So, the "just intonation" system introduces a scale where every note is aimed to be a more harmonious low-integer ratio (harmonic) relative to the fundamental. In the case of C, the scale is characterised by the following ratios (taken from Science & Music, Sir James Jean, 1937):
|Ratio of frequency to C
A problem that arises from this system is that if we want to modulate to another note such as G, the A note will no longer have 5/3 times the frequency of C, but 9/8 times the frequency of G, and so 27/16 times the frequency of C, meaning that A would need to change its pitch. So pitches would need to vary depending on the key in which we happen to be playing at the moment. I understand this idea.
Differences in key
So, if we want to play in different keys, we need new notes, different in frequency from what we were using in a previous key.
The aforementioned book takes this fact to conclude that different scales have therefore different, special and unique characteristic qualities, that differ from just a change in absolute pitch.
It even quotes this testimony from Helmholtz:
There is decidedly different character in different keys on pianofortes and bowed instruments. C major and the adjacent Db major have different aspects. The difference is not caused by a difference of absolute pitch, as can easily be verified by comparing two instruments which are tuned to different pitches. If Db on one instrument has the same pitch as C on the other, the C major still retains its brighter and stronger character on both, and the Db its soft, veiled harmonious quality.
Googling around I found sources which also assign different qualities to different keys (when playing in "just intonation"), such as this one, where it agrees that there is a difference between C and Db (aside from the difference in pitch).
C Major Completely Pure. Its character is: innocence, simplicity, naïvety, children's talk.
Db Major A leering key, degenerating into grief and rapture. It cannot laugh, but it can smile; it cannot howl, but it can at least grimace its crying.--Consequently only unusual characters and feelings can be brought out in this key.
Where does these differences among scales which are not in pitch come from?
Even if when modulating from C to G you need different notes, both scales will consist of the same ratios. As far as I understand, in both C and G the scales are going to be:
1/1 , 9/8 , 5/4 , 4/3 , 3/2 , 5/3 , 15/8 , 2/1
So the only difference I could see between C and G is pitch.
If Db on one instrument has the same pitch as C on the other, the C major still retains its brighter and stronger character on both, and the Db its soft, veiled harmonious quality.
Why? Where are these differences coming from if it's not pitch? I'm guessing the ratios DO change from one note to another but I'm not seeing it.