First define a relation $\sim$ on $\mathbb{Z}^k$ such that for any $a,b\in \mathbb{Z}^k$ where $a=(a_1,\dotsc,a_k)$, and $b=(b_1,\dotsc,b_k)$ we write $a\sim b$ if and only if $a-b=(n,n,\dotsc,n)$ for some $n\in \mathbb{Z}$. It is easily verified that this is an equivalence relation and partitions the $k$-tuples of integers into translation classes, i.e. $$a=(a_1,\dotsc,a_k), a+1=(a_1+1,\dotsc,a_k+1),\dotsc,$$ $$a+n=(a_1+n,\dotsc,a_k+n),\dotsc$$ represents the same class, denoted $[a]$. Is it possible to define a metric on $\mathbb{Z}^k$ such that for fixed $k$-tuples of integers $a=(a_1,\dotsc,a_k)$, we have
1) $$d(a,a+n)=d(a,a)=0$$ for any $n\in \mathbb{Z}$? Additionally is it possible to define it so that
2) $$d(a,b)=0$$ if and only if $a\sim b$.
Rephrased: The distance between two $k$-tuples of integers is zero if and only if they are in the same translation class.
Examples: $$(0,4,7) \text{ and } (1,5,8)$$ should have distance zero, while $$(0,4,7) \text{ and } (1,5,9)$$ should have non-zero distance.
Context: i am a musician trying to have some fun w math as follows: i want to define a metric on $k$-tuples of integers in a way to represent the pitches of a melody and the distances between two melodies. I want melodies that are mere transpositions of each other to have "distance" zero while two melodies that cannot be obtained from one another via transposition to have non zero distance.
In my work in the $k$-tuples i had defined for any $k$-note melodies $a,b$ with pitch representation $a=(p_1,\dotsc,p_k)$ and $b=(q_1,\dotsc,q_k)$ the "metric" (i must prove it is a metric yet!) $d(a,b)=|\Delta p_i - \Delta q_i|$ Where $\Delta p_i=|p_i-p_{i+1}|$. This satisfies the property 1) i.e. that $d(a,a+n)=0$ (Using the def: $a+n=(p_1+n,\dotsc,p_k+n)$ where $n\in \mathbb{Z}$) but i am stuck showing it would satisfy property 2). Is property 2) false for my candidate metric? Is it even possible to find a metric where both 1) and 2) to hold?
(Sorry for the multiple edits, I asked a little too hastily before thinking everything through)
You can put a metric on any set. If you want a metric on the equivalence classes of $R$, where $x\equiv y \iff x-y\in Z,$ that has some relation to the algebraic structure of $R,$ I suggest $$d(x_E,y_E)=|(x-[x])-(y-[y])|$$ where $[x]$ is the largest integer not exceeding $x,$ and $x_E$ is the equivalence class containing $x$. Observe that for each $x\in R$ there is a unique $x'\in [0,1)\cap x_E$, so $d(x_E,y_E)=|x'-y'|.$