Another typology is provided by Westphal in "Franken oder Sachsen? Untersuchungen an fruhmittelalterlichen Waffen":

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He also subdivides the schmallsaxe and langsax. He doesn't go into the typology of the langsax, but he does in "Untersuchungen an Langsaxen in niederlandischen Sammlungen". Differences in the typology include fuller arrangement, method of patternwelding (if present) and the average thickness of the blade:

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