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Hahn metric / Hahn-reduced lattice basis

[Gene Ward Smith, Yahoo tuning-math message 10958 (Sun Jul 18, 2004 12:57 pm)]

First I need to explain the Hahn metric. If <O a, b c| is a 7-limit monzo, then the octave class it represents does not depend on O, and a metric on octave classes will ignore it. The Hahn metric is

|| <O a b c| ||_Hahn = max(|a|,|b|,|c|,|a+b|,|b+c|,|c+a|,|a+b+c|)

It counts the number of 7-limit consonaces needed to get to the octave class. If we have a set of 7-limit commas, they are Hahn reduced by finding the comma with smallest Hahn distance, with ties broken by Tenney distance. Then we find the comma with second smallest Hahn distance independent from the first, and so forth.

In the case of the "pontiac" commas, the Hahn distance of 4375/4374 and 65625/65536 is 7, with 4375/4374 having the smaller Tenney distance. These define the Hahn reduced comma set for pontiac, since any other commas for it, such as 32805/32768 with Hahn distance 9, would under this measure be regarded as more complex.

This gives a rather different perspective on complexity for temperaments than figuring out how many generator steps it takes to reach a consonace; it figures out how many consonances it takes to reach the commas. Pontiac would, using the measure of the largest Hahn distance needed in a reduced set, have a complexity of 7. In comparison, meantone would have a complexity of 4, but orwell or miracle would have exactly the same complexity. Hemiwuerscmidt would have a complexity of 5, as would garibaldi. Shrutar would have a complexity of 6, but so would the <0 0 34 0 54 79| temperament, with mapping [<34 54 79 95|, <0 0 0 1|], which comes out with too high a badness to bother with because of its high complexity according to the usual ways to measure complexity. I've been pointing out that for some purposes, this isn't the relevant measure, which would in fact serve to rehabilitate temperaments such as this 34&68 temperament. I don't think I've managed to get my point across to anyone as yet.

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