metastable chords, utonalities, 18-tet, and extreme precision

I was just messing around with some of these metastable intervals, and
I noticed something interesting:

If you take 4:5:6:7 and you put it into 12-tet, you will hear some
pretty heavy beating. But, if you take that same chord and you put it
into 18-tet, the beating suddenly cancels out. Here's a cent-by-cent
comparison:

12-tet: 0-400-700-1000
18-tet: 0-400-733-1000

For some reason that I can't figure out, bringing that fifth up to 733
suddenly makes it incredibly resonant with little to no beating. It's
an incredibly rich sound. I suppose that doing it in 36-tet might work
out nicely as well. But why is this? Is this the "even beating" that
you guys were referring to with the metastable intervals? I can't
figure out why this is happening. The best explanation that I can come
up with is that the 1000, which is pretty close to a metastable
interval, has a 266-cent interval directly under it in the 18-tet
version, which is pretty close to 6:7. Putting a 5:6 under that 1000
doesn't sound so great either. So with the 6:7 sort of utonal
structure under the metastable interval would then introduce a
difference tone which would be as far away from the root of the chord
as the metastable interval itself would be from 7/4, which not only
makes the chord sound less "warbly" than the usual chords with
metastable intervals, but still "crunchy." Or something.

Actually, I just did some more listening tests that disproved my own
theory here. Changing that 400 to a nice 386 makes it beat more,
however, moving that 400 UP to around 417 or so so that there's a 5:6
and a 6:7 under the metastable 1000 ALSO makes it beat more. So could
it be that the 400 is itself a metastable interval between the 4:5
over the root and a 5:6 under the 6:7 under the metastable 1000? It's
like a shadow of a shadow's shadow here.

I'm not sure what the best way to describe chords in metastable
intonation, so I'm for now just going to use just notation with
decimals.

4:5:6:7.136 is the standard metastable dominant 7 chord with the
dominant 7 around 1002 cents. That 7.136 is (7 + 9*phi)/(4 + 5*phi)
4:5:6.117:7.136 is the standard metastable dominant 7 chord with that
reinforced 6:7 under it. Note there is still beating with this one,
but it's much less.
4:5.097:6.117:7.136 is the metastable dominant 7 chord with a 5:6:7
under it. Still beating. I'm not sure how to exactly calculate some
sort of equivalent to a metastable interval between the numbers here,
since we're no longer dealing with rational numbers but rather with
some sort of metastable interval between another metastable interval
and a just interval, but from listening tests, the chord with the
least beating is somewhere around here:
4:5.05:6.117:7.136

Furthermore, tweaking it even more, by changing the third note and
making it slightly flatter than the 6:7, this chord here:

4:5.05:6.102:7.136

has even less beating. AND, to be even weirder, if you flatten the
metastable 7th by under 3 cents

4:5.05:6.102:7.125

There's a LOT less beating. If you raise it up by one cent to
4:5.05:6.102:7.143, there's a lot more beating. So it seems Kraig was
right about the precision issue.

I'm sure if I sat and tweaked with this for another 3 hours I'd be
able to find some version that has no beating at all or something, as
everything would cancel out, but it would help if there were some
mathematical way to figure out what's going on here first.

-Mike

Just did some more listening - if you listen closely, by changing that
7 around very very slightly, you will find different points of
resonance only a few cents away from each other where the beats line
up perfectly. I imagine it has something to do with the beat
frequencies from one tone cancelling out the beat frequencies of
another tone, but how that's even possible I'm not sure. Or it also
sounds like it might be that the difference tones somehow align here
in some sort of way. Either way, it seems like if there's an actual
technique and a concept behind this, then we could develop a way to
somehow build chords other than just that have no beating if the
precision is high enough. Alternately I could be way off the mark
here.

-Mike

I figured it out.

4.000:5.045:6.091:7.136

is the magic chord.

Mathematically it's the equivalent of a 4:5:6:7 chord in which every
note is stretched, basically, so it might have nothing to do with
metastable intervals at all. Here's how I calculated it:

7.136 - 4 = 3.136 to find the difference between the first and last note
3.136 / 3 = 1.0453333 to find the average spacing between notes
4 + 1.0453 = 5.0453 + 1.0453 = 6.097 + 1.0453 = 7.136

I'm falling asleep here so I hope my explanation makes sense. So now,
compare these three chords:

4:5:6:7.136
4:5:6.091:7.136
4:5.045:6.091:7.136

And use a simple sine wave kind of sound. You'll see what I mean when
you get to the last chord. What's happening is that the difference
tones, although not harmonic in and of themselves, still reinforce.
What's interesting to me here is how much can you stretch a chord
before it starts to sound like another chord? Perhaps we'll start
seeing metastable chords next.

-Mike

Gah, my post got cut off. Here was the last paragraph:

And, furthermore, this last chord:

4:5.045:6.091:7.136

sounds completely different and yet equally in tune with 4:5:6:7
(provided the right timbre is used). So perhaps when we create chords
with metastable intervals, we can use this technique to make them
sound just as "in tune" as just intervals.

-Mike

Man, I just keep posting, but I'm on a roll here.

Another "discovery": This chord:

10:12.162:15

is 0-338-702, which is the noble mediant neutral third chord. 12.162
represents the noble mediant between 5/4 and 6/5. If you stretch the
fifth slightly:

10:12.162:15.2

The chord sounds a lot more like a "minor" chord. Likewise, the same
noble third chord can be spelled like this:

4:4.865:6

And if you flat the fifth slightly:

4:4.865:5.86

You start to hear it spelled more like a "major" chord. So the other
intervals in the chord also affects the perception of the metastable
interval, as the ear can also adapt to hearing these chords as
inharmonically stretched/compressed versions of other chords. Perhaps
this accounts for some of the difference between these results and
those predicted by harmonic entropy.

I didn't alter the fifths enough to actually continue the pattern for
either of these chords. For the record, taking the neutral third chord
and flattening the fifth so that it's in line with a "major" chord
pattern yields this:

4:4.865:5.730

You should hear a very "flattened" sounding major chord, similar to
the way that 19tet's major chord is "flattened" but even more so.
Alternately, if you stop for a second and try to hear it as an
isoharmonic diminished chord, that's what you'll hear.

My question with all of this is, when we're talking about how you
perceive these intervals, are we talking about how you LABEL them?
Because that seems to have to do with a number of psychological
factors that have nothing to do with acoustics. In this case, the
resonance of the chord and the proportion that the tones are in
reminds me of major, but the notes themselves remind me of diminished.
It's just a matter of two clashing schemas - this chord is where two
conflicting schemas both represent part of it. My question is, do
noble intervals in general represent a clash of some kind of
psychoacoustic field of attraction, or a clash of a schematic approach
to cognitively labeling intervals? Because in this case it seems to be
the latter.

The minor version of the chord is this:

10:12.162:15.405

This one sounds like a stretched minor chord if you give it a second -
there's no beating, but you might be apt to hear that outside interval
as a minor sixth. In fact, you might be apt to hear the chord as a
major chord in first inversion. Or you can hear it as a stretched
minor chord. Another extremely ambiguous chord.

-Mike

Mike wrote:

> What's interesting to me here is how much can you stretch a chord
> before it starts to sound like another chord? Perhaps we'll start
> seeing metastable chords next.

I'm not able to answer this particular question but you can see similar phenomena even in chords whose difference tones lie in totally different bands far far away from the fundamentals of the hypothetical (or let's say "mistuned") epimorics-- like in the case of 4:7:10:13:16, which has the relative fundamental of 1 but the difference tones between each pair of consecutive tones all have a relative value of 3. I also made a temperament for this some time ago and there was some discussion about that on this list as well. You can try to spot it by searching for "triharmonic scale" on the Tuning group website. If you are interested, I can repost the scale again.

Petr

Mike wrote:

> Here's a cent-by-cent comparison:
>
> 12-tet: 0-400-700-1000
> 18-tet: 0-400-733-1000
>
> For some reason that I can't figure out, bringing that fifth
> up to 733 suddenly makes it incredibly resonant with little
> to no beating.

Wow, with the timbre I'm using, both chords beat like
angry drunks. The chords sound pretty similar
dissonace-wise, owing to the creation of a just 7/6
in the 2nd chord in exchange for the worse 3/2.
The overall pattern of stretching may help -- jury's
out on that one -- but it's not of much significance
here.

-Carl

Carl wrote:

> Wow, with the timbre I'm using, both chords beat like
> angry drunks. The chords sound pretty similar
> dissonace-wise, owing to the creation of a just 7/6
> in the 2nd chord in exchange for the worse 3/2.
> The overall pattern of stretching may help -- jury's
> out on that one -- but it's not of much significance
> here.

Mike is probably using a timbre with very soft overtones, which can explain the question. The actual answer is the fact that the linear factor for 1000 cents is ~1.781797. That means that the difference tone between the sounding tones is ~0.781797. Dividing this by 3 makes ~0.260599. If you stack three intervals (i.e. four tones) in such a way that this is the relative difference tone between consecutive tones, then you get, counting from the lowest pitch upwards, cent sizes of ~400.931508, ~726.249868, and 1000.

Petr

Petr wrote:

> > Wow, with the timbre I'm using, both chords beat like
> > angry drunks. The chords sound pretty similar
> > dissonace-wise, owing to the creation of a just 7/6
> > in the 2nd chord in exchange for the worse 3/2.
> > The overall pattern of stretching may help -- jury's
> > out on that one -- but it's not of much significance
> > here.
>
> Mike is probably using a timbre with very soft overtones,
> which can explain the question. The actual answer is the
> fact that the linear factor for 1000 cents is ~1.781797.
> That means that the difference tone between the sounding
> tones is ~0.781797. Dividing this by 3 makes ~0.260599.
> If you stack three intervals (i.e. four tones) in such a
> way that this is the relative difference tone between
> consecutive tones, then you get, counting from the lowest
> pitch upwards, cent sizes of ~400.931508, ~726.249868,
> and 1000.

Nothing to do with difference tones could explain the
disappearance of beating that Mike was reporting. Meanwhile,
difference tones ALMOST NEVER have a significant effect
on the con/dissonance of chords in musical contexts.
The simple reason is that their amplitudes are simply
too low.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Petr wrote:
>
> > > Wow, with the timbre I'm using, both chords beat like
> > > angry drunks. The chords sound pretty similar
> > > dissonace-wise, owing to the creation of a just 7/6
> > > in the 2nd chord in exchange for the worse 3/2.
> > > The overall pattern of stretching may help -- jury's
> > > out on that one -- but it's not of much significance
> > > here.
> >
> > Mike is probably using a timbre with very soft overtones,
> > which can explain the question. The actual answer is the
> > fact that the linear factor for 1000 cents is ~1.781797.
> > That means that the difference tone between the sounding
> > tones is ~0.781797. Dividing this by 3 makes ~0.260599.
> > If you stack three intervals (i.e. four tones) in such a
> > way that this is the relative difference tone between
> > consecutive tones, then you get, counting from the lowest
> > pitch upwards, cent sizes of ~400.931508, ~726.249868,
> > and 1000.
>
> Nothing to do with difference tones could explain the
> disappearance of beating that Mike was reporting. Meanwhile,
> difference tones ALMOST NEVER have a significant effect
> on the con/dissonance of chords in musical contexts.
> The simple reason is that their amplitudes are simply
> too low.
>
> -Carl
>

But we're not actually talking about consonance and dissonance per
se, rather some kind of unity. I'm using pretty gnarly timbres
experimenting with these things by the way.

Cameron wrote:
>
> But we're not actually talking about consonance and dissonance per
> se, rather some kind of unity. I'm using pretty gnarly timbres
> experimenting with these things by the way.

Define your terms. Mike was talking about beating, which
has a widely-accepted precise definition. I additionally
brought "consonance" into play, and as I have defined it
many times, it includes beating/roughness AND a "unity" or
"cohesiveness" or "rootedness" or "tonalness" component.

-Carl

Carl wrote:

> Nothing to do with difference tones could explain the
> disappearance of beating that Mike was reporting. Meanwhile,
> difference tones ALMOST NEVER have a significant effect
> on the con/dissonance of chords in musical contexts.
> The simple reason is that their amplitudes are simply
> too low.

Correct, except that this time I'm not talking about beats in similarly pitched overtones but beats in similarly pitched difference tones. You can't say their amplitudes are "simply too low" if three of them have identical frequencies. If you want me to prove what I've said, I'll pick up pure sine waves and make a short audio clip demonstrating this.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
> > Nothing to do with difference tones could explain the
> > disappearance of beating that Mike was reporting. Meanwhile,
> > difference tones ALMOST NEVER have a significant effect
> > on the con/dissonance of chords in musical contexts.
> > The simple reason is that their amplitudes are simply
> > too low.
>
> Correct, except that this time I'm not talking about beats in
> similarly pitched overtones but beats in similarly pitched
> difference tones. You can't say their amplitudes are "simply
> too low" if three of them have identical frequencies. If you
> want me to prove what I've said, I'll pick up pure sine waves
> and make a short audio clip demonstrating this.

What I say above is entirely correct. Difference tones
can neither explain Mike's reported observation nor have a
major role in musical consonance/dissonance at large.
They _can_ have a big impact in specific conditions, and
yes I'd love to hear an audio example. Using sine tones
will only increase the importance of the difference tones
you're calculating, so if you want a harder test use more
typical timbres.

-Carl

Carl wrote:

> Using sine tones
> will only increase the importance of the difference tones
> you're calculating, so if you want a harder test use more
> typical timbres.

Didn't I say "Mike is probably using a timbre with very soft overtones"? The reason why I want to use pure sines is that the phenomenon Mike describes is more or less inaudible in sounds rich in overtones.

Okay, let's do the sounds. Hope I'll make the clip quickly enough to let you hear it soon.

Petr

On Fri, Jun 20, 2008 at 4:34 PM, Carl Lumma <carl@...> wrote:
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>> > Nothing to do with difference tones could explain the
>> > disappearance of beating that Mike was reporting. Meanwhile,
>> > difference tones ALMOST NEVER have a significant effect
>> > on the con/dissonance of chords in musical contexts.
>> > The simple reason is that their amplitudes are simply
>> > too low.
>>
>> Correct, except that this time I'm not talking about beats in
>> similarly pitched overtones but beats in similarly pitched
>> difference tones. You can't say their amplitudes are "simply
>> too low" if three of them have identical frequencies. If you
>> want me to prove what I've said, I'll pick up pure sine waves
>> and make a short audio clip demonstrating this.
>
> What I say above is entirely correct. Difference tones
> can neither explain Mike's reported observation nor have a
> major role in musical consonance/dissonance at large.
> They _can_ have a big impact in specific conditions, and
> yes I'd love to hear an audio example. Using sine tones
> will only increase the importance of the difference tones
> you're calculating, so if you want a harder test use more
> typical timbres.

I think the whole thing has to do with difference tones. Note these
three chords here:

4:5:6:7.136
4:5:6.091:7.136
4:5.045:6.091:7.136

You'll notice that the last one, which IMO sounds very resonant and
strong, is isoharmonic. So the difference tones between each note will
reinforce each other. If you use a sine wave you'll hear the effect
most pronounced. If you use a more complex timbre, and the partials in
that timbre aren't stretched by a comparable amount, then the partials
of each tone will beat with the notes in the chord.

Hi Mike,

Well figured out. I think you're right, that it has everything to do
with coinciding difference tones. I don't think it has anything to do
with metastability.

The fact that you are using sine tones is the giveaway. Metastability
should only really be relevant to tones rich in harmonics.

I think you will find that the effect you found will work equally with
_any_ amount of stretch.

Experiments with very accurate sine tones through high quality
headphones _quietly_, don't find anything special at any ratios.
_Everything_ is concordant. But as soon as the slightest non-linearity
appears in the system, e.g. less than perfect speakers or slightly
louder tones causing the ear mechanism to go into a less linear
region, then we start to get sum and difference tones being generated
and will hear the coincidence of these as special.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> I figured it out.
>
> 4.000:5.045:6.091:7.136
>
> is the magic chord.
>
> Mathematically it's the equivalent of a 4:5:6:7 chord in which every
> note is stretched, basically, so it might have nothing to do with
> metastable intervals at all. Here's how I calculated it:
>
> 7.136 - 4 = 3.136 to find the difference between the first and last note
> 3.136 / 3 = 1.0453333 to find the average spacing between notes
> 4 + 1.0453 = 5.0453 + 1.0453 = 6.097 + 1.0453 = 7.136
>
> I'm falling asleep here so I hope my explanation makes sense. So now,
> compare these three chords:
>
> 4:5:6:7.136
> 4:5:6.091:7.136
> 4:5.045:6.091:7.136
>
> And use a simple sine wave kind of sound. You'll see what I mean when
> you get to the last chord. What's happening is that the difference
> tones, although not harmonic in and of themselves, still reinforce.
> What's interesting to me here is how much can you stretch a chord
> before it starts to sound like another chord? Perhaps we'll start
> seeing metastable chords next.
>
> -Mike
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Nothing to do with difference tones could explain the
> disappearance of beating that Mike was reporting. Meanwhile,
> difference tones ALMOST NEVER have a significant effect
> on the con/dissonance of chords in musical contexts.
> The simple reason is that their amplitudes are simply
> too low.

Right. But if you start with sine tones and then put them through some
distortion, whether deliberate or accidental, difference tones are
everything.

-- Dave Keenan

> Hi Mike,
>
> Well figured out. I think you're right, that it has everything to do
> with coinciding difference tones. I don't think it has anything to do
> with metastability.
>
> The fact that you are using sine tones is the giveaway. Metastability
> should only really be relevant to tones rich in harmonics.

Well, it's not metastable in nature, but I have found that it has
interesting applications to metastable intervals in a few ways. For
example:

1. This chord, 4:4.865:5.730, is generated by taking a noble neutral
third and making it isoharmonic. The whole chord is actually ambiguous
between a major chord and a 5:6:7 diminished chord. I don't think that
outer dyad is metastable by any means, and it might not even be in a
zone of maximum harmonic entropy. So when you make chords with
metastable intervals, there are other ways to work than making sure
every dyad is metastable in and of itself. You can harmonize justly
with the just notes and have one metastable interval, you can
harmonize justly with the metastable interval, or you can harmonize in
this way and stretch the whole chord out, which distributes the
metastability of the original interval out over the whole chord. There
are a two different ways to do this - the 1002 seventh is between 7/4
and 9/5 for example, and you can distribute the chord so that the
result becomes a stretched 4:5:6:7, or a compressed 20:25:30:36, not
to mention all of the intermediant mediants (hmm) used to calculate
the noble mediant.

2. Taking the 4:5.045:6.091:7.136 stretched 4:5:6:7 might work even
better as a dominant 7 chord than just using 4:5:6:7.136, as each of
the notes in the chord is going to be slightly more dissonant and
contain slightly more harmonic entropy than if it were just a
4:5:6:7.136 chord, thus making the tendency to resolve even stronger.
Note that the 5 being stretched to 5.045 is around 401.825 cents,
which would agree with the observations of many people that a slightly
sharpened leading tone is often more effective in resolving to 1. And
the 6.091 is 26 full cents sharp of a perfect fifth, but it blends in
perfectly anyway - for a sine wave-like tone, I have to listen really
hard to tell that it isn't a perfect fifth, as the resonance of the
chord is that strong.

3. Since in this chord, you still hear a 4:5:6:7 in this kind of
chord, it's possible that when you build chords in general what you
might be hearing that is so strongly associated with the "character"
of a chord is that difference tones reinforce one another in a pattern
that you remember as well as whether or not they reinforce the
partials of each tone. So what this means is that if you have
4:5.045:6.091:7.136, you might be more likely to hear that as actually
BEING 4:5:6:7 than you are to hear that 4:7.136 as a metastable
dominant 7. So if you WANT that metastable dominant 7 over a JI major
chord, and the chord is stretched like this, you'll have to stretch
that 7.136 even FURTHER to get the equivalent "stretched" chord. What
this means is that there might be a general tendency for regular
"non-stretched" JI intervals to affect the perception of other JI
intervals in this way, so that the most "metastable" version of some
interval you want might actually change depending on the other notes
in the chord, especially if you use a pure timbre.

and,

> I think you will find that the effect you found will work equally with
> _any_ amount of stretch.

4. If you compress a major chord enough, you will eventually end up at
5:6:7 rather than 4:5:6. So there are places of ambiguity for whole
chords as well that play a similar role that the metastable intervals
themselves do. If you make an isoharmonic triad with the third between
4:5 and 5:6, and you make an isoharmonic triad with the fifth between
2:3 and 5:7, you end up with different chords. And if you stretch and
compress the chord enough, you'll eventually get at something that
sounds like a completely different chord. What's the cutoff? Is it the
metastable interval on the outer dyad, or the inner one, or do they
all affect each other in some fashion...?

> Experiments with very accurate sine tones through high quality
> headphones _quietly_, don't find anything special at any ratios.
> _Everything_ is concordant. But as soon as the slightest non-linearity
> appears in the system, e.g. less than perfect speakers or slightly
> louder tones causing the ear mechanism to go into a less linear
> region, then we start to get sum and difference tones being generated
> and will hear the coincidence of these as special.

Well, if you use a sine tone and you play 4:5.045:6:7.136, you'll hear
some pretty heavy beating. But if you throw in 4:5.045:6.091:7.136,
the beating will stop. Then again, I'm using a sound that's close but
not quite a sine wave, so that might have something to do with it -
although, using that approach, the 6 should sound more consonant as
the partials would collide, but I find the 4:5.045:6.091:7.136 sounds
better unless the sound is really complex.

-Mike

Difference tones are produced in the mind by tracking. if you put different pitches is different ears with headphones you can hear them even though they don't occur in the air.

Helmholtz did work on the inversions of triads which is explain with difference tones.
likewise i took all the four note combinations of the harmonics up to 11 and the inversions where none exceeded an octave, calculated this according to
difference tones and have had it up here for years of which one has only to play through it to see that indeed it does have something to do with it.
http://anaphoria.com/lullaby.html

then you can tune up any of the first set of meru scales.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
>
>
> Nothing to do with difference tones could explain the
> disappearance of beating that Mike was reporting. Meanwhile,
> difference tones ALMOST NEVER have a significant effect
> on the con/dissonance of chords in musical contexts.
> The simple reason is that their amplitudes are simply
> too low.
>
> -Carl
>
>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> Hi Mike,
>
> Well figured out. I think you're right, that it has everything to do
> with coinciding difference tones. I don't think it has anything to
> do with metastability.

I don't think it has much to do with either.

> The fact that you are using sine tones is the giveaway.

It would increase the likelihood it has anything to
do with difference tones.

> Metastability
> should only really be relevant to tones rich in harmonics.

I strongly disagree, Dave. Why do you say this?

> Experiments with very accurate sine tones through high quality
> headphones _quietly_, don't find anything special at any ratios.

What about Vos' experiments? The virtual pitch system
still works on sine tones.

> _Everything_ is concordant.

?

-Carl

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
> Difference tones are produced in the mind by tracking. if you put
> different pitches is different ears with headphones you can hear
> them even though they don't occur in the air.

They're produced by nonlinear effects in the ear. You may be
thinking of virtual pitch or binaural beats. The wikipedia
page on combination tones mentions a binaural experiment, but
is lacking a citation.

-Carl

On Fri, Jun 20, 2008 at 10:07 PM, Carl Lumma <carl@...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>>
>> Hi Mike,
>>
>> Well figured out. I think you're right, that it has everything to do
>> with coinciding difference tones. I don't think it has anything to
>> do with metastability.
>
> I don't think it has much to do with either.

Why not? It's an isoharmonic chord. Furthermore, it's just a stretched
4:5:6:7. If you flatten 4:5:6 enough, you get to 5:6:7. The chord will
stay resonant the whole time. It doesn't seem like anything else could
be the culprit except for difference tones.

-Mike

> > I don't think it has much to do with either.
>
> Why not? It's an isoharmonic chord.

What does that mean?

> Furthermore, it's just a stretched
> 4:5:6:7. If you flatten 4:5:6 enough, you get to 5:6:7. The chord will
> stay resonant the whole time. It doesn't seem like anything else could
> be the culprit except for difference tones.

If you stretch or compress the tones of any just chord,
you will keep the inner dyads of the chord just, so the
increase in dissonance will be smaller than with some
other kinds of perturbations. If you start with a
tempered chord whose inner dyads are already off and
improve some of them at the cost of others -- as you did --
and wind up with two chords with approximately the
same level of dissonance -- as you did -- what wants to
be explained?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> > Metastability
> > should only really be relevant to tones rich in harmonics.
>
> I strongly disagree, Dave. Why do you say this?

Because it's about maximaly non-coinciding partials, at least that's
what the noble numbers are about. If there aren't any partials what
difference could their coincidence or non-coincidence make?

> > Experiments with very accurate sine tones through high quality
> > headphones _quietly_, don't find anything special at any ratios.
>
> What about Vos' experiments? The virtual pitch system
> still works on sine tones.

Isn't that just about whether it is heard as one tone or two? Yes. I
guess that's something special, so I shouldn't have said that. But
surely there is no beating or roughness for any interval when using
accurate enough, and quiet enough, sine tones?

Kraig reminded us that difference tones can be produced in the brain,
not just the ear, since feeding one frequency to each ear still
elicits them. Does anyone know if that depends strongly on the volume
level? I would still expect it to.

-- Dave Keenan

> > Why not? It's an isoharmonic chord.
>
> What does that mean?

It means that the difference between successive tones is the same.
4:5:6:7 itself is an isoharmonic chord, and the difference between
each note in the chord is "1." That being the case helps to form the
extremely resonant "bottom" that you hear in a major chord.
4:5.045:6.091:7.136 is also isoharmonic, as each tone has a difference
of 1.0454 or so (rounded off). This means that the difference tone
formed on the bottom will be reinforced by three separate dyads. This
makes for a pretty resonant chord, with little to no beating, as
there's really nothing to beat. Stretching a chord like 10:12:15 means
the difference tones will be in a 2:3 relationship with one another,
and that will be stretched as well as the chord, but if you're using
simple sine patch or something that has stretched partials itself, it
won't be so noticeable.

> > Furthermore, it's just a stretched
> > 4:5:6:7. If you flatten 4:5:6 enough, you get to 5:6:7. The chord will
> > stay resonant the whole time. It doesn't seem like anything else could
> > be the culprit except for difference tones.
>
> If you stretch or compress the tones of any just chord,
> you will keep the inner dyads of the chord just, so the
> increase in dissonance will be smaller than with some
> other kinds of perturbations. If you start with a
> tempered chord whose inner dyads are already off and
> improve some of them at the cost of others -- as you did --
> and wind up with two chords with approximately the
> same level of dissonance -- as you did -- what wants to
> be explained?

Read further into the thread. If you start with 4:5:6:7 and stretch
that major chord until the 967 cent 7 reaches the 1002 noble 7, then
you get the same effect.

-Mike

it would have to be binaural since the pitches never meet in air to cause any effect in the ear.
It is mentioned in Roderer's book.
regardless of how, we do hear them

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> > Difference tones are produced in the mind by tracking. if you put
> > different pitches is different ears with headphones you can hear
> > them even though they don't occur in the air.
>
> They're produced by nonlinear effects in the ear. You may be
> thinking of virtual pitch or binaural beats. The wikipedia
> page on combination tones mentions a binaural experiment, but
> is lacking a citation.
>
> -Carl
>
>

This is what the scales of Mt Meru do ( except the difference tones will be other notes in the scale which may or may not be sounded). thereby making them acoustically self referential. Pretty much the example i have been using to point out that dissonant/consonant can not be determined by the nature of dyads.
hence why i state that con/dis is determined by 'acoustical coincidence'

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Mike Battaglia wrote:
>
> > > Why not? It's an isoharmonic chord.
> >
> > What does that mean?
>
> It means that the difference between successive tones is the same.
> 4:5:6:7 itself is an isoharmonic chord, and the difference between
> each note in the chord is "1." That being the case helps to form the
> extremely resonant "bottom" that you hear in a major chord.
> 4:5.045:6.091:7.136 is also isoharmonic, as each tone has a difference
> of 1.0454 or so (rounded off). This means that the difference tone
> formed on the bottom will be reinforced by three separate dyads. This
> makes for a pretty resonant chord, with little to no beating, as
> there's really nothing to beat. Stretching a chord like 10:12:15 means
> the difference tones will be in a 2:3 relationship with one another,
> and that will be stretched as well as the chord, but if you're using
> simple sine patch or something that has stretched partials itself, it
> won't be so noticeable.
>
> > > Furthermore, it's just a stretched
> > > 4:5:6:7. If you flatten 4:5:6 enough, you get to 5:6:7. The chord will
> > > stay resonant the whole time. It doesn't seem like anything else could
> > > be the culprit except for difference tones.
> >
> > If you stretch or compress the tones of any just chord,
> > you will keep the inner dyads of the chord just, so the
> > increase in dissonance will be smaller than with some
> > other kinds of perturbations. If you start with a
> > tempered chord whose inner dyads are already off and
> > improve some of them at the cost of others -- as you did --
> > and wind up with two chords with approximately the
> > same level of dissonance -- as you did -- what wants to
> > be explained?
>
> Read further into the thread. If you start with 4:5:6:7 and stretch
> that major chord until the 967 cent 7 reaches the 1002 noble 7, then
> you get the same effect.
>
> -Mike
>
>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
> > > Metastability
> > > should only really be relevant to tones rich in harmonics.
> >
> > I strongly disagree, Dave. Why do you say this?
>
> Because it's about maximaly non-coinciding partials, at least
> that's what the noble numbers are about. If there aren't any
> partials what difference could their coincidence or
> non-coincidence make?

You seem to be ignoring virtual pitch, which is more import
than roughness in musical consonance.

> > > Experiments with very accurate sine tones through high
> > > quality headphones _quietly_, don't find anything special
> > > at any ratios.
> >
> > What about Vos' experiments? The virtual pitch system
> > still works on sine tones.
>
> Isn't that just about whether it is heard as one tone or two?

That's part of it.

The ability to recognize formant filtering of a single
source as such is at the heart of our ability to understand
speech. Western harmony, as exemplified in evangelical
hymnody, is based on the exploitation of this mechanism.
I use the term "virtual pitch mechanism" to refer to this,
for want of a better term.

Vos found that listeners rated the "purity" of just
intervals higher than others, even with sine tones. We
can hypothesize that such an effect can only be more
pronounced with larger chords, as this becomes
indistinguishable from additive synthesis as more tones
are added.

> Yes. I
> guess that's something special, so I shouldn't have said that.
> But surely there is no beating or roughness for any interval
> when using accurate enough, and quiet enough, sine tones?

As long as you stay out of the critical band!

> Kraig reminded us that difference tones can be produced in the
> brain, not just the ear, since feeding one frequency to each
> ear still elicits them.

It's possible and I'm interested in a citation. But this
is neither exclusive of the per-ear intermodulation explanation
nor necessarily a mechanism by which difference tones take on
greater significance in music.

-Carl

> > > Why not? It's an isoharmonic chord.
> >
> > What does that mean?
>
> It means that the difference between successive tones is the same.
> 4:5:6:7 itself is an isoharmonic chord, and the difference between
> each note in the chord is "1." That being the case helps to form the
> extremely resonant "bottom" that you hear in a major chord.
> 4:5.045:6.091:7.136 is also isoharmonic, as each tone has a
> difference of 1.0454 or so (rounded off). This means that the
> difference tone formed on the bottom will be reinforced by three
> separate dyads. This makes for a pretty resonant chord, with
> little to no beating, as there's really nothing to beat.

Mike, this chord beats like hell.

-Carl

>> It means that the difference between successive tones is the same.
>> 4:5:6:7 itself is an isoharmonic chord, and the difference between
>> each note in the chord is "1." That being the case helps to form the
>> extremely resonant "bottom" that you hear in a major chord.
>> 4:5.045:6.091:7.136 is also isoharmonic, as each tone has a
>> difference of 1.0454 or so (rounded off). This means that the
>> difference tone formed on the bottom will be reinforced by three
>> separate dyads. This makes for a pretty resonant chord, with
>> little to no beating, as there's really nothing to beat.
>
> Mike, this chord beats like hell.
>
> -Carl

Are you using a sine wave? I hear little to no beating on my end.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Cameron wrote:
> >
> > But we're not actually talking about consonance and dissonance
per
> > se, rather some kind of unity. I'm using pretty gnarly timbres
> > experimenting with these things by the way.
>
> Define your terms. Mike was talking about beating, which
> has a widely-accepted precise definition. I additionally
> brought "consonance" into play, and as I have defined it
> many times, it includes beating/roughness AND a "unity" or
> "cohesiveness" or "rootedness" or "tonalness" component.
>
> -Carl
>

Some crossed wires here I guess- don't figured out what Mike is doing
here. As I mentioned before I also use very rich timbres so obviously
whatever I'm hearing- even if it imaginary- doesn't depend on using
sines. In fact I'd say that being tangible using all kinds of
timbres, even only roughly harmonic ones, is a requirement. JI
works that way and I think any kinds of radical extensions to JI or
shadows of JI or whatever should have to, too.

> Are you using a sine wave? I hear little to no beating on my end.

I'm not, but if I was, the 12-ET version of the chord
wouldn't beat either.

-Carl

> I'm not, but if I was, the 12-ET version of the chord
> wouldn't beat either.
>
> -Carl

> Well, if you use a sine tone and you play 4:5.045:6:7.136, you'll hear
> some pretty heavy beating. But if you throw in 4:5.045:6.091:7.136,
> the beating will stop.

[ Attachment content not displayed ]

Ah, sorry about the attachment in my last post - Gmail for some reason
sent it as an "invitation" to something. Not sure what that means.

Alright, my last post went straight to hell, so I'm posting it again:

Cameron wrote:
> Some crossed wires here I guess- don't figured out what Mike is doing
> here. As I mentioned before I also use very rich timbres so obviously
> whatever I'm hearing- even if it imaginary- doesn't depend on using
> sines. In fact I'd say that being tangible using all kinds of
> timbres, even only roughly harmonic ones, is a requirement. JI
> works that way and I think any kinds of radical extensions to JI or
> shadows of JI or whatever should have to, too.

A logical question here is, when we're talking about the "fields of
attraction" that these noble mediants fall between, are we talking
about the fields of attraction between two intervals in the just
harmonic series, or rather should we be talking about the field of
attraction between two intervals in the harmonic series of a timbre? I
am inclined to believe the latter, though perhaps raw frequencies
might play into it a bit, but I would expect that effect to be most
likely if there is some nonlinear effect such as distortion on a
timbre.

Actually I'm pretty sure it's the timbre, and I'm going to post some
examples about it shortly.

So what I'm starting to realize about the requirement that tuning be
independent of timbre is that it's a false requirement. The reason JI
works is because MOST timbres are harmonic, and if they're slightly
inharmonic then we're already so used to hearing that in each
individual note that we still hear chords out of tune. Of course, if
you mic a piano and run it through a distortion effect, even single
notes will sound "out of tune" due to that effect, but for the most
part, chords in real life sound fine.

But even more so, if a timbre is inharmonic then stretched octave
tunings that are stretched to match that inharmonicity will sound
"more in tune" than using normal JI. This is why the tuning is
stretched on an acoustic piano, like we were talking about in the
other thread.

Either way, trying to separate the timbre from the tuning and coming
up with a tuning that will work with all timbres seems like a good
idea, but I'm not sure it's going to work out all the way. Didn't
someone on here mention someone who used timbres with the partials
tuned to 10-tet so that 10-tet was in tune? That's sort of what I'm
getting at.

Carl wrote:

> Mike, this chord beats like hell.

Carl, specifically for your request, I posted the link to my short recording yesterday and I explained how one of the pitches was changing gradually. Are you telling me you don't hear the beats becoming slower and slower until they stop?

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
> Carl, specifically for your request, I posted the link to my
> short recording yesterday

That was still today for me. :) Part of the problem is that
I can't easily listen to audio at work, or before the kids go
to bed, and then after that I'm basically restricted to
headphones. But I did just reply.

-Carl

Carl wrote:

> What's the registration of this chord?

All of the tones are pure sines.

> Did you intend
> the left channel to be much louder than the right?

The mp3 stream was encoded in mono format so there should be nothing like that.

> Through headphones I hear a beat that slows and stops,
> presumably due to critical band effects between combination
> tones and original tones. When I play the left channel only
> at low volume through a single speaker, the beating is
> almost inaudible, even at the beginning of the file.

That's interesting. I hear it more or less the same whether I'm listening through headphones or loudspeakers. Maybe I should also suggest that the effect in question is more pronounced when you play the sound faster, like 150% or 200% of the original speed, which will raise the pitches.

Petr

I did my own listening example too. I purposely made it clip a little
bit to add some intermodulation distortion and emphasize the
difference tones.

http://www.box.net/shared/ncj8yacook

The chords are, in this order:

4:5:6:7
4:5:6.091:7.136
4:5.05:6:7.136
4:5.05:6.091:7.136
4:5:6:7

You'll note the fourth chord has almost no beating even though the
major third is 17 cents sharp, the perfect fifth is 26 cents sharp,
and the perfect seventh is 34 cents sharp. Then I go back to 4:5:6:7
just to compare at the end.

-Mike

i like the sequence and could imagine a really quick successions of just such chords as a piece all in itself.

Can you please explain again and give the formula exactly to make sure i am understanding what is going on correctly.
it seems like expanding the octave out and keeping the tetrad in the same proportion.
The next question would be octoad up to 11and how far one could stretch it. at a certain point it will approach the size of lower harmonics.

There is a whole set of formulas Erv worked on with these type of continuums with or informed by Hanson, who might have given Erv the G formulas for the Mt Meru scales.
i will try to get these up. the math is beyond me which why i haven't put it up along with it being replaced by the Meru scales

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Mike Battaglia wrote:
>
> I did my own listening example too. I purposely made it clip a little
> bit to add some intermodulation distortion and emphasize the
> difference tones.
>
> http://www.box.net/shared/ncj8yacook > <http://www.box.net/shared/ncj8yacook>
>
> The chords are, in this order:
>
> 4:5:6:7
> 4:5:6.091:7.136
> 4:5.05:6:7.136
> 4:5.05:6.091:7.136
> 4:5:6:7
>
> You'll note the fourth chord has almost no beating even though the
> major third is 17 cents sharp, the perfect fifth is 26 cents sharp,
> and the perfect seventh is 34 cents sharp. Then I go back to 4:5:6:7
> just to compare at the end.
>
> -Mike
>
>

Here's the extended 4:5:6:7:8:9:10:11 chord, then the normal one after it.

4:5.045:6.091:7.136:8.181:9.227:10.272:11.317 is what it works out to.

Cents wise you have 402-728-1002-1238-1447-1633-1801. The 5/4 is 16
cents sharp, 3/2 is 26 cents sharp, 7/4 34 cents sharp, the octave is
39 cents sharp, 9/4 is 44 cents sharp, 5/2 is 46 cents sharp, and 11/4
is a full 49 cents sharp.

http://www.box.net/shared/5zqasp60og

What needs to be invented is a musical instrument in which the
inharmonicity can be adjusted and the partials of the timbre itself
stretched so that the nature of JI in that instrument itself changes a
bit.

-Mike

On Sat, Jun 21, 2008 at 8:48 AM, Kraig Grady <kraiggrady@...> wrote:
> i like the sequence and could imagine a really quick successions of just
> such chords as a piece all in itself.
>
> Can you please explain again and give the formula exactly to make sure i
> am understanding what is going on correctly.
> it seems like expanding the octave out and keeping the tetrad in the
> same proportion.
> The next question would be octoad up to 11and how far one could stretch
> it. at a certain point it will approach the size of lower harmonics.
>
> There is a whole set of formulas Erv worked on with these type of
> continuums with or informed by Hanson, who might have given Erv the G
> formulas for the Mt Meru scales.
> i will try to get these up. the math is beyond me which why i haven't
> put it up along with it being replaced by the Meru scales
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> Mike Battaglia wrote:
>>
>> I did my own listening example too. I purposely made it clip a little
>> bit to add some intermodulation distortion and emphasize the
>> difference tones.
>>
>> http://www.box.net/shared/ncj8yacook
>> <http://www.box.net/shared/ncj8yacook>
>>
>> The chords are, in this order:
>>
>> 4:5:6:7
>> 4:5:6.091:7.136
>> 4:5.05:6:7.136
>> 4:5.05:6.091:7.136
>> 4:5:6:7
>>
>> You'll note the fourth chord has almost no beating even though the
>> major third is 17 cents sharp, the perfect fifth is 26 cents sharp,
>> and the perfect seventh is 34 cents sharp. Then I go back to 4:5:6:7
>> just to compare at the end.
>>
>> -Mike
>>
>>
>

Sorry, wrong link. Use this one instead.

http://www.box.net/shared/4jawte1cs8

On Sat, Jun 21, 2008 at 9:16 AM, Mike Battaglia <battaglia01@...> wrote:
> Here's the extended 4:5:6:7:8:9:10:11 chord, then the normal one after it.
>
> 4:5.045:6.091:7.136:8.181:9.227:10.272:11.317 is what it works out to.
>
> Cents wise you have 402-728-1002-1238-1447-1633-1801. The 5/4 is 16
> cents sharp, 3/2 is 26 cents sharp, 7/4 34 cents sharp, the octave is
> 39 cents sharp, 9/4 is 44 cents sharp, 5/2 is 46 cents sharp, and 11/4
> is a full 49 cents sharp.
>
> http://www.box.net/shared/5zqasp60og
>
> What needs to be invented is a musical instrument in which the
> inharmonicity can be adjusted and the partials of the timbre itself
> stretched so that the nature of JI in that instrument itself changes a
> bit.
>
> -Mike
>
> On Sat, Jun 21, 2008 at 8:48 AM, Kraig Grady <kraiggrady@...> wrote:
>> i like the sequence and could imagine a really quick successions of just
>> such chords as a piece all in itself.
>>
>> Can you please explain again and give the formula exactly to make sure i
>> am understanding what is going on correctly.
>> it seems like expanding the octave out and keeping the tetrad in the
>> same proportion.
>> The next question would be octoad up to 11and how far one could stretch
>> it. at a certain point it will approach the size of lower harmonics.
>>
>> There is a whole set of formulas Erv worked on with these type of
>> continuums with or informed by Hanson, who might have given Erv the G
>> formulas for the Mt Meru scales.
>> i will try to get these up. the math is beyond me which why i haven't
>> put it up along with it being replaced by the Meru scales
>>
>> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
>> Mesotonal Music from:
>> _'''''''_ ^North/Western Hemisphere:
>> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>>
>> _'''''''_ ^South/Eastern Hemisphere:
>> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>>
>> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>>
>> Mike Battaglia wrote:
>>>
>>> I did my own listening example too. I purposely made it clip a little
>>> bit to add some intermodulation distortion and emphasize the
>>> difference tones.
>>>
>>> http://www.box.net/shared/ncj8yacook
>>> <http://www.box.net/shared/ncj8yacook>
>>>
>>> The chords are, in this order:
>>>
>>> 4:5:6:7
>>> 4:5:6.091:7.136
>>> 4:5.05:6:7.136
>>> 4:5.05:6.091:7.136
>>> 4:5:6:7
>>>
>>> You'll note the fourth chord has almost no beating even though the
>>> major third is 17 cents sharp, the perfect fifth is 26 cents sharp,
>>> and the perfect seventh is 34 cents sharp. Then I go back to 4:5:6:7
>>> just to compare at the end.
>>>
>>> -Mike
>>>
>>>
>>
>

Here's an even further extended chord where 11 gets extended out to 12
and all of the harmonics inside stretched accordingly. This is quite
an interesting chord, because it almost sounds like an augmented major
7 chord, but it's as resonant as a 4:5:6:7:8:9:10:11 chord. I don't
know what to make of it.

What if we retuned the partials of instruments to 12-equal? Or
19-equal? Would it then sound amazing? Hmmm.

4:5.143:6.286:7.429:8.571:9.714:10.857:12

0-435-783-1072-1319-1536-1729-1902

http://www.box.net/shared/qlz13h7sok

-Mike

And by far the most interesting one of them all:

I took the 4:5:6:7:8:9:10:11 and compressed it so the outer dyad is
8/3. Here are the values:

4:4.953:5.906:6.858:7.811:8.764:9.717:10.670
0-370-675-933-1159-1358-1537-1699

http://www.box.net/shared/m12vhy0e88

So I still don't know why C-E-G-Bb-D-F on a piano is such a weak sounding chord.

This is a great example of what I'm talking about - that top note
sounds to me just like 11/4, even though it's supposedly going to get
caught in the huge field of attraction of 8/3. Although, if I focus on
just the outer dyad, I can hear it as a C-F 8/3. But, between the
pseudo-8 and the pseudo-11 you have a 540 cent interval, which is
closer to 11/8 than 4/3.

So here's an even more shrunk one:

4:4.9:5.8:6.7:7.6:8.5:9.4:10.3
0-351-643-893-1111-1305-1480-1637

The outer dyad is flatter than 8/3 and yet it still sounds like 11/8.
The dyad between the pseudo-8 and the pseudo-11 is 526 cents, which is
comfortably within 4/3 range.

It's interesting, once you hear it, and then once you hear the normal
major, it's like you're suddenly reminded of what major sounds like
and then you're tempted to hear the first chord like it's closer to
5:6:7:8:9:10:11:12 transposed down.

Note that with this last one we're getting pretty close to 9:11:13:15:17:19:21.

http://www.box.net/shared/hitr8ank04

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
> Carl wrote:
>
> > What's the registration of this chord?
>
> All of the tones are pure sines.

What is the registration of the tones? In what octaves
are they sounding?

> > Did you intend
> > the left channel to be much louder than the right?
>
> The mp3 stream was encoded in mono format so there should
> be nothing like that.

Heh, it was channel leakage in my headphones. My right
speaker is silent. Looks like my setup is sending mono
to left instead of mixing it out.

> > Through headphones I hear a beat that slows and stops,
> > presumably due to critical band effects between combination
> > tones and original tones. When I play the left channel only
> > at low volume through a single speaker, the beating is
> > almost inaudible, even at the beginning of the file.
>
> That's interesting. I hear it more or less the same whether
> I'm listening through headphones or loudspeakers. Maybe I
> should also suggest that the effect in question is more
> pronounced when you play the sound faster, like 150% or 200%
> of the original speed, which will raise the pitches.

I was going to suggest a higher registration. Headphones
can accentuate combination tones for some reason, probably
because they fire directly into the ear.

-Carl

Mike wrote:
> What needs to be invented is a musical instrument in which the
> inharmonicity can be adjusted and the partials of the timbre itself
> stretched so that the nature of JI in that instrument itself
> changes a bit.

I've gotta host my kid's birthday party today, so I'm going to
get behind. But you know Bill Sethares has set up instruments
to do this, right? And you've seen...

http://youtube.com/watch?v=Nd4h8vmEsQM

And you know that Hammond organs (in good condition) have
12-ET partials, right? Which is why many chords on them
are beatless. Of course, with great irony, we then throw
every kind of tremolo, harmonic amp distortion, and f***
rotating speakers on top and completely ruin the effect. :)

-Carl

Carl wrote:

> What is the registration of the tones? In what octaves
> are they sounding?

The lowest tone is 250Hz and the other three make the intervals as I said in the post with the link -- i.e. as long as I také all pitches related to the lowest one and round them to nearest cents.

Petr

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> I did my own listening example too. I purposely made it clip a little
> bit to add some intermodulation distortion and emphasize the
> difference tones.
>
> http://www.box.net/shared/ncj8yacook
>
> The chords are, in this order:
>
> 4:5:6:7
> 4:5:6.091:7.136
> 4:5.05:6:7.136
> 4:5.05:6.091:7.136
> 4:5:6:7
>
> You'll note the fourth chord has almost no beating even though the
> major third is 17 cents sharp, the perfect fifth is 26 cents sharp,
> and the perfect seventh is 34 cents sharp. Then I go back to 4:5:6:7
> just to compare at the end.
>
> -Mike

I don't understand why you think this has anything to do with
metastable intervals. _Any_ such chord made up of sine tones where the
intermodulation products (sum and difference tones) coincide (or are
sufficently far apart), will be beatless. As you said, you could
stretch 4:5:6:7 by _any_ amount. And, as you suggested, you could even
give the tones inharmonic partials, a la Sethares, that coincide with
the intermodulation products, and they would remain beatless (but it
would all sound like bells).

Yes, you can use these methods to make metastable intervals consonant.
You can use them to make _any_ intervals consonant.

I feel that the point of giving metastable intervals a name, is to
recognise and use them for what they are, harmonics and all, e.g. "the
worstest of the worst, but somehow with divinity imbued" (Erv Wilson
regarding the phi interval).

-- Dave Keenan

Petr wrote:

>>> Okay. In this example, the starting four-voiced chord is made
>>> of cent sizes (counting from the bottom tone and rounding to
>>> the nearest cent) of 401, 702, and 1000 cents. One of the tones
>>> is rising gradually and the other three have steady pitches.
>>> Because of this, the final intervals of the chord are 401, 726,
>>> and 1000 cents. Here is the link:
>>> https://download.yousendit.com/7905F2A906165980
>>
>> What is the registration of the tones? In what octaves
>> are they sounding?
>
> The lowest tone is 250Hz and the other three make the intervals
> as I said in the post with the link -- i.e. as long as I take
> all pitches related to the lowest one and round them to nearest
> cents.
>
> Petr

Thanks. So the 1st chord is:

250.0 Hz.
* 1.2606 = 315.15
* 1.5000 = 375.0
* 1.7818 = 445.45

So the 1st-order difference tones are:

65.15 125.0 195.45
70.45 59.85
130.3 Hz.

In the 2nd chord, 375.0 Hz. is replaced by 380.24 Hz., making
the following 1st-order difference tones:

65.15 130.24 195.45
65.20 65.10
130.3 Hz.

So yup, that's the explanation alright. If you use sine
tones to make music, you've got to be aware of this stuff.
If you don't, you can ignore it.

-Carl

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> I did my own listening example too. I purposely made it clip a little
> bit to add some intermodulation distortion and emphasize the
> difference tones.
>
> http://www.box.net/shared/ncj8yacook
>
> The chords are, in this order:
>
> 4:5:6:7
> 4:5:6.091:7.136
> 4:5.05:6:7.136
> 4:5.05:6.091:7.136
> 4:5:6:7
>
> You'll note the fourth chord has almost no beating even though the
> major third is 17 cents sharp, the perfect fifth is 26 cents sharp,
> and the perfect seventh is 34 cents sharp. Then I go back to 4:5:6:7
> just to compare at the end.

There are six chords in the file, but only five shown above.

-Carl

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
> A logical question here is, when we're talking about the "fields of
> attraction" that these noble mediants fall between, are we talking
> about the fields of attraction between two intervals in the just
> harmonic series, or rather should we be talking about the field of
> attraction between two intervals in the harmonic series of a timbre?

Of course they are usually the same, or approximately so. But when you
start using metallophones or synthesising inharmonic tibres a la
Sethares, then it may be best to think in terms of having shifted both
the Just intervals and the metastable intervals between them. So the
Just intervals are no longer at simple rationals, nor the metastables
at simple nobles. However, when this is done, some say there are also
still "ghost" versions of the original simple rational Just intervals,
irrespective of the inharmonic partials.

> I
> am inclined to believe the latter, though perhaps raw frequencies
> might play into it a bit,

Yes. It seems to be a bit of both.

> but I would expect that effect to be most
> likely if there is some nonlinear effect such as distortion on a
> timbre.

Possibly so.

> Actually I'm pretty sure it's the timbre, and I'm going to post some
> examples about it shortly.
>
> So what I'm starting to realize about the requirement that tuning be
> independent of timbre is that it's a false requirement. The reason JI
> works is because MOST timbres are harmonic, and if they're slightly
> inharmonic then we're already so used to hearing that in each
> individual note that we still hear chords out of tune. Of course, if
> you mic a piano and run it through a distortion effect, even single
> notes will sound "out of tune" due to that effect, but for the most
> part, chords in real life sound fine.
>
> But even more so, if a timbre is inharmonic then stretched octave
> tunings that are stretched to match that inharmonicity will sound
> "more in tune" than using normal JI. This is why the tuning is
> stretched on an acoustic piano, like we were talking about in the
> other thread.

Sure. But the other thing that happens is, the more inharmonic each
tone is, the less it has a well defined pitch. A tone consisting of
only noble partials might not have any "pitch" at all.

> Either way, trying to separate the timbre from the tuning and coming
> up with a tuning that will work with all timbres seems like a good
> idea, but I'm not sure it's going to work out all the way. Didn't
> someone on here mention someone who used timbres with the partials
> tuned to 10-tet so that 10-tet was in tune? That's sort of what I'm
> getting at.

You're quite right. See William Sethares, 'Tuning Timbre Spectrum Scale'.
http://eceserv0.ece.wisc.edu/~sethares/ttss.html

-- Dave Keenan

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
> What if we retuned the partials of instruments to 12-equal?

That's essentially what happens in the celebrated Hammond Organ.

Search on "Hammond" in this Yahoo group's archive.

-- Dave Keenan

As far as i can evaluate the situation. Mike is illustrating the con/dis is determined by 'acoustical coincidence'.
Surprise, Surprise
guess we will have to throw a bunch a theories in the trash. ( dyads to the front of the line please! )

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Dave Keenan wrote:
>
>
>
> I don't understand why you think this has anything to do with
> metastable intervals. _Any_ such chord made up of sine tones where the
> intermodulation products (sum and difference tones) coincide (or are
> sufficently far apart), will be beatless. As you said, you could
> stretch 4:5:6:7 by _any_ amount. And, as you suggested, you could even
> give the tones inharmonic partials, a la Sethares, that coincide with
> the intermodulation products, and they would remain beatless (but it
> would all sound like bells).
>
> Yes, you can use these methods to make metastable intervals consonant.
> You can use them to make _any_ intervals consonant.
>
> I feel that the point of giving metastable intervals a name, is to
> recognise and use them for what they are, harmonics and all, e.g. "the
> worstest of the worst, but somehow with divinity imbued" (Erv Wilson
> regarding the phi interval).
>
> -- Dave Keenan
>
>

Mike wrote:

> Sorry, wrong link. Use this one instead.
>
> http://www.box.net/shared/4jawte1cs8
>
//
> > Here's the extended 4:5:6:7:8:9:10:11 chord, then the normal
> > one after it.
> >
> > 4:5.045:6.091:7.136:8.181:9.227:10.272:11.317 is what it
> > works out to.
> >
> > Cents wise you have 402-728-1002-1238-1447-1633-1801.

Here's my version:

http://lumma.org/stuff/comparison.wav

-Carl

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> As far as i can evaluate the situation. Mike is illustrating
> the con/dis is determined by 'acoustical coincidence'.

Nope. It's just a lack of beating in mistuned otonalities
via atypical timbres.

> Surprise, Surprise
> guess we will have to throw a bunch a theories in the trash.

Hardly.

-Carl

yours is funnier Carl:)!

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> Mike wrote:
>
> > Sorry, wrong link. Use this one instead.
> >
> > http://www.box.net/shared/4jawte1cs8 > <http://www.box.net/shared/4jawte1cs8>
> >
> //
> > > Here's the extended 4:5:6:7:8:9:10:11 chord, then the normal
> > > one after it.
> > >
> > > 4:5.045:6.091:7.136:8.181:9.227:10.272:11.317 is what it
> > > works out to.
> > >
> > > Cents wise you have 402-728-1002-1238-1447-1633-1801.
>
> Here's my version:
>
> http://lumma.org/stuff/comparison.wav > <http://lumma.org/stuff/comparison.wav>
>
> -Carl
>
>

If it doesn't sound dissonant regardless of the reasons, then i am afraid it isn't.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > As far as i can evaluate the situation. Mike is illustrating
> > the con/dis is determined by 'acoustical coincidence'.
>
> Nope. It's just a lack of beating in mistuned otonalities
> via atypical timbres.
>
> > Surprise, Surprise
> > guess we will have to throw a bunch a theories in the trash.
>
> Hardly.
>
> -Carl
>
>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> As far as i can evaluate the situation. Mike is illustrating the
con/dis
> is determined by 'acoustical coincidence'.
> Surprise, Surprise
> guess we will have to throw a bunch a theories in the trash. (
dyads to
> the front of the line please! )

Huh? We've known about this since at least Helmholtz. And why would we
throw theories of dyads into the trash. We just need some good
theories of triads and higher, in addition.

Yes, for those people whose thing is sine waves or metallophones, we
probably need to take more notice of sum and difference tones.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> If it doesn't sound dissonant regardless of the reasons, then i am
> afraid it isn't.

Yup, and that agrees with my theory perfectly so far. I'm not
sure what your theory is, but if it's based on difference tones
lining up it's going to fail in a great many cases. Such as
the one I just posted. Glad you found it funny though.

-Carl

yes triads is where we should start.
i work as much with reed organs and square waves ( i am not a big fan of sines) and i haven't noticed that much difference.
dyads lead us down the wrong path say in comparing a 6-7-8 with a 43-50-57

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Dave Keenan wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > As far as i can evaluate the situation. Mike is illustrating the
> con/dis
> > is determined by 'acoustical coincidence'.
> > Surprise, Surprise
> > guess we will have to throw a bunch a theories in the trash. (
> dyads to
> > the front of the line please! )
>
> Huh? We've known about this since at least Helmholtz. And why would we
> throw theories of dyads into the trash. We just need some good
> theories of triads and higher, in addition.
>
> Yes, for those people whose thing is sine waves or metallophones, we
> probably need to take more notice of sum and difference tones.
>
> -- Dave Keenan
>
>

i just posted one in response
43-50-57 compared to 6-7-8

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > If it doesn't sound dissonant regardless of the reasons, then i am
> > afraid it isn't.
>
> Yup, and that agrees with my theory perfectly so far. I'm not
> sure what your theory is, but if it's based on difference tones
> lining up it's going to fail in a great many cases. Such as
> the one I just posted. Glad you found it funny though.
>
> -Carl
>
>

Kraig wrote...

> i just posted one in response
> 43-50-57 compared to 6-7-8

Sorry I didn't see it. It's been very hard to keep up
with message volume lately.

I just tried the two chords, and 6-7-8 is more consonant
and 43-50-57 does beat. But 43-50-57 ain't a bad chord.

I then quantized it into random ETs including 13, 14, 15,
17, 18, 19, 22, 23, and 24-ET. Only 17 and 22 came close
to being as consonant. Clearly 17 was less consonant, but
22 was very close. I would say your chord was slightly
cleaner though.

And lo and behold, when I measure the RMS error from
6-7-8 of 43-50-57 and then 22-ET, the former wins 7.43
to 7.82. Believe it or not I wrote this message as I was
this, so it's a pseudo-blind test. And the standard
party line correctly explains everything yet again.

-Carl

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> yes triads is where we should start.
> i work as much with reed organs and square waves (i am not
> a big fan of sines) and i haven't noticed that much difference.

Kraig- a well-built metallophone is one of the best acoustic
sources of sines (there's a strong fundamental, reinforced by
the resonators, and some relatively low-amplitude non-harmonic
partials that quickly decay). Those transients go a long way
to making them more musical than plain synthesize sines though.
And a good set of resonators will of course sound better in
person than speakers!

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Kraig wrote...
>
> > i just posted one in response
> > 43-50-57 compared to 6-7-8
>
> Sorry I didn't see it. It's been very hard to keep up
> with message volume lately.
>
> I just tried the two chords, and 6-7-8 is more consonant
> and 43-50-57 does beat. But 43-50-57 ain't a bad chord.
>
> I then quantized it into random ETs including 13, 14, 15,
> 17, 18, 19, 22, 23, and 24-ET. Only 17 and 22 came close
> to being as consonant. Clearly 17 was less consonant, but
> 22 was very close. I would say your chord was slightly
> cleaner though.
>
> And lo and behold, when I measure the RMS error from
> 6-7-8 of 43-50-57 and then 22-ET, the former wins 7.43
> to 7.82.

Sorry, it's actually 7.something to 9.something, or about
2 averaged cents difference. 0.4 cents probably would be
too hard for me to hear.

-Carl

> I don't understand why you think this has anything to do with
> metastable intervals. _Any_ such chord made up of sine tones where the
> intermodulation products (sum and difference tones) coincide (or are
> sufficently far apart), will be beatless. As you said, you could
> stretch 4:5:6:7 by _any_ amount. And, as you suggested, you could even
> give the tones inharmonic partials, a la Sethares, that coincide with
> the intermodulation products, and they would remain beatless (but it
> would all sound like bells).

I cane up with this specific chord by stretching the chord by just
enough so that the 7 becomes a 1002 cent noble 7. And rather than
leave the 5 and 6 in place, I stretched those notes out as well so
that the whole chord sounds like 4:5:6:7. And the chord has some
interesting properties that reflect back on metastable intervals:

1. Metastable intervals, to be REALLY metastable, might be best
calculated between two ratios of the harmonic series of the timbre
being used, rather than a perfect harmonic series.

2. There might be an interesting musical function for the chord
4:5.045:6.091:7.136. Does that dominant 7 chord resolve more strongly
to 1 than 4:5:6:7.136? If you compare the two chords even with a
complex harmonic timbre, the stretched chord has an interesting
resonant sound to it while still maintaining the "metastable" warbly
kind of character to it. Here's a sound clip of the two functioning as
V7 -> I: http://www.box.net/shared/3g39p7as8o

> Yes, you can use these methods to make metastable intervals consonant.
> You can use them to make _any_ intervals consonant.

So the next question is, what use do we have for metastable intervals
by using these methods to make them consonant? What if we have a whole
song in a sine wave tone, and suddenly we introduce a stretched chord
stretched just enough that one of the intervals becomes metastable?
Or, if we could dynamically adjust the timbre of a sound and stretch
7/4 so it becomes 1002 cents, what effect would that chord have as a V
chord in relation to a I chord in a normal harmonic series?

> I feel that the point of giving metastable intervals a name, is to
> recognise and use them for what they are, harmonics and all, e.g. "the
> worstest of the worst, but somehow with divinity imbued" (Erv Wilson
> regarding the phi interval).

If that's the main function of metastable intervals for you, then
check out the stretched V-I chord, as I feel it resolves more strongly
to I, although that V might be a little too high.

-Mike

On Sat, Jun 21, 2008 at 9:42 PM, Kraig Grady <kraiggrady@...> wrote:
> As far as i can evaluate the situation. Mike is illustrating the con/dis
> is determined by 'acoustical coincidence'.
> Surprise, Surprise
> guess we will have to throw a bunch a theories in the trash. ( dyads to
> the front of the line please! )

Yeah. So I'm not the only one who thought the current theory of dyads
fell a little short of the mark?

-Mike

> Yeah. So I'm not the only one who thought the current theory of dyads
> fell a little short of the mark?

What's the "theory of dyads" is and how does it fall short.
Bonus points if you can do it in 3K or less.

-Carl

On Sun, Jun 22, 2008 at 2:21 AM, Carl Lumma <carl@...> wrote:
>> Yeah. So I'm not the only one who thought the current theory of dyads
>> fell a little short of the mark?
>
> What's the "theory of dyads" is and how does it fall short.
> Bonus points if you can do it in 3K or less.

The approach where one treats each dyad as fitting into a field of
attraction independent of factors such as the other dyads in the
chord, the timbre of the instrument, what chords were played before
that chord, and all of that. All of the other factors I've been
mentioning since I've gotten here - even psychology plays a factor
into it - count towards what a person will consciously label an
interval or chord to be. The way they label that interval or chord
leads to expectations of how it functions and how it resolves and
such, and that as a whole determines what most people experience as
"chord quality." It was touched on recently under something like "the
sharp knife has a white handle" a while ago. The theory falls chord of
describing complex chords, and sometimes doesn't predict correctly
what a chord will be perceived as.

The two revelations I've gotten from this is that not only do the
other intervals in a chord alter the perception of one specific
interval, but the timbre of the instrument as well. Not only are you
less likely to hear an interval as a "mistuning" of some other
interval if the interval matches up with the timbre, you are also
extremely unlikely to hear an inharmonic timbre as being out of tune
in and of itself, except for maybe in an extremely low register. The
notion of dyadic interpretation that we've been discussing for a
while, in the form that it is in, certainly has something to do with
it, but it certainly doesn't explain all of harmony.

my problems with dyads is more direct with the 57/43 we have a fourth that is 10 cents out you add the 50 in between and it is almost as god as an 6-7-8.
The dyad told us the wrong!

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Mike Battaglia wrote:
>
> On Sun, Jun 22, 2008 at 2:21 AM, Carl Lumma <carl@... > <mailto:carl%40lumma.org>> wrote:
> >> Yeah. So I'm not the only one who thought the current theory of dyads
> >> fell a little short of the mark?
> >
> > What's the "theory of dyads" is and how does it fall short.
> > Bonus points if you can do it in 3K or less.
>
> The approach where one treats each dyad as fitting into a field of
> attraction independent of factors such as the other dyads in the
> chord, the timbre of the instrument, what chords were played before
> that chord, and all of that. All of the other factors I've been
> mentioning since I've gotten here - even psychology plays a factor
> into it - count towards what a person will consciously label an
> interval or chord to be. The way they label that interval or chord
> leads to expectations of how it functions and how it resolves and
> such, and that as a whole determines what most people experience as
> "chord quality." It was touched on recently under something like "the
> sharp knife has a white handle" a while ago. The theory falls chord of
> describing complex chords, and sometimes doesn't predict correctly
> what a chord will be perceived as.
>
> The two revelations I've gotten from this is that not only do the
> other intervals in a chord alter the perception of one specific
> interval, but the timbre of the instrument as well. Not only are you
> less likely to hear an interval as a "mistuning" of some other
> interval if the interval matches up with the timbre, you are also
> extremely unlikely to hear an inharmonic timbre as being out of tune
> in and of itself, except for maybe in an extremely low register. The
> notion of dyadic interpretation that we've been discussing for a
> while, in the form that it is in, certainly has something to do with
> it, but it certainly doesn't explain all of harmony.
>
>

Mike wrote:
> >> Yeah. So I'm not the only one who thought the current theory
> >> of dyads fell a little short of the mark?
> >
> > What's the "theory of dyads" is and how does it fall short.
> > Bonus points if you can do it in 3K or less.
>
> The approach where one treats each dyad as fitting into a field
> of attraction independent of factors such as the other dyads in
> the chord, the timbre of the instrument, what chords were played
> before that chord, and all of that.

I haven't seen anyone advocating this. Have you?

-Carl

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
> The dyad told us the wrong!

Yeah, the dyad told us the wrong! Say it enough times and
maybe it'll become true. -Carl

486 cents

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> > The dyad told us the wrong!
>
> Yeah, the dyad told us the wrong! Say it enough times and
> maybe it'll become true. -Carl
>
>

> Nope. It's just a lack of beating in mistuned otonalities
> via atypical timbres.

Because the partials coincide, which is what Kraig's saying.

Furthermore, I don't understand how beat frequencies are because of
nonlinearities in the ear, the air, or the brain, or how they're any
different from difference tones. They aren't. Listen to this one:

http://www.box.net/shared/lcqjdwtss0

and then take a look at this:

http://www.box.net/shared/1golx754oo

The two tones are 261 and 262 Hz, roughly. Seems obvious, right? That
beat pattern is in the wave before it enters in to the air, the ear,
the brain, or anything. So either there is some nonlinearity of
existence, or it has to do with constructive and destructive
interference between the two waves.

Beat frequencies are just really low frequency difference tones - they
haven't reached the hearing threshold, and so we don't hear them as
tones. I'm inclined to believe lately that the phenomenon where beat
frequencies are produced mentally even if the waves are in two
separate ears has to do with the same phenomenon, but simply occuring
in the brain as the information gets processed together - the notion
that the left and right ears have to be entirely "separate" is the
fallacy leading one to assume otherwise.

-Mike

> my problems with dyads is more direct with the 57/43 we have a fourth
> that is 10 cents out you add the 50 in between and it is almost as god
> as an 6-7-8.
> The dyad told us the wrong!
>

You think that one's bad? Check out
http://www.box.net/shared/m12vhy0e88 - I have 11/4 shrunken to 8/3,
and it STILL sounds like 11/4. 50 full cents flat.

And yet, ironically, if you focus on just the outer dyad and ignore
everything on the inside, it sounds like 8/3. And if you tell yourself
that outer dyad is 8/3, you'll hear 8/3.

Carl wrote:

> So yup, that's the explanation alright. If you use sine
> tones to make music, you've got to be aware of this stuff.
> If you don't, you can ignore it.

Maybe there's even something more to that. I don't know if you've tried
anything like this but let's take, for example, a large set of sine tones
ranging from 200Hz to 300Hz in steps of 2Hz -- i.e. 51 tones in total. When
you mix all of these with identical amplitudes, what comes out is actually
some sort of beats with a clearly audible frequency of 2Hz -- I mean, even
though some of the difference tones have frequencies like 4Hz or 6Hz and so
on.

Petr

[ Attachment content not displayed ]

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> > Nope. It's just a lack of beating in mistuned otonalities
> > via atypical timbres.
>
> Because the partials coincide,

Incorrect. The fact that there are partials coinciding
does NOT mean a complex will be beatless. Also there are
beatless complexes with NO coinciding partials. I've
already explained this about 10 times this month so far.
If you have any questions I'd be glad to try to answer them,
but as it stands you haven't done anything to convey you're
even reading my replies.

> Furthermore, I don't understand how beat frequencies are because
> of nonlinearities in the ear, the air, or the brain, or how
> they're any different from difference tones. They aren't.

Mike, beating is amplitude modulation. Difference tones are
intermodulation products.

> Beat frequencies are just really low frequency difference tones

No.

> they haven't reached the hearing threshold, and so we don't
> hear them as tones.

Tones are variations in air pressure. Beating tones are tones
with amplitude modulation.

-Carl

Hi Chris,

> That was interesting Carl,
>
> how many notes? 7?

7 in each chord, that's right.

> In cool edit spectral view the beating in the first chord i
>s very evident how did you generate the chords?

With Scala and a generic GM wavetable synth.

-Carl

> > > Nope. It's just a lack of beating in mistuned otonalities
> > > via atypical timbres.
> >
> > Because the partials coincide,
>
> Incorrect. The fact that there are partials coinciding
> does NOT mean a complex will be beatless. Also there are
> beatless complexes with NO coinciding partials. I've
> already explained this about 10 times this month so far.
> If you have any questions I'd be glad to try to answer them,
> but as it stands you haven't done anything to convey you're
> even reading my replies.

What you've explained "10 times this month" is that if the partials
between two tones fall within the critical band, they will be
perceived as a warbly beating version of one tone. Clearly if the
partials coincide, this won't happen, will it? If a tone has stretched
partials, and a stretched major chord is played, the chord will beat
much less than if a harmonic major chord is played on the stretched
partial timbre.

> > Furthermore, I don't understand how beat frequencies are because
> > of nonlinearities in the ear, the air, or the brain, or how
> > they're any different from difference tones. They aren't.
>
> Mike, beating is amplitude modulation. Difference tones are
> intermodulation products.

Sum and difference tones are produced by amplitude modulation when the
amplitude of a wave is modulated at a frequency that is within hearing
range. That's how AM synthesis works, and to a certain point, ring
modulation.

> > Beat frequencies are just really low frequency difference tones
>
> No.

The audio clip I just posted has two tones 1 Hz away and a beat
frequency of 1 Hz. Seems like that tone is the difference of the first
two tones to me.

> > they haven't reached the hearing threshold, and so we don't
> > hear them as tones.
>
> Tones are variations in air pressure. Beating tones are tones
> with amplitude modulation.
>
> -Carl

We perceive the result as being a tone with amplitude modulation
(which is itself a variation in air pressure), but if the second tone
were to slowly increase, you would hear the beating speed up until it
reaches about 20 Hz, in which case it would become a very, very faint
difference tone. Nonlinear effects and IMD can bring these tones out,
but they are always there at least faintly just due to good ol'
constructive and destructive interference.

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
> Furthermore, I don't understand how beat frequencies are because of
> nonlinearities in the ear, the air, or the brain,

I don't recall anyone saying that beat frequencies had anything to do
with nonlinearity. But difference tones certainly do.

> or how they're any
> different from difference tones. They aren't. Listen to this one:
>
> http://www.box.net/shared/lcqjdwtss0
>
> and then take a look at this:
>
> http://www.box.net/shared/1golx754oo
>
> The two tones are 261 and 262 Hz, roughly. Seems obvious, right? That
> beat pattern is in the wave before it enters in to the air, the ear,
> the brain, or anything. So either there is some nonlinearity of
> existence, or it has to do with constructive and destructive
> interference between the two waves.

It's the latter. Beating doesn't depend on non-linearity. No one ever
said it does. Like Carl said, beating is when we perceive two
frequencies close together as if they were a single frequency halfway
between the two, increasing and decreasing in loudness.

> Beat frequencies are just really low frequency difference tones - they
> haven't reached the hearing threshold, and so we don't hear them as
> tones.

No. That's completely wrong I'm afraid. When beats get faster than
about 24 Hz the experience is generally described as "roughness".
There is no actual sound present at the beat frequency. If you put it
into a spectrum analyser (or do a fourier analysis on it in a
spreadsheet) you'll still just see the two original frequencies. If
you put it through a low pass filter to eliminate the two original
frequencies, there will be nothing left.

But if you first pass it through some nonlinear function such as this
logistic function
f(x) = (1-exp(-x))/(1+exp(-x))
suitably scaled,
then you will actually create a real _tone_ at the difference
frequency that will show up in a fourier analysis and will remain
after low pass filtering. Of course with this example it will not be
audible.

You will also create a tone at the sum of the original freqencies and
you will create harmonics of the two original frequencies, and sums
and differences of those. But most will be of very low amplitude.

> I'm inclined to believe lately that the phenomenon where beat
> frequencies are produced mentally even if the waves are in two
> separate ears has to do with the same phenomenon, but simply occuring
> in the brain as the information gets processed together

Sure. I don't have a problem with that, whether involving beating or
difference tones. But beating and difference tones are very different
things.

> - the notion
> that the left and right ears have to be entirely "separate" is the
> fallacy leading one to assume otherwise.

I don't understand what you mean by "the fallacy that the left and
right ears have to be entirely separate". Can you state the fallacy as
a proposition?

-- Dave Keenan

> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
> > Furthermore, I don't understand how beat frequencies are because of
> > nonlinearities in the ear, the air, or the brain,
>
> I don't recall anyone saying that beat frequencies had anything to do
> with nonlinearity. But difference tones certainly do.

I think we're talking about two different things. There are sum and
difference tones that have to do with nonlinearity, and there are sum
and difference tones that have to do with high speed amplitude
modulation. The latter has nothing to do with nonlinearity and is
entirely a function of constructive vs. destructive interference on
the waves. And since amplitude modulation is what ends up happening
when two frequencies that are close together play at the same time,
and that the frequency of this modulation is exactly the difference of
the original two frequencies, it makes sense that as the frequency
increases, the frequency of amplitude modulation would increase, until
legitimate sum and difference tones are produced, although they may be
very low in volume.

Perhaps I shouldn't call them sum and difference tones. Maybe they're
sum and difference beats, I don't know. Is there something about this
process that I am misunderstanding?

> Beating doesn't depend on non-linearity. No one ever
> said it does. Like Carl said, beating is when we perceive two
> frequencies close together as if they were a single frequency halfway
> between the two, increasing and decreasing in loudness.

Maybe I'm confusing this thread with one going on in MMM in which a
lot of people said those very things. Sorry, it's hard to keep them
all straight at times :P

> > Beat frequencies are just really low frequency difference tones - they
> > haven't reached the hearing threshold, and so we don't hear them as
> > tones.
>
> No. That's completely wrong I'm afraid. When beats get faster than
> about 24 Hz the experience is generally described as "roughness".
> There is no actual sound present at the beat frequency. If you put it
> into a spectrum analyser (or do a fourier analysis on it in a
> spreadsheet) you'll still just see the two original frequencies. If
> you put it through a low pass filter to eliminate the two original
> frequencies, there will be nothing left.
>
> But if you first pass it through some nonlinear function such as this
> logistic function
> f(x) = (1-exp(-x))/(1+exp(-x))
> suitably scaled,
> then you will actually create a real _tone_ at the difference
> frequency that will show up in a fourier analysis and will remain
> after low pass filtering. Of course with this example it will not be
> audible.
>
> You will also create a tone at the sum of the original freqencies and
> you will create harmonics of the two original frequencies, and sums
> and differences of those. But most will be of very low amplitude.

Ah, I see what you're saying. It might be that part of the effect I've
been hearing is actually due to the generation of another sinusoidal
tone like that. But then again, why couldn't part of it also be simple
constructive and destructive interference?

> > I'm inclined to believe lately that the phenomenon where beat
> > frequencies are produced mentally even if the waves are in two
> > separate ears has to do with the same phenomenon, but simply occuring
> > in the brain as the information gets processed together
>
> Sure. I don't have a problem with that, whether involving beating or
> difference tones. But beating and difference tones are very different
> things.

I'm referring to the concept that the frequency of beating is equal to
the difference between the two tones. This could occur in the brain as
well.

I'm not sure how exactly it would work, but I think it has something
to do with the fact that when the hair cells vibrate in the ear in
response to an impulse, they might not be merely switching "on" and
"off" tonal centers in the brain as a digital switch would - rather
they may be actually SENDING the vibrations as nerve impulses to the
brain, so that even in the brain the tones are still waveforms
changing in time. Or it could not be, this is just my own speculation.

> > - the notion
> > that the left and right ears have to be entirely "separate" is the
> > fallacy leading one to assume otherwise.
>
> I don't understand what you mean by "the fallacy that the left and
> right ears have to be entirely separate". Can you state the fallacy as
> a proposition?

1) The information entering the right ear, if isolated from the left
ear (save any potential resonance occuring in the head), will not
enter the left ear.
2) Thus the two ears appear to work somewhat independently, as you can
see from how we use the independent information mentally to spatialize
the incoming data.
3) It is fallacious to jump from that to that the actual information
from the two ears remains independent all the way until the end of the
signal processing. It does not follow.

And as we can see, it isn't the case after all, since binaural beating
does exist.

-Mike

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> > > > Nope. It's just a lack of beating in mistuned otonalities
> > > > via atypical timbres.
> > >
> > > Because the partials coincide,
> >
> > Incorrect. The fact that there are partials coinciding
> > does NOT mean a complex will be beatless. Also there are
> > beatless complexes with NO coinciding partials. I've
> > already explained this about 10 times this month so far.
> > If you have any questions I'd be glad to try to answer them,
> > but as it stands you haven't done anything to convey you're
> > even reading my replies.
>
> What you've explained "10 times this month" is that if the partials
> between two tones fall within the critical band, they will be
> perceived as a warbly beating version of one tone. Clearly if the
> partials coincide, this won't happen, will it?

Nope. But the fact that a complex has coinciding partials
is neither necessary or sufficient for consonance, as you
implied.

> If a tone has stretched
> partials, and a stretched major chord is played, the chord will beat
> much less than if a harmonic major chord is played on the stretched
> partial timbre.

Yes.

> > > Furthermore, I don't understand how beat frequencies are because
> > > of nonlinearities in the ear, the air, or the brain, or how
> > > they're any different from difference tones. They aren't.
> >
> > Mike, beating is amplitude modulation. Difference tones are
> > intermodulation products.
>
> Sum and difference tones are produced by amplitude modulation
> when the amplitude of a wave is modulated at a frequency that
> is within hearing range. That's how AM synthesis works, and
> to a certain point, ring modulation.

Yes. And ring modulation and AM are synonymous.

> > > Beat frequencies are just really low frequency difference
> > > tones
> >
> > No.
>
> The audio clip I just posted has two tones 1 Hz away and a beat
> frequency of 1 Hz. Seems like that tone is the difference of the
> first two tones to me.

A beating is not a "tone". It's a common myth that AM of a
certain frequency becomes a pitch or a tone. It does not.

> > > they haven't reached the hearing threshold, and so we don't
> > > hear them as tones.
> >
> > Tones are variations in air pressure. Beating tones are tones
> > with amplitude modulation.
> >
> > -Carl
>
> We perceive the result as being a tone with amplitude modulation
> (which is itself a variation in air pressure),

It's a variation of a variation, and it is never perceived as
a pitch no matter how fast or slow it gets.

> Nonlinear effects and IMD can bring these tones out,
> but they are always there at least faintly just due to good ol'
> constructive and destructive interference.

No. Destructive interference is a different mechanism.

-Carl

>> > > Furthermore, I don't understand how beat frequencies are because
>> > > of nonlinearities in the ear, the air, or the brain, or how
>> > > they're any different from difference tones. They aren't.
>> >
>> > Mike, beating is amplitude modulation. Difference tones are
>> > intermodulation products.
>>
>> Sum and difference tones are produced by amplitude modulation
>> when the amplitude of a wave is modulated at a frequency that
>> is within hearing range. That's how AM synthesis works, and
>> to a certain point, ring modulation.
>
> Yes. And ring modulation and AM are synonymous.
//
> A beating is not a "tone". It's a common myth that AM of a
> certain frequency becomes a pitch or a tone. It does not.
//
>> We perceive the result as being a tone with amplitude modulation
>> (which is itself a variation in air pressure),
>
> It's a variation of a variation, and it is never perceived as
> a pitch no matter how fast or slow it gets.

I get it. So the nonlinear effects serve to actually leak the
modulator into the signal, which is what produces the sum and
difference tones?

What I am noticing here is that there is an AM envelope that is equal
to the actual difference of the two frequencies. Does the frequency
that is equal to the sum of the two frequencies have its own envelope
or play some role in the resulting sound? Also, does the frequency
that appears to be an "average" of the two frequencies really only
correspond to a failure to resolve the two frequencies within the same
critical band?

And a question about critical band effects: is there a well defined
cutoff, frequency-wise, for when two tones will start to be heard as
two distinct tones rather than as one fused tone with beating? I hear
two distinct tones at even 261 and 271 Hz, although it might be
actually like one tone that has a very small degree of FM happening -
when the amplitude envelope from the beating is loud, it sounds
sharper, and when it's soft, it sounds flatter. At 13 Hz I definitely
hear two distinct but somewhat fused tones. Why 10 and 13 Hz? That
doesn't correspond to a threshold of hearing, nor the width of a
critical band, or anything that I know of.

-Mike

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
> I think we're talking about two different things. There are sum and
> difference tones that have to do with nonlinearity, and there are sum
> and difference tones that have to do with high speed amplitude
> modulation. The latter has nothing to do with nonlinearity and is
> entirely a function of constructive vs. destructive interference on
> the waves. And since amplitude modulation is what ends up happening
> when two frequencies that are close together play at the same time,
> and that the frequency of this modulation is exactly the difference of
> the original two frequencies, it makes sense that as the frequency
> increases, the frequency of amplitude modulation would increase, until
> legitimate sum and difference tones are produced, although they may be
> very low in volume.
>
> Perhaps I shouldn't call them sum and difference tones.

You got it.

> Maybe they're
> sum and difference beats, I don't know. Is there something about this
> process that I am misunderstanding?

It seems you may understand the process, you may just be confusing us
by using non-standard terminology. Yes beats and difference tones both
have a frequency which is the difference of the two original
frequencies. But with beating there is nothing actually vibrating at
that frequency. It's just that the faster vibration is varying how
_far_ it vibrates each time. Yes this is purely the result of
constructive/destructive interference, but it is never referred to as
a difference _tone_ because it isn't a tone. Actually I prefer to
refer to "beat rate" rather than a beat frequency, to further limit
the possibility of confusion.

Play sines quietly at 200 Hz and 324 Hz. Do you hear a tone at 124 Hz?
If you do, it would only be very faint and can only be because the
signal processing chain (maybe including inside your head) isn't clean
enough.

Introduce some deliberate distortion and the 124 Hz tone will be there.

With a difference tone we actually have energy in a new part of the
spectrum. There is an actual vibration going on at that frequency,
superposed on the original frequencies.

Nonlinearities can cause the energy to shift to frequencies f1-f2
f1+f2, 2*f1-f2, 2*f2-f1, 2*f1, 2*f2, 3*f1, 3*f2 etc. etc.

Beats only ever occur at the difference of the frequencies, not the
sum, nor anything else.

> Ah, I see what you're saying. It might be that part of the effect I've
> been hearing is actually due to the generation of another sinusoidal
> tone like that. But then again, why couldn't part of it also be simple
> constructive and destructive interference?

When the difference frequency is well below 24 Hz then you can _only_
be hearing beating, i.e. constructive and destructive interference.

Here's an idea. If you've got a good subwoofer, find a low frequency
that you can feel in your guts. Then play two mid to high frequencies
that differ by that gut-vibrating frequency. You'll hear the beating
shading into roughness, but there will be no gut vibration unless
you're driving the amp or speakers into distortion. Then put it
through a fuzz box and try it again.

Maybe this isn't a practical experiment given the amplitude of bass
you may need in order to feel it, but you can probably come up with
some other experiment of this kind.

> > I don't understand what you mean by "the fallacy that the left and
> > right ears have to be entirely separate". Can you state the fallacy as
> > a proposition?
>
> 1) The information entering the right ear, if isolated from the left
> ear (save any potential resonance occuring in the head), will not
> enter the left ear.
> 2) Thus the two ears appear to work somewhat independently, as you can
> see from how we use the independent information mentally to spatialize
> the incoming data.
> 3) It is fallacious to jump from that to that the actual information
> from the two ears remains independent all the way until the end of the
> signal processing. It does not follow.
>
> And as we can see, it isn't the case after all, since binaural beating
> does exist.

I totally agree with you here. There's nothing special about where air
vibration leaves off and neural firing begins. It's just another
interface in the signal chain.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@> wrote:
> > > Mike, beating is amplitude modulation. Difference tones are
> > > intermodulation products.
> >
> > Sum and difference tones are produced by amplitude modulation
> > when the amplitude of a wave is modulated at a frequency that
> > is within hearing range. That's how AM synthesis works, and
> > to a certain point, ring modulation.
>
> Yes. And ring modulation and AM are synonymous.

AM synthesis is beating _in_reverse_.

You take one frequency and apply a beat to it by amplitude modulating
it. This creates two real frequencies where there was one (assuming
all sines).

Amplitude modulating one signal by another is the ultimate
nonlinearity. You are actually _multiplying the two signals instead of
adding them. This is equivalent to puting each through a log function,
then adding them and then putting the result through an exponential
function.

-- Dave Keenan

Just trying to tidy up some loose ends.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> You seem to be ignoring virtual pitch, which is more import
> than roughness in musical consonance.
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> > > > Experiments with very accurate sine tones through high
> > > > quality headphones _quietly_, don't find anything special
> > > > at any ratios.
> > >
> > > What about Vos' experiments? The virtual pitch system
> > > still works on sine tones.
> >
> > Isn't that just about whether it is heard as one tone or two?
>
> That's part of it.
>
> The ability to recognize formant filtering of a single
> source as such is at the heart of our ability to understand
> speech. Western harmony, as exemplified in evangelical
> hymnody, is based on the exploitation of this mechanism.

Good point.

> I use the term "virtual pitch mechanism" to refer to this,
> for want of a better term.

OK.

> Vos found that listeners rated the "purity" of just
> intervals higher than others, even with sine tones. We
> can hypothesize that such an effect can only be more
> pronounced with larger chords, as this becomes
> indistinguishable from additive synthesis as more tones
> are added.

Assuming there is such an effect, then I agree it is likely to become
more pronounced with larger chords. But I can only imagine the effect
to be due to beating between intermodulation products, primarily
difference tones.

When I listen to the diverging sine tone example here
http://en.wikipedia.org/wiki/Consonance
at low volume, I hear nothing special at the fifth or octave. When I
play it loudly I hear beating near the octave and possibly the fifth.

Is there any possibility that people differ widely in their internal
production of difference tones, whether mechanical or neural?

> > Yes. I
> > guess that's something special, so I shouldn't have said that.
> > But surely there is no beating or roughness for any interval
> > when using accurate enough, and quiet enough, sine tones?
>
> As long as you stay out of the critical band!

Yes. Sorry. Quite right. There is beating getting faster as we move
away from the unison, going to roughness at about the semitone and
then gone by about the minor third, nothing after that, even near the
octave.

-- Dave Keenan

Carl wrote:
>> That's part of it.
>>
>> The ability to recognize formant filtering of a single
>> source as such is at the heart of our ability to understand
>> speech. Western harmony, as exemplified in evangelical
>> hymnody, is based on the exploitation of this mechanism.

How so?

Dave wrote:
>
> > Vos found that listeners rated the "purity" of just
> > intervals higher than others, even with sine tones. We
> > can hypothesize that such an effect can only be more
> > pronounced with larger chords, as this becomes
> > indistinguishable from additive synthesis as more tones
> > are added.
>
> Assuming there is such an effect, then I agree it is likely
> to become more pronounced with larger chords. But I can only
> imagine the effect to be due to beating between
> intermodulation products, primarily difference tones.

The above was based on a review by Paul E. of one of his
papers. The paper in question doesn't seem to be available
on the internet, however. I'm going to get it at the
library if nobody else beats me to it. Vos was very well
aware of difference tones and would have controlled for them.
However, don't forget OAE interactions!

-Carl