Step 1: Identifying the Ragas:
Let us analyze the three distinctive melodic phrases provided:
• Phrase (i): $\underline{Ga} \, Ma \, Dha \, \underline{Ni} \, Dha, \, Ma \, Pa \, Dha \, \underline{Ga}$.
This phrase contains flat Gandhar ($\underline{Ga}$), flat Nishad ($\underline{Ni}$), and the signature crooked, weak use of Pancham ($Pa$). This is clearly Raga Bageshri.
• Phrase (ii): $Ga \, Ma \, \underline{Dha} \, Pa, \, \underline{Re} \, Ni \, Sa$.
This phrase contains flat Rishabh ($\underline{Re}$), flat Dhaivat ($\underline{Dha}$), and Shuddha $Ga, Ma, Pa, Ni$. The slow, heavy oscillations on $\underline{Re}$ and $\underline{Dha}$ characterize this catchphrase. This is clearly Raga Bhairav.
• Phrase (iii): $\underline{Dha} \, \underline{Ni} \, \underline{Dha} \, Ma, \, \underline{Ga} \, \underline{Ni} \, Sa$.
This phrase has completely omitted Rishabh ($Re$) and Pancham ($Pa$), leaving exactly 5 notes: flat $\underline{Ga}$, flat $\underline{Dha}$, and flat $\underline{Ni}$. This is clearly Raga Malkauns.
Step 2: Melodic Swara-Expansion (Vistar/Alaap) of Raga Bageshri (80 Swaras):
We expand the raga step-by-step from the lower octave up to the higher octave:
• $Sa, \, \underline{Ni}_{\bullet} \, Dha_{\bullet} \, Sa, \, \underline{Ni}_{\bullet} \, Dha_{\bullet} \, Sa$ (7)
• $Sa, \, \underline{Ga} \, Ma, \, Ma \, \underline{Ga} \, Re, \, Sa$ (7)
• $Sa, \, \underline{Ni}_{\bullet} \, Dha_{\bullet} \, Sa, \, \underline{Ga} \, Ma, \, Dha, \, Ma, \, \underline{Ga} \, Ma \, \underline{Ga} \, Re, \, Sa$ (14)
• $Sa, \, \underline{Ga} \, Ma, \, Dha, \, \underline{Ni} \, Dha, \, Ma, \, Dha, \, Ma \, Pa \, Dha \, \underline{Ga}, \, Ma \, \underline{Ga} \, Re, \, Sa$ (17)
• $Ma, \, Dha, \, \underline{Ni}, \, \dot{S}a, \, \dot{S}a \, \underline{Ni} \, Dha, \, Ma \, Pa \, Dha, \, Ma, \, \underline{Ga} \, Ma, \, \underline{Ga} \, Re, \, Sa$ (17)
• $Sa, \, \underline{Ga} \, Ma, \, Dha, \, \underline{Ni}, \, \dot{S}a, \, \dot{Re} \, \dot{S}a, \, \underline{Ni} \, Dha, \, Ma \, Pa \, Dha \, \underline{Ga}, \, Ma \, \underline{Ga} \, Re, \, Sa$ (18)
Total: $7 + 7 + 14 + 17 + 17 + 18 = 80$ Swaras.
Step 3: Melodic Swara-Expansion (Vistar/Alaap) of Raga Bhairav (80 Swaras):
We expand using the slow, meditative, devotional phrases of Bhairav:
• $Sa, \, \underline{Re} \, \underline{Re} \, Sa, \, \underline{Re}_{\bullet} \, \underline{Re}_{\bullet} \, Sa$ (6)
• $Sa, \, \underline{Re}, \, Ga, \, Ma, \, Ga, \, \underline{Re}, \, Sa$ (7)
• $Sa, \, \underline{Re}, \, Ga, \, Ma, \, Pa, \, Ma, \, Ga, \, \underline{Re}, \, Sa$ (9)
• $Ga, \, Ma, \, \underline{Dha} \, \underline{Dha} \, Pa, \, Ma, \, Pa, \, \underline{Dha}, \, \underline{Dha} \, Pa, \, Ma, \, Ga, \, \underline{Re}, \, Sa$ (14)
• $Pa, \, \underline{Dha}, \, Ni, \, \dot{S}a, \, \dot{S}a \, Ni, \, \underline{Dha}, \, Pa, \, Ma, \, Pa, \, Ga, \, Ma, \, \underline{Re}, \, Sa$ (14)
• $Sa, \, \underline{Re}, \, Ga, \, Ma, \, Pa, \, \underline{Dha}, \, Ni, \, \dot{S}a, \, \dot{Re} \, \dot{S}a, \, Ni, \, \underline{Dha}, \, Pa, \, Ma, \, Ga, \, \underline{Re}, \, Sa$ (17)
• $Ga, \, Ma, \, Pa, \, \underline{Dha}, \, \underline{Dha} \, Pa, \, Ma, \, Ga, \, \underline{Re}, \, \underline{Re} \, Sa$ (10)
Total: $6 + 7 + 9 + 14 + 14 + 17 + 10 = 80$ Swaras.
Step 4: Melodic Swara-Expansion (Vistar/Alaap) of Raga Malkauns (80 Swaras):
We expand Malkauns using pentatonic movements, emphasizing the Madhyam ($Ma$):
• $Sa, \, \underline{Ni}_{\bullet} \, \underline{Dha}_{\bullet} \, \underline{Ni}_{\bullet} \, Sa, \, \underline{Ga} \, Ma, \, \underline{Ga} \, Sa$ (9)
• $Sa, \, \underline{Ga} \, Ma, \, \underline{Dha}, \, Ma, \, \underline{Ga} \, Ma, \, \underline{Ga} \, Sa$ (9)
• $Sa, \, \underline{Ga} \, Ma, \, \underline{Dha}, \, \underline{Ni}, \, \underline{Dha}, \, Ma, \, \underline{Ga} \, Ma, \, \underline{Ga} \, Sa$ (11)
• $\underline{Ga} \, Ma, \, \underline{Dha}, \, \underline{Ni}, \, \dot{S}a, \, \dot{S}a \, \underline{Ni} \, \underline{Dha}, \, Ma, \, \underline{Ga} \, Ma, \, \underline{Ga} \, Sa$ (12)
• $Ma, \, \underline{Dha}, \, \underline{Ni}, \, \dot{S}a, \, \dot{Ga} \, \dot{S}a, \, \underline{Ni} \, \underline{Dha}, \, Ma, \, \underline{Ga} \, Ma, \, \underline{Ga} \, Sa$ (12)
• $Sa, \, \underline{Ga} \, Ma, \, \underline{Dha}, \, \underline{Ni}, \, \dot{S}a, \, \dot{Ga} \, \dot{Ma} \, \dot{Ga} \, \dot{S}a, \, \underline{Ni} \, \underline{Dha}, \, Ma, \, \underline{Ga} \, Ma, \, \underline{Ga} \, Sa$ (17)
• $\underline{Dha}, \, \underline{Ni}, \, \underline{Dha}, \, Ma, \, \underline{Ga} \, Ma, \, \underline{Dha} \, Ma, \, \underline{Ga} \, Sa, \, Sa$ (10)
Total: $9 + 9 + 11 + 12 + 12 + 17 + 10 = 80$ Swaras.