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According to this page, ukulele natural harmonics can be played on the 12th (octave), 9th, 7th (octave+fifth), 5th (2octaves) and 3rd frets.

Which interval is obtained at the 9th and 3rd frets?

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    At least closely related, if not dupe - 'what are the actual notes of each natural harmonic?'.
    – Tim
    Commented Aug 19, 2022 at 7:58

3 Answers 3

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Harmonics can be played at any position where overtones have a "node", which is all integral divisions of the string. The 12th fret is at 1/2 of the string, the 7th and the 19th fret correspond to 1/3 and 2/3, The 5th fret and the 24th fret correspond to 1/4 and 3/4, slightly below 4th freth, 9th fret, 16th fret correspond to 1/5, 2/5, 3/5. 3rd corresponds to 1/6. Between 2/3, slightly under 6, slightly under 10, 15, 22 is 1/7, 2/7, 3/7, 4/7, 5/7. Slightly above 2nd fret, 8th fret, 16th fret correspond to 1/8, 3/8, 5/8. And so it goes on, but the higher you get the harder it becomes to pinpoint a node and the less energy is there, so it gets harder to get a nice harmonic.

1/2 then produces multiple of the first overtone, which is an octave. 1/3, 2/3 produce the second overtone, which is a fifth plus one octave. 1/4, 3/4 produce multiples of the third overtone, which is two octaves. 1/5, 2/5, 3/5, 4/5 produce the fourth overtone which is a (pure) major 3rd plus two octaves. 1/6, 5/6 produce the fifth overtone, which is a fifth plus two octaves, 1/7 - 6/7 produce the sixth overtone, which is a natural seventh. 1/8, 3/8, 5/8, 7/8 produce the seventh overtone, which is three octaves. And so on.

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    "integral divisions" - do you mean integer?
    – Vegard
    Commented Aug 19, 2022 at 13:07
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The 3rd fret is two octaves plus a fifth.
(see https://liveukulele.com/lessons/techniques/harmonics/#natural)

The 9th fret is two octaves plus a major third.
(see https://fretsource-guitar.weebly.com/natural-harmonics-map.html. Although the source regards guitar, it will be the same for ukulele.)

As a supplement, the Wikipedia page regarding the harmonic series may be of interest.

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The ukulele harmonic at the 12th fret corresponds to the same pitch as the open string, but one octave higher. The harmonic at the 7th fret corresponds to the same pitch as the 5th fret note, while the harmonic at the 5th fret corresponds to the same pitch as the 7th fret note.

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