RE: [tuning] Re: Part 2: Definition Question for Paul Erlich: "Tr ue" Interval

Jacky wrote,

>So the ear could percieve a 5/4 as a 9/7 or 16/13 etc...?

With very low probability, yes.

>Also: "Combination tones complicate the matter but with a knowledge
>of the amplitudes and frequencies of all combination tone components,
>Plomp's algorithm can be still applied. "

>Do the harmonic entropy computations include first, second ,third
>(etc..) order combination and difference tones supplied by
>neurological processing of intervals, or is anything beyond the first
>order negligible?

To be a bit oversimplistic, combination tones occur in the ear, not in the
neurological processing. Absolutely no combination tones of any order have
been included in my harmonic entropy calculations.

>What are some compositional uses of this theory?

Well, one might first want to construct theories of progressions and so on
that are founded on this theory, or you could just go off and write stuff in
some of the scales I've been finding . . .

Monz wrote,

>But what's *more* important, the theory says that the ear
>(more accurately, the ear/brain system) will *tend* to do
>the opposite: it will tend to perceive the 9/7, 16/13, etc.,
>as a 5/4, because the 5/4 falls at the point of one of the
>lower (i.e., more powerful) minima. The minima describing
>the other intervals are higher (less powerful), so perception
>will gravitate towards the 5/4.

Monz, first of all, in the results I've been showing and using so far, 9:7
has its own minimum, which means you can be pretty certain that 9:7 will
tend to be perceived primarily as 9:7. Also, there are ratios which are the
most likely interpretations of ranges of certain intervals but which don't
get local minima of their own.

I think you're confusing the input assumptions with features of the output
of the entropy calculation.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

> Monz wrote,
>
> Monz, first of all, in the results I've been showing and using
> so far, 9:7 has its own minimum, which means you can be pretty
> certain that 9:7 will tend to be perceived primarily as 9:7.

Oops... my bad. Should've looked at the chart before I took
a stab at answering this.

> Also, there are ratios which are the most likely interpretations
> of ranges of certain intervals but which don't get local minima
> of their own.
>
> I think you're confusing the input assumptions with features of
> the output of the entropy calculation.

Hmmm... can you explain that first paragraph in a little more
detail?

(Thanks Paul. Next time someone asks you a question, I'll wait
for *you* to answer it! :)

-monz
http://www.ixpres.com/interval/monzo/homepage.html

I wrote,

>> Also, there are ratios which are the most likely interpretations
>> of ranges of certain intervals but which don't get local minima
>> of their own.
>
>> I think you're confusing the input assumptions with features of
>> the output of the entropy calculation.

>Hmmm... can you explain that first paragraph in a little more
>detail?

For every true interval, there will be one ratio with the highest
probability value, but that ratio does not necessarily have a local minimum
of its own. I can't be bothered to look for it right now, but I posted a
bunch of harmonic entropy calculations (maybe around wintertime), primarily
addressing Dan Stearns, in which I gave the ranges for most likely
interpretations, and not all the ratios which had ranges corresponded to
minima. Perhaps Dan has this handy?

Paul H. Erlich wrote,

> For every true interval, there will be one ratio with the highest
probability value, but that ratio does not necessarily have a local
minimum of its own. I can't be bothered to look for it right now, but
I posted a bunch of harmonic entropy calculations (maybe around
wintertime), primarily addressing Dan Stearns, in which I gave the
ranges for most likely interpretations, and not all the ratios which
had ranges corresponded to minima. Perhaps Dan has this handy?

Hmm, I'm not sure, but I'll look around and see... I think that was
part of a thread where I was wondering if an algorithm I had used in
the past could perhaps be said to be something like a "poor man's"
Harmonic Entropy; Where if you let:

[((x1*y)*2)+1]
--------------
[((x2*y)*2)+1]

assume something like a margin of gravity to the left (i.e., flat of a
given ratio), and:

[((x1*y)*2)-1]
--------------
[((x2*y)*2)-1]

assume a margin of gravity to the right (i.e., sharp of a given
ratio); where "x1" and "x2" are x1=[N/(D/a)/b] and x2=[D/(D/a)/b], and
"a" and "b" are a=N*D and b=N+D, and "y" is
y=[(log(N)-log(D))*(n/log(2))], where "N" and "D" are the numerator
and denominator, and "n" is the variable that dictates the overall
range of gravity... you could achieve a slightly more 'bumpy' ride
across a minima and maxima field by allowing the ratios that comprise
a given "n" to occupy a 'height' that is N*D (which should also
theoretically give something along the lines of a Helmholtzian
"intensity of influence" measure across a given "n"). For example, if
n=9 you'd have something on the order of:

1/1 1132.10 0.00 76.96 1 1/1
22/21 74.90 80.54 87.09 462 22/21
19/18 87.07 93.60 101.19 342
17/16 97.65 104.96 113.44 272
15/14 111.16 119.44 129.05 210
13/12 129.01 138.57 149.66 156
12/11 140.28 150.64 162.65 132
11/10 153.70 165.00 178.11 110
10/9 169.96 182.40 196.82 90
9/8 190.07 203.91 219.93 72
8/7 215.59 231.17 249.20 56
7/6 249.03 266.87 287.47 42 7/6
13/11 279.26 289.21 299.90 143 13/11
6/5 294.78 315.64 339.69 30 6/5
11/9 335.62 347.41 360.06 99
16/13 351.27 359.47 368.07 208 16/13*
5/4 361.18 386.31 415.22 20 5/4
14/11 408.11 417.51 427.35 154 14/11
9/7 420.61 435.08 450.59 63 9/7
13/10 444.07 454.21 464.83 130
17/13 456.62 464.43 472.50 221 17/13*
4/3 466.41 498.04 534.32 12 4/3
15/11 528.07 536.95 546.14 165 15/11
11/8 539.26 551.32 563.93 88
7/5 563.73 582.51 602.60 35 7/5
17/12 595.10 603.00 611.11 204 17/12*
10/7 604.17 617.49 631.42 70 10/7
13/9 626.30 636.62 647.28 117
16/11 640.26 648.68 657.33 176
19/13 649.88 656.99 664.25 247 19/13
3/2 659.11 701.96 750.86 6 3/2
17/11 745.63 753.64 761.82 187 17/11
14/9 755.21 764.92 774.88 126
11/7 770.15 782.49 795.24 77 11/7
19/12 788.36 795.56 802.89 228 19/12*
8/5 796.77 813.69 831.36 40 8/5
13/8 830.00 840.53 851.33 104
18/11 844.95 852.59 860.37 198 18/11
5/3 857.38 884.36 913.13 15 5/3
17/10 910.50 918.64 926.93 170 17/10
12/7 921.62 933.13 944.94 84 12/7
19/11 938.88 946.20 953.63 209 19/11
7/4 949.19 968.83 989.32 28 7/4
16/9 987.37 996.09 1004.97 144 16/9
9/5 1002.16 1017.60 1033.53 45 9/5
20/11 1027.97 1035.00 1042.12 220 20/11
11/6 1036.65 1049.36 1062.41 66 11/6
13/7 1060.89 1071.70 1082.74 91
15/8 1078.87 1088.27 1097.84 120
17/9 1092.73 1101.05 1109.49 153
19/10 1103.74 1111.20 1118.76 190
21/11 1112.70 1119.46 1126.31 231
23/12 1120.14 1126.32 1132.57 276 23/12
2/1 1132.10 1200.00 1276.96 2 2/1

The "maxima" marked with an asterisk are all extremely weak, or
comprise such a narrow span that they could perhaps be called trivial,
in which case some of the weaker peaks and valleys (11/7 for instance)
would actually just represent a part of a given slope...
unfortunately, all of this is of course very contingent on the value
"n", but for a given small range "n" this will pretty convincingly
mimic the entropy minima and maxima of a Farey series where N is both
set very low and up on to the range where N starts to more or less
stabilize.

Dan

Paul H. Erlich wrote,

> Perhaps Dan has this handy?

Well I looked, and I looked some more... and apparently I don't have
it anymore... either that or it's tucked away in some obscure subject
line that I'm just not seeing.

Dan

I wrote,

>> Perhaps Dan has this handy?

Dan wrote,

>Well I looked, and I looked some more... and apparently I don't have
>it anymore... either that or it's tucked away in some obscure subject
>line that I'm just not seeing.

OK, I looked and found:
http://www.egroups.com/message/tuning/6250

It's a perfect illustration of what I'm talking about.