Yahoo Tuning Groups Ultimate Backup tuning Weighted Error and Concordance

In "A Middle Path", Paul says this about TOP error:

"I then realized that this method of tuning was optimal in a certain sense. Most models of discordance predict that the simplest ratios are most sensitive to mistuning, more complex ratios are less sensitive to mistuning, and still more complex ratios are essentially insensitive to mistuning (as they are not local minima of discordance in the first place). To evaluate the damage to concordance caused by the mistunings in a temperament, then, it makes sense to scale them so that mistuning a simple ratio by a given amount (in cents) corresponds to more damage than mistuning a complex ratio by the same amount. A straightforward way of doing this is to divide each mistuning by the Harmonic Distance of the JI interval mistuned."

I've read this paper about 400 times and it's only now that I'm starting to think critically about this passage. It's basically become taken as gospel here in this group that this passage is correct and this method of weighting error is the best.

But it's not necessarily true that concordance is proportional to Tenney Harmonic Distance. Consider this HE curve:

https://fbcdn-sphotos-a.akamaihd.net/hphotos-ak-snc7/305278_10150327283689367_766299366_7800075_1184789201_n.jpg

Paul's commentary on it:
"Dyadic harmonic entropy for a pretty fine ear. Here we seed with a Farey series, resulting in a downward overall slope. Some may like this slope because, like critical band roughness, it rates all very large intervals as low in discordance (which might be helpful as a concept for those who like Benjamin Britten work with "registral polytonality" or setting up independent tonal movements/tonal centers within different pitch registers, where they may be able to operate without interfering with one another as much). "

Notice here that 3/1 is actually *lower* than 2/1!

My question is, what if we wanted error weighting to reflect the above curve, rather than the standard curves that show 2/1 as lower in discordance than 3/1? Tenney weighting won't do this.

-Igs

🔗cityoftheasleep <igliashon@...>

Also, I have to ask: if we're weighting error to model concordance, and concordance is what we're really interested in (as opposed to, say, "integrity of intervallic identity" or something) why even bother looking at error and JI approximation at all? Why not just do what Keenan did, and look directly for scales that minimize average HE (for some formulation of HE)? Seems like the whole "error from JI" approach introduces an unnecessary middle-man to the equation.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> In "A Middle Path", Paul says this about TOP error:
>
> "I then realized that this method of tuning was optimal in a certain sense. Most models of discordance predict that the simplest ratios are most sensitive to mistuning, more complex ratios are less sensitive to mistuning, and still more complex ratios are essentially insensitive to mistuning (as they are not local minima of discordance in the first place). To evaluate the damage to concordance caused by the mistunings in a temperament, then, it makes sense to scale them so that mistuning a simple ratio by a given amount (in cents) corresponds to more damage than mistuning a complex ratio by the same amount. A straightforward way of doing this is to divide each mistuning by the Harmonic Distance of the JI interval mistuned."
>
> I've read this paper about 400 times and it's only now that I'm starting to think critically about this passage. It's basically become taken as gospel here in this group that this passage is correct and this method of weighting error is the best.
>
> But it's not necessarily true that concordance is proportional to Tenney Harmonic Distance. Consider this HE curve:
>
> https://fbcdn-sphotos-a.akamaihd.net/hphotos-ak-snc7/305278_10150327283689367_766299366_7800075_1184789201_n.jpg
>
> Paul's commentary on it:
> "Dyadic harmonic entropy for a pretty fine ear. Here we seed with a Farey series, resulting in a downward overall slope. Some may like this slope because, like critical band roughness, it rates all very large intervals as low in discordance (which might be helpful as a concept for those who like Benjamin Britten work with "registral polytonality" or setting up independent tonal movements/tonal centers within different pitch registers, where they may be able to operate without interfering with one another as much). "
>
> Notice here that 3/1 is actually *lower* than 2/1!
>
> My question is, what if we wanted error weighting to reflect the above curve, rather than the standard curves that show 2/1 as lower in discordance than 3/1? Tenney weighting won't do this.
>
> -Igs
>

🔗Carl Lumma <carl@...>

Either approach is fine, and they give similar results,
which is the point. Starting with JI lets you invoke group
theory and gives you RMP. It's also a lot easier to do
the calculations.

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Also, I have to ask: if we're weighting error to model
> concordance, and concordance is what we're really interested
> in (as opposed to, say, "integrity of intervallic identity"
> or something) why even bother looking at error and JI
> approximation at all? Why not just do what Keenan did, and
> look directly for scales that minimize average HE (for some
> formulation of HE)? Seems like the whole "error from JI"
> approach introduces an unnecessary middle-man to the equation.
>
> -Igs

🔗gbreed@...

It's great being an armchair theorist because you can simplify the problem by adding the word "just". In practice we don't "just" do what Keenan did because it's easier to use ratios. Except Keenan apparently did use harmonic entropy and he's one of us. So we must have done that after all.
Showing errors in target intervals also makes it clear what you got relative to what you wanted. And make it easier to say what you want to start with. And measure complexity.
That should give you some pointers. If you aren't convinced, you can just develop Keenan's work into exactly what you want.

Graham

------Original message------
From: cityoftheasleep <igliashon@sbcglobal.net>
To: <tuning@yahoogroups.com>
Date: Monday, January 16, 2012 5:55:43 PM GMT-0000
Subject: [tuning] Re: Weighted Error and Concordance

-Igs

------------------------------------

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.
Yahoo! Groups Links

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Notice here that 3/1 is actually *lower* than 2/1!
>
> My question is, what if we wanted error weighting to reflect the above curve, rather than
> the standard curves that show 2/1 as lower in discordance than 3/1? Tenney weighting
> won't do this.

After discussing this with Paul, I'm realizing that I was a bit confused on this. Tenney weighting doesn't reflect absolute concordance, but rather increase in discordance per cent of mistuning, which doesn't change in the downward-sloping curve. So even if 3/1 is a deeper minimum than 2/1, it can still be less sensitive to mistuning than 2/1, on account of it still being a *relatively* shallow minimum (i.e. compared to the adjacent maxima).

-Igs

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

🔗cityoftheasleep <igliashon@...>

Also, when did you ever suggest that Tenney weighting doesn't reflect absolute concordance? Had you suggested that directly at the outright, I could've been spared a lot of confusion.

And aren't you supposed to be in Orlando??

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> > >
> > > After discussing this with Paul, I'm realizing that I was
> > > a bit confused on this. Tenney weighting doesn't reflect
> > > absolute concordance, but rather increase in discordance
> > > per cent of mistuning, which doesn't change in the downward-
> > > sloping curve.
> >
> > <pulls hair out> What'd Paul do, give you a hot stone massage
> > to lubricate the notion?
>
> No, he just explained himself patiently, and repeatedly requested me to clarify what I was asking when he didn't understand. I've never seen Paul get snarky in response to foolishness. The man has the patience of a saint.
>
> -Igs
>