A taxpayer shall pay no less than one-quarter of the estimated tax to the tax commissioner on April fifteenth, June fifteenth, and September fifteenth of the taxable year, and January fifteenth of the following taxable year; provided, that a taxpayer having a taxable year other than a calendar year shall pay the estimated tax on the fifteenth day of the fourth, sixth, and ninth months of the taxable year, and the fifteenth day of the first month of the following taxable year. In the case of a tax year that is for a period of less than one year, and the short tax year ends prior to any remaining due dates under this section, the final estimated tax payment is due on the fifteenth day of the last month of the short tax year. In the case of a tax year that is for a period of less than one hundred twenty days, no estimated tax payment is due.