efficiency

Whoops, strike that last message

I just wrote a program to calculate efficiency. Unfortunately, the time it
takes the program to run increases exponentially with the cardnality of the
scale and it already takes maybe half an hour with 9 notes. But here are
values for all the low cardinality 1-2 MOS's, for any interested:

_Generator_____Step Pattern____Efficiency
1/4 221 1.0
2/5 211 0.889

1/5 2111 1.0
2/7 2221 0.833
3/7 21211 0.86
3/8 22121 0.82

1/6 21111 1.0
2/9 22221 0.8
3/10 2121211 0.797
3/11 2212121 0.777
4/9 2112111 0.853
5/12 2221221 0.769
4/11 21211211 0.793
5/13 22122121 0.759

1/7 211111 1.0
2/11 222221 0.778

Anyone hazard a guess as to what we can correlate high efficiency with?
(This is another one of those are-you-reading-the-numbers quizes)

jason

Jason Yust wrote...

>_Generator_____Step Pattern____Efficiency
>1/4 221 1.0
>2/5 211 0.889
>
>1/5 2111 1.0
>2/7 2221 0.833
>3/7 21211 0.86
>3/8 22121 0.82
>
>1/6 21111 1.0
>2/9 22221 0.8
>3/10 2121211 0.797
>3/11 2212121 0.777
>4/9 2112111 0.853
>5/12 2221221 0.769
>4/11 21211211 0.793
>5/13 22122121 0.759
>
>1/7 211111 1.0
>2/11 222221 0.778
>
>Anyone hazard a guess as to what we can correlate high efficiency with?

Um... better just tell us. BTW, Scala calculates efficiency. Manual,
how's your algorithm doing? Were you ever able to figure out or make use
of Rothenberg's crazy "strategy"?

-Carl

Carl,

I wrote...
>
>>_Generator_____Step Pattern____Efficiency
>>1/4 221 1.0
>>2/5 211 0.889
>>
>>1/5 2111 1.0
>>2/7 2221 0.833
>>3/7 21211 0.86
>>3/8 22121 0.82
>>
>>1/6 21111 1.0
>>2/9 22221 0.8
>>3/10 2121211 0.797
>>3/11 2212121 0.777
>>4/9 2112111 0.853
>>5/12 2221221 0.769
>>4/11 21211211 0.793
>>5/13 22122121 0.759
>>
>>1/7 211111 1.0
>>2/11 222221 0.778
>>
>>Anyone hazard a guess as to what we can correlate high efficiency with?
>
>Um... better just tell us.

Alright, alright. There are two correlations: the stronger correlation is
between high efficiency scales and scales with fewer gaps (notes in the
system but not in the scale). The weaker correlation is between high
efficiency and low cardnality. One interesting point is that for stricty
proper MOS such that the cardinality is smaller than half the cardnality of
the system (with step sizes 2-3 or 3-5, et c--that is, more scale degrees
than gaps), the weaker correlation is the only one. Another interesting
point is that if we arrange the following kind of scale-tree:

1/3
1/4 2/5
1/5 2/7 3/7 3/8
1/6 2/9 3/10 3/11 4/9 5/12 4/11 5/13

The scales can be grouped into families which will all have a similar
efficiency by taking up-right to down-left diagonals. So 1/3, 1/4, 1/5,
and 1/6 all have the form 2111 . . ., 2/5, 2/7, 2/9 have the form . . .
2221. The first family with numerator 3 has a pattern of 21's and one 211,
the second has 21's and one 221. And so on. The new families will always
begin on the right half of the tree.

>BTW, Scala calculates efficiency. Manual,
>how's your algorithm doing? Were you ever able to figure out or make use
>of Rothenberg's crazy "strategy"?

I'd be interested to know what this strategy is. I used a brute force
method suggested to me by John Chalmers looking for sufficient sets in the
interval matrix. I could have speed up the program considerably by having
it recogise that one suf. set was good enough for every permutation that
begins with it, but I lost my original code and don't want to rewrite the
program. How is scala with 11 - 12 and higher cardinality scales?

jason

>

>>>_Generator_____Step Pattern____Efficiency
>>>1/4 221 1.0
>>>2/5 211 0.889
>>>
>>>1/5 2111 1.0
>>>2/7 2221 0.833
>>>3/7 21211 0.86
>>>3/8 22121 0.82
>>>
>>>1/6 21111 1.0
>>>2/9 22221 0.8
>>>3/10 2121211 0.797
>>>3/11 2212121 0.777
>>>4/9 2112111 0.853
>>>5/12 2221221 0.769
>>>4/11 21211211 0.793
>>>5/13 22122121 0.759
>>>
>>>1/7 211111 1.0
>>>2/11 222221 0.778
>>>
>>>Anyone hazard a guess as to what we can correlate high efficiency with?
>>
>>Um... better just tell us.
>
>Alright, alright. There are two correlations: the stronger correlation is
>between high efficiency scales and scales with fewer gaps (notes in the
>system but not in the scale). The weaker correlation is between high
>efficiency and low cardnality.

Isn't it just with scales whose pattern of seconds is smooth?

>One interesting point is that for stricty proper MOS such that the
>cardinality is smaller than half the cardnality of the system (with step
>sizes 2-3 or 3-5, et c--that is, more scale degrees than gaps), the weaker
>correlation is the only one.

That makes it the weaker correlation?

>>BTW, Scala calculates efficiency. Manual, how's your algorithm doing?
>>Were you ever able to figure out or make use of Rothenberg's crazy
>>"strategy"?
>
>I'd be interested to know what this strategy is.

John Chalmer's, Manuel, or I will send it to you. I don't think any of us
understand it (guys?). Good luck!

-Carl

Carl Lumma wrote:
>>BTW, Scala calculates efficiency. Manuel, how's your algorithm doing?
>>Were you ever able to figure out or make use of Rothenberg's crazy
>>"strategy"?

My algorithm is fine. Only slowly for larger scales so I don't execute it
for scales with more than 11 notes (some exceptions like Pythagorean scales
for which I derived the result analytically). But because of "Miller's law"
I
think RE is not so meaningful for larger scales anyway.
Rothenberg's papers don't describe the algorithm in detail and I don't know
how it works. I only know that it's limited to subsets of equal
temperaments,
which my algorithm is not.
However Robert Walker and I have held a little contest. Robert invented
another algorithm which he claimed would be faster than mine. I suspect
that his is similar to Rothenberg's approach. It was indeed faster, but
also limited to subsets of equal temperaments (with less than 64 tones),
and
it consumes more memory.
I planned to incorporate his routine in Scala, but other additions to be
made
have higher priority.
Robert or I can email his or my code to anyone interested.

Manuel Op de Coul coul@ezh.nl

>>>BTW, Scala calculates efficiency. Manuel, how's your algorithm doing?
>>>Were you ever able to figure out or make use of Rothenberg's crazy
>>>"strategy"?
>
>My algorithm is fine. Only slowly for larger scales so I don't execute it
>for scales with more than 11 notes (some exceptions like Pythagorean scales
>for which I derived the result analytically). But because of "Miller's law"
>I think RE is not so meaningful for larger scales anyway.

Agreed.

>Rothenberg's papers don't describe the algorithm in detail and I don't know
>how it works. I only know that it's limited to subsets of equal >temperaments, which my algorithm is not.

The actual implementation he used may be limited to subsets of ETs, but
is the strategy? Rothenberg's paper on the strategy struck me as using some
very specialized language, and its application in an algorithm for
calculating efficiency seemed non-trivial, but it certainly didn't strike
me as not being detailed enough!

>However Robert Walker and I have held a little contest. Robert invented
>another algorithm which he claimed would be faster than mine. I suspect
>that his is similar to Rothenberg's approach. It was indeed faster, but
>also limited to subsets of equal temperaments (with less than 64 tones),
>and it consumes more memory. I planned to incorporate his routine in
>Scala, but other additions to be made have higher priority. Robert or I
>can email his or my code to anyone interested.

Groovy. You guys are a great asset to our community! I request that
both algorithms be posted to the list, in pseudocode if possible, if
they aren't too long. If they are too long, send them to me off-list.
Thanks!

-Carl

Please take me off the mailing list.
Kevin

>
>________________________________________________________________________
>________________________________________________________________________
Carl,

Let me get back to your questions on my comments about efficiency,

I said:
>>There are two correlations: the stronger correlation is
>>between high efficiency scales and scales with fewer gaps (notes in the
>>system but not in the scale). The weaker correlation is between high
>>efficiency and low cardnality.

You replied:
>Isn't it just with scales whose pattern of seconds is smooth?

I don't know what you mean by smooth. It is limited to scales with
generators which are subsets of an ET. Also, it's limited to Ls ratio of
2, but as I said, efficiencies for MOS with s > 1 (these must represent
fewer than half the notes of the ET they're in) can be calculated more
simply. That leaves only Ls ratios 3:1, 4:1, et c as interesting cases for
MOS. I'll try some examples of those.

>>One interesting point is that for stricty proper MOS such that the
>>cardinality is smaller than half the cardnality of the system (with step
>>sizes 2-3 or 3-5, et c--that is, more scale degrees than gaps), the weaker
>>correlation is the only one.
>
>That makes it the weaker correlation?

It's weak for L=2, s=1 scales, the ones I was discussing. What I called
the strong correlation reaches a plateau, you could say, when the number of
notes in the scale equals the number of gaps in the scale. So in a sense
it still is present as the stronger factor, but it no longer admits of
degree.
Let me re-list the efficiencies in order next to the ratio of tones in the
scale to gaps:

_notes:gaps____Step Pattern____Efficiency
3:1 211 1.0
4:1 2111 1.0
5:1 21111 1.0
6:1 211111 1.0
3:2 221 0.889
5:2 21211 0.86
7:2 2112111 0.853
9:2 21121111 0.853
4:3 2221 0.833
5:3 22121 0.82
5:4 22221 0.8
7:3 2121211 0.797
8:3 21211211 0.793
10:3 2112112111 0.788
6:5 222221 0.778
7:4 2212121 0.777
7:5 2221221 0.769
8:5 22122121 0.759
9:7 222212221 0.744
10:7 2221221221 0.737

jason

To cover those other interesting cases of ET MOS with integral L:s > 2,
even for L:s = 3:1, scales which don't hit the "efficiency plateau" I
mentioned earlier are relatively rare. Actually, let's call it a
"flatline" since its the lowest possible efficiency of the scales of a
given cardinality. That flatline I said occurs when the scale covers half
or fewer of the pitches of the ET. Actually it occurs one step earlier,
when (cardnality) <= (number of gaps) + 1. Look at the complete list for
the first few groups of 3-1 scales:

Generator______Step Pattern__card:gaps_efficiency_____2(n+1)/3n
1/4 31 2:2 1.0 1.0
1/5 311 3:2 0.889 0.889
3/7 331 3:4 0.889 0.889
1/6 3111 4:2 0.875 0.833
3/10 3331 4:6 0.833 0.833
4/9 31311 5:4 0.8 0.8
4/11 33131 5:6 0.8 0.8
1/7 31111 5:2 0.86 0.8
3/13 33331 5:8 0.8 0.8
4/13 3131311 7:6 0.762 0.762
4/15 3313131 7:8 0.762 0.762
5/11 3113111 7:4 0.777 0.762
7/17 3331331 7:10 0.762 0.762
5/14 31311311 8:6 0.754 0.75
7/18 33133131 8:10 0.75 0.75

All but a few have efficiency equal to the flatline value for their
cardnality: 2(n + 1)/3n where n = cardinality. These nontrivial cases line
up like so:

Generator______Step Pattern__card:gaps_efficiency_____2(n+1)/3n
1/6 3111 4:2 0.875 0.833
1/7 31111 5:2 0.86 0.8
1/8 311111 6:2 0.856 0.778
5/11 3113111 7:4 0.777 0.762
6/13 311131111 9:4 0.763 0.741
5/14 31311311 8:6 0.754 0.75

As you can see, the correlations are the same as the L:s = 2:1 cases I
posted. For higher L:s, nontrivial cases become even more rare, of course.
Some 4:1 cases with similar cardnality are:

Generator______Step Pattern__card:gaps_efficiency_____2(n+1)/3n
1/8 41111 5:3 0.82 0.8
1/9 411111 6:3 0.806 0.778
7/15 411141111 9:6 0.746 0.741

jason