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I have been looking into creating the sound of a rainbow, (having taken four proposed ranges for each colour band, translating Terahertz to Hertz and taking the mean average mid-points of each colour to the closest natural note), I have arrived at the following set of notes:

445 Hz Fourth A

496 Hz Fourth B

524 Hz Fifth C

560 Hz Fifth D flat

630 Hz Fifth E flat

685 Hz Fifth F

744 Hz Fifth G Flat

I have built a working seven-string harp/lyre thing, designed for acoustic Guitar strings with an approximate resonating length of 56 cm/22".

I understand enough to know that there are pitch/tension calculations involved, but I have no idea of how to do these calculations.

If anyone feels kind enough to do the maths for a music beginner I would be very grateful.

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  • I think you might want to try tuning them by ear; I'm sure there's a way (but don't have time to look for it now) to generate the pitches you seek on your computer.
    – Dave
    Mar 14, 2014 at 15:54
  • I’m guessing he’d like to know what gauge he should buy, before tuning.
    – Édouard
    Mar 14, 2014 at 16:21
  • The smallest string on a standard acoustic guitar plays E4. To get all the way up to Gb5 with a scale length of 22" will require either a much smaller string or a much higher tension than is normally on a guitar.
    – Ben Miller
    Mar 14, 2014 at 17:20
  • You may find a perusal of Scriabin's study of the subject interesting. For a start, see "Philosophical influences and the influence of colour" at en.wikipedia.org/wiki/Alexander_Scriabin . I will be curious to see whether your analysis of relative frequencies derives anything similar to his scheme.
    – BobRodes
    Mar 14, 2014 at 19:58
  • I had a look at the Scriabin page, my research has kind of finished now that I have the seven notes that I need. The key difference between us is that I started from scientific and modern note values and put my, 'red,' as an A, he decided that his red was a C. I also looked for a set of seven notes, while he had two related sets of six. I have no plans to look into this further, but you never know.
    – Windfire
    Mar 16, 2014 at 15:58

3 Answers 3

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Wikipedia is a good source for this kind of information. From the vibrating string page, you can see that the formula for the frequency of a (idealised) string is

f = 1 ∕ L ⨉ square_root( T / µ )

  • f: the frequency (in Hertz)
  • L: the length of the string (in metres)
  • µ: the linear mass of the string (in kilograms / metre)
  • T: the tension of the string, i.e. the force which is applied to it (in Newtons)

In your case the length of the string is determined by your instrument. You thus have two parameters on which you can play: the linear mass (by choosing your string) and the tension.

Obviously, you will adjust the tension through tuning. Depending on the force your harp can handle, you may want to choose different string, as you would have to apply more tension to heavier strings.

Moreover, I’m not sure the linear mass of guitar string is readily available on packages. If you already have strings, measuring the linear mass is straightforward: weigh a (not too short) piece of string, divide by its length.

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  • I have my spreadsheet programmed, and if I install a sloped metal bar inside the sound chamber I can get control over the length as well. Thank you
    – Windfire
    Mar 14, 2014 at 19:57
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Given that the first string in a guitar (the high E string) is E4 which is about 329Hz, maybe you'll only be able to use a bunch of .009 strings and work on their length to match the notes you want.

FWIW you can play the notes you mention on a guitar all on the first string, using frets 5, 7, 8, 9, 11, 13 and 15. Measure the length from the bridge to those frets. If you'd rather use the second string (0.011 gauge) then they're frets 10, 12, 13, 14, 16, 18 and 20 (just add 5 to every fret in the first string).

Of course you can vary the tension in some cases, since the notes are all a semitone or a whole tone apart.

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The D'Addario String Company publishes a large PDF guide that shows tables of exactly the kind of information you are looking for. It is designed for instruments in the guitar family, but I believe you will find it adaptable to your project.

It lists all the types and models of strings that they sell.

It explains the math involved in calculating certain tunings, but it also makes it convenient for you to start with a certain scale length and a certain pitch, and look up the exact model and formulations of strings that will work for your tuning, at which point it also lists the tension for that string at that pitch.

The guide is designed for various instruments in standard scale lengths from the bass guitar through the acoustic guitar and down to the mandolin.

I think there's valuable information for you here because you can learn a lot about the properties of various kinds of wound strings, not just plain strings. You can also learn about all the kinds of standard mass-produced strings on the market, and which might be suitable for you to buy for use in your project.

D'Addario String Tension Guide

They also have another chart and guide for bowed string instruments.

enter image description here

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