What is the relationship between “do re mi” and note letter names?

Answer

Do-re-mi-etc. is "sol-fa" or "solfege".
Sol-fa represents a major scale, with Do being the first note, Re being the second, and so on. I'm sure you can sing that scale.
The A-G note names are absolute names for a certain note. An 'A' is an 'A' no matter what key you are performing in.
There are two variants of sol-fa. Fixed do and Movable do.
Fixed do is used in China, France, Italy, Portugal, Spain, Romania, Russia, South America and parts of North America, Japan, and Vietnam.
In fixed-do, Do is always equivalent to C; Re is always equivalent to D; and so on.
Movable Do is used in Britain, Germany, Indian classical music, and the United States.
In movable-do, you can pick a different pitch to start on, and sing Do-re-mi-... starting at that note. What you are doing when you do that is singing the major scale in different keys.
If you are singing in C major, d,r,m,f,s,l,t,d is C,D,E,F,G,A,B,C
If you are singing in D major, d,r,m,f,s,l,t,d is D,E,F♯,G,A,B,C♯,D
... and so on.

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