This is just painful. Instruments do not have scales. Music is written in different keys, and often modulates to another key and back again during the piece. In order to cover the range of pitches, different clefs are used to make the notation easier to read. Piano music is written in two clefs, bass and treble, centered around "middle C". This is called the grand staff. Depending upon an instrument's range different clefs are used. Violoncellos use bass, tenor, and treble clef. Violas use alto clef.

Scales are a function that acts upon the set of pitches used in music. A C major scale has the pitches C, D, E, F, G, A, and B. A G major scale (function) has the pitches G, A, B, C, D, E, and F#.

"Mapping relative pitches to a scale" is nonsensical. A scale (function) is an absolute set of pitches, immutable.

There are two forms of solfège (do, re, mi, ...): fixed and moveable. In moveable the first pitch of the scale is always do. In fixed the pitch called C is always do. Watch "The Sounds of Music" to learn the solfège for a major scale, it's the do-re-mi song. There are other syllables used for the three minor scale forms. To my knowledge (admittedly not complete) Italy, France, and perhaps England used a fixed-do system, whereas the rest of the world uses a moveable-so system.