Ratio Scale transformation

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I am in a pickle, and I would genuinely appreciate it if you could guide me.

I spent 2 years to find a way to develop a ratio scale, and I did it; however, it cannot be used the way it is.

The scale consists of 17 values 1/9, 1/8 , 1/7, 1/6, 1/5 , 1/4, 1/3, 1/2, 0 , 2, 3, 4, 5, 6,7, 8, 9, and I need to transform the 0 to 1 in way that ratios among all values remains the same after transformation. It possible?

Thanks in advance for your valuable time