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We call any series of frequencies which are multiples of 1,2,3,4,5... etc a harmonic series, and they define the notes (for example) C0-C1-G1-C2-E2-G2-Bb2-...

Is there a name for the series of frequencies 1,2,4,8,16... multiples of which would define the octaves C0-C1-C2-C3-C4-C5...?

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Not sure I'm interpreting your question correctly, but you seem to be asking about about the mathematical relationship between the frequency (Hz) of a pitch and the sequences of pitches that we organize into scales and octaves.

So at the risk of being too pedantic, we map frequencies to pitches using a logarithmic scale (in base 2).

Because our ears don't perceive pitch distance linearly with frequency, as the wikipedia article for "12-tone equal temperament" says:

the ratios of the frequencies of any adjacent pair of notes is the same, and, as pitch is perceived roughly as the logarithm of frequency, equal perceived "distance" from every note to its nearest neighbor.

So to answer your question, an octave is a unit of frequency used with logarithm base-2. (A semitone is one twelfth of an octave. A cent is one hundredth of a semitone.) The "series" of octaves itself would just be referred to as a base-2 logarithmic scale.

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If I understand your question right these are the names of the different octavas:

C0 - B0: sub-contra octave (A0 is the lowest pitch on a full piano) C1 - B1: contra octave C2 - B2: great octave C3 - B3: small octave C4 - B4: one-line octave, or 2nd small octave (contains both middle C and A440) C5 - B5: two-line octave, or 3rd small octave C6 - B6: three-line octave, or 4th small octave C7 - B7: four-line octave, or 5th small octave C8 - B8: five-line octave, or 6th small octave (C8 is the highest pitch on a full piano)

source:

https://www.liveabout.com/pitch-notation-and-octave-naming-2701389

As the following table shows there are different pitch names used in different countries (languages) and the naming in midi is a special kind of numbering 1-128

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