You must be paid at least 2 times a month. While there can be issues determining the correct pay if hours are not accounted for, and that might legitimately delay some pay, withholding your entire pay because of a date error is not permitted. You can contact the NYS Dept. of Labor and they can pursue a claim for unpaid wages for you at no cost.
You state that you are a private fitness contractor, which implies that you are not an employee. The distinction matters because the payment of wage provisions apply to employees. If you are an independent contractor your claim is contractual because the dance studio is not your employer.
Section 191 of the nys labor law governs payment of wages and requires that a manual worker be paid weekly and no later than the end of the week following the week in which the money was earned. There is an exception that permits an employer to pay semi monthly but the employer must be granted permission by the commissioner of labor.
Calling the dol will probably resolve your problem.