Help with meaning of "the difference narrowing the longer women had the vote"

Question

In the quoted passage below, I came across a sentence I don't understand:

In 1955 Maurice Duverger published The Political Role of Women, the first behavioralist, multinational comparison of women’s electoral participation ever to use election data and survey data together. His study analyzed women’s patterns of voting, political candidacy, and political activism in four European countries during the first half of the twentieth century. Duverger’s research findings were that women voted somewhat less frequently than men (the difference narrowing the longer women had the vote) and were slightly more conservative.

Does the italicized sentence mean the difference of men and woman voting rates narrows the time that women have for voting?

Why shouldn't the sentence be "the difference narrower, the longer women had the vote"?

Thank you so much.

Answer 1

The parenthetical

(the difference narrowing the longer women had the vote)

is not a complete sentence; it's an aside describing the gap between the number of women who vote and the number of men. Women's participation is lower than men's, but the difference is larger just after women attain suffrage. Thereafter, the difference becomes less. The meaning is

(starting with the grant of female suffrage, the difference in voting frequency narrows as time goes along)

What's being described is a proportional relationship between two variables (which I'll call X and Y). English has an idiomatic way of phrasing this with the comparative form of adjectives:

the [comparative] X, the [comparative] Y

where [comparative] is the comparative of an adjective of magnitude. If the comparatives are of the same order -- both smaller (-) or both larger (+) -- then we have a direct proportionality; if they're of different order -- one smaller (-), one larger (+) -- then we have an indirect proportionality. For example, direct:

(+/+): The harder you work, the more you'll succeed.
(-/-): The smaller your salary, the less you can save.

For example, indirect:

(+/-): The steeper the grade, the slower the pace of the runners.
(-/+): The smaller the dog, the louder the bark.

X is the independent variable, the one you can think about varying freely. Y is the dependent variable, the one with values that follow the change in the X variable. Your suggestion isn't ungrammatical; it just violates the idiomatic expectation of

The longer women have the vote, the narrower the difference.

Answer 2

It means that as of 1955, in regions with female suffrage, women used their franchise less often than men, but the more years women's franchise had existed somewhere, the more often women voted there.

If we were to draw an X,Y graph with X=(4 year intervals) and Y=(votes by gender), the women's line would be drawn below the men's line, and the up/down gap between both lines would be wider on the left side, and narrower on the right side.

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Parsing it out:

1. Past Collection A (Men's past voting rates )
2. Present Collection A (Men's present voting rates)
3. Past Collection B (Women's past voting rates)
4. Present Collection B (Women's present voting rates)

Unrolling it, we get:

the [line-graphed] difference [between 1. and 3. vs 2. and 4., i.e. a 2nd order difference between two 1st order differences] narrowing the longer women had the vote

Think of it like the phrase "the more the merrier", but inverted, and with a four term 2nd order comparative in the first position.