One may make equal temperament more harmonious.
Musical ratios are on the harmonic series. Raw equal temperament iz not. Dijits after decimals in my second column reflect error. I use naturals like those in column four almost exclusively in my synthesizer. Those naturals make it easier to compare one note to any other without comparison to a reference: Reduce a fraction. For example G#/G = 1152:1080 = (1152:1080)/72 = 16:15, which is a minor semitone. One may make equal temperament more harmonious.
You might be inclined to form a just intonation table like mine, only with your chosen ratios, if you do not want to use a tritone, for instance. I arrived at seven hundred twenty az a starting point from multiplying the largest power of any prime factor in all denominators of my ratios. Perhaps you would prefer to use 9:5 for a minor seventh. In detail, sixteen iz the largest power of two az a factor in my denominators: Nine iz the largest power of three az a factor in my denominators: Five iz the largest power of five az a factor in my denominators. So, 16*9*5 = 720. Using 720 az a starting point enabled me to avoid fractions in my naturals. While I can see that my method will always work, I do not know how to prove it.
Just Intonation iz usually another kettle of beans.
Before I knew what my tuning system was usually called, I started writing about sixty short pieces, median less than twenty seconds, and the vast majority of them use naturals below one hundred. My works that stretch into four part harmony, however, add up to about four. That iz probably because I was not willing to use numbers approaching a thousand. One can find a common factor in all notes of a tune, and then simply divide all of those naturals by a constant. This will be my future mode of approach; to use large naturals in the first place and then hope to reduce them when my work iz finished. Norman Lloyd wrote most of these ratios (with 9:5 for a minor seventh) into “The Golden Encyclopedia of Music” (1968), on page 256. Terry Riley used them (with 64:45 for 25:18) in The Harp of New Albion.
Low numbers? Good. Low ratios? Better.
Putting a major semitone into my table boosted my base natural (which should be C-sharp, according to Terry Riley) from 360 to 2520. Three choices are for minor sevenths, and you can use them all in the right context. Various choices for minor and major seconds are in my table. I think it iz more important to avoid a grave fifth (40:27) in concert. A grave fifth does not howl in series. Just Intonation iz very different from Equal Temperament.
Verify my lack of backward compatibility. FROG PROVERB: Time iz fun when yoh're having fliiz. .