Gear normal pitch
= Gear normal pitch
{title2=$p_n=\pi m_n$}
<Gear normal pitch> $p_n$ is the repeat spacing of corresponding <helical gear> tooth flanks normal to their tooth direction on the developed reference cylinder. It equals $\pi$ times <gear normal module> $m_n$. It is a different geometric spacing from <gear transverse pitch> $p_t$, measured circumferentially, and from <gear axial pitch> $p_x$, measured parallel to the rotation axis.<KHK Gear Nomenclature>[Technical reference §4.3, equations(4.13)–(4.14) and Figure4.7, normal and transverse systems]{ref}
Unwrap the reference cylinder into a plane. The equally spaced tooth traces become parallel straight lines making <gear helix angle> $\beta$ with the axis. Projecting the transverse spacing onto the normal to those traces gives
$$
p_n=p_t\cos\beta=\pi m_t\cos\beta=\pi m_n.
$$
For reference diameter $d=60\,\mathrm{mm}$, $N=30$ teeth and $\beta=30\,{}^{\circ}$, $p_t=\pi d/N=2\pi\,\mathrm{mm}$, whereas $p_n=\sqrt3\pi\,\mathrm{mm}\approx5.4414\,\mathrm{mm}$. Multiplying $p_n$ by $N$ gives about $163.24\,\mathrm{mm}$, not the $188.50\,\mathrm{mm}$ circumference: normal pitch is not the circumferential step.
At $\beta=0$, normal and transverse pitches agree, as expected for a <spur gear>. For nonzero <gear helix angle>, using one in place of the other introduces a factor $\cos\beta$. Normal pitch is also distinct from <gear base pitch>, which refers to the base-circle geometry of an <involute>. Matching a pitch value without its measurement convention, pressure angle and tooth geometry is insufficient to establish a compatible mesh.