Narada Purana (Jyotisha Selections) · by Vyasa
Chapter 6 · Calendar, Muhurta Elements
कालमानं मुहूर्ताङ्गानि च
शुभाशुभे कर्मणि च प्रशस्तौ भवतः सदा ।क्रमात्तिथीनां ब्रह्माग्नी विरिंचिविष्णुशैलजाः ॥ १३३ ॥
In the full Narada Purana (Purva Bhaga) this is chapter 56, verse 133.
Both are always praised for acts, good and bad. The tithi (moon day) lords in order are Brahma, Agni, Virinchi, Vishnu and Shailaja.
Simple meaning: 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.
नंदाभद्राजयारिक्ता पूर्णाः स्युस्तिथयः पुनः ।
Five kinds of moon day go round three times. In the bright fortnight the three rounds are poor, middling, then good.