When the people sing the song, we use do, re, mi, fa, sol, la, ti, si, ect, but some time I hear that the music people say A, B, C, C flat, ect.
Is their relationship or are they two different things?
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.
In both forms, there are other phonetic names for sharp and flat names. The complete list is on Wikipedia.
On a Fixed Do scale the A Note is assigned to La, the A was assigned because it has the precision of the frequency (440 Hz) which doesn't have decimals, so it's easier to remember. So you end up with a correspondence as follows
A = La
B = Si
C = Do
D = Re
E = Mi
F = Fa
G = Sol
Take into account that while on the english notation (A,B,C,D) you start on the 400 Hz note which is A4, on fixed Do you start on the first discernible note (which is C0 at 16.3516 Hz)
By the way the note names on the Fixed Do scale come from a poem of the benedictine monk "Paul the Deacon" which goes like this:
Ut queant laxis
And the Ut was replaced by Do to make it easier to say on the Romance languages
Solfege (do, re, mi, ...) is a numbering system which starts from the first note of the scale (do=1, re=2, ...). Solfege identifies the position of the note in the scale.
Pitch names are a numbering system which starts from a particular sound (A440 or concert A=1, B=2, ...). Pitch names identify the sound of the note in the scale.
Take an example. Say you are singing a song that’s in F major. The pitch names for that scale will be F G A B♭ C D E F, and the solfege for that scale will be do re mi fa so la ti do.
Now, sing that song in G major. The pitch names will change to G A B C D E F♯ G, but the solfege for that scale will still be do re mi fa so la ti do.
If you have ever looked at pop sheet music in Spanish, you'll see that they don't use "C, D, E, F, G, A, B" at all. They only call notes by their Fixed Do names.
In Spanish sheet music, a C dominant 7 chord is called "Do7". A G# dominant 7 chord is called "So#7".
Thank you for your interest in this question.
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