EDOs & Rational Intonation

Hi Gang,
I think that I am going to have to give up on describing EDOs in terms of ratios and harmonics they approximate in the primer at which I'm slowly chipping away. Here's why:

1. Too many conflicting theories about estimating the concordance of ratios. Consider the difference between a 12-tET major 3rd and a 17-tET major third. If we allow a plus/minus 8 cent margin of error around a ratio, the lowest-limit ratio we can give 400 cents is 24/19, and the lowest for 17-tET is 14/11. From a periodicity stand-point, the 17-tET 3rd is more concordant. Also, if we look at the interval between the 5th partial of a root tone and the 4th partial of the note a 3rd above it, the 12-tET 3rd is higher in critical-band roughness than the 17-tET 3rd (it has a more audible beating to it). Yet from a harmonic entropy perspective, 400 cents is judged as more concordant because it is "closer" to 5/4. So in some regards the 12-tET third is more concordant, and in some regards it isn't...and this definitely reflects my experience with 17-tET. So it's really impossible say which one is absolutely more concordant.

2. Too many ways to interpret many tempered intervals as ratios. Do we go for the lowest-limit ratio within 8 cents? Do we treat them according to some regular mapping, and allow for errors larger than the 8-cent threshold of indistinguishability? Do we simply ignore ratios above a certain Tenney Height? Do we look at harmonic series chords that are high up in the series? I just don't know.

To do justice to all possible interpretations would be vastly too time-consuming (for me, and for the reader), and to single out any one interpretation is too biased for my liking. To switch interpretations depending on temperament (i.e. look at 19-EDO in terms of JI approximations, 14-EDO in terms of critical band roughness between partials, etc.) would make it too hard to compare between them in musically-meaningful ways.

Basically, I'm torn by this dilemma: on the one hand, it does seem important when some EDO well-approximates some simple-integer ratio (or ratios), but on the other hand, if an EDO *doesn't* well-approximate any simple-integer ratios, that doesn't mean it sounds like shite. 8-EDO, for instance--about the simplest thing it approximates well is 11/6, but I swear it sounds more relaxed than 11 or even 14 if you use it right. And furthermore, I firmly believe intervals high in harmonic entropy--i.e. describable in terms of multiple ratios of similar complexity--are tunable by ear and musically-useful, and possibly even stable or pleasant in the right context, but they make messes out of any attempt at definition in terms of ratios.

So I'm giving up, I'm leaving psychoacoustics out of it and taking the Blackwood approach of "these are some nice-sounding chords I've found, and this is how "smooth" typical major/minor triads sound". Most musicians probably wouldn't want to be bothered with the psychoacoustics anyway, I'm sure.

-Igs

On Wed, Nov 3, 2010 at 6:20 PM, cityoftheasleep <igliashon@...> wrote:
>
> Yet from a harmonic entropy perspective, 400 cents is judged as more concordant because it is "closer" to 5/4. So in some regards the 12-tET third is more concordant, and in some regards it isn't...and this definitely reflects my experience with 17-tET.

HE is just a model. It's just a really simple but powerful consonance
model. I don't think that it's worth getting into the absolute
specifics. In the year 3000 AD, we'll be laughing at how simple HE
was. Ha ha ha ha ha, we'll say. And double ha.

One thing I'm learning as I develop this "harmonic laplace transform"
function is that the resultant entropy calculation will yield a
slightly different curve from HE -- less consonant dyads will have
"narrower" fields of attraction than more consonant ones. This means
that there will be more minima than with HE, but that the "valleys"
around the minima will be much narrower, indicating that these
intervals are extremely sensitive to mistuning. I haven't yet decided
if this is a bug or a feature.

But the HE curve comes directly out of the assumptions that it makes,
some of which are a bit mathematically arbitrary or obviously
oversimplified just to come up with a "very simple" model to get the
ball rolling.

> 2. Too many ways to interpret many tempered intervals as ratios. Do we go for the lowest-limit ratio within 8 cents? Do we treat them according to some regular mapping, and allow for errors larger than the 8-cent threshold of indistinguishability? Do we simply ignore ratios above a certain Tenney Height? Do we look at harmonic series chords that are high up in the series? I just don't know.

Maybe mistuning an interval is a feature instead of a bug. I keep
throwing this 0-350-702 example around. Play that chord and it's
pretty dissonant. Then add a just 7/4, 9/4, 11/4, etc on top of it and
it becomes a very palatable mix of consonance and dissonance. The same
applies to really sharp 5ths a la 13-tet or whatever - go with
0-384-738, then add 7/4, 8/4, 9/4, whatever, on top of it. Suddenly
the 1/3-tone sharp fifth becomes much more musically useful.

-Mike

Sounds good to me!
--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi Gang,
> I think that I am going to have to give up on describing EDOs in terms of ratios and harmonics they approximate in the primer at which I'm slowly chipping away. Here's why:
>
> 1. Too many conflicting theories about estimating the concordance of ratios. Consider the difference between a 12-tET major 3rd and a 17-tET major third. If we allow a plus/minus 8 cent margin of error around a ratio, the lowest-limit ratio we can give 400 cents is 24/19, and the lowest for 17-tET is 14/11. From a periodicity stand-point, the 17-tET 3rd is more concordant. Also, if we look at the interval between the 5th partial of a root tone and the 4th partial of the note a 3rd above it, the 12-tET 3rd is higher in critical-band roughness than the 17-tET 3rd (it has a more audible beating to it). Yet from a harmonic entropy perspective, 400 cents is judged as more concordant because it is "closer" to 5/4. So in some regards the 12-tET third is more concordant, and in some regards it isn't...and this definitely reflects my experience with 17-tET. So it's really impossible say which one is absolutely more concordant.
>
> 2. Too many ways to interpret many tempered intervals as ratios. Do we go for the lowest-limit ratio within 8 cents? Do we treat them according to some regular mapping, and allow for errors larger than the 8-cent threshold of indistinguishability? Do we simply ignore ratios above a certain Tenney Height? Do we look at harmonic series chords that are high up in the series? I just don't know.
>
> To do justice to all possible interpretations would be vastly too time-consuming (for me, and for the reader), and to single out any one interpretation is too biased for my liking. To switch interpretations depending on temperament (i.e. look at 19-EDO in terms of JI approximations, 14-EDO in terms of critical band roughness between partials, etc.) would make it too hard to compare between them in musically-meaningful ways.
>
> Basically, I'm torn by this dilemma: on the one hand, it does seem important when some EDO well-approximates some simple-integer ratio (or ratios), but on the other hand, if an EDO *doesn't* well-approximate any simple-integer ratios, that doesn't mean it sounds like shite. 8-EDO, for instance--about the simplest thing it approximates well is 11/6, but I swear it sounds more relaxed than 11 or even 14 if you use it right. And furthermore, I firmly believe intervals high in harmonic entropy--i.e. describable in terms of multiple ratios of similar complexity--are tunable by ear and musically-useful, and possibly even stable or pleasant in the right context, but they make messes out of any attempt at definition in terms of ratios.
>
> So I'm giving up, I'm leaving psychoacoustics out of it and taking the Blackwood approach of "these are some nice-sounding chords I've found, and this is how "smooth" typical major/minor triads sound". Most musicians probably wouldn't want to be bothered with the psychoacoustics anyway, I'm sure.
>
> -Igs
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> One thing I'm learning as I develop this "harmonic laplace transform"
> function is that the resultant entropy calculation will yield a
> slightly different curve from HE -- less consonant dyads will have
> "narrower" fields of attraction than more consonant ones. This means
> that there will be more minima than with HE, but that the "valleys"
> around the minima will be much narrower, indicating that these
> intervals are extremely sensitive to mistuning. I haven't yet decided
> if this is a bug or a feature.

Sounds like a feature to me, for that sounds more like what I hear in real life. As I've been saying here for several years now, I don't hear a smooth curve between the HE minima (minima which I agree exist,
more or less in the proportions graphed as far as I can tell), but
little dips and "sweet spots" between them.

And you do need to make it SHE, not just HE, for reasons I earlier gave.

I predict that you're going to get some odd things, like a dip at 11:9
that asymmetrically favors the lower side. Looking forward to what you do here and thanks for taking it on!
>
> But the HE curve comes directly out of the assumptions that it >makes,

I think that's clear.

-Cameron Bobro

On Wed, Nov 3, 2010 at 7:27 PM, cameron <misterbobro@...> wrote:
>
> Sounds like a feature to me, for that sounds more like what I hear in real life. As I've been saying here for several years now, I don't hear a smooth curve between the HE minima (minima which I agree exist,
> more or less in the proportions graphed as far as I can tell), but
> little dips and "sweet spots" between them.

I assumed that you'd think it's a feature. Once I finish working it
out, I'll come up with two versions of the curve: one that looks like
I described above, and one in which there's an additional smoothing
curve that represents the critical band. We'll see which matches
hearing better in a blind listening test.

What we're really figuring out is how the place and periodicity
mechanisms interact.

> And you do need to make it SHE, not just HE, for reasons I earlier gave.

Haha, I don't remember seeing that. I was going to come up with a
"Scene Entropy" calculation that would account for there being some
kind of noise channel. Maybe I'll call it "Scene Harmonic Entropy" so
that it spells SHE and HE. Then this other other type of entropy I'm
talking about could be called WE.

> I predict that you're going to get some odd things, like a dip at 11:9
> that asymmetrically favors the lower side. Looking forward to what you do here and thanks for taking it on!

They will all very slightly most likely asymmetrically favor the lower
side, since we're looking at a curve that would be symmetrical on a
linear plot, but on a log frequency plot will be skewed to the lower
side.

-Mike

Cameron (about Mike's new variation on HE)>"I predict that you're going to get
some odd things, like a dip at 11:9 that asymmetrically favors the lower side.
Looking forward to what you do here and thanks for taking it on!"

Now this sounds exactly like what the doctor ordered...something that makes
"sub curves" to denote the kind of small troughs in the curve formed by
higher-limit ratios that Cameron and I have noted repeatedly by ear...
And maybe this could also be applied to 3D Harmonic Entropy and actually
have a fair chance at analyzing the kind of higher limit chords possible in
things like Ozan's chord-intensive Maqam theories and perhaps helping us design
strong both "Persian and Western" compatible tuning systems...sweet deal!