From the Gobhila Grihya Sutra

Through the variations of Paushyanji and Glipsa, the collections Samhitas were multiplied. || 5 ||
And the disciple of Hiranyanabha, the great-minded one named Kriti, taught twenty-four collections to his students. || 6 ||
Through them also, this Samaveda was expanded into many branches. || 7 ||" Vishnu Purana 3.6

Did the branches of the Vedas originate from Vyasa-ji?

Many people say that Lord Krishna Dvaipayana Another name for the sage Vyasa, who is traditionally credited with arranging the Vedas. (Maharishi Vyasa) was the divider of the Vedas, the seer mantra-drashta of the hymns, and the originator of the branches shakhas. It is also written this way in the Puranas (such as the Vishnu Purana and Bhagavata Purana), but this is not entirely correct.

Maharishi Vyasa was a peerless scholar of the Veda. He collected the Vedic mantras that were scattered here and there in an even better manner; he placed the mantras of each Veda in their proper locations and taught them to many disciples according to various collections Samhitas. He also used great effort to bring to light the branches that existed before his time but had become dormant or lost due to the passage of time, discovering them from various places and teaching them to his four disciples. In his era, the Veda reached its ultimate peak of development, and he was the unique promoter of the Veda at that time. Through his intellect, penance, and diligence, he traced and systematized all the branches of the Veda as far as possible. Because of this, it is said that the Veda "originated" from "Veda Vyasa" The name "Veda Vyasa" literally means "the one who divided or arranged the Veda.".

It is written: "The composition of the branches occurred in the Dvapara age" original: "शाखाप्रणयनं चैव द्वापरे सम्भूदिदम्" (śākhāpraṇayanaṃ caiva dvāpare sambhūdidam). This means that the branches of the Vedas continued to proliferate until the Dvapara Yuga In Hindu cosmology, the third of the four world ages.. We know this because even before the Dvapara, during the Treta Yuga, these branches existed. For instance, the Ayodhya Kanda of Valmiki’s Ramayana mentions "the teachers of the Taittiriyas" original: "आचार्यां स्तैत्तिरीयाणाम्" (ācāryāṃ staittirīyāṇām) and again, "these Kathas and Kalapas" original: "एतेकठाः कलापाश्च" (etekaṭhāḥ kalāpāśca). However, the Puranas state the opposite—that branches like the Katha and Kalapa originated among the disciples of Vyasa. The explanation for this contradiction has already been provided i.e., Vyasa revitalized and organized existing traditions rather than inventing them anew..

The Lineage of Teachers of the Samaveda

The list of teachers who promoted the tradition of the Samaveda * is recorded as follows: Brahma taught Brihaspati, he taught Narada, from him it went to Vishvaksena, he taught Vyasa the son of Parashara, and Vyasa taught Jaimini...

* Now, the author enumerates the teachers who promoted the tradition of this Samavidhana. This is the method of Prajapati. Prajapati spoke this to Brihaspati; Brihaspati to Narada; Narada to Vishvaksena; Vishvaksena to Vyasa, the son of Parashara; Vyasa to Jaimini; Jaimini to Paushpindya; Paushpindya to Parasharyayana; Parasharyayana to Badarayana; Badarayana to Tarkshya and Shalyani; and Tarkshya and Shalyani to many others.