By applying findpeaks function (with appropriate option settings) to the average intensity profile, you can detect locations of dark bands.
I'm not sure what is the definition of the "average pixel-width" of dark band, but I believe one possible (and simple) solution would be detectiong width at the half-prominence point for each nitch in the the average intensity profile.
% Read the image
I = imread('arsringsbild2.png');
I = rgb2gray(I);
% Calculate average intensity and dark positions
avgIntensity = mean(double(I));
[val,pos] = findpeaks(-1*avgIntensity,...
'MinPeakProminence', 40,...
'MinPeakHeight', -100);
val = -1*val;
% Visualize the result
figure
subplot('Position',[0.1 0.8 0.8 0.15])
imshow(I)
subplot('Position',[0.1 0.1 0.8 0.7])
plot(avgIntensity)
hold on
scatter(pos,val,'^','filled')
legend({'Avg. Intensity','Detected dark position'},'FontSize',12)
xlim([1 size(I,2)])
ylabel('Average intensity for each horizontal position')
