Understanding the concepts of isometric basis and musical isomorphism

[KungFu]

TL;DR

I want to find the isometric basis corresponding to the musical isomorphism between the vector space E = $R^3$ and its dual space $E^* = (R^3)^*)$ .

Im very new to the terminologies of isometric basis and musical isomorphism, will appreciate a lot if someone could explain this for me in a simple way for a guy with limited experience in this field.

The concrete problem I want to figure out is this:
Given:
Let $v_1 = (1,0,0) , v_2 = (1,1,0), v_3 = (0,1,1)$ be a basis in the vector space $E = R^3$

problem:
Find a dual basis to ${v_1,v_2,v_3}$ in $E^*$ = $(R^3)^*$ and the isometric basis corresponding to the musical isomorphism between E and E*.

I have found the dual basis: I did this by using the property $(v^*)^i(v_j) = \delta_{ij}$, where $(v^*)^i$ is the ith covector in the dual basis.
I represent the dual basis as a linear combination of the orthonormal basis in $(R^3)^*$, the dual basis is then
$(v_1)^* = (1,-1,1)$, $(v_2)^* = (0,1,-1)$ and $(v_3)^* = (0,0,1)$

now, to my question : how is the isometric basis corresponding to the musical isomorphism between E and $E^*$ defined ?
and even more basic : what do we mean be an isometric basis, and what do we mean by a musical isomorphism, you can as well try to explain what an isomorphism is first, in simple words ;)

 

[fresh_42]

What is a musical isomorphism?

 

[Stephen Tashi]

What is a musical isomorphism?

Perhaps it's a "cannonical" isomorphism, seeing as a "cannon" is a musical form.

 

[martinbn]

What is a musical isomorphism?

What the physicists call the rising and lowering indices. The two maps are often denoted by the sharp and flat symbols.