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  • For any harmonic, you first need to get a sound source of that frequency such as a tuning fork or speaker. It's best to place the source where you expect an antinode to be. You can try to just pluck the string at that point but that will probably also produce a lots of harmonics you don't want.

    For the kth harmonic, there are k antinodes at (2i-1)/(2k) of the string length, where ik; i∈ℕ.

    Fundamental (A1):

    • ½ the string length (fret 12)

    2nd harmonic (1 octave up, A2):

    • ¾ the string length (fret 4.98)
    • ¼ the string length

    3rd harmonic (perfect fifth from A2 or approx. E3):

    • ⅚ the string length (fret 3.15)
    • ½ the string length (fret 12)
    • ⅙ the string length

    4th harmonic (2 octaves up, A3):

    • ⅞ the string length (fret 2.32)
    • ⅝ the string length (fret 8.14)
    • ⅜ the string length
    • ⅛ the string length

    8th harmonic (4 octaves up, A4):

    • 15/16 the string length (fret 1.12)
    • 13/16 the string length (fret 3.59)
    • 11/16 the string length (fret 6.49)
    • 9/16 the string length (fret 9.96)
    • 7/16 the string length
    • 5/16 the string length
    • 3/16 the string length
    • 1/16 the string length

    Using fractional frets is cumbersome because they are non-linear. You're probably better off with a tape measure or ruler.

  • the 12th fret is 1 octave up

    Yup. 12 semitones is 1 octave so A2 on the bass guitar's A string. The frequency ratio to A1 is 2:1.

    5th is 2 octaves

    What? No. That's 5 semitones or 500 cents from A1, which is D2, close to a perfect fourth from A1 (frequency ratio 4:3 or 498 cents).

    Two octaves would be 24 semitones or 24 frets (not available on most fretted instruments) for a frequency ratio of 4:1, or A3.

    just past the second fret is 3 octaves

    No! The 2nd fret is 2 semitones or 200 cents above A1, which is B1, close to a major second from A1 (frequency ratio 9:8 or 196 cents).

    3 octaves would be 36 semitones or 3600 cents for a frequency ratio of 8:1, or A4.

  • As Frozengyro@lemmy.world mentioned, you can have interesting results by using harmonics - tones that are a whole (k) multiple of the base frequency because then the string vibrates in a standing wave forming a series of k+1 nodes (including ends) and k antinodes equally spaced across its length. Such notes are:

    Closest noteFreq.Harm.Relation to A
    A155 Hzbase(aka fundamental or open string frequency)
    A2110 Hz2ndoctave above A1
    E3 + 2 cents165 Hz3rdperfect fifth from A2
    A3220 Hz4thoctave above A2
    C#4 - 14 cents275 Hz5thmajor third from A3
    E4 + 2 cents330 Hz6thperfect fifth from A3
    G4 - 31 cents385 Hz7thfar from a note on the chromatic scale
    A4440 Hz8thoctave above A3
    B4 + 4 cents495 Hz9thmajor second from A4
    C#5 - 14 cents550 Hz10thmajor third from A4
    D#5 - 49 cents605 Hz11thvery far from a note on the chromatic scale
    E5 + 2 cents660 Hz12thperfect fifth from A4
    F5 + 41 cents715 Hz13thvery far from a note on the chromatic scale
    G5 - 31 cents770 Hz14thfar from a note on the chromatic scale
    G#5 - 12 cents825 Hz15thminor second below A5
    A5880 Hz16thoctave above A4

    Frequencies and relations are exact, closest chromatic (piano) notes other than A are approximate, the deviation is expressed in whole cents (hundreths of semitones). Notes more than 20 cents off the chromatic scale will probably sound off so they are discouraged. You could continue forever but frequencies above that will have a very weak response.

    Yes, you will get some resonance on non-integer multiples but way less.

  • I think that would make a standing wave with a series of nodes/antinodes on the string, and how well it works would strongly depend on where the tuning fork is along the string. This has the potential to be more interesting but it's not as easy. See my other comment for a table at which frequencies a standing wave occurs on the A1 string.

  • For context, bass guitar strings are tuned 3 octaves lower than that. The frequency of the A string is 55 Hz. You can't even reach 220 Hz using the 12th fret on the highest (G) string. Tuning a bass A string to 8 times the frequency would require increasing its tension almost 3 times 64 times. The guitar body should will not survive such forces but the string will snap long before you reach 110 Hz.

    Edit: got the quadratic formula the other way around

    You can do the experiment on a non-bass guitar: "shorten" the high E string (330 Hz) by 5 frets to reach close to 440 Hz. It's a chromatic scale and not a perfect fifth (error of +0.02 semitones) but that can be corrected without damage by holding the 5th fret and tuning the string to exactly 440 Hz. This shorter string will then react to the tuning fork as intended.