Bhashya
विमर्शे हि कृते सति, प्रतिवचनोक्तिरुपपन्ना-- वागवऋक् प्राण साम ओमित्येतदक्षरमुद्गीथ इति ।वागृचोरेक-त्वेऽपि न अष्टमत्वव्याघात , पूर्वस्मात् वाक्यान्तरत्वात् , आ-प्तिगुणसिद्धये हि ओमित्येतदक्षरमुद्गीथ इति ।वाक्प्राणौऋक्सामयोनी इति वागेव ऋक् प्राण साम इत्युच्यत , यथा-क्रमम् ऋक्सामयोन्योर्वाक्प्राणयोर्ग्रहणे हि सर्वासामृचा स-र्वेषा च साम्नामवरोध कृत स्यात् , सर्वर्क्सामानवरोधे चऋक्सामसाध्याना च सर्वकर्मणामवरोध कृत स्यात् , तद-वरोधे च सर्वे कामा अवरुद्धा स्यु ।ओमित्येतदक्षरम्उद्गीथ इति भक्त्याशङ्का निवर्त्यते ।तद्वा एतत् इति मिथुनंनिर्दिश्यते ।कि तन्मिथुनमिति, आह-- यद्वाक्च प्राणश्चसर्वर्क्सामकारणभूतौ मिथुनम् , ऋक्च साम चेति ऋक्सा-मकारणौ ऋक्सामशब्दोक्तावित्यर्थ , न तु स्वातन्त्र्येण ऋक्चसाम च मिथुनम् ।अन्यथा हि वाक्प्राणश्च इत्येकं मिथुनम् , ऋक्साम च अपरम् , इति द्वे मिथुने स्याताम् , तथा च तद्वा एतन्मिथुनम् इत्येकवचननिर्देशोऽनुपपन्न स्यात् , तस्मात् ऋक्सामयोर्न्योर्वाक्प्राणयोरेव मिथुनत्वम् ॥
In plain words
Once the inquiry is made, the reply naturally follows: speech is the ṛk, prāṇa is the sāman, and the syllable OM is the udgītha. Although speech and the ṛk are identical, the eighth position remains undisturbed because this is a separate sentence, uttered to secure the fruit of attainment. Because speech and prāṇa are the underlying sources of both ṛk and sāman, the text states 'speech is ṛk, prāṇa is sāman.' By grasping speech and prāṇa as the sources of ṛk and sāman, one implicitly includes all ṛks and sāmans, thereby including all rituals performed through them, and through that, fulfilling all desires. This dispels any notion that calling OM the udgītha is merely figurative. The text points to a pair: 'which are speech and prāṇa.' They are called the sources of ṛk and sāman, referred to by those very names, rather than ṛk and sāman forming an independent pair by themselves. Otherwise, speech and prāṇa would form one pair, and ṛk and sāman another, resulting in two pairs—making the singular phrase 'that indeed is this pair' contradictory. Therefore, the pairing is exclusively between speech and prāṇa as the sources of ṛk and sāman.
Close to the Sanskrit
When deliberation is made, the utterance of the reply is appropriate: 'speech is ṛk, prāṇa is sāman, this syllable om is the udgītha.' Even though speech and ṛk are identical, there is no contradiction of eighthness, because it is a different sentence; and for the accomplishment of the quality of attainment, 'this syllable om is the udgītha' is stated. Because speech and prāṇa are the sources of ṛk and sāman, 'speech is ṛk, prāṇa is sāman' is spoken—respectively, by grasping speech and prāṇa as the sources of ṛk and sāman, the inclusion of all ṛks and all sāmans would be accomplished; and with the inclusion of all ṛks and sāmans, the inclusion of all actions accomplished by ṛk and sāman would be accomplished; and with their inclusion, all desires would be attained. The doubt that 'om is this syllable, the udgītha' is a mere secondary designation is removed. 'That indeed is this' indicates a pair. What is that pair? He says: 'which are speech and prāṇa'—they are the causes of all ṛks and sāmans, being called by the words ṛk and sāman, meaning they are designated by the words ṛk and sāman, not that ṛk and sāman independently form a pair. Otherwise, speech and prāṇa would be one pair, and ṛk and sāman another, thus there would be two pairs, and then the singular expression 'that indeed is this pair' would be untenable. Therefore, the pairing belongs solely to speech and prāṇa, which are the sources of ṛk and sāman.