How can make the differences between these values less extreme while still allowing them to sum to 100%?

Question

I am designing a digital fretboard. The frets on fretted instruments decrease in width as you get further down the fretboard.

I looked up a fretboard width calculator online. I don't know the algorithm this calculator used. The calculator gave me the inch widths of each fret and I converted the values to percentages so that I could use them in a CSS stylesheet (styling for websites).

Here are the values it gave me:

https://i.stack.imgur.com/Tge3w.jpg

Here they are for copy and pasting

    8.42384106%
    7.95584989%
    7.505518764%
    7.081677704%
    6.684326711%
    6.313465784%
    5.960264901%
    5.624724062%
    5.306843267%
    5.006622517%
    4.72406181%
    4.459161148%
    4.21192053%
    3.973509934%
    3.752759382%
    3.540838852%
    3.346578366%
    3.152317881%
    2.975717439%

My problem is that the first frets are too wide and the final widths are to narrow. Is there an equation I can use that will allow me to smooth out the differences between these values, while allowing them to still sum to 100%?

An equation where there are n frets and then maybe some coefficient C that lets me play around with getting a distribution of widths like this?

Answer 1

I don't think there is a solution to your problem. If you smooth out the differences, you'll end up with some weird sort of tuning. You can see this because one "solution" would be to make all the frets equal, and you know that wouldn't work. The frets are the widths they are so that your instrument can be well-tempered. You want your 3rds and 4ths to sound like 3rds and 4ths. Not like 17/5 ths, eh?