From sine waves to complex tones

Michael said: "I think, at least with fairly low prime limit ratios your formula should work pretty well...but I'm pretty sure temperaments with high-limit fractions fairly near low-limit ones will confuse it (due to things like the "so close to the octave" mis-match you ran into)."

I have an idea (possibly not new) that any complex interval (e.g. 473/331) that is within 6.78 cents (256/255) of a simple interval (e.g. 10/7) then the strength of the complex interval is the strength of the simple interval multiplied by
(6.78 - x)/6.78 where 'x' is the deviation in cents of the complex interval from the simple interval.

In this case the difference between the 473/331 and the 10/7 is 0.52 cents. If my formula (using only sine wave tones with no overtones) is correct then the 10/7 simple interval has a value of 0.7714.
0.7714 * ((6.78 - 0.52)/6.78) = 0.7122
which is the strength value of the complex interval. The 256/255 threshold is an educated guess. This idea should be included in my formula (which I know now is wrong for notes that have a harmonic series, i.e. not pure sine waves and may still be wrong even *with* pure sine waves but I'm reluctant to let go of it just yet).

Michael, you said that the 47/19 interval is VERY close to the 5/2 interval. I checked this and it looks like there are 18.3 cents between them, a noticable difference IMVHO.

Complex tones...the following is an idea I used for years but discarded a few months ago and am now resurrecting it. How strong is each harmonic in a musical note? (assuming that the harmonic series is exactly x, 2x, 3x, 4x, 5x etc. and the 'initial' strength value of each harmonic is 1, 0.5, 0.333, 0.25, 0.2 etc.
I say 'initial' strength value because when each harmonic is paired with all the others their 'overall' strength values are greater.

Start with the fundamental and pair it with, say, the next fifteen overtones using the (2 + 1/x + 1/y - x/y)/2 formula.

1 and 2 yields 1.5. The second harmonic is (initially) only half as strong so just as a chain is only as strong as its weakest link, so the strength of an interval is only as strong as the weaker element of the pair (in this case 1.0 and 0.5, 0.5 is weaker). So the 1.5 must be "weighted".
1.5 * 0.5 = 0.75

1 and 3 also yields 1.5 but the weakest element has an 'initial' strength value of 0.333.
1.5 * 0.333 = 0.5.

Repeat the process until all 15 intervals (that contain the fundamental are worked out and add these together. Finally add 1.0 to the total because this is the 'initial' strength value of the fundamental on it's own. The overall result for the first harmonic is 4.57109.

Next pair the second harmonic with the 1st, 3rd, 4th, 5th etc and add the 15 results together. Then add 0.5 to the result as this is the 'initial' strength value of the second harmonic. The overall result for the second harmonic is 3.43374

Next pair the 3rd harmonic with the 1st, 2nd, 4th ,5th etc...

The completed list is as follows...
H...value
1...4.57109
2...3.43374
3...2.7765
4...2.35388
5...2.054
6...1.82728
7...1.64844
8...1.50306
9...1.38225
10...1.28019
11...1.19287
12...1.1174
13...1.05168
14...0.994097
15...0.943403
16...0.898617

The 'initial' strength values of each harmonic (1 + 0.5 + 0.333 + 0.25 +.....+ 0.0625) add up to 3.38073. The paired harmonics add up to 25.64777.
3.38073 + 25.64777 = 29.0285. I think that if you do an infinite number of iterations then the total would have no limit (i.e. it's infinite).

Note that the total here (29.0285) is misleading because when working out the values of each pair of harmonics, each pair was considered twice (e.g. 5 was paired with 7 and later 7 was paired with 5. So the 'true' overall value is...
3.38073 + (25.64777)/2 = 16.204615. This is perhaps irrelevant and all that matters is how strong each harmonic is in relation to all the others.

If we tweak things so that the strength values when added up equal 1.0 exactly the list looks like this...

H...value
1...0.157469
2...0.118289
3...0.0956476
4...0.0810888
5...0.070758
6...0.062948
7...0.056787
8...0.0517787
9...0.0476171
10...0.0441013
11...0.041093
12...0.0384933
13...0.0362293
14...0.0342456
15...0.0324992
16...0.0309564

Total = 1.0.

Now consider the 3/5 interval whose harmonic series is similar to the mathematical ideal (complex tones). List all sixteen harmonics of both notes and pair each one of the 3's harmonics with each one of the 5' harmonics using the (2 + 1/x + 1/y - diss)/2 formula but *weight* each result according to the numbers in the list above.

Add the 256 (16*16) "weighted" results and add them up. If the result is greater than 1.0 then the interval should be Major. (I need to check this last point.)

All of the above is a rough guess but is hopefully going in the right direction.

John.

Funny, I had already sent this message/reply but it never came through. Anyhow, here is my (re-sent) response:

>"Michael, you said that the 47/19 interval is VERY close to the 5/2
interval. I checked this and it looks like there are 18.3 cents between
them, a noticable difference IMVHO."

You're right, at a quick glance it looked to me more like 13-14 cents or so before. Still, you have to remember that the huge space between the two tones (IE huge critical band difference) makes the beating causes by that 18 cent lack of periodicity far less intense than would be otherwise.

>"In this case the difference between the 473/331 and the 10/7 is 0.52
cents. If my formula (using only sine wave tones with no overtones) is
correct then the 10/7 simple interval has a value of 0.7714.
0.7714 * ((6.78 - 0.52)/6.78) = 0.7122"
That's a pretty significant amount of supposed consonance loss for something less than 7 cents away...I still think your formula would work better with more slack toward de-tuning/tempering IE creating nearly a flat curve between the 0 and 6.78 cent difference. I think "deviating from the curve" is akin to adding a chorus/de-tune effect on your instruments: the net effect is virtually nothing.

>"The completed list is as follows...
H...value
1...4.57109
2...3.43374
3...2.7765......."

The problem I see here is that the list just falls in a fairly straight-line fashion with a slight curvature making harmonics further down the series less different than each other.

It brings me back to your "old" version of the theory that has the smallest "strongly consonant" interval at about 6/5...and, granted, from 6/5 to about 16/15 (IE 7/6,8/7,9/8...) the amount of consonance changes dramatically. I'm convinced by lots of listening there are "bumps" where consonance changes more quickly around 12/11, 5/4, and 3/2...but beyond that fairly little to gain. I seriously doubt it all follows such a convenient line.

I do, however, agree that a way to take temperament into account when judging consonance of imperfect intervals is a necessary concept and am glad you are taking that challenge.
I do tend to think you emphasis on "harmonic strength" is a bit too extreme. Even with overtone/non-sine-wave clashing issues...you usually only have to worry about clashes between the first 5 or so overtones' clashing and ruining a chord's "strength" since they are usually by far the loudest ones. If you need a way to do that I would use Plompt and Llevelt's dyadic dissonance curve to

1) Compare all dyads in a chord to each other and
2) To their overtones number 2-5 IE root*2, root*3
3) Compare dissonance of overtones and other overtones
and then subtract some multiple of that total from your consonance rating for that chord....this is all somewhat covered ground by Sethares and (sadly) when he solved for 1 and 3 he came up with the plain old 7-tone JI diatonic scale. Although, of course, you may come up with a different scale if you weigh harmonics 1 to 5 more heavily (which I would recommend). You can also do things like add factors to handle things like "how more periodicity any overtone clashes have" and skew the results accordingly...though that would be one seriously complex algorithm which would likely combine Sethares work and Rick Ballan's work on rounding tempered tones to the nearest JI fraction(s) via LCD calculations (as I understand it).

The only true way I can imagine to make an educated sum like this (without taking huge amounts of time to calculate it) is to write a computer program to do that for you...which is pretty much what William Sethares did. The other solution is to do the reverse and match your timbre to fit the scale, something I'd highly recommend as it brings you many more options.
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Another bizarre thing I noted in working on my latest scale is that certain super-particular ratios can be squared to produce near-perfect JI ratios in triads. For example 10/9 * 10/9 = 100/81 which is VERY near 5/4. Same goes with 11/10 * 11/10 = 121/100 being very near 6/5. Sure there is major and minor IE 4:5:6 = major and 10:12:15 (reverse interval order) = minor...but what do you call 9:10:(45/4) AKA (36:40:45)? This is another weird coincidence as the periodicity of the chord appears low but the dyads that make up the chord are all simple ratios (IE 10/9, 10/9, and 5/4)

I just figure you may find some good use for this "hack".

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Another bizarre thing I noted in working on my latest scale is that certain super-particular ratios can be squared to produce near-perfect JI ratios in triads. For example 10/9 * 10/9 = 100/81 which is VERY near 5/4. Same goes with 11/10 * 11/10 = 121/100 being very near 6/5.

Not bizarre but explained by math:
(N/(N-1))^2 = N^2/(N^2-2N+1) ~= N/(N-2) [first example]

((N+1)/N)^2 = (N^2+2N+1)/N^2 ~= (N+2)/N [second example].

Having said that, 100/81 is still a syntonic comma away from 5/4, i.e. 21+ cents, so not VERY near ...

Steve M.

Martin>"Having said that, 100/81 is still a syntonic comma away from 5/4, i.e.
21+ cents, so not VERY near ..."

True, it's about 22 cents off and the result of the arithmetic mean between 100/81 and 5/4 is about 11 cents off.
You're right though there's a fairly strong resemblance it's not "very very near"...some tempering is required to make it really work.

I guess that leaves me with the option of tempering the second 100/81 into 1.2269 or approximately 27/22 to get the error down to around 11 cents. Or doing 100/81 vs. 5/4 = 400/324 vs. 405/324 or 800/648 vs. 810/648 which = 805/648 or 1.24228 AKA 41/33 (which, I think, is the harmonic mean).

________________________________
From: martinsj013 <martinsj@...m>
To: tuning@yahoogroups.com
Sent: Wed, April 14, 2010 11:07:05 AM
Subject: [tuning] Re: From sine waves to complex tones

--- In tuning@yahoogroups. com, Michael <djtrancendance@ ...> wrote:
> Another bizarre thing I noted in working on my latest scale is that certain super-particular ratios can be squared to produce near-perfect JI ratios in triads. For example 10/9 * 10/9 = 100/81 which is VERY near 5/4. Same goes with 11/10 * 11/10 = 121/100 being very near 6/5.

Not bizarre but explained by math:
(N/(N-1))^2 = N^2/(N^2-2N+ 1) ~= N/(N-2) [first example]

((N+1)/N)^2 = (N^2+2N+1)/N^ 2 ~= (N+2)/N [second example].

Having said that, 100/81 is still a syntonic comma away from 5/4, i.e. 21+ cents, so not VERY near ...

Steve M.