Natural harmonics exist everywhere.
It is almost impossible to produce a sound that is so pure that only its fundamental exists without any overtones. The presence of overtones is what gives a sound its timbre, its color. The louder the overtones, the brighter, more nasal, is its timbre. The softer the overtones, the darker, purer, is its timbre.

On certain instruments, like the wind and string instruments, certain overtones can be extracted from the fundamental and produced in their own right. The military bugle performs uniquely with its overtones (numbers 3, 4, 5, 6 to be precise) and this was also formerly the case with trumpets and horns (but with a greater number - the horns playing from 2 to 16). String instruments use them more sparingly, as a special fluted effect.

Two questions beg immediate attention -
1. How important a role do natural harmonics play in the generation of musical phenomena?
     Actually, none.
          Harmony and Harmonics both use the same interval-ratios.
               However, Harmonics use them indiscriminately and unidirectionally,
                    whereas, Harmony uses them parsimoniously, prioritarily, and symmetrically.

2. How "in tune" are they?
     Which ones are in tune and which ones are not?
          Are there objective criteria which we can apply,
               or is there considerable cultural prejudice involved?

What are they and how does one calculate their pitch?
From time to time it might be useful to glance at the Natural Harmonics Table
     where the first 25 harmonics are listed from bottom to top with their
          Number,
          (Name of) Note, (sometimes approximate)
          Proportion (in fractions and decimals) and
               Distance in ML2 from the fundamental
                    (in this case the note D, to be consistant with our other tables)

All of these natural harmonics belong to the realm of Just Intonation because the frequency of each harmonic is a multiple of the frequency of the fundamental and this multiple is the number we give to each harmonic. (Harmonic 3 has a frequency 3 times that of the fundamental, or 3/2 if we remove the octave - we immediately recognize the fifth). This fraction (proportion) can be transformed into ML2 which, in turn, gives us the distance from the fundamental, always kept within an octave, the same distances we found in the previous tables.

Construction Table

Table Of Progressive Pitch

Let's start from the bottom of the
          Natural Harmonics Table
and answer our questions in reverse order as we progress.

Harmonics 1 and 2 are the same note (D) an octave apart.
     Harmonics 4, 8 and 16 will also be the same note (D) at higher octaves.
          They are all colored as COMMON TONEs.

Harmonic 3, as we have seen, is a fifth higher than Harmonic 2, the note A.
          We have here the source of Trunk Tuning.
     Harmonics 6, 12 and 24 will also be the same note (A) at higher octaves.
          They are all colored as PROPER TONEs.

Harmonic 5 is a natural major third higher than Harmonic 4, the note F#-.
          We have here the source of Short Branch Tuning.
     Harmonics 10 and 20 will also be the same note (F#-) at higher octaves.
          They are all colored as MEDIANs.

Within the restricted realm of Functional Tuning,
all of these harmonics are perfectly in tune.

Harmonic 7 is where the problems begin.
     It has always been considered too flat to be used as a MOTRIX.
          Players of brass instruments are recommended not to use it and
               to jump directly to Harmonic 8.
     (an approximate note name, C-, is placed in parentheses)
The only practical use of Harmonic 7 seems to be in the
          tuning of "drone" instruments, like the bag-pipes.
     Harmonic 14 will also be the same note (C-) at a higher octave.

It is interesting to note here that Harmonic 5 is often also considered flat
(being lower than Trunk Tuning as we well know)
and many performers (like trombonists)
are recommended to compensate for this "flatness".

Harmonic 9 is in tune, being the fifth of the fifth, the note E,
          as we well know from Trunk Tuning.
     Harmonic 18 will also be the same note (E) at a higher octave.

Harmonic 11 presents another problem,
               as all amateurs of ancient trumpets and horns well know,
          somewhere between G and G#, close to G++.
          It had to be constantly humored, with the lip and/or the hand,
          to approximately serve as a G or a G#.
     Harmonic 22 will also be the same note (G++) at a higher octave.

Harmonic 13 is also out of tune, somewhere around Bb++.

          On the other hand,
Harmonic 15, the major third of the fifth, the note C#-,
          a combination of Trunk Tuning and Short Branch Tuning, and
Harmonic 25, the major third of the major third, the note A#--,
          the source of Long Branch Tuning,
     are perfectly in tune.

Harmonics 17, 19, 21, 23, are all out of tune.
 

Finally, the only harmonics which are considered in tune in Functional Tuning are the mutiples of 3 (the fifth of Trunk Tuning) and 5 (the major third of Short Branch Tuning and Long Branch Tuning), with, of course, the multiples of 2 which merely add extra octaves.

There are places where the Natural Harmonics coincide with Trunk and Branch Tunings but it would not seem reasonable to grant them an appreciable role in the generation of musical phenomena.

Just Intonation - Preface

Just Intonation - Diagrams

Just Intonation - Tables

Just intonation - Performance