Just to add my own two cents:
To say that the 23rd partial is the first microtonal partial is inaccurate. If we can define a microtonal interval as being smaller than 100 cents (cent=1/1200 of an octave in equal spacing) a.k.a. the equal-tempered half-step, then you would first see a microtonal interval between the 17th and 18th partials, where the difference is only 99 cents. 18:19 is 94 cents, 19:20 is 88 cents, and so on.
Secondly, and much more importantly, it is misleading to label all of the odd-numbered partials as "out of tune". For example the amount that the 5th partial is "flat" (compared to equal-temperament) is the same amount by which a pure major third is lower than a third on the piano. Before the proliferation of equal-temperament and the invention of valved brass instruments, composers used these "out of tune" partials, especially 5 and 7, because they gave pure intervals. This is why composers such as Mozart would use two horns (natural horns, of course) with different crooks for different keys. If Mozart wrote in D major, for example, he might use one horn in D and the other in A, the latter being a 3:2 (perfect 5th) extension of the D major harmony.
djwesp's list of the partials is interesting but a little problematic. Although different partials behave differently on each instrument, the average will correspond fairly closely to the harmonic series (
http://en.wikipedia.org/wiki/Harmonic_s ... 28music%29" target="_blank" target="_blank). So to say that partial #10 is less out of tune than partial #5 should not (again, on average) be the case as 10 is merely an octave repetition of 5 (5x2). 7 is a full 31 cents lower than its equal-tempered counterpart and 11 is almost exactly a quarter-tone between a perfect fourth and a tritone! 13 is 41 cents higher than the piano's minor 6th but 17 is actually one of the closest to equal temperament being only 5 cents high.
All of that said, it is important to remember that these pure rational intervals (between different partials) are more in tune than equal-temperament ever could be. You can hear the differences in the sound clips in this article:
http://en.wikipedia.org/wiki/Just_intonation" target="_blank" target="_blank. As an experiment, listen to two brass players (trombones or euphs would be ideal) playing their 5th and 7th partials together. The result is a beautifully consonant tritone. This is the interval that should be heard between the 7th and the third of a dominant chord (which, expressed rationally is 4:5:6:7 and is not really a dissonance at all) or between the root and fifth of a diminished triad (5:6:7).
To make it even more interesting, these pure intervals (when perfectly in tune with the harmonic series) cause phenomena known as resultant tones. These are summation tones and difference tones, which are extra notes produced by the combination of sound waves in a sonority. Many people have heard these as low notes (difference tones) that are heard when two trumpet players are playing together (listen for a low, pure sound when trumpets play a whole step [9:8] apart). A 5:3 (pure major sixth) will have a difference tone of 2 (5 minus 3) and a summation tone of 8 (5 plus 3). On a Bb tuba, this means playing the 3rd partial F and 5th partial D (allowing it to be "flat") and hearing a low Bb. The same low Bb would be heard if you played the consonant tritone 7:5 that I described earlier. This is the same principal behind hearing the third of a chord (partial #5) when only playing the root and fifth (#3 and #2).
Lastly, using these notes gives us a greater variety of intervals. For example, 6:5 and 7:6 can both be called minor thirds but they have a very different sound. 6:5 is 16 cents wider than the piano's minor third and 7:6 is about 29 cents narrower. Practically speaking, 6:5 is the upper part of a major triad and 7:6 is the upper part of a dominant 7th chord.
If all of this sounds very modern, consider that equal-temperament is actually a the most modern and mathematically complex system that has ever been widely used. Any skilled musician singing or playing an instrument without rigidly fixed pitch will automatically play these pure intervals (at least the 7-limit ones). In Indian music, musical scales have used notes corresponding to the harmonic series for 3000 years. Mozart even described 19 notes per octave in his lessons with students.
Just my two cents (no pun intended)
~Luke Storm
