Just Intonation
The idea of just tuning comes out of the premise that musical intervals should be generated by choosing simple ratios between frequencies to represent consonant tones, and that more complex ratios will generate increasingly dissonant tones.
This premise is well supported by the fact that nearly everyone on Earth recognizes the octave as the most important consonant nonunison interval, which is the most simle ratio of 2:1. I would argue that almost everyone recognizes the perfect fifth as the next most consonant interval (ratio 3:2). But after that, it's really hard to say. Certainly the most simple ratios like 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third), etc are the easiest on my ears.
The ancient Greek thinker Ptolemy is generally credited for coming up with the intervals of the major and minor scales. Many theorists these days set limits on the largest prime number that can be used to build the interval ratios. I lean toward the plan of using simple ratios to come up with the most consonant intervals, like Ptolemy, and then disecting them to make step intervals to construct the theoretical scales based on tonality.
The western tonalities are major, dominant, minor, augmented and diminished. Nonwestern tonalities include neutral intervals and tritones (the term "tritone" is a western interpretation of three whole tones, but this is likely not the truest representation of the idea). The dominant tonality can be viewed in western music either chordally or as a hybrid of other tonalities. The steps I feel confident about using are:
diminuiative step: 41.06 cents (128/125)
half step: 111.73 cents (16/15)
greater half step: 133.24 cents (27/25)
lesser whole step: 182.40 cents (10/9)
whole step: 203.91 cents (9/8)
augmented step: 274.58 cents (75/64)
So here are the intervals I feel are important and the cents I feel work best both aurally and logically:
diminished second: 41.06 cents
minor second: 111.73 cents
neutral second: 165.00 cents
major second: 203.91 cents
augmented second: 274.58 cents
diminished third: 244.97 cents
minor third: 315.64 cents
neutral third: 347.41 cents
major third: 386.31 cents
augmented third: 478.49 cents
diminished fourth: 427.37 cents
perfect fourth: 498.04 cents
harmonic eleventh: 551.32 cents
tritone: 582.51 cents
augmented fourth: 590.22 cents
diminished fifth: 609.785 cents
perfect fifth: 701.96 cents
augmented fifth: 772.63 cents
diminished sixth: 743.01 cents
quadratone: 764.92 cents
minor sixth: 813.69 cents
neutral sixth: 840.53 cents (doubles as harmonic 13th)
major sixth: 884.36 cents
augmented sixth: 976.54 cents
diminished seventh: 925.42 cents
harmonic seventh: 968.83 cents
minor seventh: 1017.60 cents
neutral seventh: 1035.00 cents
major seventh: 1088.27 cents
augmented seventh: 1158.94 cents
I strongly suggest that these intervals be combined to form scales only within certain parameters. Stretching from a diminished tone to an augmented tone, or vice versa, does not translate well into any musicality, for instance. In the strictest sense, the prestated steps should be used as strict building blocks.
So, if these intervals sound so great, what's the downside? The downside of any just tuning is that the number of notes needed is nearly equal to the number of intervals times the number of keys needed. This works well for performance in one or two keys, but there is no closed loop of notes and keys as there is in equal temperaments.
For fretted instruments, like guitars, the tuning is difficult to utilize, as the frets need to be carefully bent and placed into curved fret slots.