1:2 (8)
2:3 (5)
3:4 (4), 3:5 (major 6)
4:5 (major 3), 4:7 (harmonic 7)
5:6 (minor 3), 5:7 (augmented 4), 5:8 (minor 6), 5:9 (minor 7)
6:7 (septimal minor 3), 6:11
7:8 (*augmented 2), 7:9 (*augmented 3), 7:10 (diminished 5), 7:11, 7:12, 7:13
8:9 (major 2), 8:11, 8:13, 8:15

The dyads with stars are tuning dependent, and any dyad with an integer larger than 10 basically do not have names since they don’t naturally occur in low order tunings.

Next, we classify consonant triads, by combining intervals

1:2:3 (P8+P5)
2:3:4 (P5+P4)
3:4:5 (P4+M3, major)
4:5:6 (M3+m3, major), 4:5:7 (M3+A4, partial 7), 4:5:8 (M3+m6), 4:6:7 (P5+sm3, partial 7)
5:6:7 (m3+sm3, diminished), 5:6:8 (m3+P4, partial major), 5:6:9 (m3+P5, partial dim7), 5:7:8 (A4+A2, partial 7), 5:7:9 (A4+A3, partial dim7), 5:8:9 (m6+M2, partial 9)
6:7:8 (sm3+A2, partial m7), 6:7:9 (sm3+A3, harmonic minor), 6:7:10 (sm3+d5, diminished), 6:8:9 (4+2, quartal triad)
7:8:9 (A2+M2, getting weird…), 7:8:10 (A2+M3, partial 7), 7:9:10 (A3+M2, partial m7)
8:9:10 (still stranger)
10:12:15 (m3+M3, “standard” minor)

Though there are countless more of these, we stop here. From the above, we begin to realize that most chords are just partial suggestions of the natural harmonic series. If we only take integers no larger than 10, we will get the harmonic series with the indicated intervals between harmonics:

1:2:3:4:5:6:7:8:9:10
(P8+P5+P4+M3+m3+sm3+A2+M2+M2)

The simplest chords just build from the base up. The more harmonics taken, the higher order the chord would appear to be. In practice, modifications are made by omitting (and implying) the lower harmonics, especially the zeroth, which is almost always omitted.

To generalize even further, omitting harmonics is a special case of applying weighting (or a spectral filter) to all the harmonics. This is perhaps what people aptly call “color” of the sound. Thus all chords and chord colors can actually be identified entirely by the weight vector on the harmonic series. (This may not be as trivial as it sounds, because it implies additional degrees of freedom in musical activity, by varying weights dynamically and continuously, for example, and generalizing discrete progression of chords.)

Depending on the psychoacoustic circuitry, these weight vectors somehow can be partitioned into equivalence classes, so there is more complication here. I haven’t thought this one through, but for instance, not only are 3:4:5 and 6:8:10 equivalent, but 3:4:5 and 4:5:6 are also somewhat equivalent as they are inversions of each other. The latter case is possibly due to psychoacoustic masking of 6 in the implied chord 3:4:5:6 in 4:5:6. Not sure.

In any case, choices not taken from the series below 10 will tend to be dissonant, because the higher harmonics, if they are not clearly resolvable in frequency psychoacoustically, will tend to combine as colored noise in the limit. In particular, the (non-harmonic) minor triad is exceptionally dissonant, but I suppose we have learned to accept it, if only for contrast.

The first paragraph of this old paper is intriguing. Don’t agree with the rest of the conclusion.

However, there is a way to figure this matter of structure from first principles (and perhaps generate a more unique style as a result), albeit with the tradeoff that you cannot be quick, you must be careful.

The first principle for aesthetics is that the character must stand … this is something my old man told me, actually, so I didn’t figure this out myself, but it is very true. If you hold up the piece of paper and look at the strokes as struts of a building, it must look like the character is architecturally sound, i.e. reasonably symmetric if need be, balanced in weight so will not tip over, is not poorly supported with too small a bottom and too big a top, etc. This isn’t too difficult if the character is mechanically drawn, but the trick is to do it even with asymmetric calligraphic strokes and multi-part characters with asymmetric radicals and caps.

The second principle for aesthetics is about spacing, and this is much like optimal typography and typesetting. The strokes should be spread out evenly so that where they appear parallel, they appear to have nearly identical spacing as other such spaces. Otherwise there will be ugly bunching and voids. This is very difficult because the strokes are written in order so there is a pre-commitment issue. Once you commit to a particular stroke, it also commits the spacing requirements for the rest of the character. So one slightly off stroke and you are screwed. This is more a problem for large writing, since bigger mistakes are possible.

Then is the issue of multiple character layout. This wouldn’t be so much of an issue if all characters were the same shape and complexity, but they are not. Some are extremely sparse, and some are very dense. Some are tall and some are fat. They all have to be laid out on paper to look like they take up the same space and also evenly spaced from each other. There is also the compromise of making inter-stroke space appear similar in multiple characters. So one needs to deal with some visual artifacts and vision tricks. As a result, the characters will not all be the same size and will not be spaced evenly, so this is a very tricky thing to get right. You can have perfectly written individual characters but still a terrible collection.

And finally here is a side point: people say Simplified characters are uglier than Traditional characters for calligraphy. In fact this cannot be true. What happens is Simplified characters are sparser and sparser characters writ large are the most difficult to get correctly (not to mention there are no classic master’s character books to trace in Simplified). They are ugly only because (or to the extent that) they are not written well. The bastion of poor practioners (like me) is in small dense characters that distract from scrutiny and generally look pretty good no matter how you write them.