The use of a seven-note diatonic musical scale is ancient, though originally it was played in descending order.
In the eleventh century, the music theorist Guido of Arezzo developed a six-note ascending scale that went as follows: ut, re, mi, fa, sol, and la. A seventh note, “si” was added shortly after. The names were taken from the first verse of the Latin hymn Ut queant laxis, where the syllables fall on their corresponding scale degree.

Sheet Music for Ut Queant Laxis

Ut queant laxis resonāre fibris
Mira gestorum famuli tuorum,
Solve polluti labii reatum,
Sancte Iohannes.

The hymn (The Hymn of St. John) was written by Paulus Diaconus in the 8th century. It translates as:
So that these your servants can, with all their voice, sing your wonderful feats, clean the blemish of our spotted lips, O Saint John!

“Ut” was changed much later in Italy to the open syllable Do, and Si was added to complete the diatonic scale. In Anglo-Saxon countries, “si” was changed to “ti” by Sarah Glover in the nineteenth century so that every syllable might begin with a different letter. “Ti” is used in tonic sol-fa and in the song “Do-Re-Mi”.
In the Elizabethan era, England and its related territories used only four of the syllables: mi, fa, sol, and la. “Mi” stood for modern ti, “fa” for modern do or ut, “sol” for modern re, and “la” for modern mi. Then, fa, sol and la would be repeated to also stand for their modern counterparts, resulting in the scale being fa, sol, la, fa, sol, la, mi, fa. This was eventually eliminated by the 19th century, but it was (and still is in a few rare circumstances) used in the shape note system, which gives each solfège syllable a diffferent shape.

déduite de la considération des nombres relatifs de vibrations

par D. Deloche

Etienne Giraud, Libraire Editeur, 16 Rue Dauphine, Paris , 1857

Theorie de la Musique (1857)

… Au dire de Plutarque, les deux plus célèbres musiciens de l’Antiquité, Olympe et Terpandre, ne se servaient jamais que de trois notes…. (p22)

The above association applies to any note of our musical system.

When we play the C for example as a C7 chord it in turn becomes unstable, it assumes to our ear the role of Overtone No.3, and resolves to its Fundamental tone a perfect fifth down, the F.

The F in turn as an F7 chord resolves a fifth down to Bb.

Bb7 resolves a fifth down to Eb, and so on.

Continuing this process from chord to chord we eventually return back to our starting point.

This circle is called the Circle of Fifths.

The Circle of Fifths is sometimes also referred to as the Circle of Fourths.

For moving a perfect 5th downwards (from G down to the C below) is the same as moving a perfect 4th upwards (from G up a 4th to the C above).

Applications and Exemples

Just about every Jazz Standard, exhibits the same principle.

Here a few examples :

Fools Rush In                             All of Me                         Heart and Soul

• “Fools Rush In” by Johnny Mercer and Rube Bloom starts with :

Fmaj7 – Bm7b5 – Em7 – Am7 – Dm7 – G7 – Cmaj7 –

• “All of Me”  by Simons and Marks contains this progression :

Cmaj7 – E7 – A7 – Dm7 – G7 –

• “Heart and Soul” by Hoagy Carmichael (similar to I Got Rhythm by George Gershwin, Oleo by Sonny Rollins, and several other Jazz Standards) goes :

F – Dm – Gm – C – F – Dm – Gm – C –

• The three chords for the 12 bar Blues also are three adjacent members on the Circle of Fifths :Blues in C : V = G , I = C , IV = F

Blues in F : V = C , I = F , IV = Bb

Blues in A : V = E , I = A , IV = D

etc.

Source: JazClass from Michael Furstner, who developped a very comprehensive Jazz Class course. Thanks to him.

The Whole-tone scale

All the notes of the chromatic scale are spaced at equal (‘symmetrical’) , semitone, intervals. This scale is therefore a called a symmetric scale. One can also form a symmetric scale by selecting notes spaced at whole tone (2 semitones) intervals. This scale consists of six notes only and is called the Whole-tone scale.

Here the C whole-tone scale.

Any of the above notes can form the tonic of the whole-tone scale using the same six notes.

Jazz improvisers were quick to take up this new sound for improvisation over augmented triads and augmented 7th chords (+7 or 7^aug).

8-note Symmetric scales

Other symmetric scales can be formed by alternating note spacings of tones and semitones. There are two possibilities.

8-note Dominant scale (semitone – tone – semitone – tone -)
This scale starts with a semitone between the tonic and 2nd note, then a tone between the 2nd and 3rd notes, again a semitone between the 3rd and 4th note, and so on.

In Jazz this scale is used as an alternative choice for improvisation over dominant 7th and altered dominant 7th chords .

8-note Diminished scale (tone – semitone – tone – semitone – )
This scale starts with a tone between the tonic and 2nd note, then a semitone between the 2nd and 3rd notes, again a tone between the 3rd and 4th note, and so on.

In Jazz this scale is used for improvisation over diminished chords