Re: complexity formulae

Paul Hahn wrote:
> 126/125 should be 7.451211.
> 50/49 should be 5.614709.
[snip]
> Which means that my algorithm doesn't work for the weighted version,
> only the integer one.

I see, I have changed the algorithm and decided to follow the shortest
route in
the triangular lattice but at each step taking the log of the highest prime
instead
of the lowest one. Otherwise one would get the strange result of 7/6 and
6/5 having a
lower complexity than 5/4.
So now the results are

225/224 8.299208
126/125 7.451211
128/125 6.965784
81/80 7.076815
64/63 5.977279
50/49 5.614709 |

These are the values for the tones in genus [33557]
1: 21/20 4.392317
2: 16/15 3.906890
3: 35/32 5.129283
4: 7/6 2.807354
5: 6/5 2.321928
6: 5/4 2.321928
7: 21/16 4.392317
8: 4/3 1.584962
9: 7/5 2.807354
10: 35/24 5.129283
11: 3/2 1.584962
12: 8/5 2.321928
13: 105/64 6.714245
14: 5/3 2.321928
15: 7/4 2.807354
16: 28/15 4.392317
17: 15/8 3.906890
18: 2/1 0.000000

Does this look about right?

Manuel Op de Coul coul@ezh.nl

On Thu, 11 Mar 1999 manuel.op.de.coul@ezh.nl wrote:
> I see, I have changed the algorithm and decided to follow the shortest route in
> the triangular lattice but at each step taking the log of the highest prime instead
> of the lowest one. Otherwise one would get the strange result of 7/6 and 6/5 having a
> lower complexity than 5/4.

You're right, this makes more of a difference than I'd realized. I'm
more used to using the unweighted version, which allows (as we've seen)
certain shortcuts that the weighted version doesn't. There were other
errors in your list which you have corrected now even though I missed
them before.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "

NOTE: dehyphenate node to remove spamblock. <*>

In an ancient Tuning Digest (from 5 years ago), I found
a very interesting discussion of harmonic complexity measures.

ftp://ella.mills.edu/ccm/tuning/list/archive/apr942

Approximately 2/3 of the way thru this file, in a discussion
about Euler's Gradus Suavitatis, Georg Hajdu says:

> My guess is that harmonic hearing depends so much on whether
> a strong virtual fundamental can be produced by our nervous
> system or not.

This falls right in line with some of the conclusions that
have been made on this subject here recently.

Adapting this line of reasoning to my idea of sonance
(i.e., a consonance-dissonance continuum rather than a sharp
break between the two), and the attempts we've made to find
workable formulae to measure this property, I'd replace
the dichotomy of 'whether a strong virtual fundamental can
be produced by our nervous system or not' with the idea of
measuring the *degree* to which that virtual fundamental is
perceivable.

It seems to me that this is the most workable method for
determining the sonance, or what should perhaps be called
the 'harmonic fit', of chords (i.e., tri- and higher-ads).

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Monz wrote,

>In an ancient Tuning Digest (from 5 years ago), I found
>a very interesting discussion of harmonic complexity measures.

>ftp://ella.mills.edu/ccm/tuning/list/archive/apr942

>Approximately 2/3 of the way thru this file, in a discussion
>about Euler's Gradus Suavitatis, Georg Hajdu says:

>> My guess is that harmonic hearing depends so much on whether
>> a strong virtual fundamental can be produced by our nervous
>> system or not.

>This falls right in line with some of the conclusions that
>have been made on this subject here recently.

>Adapting this line of reasoning to my idea of sonance
>(i.e., a consonance-dissonance continuum rather than a sharp
>break between the two), and the attempts we've made to find
>workable formulae to measure this property, I'd replace
>the dichotomy of 'whether a strong virtual fundamental can
>be produced by our nervous system or not' with the idea of
>measuring the *degree* to which that virtual fundamental is
>perceivable.

>It seems to me that this is the most workable method for
>determining the sonance, or what should perhaps be called
>the 'harmonic fit', of chords (i.e., tri- and higher-ads).

Since there is always some ambiguity as to _which_ virtual fundamental is
the right one, and as each possibility will have a different degree of
perceptibility, I suggest that my harmonic entropy concept is the best
approach for quantifying what you and Hajdu are talking about. The earliest
prominent proponent of this type of concept was Rameau.

But I think we've established beyond a reasonanble doubt that there is a
second component of sonance. That component is what Sethares calls "sensory
dissonance" and has to do with clashing partials. The earliest prominent
proponent of this type of concept was Helmholtz.

A correct evaluation of either of these two components should include not
only the played notes but also all the combination tones created in the ear
(as well as other auditory effects such as masking and pitch-shifts).

Finally, there appears to be an effect which makes chords with a power of
two times the virtual fundamental in the bass part more stable than those
with some other harmonic of the fundamental in the bass part. This may be a
separate component of sonance, or an indication that the harmonic entropy
concept needs to be modified to take this type of octave-equivalence into
account.