What type of tenancy has a duration that is indefinite and determined by the frequency of rent payments?

The correct answer is periodic tenancy, which is characterized by its indefinite duration and the frequency of rent payments. In a periodic tenancy, the lease automatically renews at the end of each rent period, whether that's weekly, monthly, or annually, until either the landlord or tenant gives notice to terminate the tenancy. This creates a flexible arrangement where the contract is not fixed in terms of duration but continues based on the cycle of rent payments.

In contrast, an estate for years or lease for years is a tenancy for a specified, fixed term. This type of arrangement has defined start and end dates and does not automatically renew once that term has ended. A tenancy at sufferance arises when a tenant remains in possession of a property after their lease has expired, without the landlord's consent. This situation does not involve a regular schedule of rent payments and can be more precarious for the tenant. Therefore, periodic tenancy is the only option that fits the criteria of being indefinite and based on rent payment frequency.