I'm starting to study music, and I'm reading about semitones and tones.
Looking at a piano, I know there are 8 notes in an octave:
Do – Re – Mi – Fa – Sol – La – Si – Do
Here the Do is counted twice.
Counting the semitones, it looks something like this (homophonic semitones grouped together):
Do – Do# (Re♭) – Re – Re# (Mi♭) – Mi – Fa – Fa# (Sol♭) – Sol – Sol# (La♭) – La – La# (Si♭) – Si
Which gives 12 semitones.
But why is Do not counted again in the end?
An octave is an interval composed of 12 semitones. A semitone is an interval, so they are:
C → C#, C# → D, D → D#, D# → E, E → F, F → F#, F# → G, G → G#, G# → A, A → A#, A# → B, B → C.
In Solfège:
Do → Do#, Do# → Re, Re → Re#, Re# → Mi, Mi → Fa, Fa → Fa#, Fa# → Sol, Sol → Sol#, Sol# → La, La → La#, La# → Ti, Ti → Do.
See how you count the intervals, not the notes? The C# is not a semitone, it is a semitone above C.
It's almost correct to say that there are eight notes in an octave. Really, there are eight notes of a major scale within the interval of an octave. This is also true of minor scales, but it's not true of some other types of scale (e.g. pentatonic, blues). The major scale itself only has seven unique notes, but the within an octave, the first gets repeated, so you end up with eight.
So it seems that there's a small error in your count of semitones, and a slight misunderstanding of the relationship between an octave and a scale.
This is just a weird historic mistake. Apparently some people in the middle ages didn't know zero as a number, and hence labelled the zero-interval with 1 (unison). Continuing this through the diatonic scale ends you up with the label 8 (octave) on the equivalence-class interval. But there are not eight notes in an octave of diatonic scale / white keys, despite the name; there are in fact only seven (the diatonic scale is a heptatonic scale).
On the more recent 12-edo scale which most modern western instruments use to approximate diatonic scales, this mistake was not repeated: the unison in 12-edo consists of zero semitone steps. Hence the octave has the correct label of 12, corresponding to the fact that the 12-edo scale really divides the octave in twelve steps.
Count two octaves:
12345678-12345678 = 16 (wrong) Most make the mistake of counting the first number eight twice using it as number one on the second octave only count one number eight note in the middle. That should equal 15 notes.
12345678-2345678 = 15. (correct)
You have chosen to count do twice in the major scale, only once in the chromatic scale. Count them in the same way, there's no problem.
The major scale spans 8 notes, contains 7 differently-named notes.
The chromatic scale spans 13 notes, contains 12 differently-named ones.
The historical naming of intervals has mostly to do with the idea that things that are the same are a unity (like the United States of America). A unison (two middle Cs for example) are a unity. The distance between two middle Cs is zero steps but the Cs are called a unison. Calling it a "zero" or "nihil" or "nothing" or something similar doesn't really help. The next note added also doesn't seem to be the first note, it's the second note.
There is an inherent problem between names of objects and names of distances between ordered arrangements of those objects. It's easy to count one, two, three, four for 4 notes or zero, one, two, three for the same 4 notes. Some mathematical representations of music do this. However, I once angered a math teacher (who was saying that "first" and "one" are not related words so why shouldn't zero be called the first number; I just asked how that explained the relationship between four and fourth. and five and fifth, six and sixth, etc.)
The logical problem is that a set of N points in a line form N-1 first differences. If the same set of names is used, something won't match. This problem surfaces in some languages: https://spanish.stackexchange.com/questions/13014/why-does-every-eight-days-mean-once-a-week
