Yahoo Tuning Groups Ultimate Backup tuning-math Re: Entropic efficiency of a scale (from Tuning)

>>> The efficency of an alphabet of n letters can be defined as the ratio
>>> of the entropy of the alphabet over the maximum entropy for such an
>>> alphabet. Here each letter ni appears with proabilbility pi, and the
>>> the entropy of the alphabet is -sum_i pi log2(pi). If each pi is 1/n,
>>> we get the maximum entropy, which is just log2(n). Hence the efficiency
>>> can also be written -sum_i pi log_n(pi).
>>>
>>> We may consider the scale steps si of an octave-repreating scale to be
>>> an alphabet, if we assign a probability of log2(si) to each scale
>>> step. While larger steps are not proportionally more likely to occur,
>>> they do take up more room, so there's some idea of efficiency involved
>>> here. Then the entropic efficency of a scale with scale steps si would
>>> be -sum_i log2(si) log2(log2(si)) / log2(n). It is a measure of how
>>> nearly equal the step sizes are, achieving the value of 1 in the case
>>> of equal temperaments.
>>
>> How is this different from Rothenberg efficiency?
>
> Rothenberg efficiency is a totally different thing, isn't it?

I'm not clear on what si is above. I would think pi = si/1200 if
si were in cents.

But I can tell you that they're both called efficiency, both related
to entropy, and you both happened to use the variable si for the thing
being summed over. Though you're just (apparently) looking at size
while Rothenberg considers "sufficient subsets", his efficiency does:

>Efficiency measures the complexity of the rank-order matrix in terms of
>the modes. Specifically, it measures how alike the modes are. When a
>listener is first exposed to a pitch set, he checks to see if he already
>knows its rank-order matrix. Efficiency measures how long it takes him
>to do this -- on average, what portion of the scale must he hear before
>he can establish which pitch is the tonic and start tracking scale
>degrees? If the modes are similar in structure, he must wait longer.
>For scales whose modes are identical (scales with mean variety = 1),
>such as the "whole-tone" scale (6-tET), there will be no way to assign
>a tonic at all. Rothengberg considers such scales minimally efficient,
>but they could just as well be considered maximally efficient.

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> I'm not clear on what si is above. I would think pi = si/1200 if
> si were in cents.

Sorry, these are subscripts; I should have written s_i and p_i I suppose.

EE = -sum_i log2(s_i) log2(log2(s_i)) / log2(n)

> But I can tell you that they're both called efficiency, both related
> to entropy, and you both happened to use the variable si for the thing
> being summed over.

I'm just using a terminology that sometimes occurs in information
theory; nothing to do with Rothenberg per se, though as you point out
what I'm calling "efficency" is inversely related to what Rothenberg
called "efficiency". It's really a measure of regularity.

If you want to work it so it is directly correlated, we could take
this instead:

E = 1 + sum_i log2(s_i) log2(log2(s_i)) / log2(n)

We then have for example E=0 for equal divisions, E=0.011 for
1/4-comma diatonic, E=0.014 for Zarlino/Ptolemy, and E=0.02 for 12-et
diatonic. Maybe multiplying by 100 would be a good idea.

🔗Carl Lumma <ekin@lumma.org>

>> I'm not clear on what si is above. I would think pi = si/1200 if
>> si were in cents.
>
>Sorry, these are subscripts;

Right, but are they? Scale degrees in Hz. or something?

>I should have written s_i and p_i I suppose.
>
>EE = -sum_i log2(s_i) log2(log2(s_i)) / log2(n)

No, I could read it. It's the extra log2 on top that throws me.

>> But I can tell you that they're both called efficiency, both related
>> to entropy, and you both happened to use the variable si for the thing
>> being summed over.
>
>I'm just using a terminology that sometimes occurs in information
>theory; nothing to do with Rothenberg per se,

Rothenberg's is also from information theory.

>though as you point out what I'm calling "efficency" is inversely
>related to what Rothenberg called "efficiency".

Is it? If you read what I wrote, Rothenberg calls ETs minimally
efficient, but that's an arbitrary choice. The sufficient sets could
just as well be the entire scale rather than a single element as
Rothenberg has it.

>It's really a measure of regularity.

So is Rothenberg's.

-Carl