The scales of Raga Hameer (or Hamir) Kalyani are as follows :
Aarohanam (Ascending) : S P M2 P D2 N3 S’
Avarohanam (Descending) : S’ N3 D2 P M2 M1 G3 P M1 R2 S
It is a Janya raga, derived from Mecha Kalyani which is 65th on the Melakarta Scale (see below).
This is an import from Hindustani Music but has become an integral part of Carnatic Music now. An elegant raga, it has a weave and flow built into it which, I think, makes it very pleasing indeed. It can be sung to show a sense of supplication; it can also convey a sense of gentleness and peace. There are some beautiful compisition in this raga, such as the Venkata Shaila by Subbaraya Shastri, Manamu Leda by Tyagaraja, Parimala Ranganatham by Muthuswami Dikshithar and Gangeya Vasana by Swathi Thirunal. But the first song I always remember is Thoomani Madathu, the Thiruppavai song set to Hameer Kalyani by the great Ariyakudi Ramanuja Iyengar.
Tamil speakers may enjoy this excellent episode on Hamir Kalyani by Charulata Mani.
Note : The 12 notes in the octave are named as below. Please note that C is used as Sa for the sake of simplicity as the scale is relative in Carnatic Music. Also note that the scales paint only a superficial picture of the raga as the gamakas(ornamentations) and prayogas (signature phrases) are a very important part of a raga.