This music book I'm reading taught me about frequency ratios as the basis for scales, but it doesn't explain it well enough for me, so I'm trying to learn more about it on my own. I want to know how they're applied to strings. I thought that since 3:2 (P5) is 1.5:1, that it would appear exactly halfway between the end of the string and the halfway point (octave), but that isn't the case, which I realized is due to the fact that pitches aren't evenly spaced throughout the string, a difference of pitch corresponds to a shorter length of string the farther "up" the string you get. And what actually does appear to be located halfway between the end and the octave is actually 4:3 (P4). And I'm really struggling to find a way to convert ratios, starting with 1:1 at the end of the string and ending at 2:1 halfway, to string length, starting at 0 and ending at the octave, which on my instrument seems to be around 15 1/2 inches. Why does 3:2 seem to appear at 2/3s of the string (up to the octave), while 4:3 shows up at 1/2? What's the conversion process I'm supposed to use here?

I'm really not very knowledgeable about math and I have trouble understanding things a lot of the time, just as a warning.

Edit: I should mention that the instrument I'm using for reference is a shamisen, which has no frets.

  • 1
    Maybe this can provide the beginning of an answer? en.wikipedia.org/wiki/…. For instance, for the fifth: that's the third harmonic. What you get is three time the fundamental pitch of the string. Scale it down to the initial octave (divide by two) ends with a ratio of frequencies of 3/2, which is P5.
    – Tom
    Nov 17, 2021 at 8:38
  • This actually gives me another question. I thought that strings vibrated in half, then in 4ths, then in 8ths, etc. But this article seems to be saying it vibrates in half, then in thirds, in fourths, etc. Nov 17, 2021 at 8:43
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    Related question.
    – guidot
    Nov 17, 2021 at 8:44
  • Thanks. I don't think I can read through all of that to find whatever helpful information might possibly be there though. I'm really just looking to have my specific question answered. (This comment is meant genuinely, and not with some kind of smug or dismissive undertone; I have Asperger's.) Nov 17, 2021 at 8:50
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    Faulty maths! The point halfway between the end of the string and its halfway point is actually 3/4, not 2/3. There lies the 2nd octave. The string 'vibrates' in 1/2, 1/3, 1/4, 1/5 etc. At those points, you'll find each node for the harmonic. Found simplest from each end of the string.
    – Tim
    Nov 17, 2021 at 14:43

1 Answer 1


Frequency is inversely proportional to string length (f=C/l where C is constant for a given string at a given tension), so halving the length doubles the frequency (an octave). Halfway to the octave leaves you with 3/4 of the string length, so 4/3 of the frequency (a perfect fourth). If you want a perfect fifth (3/2) you need a string length of 2/3, so you finger at 1/3 of the length.

  • Ok, would this also hold to something like 3/5ths of the length being 5:3? Nov 17, 2021 at 9:08
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    @コナーゲティ That's correct
    – PiedPiper
    Nov 17, 2021 at 9:11
  • +1 for the clarity of this answer and boiling it down to simple but accurate terms, per the background knowledge of the asker
    – jdjazz
    Dec 6, 2021 at 16:57

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