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.

A description: http://www.albany.edu/piporg-l/meantone.html

And a demonstration:


The just intonation for the white piano keys starting from C are in ratios of
9/8, 10/9, 16/15, 9/8, 10/9, 9/8, 16/15

Note that a pure major third has ratio 5/4 and a pure minor third has ratio 6/5. On the other hand, in equal temperament the major third has ratio 2^(4/12) = 1.2599 > 5/4 and a minor third has ratio 2^(3/12) = 1.1892 < 6/5.

Intuitive experience is that the sharper the interval, the brighter the tone, while the flatter the interval, the sadder the tone. So while a minor third being flatter than a major third obviously generates the characteristic associations of emotional qualities to major vs. minor keys, it is interesting to find that pure intonation intervals themselves don't actually make these qualities more pronounced... Indeed, a pure major third is supposed to be flatter than the equal temperament equivalent and a pure minor third sharper than its equal temperament equivalent. Perhaps counter-intuitively, bending towards Pythagorean tonal purity within the scale lessens major/minor third affinity to their respective qualities. It’s the wrong knob to turn!

However, what pure intonation suggests is that certain keys in pure intonation (and derived mean-tone) tunings have pronounced major/minor qualities due to exaggerated thirds. D minor begins with the rather flat D-F minor third interval with ratio 1.1852 < 2^(3/12) < 6/5, so D minor is a reasonable candidate for the saddest of all keys in my book. Probably not a coincidence that, in some traditions, the minor scale developed from the (degree 2) Dorian mode to begin with.

Here are some supposed key qualities that have become nonsense in the equal temperament world.

From this, I’m just guessing that in common tunings, the D-F# major third interval was sharp, which contributed to the D major key’s qualities. Maybe.