The big picture (was Re: [tuning] Re: 4-tone chords and harmonic entropy: when odd limit fails)

On Mon, Aug 23, 2010 at 9:04 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Okay, Mike, here goes. Please note that I don't actually have this all worked out, I'm just sort of spinning hypotheses here as I go. But this is getting kind of more long-winded than it should, so maybe we can splinter some of these ideas off into separate threads?

OK, I edited the subject line in my response. Let's see if that does the trick.

> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Right, but what about 7/6? 7/6 is not near enough to the gravity-well
> > of 6/5 to get sucked in. Compare the sound of 22-tet superpyth aeolian
> > to 19-tet superpyth aeolian, and that still also "works." And to my
> > ears, at least, 7/6 and 6/5 are two different intervals.
>
> Well, the H.E. curve is much flatter between 6/5 and 7/6 than it is between 6/5 and 5/4. Below 6/5, it seems like the transition between "minor third" and "major second" is a very smooth one, without a significant peak of "what the hell IS that interval?". IOW, that whole region is already pretty high in discordance. Using the "adjusted and inverted scale" of H.E. that Steve M. just helped me work out, where a unison is 100 and a 51-cent quartertone is 0 (and an octave is 85, a fifth 50, etc.), 7/6 is about 23.74, 6/5 is about 25.91, and the "dip" between them is about 22.24 at about 288 cents. So the difference between local maxima and minima of H.E. in that region is very slight. OTOH, 5/4 scores about 30.3, and the valley (or peak, with the original H.E. numbers) between that and 6/5 is 18.73 at 350 cents; much more significant.

One thing that I think needs to be said here is - that HE was
formulated from an information theory standpoint. It is an excellent
model of concordance, or consonance, or whatever the word du jour is,
and it's the first of its kind too. But I think there might be a few
practical limitations that really prevent us from taking the leap that
the graph is a perfect representation of concordance as it stands.

The first is that the graph changes with that s parameter. So if you
make s low enough (or is it high enough? I never remember), you will
indeed see distinct notches for 7/6, 5/4, and 11/9 too. It changes the
shape of the curve.

The second thing is that the way that HE is currently formulated has a
bit of arbitrariness to it. First you generate a Farey series up to
some arbitrary limit, and then you arbitrarily decide that every
interval has a "domain" equal to the distance between the mediant of
the interval and it's lower neighbor, and the mediant of the interval
and it's upper neighbor. Then you decide that the dyad coming in has a
probability of being heard as ANY dyad and that that probability is
represented with a Gaussian curve, and then you do the classical
information theory entropy calculation.

So while it's an elegant idea, there are enough arbitrary aspects to
it that I would take the curve with a grain of salt. It's good for
getting a general idea of what's going on - that 5/4 is concordant,
and less concordant than 3/2; and that 6/5 is less concordant than
5/4, and the space between them is some kind of local entropy maximum,
etc. But when you start getting into very small and fine details like
that - keep in mind it's really just a very basic, rough model
(although I can't praise Paul enough for coming up with it).

A while ago I came up with an alternative formulation for the HE curve
that would model a filterbank of combination-sensitive neurons in the
brain, and see how many fire for any given dyad (more firing strongly
= higher entropy). ome day I'll get it finished. I'm interested to see
how it would match up with Paul's. I expect they'd look extremely
similar, but that there might be small differences.

You might also note that sometimes, as I said earlier - the "flip
flopping" between intervals at an area of maximum entropy might be
because we're using different diatonic (or whatever) maps to try and
place the interval. Hence 9/7 "sounds like" 5/4, because they both
"sound like" a major third in the LLsLLLs sense. A good way to do an
HE test without this cultural bias coming into it would be to create a
timbre that consists only of partials 9 and 7, and play some chords
and fast runs with it and see what fundamental comes out. I might come
up with some examples later to see how 9/7, 11/9, 6/5, and 7/6 behave.
And 19/16 too.

-Mike

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

> You might also note that sometimes, as I said earlier - the "flip
> flopping" between intervals at an area of maximum entropy might be
> because we're using different diatonic (or whatever) maps to try and
> place the interval. Hence 9/7 "sounds like" 5/4, because they both
> "sound like" a major third in the LLsLLLs sense. A good way to do an
> HE test without this cultural bias coming into it would be to >create a
> timbre that consists only of partials 9 and 7, and play some chords
> and fast runs with it and see what fundamental comes out. I might come
> up with some examples later to see how 9/7, 11/9, 6/5, and 7/6 behave.
> And 19/16 too.
>
> -Mike
>

The perception of middle thirds as flip-flopping between major and minor is surely a result of cultural conditioning. Your ability to distinguish between pitches probably far exceeds mine, yet I, and an enormous part of the world's population, hear middle thirds simply as middle thirds.

It's true that in the lower harmonic partials, there is no direct reference for a middle third. If our frame of reference for interval perception does indeed include the proportions found in the first, say, nine, harmonic partials, the identity of a middle third could reasonably be described as "between 6/5 and 5/4". Notice, though, that in that vast swathe of the globe in which distinguishing middle thirds is perfectly normal (from about the Straits of Gibraltar over to Bosnia and through Indonesia, at least), many bright or nasal timbres are common. The harmonic proportion of 11:9 can be quite evident on such instruments, as you may verify for yourself by listening to the saz, zurna, cimbalom, various spike fiddles, and so on. So, given timbres with strong energy around about the 12th partial (ie. every damn bright middle-eastern and asian instrumental timbre I've ever measured :-) ), middle thirds can be "justified" as spectral references to 11:9.

Personally I'd extend "justification" of middle thirds to 16:13 as well, but the perception as Just of harmonic relationships this high in the series may rely on too many muscial conditions to be qualify as reasonable general spectral references. (Whereas it would be rank buffoonery to claim that "fifths" have nothing to do with the third harmonic partial and the relation between the third and second harmonic partials, for example). In my own music, with lots of 16:13, it could be argued that the Just effect of the interval is largely due to the frequent appearance of tone in question as lower tone of the octave complement and as part of a harmonic series. In other words, 16:13 is clearly a spectral proportion in my music, or Just, but the tone making the third against the tonic often appears as the lower tone of the sixth formed with the octave, so, part of a 13:8 proportion, as well as part of a harmonic series when the second of the scale is a 14:13 and the fourth may very well be lowered to an approximate 17:13, etc.

In the case of 11:9, however: if you're ever around these parts you can drop by and hear for yourself that 11:9 is as resonant and easy to find on my spike fiddle (a Chinese erhu) as a 6:5. A spectogram reveals that this is hardly surprising, as the spectrum features a very strong eleventh partial and partials multiple of 3, but no particularly strong fifth partial. This phenomenon is not a matter of body formants, as it is consistent however the open strings are tuned.

-Cameron Bobro

Sorry, the rest of my reply got cut off. Here it is:

Igliashon wrote:
> But at any rate, none of this may actually matter, and what may be more important--as I've tried to suggest--is that when you bring it back to a question of generators, in the diatonic range it's always going to be the case that (4g-2p)>(-3g+2p), whereas in Mavila, (4g-2p)<(-3g+2p). At 7-EDO, (4g-2p)=(-3g+2p). At 5-EDO, (4g-2p)=(-1g+1p), and (-3g+2p)=(2g-1p). As long as you can distinguish 4g-2p from -1g+1p, it doesn't matter what the exact ratio is. However, in order to know whether you can distinguish between two intervals, you have to look at harmonic entropy, which drags JI back into the equation.

Sorry, I'm confused here - what are g and p? The axes? Is g a fifth
and p a third or something?

> The thing that really hammers me about neutrals is that I've never found a good way to map them on a lattice, except as BOTH major and minor thirds. I have to say, I don't like doing it that way, but it sort of makes sense. I really think that with 7-EDO, there are multiple maps coinciding at once, and with the right chord progressions and melodic phrasing, you can suggest different maps.

I agree. The piece I posted on MMM to me sounds like G mixolydian.
Whether this is because I hear the m7 and a rough major 3rd and
automatically just snap it to my "mixolydian" map, or because of some
deeper inborn thing, is something I don't know...

> 5/4 and 6/5 are only important in the diatonic scale because they are psychoacoustically as far as possible from each other AND from 4/3 (which is -1g+1p) and 9/8 (which is 2g-1p). It is their psychoacoustic spacing that matters, not so much their actual low harmonic entropy. This is what makes them ideal. 12-tET could be another sort of ideal, since the space between 200, 300, 400, and 500 cents is all equal and as wide as possible. But the point is that the spacing doesn't HAVE to be ideal, it just has to be enough for us to tell the difference. As long as the minor third is above the entropic peak around 240 cents, and the major third is below the entropic peak around 453 cents, we can get a recognizable diatonic scale.

That is a good point. Span comes into it as well. But why are 4/3 and
9/8 so important then, if 5/4 and 6/5 aren't?

> It's like holograms. Anything that can generate a meantone scale contains within it all the information of the entire scale. I'm not familiar with using major and minor seconds to generate a meantone, though...how does that work? Are you talking about treating meantone as a rank-3 temperament? I don't see how you could use a major second and a minor second as sole inputs to an MOS algorithm and come out with a diatonic scale. One has to be the modulo.

No, you'd just have a large step L, which is a tempered approximation
to both 9/8 and 10/9, and a small step s, which is a tempered
approximation to both 16/15 and 27/25. Then an octave would be 5 L's
and 2 s's, and a major third would be 2 L's. and a fifth would be 4
L's and an s. I'm not sure how this would work out from an MOS
standpoint, but you could still use those intervals as your axes and
still derive the entire meantone scale. A chromatic half step would be
L-s, so you'd be able to get chromatic (and hyperchromatic) scales
too.

-Mike

Cameron: neutral thirds were difficult for me to hear as anything but
"maximally distant" from a minor and major third for a long time. Once
I stated listening to some music in 7-tet, where neutral thirds are
reachable by 3 fourths and by 4 fifths, I started to hear them as
serving more of a melodic and harmonic function.

I think the perception of a neutral third has to do, schematically,
with not just what the notes "are" - but what they "could be." That
is, how do you hear them as potentially fitting in with other
potential notes? When you hear a neutral triad, what tonal structures
could it be a part of, and what other notes could potentially be in
its "vicinity?"

To someone who's used to 12-tet, or meantone tunings in general,
nothing will come to mind, and so the best that this person can come
up with is to try to fit it as a round peg into the square "major
chord" hole, which flights of fancy will reveal that it could be a
part of the major scale or mixolydian the octatonic scale or what not
- or to fit it as a round peg into the triangular "minor chord" hole,
in which daydreaming will reveal that it could be a part of the minor
scale or dorian or phrygian.

Other cultures that are used to having some kind of tonal heirarchy or
modal system in which neutral triads are a distinct part will
definitely not have this problem.

-Mike

On Wed, Aug 25, 2010 at 5:33 AM, cameron <misterbobro@...> wrote:
>
>
> The perception of middle thirds as flip-flopping between major and minor is surely a result of cultural conditioning. Your ability to distinguish between pitches probably far exceeds mine, yet I, and an enormous part of the world's population, hear middle thirds simply as middle thirds.
>
> It's true that in the lower harmonic partials, there is no direct reference for a middle third. If our frame of reference for interval perception does indeed include the proportions found in the first, say, nine, harmonic partials, the identity of a middle third could reasonably be described as "between 6/5 and 5/4". Notice, though, that in that vast swathe of the globe in which distinguishing middle thirds is perfectly normal (from about the Straits of Gibraltar over to Bosnia and through Indonesia, at least), many bright or nasal timbres are common. The harmonic proportion of 11:9 can be quite evident on such instruments, as you may verify for yourself by listening to the saz, zurna, cimbalom, various spike fiddles, and so on. So, given timbres with strong energy around about the 12th partial (ie. every damn bright middle-eastern and asian instrumental timbre I've ever measured :-) ), middle thirds can be "justified" as spectral references to 11:9.
>
> Personally I'd extend "justification" of middle thirds to 16:13 as well, but the perception as Just of harmonic relationships this high in the series may rely on too many muscial conditions to be qualify as reasonable general spectral references. (Whereas it would be rank buffoonery to claim that "fifths" have nothing to do with the third harmonic partial and the relation between the third and second harmonic partials, for example). In my own music, with lots of 16:13, it could be argued that the Just effect of the interval is largely due to the frequent appearance of tone in question as lower tone of the octave complement and as part of a harmonic series. In other words, 16:13 is clearly a spectral proportion in my music, or Just, but the tone making the third against the tonic often appears as the lower tone of the sixth formed with the octave, so, part of a 13:8 proportion, as well as part of a harmonic series when the second of the scale is a 14:13 and the fourth may very well be lowered to an approximate 17:13, etc.
>
> In the case of 11:9, however: if you're ever around these parts you can drop by and hear for yourself that 11:9 is as resonant and easy to find on my spike fiddle (a Chinese erhu) as a 6:5. A spectogram reveals that this is hardly surprising, as the spectrum features a very strong eleventh partial and partials multiple of 3, but no particularly strong fifth partial. This phenomenon is not a matter of body formants, as it is consistent however the open strings are tuned.
>
> -Cameron Bobro

Hi Mike. Was wondering why the bulk of my post didn't seem to be addressed in your reply! Anyway:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Sorry, the rest of my reply got cut off. Here it is:
>
> Igliashon wrote:
> > But at any rate, none of this may actually matter, and what may be more important--as I've tried to suggest--is that when you bring it back to a question of generators, in the diatonic range it's always going to be the case that (4g-2p)>(-3g+2p), whereas in Mavila, (4g-2p)<(-3g+2p). At 7-EDO, (4g-2p)=(-3g+2p). At 5-EDO, (4g-2p)=(-1g+1p), and (-3g+2p)=(2g-1p). As long as you can distinguish 4g-2p from -1g+1p, it doesn't matter what the exact ratio is. However, in order to know whether you can distinguish between two intervals, you have to look at harmonic entropy, which drags JI back into the equation.
>
> Sorry, I'm confused here - what are g and p? The axes? Is g a fifth
> and p a third or something?

Sorry, I should have clarified: g = generator, p = period; in the case of diatonic scales, g = ~3/2, p = 2/1.

> > 5/4 and 6/5 are only important in the diatonic scale because they are psychoacoustically as far as possible from each other AND from 4/3 (which is -1g+1p) and 9/8 (which is 2g-1p). It is their psychoacoustic spacing that matters, not so much their actual low harmonic entropy. This is what makes them ideal. 12-tET could be another sort of ideal, since the space between 200, 300, 400, and 500 cents is all equal and as wide as possible. But the point is that the spacing doesn't HAVE to be ideal, it just has to be enough for us to tell the difference. As long as the minor third is above the entropic peak around 240 cents, and the major third is below the entropic peak around 453 cents, we can get a recognizable diatonic scale.
>
> That is a good point. Span comes into it as well. But why are 4/3 and
> 9/8 so important then, if 5/4 and 6/5 aren't?

Because they are the values that you get when you go up by two generators and down an octave (9/8) or down by one and up an octave (4/3), because our generator has to be AT LEAST an approximate 3/2 for the scale to have any possibility to sound diatonic. They are important because they are the values the thirds have to be distinguished from in order NOT to sound like 5-EDO. But of course I'm not talking about exact 4/3 or 9/8, because the diatonic scale only requires an approximate 3/2 as its generator. Of course, like I said, because the period is an octave, we can also use the octave-inversion of a 3/2, which will invert the generator/period coordinates.

> > It's like holograms. Anything that can generate a meantone scale contains within it all the information of the entire scale. I'm not familiar with using major and minor seconds to generate a meantone, though...how does that work? Are you talking about treating meantone as a rank-3 temperament? I don't see how you could use a major second and a minor second as sole inputs to an MOS algorithm and come out with a diatonic scale. One has to be the modulo.
>
> No, you'd just have a large step L, which is a tempered approximation
> to both 9/8 and 10/9, and a small step s, which is a tempered
> approximation to both 16/15 and 27/25. Then an octave would be 5 L's
> and 2 s's, and a major third would be 2 L's. and a fifth would be 4
> L's and an s. I'm not sure how this would work out from an MOS
> standpoint, but you could still use those intervals as your axes and
> still derive the entire meantone scale. A chromatic half step would be
> L-s, so you'd be able to get chromatic (and hyperchromatic) scales
> too.

But just having L & s doesn't tell you anything about the order of them; you could end up with LsLLLLs or LssLLLL. Also, you have to stipulate that the scale will repeat at the octave, or else there's nothing to stop you from making scales like ssLsLssL. Also, defining sizes of L & s according to JI doesn't work. For instance, 22-EDO would be out of the question as a diatonic scale, since its L is very far from 10/9 (it more approximates 9/8 and 8/7) and its s does not REMOTELY approximate 16/15. Looking at just those values, you'd think that 2\22-EDO would be a better s, but then you'd end up with 11-EDO. If you really wanted to use L & s to generate the scale, you'd have to specify their size such that 5L+2s equals a 2/1. But then, you'd have no way to rule out a scale where L = 239 cents and s = 2.5 cents (or something), which definitely wouldn't sound diatonic, unless you started stipulating JI approximations for other combos of L & s (like 2L = ~5/4, or 3L+s = ~3/2)...but you want to avoid that, right?

At least with the MOS-based approach I outlined above, relationship to JI basically becomes incidental. If you define scales according to how different numbers of generators, modulo the period, relate to each other pitch-wise, you all you have to stipulate is how close you will allow the note produced by one set of generator/period coordinates to come to the note produced by another set of generator/period coordinates. The whole thing can be done with cents values, really, and JI only enters the picture when we want to start evaluating the harmonic concordance of chords produced in various scales.

So on those lattices I posted? You can forget everything I said about JI ratios. The "/" just means "4gen-2per" and the "\" means "-3gen+2per", and the "---" means "1gen+0per". Of course, you don't have to think this way to use lattices, and yes, you can certainly represent the diatonic scale on a lattice using major and minor 2nds. The point is for it to be a map and to be helpful. But I'm pretty sure that, as far as what gives the diatonic scale its "character" is the fact that 1) it's generator is somewhere around a fifth and its period is an octave, 2) we can tell the difference between -1g+1p and 4g-2p, and between -3g+2p and 4g-2p (etc).

-Igs