User:Kaiveran/OLITTA Distance

Odd-limit-integrating triangularized taxicab distance (or OLITTA Distance, for short) is a modified form of triangularized taxicab distance, developed by Kaiveran in response to an issue raised by Kite Giedraitis in his original formulation, namely that the procedure can designate certain intervals as "2-steps" (e.g 9/7) which are in fact less complex both in prime- and odd-limit than some "1-steps" (e.g 11/7).

Procedure

(bleh, odd-number step down, include adjacent non-primes [e.g 13-prime-limit automatically extends to 15], open with comparison to Kite)

Advantages & Disadvantages

(simplify and gives you more moves to use)

The chief disadvantage is that OLITTA distance (as formulated above) is not a deterministic measure, unlike both regular and Kite-triangularized taxicab distance. Or, put another way, OLITTA measure has a designated odd-limit, just like your piece of music does.

This might be interesting for certain purposes, but most people would prefer a measure that does not change arbitrarily, and whose results depend only on the prime factors involved in the ratio.

(prime gap splitting method, with adjacent non-primes and midpoints between prime gaps always included:

7-limit automatically includes ratios of 9

13-limit automatically includes ratios of 15

23-limit automatically includes ratios of 25

89-limit automatically includes ratios of 91 and 93)

Examples

(comparative btw all 3 methods. you might not always want OLITTA, esp. for comma pump progressions)