How to remove a theme's default menu class from a specific menu?
You can remove all unwanted classes and add your own classes using the code below, Its for a rough idea, use dpm or any debugger to locate exact tool

function THEME_menu_link(array $variables) {
  //unset all the classes
  unset($variables['element']['#attributes']['class']);

  $element = $variables['element'];

  if($variables['element']['#attributes'])

  $sub_menu = '';


  if ($element['#below']) {
    $sub_menu = drupal_render($element['#below']);
  }
  $output = l($element['#title'], $element['#href'], $element['#localized_options']);
  return '&lt;li' . drupal_attributes($element['#attributes']) . '&gt;' . $output . $sub_menu . "&lt;/li&gt;\n";
}