Narada Purana (Jyotisha Selections) · by Vyasa
Chapter 6 · Calendar, Muhurta Elements
कालमानं मुहूर्ताङ्गानि च
तन्नक्षत्राह्वयो मासः पौर्णमासी तदाह्वया ।तत्पक्षौ दैवपित्राख्यौ शुक्लकृष्णौ तथापरे ॥ १३२ ॥
In the full Narada Purana (Purva Bhaga) this is chapter 56, verse 132.
... takes the name of that nakshatra (star). The full moon takes that name too. Its two fortnights are called the gods' and the ancestors'. Others call them the bright and the dark.
Simple meaning: A month is named after the star its full moon sits in. Its two fortnights are the bright and the dark.
शुभाशुभे कर्मणि च प्रशस्तौ भवतः सदा ।
The two fortnights are both praised for acts. The first five moon days are ruled by Brahma, Agni, Virinchi, Vishnu and Shailaja.
विनायकयमौ नागचंद्रौ स्कंदोऽर्कवासवौ ।
This verse lists more gods who rule the moon days in order.
शिवविष्णूहरिरवीकामःसर्वःकलीततः ।
The list of moon-day gods goes on. The new moon day is ruled by the ancestors.