DAC Question: Delta-Sigma vs R2R

The critical thing is how the DAC sounds. Technology is important for the engineers designing the component, but implementation is everything. 
 

So, try and listen to as many DACs as you can, and read professional reviews of them. See if you can establish the kind of sound you like in standard audio terminology it will make finding a unit appropriate for you much easier. 
 

I got into high end audio fifty years ago and quickly changed my focus from technology to the sound of the unit. A good engineer uses a collection of technology to achieve a goal. 

If I were to point you in a direction it would be towards a tubed DAC. Tubed equipment tend (also highly dependent on the implementation) to be more natural sounding, especially in the budget category. The Border Patrol is immediately appealing for that reason and not the R2R thing. If you can find a used Audio Research DAC… also tubed… it would have the character it sounds like you are looking for. What is your other equipment… this makes a huge difference in the ultimate output. 

@big_greg 

+1

As @tcadden brings up… learning about sound and what you want is critical. I highly recommend this book:

The Complete Guide to High-End Audio

Book by Robert Harley

 Knowing the terminology and match your observations to each of them makes learning much quicker. This allows you to identify and articulate what you want and to read professional reviews.

OP,

Congratulations . You said what is important to you is: “but the natural and accurate tone of the instrument I am listening to” 

Yes… I second that. Every component you own effects the output and not concentrating on just detail can lead to a more satisfying system sound.

@mdrone 

I don’t find thinness an attribute of any particular design, but of the overall quality (and typically expense) of the unit. A really good designer will carefully select all the components as well as the design so as to optimize the sound. Typically as you get into better DACs… let’s just say >$6 or $7K the overall presence, bass, and natural sound quality get better as well as the noice floor getting lower… revealing more detail.