!!!!!!!!! THEORY WARNING !!!!!!!!!!!

Dear tuning folk,

I'd like to back up a little here. Here's my view on what we are trying to

do in this thread.

First a reminder that what we are talking about here is just a small part

of the attempt to mathematically model what we hear when several notes are

played together. We know our models will always be inadequate, not least

because people don't agree about what they hear. But there are various

reasons to persist, and various tradeoffs available between the accuracy of

a model and its user-friendliness.

We want to reduce any chord to a (hopefully short) list of numbers. But not

just any bunch of numbers, that's easy. Numbers such that chords that sound

similar in some way have numbers that are close together. Each number

should approximate a "dimension" of our perceptual space. We've easily

agreed to factor out loudness and pitch as the most salient dimensions and

can, and have been, ignoring them in this discussion.

There are plenty of dimensions left, and we could take them as

prime-affects such as Carl Lumma's

3=openess

5=sweetness

7=florescent lightingness,

and go on from there. But in this discussion we are attempting to first

factor out what seems to many to be the next most salient dimension,

consonance/dissonance. Joe Monzo wants to call this dimension "sonance", to

make it clear that these are two polarities of the same dimension, but then

I don't know what increasing numbers mean. It seems mathematically simpler

to give the lowest value to the most consonant "chord", i.e. a single note

(or silence?), so I want increasing numbers to model increasing *dissonance*.

We are also, at first, only considering the case of 2-note chords, also

called intervals or dyads. Partly because modelling dissonance for more

than 2 notes looks hard and partly because we expect we have to agree on

dyads (intervals) first if that project is to have any hope of success. An

interval may be described by its width in cents or as a ratio.

We mostly seem to agree that dissonance can be approximated by composing

two functions. The first, I think we should call complexity (after Wilson).

Although Barlow calls a reciprocal measure "harmonicity" I think that

sounds too much like "consonance", and I'd prefer to make it clear that we

don't expect this function to approximate consonnance or dissonance on its

own. For one thing, it gives far too much importance to high primes (at

least in dyads).

The second function has been called tolerance. This is some kind of

blurring function where the dissonance of a complex ratio will depend more

on its proximity to nearby simpler ratios, thus limiting the significance

of higher primes. Paul Erlich's Harmonic Entropy and the Plomp/Levelt

dissonance curve for two sine tones, perform this function in different

models.

Bill Sethares' model works well but is rather computationally intensive. We

agree that there are simpler models of dissonance that work well enough for

timbres with harmonic or near-harmonic partials. The two most popular

do-it-in-your-head dissonance measures are odd-limit and prime-limit. These

are really only complexity measures, and tolerance is often ignored or

treated somewhat ad hoc.

Mostly what we expect from a complexity measure is that it will at least

rank the intervals in the correct order of increasing dissonance.

Complexity is only defined for ratios, not irrationals or cent values (the

tolerance or blurring function takes care of them). The complexity of a

ratio is typically described as some function of the complexities of its

numerator and denominator. In the case of odd-limit and prime-limit this is

just the maximum value (this is part of the appropriateness of the word

"limit"). My spreadsheet has so far only considered whole numbers.

Partch's Odd-limit ranks the whole numbers up to 19 as

(1,2,4,8,16),(3,6,12),(5,10),(7,14),(9,18),11,13,15,17,19

while Prime-limit gives

(1,2,4,8,16),(3,6,9,12,18),(5,10,15),(7,14),11,13,17,19

where those in parenthesis are given equal rank.

These two complexity measures disagree most importantly about the

dissonance of ratios of 9, and less importantly about 15.

There are formulae for musical complexity whose computational complexity is

intermediate. Barlow's is one such. Note that Barlow calls it

indigestibility when it is applied to whole numbers but flips it over and

calls it harmonicity (a consonance measure) when applied to ratios.

Barlow's ranking is

1,2,4,3,8,6,16,12,9,18,5,10,15,7,14,11,13,17,19.

This is hardly different from that of prime-limit, except in placing 3 and

6 below 8 and 16 respectively.

Another is Wilson's Harmonic Complexity, giving

(1,2,4,8,16),(3,6,12),(5,10),(9,18),(7,14),15,11,13,17,19

In its treatment of 9 and 15 it is intermediate between odd-limit and

either prime-limit or Barlow's, and as such it agrees better with my

experience.

Just as there are both linear and logarithmic measures of loudness (pascals

vs. decibels) and pitch (hertz vs. cents) there are linear and logarithmic

measures of dissonance. We could call the respective units the partch and

the wilson since Partch's odd-limit (and the so called integer-limit) is a

linear measure while Wilson's complexity (and Barlow's and prime-limit) is

logarithmic. The easiest way to tell is by the value they assign to 1. If

this is zero, it is most likely logarithmic. The other way is just by

comparing their curves. By the way, the curves imply that the prime-limit

of 1/1 may be safely defined as 0 even though its odd-limit is 1.

Of course taking the log (or its inverse the exponential) of any complexity

measure does not alter its ranking of the intervals. However, I'm inclined

to say that the logarithmic measures correspond more closely to our

experience (as they do for loudness and pitch).

When I entered this thread I proposed a complexity measure that had a

variable parameter (an exponent weight) for each prime, so that it could be

made to approximate the various existing measures and allow for others in

between, in the hope of enabling a consensus.

I now understand that Manuel Op de Coul has already done this in Scala by

generalising Barlow's harmonicity to allow the prime exponent weights to be

changed with the SET HARMCONST comand. This is good, except I'd prefer to

stick to a *dissonance* measure (not consonnance) and would also ignore the

sign (+ or -). The absolute value of the reciprocal of Manuel's harmonicity

is entirely equivalent to the the logarithmic version of my complexity when

my weights are multiplied by the log of their prime. So I'd like to use

this formulation from now on to explore the possibility for some kind of

consensus on the weights.

Given a whole number whose prime factorisation is

n = 2^a * 3^b * 5^c * ...

its logarithmic complexity is now

log_complexity(n) = k_2*a + k_3*b + k_5*c + ...

where the k_p's are the prime exponent weights (Manuel's HARMCONSTs).

The logarithmic complexity of a ratio n/d in lowest terms is then just the

sum of the log complexities of numerator and denominator.

Alternatively and equivalently a ratio n/d (which need not be in lowest

terms) may be expressed as a single prime factorisation where exponents may

be positive or negative and its log complexity is

k_2*|a| + k_3*|b| + k_5*|c| + ...

where |x| is the absolute value of x.

The linear complexity is therefore

lin_complexity = e^log_complexity

= e^(k_2*|a| + k_3*|b| + k_5*|c| + ...)

= e^(k_2*|a|) * e^(k_3*|b|) * e^(k_5*|c|) * ...

where e is the natural log base. Note that since my first proposal, the e's

have replaced the primes themselves, however the equivalence is explained

by the fact that

2^(k_2*a) = e^(ln(2)*k_2*a) and

3^(k_3*b) = e^(ln(3)*k_3*b) etc.

where ln(x) is the natural log of x.

I'll let you know when I've updated my spreadsheet, and I promise to answer

any earlier questions that weren't answered above, when I can find the time.

Regards,

-- Dave Keenan

http://dkeenan.com

Dave Keenan wrote,

>The second function has been called tolerance. This is some kind of

>blurring function where the dissonance of a complex ratio will depend

more

>on its proximity to nearby simpler ratios, thus limiting the

significance

>of higher primes. Paul Erlich's Harmonic Entropy and the Plomp/Levelt

>dissonance curve for two sine tones, perform this function in different

>models.

Did you mean the Plomp/Levely dissonance curve for two non-sine harmonic

tones?

P.S. Dave: I don't like it if 15/8 is given the same complexity as 6/5,

assuming factors of 2 are ignored. This last post of yours seems to

propose a family of formulae that do give the two the same complexity if

factors of 2 are ignored. Did you see Paul Hahn's post on using

triangular lattices?

Message text written by Paul Erlich

>P.S. Dave: I don't like it if 15/8 is given the same complexity as 6/5,

assuming factors of 2 are ignored.<

P.E.:

If you don't mind my jumping into your postscript, what do you imagine a

complexity formula to look like where the factors of 2 are not ignored?

Might this be a useful, if not necessary, component in a complexity theory

in which register is taken into account? I am thinking here of something

in addition to a roughness/critical band curve.

DJW

Dave Keenan wrote,

>The second function has been called tolerance. This is some kind of

>blurring function where the dissonance of a complex ratio will depend

more

>on its proximity to nearby simpler ratios, thus limiting the

significance

>of higher primes.

Have you noticed that this argument is far more valid if you replace

"primes" with "odds" at the end of the sentence, showing that a strict

lattice approach like all those we've been discussing is much more

likely to be meaningful if we restrict ourselves to, say, the

11-odd-limit, than if we restrict ourselves to the 11-prime-limit, the

7-prime-limit, the 5-prime-limit, or even the 3-prime-limit? The

11-odd-limit seems about right given ideal conditions for pitch

discrimination. And again, a triagular lattice would be more

appropriate.

>>P.S. Dave: I don't like it if 15/8 is given the same complexity as

6/5,

>assuming factors of 2 are ignored.<

>P.E.:

>If you don't mind my jumping into your postscript, what do you imagine

a

>complexity formula to look like where the factors of 2 are not ignored?

>Might this be a useful, if not necessary, component in a complexity

theory

>in which register is taken into account? I am thinking here of

something

>in addition to a roughness/critical band curve.

>DJW

Yes, factors of 2 should be important in an octave-specific theory. I'm

not sure what you mean by "in addition to"; a true psychoacoustical

dissonance curve (including both roughness and harmonic entropy) would

be a refinement over a lattice-based complexity measure; the latter is

desireable as an approximation which allows actual _visualization_ of

scales with "closeness" representing "consonance" to a good

approximation.

Message text written by P. Erlich

>Yes, factors of 2 should be important in an octave-specific theory. I'm

not sure what you mean by "in addition to"; a true psychoacoustical

dissonance curve (including both roughness and harmonic entropy) would

be a refinement over a lattice-based complexity measure; the latter is

desireable as an approximation which allows actual _visualization_ of

scales with "closeness" representing "consonance" to a good

approximation.

<

With my students, I've always explained lattices as analogous to flat

geographic maps. That is, that they represent distances (harmonic and

physical, resp.) quite precisely in the center but with increased distorted

in the extremes. For the analysis or composition of a piece of tonal music,

in a typical chorale range and voicing, the lattice will be close enough.

The 'in addition to' meant precisely the inclusion of roughness, and of 2^n

phenomena (which I have previously called remoteness, but I gather is

covered by your harmonic entropy; I'm pleased to find the better term).

These are more than details in real musical situations. For example, if I

wanted to write a choral piece with a minor third between the tenors and

basses, how low could it be voiced before becoming too rough? In the past I

had always orchestrated with the rule of thumb than the lowest 'clean'

voicings would follow the intervals of a harmonic series two octave below

Bb. (So, for example, Bb - d, was the lowest 'clean' M3rd etc.). Recently,

in preparation for a large orchestra piece, I have been refining this more

systematically by subjective analysis of sampled sounds. So, not only would

my ideal complexity formula include roughness, but have the roughness

classified further by timbre.

Not directly related, I have also used variations on a formula, called

'effort', to help in the generation of melodic lines with a vocal charcter.

This is based on the notion that the larger the interval, the greater the

vocal 'effort' to produce the leap, while leaps of simpler ratios reduce

the error. Most often, I have have calculated effort simply by multiplying

the Wilson complexity by cents. Here are some effort values, calculated off

the top of my head:

2/1 1200

15/8 8704

11/6 14686

9/5 11209

16/9 5976

7/4 6783

5/3 7072

13/8 10933

8/5 4070

14/9 9945

3/2 2106

11/8 6061

4/3 1494

9/7 5655

81/64 4896

5/4 1930

11/9 5899

6/5 2528

7/6 2670

8/7 1617

9/8 1224

10/9 2002

16/15 672

25/24 923

1/1 0