Contribution Page payment not processed if no membership selected in price set

On a Contribution Page intended for membership renewal plus an optional donation, which uses a membership price set containing a Select field for membership type, and a Text/Numeric Quantity field for an additional optional donation, if no membership type is selected then the contribution will not be processed after clicking Contribute, but the next page will indicate that the payment was processed successfully. However no contribution will be recorded and no email is sent to the contributor.

This only happens when a membership price set it used.

It is a particular problem for lifetime members, being a case where no membership type will be selected.

To reproduce:

• Create a price set, used for Memberships
  • Add a select field, "Membership type" and add options to it including lifetime member (set to not a required field)
  • Add a text/numeric field, "Optional donation" and configure with unit price of $1.00 (set to not a required field)
• Create a contribution page, financial type Member Dues
  • In the Memberships tab select the price set created above
  • Unselect Require Membership Signup
  • Set up Receipt Tab to send email receipt
• Add a lifetime membership to a contact
• Open the contribution page created above for the contact with a lifetime membership
• Enter a donation amount in the membership price set's text/numeric field
• Don't select a membership type
• Fill in credit card details etc and click Contribute and go through the confirmation page
• The Thank You page will appear and indicate that the payment was processed. However no email is sent to the contributor and no contribution will be recorded.

Originally found on Joomla 3.8.3 + CiviCRM 4.7.29. Tested on wpmaster.demo.drupal.org ("4.7.31") and the problem also occurs there.