5-limit Just Intonation not an Ideal Tuning

Just intonation is not an ideal tuning: there is no perfect assignment of pitches to note names. In all but the simplest pieces there will be intervals (melodic and/or harmonic) with pitch ratios other than ideal. One simple example is the common succession of the chords F-A-C and D-G-B, such as in the last measure below:

FAC DGB

The pitches involved have a grey background in the following matrix section:

C♯	G♯	D♯	A♯	E♯	B♯	F♯♯	C♯♯	G♯♯
A	E	B	F♯	C♯	G♯	D♯	A♯	E♯
F	C	G	D	A	E	B	F♯	C♯
D♭	A♭	E♭	B♭	F	C	G	D	A
B♭♭	F♭	C♭	G♭	D♭	A♭	E♭	B♭	F

(A-D pitch ratio 40/27)

Checking the interval matrix, you can deduce that the melodic interval A-D (here a perfect fifth down) has a pitch ratio 40/27 which is slightly different from the ideal 3/2.

There is an alternative assignment of pitches:

C♯	G♯	D♯	A♯	E♯	B♯	F♯♯	C♯♯	G♯♯
A	E	B	F♯	C♯	G♯	D♯	A♯	E♯
F	C	G	D	A	E	B	F♯	C♯
D♭	A♭	E♭	B♭	F	C	G	D	A
B♭♭	F♭	C♭	G♭	D♭	A♭	E♭	B♭	F

(C-G pitch ratio 27/20)

But now the melodic interval C-G (here a perfect fourth down) has a pitch ratio 27/20 which is slightly different from the ideal 4/3.

Hidden in chords, melodic intervals with ratios 40/27 and 27/20 could be acceptable. Note that there are two things here that make the 40/27 and 27/20 pitch ratios sound even worse. 1) Reverberation effectively turns each melodic interval into a harmonic one: tuning imperfections are easier to hear in harmonic intervals. 2) An organ stop of pitch 2 2/3' is used here: it amplifies the tone's third partial sine wave and thereby emphasises the impurity of the 40/27 and 27/20 pitch ratios.

There are other awkward intervals which cannot always be avoided, such as the syntonic comma. The most we can hope for is:

That the number of awkward intervals is small (hence the optimization algorithm that minimizes the number).
That we may be able to hide many of the remaining awkward intervals in voices/positions where they harm our ears the least.
That we may adapt the sound quality so beating intervals annoy us less (see Sleep by E. Whitacre).
That we can find minor and artistically justifiable changes to the notes to remove other awkward intervals.
That we will get used to what still remains or even start to appreciate.

The latter may sound too good to be true, but actually a situation occurred where a melodic syntonic comma had a good effect: it neatly marked an accent in the melody and it was as if the virtual musicians attentively listened to one another and adapted their tuning, rather than concentrating only on their own part.