complexity weighting triads

For those of you who are satisfied by the
n*d complexity weighting for diads,
there is (imho) an equivalent for complexity
weighting for multi-ads.

What do you think about:
n*d also represent the distance between the virtual
fundamental (first common subharmonic) and the
guide tone (first common harmonic)

(1) : 4 : 5
I 1/5 : 1/4 : (1)
I I
I I
I<----------(4*5)--------->I
I<----------( 20)--------->I

now you can do something similar for a triad:

(1) : 3 : 4 : 5
I 1/20 : 1/15 : 1/12 : (1)
I I
I I
I<-----------------(3*20)------------->I
I<-----------------(4*15)------------->I
I<-----------------(5*12)------------->I
I<-----------------( 60 )------------->I

Here are some calculated examples so you can
find out if all this make sense.

3:4:4 12
3:4:5 60
3:4:6 12
3:4:7 84
3:4:8 24
3:4:9 36
4:5:6 60
4:5:7 140
4:5:8 40
4:5:9 180
3:5:6 30
3:5:7 105
3:5:8 120
3:5:9 45

12:15:20 60
12:21:28 84

To consider:

Peter Mulkers
Belgium

Peter,

I don't think most people would agree that, for example,
3:5:7 is more consonant than 3:5:8, which is why many
people on the list aren't satisfied with the use of n*d
as a consonance rating.

If you're going to go this route, better to go with something
like Euler's method, where it's the size of the primes that
matters more than the size of the least common multiple.

For those not familiar with Euler's method for calculating
the _gradus suavitatis_:

where the least common multiple is equal to
2^a * 3^b * 5^c * 7^d..., the _gradus suavitatis_ will be
equal to (2a + 3b + 5c + 7d...) - (a + b + c + d... - 1).

So, for the triads you gave:

triad LCM GS
3:4:4 12 5
3:4:5 60 9
3:4:6 12 5
3:4:7 84 11
3:4:8 24 6
3:4:9 36 7
4:5:6 60 9
4:5:7 140 13
4:5:8 40 8
4:5:9 180 11
3:5:6 30 8
3:5:7 105 13
3:5:8 120 10
3:5:9 45 9
12:15:20 60 9
12:21:28 84 11

There are problems with the Euler method too. It doesn't give fine
enough distinctions of consonance, and it implies a kind of idealism
(as does n*d), whereby higher ratios are sometimes best interpreted
as approximations to lower ones. And there are still some cases in
which, like n*d, it seems just plain wrong (3:5:9 more consonant than
3:5:8?). To give finer distinctions, you could find the GS and then
take a ratio with a smaller LCM as being more consonant than a ratio
with a larger one. But even this wouldn't work in some cases (3:5:9
more consonant than 12:15:20?), so I durno. Never mind. :P

Dale Scott

P.S. Does anybody know offhand the "improved Euler method" given by
Martin Vogel in _On the Relations of Tone_? I seem to remember it
provided for finer distinctions of consonance (i.e., values of whole
numbers with added decimal fractions) by taking into account things
such as inversions and varying degrees of completeness of sonorities.

Peter Mulkers wrote,

>For those of you who are satisfied by the
>n*d complexity weighting for diads,
>there is (imho) an equivalent for complexity
>weighting for multi-ads.

>What do you think about:
>n*d also represent the distance between the virtual
>fundamental (first common subharmonic) and the
>guide tone (first common harmonic)

>now you can do something similar for a triad:

Yes, this has been done before, by Euler, by Marion on this list, and many
others. Numerically, it is the LCM (least common multiple) of the numbers
making up the chord in a smallest-integers representation. One flaw, as I
see it, is that it makes utonal and otonal chords equally consonant. Compare
the sound of 4:5:6:7:9:11 with 1/11:1/9:1/7:1/6:1/5:1/4. Are they equally
consonant?

D. Wolf wrote,
> This difficulty here is that harmonic chords and their subharmonic
> inversions will have the same complexity quantity.
> Most listeners would probably find the latter to be more complex.

Dale Scott wrote,
> I don't think most people would agree that, for example,
> 3:5:7 is more consonant than 3:5:8, which is why many
> people on the list aren't satisfied with the use of n*d
> as a consonance rating.

Paul H. Erlich wrote,
> Compare the sound of 4:5:6:7:9:11 with 1/11:1/9:1/7:1/6:1/5:1/4.
> Are they equally consonant?

I'm sorry. I see you are right about consonance. But please wait.
In fact I was just talking about complexity. Not consonance.
I thought complexity and consonance were two different things.
Complexity is just a matter of how complex a range of numbers
behave to each other, a matter of Maths (and maybe Physics).
And consonance as a matter of how we hear and feel all this,
a matter of Human Perception and anatomy of the human ear.
No ?

(I'm afraid Paul is going to tell me, as usual,
this is all discussed before)
;-)

Peter Mulkers
Belgium

To Paul and the other mathematicians on the List:

I'm interested in seeing what happens when you compute for
tones of a given 'multi-ad', the probability bell-curves
comparing the given pitches to those of the harmonics of
the virtual fundamental implied by the set.

Wouldn't this be a sort of Harmonic Entropy for the entire
chord, as opposed to that formulated already for dyads?

-monz

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

>To Paul and the other mathematicians on the List:

>I'm interested in seeing what happens when you compute for
>tones of a given 'multi-ad', the probability bell-curves
>comparing the given pitches to those of the harmonics of
>the virtual fundamental implied by the set.

>Wouldn't this be a sort of Harmonic Entropy for the entire
>chord, as opposed to that formulated already for dyads?

As I've told you before, and am discussing with Carl Lumma at the moment,
you have the right idea, but the problem of generalizing mediants to
'multi-ads' appears intractable. I discussed it with eminent mathematics
professors while at Yale and none had a clue. I'm always interested in
pursuing this further, but it would take a number theorist of formidable
abilities to tackle this one.