Inverse-complexity-prescaled complexity: Difference between revisions

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This article is a cautionary tale for anyone who (as I, [[Douglas Blumeyer]], did) got temporarily seduced and totally confused about: using the inverse of a complexity prescaler with a [[interval complexity|complexity]] function, i.e. one that is [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_all-interval_tuning_schemes#Normifying_complexities|in (prescaled) norm form]]. Inverse-complexity-prescaled complexity functions: don't use them! Now that I have good enough terminology for the constituent parts, the name itself seems as self-contradictory as the concept.
This article is a cautionary tale for anyone who (as I, [[Douglas Blumeyer]], did) got temporarily seduced and totally confused about: using the inverse of a complexity prescaler with a [[interval complexity|complexity]] function, i.e. one that is [[Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes#Normifying complexities|in (prescaled) norm form]]. Inverse-complexity-prescaled complexity functions: don't use them! Now that I have good enough terminology for the constituent parts, the name itself seems as self-contradictory as the concept.


So why would someone ever want to try this? Well, I had looked into it because I was curious about the limitation of [[all-interval tuning scheme]]s whereby they only work with [[simplicity-weight]] [[damage]]. I'd wondered if there was nonetheless a way to achieve complexity-weight-like effects anyway. As you will see from this article, the answer is a very slight "sort of", but at such a cost of reasonableness that there's no way it could be worth it.
So why would someone ever want to try this? Well, I had looked into it because I was curious about the limitation of [[all-interval tuning scheme]]s whereby they only work with [[simplicity-weight]] [[damage]]. I'd wondered if there was nonetheless a way to achieve complexity-weight-like effects anyway. As you will see from this article, the answer is a very slight "sort of", but at such a cost of reasonableness that there's no way it could be worth it.
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Let's see what we'd get if we chose our default complexity, [[log-product complexity|log-product]] <math>\text{lp-C}()</math> in this case. This means substitute in:
Let's see what we'd get if we chose our default complexity, [[log-product complexity|log-product]] <math>\text{lp-C}()</math> in this case. This means substitute in:
* <math>1</math> in place of our [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_all-interval_tuning_schemes#Power_norms|norm power]] <math>q</math>,  
* <math>1</math> in place of our [[Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes#Power norms|norm power]] <math>q</math>,  
* the log-prime matrix <math>L</math> in place of <math>X</math>, and  
* the log-prime matrix <math>L</math> in place of <math>X</math>, and  
* its inverse <math>L^{-1}</math> in place of <math>X^{-1}</math>.  
* its inverse <math>L^{-1}</math> in place of <math>X^{-1}</math>.