>The set of notes defined by (1) are typically referred to as "Just Intonation", and was the basis for most western music (and some non-western music) until somewhere between about 1400 and 1600 (lots of room for discussion/debate there).
Using the harmonic series to derive the notes of the scale is a distinctly modern phenomenon, and it does not represent how musicians thought about the issue in history. There are already presented plenty of criticisms to this perspective, so I will simply give the historical view of deriving the diatonic notes(or "Just Intonation") via ratios alla Galilei. Here the larger interval is always divide into two ratios which when multiplied together gives back the original ratio. The numbers in the smaller ratios are selected to be as small as possible. This is repeated until you have all the intervals. We start with the 8ve (2:1) since that's the simplest possible ratio which is not unity.
The 8ve (2:1) is divided into two unequal portions: the 4th (4:3) and the 5th (3:2).
The 5th (3:2) is divided into two unequal portions: the major 3rd (5:4) and the minor 3rd (6:5).
The major 3rd (5:4) is divided into two unequal portions: the major tone (9:8) and the minor third (10:9). Note that there are actually two different tones!
The major tone (9:8) is divided into two unequal portions: the major semitone (16:15) and the minor semitone (25:24). There are also two different semitones!
Now the rest of the intervals in the octave can be obtained by adding up the intervals:
The major 6th (5:3) is the fourth (4:3) plus the major third (5:4).
The minor 6th (8:5) is the fifth (3:2) plus the major semitone (16:15).
The minor 7th (15:8) is the fifth (3:2) plus the major third (5:4).
The minor 7th (9:5) is the fifth (3:2) plus the minor third (6:5).
Note that Galilei takes all the ratios for the intervals as granted, so these "derivations" are no less arbitrary than those from the harmonic series. But it at least has historical standing and is in my opinion more aesthetically pleasing. If you keep on going and dividing the ratios derived with each other you'll sometimes end up with ratios already derived, and sometimes ratios which are slightly different, this is the root of the issue with Just Intonation, and Galilei spends the second half of his treatise explaining precisely why nobody uses it in real life.
Using the harmonic series to derive the notes of the scale is a distinctly modern phenomenon, and it does not represent how musicians thought about the issue in history. There are already presented plenty of criticisms to this perspective, so I will simply give the historical view of deriving the diatonic notes(or "Just Intonation") via ratios alla Galilei. Here the larger interval is always divide into two ratios which when multiplied together gives back the original ratio. The numbers in the smaller ratios are selected to be as small as possible. This is repeated until you have all the intervals. We start with the 8ve (2:1) since that's the simplest possible ratio which is not unity.
The 8ve (2:1) is divided into two unequal portions: the 4th (4:3) and the 5th (3:2).
The 5th (3:2) is divided into two unequal portions: the major 3rd (5:4) and the minor 3rd (6:5).
The major 3rd (5:4) is divided into two unequal portions: the major tone (9:8) and the minor third (10:9). Note that there are actually two different tones!
The major tone (9:8) is divided into two unequal portions: the major semitone (16:15) and the minor semitone (25:24). There are also two different semitones!
Now the rest of the intervals in the octave can be obtained by adding up the intervals:
The major 6th (5:3) is the fourth (4:3) plus the major third (5:4).
The minor 6th (8:5) is the fifth (3:2) plus the major semitone (16:15).
The minor 7th (15:8) is the fifth (3:2) plus the major third (5:4).
The minor 7th (9:5) is the fifth (3:2) plus the minor third (6:5).
Note that Galilei takes all the ratios for the intervals as granted, so these "derivations" are no less arbitrary than those from the harmonic series. But it at least has historical standing and is in my opinion more aesthetically pleasing. If you keep on going and dividing the ratios derived with each other you'll sometimes end up with ratios already derived, and sometimes ratios which are slightly different, this is the root of the issue with Just Intonation, and Galilei spends the second half of his treatise explaining precisely why nobody uses it in real life.