User:Kaiveran/OLITTA Distance: Difference between revisions

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'''Odd-limit-integrating triangularized taxicab distance''' (or '''OLITTA Distance''', for short) is a modified form of triangularized [[Commas by taxicab distance#Triangularizing proposal|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]]).

'''Odd-limit-integrating triangularized taxicab distance''' (or '''OLITTA Distance''', for short) is a modified form of triangularized [[Commas by taxicab distance#Triangularizing proposal|taxicab distance]], developed by [[User:Kaiveran|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]]).

== ~~Default~~ Procedure ==

== Procedure ==

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

== ~~Tweaks~~ ==

== Advantages & Disadvantages ==

(you ~~can include~~ more ~~or less odd non-primes into the OLITTA measure, like a 7-limit that doesn't include 9s, or a 23-limit that includes both 25 and 27~~

(simplify and gives you more moves to use)

The chief disadvantage ~~of this case~~ is that ~~there then ceases to~~ be a ~~singular OLITTA~~ measure.)

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 ==

(~~yah~~)

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

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-'''Odd-limit-integrating triangularized taxicab distance''' (or '''OLITTA Distance''', for short) is a modified form of triangularized [[Commas by taxicab distance#Triangularizing proposal|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]]).
+'''Odd-limit-integrating triangularized taxicab distance''' (or '''OLITTA Distance''', for short) is a modified form of triangularized [[Commas by taxicab distance#Triangularizing proposal|taxicab distance]], developed by [[User:Kaiveran|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]]).
-== Default Procedure ==
+== Procedure ==
 (bleh, odd-number step down, include adjacent non-primes [e.g 13-prime-limit automatically extends to 15], open with comparison to Kite)
-== Tweaks ==
+== Advantages & Disadvantages ==
-(you can include more or less odd non-primes into the OLITTA measure, like a 7-limit that doesn't include 9s, or a 23-limit that includes both 25 and 27
+(simplify and gives you more moves to use)
-The chief disadvantage of this case is that there then ceases to be a singular OLITTA measure.)
+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:
+-limit automatically includes ratios of 9
+-limit automatically includes ratios of 15
+-limit automatically includes ratios of 25
+-limit automatically includes ratios of 91 and 93)
 == Examples ==
-(yah)
+(comparative btw all 3 methods. you might not always want OLITTA, esp. for comma pump progressions)