\right)^n$$

\end{document}

For comparison, here is the original \frac version.

Answer 2

The following is primarily an attempt to replicate your screenshot, which is based on a proprietary setup, with non-proprietary tools. Hopefully, the answer also provides a usable suggestion for how the issue of the excessive distance of the numerator and denominator terms from the fraction bars may be addressed.

My code uses LuaLaTeX as I don't have access to a non-Opentype version of the Dante font; specifically, I use Dante MT Std. I also use the Cambria Math math font, as I find it to be a much better match to Dante than, say, TeX Gyre Pagella Math is. The match isn't perfect; Cambria Math is simply the best match I was able to come up with. Fortunately, the match can be improved considerably by using various \setmathfont directives to instruct the system to use Dante MT glyphs for all digits and letters as well as for various symbols.

Note that the numerator and denominator terms are not nearly as far above and below the fraction bars as they are in your proprietary setup.

Summing up: Is switching from pdfLaTeX to LuaLaTeX and using the Cambria Math math font family -- along with using either Dante MT Std or Dante MT Pro as the text font -- an option for you?

% !TEX TS-program = lualatex
\documentclass{article}

% Set text font
\usepackage[no-math]{fontspec}
\setmainfont[Numbers=OldStyle]{Dante MT Std} 

\usepackage{amsmath} % for 'gather*' environment

% Set default math font
\usepackage{unicode-math}
\setmathfont[Scale=MatchLowercase]{Cambria Math}

% Use Dante MT for numerals, all letters, and various symbols
\setmathfont[range={"0030-"0039},Numbers=OldStyle]{Dante MT Std}
\setmathfont[range=\mathrm]{Dante MT Std}
\setmathfont[range=\mathit]{Dante MT Std Italic}
\setmathfont[range=\mathbfup]{Dante MT Std Bold}
\setmathfont[range=\mathbfit]{Dante MT Std Bold Italic}
\setmathfont[range={"0025,"002A-"002F,
     "003A-"003F,"005B-"005D,"007B-"007D,
     "03A9,"00B0-"00B7,"00BC-"00BE,"00D7,
     "00F7,"03A9,"03BC,"03C0,"2153,"2154,
     "215B-"215E,"2202,"2206,"2212,"2215,
     "2219,"221E,"2248,"2260,"2264,"2265}]{Dante MT Std}

% A couple of user-defined shorthand macros
\newcommand{\md}{\mathrm{d}}
\newcommand{\mt}{\mathbf{t}}

\begin{document}
\begin{gather*}
y_0(t)=1\\
y_1(t)=1+\int^t_0 \mt\,\md\mt=1+\frac{t^2}{2}\\
y_2(t)=1+\int^t_0 \mt\left(1+\frac{\mt^2}{2}\right)\md\mt=1+\frac{t^2}{2}+\frac{t^4}{8}\\
\vdots \\
y_n(t)=1+\int^t_0 \mt\,y_{n-1}(\mt)\,\md\mt=
1+\frac{t^2}{2}+\frac{1}{2!}\left(\frac{t^2}{2}\right)^{\!2}+\dots+
\frac{1}{n!}\left(\frac{t^2}{2}\right)^{\!n}
\end{gather*}
\end{document}

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Addendum: Speaking for myself, I find the use of oldstyle numbers in mathematics rather questionable. Here's the look that results from omitting the Numbers=OldStyle option in the second \setmathfont directive of the MWE. (You are, of course, free to decide for yourself if you prefer oldstyle or lining numeral in mathematics.)