What sounds "good" is a very subjective call. Some people describe harmonies in Just Intonation as being "full" or "rich", while others describe them as "bland" or "lifeless". However, most people would agree that JI harmonies are considerably smoother than tempered harmonies. There are several reasons for this.
First of all, if I may be allowed to get away with a few over-simplifications, our auditory systems find it easier to process simple harmonic information than complex harmonic information. As the sweetest JI intervals are those where two pitches are tuned in simple harmonic relation, these are much easier for our ears to "digest". For example, a pure major third has the two tones in the ratio of 4:5, whereas a major third in 12 Tone Equal Temperament has them in a ratio which is approximately 504:635. Our brains tell us that the first interval is more consonant, simply because it requires less effort to process than the second one. Of course, in the real world, things are rarely absolutely perfect and two tones in the ratio of 40001:50001 will be heard as "near enough" to 4:5. However, the further an interval strays from a nice simple ratio, the harsher it will tend to sound. Intervals that do not form perfect small number ratios with each other produce beats. I won't go into the physics behind this here (interested parties are referred to this page for more information), but the phenomenon should be well known to most harmonica players. The tremolo harmonica deliberately has each pair of notes slightly out of tune with each other, producing the well known effect:
Beats are also quite obvious when playing octaves on a harmonica that is not quite in tune:
This effect is not just limited to unisons and octaves, but happens when any interval deviates from a perfectly Just relationship. The following audio clip has three examples of a major third interval. The first is a 12TET major third (about 14 cents sharper than the JI version); the second one is midway between the 12TET version and the JI version; the third one is in JI, tuned in a perfect 4:5 relationship. You should be able to hear that the first example beats very harshly, the second beats at a moderate rate and the final one is almost totally beatless.
The ability to hear beats is an essential skill for tuning any instrument in JI. Rather than merely adjusting the pitch of each individual note to a certain number of cents plus or minus on an electronic device, a skilled instrument tuner will play certain combinations of notes and listen for beats, adjusting the notes until the beating slows down to an imperceptible rate.
Then there is the issue of overtones. When any note is sounded on a musical instruments, as well as the fundamental pitch, there are also overtones produced. In text books, these overtones (also known as harmonics or partials) occur as whole number multiples of the fundamental. For example, if you play a pitch of 1000Hz, it will produce overtones of 2000Hz, 3000Hz, 4000Hz, 5000Hz, etc. In the real world, these overtones often deviate from this ideal, but in the case of the harmonica, we are blessed with overtones which are almost perfectly harmonic. This means that when we play our pure major chord tuned in the ratio of 4:5:6, certain overtones of each note will line up in perfect tune with each other. If our major chord consists of the notes C and E, then the fifth harmonic of the C will be exactly the same pitch as the 4th harmonic of the E; the sixth harmonic of the E will be exactly the same pitch as the fifth harmonic of the G; the fifteenth harmonic of the C, the twelfth harmonic of the E and the tenth harmonic of the G will be exactly the same pitch; etc. Other harmonics fall into pure harmonies with each other and the end result is a very strong cohesive overtone structure. Any tempering of the fundamental chord will mess up all the relationships between the overtones, causing high frequency noise. On an instrument like the piano, where the strings posses a notable degree of inharmonicity (due to the length and stiffness of the strings, the overtones become progressively sharp as you run up the harmonic series), this is much less of a problem than it is with the harmonica - in fact, the piano has to have its tuning tempered in order to reduce this problem in its upper ranges.
Finally, we have difference tones. The basic idea of difference tones is explained on this page. When we play that perfectly tuned major triad in the ratio of 4:5:6, the difference tone produced by the root and the major third is pitched exactly two octaves below the root of the chord. The difference tone produced by the major third and the fifth is also exactly two octaves below the root. The difference tone produced the root and the fifth is exactly one octave below the root. In turn, these difference tones produce secondary difference tones, all of which converge to form a strong bass tone two octaves under the root of the chord and summational tones are also produced which further strengthen the harmonic structure. However, with a major triad tuned in 12TET, all the difference tones are out of tune with each other. If you play a 12TET major triad, the difference tone generated by the root and the major third is more than 70 cents sharper than the pitch two octaves below the root of the chord; the difference tone generated by the major third and the fifth is more than 80 cents flatter than the pitch two octaves below the root of the chord; the difference tone generated by the root and the fifth isn't quite so bad, being only about 6 cents flatter than the pitch of the note one octave below the root of the chord. These difference tones all clash against each other and the secondary difference tones further increase the unpleasant rumbling in the bass register, with the out-of-whack summational tones adding noise in the higher ranges. On an instrument like the piano, where the tones it produces die away comparatively quickly, this is not too much of a problem. With long sustained tones on the harmonica, it is another thing altogether.
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