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Fo is an harmonic component, and it is the first of the sequence, so in my opinion we can call it h1, and so on, h2, h3,...
In this way their numbering become very obvious and simple, and math help us.

If we consider the octave intervals of the harmonic series, we note that their numbers are:
h1 = Fo, base frequency
h2 = first octave
h4 = second octave
h8 = third octave
h16 = fourth octave

so, obviously

the first octave includes 1 harmonic component
the second octave includes 2 harmonic components
the third octave includes 4 harmonic components
the fourth octave includes 8 harmonic components

Not only, for example the octave of h3 is h6, the octave of h5 is h10, and so on.
it is very obvious but really charming for me!

What do you think?

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### Replies to This Discussion

Yes!
For me it is also charming visibility. I love this system.
Filip
hi friends,
just to continue, here a few relations you can read in the harmonic scale.

simplified, the diatonic intervalls are defined by the following relations (geometrically as string lenghts):

minor second 16/15
major second 9/8
minor third 6/5
major third 5/4
perfect fourth 4/3
tritone 7/5 (simpliest, contents 7 as higher irregularity)
perfect fifth 3/2 ...... (o.k., very old hats)

all these relations you can find as intervals between harmonics with the same couple of numbers but only when the fundamental pitch is called 1! (the minor second between 15th and 16th harmonic...)

as mentined by marco the first and framing regularity is factor 2 (the simpliest possible). if you do so with factor 3 you will find the pythagorean intervals 3/2, 9/8, 27/16, 81/64... means the 3th, 9th, 27th, 81th... harmonic in relation to the fundamental pitch or his octaves. strange: the fifth harmonic (5x1) is very closed to 3x3x3x3x3. the difference between both intervals (81/80) is exactly the same between the two diatonic seconds (9/8 and 10/9) and all other two variants of diatonic/pythagorean intervals. here the first steps:

prime 0th power of 3 / 0th power of 2 = 1/1
perfect fifth 1st power of 3 / 1st power of 2 = 3/2
major second 2nd... 9/8
major sixth 27/16
major third 81/64
major seventh 243/128
tritone 729/512.....

(by the way 5x5 and higher does'nt fit to this context)
so we have two systems of "diatonic" intervals. the first is based on the simpliest relations of small numbers (1-5) and their combinations (excepting the tritone). the second is based only on powers of 2 and 3 but reaches up to numbers like 531441/524288. both of them are ambiguous in detail but theoretically included in the harmonic scale! both types of intervals are used by musicians and instrument makers (it's sometimes really confusing and sounds awfully together).

in the harmonic scale each interval is different to it's neighbour. but lower octaves do exist for any harmonic wich contents factor 2, other intervals are repeated in any other multiple relation.

the diatonic scale basically contents only two types of intervals (major and minor seconds). but again each of these seconds is slightly different.

all these regularities are perceivably as distinct qualities or degrees of consonance for human ears and minds. we would recognise any aberrance. (there are some more options. equal intervals are another audible criterion. it leads among others to the well-tempered chromatic scale...
Yes,
this is also very nice feature of the "scientific" system.

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