First note that the author of this books uses "tone" and "semitone" in a slightly non standard way. The author is using "tone" for intervals between neighboring scale notes (such as C→D) and "semitone" for a scale tone to its higher alterations such as C→C#.
The standard use of the term "tone" goes back to the tónos of Aristoxenos of Tarantinos. This denotes a specific interval which he divides in half tones, third tones, &c. and uses these to form other intervals, such as a fifths being 3 tones and 1 half tone.
And this is exactly the usual meaning of tone and semitone. A tone is a big step such as in C→D and a semitone is a small step such as E→F.
Now, in modern music (modern in the sense of non ancient) we have no use for third tones and lower, we just use intervals explainable by whole tones and half tones (as long as we ignore tuning systems and such, which immediately makes things more complicated).
But not all notes reachable from the root by tones and semitones where actually used (which would form our modern "chromatic scale"). To understand this we need again consider ancient greek music theory. Here scales are built by using so called "Tetrachords". These are stringed instruments with four chords, which where tuned in different systems or genera. A diatonic tetrachord would have most classically been one made up from two whole tones and one semitone.
Scales built using two diatonic tetrachords would then be diatonic scales. So a diatonic scale is a scale made of of whole tones except of two semitones (usually also demanding that there are either two or three whole tones between the semitones).
And this system forms the basis of medieval music theory, just instead of tetrachords one would extend to hexachords (i.e. six strings). Medieval music theory then used three different hexachords, the hexachordum naturale which would be the first six notes of a major scale, the hexachordum molle, which is the same but a fifth higher and the hexachordum durum which is a fifth higher.
So in note names we’d have
- C D E F G A
- F G A Bb C D
- G A B C D E
So if you pay attention in total you get the whole major scale plus a minor 7th:
This system is the basis for both note names and solfeggio. Guido of Arezzo assigned to each step of the hexachord a syllable (ut, re, mi, fa, so, la, from the hymn:
Ut queant laxis
resonare fibris
mira gestorum
famuli tuorum
solve polluti
labii reatum
Sancte Iohannes.
with each verse starting one step higher). This is the origin of solfeggio. And he assigned a letter to every note used in this medieval system, which where Gamma, A, B, C, D, E, F, G, a, b, b*, c, d, e, f, g, aa, bb, bb*, cc, dd, ee.
Here we have two different (small) bs, one of the mollum hexachord and one of the durum one, so a lower one and a higher one. This was represented by using a round b for mollum and a rectangular b for durum.
Later people started to fill up the gaps with semi tones. These letters then developed into our common note names (C D E F G A B) and the syllables to solfeggio. The two different bs turned into ♭ and ♮ as well as ♯ (these have quite a lot of history until they arrived at what we have today though). They were then applied to signify that a note should be played a semitone lower or higher, giving next to [C D E F G A B] also the alterations [C♯ D♯ (E♯) F♯ G♯ A♯ (B♯)] and [(C♭) D♭ E♭ (F♭) G♭ A♭ B♭]. And from there these got adopted to English note naming.
But in fact there are different note naming systems. In northern europe it is quite common to refer to notes by names, but add "is" or "es" for "#" and "b". In the Romanic languages it is quite common to use solfeggio instead with using a variant of diesis and b mollum for "#" and "b". The latter comes directly from the hexachordum mollum, the first one is a bit different (it comes from a thing called diesis, which is the distance between an Octave and either three big thirds or four small thirds. I suppose since B# and C are a diesis this names got adopted).
