Some more periodicity buzz tests - with chords! Please listen!

OK, I think we've finally reached an important, concrete result here.

Each one of these examples contains a chord played with sines. The
first quarter of each example has all of the sines phase-locked. The
next three have the phases of each sine staggered so that they are all
out of phase with one another, but in different ways. If this doesn't
make sense, it will when you see the images. Finally, I play the first
example again at the end for comparison.

Your task is to tell me if you think that the periodicity buzz is
changing between the four sections of the audio.

Sound: http://www.mikebattagliamusic.com/music/567buzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/567buzz.png

Sound: http://www.mikebattagliamusic.com/music/78910buzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/78910buzz.png

Sound: http://www.mikebattagliamusic.com/music/16171819buzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/16171819buzz.png

I have also created "alternate" versions of the 5:6:7 and 7:8:9:10
examples, but I'm not going to tell you just how alternate - yet.
Suffice to say they have to be at least kind of alternate, or else I
wouldn't have said they're alternate. I personally enjoy alternating
between them a lot, but maybe that's just me. Anyway:

Sound: http://www.mikebattagliamusic.com/music/alt567buzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/alt567buzz.png

Sound: http://www.mikebattagliamusic.com/music/alt78910buzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/alt78910buzz.png

I have also created three "mystery" examples. Let's have some fun here
- see if you can guess what these chords are!

Sound: http://www.mikebattagliamusic.com/music/mysterybuzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/mysterybuzz.png

Sound: http://www.mikebattagliamusic.com/music/mystery2buzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/mystery2buzz.png

Sound: http://www.mikebattagliamusic.com/music/mystery3buzz.wav
Gammatone: http://www.mikebattagliamusic.com/music/mystery3buzz.png

Once you've made your guess, check the next message in this thread to
see both my personal results for all of these examples, my
interpretation of them, what was so alternate about the alternate
examples, and answer to the mystery chords, which I'm certain you
probably didn't get right anyway. No cheating!

-Mike

First off, if you didn't listen to all the examples - just go back and
listen to them. Jesus, they're all like 8 seconds long, and there's
only 8 of them. This takes what, 64 seconds of your time? I spent like
2 hours making all of this. Yes, I am bitter.

OK, anyways, here's my results, as well as the answer to the mystery chords:

-=-PERIODICITY BUZZ RESULTS-=-
Each chord has 4 different choices of phase shift. Of those choices, I
heard the first choice (phases all aligned) as having by far the most
obvious periodicity buzz for ALL the chords played, every time. Of the
remaining 3, I usually heard the second or third as having much less
periodicity buzz than the others; usually the second was the least.

Periodicity buzz was not constant for all the examples - I heard more
buzz when the intervals were closer together. I also sometimes heard
weird pitch shifting effects taking place for different phase shifts,
especially when the intervals were really close together.

-=-ALTERNATE EXAMPLES-=-
The alternate examples are non-periodic, linearly stretched or
compressed versions of the chord they're representing. In each case,
the outer dyad has been changed to sqrt(2), and the intervals have
been uniformly stretched with respect to LINEAR frequency (not cents)
to even out the difference.The alternate 5:6:7 one is actually
1:sqrt(2)/2-1/2:sqrt(2). The alternate 7:8:9:10 one is the same thing
- the 7:10 has been compressed to be sqrt(2). So this chord is
1:sqrt(2)/3-2/3:2*sqrt(2)/3-1/3:sqrt(2). Note the pattern here - every
interval is separated from its nearest neighbor by a constant
frequency difference.

Note that there is still periodicity buzz, and so the buzz is
apparently not "periodicity" buzz at all.

And note also that the VF mechanism still manages to do its job. I
also chose these examples to demonstrate a side point about the VF
mechanism: when people complain about sqrt(2) being a really crappy
approximation for 7/5 and 10/7, they should note that it might not be
too bad - as long as all of the other intervals in a sonority like
7:8:9:10 are compressed linearly and evenly. The same applies even for
the 5-equal fifth, as we'll see. This also suggests a new method of
error optimization that I'd like to explore, but later.

-=-MYSTERY EXAMPLES-=-
The first example has the outer dyad as phi, and creates a pentad with
a constant frequency difference between the notes. So you have
1:phi/5-4/5:2*phi/5-3/5:3*phi/5-2/5:4*phi/5-1/5:phi. This is one of
the least periodic signals ever, but I still hear "periodicity" buzz,
and it's still phase-dependent. You can see a plot of the waveform
here:

http://www.mikebattagliamusic.com/music/phiwaveform.png

Note that when the phase is constant, the uniform group delay causes
huge "spikes" in the waveform. The spikes are recurring, but the
signal isn't actually periodic. So it's not periodicity buzz at all.
We'll come back to this later for an explanation. The point is that
periodicity buzz doesn't actually have to do with periodicity, but the
frequency differences between adjacent notes; I have termed this
property of a chord "linear evenness," because difference "tones"
aren't really involved here.

The other two mystery examples were all linearly-even as well:
mystery2buzz was a linearly-even tetrad in which the outer dyad was
sqrt(phi), and mystery3buzz was a tetrad in which the outer dyad was
1000 cents. I did this in response to Gene's question in this thread:
http://launch.dir.groups.yahoo.com/group/makemicromusic/message/24727

The last one is notable because this is basically a stretched 4:5:6:7;
the cents values are 0-400.9-726.25-1000. 30-equal approximates this
triad really well.

-=-PSYCHOACOUSTIC EXPLANATION-=-

There are a lot of ways to explain this from a signal processing
standpoint, most involving the mixed time-frequency response of the
cochlea due to nonuniform damping. However, this same concept is
something most of us already intuitively understand, and it has to do
with the varying width of the auditory filter, and hence the critical
band, along the frequency spectrum. Let's look at the stretched
4:5:6:7 example again (the phi one):

http://www.mikebattagliamusic.com/music/stretched4567labeled.png

The approximate frequencies at which 4, 5, 6, and 7 actually occur are
labeled. Note that the buzzing is taking place BETWEEN these
frequencies - not at them. So what's going on here?

Let's look at the stretched 6:7 dyad, since that's the smallest, and
hence has the most buzz. The frequencies of the stretched 6:7 are 397
and 465 Hz, which is a difference of 68 Hz. Here are the ERB's at 397
Hz and 465 Hz:

397 Hz ERB = 24.7*(4.37*0.397 + 1) = 67.55 Hz = 33.78 Hz in either direction
465 Hz ERB = 24.7*(4.37*0.465 + 1) = 74.89 Hz = 37.45 Hz in either direction

The problems with ERB's aside, this is clearly not wide enough for the
465 Hz to fall into 397 Hz's auditory filter, or vice versa. So what
IS going on?

A clue can be found by going back to the gammatone plot for 4:5:6:7 -
note that there is a minimum of AM at the actual frequencies that are
being played, but lots of AM between the frequencies between them.
This suggests that while 465 is out of 397 Hz's critical band, and
while 397 is out of 465 Hz's critical band, both fall within
(465+397)/2 = 431 Hz's critical band. Let's work out the ERB here:

431 Hz ERB = 24.7*(4.37*0.431 + 1) = 71.222 Hz = 35.61 Hz in either direction
431 Hz max ERB cutoff: 431 Hz + 35.61 Hz = 466.61 Hz
431 Hz min ERB cutoff: 431 Hz - 35.61 Hz = 395.38 Hz

And there's your answer - the two tones just barely fit in there. So
why is it that you see beating all along the range between the two
frequencies - the above would suggest that this is really fragile, and
should only occur at an extremely narrow range around the midpoint,
right? The reason is that although we like to talk about ERBs around
here - that the bandwidth at this or that frequency is xxx Hz -
they're just a simplification. The auditory filter doesn't actually
have a rectangular response; its bandwidth is infinite and its
frequency response extends outward in all directions, tapering off
slowly. So the max beating occurs at the midpoint, and the beating
lessens as you diverge away from the midpoint. Thus the fact that the
gammatone filters don't exactly match the above are probably due to
limitations in the accuracy of the frequency spreading of the gamma
distribution as it applies to the cochlea, and due to limitations in
the accuracy of using ERBs to represent the actual bandwidth of the
auditory filter.

Phew! So the next step, then, is to come up with "faux-periodicity
buzz" examples for harmonic waveforms - I suspect that we'll discover
that the "ideal" frequencies for faux-periodicity buzz, when harmonic
waveforms are involved, are related to rational intonation.

-Mike

Interesting experiments. Thanks !

> (Mike wrote) :
> Phew! So the next step, then, is to come up with "faux-periodicity
> buzz" examples for harmonic waveforms - I suspect that we'll discover
> that the "ideal" frequencies for faux-periodicity buzz, when harmonic
> waveforms are involved, are related to rational intonation.

Also what about a [26 29 32 35 38] chord ? :
would we hear a buzz of "1", or of "3" ? or both ?
- - - - - - - -
Jacques

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

> And note also that the VF mechanism still manages to do its job. I
> also chose these examples to demonstrate a side point about the VF
> mechanism: when people complain about sqrt(2) being a really crappy
> approximation for 7/5 and 10/7, they should note that it might not be
> too bad - as long as all of the other intervals in a sonority like
> 7:8:9:10 are compressed linearly and evenly. The same applies even for
> the 5-equal fifth, as we'll see. This also suggests a new method of
> error optimization that I'd like to explore, but later.

Oh man, if it is really true that linear evenness causes "periodicity"
buzz (which I think you have not shown conclusively because perhaps
some not so even chords have this buzz too) then I think this
is a major discovery!

I wonder about the new method of error optimization, would you aim for
linear evenness and low error at the same time or what?

Kalle

MikeB>"when people complain about sqrt(2) being a really crappy approximation
for 7/5 and 10/7, they should note that it might not be
> too bad - as long as all of the other intervals in a sonority like 7:8:9:10 are
>compressed linearly and evenly.

I tried this theory out a bit. Say you have a 5:7:9 chord. Making the
7:5 into square root of two would create an interval that would multiply 9/5 to
form an "equivalently stretched" 20/11. I tried a 1/1 17/12 20/11 (where 17/12
is very close to sqrt(2)) and it still sounds pretty sour to me...I'm actually
pretty sure I even like the (non-stretch) 1/1 17/12 9/5 better. But, to be
honest, the stretched version sounds brighter, shaky and nervous, but brighter
(if you equate "brightness" to a lot of the feel of "consonance" it might work
for you...I could see how it may work for some people).
But (Mike B) maybe you could post some test/examples of your theory and see
what others think?

>"I wonder about the new method of error optimization, would you aim for
linear evenness and low error at the same time or what? "
Same here.

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

> Sound: http://www.mikebattagliamusic.com/music/567buzz.wav
> Gammatone: http://www.mikebattagliamusic.com/music/567buzz.png
>
> Sound: http://www.mikebattagliamusic.com/music/78910buzz.wav
> Gammatone: http://www.mikebattagliamusic.com/music/78910buzz.png
>
> Sound: http://www.mikebattagliamusic.com/music/16171819buzz.wav
> Gammatone: http://www.mikebattagliamusic.com/music/16171819buzz.png

Yup, they change. So do your next examples. I don't have the energy to play guessing games about chords composed of sine waves this AM.

On Thu, Jan 20, 2011 at 2:40 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Yup, they change. So do your next examples. I don't have the energy to play guessing games about chords composed of sine waves this AM.

I'm sorry. I hope we can still be friends. Anyway, had you played
along, you'd have discovered that - the next chords aren't periodic at
all. The alternate versions of each chord are stretched and compressed
versions of 5:6:7 and 7:8:9:10 such that the outer dyad is sqrt(2) in
both cases.

The mystery chords are all "linearly even" chords (see the above
messages for a definition for this). Mystery1 is a linearly even
pentad in which the outer dyad is phi. Mystery2 is a linearly even
tetrad in which the outer dyad is sqrt(phi), and mystery3 is a
linearly even tetrad in which the outer dyad is 1000 cents.

Mystery3 is basically a linearly stretched 4:5:6:7 in which the outer
dyad is 1000 cents. It was inspired by your question here:

http://launch.dir.groups.yahoo.com/group/makemicromusic/message/24727

So the answer is - logarithmically evenly stretching 4:5:6:7 probably
would make it sound like crap, but linearly evenly doesn't.

CLIFFNOTES: periodicity buzz has nothing to do with periodicity.

-Mike

On Thu, Jan 20, 2011 at 11:46 AM, Kalle Aho <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > And note also that the VF mechanism still manages to do its job. I
> > also chose these examples to demonstrate a side point about the VF
> > mechanism: when people complain about sqrt(2) being a really crappy
> > approximation for 7/5 and 10/7, they should note that it might not be
> > too bad - as long as all of the other intervals in a sonority like
> > 7:8:9:10 are compressed linearly and evenly. The same applies even for
> > the 5-equal fifth, as we'll see. This also suggests a new method of
> > error optimization that I'd like to explore, but later.
>
> Oh man, if it is really true that linear evenness causes "periodicity"
> buzz (which I think you have not shown conclusively because perhaps
> some not so even chords have this buzz too) then I think this
> is a major discovery!

I do think that I have shown conclusively that periodicity buzz isn't
related to periodicity though. As for linear evenness: you're right. A
chord like 16:17:19 exhibits periodicity buzz, despite the fact that
11-9 is 2 and 9-8 is 1. I ran this chord through the gauntlet and this
is what I got:

http://www.mikebattagliamusic.com/161719buzz.wav
http://www.mikebattagliamusic.com/161719buzz.png

I wasn't as successful at making the buzz vary across the spectrum
because this chord is so smushed together, but if you listen
carefully, you can still hear some change between the fourth example
as it works back to the first one (almost sounds like FM is taking
place... ?!). A look at the gammatone plot reveals somewhat what's
going on here, in that the bottom dyad is buzzing half as fast as the
upper dyad, but only if the phases are out of alignment (haven't done
the math yet to see why this is).

However, since the difference between 16 and 17 and 17 and 19 are
harmonically related (one is twice as fast as the other), the buzz
still ends up being in a recurring pattern. Here's another example,
32:34:37, which means that the buzz ends up being in a 3 against 2
polyrhythm:

http://www.mikebattagliamusic.com/323437buzz.wav
http://www.mikebattagliamusic.com/323437buzz.png

You also hear a bit of pitch fluctuation going on, especially when the
phases are staggered - looking at the gammatone filter plot can be
somewhat revealing as to why this happens.

So I do still think that linear evenness is what's important, but a
generalized version of linear evenness: as long as the difference in
frequencies between the notes in a chord end up being harmonically
related to each other, there will be buzz (provided the notes are
close enough together in pitch for this to happen). This is what's
going on when you deal with buzz in harmonic waveforms: if you have
500 Hz and 400 Hz playing together, you'll get buzz at 100 Hz, but
then you have 1000 Hz and 800 Hz as well, so you get buzz at 200 Hz,
etc. The whole thing forms a 1:2:3:4:5:6:7:8:... polyrhythm. An
omnipolyrhythm, I guess.

The next step is to figure out if faux-periodicity buzz can occur for
harmonic waveforms that doesn't sound like crap. If you play a
linearly stretched 4:5:6:7 with a harmonic waveform, the overtones
themselves aren't stretched, so you're going to get a multiplicity of
buzz rates as the 2 of the stretched 3 beats against the 3 of the
stretched 2, and these rates will NOT be in a coherent polyrhythm as
described above. Rather, I think they'll be in some kind of polyrhythm
with all of the buzz rates offset by a linear frequency, which is no
good (just a hunch). 4:5:6 is a nicer-sounding polyrhythm than
4.13:5.13:6.13 :)

So here's the fundamental question: is there a way to put the buzzing
in a coherent relationship, between both the root notes and all of
their harmonics, even if the notes aren't in a rational relationship?
That is, Jacques has done some pioneering work on equal beating
chords. Is there some additional constraint that can be added to the
construction of these chords such that the result sounds like
periodicity buzz? I think so, but let's find out! :)

> I wonder about the new method of error optimization, would you aim for
> linear evenness and low error at the same time or what?

Something like that I think, yes, although since it looks like the
generalized definition of linear evenness I wrote above is what
actually matters, there will be more than one option for each chord to
be optimized, I think. We're getting back to the concept of a "brat"
now.

-Mike

MikeB>"So the answer is - logarithmically evenly stretching 4:5:6:7 probably
would make it sound like crap, but linearly evenly doesn't."
Ah, ok....got it.

Sweeping question though...suppose we can use a sort of equal beating to align
difference tones to create polyrhythmic "periodicity buzz". Pretty cool
but....what's the point of generating the buzz in the first place? To help us
simply tune our instruments more easily by saying "I hear the buzz, ergo it must
be in tune"...or something on a greater scale?

>"CLIFFNOTES: periodicity buzz has nothing to do with periodicity."
Got it, and maybe a change in the name of the term is in order... Maybe
linear proportionality buzz (LPB) would make more sense....

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

> The next step is to figure out if faux-periodicity buzz can occur for
> harmonic waveforms that doesn't sound like crap.

I don't think synch beating tetrads sound like crap.

On Thu, Jan 20, 2011 at 1:41 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"when people complain about sqrt(2) being a really crappy approximation for 7/5 and 10/7, they should note that it might not be
> > too bad - as long as all of the other intervals in a sonority like 7:8:9:10 are compressed linearly and evenly.
>
>      I tried this theory out a bit.   Say you have a 5:7:9 chord.  Making the 7:5 into square root of two would create an interval that would multiply 9/5 to form an "equivalently stretched" 20/11.  I tried a 1/1 17/12 20/11 (where 17/12 is very close to sqrt(2)) and it still sounds pretty sour to me...I'm actually pretty sure I even like the (non-stretch) 1/1 17/12 9/5 better.

The thing is that the chord won't actually be 1/1 17/12 20/11 - that
is a "tempered" version of the actual linearly even chord. The actual
linearly even chord would be 1:sqrt(2):2*sqrt(2)-1. The stretching
isn't what causes it to be sour, but rather your rationalized
tempering of it.

Here's a sound clip of the two to compare:

http://www.mikebattagliamusic.com/music/buzztempering.wav

It starts out with a just 5:7:9, then a linearly stretched 5:7:9 such
that the 5:7 becomes sqrt(2), and then finally shifts to your
rationalized version, in which the buzz is "tempered." You can hear
for yourself how warbling starts in the last one.

Here's the gammatone filter plot:

http://www.mikebattagliamusic.com/music/buzztempering.png

The vertical lines correspond to the period of buzzing for the lower
dyad. Note that in the first two examples, the upper dyad's buzz is in
phase with the lower dyad's, but in the last one, the upper dyad's
buzz is at a slightly different frequency and slowly crawls out of
phase with the lower one. And this is over 250 ms, or about a quarter
of a second, so you should hear very slow warbling as the beating of
the one dyad slowly drifts in and out of phase with the other, and
this is exactly what I hear.

> But, to be honest, the stretched version sounds brighter, shaky and nervous, but brighter (if you equate "brightness" to a lot of the feel of "consonance" it might work for you...I could see how it may work for some people).
>     But (Mike B) maybe you could post some test/examples of your theory and see what others think?

What further examples do you want...? All of the examples I've posted
so far are examples of this theory, and the theory stemmed right out
of the examples. Did you read the message revealing what the mystery
chords are?

-Mike

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

> The thing is that the chord won't actually be 1/1 17/12 20/11 - that
> is a "tempered" version of the actual linearly even chord. The actual
> linearly even chord would be 1:sqrt(2):2*sqrt(2)-1. The stretching
> isn't what causes it to be sour, but rather your rationalized
> tempering of it.

I dunno. I liked the last one better than the middle one, warble and all.

On Thu, Jan 20, 2011 at 5:08 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The thing is that the chord won't actually be 1/1 17/12 20/11 - that
> > is a "tempered" version of the actual linearly even chord. The actual
> > linearly even chord would be 1:sqrt(2):2*sqrt(2)-1. The stretching
> > isn't what causes it to be sour, but rather your rationalized
> > tempering of it.
>
> I dunno. I liked the last one better than the middle one, warble and all.

I'm not making claims as to which one people will "like" more. Well,
at least I'm not unless we're dealing with harmonic timbres, where I
postulated above that some kind of omni-sync beating where all of the
partials beat in alignment would be generally more pleasant than the
alternative. That was just speculation. But people like the sound of
beating just fine, it generally boils down to personal preference. I'm
simply elucidating on a psychoacoustic explanation for all of this.

-Mike

On Thu, Jan 20, 2011 at 4:07 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The next step is to figure out if faux-periodicity buzz can occur for
> > harmonic waveforms that doesn't sound like crap.
>
> I don't think synch beating tetrads sound like crap.

They don't sound like crap, but the problem is that the effect is
destroyed by the harmonics of the timbre you use. You have a ton of
different beating rates going on as the harmonics of each note beat
against the harmonics of other notes, and these beating rates aren't
themselves in sync with the beating rates of the fundamentals.

If we could place the restriction that the beating rates of the
harmonics with other harmonics also has to be in sync with the beating
rates of the fundamental, I think we'd start to see the true strength
of equal beating chords start to jump through. If this is done, I
think the whole thing probably will probably lead to a discrete set of
frequencies for the fundamental that correspond to something like
w-coherent brats, or "super-brats," a super-brat being a brat in which
all of the harmonics beat in sync with the fundamentals as well (or
beat in a polyrhythm with the fundamentals). I'm not sure this
discrete set is a subset of the rationals - it might be possible to
construct super-brats with irrational intervals as well. Maybe.

It would probably be easy to do this by tempering the octaves in your
tuning, but a very, very interesting subset of this would be
super-brats where the octaves are pure.

To further demonstrate, here are two examples of "5-limit" chords that
have w-coherent brats: 10:12:15 and 16:19:24. 16:19:24 is close enough
to be mapped to the 5-limit that we can consider it to actually be a
tempered version of the 5-limit. The brats for 10:12:15 and 16:19:24
are different - you'll see periodicity buzz in a 2:3 for ratio for the
first chord, and in a 3:5 ratio for the second chord. However, as far
as the VF mechanism is concerned, these two are probably similar
enough to be a part of the 5-limit.

I think that when people distinguish between 16:19:24 and 10:12:15, or
they claim to be able to hear 23-limit harmonies or whatever, they're
actually going by the periodicity buzz imprint, NOT the VF mechanism's
output.

I also think that this type of concordance might be more important
than VF concordance, as far as the overall pleasantness of the end
sound is concerned, when triads and tetrads and the like are
concerned.

-Mike

>"Here's a sound clip of the two to compare:
http://www.mikebattagliamusic.com/music/buzztempering.wav
It starts out with a just 5:7:9, then a linearly stretched 5:7:9 such
that the 5:7 becomes sqrt(2), and then finally shifts to your
rationalized version, in which the buzz is "tempered." You can hear
for yourself how warbling starts in the last one."

You're right, the second one maintains the "beatless" feel of the
first...even if it sounds a bit more tense (though not much so). You could
really be on to something here...I'm convinced. :-)

Now how did you calculate what the 9 becomes when linearly stretched? I
want to try this for a few examples myself...maybe even make a new scale using
this idea.

On Thu, Jan 20, 2011 at 9:01 PM, Michael <djtrancendance@...> wrote:
>
> >"Here's a sound clip of the two to compare:
> http://www.mikebattagliamusic.com/music/buzztempering.wav
> It starts out with a just 5:7:9, then a linearly stretched 5:7:9 such
> that the 5:7 becomes sqrt(2), and then finally shifts to your
> rationalized version, in which the buzz is "tempered." You can hear
> for yourself how warbling starts in the last one."
>
>    You're right, the second one maintains the "beatless" feel of the first...even if it sounds a bit more tense (though not much so).  You could really be on to something here...I'm convinced. :-)

Holla!

>     Now how did you calculate what the 9 becomes when linearly stretched?  I want to try this for a few examples myself...maybe even make a new scale using this idea.

Note that 5:7:9 has a constant frequency difference between
consecutive notes. If the lower dyad is supposed to be 1:sqrt(2), then
the difference is sqrt(2)-1. So the upper dyad must have the same
difference, meaning you get 1:sqrt(2):sqrt(2)+sqrt(2)-1 =
1:sqrt(2):2*sqrt(2)-1

-Mike

Me>> You're right, the second one maintains the "beatless" feel of the
first...even if it sounds a bit more tense (though not much so). You could
really be on to something here...I'm convinced. :-)
MikeB>Holla!
Yep...keep running with it.... :-)

>"Note that 5:7:9 has a constant frequency difference between consecutive notes.
>If the lower dyad is supposed to be 1:sqrt(2), then the difference is
>sqrt(2)-1."

The thing is (if I have it right)...making entire scales using this would
imply either
A) The scale is a standard straight harmonic series and recognizable as such
B) The scale is a stretched version of one...with the scale NOT repeating on
the octave but instead some bizarre stretched interval high enough in the
harmonic series it would be considered non-harmonic. In other words, make a
point of using odd/"bad" intervals. :-D

Actually with sqrt(2), if I have it right, I could chop it down
into......difference = 0.414 half difference = 0.207 quarter difference =
0.1035. So using combinations of these as additive intervals I can get crazy
(wonderful) "anti-JI" stuff like

Mike's "Bad Intervals sound better when Buzzed" scale :-D

1
1.1035
1.207
1.3105
1.414
1.5175
1.621
1.828
2.035

It sounds....much better than expected...normally I can't stand intervals
near 13/8, the 12TET tritone, and 21/16...but they sound strangely decent with
the "buzz/tint" this scale gives them. :-)

Another note....if I have this theory right, it could have huge
implications to Adaptive JI

An Adaptive JI algorithm could not only

A) Align notes to chords that fit a straight harmonic series
....but, if such a chord is unable to be found without significant
comma-shifting..........
B) Search for nearby tones that align for equal beating without moving any one
tone more than, say, 8 cents.

For example
1........1 1.18 1.414 could become 1 1.207 1.414 or
2........1 1.05 1.125 1.25 could become 1 1.063 1.126
1.252

There's lots of implications for lots of things. First off, I think
that this is much more sensitive to mistuning than the VF mechanism,
as per your example.

Secondly, I think that all of the people who hate vibrato, prefer
extremely accurate tunings, and hate things like blackwood generally
are attracted to -this- sound, although they might attribute what
they're liking to that they're hitting sonorities that minimize
harmonic entropy. Conversely, I think that the people who enjoy using
vibrato, tremolo, specially-crafted timbres, timbral envelopes, and
generally other stuff that Igs does, enjoy temperaments as far out as
father and bug temperament. This is because specially crafted timbres
can probably hide what's going on here, smearing all of this out such
that there's not really much uniform buzzing anywhere.

What needs to be done is to construct a chord that sounds like it has
periodicity buzz, even when played with harmonic timbres, that isn't
actually periodic.

-Mike

On Thu, Jan 20, 2011 at 10:06 PM, Michael <djtrancendance@...> wrote:
>
>
>
>       Another note....if I have this theory right, it could have huge implications to Adaptive JI
>
> An Adaptive JI algorithm could not only
>
> A) Align notes to chords that fit a straight harmonic series
>             ....but, if such a chord is unable to be found without significant comma-shifting..........
> B) Search for nearby tones that align for equal beating without moving any one tone more than, say, 8 cents.
>             For example
>                     1........1  1.18 1.414 could become 1 1.207 1.414 or
>                     2........1   1.05 1.125 1.25 could become 1  1.063 1.126  1.252

On Thu, Jan 20, 2011 at 4:07 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The next step is to figure out if faux-periodicity buzz can occur for
> > harmonic waveforms that doesn't sound like crap.
>
> I don't think synch beating tetrads sound like crap.

Gene, quick question: I was going back through your conversation with
Jacques Dudon on brats, and I feel like I'm reinventing the wheel
here. When you guys constructed equal beating meantones with
irrational generators, the aim was for the the partials to beat in a
harmonic, polyrhythmic relationship with one another, right?

So since all of the partials are in harmonic relationships with one
another, this means that if you have a rational brat (like 1.5 or
something), all of the beating between all of the partials will be
harmonically related, correct?

And did this work out such that the frequencies of chords ended up
being separated by a constant difference in frequency, i.e. the
meantone equal-beating 4:5:6 would be something like 4:4+d:4+2d, where
d could be some irrational interval?

So in lieu of all of this gammatone filter stuff, it looks like that
this approach would yield two different beating rates going on: the
polyrhythmic beating of the partials, represented elegantly in "brat"
form, and then the "buzzing" of the actual chord itself, which for
a:b:c has to do with the brat between (c-b) and (b-a). So if you
wanted to have THOSE two brats in sync, and make some kind of
omni-brat, seems like you'd have to work in rational intonation,
because 2a-b would have to equal k(b-a) for some rational k

let d = b-a, the difference between b and a
2a-b = kd
a+(a-b)=a-d=kd
a = (k+1)d
d = a/(k+1)
normalize a to 1 and you get d = 1/(k+1)

So it looks like if you want k to be rational, and also have the
periodicity buzz sync up with the beating, the difference d would also
have to be rational. So perhaps there's some use for chords like
11:14:17:20 or 15:19:23:27 when targeting an optimal 4:5:6:7 for a
really wide-fifth temperament (except as soon as you double the root
down an octave the whole thing goes to hell).

Would you agree with this analysis?

-Mike

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

> Gene, quick question: I was going back through your conversation with
> Jacques Dudon on brats, and I feel like I'm reinventing the wheel
> here. When you guys constructed equal beating meantones with
> irrational generators, the aim was for the the partials to beat in a
> harmonic, polyrhythmic relationship with one another, right?
>
> So since all of the partials are in harmonic relationships with one
> another, this means that if you have a rational brat (like 1.5 or
> something), all of the beating between all of the partials will be
> harmonically related, correct?
>
> And did this work out such that the frequencies of chords ended up
> being separated by a constant difference in frequency, i.e. the
> meantone equal-beating 4:5:6 would be something like 4:4+d:4+2d, where
> d could be some irrational interval?

Here is one such "4:5:6" : 181:226:271
- both thirds (and fifth) equal-beating of harmonics = 1
- both thirds difference tones = 45
- diff.tone/beating ratio = 45:1

and another one : 183:228:273
- both thirds (and fifth) equal-beating of harmonics = 3
- both thirds difference tones = 45
- diff.tone/beating ratio = 15:1

- - - - -
jacques

> So in lieu of all of this gammatone filter stuff, it looks like that
> this approach would yield two different beating rates going on: the
> polyrhythmic beating of the partials, represented elegantly in "brat"
> form, and then the "buzzing" of the actual chord itself, which for
> a:b:c has to do with the brat between (c-b) and (b-a). So if you
> wanted to have THOSE two brats in sync, and make some kind of
> omni-brat, seems like you'd have to work in rational intonation,
> because 2a-b would have to equal k(b-a) for some rational k

> -Mike

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

> So since all of the partials are in harmonic relationships with one
> another, this means that if you have a rational brat (like 1.5 or
> something), all of the beating between all of the partials will be
> harmonically related, correct?

Except that with tetrads you'd need two brats to define things, etc.

> And did this work out such that the frequencies of chords ended up
> being separated by a constant difference in frequency, i.e. the
> meantone equal-beating 4:5:6 would be something like 4:4+d:4+2d, where
> d could be some irrational interval?

The classic example is the Wilson fifth, a root of f^4-2f-2. If if we set e = 3-2f, then the 3 is moved to 3-e, and the 5 to 5-e.

> So in lieu of all of this gammatone filter stuff, it looks like that
> this approach would yield two different beating rates going on: the
> polyrhythmic beating of the partials, represented elegantly in "brat"
> form, and then the "buzzing" of the actual chord itself, which for
> a:b:c has to do with the brat between (c-b) and (b-a).

Not sure what you mean by the brat between c-b and b-a, or what two brats you are talking about.

On Fri, Jan 21, 2011 at 1:11 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > And did this work out such that the frequencies of chords ended up
> > being separated by a constant difference in frequency, i.e. the
> > meantone equal-beating 4:5:6 would be something like 4:4+d:4+2d, where
> > d could be some irrational interval?
>
> The classic example is the Wilson fifth, a root of f^4-2f-2. If if we set e = 3-2f, then the 3 is moved to 3-e, and the 5 to 5-e.

So you'd get something like 4:4.9:5.9 then? Not 4:4.9:5.8?

> > So in lieu of all of this gammatone filter stuff, it looks like that
> > this approach would yield two different beating rates going on: the
> > polyrhythmic beating of the partials, represented elegantly in "brat"
> > form, and then the "buzzing" of the actual chord itself, which for
> > a:b:c has to do with the brat between (c-b) and (b-a).
>
> Not sure what you mean by the brat between c-b and b-a, or what two brats you are talking about.

The second brat, which maybe shouldn't be called a brat for
terminology's sake, is a new concept I'm defining that stems out of
all of the work I just did in this thread. "Periodicity buzz" will
occur if, for some triad a:b:c, (c-b) and (b-a) are in some kind of
harmonic relationship. And by "harmonic" relationship, I mean that
(c-b) and (b-a) are related by some kind of integer ratio, not that
the notes themselves are in a harmonic relationship. The simpler the
relationship, the more coherent the "periodicity" buzz is.

This also works for irrational ratios, so long as (c-b) and (b-a) are
harmonically related. So It doesn't actually have to do with
periodicity, but with this harmonic difference relationship type of
thing, which I have termed "linear evenness." This concept generalizes
to n-ads as you'd expect, and check out the examples and gammatone
filter plots posted earlier in this thread for proof of this. So in a
sense, periodicity buzz is synchronized roughness between the notes of
an n-ad, and roughness is a special form of beating.

Anyway, assuming you've followed all that, my goal was to come up with
examples that buzz coherently even for harmonic timbres. This means
that not only do (c-b) and (b-a) have to be in a harmonic
relationship, but (2c-a) and (c-2a) and (3a-2c) and (5a-4b) and, in
general, (n*a-m*b) for any set of integers n and m have to be in a
harmonic relationship. This means that we want to apply equal beating
techniques here, but also make it such that the rate of beating in the
partials is harmonically synchronized with the rate of buzzing itself.
The only solution I can find for this is to use rational intonation,
the simplest example of which is recurrent sequences. Maybe another
exists, but I'm not seeing it.

So if you're linearly stretching 4:5:6:7, for example, with sines, any
linearly stretched 4:5:6:7 (e.g. 4:4+d:4+2d:4+3d) will buzz evenly.
For complex timbres you'll find that the maximum points of
synchronized buzz occur at chords like

0.000 409.244 740.006 1017.596
0.000 404.442 732.064 1007.442
0.000 401.303 726.865 1000.788
0.000 399.090 723.197 996.090

The top chord has the simplest buzz ratio but is also pretty severely
tempered from a VF standpoint. As you go down more, the buzz becomes
more complex (and slower), but the chord moves closer to the ideal
4:5:6:7. To strike a midpoint between the two would be really
interesting. In some ways I actually enjoy the slower buzz - the
bottom chord might make for an excellent 7-limit blackwood
optimization.

These chords also happen to be

15:19:23:27
19:24:29:34
23:29:35:41
27:34:41:48

This provides some kind of incentive for us (or maybe just me :P) to
track how higher primes are represented in tunings - we're in the
41-limit here, although we are actually treating it like a tempered
version of the 7-limit. Although we don't actually perceive 41-limit
harmony in the VF sense, I think that there is some utility in keeping
track of things like these if an approach like this is desired.

-Mike

On Fri, Jan 21, 2011 at 2:26 PM, Mike Battaglia <battaglia01@...> wrote:
> 0.000 399.090 723.197 996.090
>
//snip
> In some ways I actually enjoy the slower buzz - the
> bottom chord might make for an excellent 7-limit blackwood
> optimization.

I should add that the perception of this, as with regular old just
periodicity buzz, is extremely fragile. 30-equal represents every note
in this chord to within 4 cents, but the 30-equal version sounds
completely different. The 30-equal version buzzes much more slowly,
which isn't really that unpleasant, and there is surely much more
analysis to be done here, but to note - a difference of 4 cents can
completely change the buzz of a chord.

There are also other ways to optimize this I am no doubt missing - the
notes in a:b:c:d don't actually have to be separated by a constant
difference. d-c can be 4/3 * the difference of b-a, for example. Also,
it doesn't really matter what the beating rates are around the 7th and
8th harmonics, because they get pretty quiet, so maybe they can be
ignored for all intents and purposes.

There's also the fact that doubling the root down an octave for the
chords mentioned above makes the chord sound -awful-, so the tuning
should probably be optimized for things like 3/2*a - c and so on.

And then there's the question of whether or not you can split the buzz
into independent "cycles" that aren't harmonically related at all, and
what kind of aural effect that would have.

-Mike

On Fri, Jan 21, 2011 at 9:47 AM, Jacques Dudon <fotosonix@...> wrote:
>
> Here is one such "4:5:6" : 181:226:271
> - both thirds (and fifth) equal-beating of harmonics = 1
> - both thirds difference tones = 45
> - diff.tone/beating ratio = 45:1

This is great! Thanks for this! So a great approach, then, is to
maximize the diff/beating ratio? I was going for the "simplest" buzz
before, but perhaps a nice, slow, rhythmic buzz is what's actually
preferable here.

I am very hesitant to call it a difference "tone," because it's not
actually a tone at all - the difference is 45, but that doesn't mean a
pitch at 45 Hz is generated. Rather, it ends up meaning that the notes
end up beating 45 times a second, or more precisely put that amplitude
modulation occurs at the critical band between each successive pair of
notes at 45 times a second.

> and another one : 183:228:273
> - both thirds (and fifth) equal-beating of harmonics = 3
> - both thirds difference tones = 45
> - diff.tone/beating ratio = 15:1

This one is good, but I liked the above better. So here's a
theoretical question: does this approach basically invalidate regular
temperament, then? I don't think that there's a way to set this up
such that you have a rational, equal beating meantone, set such that
for a:b:c ~ 4:5:6, c^4 = 4b and c-b = k*(b-a), where k is some kind of
rational number.

You'd have to solve the following equation:
(k+1)*(c^4)/4 - c - k = 0

And find two rationals k and c that solve the equation, with c being
sufficiently close to 3/2. Perhaps you could run it through an integer
relation algorithm and get a series of decent approximations, all of
which end up sounding "close enough" to true periodicity buzz so as to
not matter by the time you hit the end of it, and yet still have the
whole thing be meantone.

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > So since all of the partials are in harmonic relationships with one
> > another, this means that if you have a rational brat (like 1.5 or
> > something), all of the beating between all of the partials will be
> > harmonically related, correct?
>
> Except that with tetrads you'd need two brats to define things, etc.
>
> > And did this work out such that the frequencies of chords ended up
> > being separated by a constant difference in frequency, i.e. the
> > meantone equal-beating 4:5:6 would be something like 4:4+d:4+2d, where
> > d could be some irrational interval?
>
> The classic example is the Wilson fifth, a root of f^4-2f-2. If if we set e = 3-2f, then the 3 is moved to 3-e, and the 5 to 5-e.
>
> > So in lieu of all of this gammatone filter stuff, it looks like that
> > this approach would yield two different beating rates going on: the
> > polyrhythmic beating of the partials, represented elegantly in "brat"
> > form, and then the "buzzing" of the actual chord itself, which for
> > a:b:c has to do with the brat between (c-b) and (b-a).
>
> Not sure what you mean by the brat between c-b and b-a, or what two brats you are talking about.

I think the second brat Mike talks about is the ratio between the 1st order difference tones of the successive intervals of the chord, that we want to be equal, or of simple ratios. But what we understand by "brat" usually is rather the harmonic beating ratio, so it would be more clear to call the other "differential ratio" (or why not, the "drat"...?)
For example, the "equal-differential" property of the Wilson fifth is expressed by 4f - f^4 = f^4 - 4.

On Fri, Jan 21, 2011 at 2:53 PM, Jacques Dudon <fotosonix@...> wrote:
>
> I think the second brat Mike talks about is the ratio between the 1st order difference tones of the successive intervals of the chord, that we want to be equal, or of simple ratios. But what we understand by "brat" usually is rather the harmonic beating ratio, so it would be more clear to call the other "differential ratio" (or why not, the "drat"...?)
> For example, the "equal-differential" property of the Wilson fifth is expressed by 4f - f^4 = f^4 - 4.

Ha! I like that! "drat" it is!

-Mike

On Fri, Jan 21, 2011 at 2:48 PM, Mike Battaglia <battaglia01@...> wrote:
>
> This one is good, but I liked the above better. So here's a
> theoretical question: does this approach basically invalidate regular
> temperament, then? I don't think that there's a way to set this up
> such that you have a rational, equal beating meantone, set such that
> for a:b:c ~ 4:5:6, c^4 = 4b and c-b = k*(b-a), where k is some kind of
> rational number.

I'm being silly, I just realized - of course this is possible. Pick a
rational approximation to the meantone fifth, and then the rational
third you end up getting would give you a harmonic relationship
between ~4:5 and ~5:6 by definition, since the whole thing will be
rational. So it's just a matter of coming up with the simplest
relationship between the ~4:5 and ~5:6 possible. I'm not sure if it's
possible for all k. Or, if it is, I can't find a rational solution for
k = 1. This page says none exists:

http://www.numberempire.com/equationsolver.php?function=c^4-2c-2%3D0&var=c&answers=

So I'm not sure for what rationals k a rational solution for c exists.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Jan 21, 2011 at 9:47 AM, Jacques Dudon <fotosonix@...> wrote:
> >
> > Here is one such "4:5:6" : 181:226:271
> > - both thirds (and fifth) equal-beating of harmonics = 1
> > - both thirds difference tones = 45
> > - diff.tone/beating ratio = 45:1
>
> This is great! Thanks for this! So a great approach, then, is to
> maximize the diff/beating ratio? I was going for the "simplest" buzz
> before, but perhaps a nice, slow, rhythmic buzz is what's actually
> preferable here.
>
> I am very hesitant to call it a difference "tone," because it's not
> actually a tone at all - the difference is 45, but that doesn't mean a
> pitch at 45 Hz is generated. Rather, it ends up meaning that the notes
> end up beating 45 times a second, or more precisely put that amplitude
> modulation occurs at the critical band between each successive pair of
> notes at 45 times a second.
>
> > and another one : 183:228:273
> > - both thirds (and fifth) equal-beating of harmonics = 3
> > - both thirds difference tones = 45
> > - diff.tone/beating ratio = 15:1
>
> This one is good, but I liked the above better. So here's a
> theoretical question: does this approach basically invalidate regular
> temperament, then? I don't think that there's a way to set this up
> such that you have a rational, equal beating meantone, set such that
> for a:b:c ~ 4:5:6, c^4 = 4b and c-b = k*(b-a), where k is some kind of
> rational number.

Hmmm, you mean (c/a)^4 = 4(b/a), right ?

And I 'm afraid for a major triad the only simple k you can find is 1/1 - the only linear meantone having this property is the Wilson meantone.
Otherwise you can find tons of equal or polyrhythmic beatings meantones whether in rational or irrational form, but none other with c-b = b-a.
I take an example: here is a meantone algorithm having a 3/2 brat between [EG] and [CE] :
6f^4 = 10f + 15
ex. C,E,G = [2256 2815 3372]
but the "drat" G-E / E-C here is ... 557/559

> You'd have to solve the following equation:
> (k+1)*(c^4)/4 - c - k = 0
>
> And find two rationals k and c that solve the equation, with c being
> sufficiently close to 3/2. Perhaps you could run it through an integer
> relation algorithm and get a series of decent approximations, all of
> which end up sounding "close enough" to true periodicity buzz so as to
> not matter by the time you hit the end of it, and yet still have the
> whole thing be meantone.
>
> -Mike

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

> > The classic example is the Wilson fifth, a root of f^4-2f-2. If if we set e = 3-2f, then the 3 is moved to 3-e, and the 5 to 5-e.
>
> So you'd get something like 4:4.9:5.9 then? Not 4:4.9:5.8?

Nope; 2*(3-e) = 6-2e. What you get is 4, 4.989 and 5.978.

> > Not sure what you mean by the brat between c-b and b-a, or what two brats you are talking about.
>
> The second brat, which maybe shouldn't be called a brat for
> terminology's sake, is a new concept I'm defining that stems out of
> all of the work I just did in this thread. "Periodicity buzz" will
> occur if, for some triad a:b:c, (c-b) and (b-a) are in some kind of
> harmonic relationship. And by "harmonic" relationship, I mean that
> (c-b) and (b-a) are related by some kind of integer ratio, not that
> the notes themselves are in a harmonic relationship. The simpler the
> relationship, the more coherent the "periodicity" buzz is.

So in the case of the Wilson meantone, (6-2e)-(5-e)= 1-e and (5-e)-4 = 1-e, and the ratio is 1. In general, if f is the fifth and t is the major third, it's (f-2t)/(2(t-1)). It's not a rational function of the brat, so it's a new kid on the block.

> 15:19:23:27
> 19:24:29:34
> 23:29:35:41
> 27:34:41:48

You want chords in an arithmetic progression, but I don't see why it should matter that the progression is defined in terms of rationals. This suggests to me looking at chords like 1, 3+e, 5+2e, 7+3e; 4, 5+e, 6+2e, 7+3e; 3-e, 4, 5+e. Sort of synch tuning plus.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
So it's just a matter of coming up with the simplest
> relationship between the ~4:5 and ~5:6 possible. I'm not sure if it's
> possible for all k. Or, if it is, I can't find a rational solution for
> k = 1.

What do you mean ? I just gave you two rational examples.

This page says none exists:
>
> http://www.numberempire.com/equationsolver.php?function=c^4-2c-2%3D0&var=c&answers=

It must be because of the error in your formula.
You should get c/a = 1.49453018047967

> So I'm not sure for what rationals k a rational solution for c exists.
>
> -Mike
>

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

> This one is good, but I liked the above better. So here's a
> theoretical question: does this approach basically invalidate regular
> temperament, then?

I don't understand your bee in the bonnet about rational numbers in this business. It does seem to direct us to codimenion 1 (one comma) temperaments.

On Fri, Jan 21, 2011 at 3:34 PM, Jacques Dudon <fotosonix@...> wrote:
>
> Hmmm, you mean (c/a)^4 = 4(b/a), right ?
>
> And I 'm afraid for a major triad the only simple k you can find is 1/1 - the only linear meantone having this property is the Wilson meantone.
> Otherwise you can find tons of equal or polyrhythmic beatings meantones whether in rational or irrational form, but none other with c-b = b-a.
> I take an example: here is a meantone algorithm having a 3/2 brat between [EG] and [CE] :
> 6f^4 = 10f + 15
> ex. C,E,G = [2256 2815 3372]
> but the "drat" G-E / E-C here is ... 557/559

OK, so to rigorously define this, these are the constraints:

- For omni-equal beating, the triad should be rational, I think
- Let's see if we can get it to be a meantone, e.g. fifth^4 = 4*major third
- If the drat is represented by n/d, we want to MINIMIZE n*d

So I guess the question is, what's the series of rational generators
ordered by 1/(n*d)? I'll assume that the first one is going to be
nowhere near 4:5:6, but that as you progress through the series you'll
start to move closer to 4:5:6, and maybe we can define an arbitrary
cutoff where the triad finally starts to sound like 4:5:6 based on
something like harmonic entropy.

-Mike

On Fri, Jan 21, 2011 at 3:43 PM, Jacques Dudon <fotosonix@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> So it's just a matter of coming up with the simplest
> > relationship between the ~4:5 and ~5:6 possible. I'm not sure if it's
> > possible for all k. Or, if it is, I can't find a rational solution for
> > k = 1.
>
> What do you mean ? I just gave you two rational examples.

Not for meantones, and it also looks like it isn't possible to derive
a rational generator for all k - for k=1/1, for instance, the
generator is irrational.

> This page says none exists:
> >
> > http://www.numberempire.com/equationsolver.php?function=c^4-2c-2%3D0&var=c&answers=
>
> It must be because of the error in your formula.
> You should get c/a = 1.49453018047967

Does that end up working out to the huge radical on the page?

-Mike

--- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:

> And I 'm afraid for a major triad the only simple k you can find is 1/1 - the only linear meantone having this property is the Wilson meantone.

There are alternatives to the Wilson meantone. If f is the fifth and t is the major third, we have

brat = (6f-5t)/(4t-5)
drat = (f-2t)/(2t-2)

If brat=4 in meantone, the fifth satisfies f^4+2f=8. The drat is then -2; both the brat and the drat are integers, instead of just the brat in Wilson.

On Fri, Jan 21, 2011 at 3:41 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > The classic example is the Wilson fifth, a root of f^4-2f-2. If if we set e = 3-2f, then the 3 is moved to 3-e, and the 5 to 5-e.
> >
> > So you'd get something like 4:4.9:5.9 then? Not 4:4.9:5.8?
>
> Nope; 2*(3-e) = 6-2e. What you get is 4, 4.989 and 5.978.

Oh, sorry, I misunderstood. I misread that as the 6 moving to 6-e. Whoops.

> > 15:19:23:27
> > 19:24:29:34
> > 23:29:35:41
> > 27:34:41:48
>
> You want chords in an arithmetic progression, but I don't see why it should matter that the progression is defined in terms of rationals. This suggests to me looking at chords like 1, 3+e, 5+2e, 7+3e; 4, 5+e, 6+2e, 7+3e; 3-e, 4, 5+e. Sort of synch tuning plus.
//
> I don't understand your bee in the bonnet about rational numbers in this business. It does seem to direct us to codimenion 1 (one comma) temperaments.

The reason that I'm talking about rationals is because I want to
explore equal beating chords in which the partials also beat in sync
with the actual buzzing of the chord.

If you want to make the partials beat in sync with everything else,
for some tetrad a:b:c:d, not only do you have to take into account
making d-c, c-b, and b-a harmonically coherent (if you're stretching
it linearly, then they're all equal), you also have to make 2*a-d
harmonically coherent with the others as well. 2*a-d doesn't have to
actually be equal to d-c, but rather it has to be equal to k(d-c),
where k is some integer ratio. And you have to make 3*a-2*c = k(d-c),
etc.

I don't think this is possible for anything except the rationals
because of the following proof which I posted before:

For some dyad a:b
let d = b-a, the difference between b and a
2a-b = kd
a+(a-b)=a-d=kd
a = (k+1)d
d = a/(k+1)
normalize a to 1 and you get d = 1/(k+1)

I think this method works pretty much only as a way to get otherwise
"bad" temperaments (like father temperament or 5n-equal tunings or
something) to buzz nicely instead of sounding dirty and chaotic. Even
if the buzzing is really slow, I still hear a difference between
318:397:476 and 318:397.151:476.302, for example. I guess if the
buzzing is really slow, or if it's irrational and sufficiently close
to a rational, it probably won't matter.

-Mike

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

> Does that end up working out to the huge radical on the page?

Don't get me started on the idiocy of using regular radicals rather than Chebychev radicals for that sort of thing.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@> wrote:
>
> > And I 'm afraid for a major triad the only simple k you can find is 1/1 - the only linear meantone having this property is the Wilson meantone.
>
> There are alternatives to the Wilson meantone. If f is the fifth and t is the major third, we have
>
> brat = (6f-5t)/(4t-5)
> drat = (f-2t)/(2t-2)

> If brat=4 in meantone, the fifth satisfies f^4+2f=8. The drat is then -2; both the brat and the drat are integers, instead of just the brat in Wilson.

Yes, I know well this "Skisni meantone" ... but what Mike wanted was equality for the G-E and E-C dyads, he did not refer to G-C.
And the only G-E = E-C meantone is Wilson's.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Jan 21, 2011 at 3:43 PM, Jacques Dudon <fotosonix@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > So it's just a matter of coming up with the simplest
> > > relationship between the ~4:5 and ~5:6 possible. I'm not sure if it's
> > > possible for all k. Or, if it is, I can't find a rational solution for
> > > k = 1.
> >
> > What do you mean ? I just gave you two rational examples.
>
> Not for meantones, and it also looks like it isn't possible to derive
> a rational generator for all k - for k=1/1, for instance, the
> generator is irrational.

Yes, they were just distorted meantone sequences, in order to give a drat/brat = whole number, but they were verifying the Wilson meantone sequence.
If you prefer the "non-distorted" version take :
C, E, G = [182:227:272]
harmonic beatings of [GE] = [EC] = [GC] = 2
frequency difference [G-E] = [E-C] = 45
drat/brat ratio = 45/2...
272/182 etc. ~1.49453018047967
(my prefered rational seed for a Wilson meantone sequence)

The solutions for all eq-b (and/or -c) linear temperaments will be irrational of course, but most of them will find practical rational
versions, with the advantage over the irrational versions that the beatings ratios between all intervals in the generated scales will be in JI - or polyrhythmic, as you prefer - it means periodic - for buzzing what you are looking for ;)

> > This page says none exists:
> > >
> > > http://www.numberempire.com/equationsolver.php?function=c^4-2c-2%3D0&var=c&answers=
> >
> > It must be because of the error in your formula.
> > You should get c/a = 1.49453018047967
>
> Does that end up working out to the huge radical on the page?
>
> -Mike

I was happy to find such a programm ! but I could not get it working.
Good night all, sweet dreams,
- - - -
Jacques

On Fri, Jan 21, 2011 at 4:52 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Does that end up working out to the huge radical on the page?
>
> Don't get me started on the idiocy of using regular radicals rather than Chebychev radicals for that sort of thing.

OK, but do you understand why I'm using rationals now? Is there some
irrational solution for harmonically related brats and drats that I'm
missing?

Perhaps if you temper the octaves as well?

-Mike

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

> OK, but do you understand why I'm using rationals now? Is there some
> irrational solution for harmonically related brats and drats that I'm
> missing?

Yes, but you've not been very specific as to what, exactly, you want. Your "k" can hardly be any old rational number.

On Fri, Jan 21, 2011 at 4:41 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:
>
> > And I 'm afraid for a major triad the only simple k you can find is 1/1 - the only linear meantone having this property is the Wilson meantone.
>
> There are alternatives to the Wilson meantone. If f is the fifth and t is the major third, we have
>
> brat = (6f-5t)/(4t-5)
> drat = (f-2t)/(2t-2)
>
> If brat=4 in meantone, the fifth satisfies f^4+2f=8. The drat is then -2; both the brat and the drat are integers, instead of just the brat in Wilson.

Why are you using that definition of drat? It seems like it would make
more sense if we were doing, for a:b:c, (c-b)/(b-a). So if we're
dealing with 1:t:f, in your notation, we'd have (f-t)/(t-1). If you
replace the f above with 2f, all of this is the same. Why did you go
with f instead of 2f?

Also I'm a bit confused, what's the generator here? Does this
correspond to a rational chord?

-Mike

On Sat, Jan 22, 2011 at 3:01 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > OK, but do you understand why I'm using rationals now? Is there some
> > irrational solution for harmonically related brats and drats that I'm
> > missing?
>
> Yes, but you've not been very specific as to what, exactly, you want.

I was trying to be specific here:
/tuning/topicId_95699.html#95744

> OK, so to rigorously define this, these are the constraints:
>
> - For omni-equal beating, the triad should be rational, I think
> - Let's see if we can get it to be a meantone, e.g. fifth^4 = 4*major third
> - If the drat is represented by n/d, we want to MINIMIZE n*d
>
> So I guess the question is, what's the series of rational generators
> ordered by 1/(n*d)? I'll assume that the first one is going to be
> nowhere near 4:5:6, but that as you progress through the series you'll
> start to move closer to 4:5:6, and maybe we can define an arbitrary
> cutoff where the triad finally starts to sound like 4:5:6 based on
> something like harmonic entropy.

Is something inconsistent with this?

> Your "k" can hardly be any old rational number.

That's what I was trying to figure out here:
/tuning/topicId_95699.html#95738

> So it's just a matter of coming up with the simplest
> relationship between the ~4:5 and ~5:6 possible. I'm not sure if it's
> possible for all k. Or, if it is, I can't find a rational solution for
> k = 1. This page says none exists:
>
> http://www.numberempire.com/equationsolver.php?function=c^4-2c-2%3D0&var=c&answers=
>
> So I'm not sure for what rationals k a rational solution for c exists.

My thinking about this has evolved at such a rapid rate that I've
bombarded everyone with a sea of messages. But is there a way to
figure out for what rationals this works?

-Mike

To all of you discussing the synchronous meantone topic:
/tuning/topicId_62320.html#62320

Petr

I wrote:

> To all of you discussing the synchronous meantone topic:
> /tuning/topicId_62320.html#62320

And in case you were really interested:
/tuning/topicId_66080.html#66080

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> To all of you discussing the synchronous meantone topic:
> /tuning/topicId_62320.html#62320
>
> Petr

Hi Petr, thanks !
I must have been away from the list at that time, because I never saw
this post of yours.
I confirm all these are great equal or synchronous beating meantones.
All of them belong precisely also to my own collection, which counts
(more or less, depending on what you consider as the meantone range)
about 300 hundred eq-b/h.eq-b and -c meantones.

You may be interested to know my own names for these :

syncmt1 is Michemine, 2x^4 = 20 - 3x^3

syncmt1a is Nikkad, 7x^4 = 10x + 20

syncmt2 is what we commonly call the "Wilson meantone", x^4 = 2x + 2
(in fact I also invented it, as a recurrent fractal waveform perhaps
even before Erv Wilson, but it's known by his name)

syncmt3 is Aurus, 3x^4 = 2x + 12, triple h.eq-b, quite close to the
Golden meta-temperament (696.214473955 cents)

syncmt4 is Tara, 2x^4 = 2x + 7, triple h.eq-b, close to the Golden
meta-temperament, and to the Woolhouse 7/26 syntonic comma
(696.164846 cents)

syncmt5 is Madame, 4x4 = 30 - 3x3, one of the closest sequences I
know for a 1/4 Pythagorean comma meantone

I am not surprised Michemine is among your favorites, it's one of
mine too, and one of the only 3 triple-equal beating minor triad
meantones along with Sireine and Deolia (discussed on this list).
You find rational versions of those, and many other eq-b meantones in
some of the Ethno2 scala files in my folder of the TL files :
Michemine.scl
Atlantis.scl (= Michemine extended to 7-limit neighbourg Saptlante :
x^10 = 64x - 40)
Glolden h7 eq-b.scl (= Tara) - it has been used by Torsten Anders and
Herve Fouere in two Ethno demos
Comptine.scl (one of my favorites) is hyper close to Madame, that
means many of their rational sequences would share their respective
properties.

- - - - - - -
Jacques

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

> My thinking about this has evolved at such a rapid rate that I've
> bombarded everyone with a sea of messages. But is there a way to
> figure out for what rationals this works ?
>
> -Mike

The way I see it to this point Mike, is that what would be important to test is if the ratio
d/b = differential frequency / harmonic beating frequency
for the successive intervals of a triad or a tetrad etc. (and even simple dyads)
can be made to be a whole number.
(then the differential frequencies would be periodic with the harmonic beating frequencies),
and to hear what kind of buzzing it does.
Your questions about other values than 1 for "k" (= the drat), and other temperaments could come after, once this first step is experimented.
If we already use what we have, a Wilson meantone sequence gives you by definition a drat = 1.
And it's easy to add to it an additional 7:6 of the same drat, to have a equal-diff [7:6:5:4] tetrad in option.
One of the things you said you also wanted was a real meantone sequence.
Let's find out what the Wilson meantone sequence suggests by itself as d/b :
= ((f^4)/ - 1) / (5 - f^4) = 22.6026735727
That's not a whole number, but never mind, let's find a sequence that approximates it.
One perhaps more accurate ratio could be 113/5 (I also used 45/2 in my last example),
but the simplest approaching harmonic ratio has to be 23/1 -
Here is a triad that does it :
C E G = [93 116 139]
h. beating = 1 everywhere
diff = 23
d/b = 23/1

As an option, it would perhaps not be a linear meantone anymore but it could be a buzzing tetrad, you can have :
C E G Bb = [93 116 139 162] (and even 185 and 208 as upper C and D etc. if you wish !) :
h. beating = 1 everywhere
diff = 23
d/b = 23/1

My contribution stops here, because you're the one who knows how to synthesize that and align the phases etc., if you wish !
I am curious to know if with sine waves we would hear beats. But if you add the appropriate harmonics, probably.
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> it could be a buzzing tetrad .../...

Of course for a more relaxed beating you could have also :

C E G Bb C'D'E' = [181 226 271 316 361 406 451]... etc.
h. beating = 1 everywhere
diff = 45
d/b = 45/1

- - - - -
Jacques

--- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
>
> > it could be a buzzing tetrad .../...
>
> Of course for a more relaxed beating you could have also :

or "a la Cordier"... :
C E G Bb C'D'E' = [179 224 269 314 359 404 449]... etc.
h. beating = 1 everywhere
diff = 45
d/b = 45/1

- - - - -
Jacques

Strong work! I need to analyze this more to figure out how you worked
this all out.

What are the drats and brats here?

-Mike

On Sat, Jan 22, 2011 at 4:25 AM, Petr Parízek <petrparizek2000@...> wrote:
>
>
>
> I wrote:
>
> > To all of you discussing the synchronous meantone topic:
> > /tuning/topicId_62320.html#62320
>
> And in case you were really interested:
> /tuning/topicId_66080.html#66080
>
> Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Strong work! I need to analyze this more to figure out how you worked
> this all out.

Not sure if you mean the synchronous meantones or the rational approximation to 2/7-comma; but thanks anyway.

> What are the drats and brats here?

I see I haven't been following the discussion carefully enough; okay, drats, brats, ... Now if I just knew which one is which. :-D

Petr

On Sat, Jan 22, 2011 at 6:06 PM, petrparizek2000
<petrparizek2000@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Strong work! I need to analyze this more to figure out how you worked
> > this all out.
>
> Not sure if you mean the synchronous meantones or the rational approximation to 2/7-comma; but thanks anyway.

I particularly liked the rational 2/7-comma approximation. I need to
compare to actual 2/7-comma meantone to see if I can hear a
difference. I'm using Microsoft's GM "Reed Organ" timbre, which is
great for equal beating tests because the timbre is so harsh.

> > What are the drats and brats here?
>
> I see I haven't been following the discussion carefully enough; okay, drats, brats, ... Now if I just knew which one is which. :-D

Drat = "differential ratio," brat = "beating ratio."

A brat, which was defined first, is defined as for some detuned ~4:5:6
triad, how the partials of the 6/5 beat against one another divided by
how the partials of the 5/4 beat against one another.
A drat, which people are now using in different ways, stems out of the
realization that periodicity buzz stems from the "beating" between
adjacent notes in a chord, so for a:b:c I defined it as (c-b)/(b-a).
Gene had some other related definition though.

My question then, was whether the rate of beating was synchronized
with the frequency differential between the notes, and what the ratio
of beating between the roots of 4:5 and 5:6 was.

I'm still not sure all of this matters 100%, as it doesn't take into
consideration that higher partials are lower in volume and hence
matter less than lower ones, and that if the interval is wide enough,
there won't be any audible beating at all (I don't think it occurs for
1:2:3 with sines, for example, no matter how you detune it). It also
doesn't take into consideration that if the brat is slow enough, it
probably doesn't matter as much if it's really in sync with the drat
anymore, but maybe.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Jan 22, 2011 at 6:06 PM, petrparizek2000
> <petrparizek2000@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > Strong work! I need to analyze this more to figure out how you worked
> > > this all out.
> >
> > Not sure if you mean the synchronous meantones or the rational approximation to 2/7-comma; but thanks anyway.
>
> I particularly liked the rational 2/7-comma approximation. I need to
> compare to actual 2/7-comma meantone to see if I can hear a
> difference. I'm using Microsoft's GM "Reed Organ" timbre, which is
> great for equal beating tests because the timbre is so harsh.

Hey, man, this is courageous; if you really manage to find a way to make audible a distance of 1/40 of a cent, you're a master indeed.

Aha, I see. So if the starting tone was, let's say, C, then the brat takes the beat rate of E-G divided by the beat rate of C-E, is that it?
I don't know how to figure this out myself as this time (speaking of the 2/7-comma approximation) I made an exception to my customs and used minor triads as the starting point instead of major triads. As I've said there, if you use the meantone tuning which has the same beats both in C-Eb and Eb-G, you get a minor second of ~120.952 cents, while 2/7-comma meantone has a minor second of ~120.948 cents. Since the fifths aren't pure and the brats seem to be biased towards major triads rather than minor, I'm not sure how I should find the drat and brat values.

Petr

On Sat, Jan 22, 2011 at 7:59 PM, petrparizek2000
<petrparizek2000@...> wrote:
>
> Hey, man, this is courageous; if you really manage to find a way to make audible a distance of 1/40 of a cent, you're a master indeed.

Sorry, I didn't realize it was so close. I haven't delved into it yet,
I just appreciated the work and was going to check it out later.

1/40 of a cent is a stretch, but moving the notes around by a cent or
two can make a huge difference in the end perception of this. I think
I'm going to have to read more on the perception of rhythm and
"tempered rhythms" to come up with a useful metric for this.

> Aha, I see. So if the starting tone was, let's say, C, then the brat takes the beat rate of E-G divided by the beat rate of C-E, is that it?

That's the drat, the difference ratio. The brat for C-E-G is the
beating between the "common partial" of E-G, divided by that of C-E.
So E-G have a common partial at B a few octaves up, and C-E have a
common partial at E a few octaves up. How the beating of these two
partials are related is the brat.

If you want to sync up the brat and drat, you end up limiting yourself
to the rationals.

-Mike

> (Petr) : Since the fifths aren't pure and the brats seem to be
> biased towards major triads rather than minor, I'm not sure how I
> should find the drat and brat values.
>

Hi Mike, and Petr,

This should help (TL 9th of May 2010) :

/tuning/topicId_88708.html#88984

except that you should read :
mb = b/x
as it is said here :

/tuning/topicId_88708.html#88994

This explains why eq-b minor triads have not simple brats ratios, but
may refer with pertinence to another type brat, that I named the
"minor brat",
in which the eq-b 2/7 syntonic comma meantone tuning you refer to has
a mb = -1.

Now I don't want to open a discussion that could be a huge one here,
but to resume :
The brat concept, as Gene defined it (and invented it ??) for equal-
beating applications refers to the major triad.
But each triad of any sort in a temperament would need its specific
brat definition :
tempering [10 12 15] would refer to the "minor brat",
tempering [4 6 7] would refer to a septimal brat (which I named sapnat),
tempering [6 7 8] would refer to another septimal brat (which I named
saptak),
tempering [8 9 11] would refer to a mohajira brat, etc, etc.

Personnally I didn't use the "brat" parameter to find my equal-
beating sequences, and apparently Petr didn't either, but not
surprisingly we arrived to the same results.
I think it can be a useful tool, except as I say, extended to other
triads each one would need its own specific brat definition.
To solve this a "ub" or universal harmonic brat for any triad [a, b,
c] should be defined from the beginning, it's possible but a
complicate task and in meantime expressions like for example
((GBb)) = k*((BbC')) where "((xy))" or any other symbolism would
mean "coïncident harmonic beating frequency of the xy dyad" should be
enough to give, in practice the same informations.
Perhaps Gene or other mathematicians have some better ideas here, but
the only other answer would be to define a universal triadic brat -
and if possible extensible to tetrads, which can even be more complex.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

Mike wrote:

> That's the drat, the difference ratio. The brat for C-E-G is the
> beating between the "common partial" of E-G, divided by that of C-E.
> So E-G have a common partial at B a few octaves up, and C-E have a
> common partial at E a few octaves up. How the beating of these two
> partials are related is the brat.

Well, now I'm totally confused since I thought both of us were saying the same. :-D
Okay, let's take a particular example.
We shall suppose a triad of 40Hz, 51Hz, 62Hz.
The beats in the major third are 4Hz (204 to 200), the beats in the minor third are also 4Hz (310 to 306). So, 4/4 = 1 and that I thought was the brat
If I changed it to 40:51:60Hz, then there would be 4Hz for the major third and -6Hz for the minor third (300 to 306), which means a result of -3/2.
Now if I've understood what you were saying previously, the drat would be 1 for the former and -1 for the latter, am I right?
So what I was saying that if I make a meantone where C-Eb beats the same as Eb-G, I don't see much of a point in these values unless you work with rising major thirds and falling minor thirds.

Petr

On Sun, Jan 23, 2011 at 10:01 AM, petrparizek2000
<petrparizek2000@...> wrote:
>
> Mike wrote:
>
> > That's the drat, the difference ratio. The brat for C-E-G is the
> > beating between the "common partial" of E-G, divided by that of C-E.
> > So E-G have a common partial at B a few octaves up, and C-E have a
> > common partial at E a few octaves up. How the beating of these two
> > partials are related is the brat.
>
> Well, now I'm totally confused since I thought both of us were saying the same. :-D
> Okay, let's take a particular example.
> We shall suppose a triad of 40Hz, 51Hz, 62Hz.
> The beats in the major third are 4Hz (204 to 200), the beats in the minor third are also 4Hz (310 to 306). So, 4/4 = 1 and that I thought was the brat
> If I changed it to 40:51:60Hz, then there would be 4Hz for the major third and -6Hz for the minor third (300 to 306), which means a result of -3/2.
> Now if I've understood what you were saying previously, the drat would be 1 for the former and -1 for the latter, am I right?

The drat for the former would be (62-51)/(51-40) = (11/11) = 1. The
drat for the latter would be (60-51)/(51-40) = 9/11. Perhaps my
definition is too different from the definition of the brat, and we
should make the two similar in some regard?

> So what I was saying that if I make a meantone where C-Eb beats the same as Eb-G, I don't see much of a point in these values unless you work with rising major thirds and falling minor thirds.

In what values, do you mean?

-Mike

Mike wrote:

> In what values, do you mean?

In brats/drats for major triads.
If I wanted to characterize the beating for the "almost 2/7-comma meantone", I would have to say that "if the brat compared 10:12:15 instead of 4:5:6, then it would be 1".
For example, let's say we have 101Hz, 120Hz and 148.5Hz. Then the beating for the minor third is -6Hz (600 to 606) and the beating for the major third is also -6Hz (594 to 600). So if the brat was minor-triad-biased rather than major-triad-biased, this would come out as 1. But for major triads, I have no idea what the brat would be then.

Petr

On Sun, Jan 23, 2011 at 10:32 AM, petrparizek2000
<petrparizek2000@...m> wrote:
>
> Mike wrote:
>
> > In what values, do you mean?
>
> In brats/drats for major triads.
> If I wanted to characterize the beating for the "almost 2/7-comma meantone", I would have to say that "if the brat compared 10:12:15 instead of 4:5:6, then it would be 1".
> For example, let's say we have 101Hz, 120Hz and 148.5Hz. Then the beating for the minor third is -6Hz (600 to 606) and the beating for the major third is also -6Hz (594 to 600). So if the brat was minor-triad-biased rather than major-triad-biased, this would come out as 1. But for major triads, I have no idea what the brat would be then.

This is a question more for Gene, since he's the one who coined the
term. I would think it's the same, right? You just get the beating for
the minor third and then the major third and divide the minor by the
major, regardless of which one is on top. At least that's what I think
it is.

-Mike

Mike wrote:

> This is a question more for Gene, since he's the one who coined the
> term. I would think it's the same, right? You just get the beating for
> the minor third and then the major third and divide the minor by the
> major, regardless of which one is on top. At least that's what I think
> it is.

This can't work because in C1-E1-G1 you're comparing beats in B3 to beats in E3, while in C1-Eb1-G1 you're comparing beats in G3 to other beats in G3.

Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Jan 23, 2011 at 10:32 AM, petrparizek2000
> <petrparizek2000@...> wrote:
> >
> > Mike wrote:
> >
> > > In what values, do you mean?
> >
> > In brats/drats for major triads.
> > If I wanted to characterize the beating for the "almost 2/7-comma meantone", I would have to say that "if the brat compared 10:12:15 instead of 4:5:6, then it would be 1".
> > For example, let's say we have 101Hz, 120Hz and 148.5Hz. Then the beating for the minor third is -6Hz (600 to 606) and the beating for the major third is also -6Hz (594 to 600). So if the brat was minor-triad-biased rather than major-triad-biased, this would come out as 1. But for major triads, I have no idea what the brat would be then.
>
> This is a question more for Gene, since he's the one who coined the
> term. I would think it's the same, right? You just get the beating for
> the minor third and then the major third and divide the minor by the
> major, regardless of which one is on top. At least that's what I think
> it is.
>
> -Mike

These sort of questions, after a simple inversion of the intervals in a major triad (not to mention the subtility that brats can be negative, while beats, like frequencies, are usually considered as positive numbers), illustrates well why I say that if you want to use the brat concept other than for major triads, a universal beat ratio has to be strictly defined.
It should also be able to be extended to tetrads or n-ad chords, and having worked on the subject, I quickly arrived to the conclusion that in order to be generalized, it would be a complex logical question, requiring some conventional choices.

The other alternative, perhaps easier, is to simply give a list of individual harmonic beating frequencies for each dyad.
Such as, for any tetrad of frequencies [a b c d] ->
[(h(b) - h(a)), (h(c) - h(b)), (h(d) - h(c)) ; (h(c) - h(a)), (h(d) - h(b)) ; (h(d) - h(a))], where h = most pertinent coïncident harmonics for each dyad.

(h(b) - h(a)) etc. here instead of (h(a) - h(b)) is already conventional, but would harmonize it with the differential frequency equivalent data, that in the [class1-class2-class3 intervals] order would be :
[(b - a), (c - b), (d - c); (c - a), (d - b); (d - a)]

but which in another order [1st triad intervals, then including the 4th note] could also be :
[(b - a) (c - b), (c - a); (d - c), (d - b), (d - a)]

These are just examples and I would be interested to know if anyone has some thought on the subject, because at least some agrement could be found without too much difficulty, on how these types of lists should be formulated in a convenient and mathematically correct way.
- - - - -
Jacques

--- Mike Battaglia <battaglia01@...> wrote:

> Sound: http://www.mikebattagliamusic.com/music/567buzz.wav
> Gammatone: http://www.mikebattagliamusic.com/music/567buzz.png

That's more like it. What order are the plots in the audio?

> Sound: http://www.mikebattagliamusic.com/music/78910buzz.wav
> Gammatone: http://www.mikebattagliamusic.com/music/78910buzz.png

As above, all parts of the audio exhibit periodicity buzz,
some mildly stronger than others. But I don't see any obvious
correlation to the plot.

-Carl

--- Mike Battaglia <battaglia01@...> wrote:

> -=-ALTERNATE EXAMPLES-=-
> The alternate examples are non-periodic, linearly stretched or
> compressed versions of the chord they're representing. In each
> case, the outer dyad has been changed to sqrt(2), and the
> intervals have been uniformly stretched with respect to LINEAR
> frequency (not cents) to even out the difference.The alternate
> 5:6:7 one is actually 1:sqrt(2)/2-1/2:sqrt(2). The alternate
> 7:8:9:10 one is the same thing - the 7:10 has been compressed
> to be sqrt(2). So this chord is
> 1:sqrt(2)/3-2/3:2*sqrt(2)/3-1/3:sqrt(2). Note the pattern
> here - every interval is separated from its nearest neighbor
> by a constant frequency difference.

They exhibit PB, but it's not clear if it's from the linear
spacing or from approximating JI.

> -=-MYSTERY EXAMPLES-=-
> The first example has the outer dyad as phi, and creates a
> pentad with a constant frequency difference between the notes.
> So you have 1:phi/5-4/5:2*phi/5-3/5:3*phi/5-2/5:4*phi/5-1/5:phi.
> This is one of the least periodic signals ever, but I still
> hear "periodicity" buzz, and it's still phase-dependent.
> You can see a plot of the waveform here:
> http://www.mikebattagliamusic.com/music/phiwaveform.png

There seem to be five sections in the mystery files (?).

> The other two mystery examples were all linearly-even as well:
> mystery2buzz was a linearly-even tetrad in which the outer dyad
> was sqrt(phi), and mystery3buzz was a tetrad in which the outer
> dyad was 1000 cents.

Can you make a linearly even triad sweep with the frequency
difference going from 20Hz to 1000Hz and the root constant?
Can you make three of these with roots at 150Hz 300Hz and
600Hz?

-Carl

On Mon, Apr 25, 2011 at 3:44 PM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > Sound: http://www.mikebattagliamusic.com/music/567buzz.wav
> > Gammatone: http://www.mikebattagliamusic.com/music/567buzz.png
>
> That's more like it. What order are the plots in the audio?

Top left, top right, bottom left, bottom right.

> > Sound: http://www.mikebattagliamusic.com/music/78910buzz.wav
> > Gammatone: http://www.mikebattagliamusic.com/music/78910buzz.png
>
> As above, all parts of the audio exhibit periodicity buzz,
> some mildly stronger than others. But I don't see any obvious
> correlation to the plot.

The plot will not perfectly predict the exact degree of buzz, but it
visually shows the phase coherence. The phase coherence very roughly
predicts how much buzz there'll be, with the general rule that all
coherent = most buzz, all less coherent = less buzz, this being the
same thing the paper on roughness in the other thread demonstrated.
The plots will not predict this perfectly, because the real-life
auditory filter is asymmetrical, which is also something they dealt
with in that paper.

-Mike

On Mon, Apr 25, 2011 at 4:09 PM, Carl Lumma <carl@...> wrote:
>
> They exhibit PB, but it's not clear if it's from the linear
> spacing or from approximating JI.

See this thread:

/tuning/topicId_95699.html#95717

This brings us back to the listening you example you just did, with
the tempered buzz waveform. This thread describes what the example
was. I noted you heard chorus on the beating, but said it didn't
destroy the buzz. This is in line with the prediction of the model,
which is given in the above thread. See this image

http://www.mikebattagliamusic.com/music/buzztempering.png

The first two examples are linearly spaced, and the result of this is
shown on the gammatone plots. The last example is periodic, but not
linearly spaced, which means that the buzz between the two dyads will
very slowly drift in and out of phase with one another. There should
still be buzz, but the AM components should very slowly drift in and
out of phase with one another, and you should hear a chorus effect on
the whole thing. This is precisely what I hear. This doesn't happen
for either the JI or the linearly spaced buzz examples, which have the
linear spacing property.

> > -=-MYSTERY EXAMPLES-=-
> > The first example has the outer dyad as phi, and creates a
> > pentad with a constant frequency difference between the notes.
> > So you have 1:phi/5-4/5:2*phi/5-3/5:3*phi/5-2/5:4*phi/5-1/5:phi.
> > This is one of the least periodic signals ever, but I still
> > hear "periodicity" buzz, and it's still phase-dependent.
> > You can see a plot of the waveform here:
> > http://www.mikebattagliamusic.com/music/phiwaveform.png
>
> There seem to be five sections in the mystery files (?).

There should be five sections in all of these, which may be why you
didn't see a correlation to the plot. The fifth, in each case, is a
return to the first. For most of these I hear the buzz increasing as
it goes from the fourth back to the first.

> > The other two mystery examples were all linearly-even as well:
> > mystery2buzz was a linearly-even tetrad in which the outer dyad
> > was sqrt(phi), and mystery3buzz was a tetrad in which the outer
> > dyad was 1000 cents.
>
> Can you make a linearly even triad sweep with the frequency
> difference going from 20Hz to 1000Hz and the root constant?
> Can you make three of these with roots at 150Hz 300Hz and
> 600Hz?

OK, but that'll take a bit. Somewhere on here I think I also had a
listening example where I showed that 400:500:600 vs 550:650:750 vs
500:600:700 Hz exhibit the same buzz rate, whereas 500:700:900 Hz
buzzes twice as quickly.

-Mike

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

> > > Sound: http://www.mikebattagliamusic.com/music/567buzz.wav
> > > Gammatone: http://www.mikebattagliamusic.com/music/567buzz.png
> >
> > That's more like it. What order are the plots in the audio?
>
> Top left, top right, bottom left, bottom right.

Do you repeat the first (top left) again at the end?

> > > Sound: http://www.mikebattagliamusic.com/music/78910buzz.wav
> > > Gammatone: http://www.mikebattagliamusic.com/music/78910buzz.png
> >
> > As above, all parts of the audio exhibit periodicity buzz,
> > some mildly stronger than others. But I don't see any obvious
> > correlation to the plot.
>
> The plot will not perfectly predict the exact degree of buzz,
> but it visually shows the phase coherence. The phase coherence
> very roughly predicts how much buzz there'll be, with the
> general rule that all coherent = most buzz, all less
> coherent = less buzz, this being the same thing the paper on
> roughness in the other thread demonstrated. The plots will not
> predict this perfectly, because the real-life auditory filter
> is asymmetrical, which is also something they dealt with in
> that paper.

Ok, assuming you're repeating the first segment at the end,
I *do* see an obvious correlation to the plot.

What advantage is there of using this time-frequency plot,
vs. the usual time-amplitude plot? Won't it also show (even
better) the effects of phase?

-Carl

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

> There should be five sections in all of these, which may
> be why you didn't see a correlation to the plot.

Ok good. Classic case of over-reliance on an assumption
(that there must be four parts).

> > Can you make a linearly even triad sweep with the frequency
> > difference going from 20Hz to 1000Hz and the root constant?
> > Can you make three of these with roots at 150Hz 300Hz and
> > 600Hz?
>
> OK, but that'll take a bit.

Very well.

-Carl

On Tue, Apr 26, 2011 at 2:09 AM, Carl Lumma <carl@...> wrote:
>
> > Top left, top right, bottom left, bottom right.
>
> Do you repeat the first (top left) again at the end?

Yes.

> > The plot will not perfectly predict the exact degree of buzz,
> > but it visually shows the phase coherence. The phase coherence
> > very roughly predicts how much buzz there'll be, with the
> > general rule that all coherent = most buzz, all less
> > coherent = less buzz, this being the same thing the paper on
> > roughness in the other thread demonstrated. The plots will not
> > predict this perfectly, because the real-life auditory filter
> > is asymmetrical, which is also something they dealt with in
> > that paper.
>
> Ok, assuming you're repeating the first segment at the end,
> I *do* see an obvious correlation to the plot.
>
> What advantage is there of using this time-frequency plot,
> vs. the usual time-amplitude plot? Won't it also show (even
> better) the effects of phase?

I don't think it would show the effects of phase more accurately, but
it would certainly show them more intuitively. So yes, and I had
thought about working that out at one point, but damn, that's a hell
of a project.

I had worked it out at one point but don't remember the details
anymore. I think what I figured out was that to do this, you'd have to
use the gammatone time-domain waveform as a windowing function, but I
need to double-check. This wouldn't take into account the varying
time-frequency resolution of the ear; e.g. it wouldn't model effects
that happen because we have better time resolution in upper registers
and better frequency resolution in lower registers. If we're trying to
do things a la harmonic entropy and pick a middle of the road setting
for "musical" registers, we could get around the issue, but I wanted
more accurate results first to simplify.

-Mike

On Tue, Apr 26, 2011 at 2:31 AM, Mike Battaglia <battaglia01@...> wrote:
>
> I had worked it out at one point but don't remember the details
> anymore. I think what I figured out was that to do this, you'd have to
> use the gammatone time-domain waveform as a windowing function, but I
> need to double-check.

Actually, if I really wanted to do this, I'd probably just go all the
way with it and abandon gammatone filters entirely at that point.
There's that asymmetric auditory filter model that they used in the
roughness paper I linked to in the other thread; that's one of the
most accurate models that we have, and pretty simple. All we'd need is
the impulse response of the filter and we'd be set. It wouldn't model
the effects of things all up and down the spectrum, but it would model
the middle range of hearing pretty well. You could probably completely
dominate Sethares' model of roughness if you really tweaked it. But
this is quite a project, like I said, and if I get involved in one
more project right now, I'm going to fall off the face of the Earth.

-Mike

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

> > What advantage is there of using this time-frequency plot,
> > vs. the usual time-amplitude plot? Won't it also show (even
> > better) the effects of phase?
>
> I don't think it would show the effects of phase more
> accurately, but it would certainly show them more intuitively.
> So yes, and I had thought about working that out at one point,
> but damn, that's a hell of a project.

Eh? I just mean the ordinary 'waveform' plot.

-Carl

On Tue, Apr 26, 2011 at 4:59 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > What advantage is there of using this time-frequency plot,
> > > vs. the usual time-amplitude plot? Won't it also show (even
> > > better) the effects of phase?
> >
> > I don't think it would show the effects of phase more
> > accurately, but it would certainly show them more intuitively.
> > So yes, and I had thought about working that out at one point,
> > but damn, that's a hell of a project.
>
> Eh? I just mean the ordinary 'waveform' plot.

What do you mean a waveform plot? You mean a spectrogram?

-Mike

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

> > Eh? I just mean the ordinary 'waveform' plot.
>
> What do you mean a waveform plot? You mean a spectrogram?

http://www.google.com/images?q=waveform

-Carl

I like the waveform bracelet in particular.

On Tue, Apr 26, 2011 at 2:50 PM, Carl Lumma <carl@...> wrote:

>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > Eh? I just mean the ordinary 'waveform' plot.
> >
> > What do you mean a waveform plot? You mean a spectrogram?
>
> http://www.google.com/images?q=waveform
>
> -Carl
>
>
>

On Tue, Apr 26, 2011 at 2:50 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > Eh? I just mean the ordinary 'waveform' plot.
> >
> > What do you mean a waveform plot? You mean a spectrogram?
>
> http://www.google.com/images?q=waveform

What, just the actual time-domain waveform? That doesn't give you any
information at all about what's going on in the cochlea, but you'll
find that more buzz = more "spikes" in the waveform and less buzz =
flatter.

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> > What advantage is there of using this time-frequency plot,
> > vs. the usual time-amplitude plot? Won't it also show (even
> > better) the effects of phase?
//
> What, just the actual time-domain waveform? That doesn't give
> you any information at all about what's going on in the
> cochlea, but you'll find that more buzz = more "spikes" in the
> waveform and less buzz = flatter.

Ok great. I think I've been saying this from day 1. But
we'd need to see them next to one of your gammatone quadtyches
with the same time scale to be sure.

-Carl

On Tue, Apr 26, 2011 at 3:33 PM, Carl Lumma <carl@...> wrote:
>
> > What, just the actual time-domain waveform? That doesn't give
> > you any information at all about what's going on in the
> > cochlea, but you'll find that more buzz = more "spikes" in the
> > waveform and less buzz = flatter.
>
> Ok great. I think I've been saying this from day 1.

What, exactly, have you been saying from day 1?

> But we'd need to see them next to one of your gammatone quadtyches
> with the same time scale to be sure.

This has been posted here somewhere, if you can't find it I'll upload
it again, but that's requests for like 3 examples now and I'm starting
to get behind on work.

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> > > What, just the actual time-domain waveform? That doesn't give
> > > you any information at all about what's going on in the
> > > cochlea, but you'll find that more buzz = more "spikes" in the
> > > waveform and less buzz = flatter.
> >
> > Ok great. I think I've been saying this from day 1.
>
> What, exactly, have you been saying from day 1?

That there's no need to consider critical band mixing and
the time domain plot should show the buzz, effects of phase
included. Here are some of my posts in the thread

/tuning/topicId_95321.html#95408
/tuning/topicId_95321.html#95414
/tuning/topicId_95321.html#95417
/tuning/topicId_95522.html#95536
/tuning/topicId_95522.html#95541
/tuning/topicId_95522.html#95542
/tuning/topicId_95522.html#95554
/tuning/topicId_95522.html#95557
/tuning/topicId_95561.html#95601

Because I try to be as concise as possible, it's more
trouble to follow these links than it is to read the
content. So I'll assemble the bones of it at the end of
this message.

> > But we'd need to see them next to one of your gammatone
> > quadtyches with the same time scale to be sure.
>
> This has been posted here somewhere, if you can't find it
> I'll upload it again,

I found some time domain plots, though not next to
matching gammatone plots with the same time scale
(see below). Please, nobody will begrudge you if you
take a breather for other activities!

-Carl

> Here are some examples:
> http://www.mikebattagliamusic.com/music/441HzBuzz.wav
> http://www.mikebattagliamusic.com/music/220.5HzBuzz.wav
> http://www.mikebattagliamusic.com/music/110.25HzBuzz.wav
> http://www.mikebattagliamusic.com/music/55.125HzBuzz.wav
> http://www.mikebattagliamusic.com/music/27.5625HzBuzz.wav

Hm, interesting! [see, I was listening to your examples]

> Specifically it's that a sine wave simultaneously excites
> all of the hairs nearby a certain peak frequency in the
> basilar membrane, so why wouldn't 6:7 be close enough to
> produce some kind of small perturbation?

7/6 is larger than a CB in this range and anything larger
than CB/2 is well-resolved enough not to create anything like
periodicity buzz. That's easy to test with irrational
intervals. [of similar size which do not buzz]

There's no roughness in the 7:6 here. And it's looking more
and more to me like periodicity buzz is an audible time-
domain thing...

> > In the example I posted, I definitely hear a buzz frequency
> > corresponding to the waveform period. I feel like I may also
> > be hearing some faster buzz rates superimposed, but it's
> > hard to tell.
>
> Which example that you posted?

Sorry, let me rephrase that. In every example of periodicity
buzz I've ever heard, I believe I hear a buzz rate corresponding
to the approximate (if tempered) waveform period. I feel there
may also be faster buzz rates superimposed, but it's hard to
tell for sure.

> In the frequency domain, the convolution will cancel out all
> frequencies except those which are at integer multiples of the
> fundamental. So it's a good idea to come up with periodic
> versions of the above three waveforms to evaluate their impact
> on the periodicity buzz produced.

Can you explain what you think any of this has to do with
periodicity buzz? I'm not sure I hear any in your examples,
and I certainly don't in most ordinary (harmonic) timbres.

> You seemed to have a different opinion when I played the "buzz"
> waveform a while ago, which sparked you onto this new train of
> thought that it was only the time domain that mattered.

My own investigations sparked that. I've never understood
how a "buzz waveform" is significant here.

> Did you listen to the examples?

Yes - they don't sound much like periodicity buzz.

> > > I have never heard of anything like this in the
> > > psychoacoustics literature,
> >
> > What would it have said if you did?
>
> What do you mean?

I'm trying to figure out what you're talking about, and
thought it might help if you actually said what it is.

> Here's a graph comparing the 4:
> http://www.mikebattagliamusic.com/music/567buzztest.png

Perfect, thank you!
[The first and last sounds had the strongest buzz]

/Fin

On Tue, Apr 26, 2011 at 5:08 PM, Carl Lumma <carl@...> wrote:
>
> That there's no need to consider critical band mixing and
> the time domain plot should show the buzz, effects of phase
> included.

I call this the "Rick Ballan argument." Just because the time domain
plot shows the buzz doesn't mean anything. The time domain plot is
filtered by every hair on the basilar membrane, producing thousands of
parallel time domain plots. Looking at the outputs of THESE waveforms,
e.g. the auditory filtered outputs, actually will tell you something
about the behavior of the system, and the paper I linked to in the
other thread does exactly this - looks at the envelope fluctuations
from each filtered output. If you want to see the envelope
fluctuations of each filtered output, a simple way is to use a
gammatone filterbank.

Now, if you're trying to make the argument that the brain adds up the
time domain waveform and sees the original again, and then gets buzz
from it - first off, the statement that the waveform has more spikes,
and that the waveform is phase locked, are the same thing. They are
not different. When I talk about the phase of the frequencies in the
waveform, I don't think that the Fourier artifice of "phase" is
directly what's at work. I think that the delay and offset between the
constituent sinusoids is what matters.

In the time domain, this manifests as spiky waveforms. In the
frequency domain, it manifests as non-uniform group delay. And in a
mixed time-frequency plot like a gammatone filter bank, it manifests
as the buzz being placed in complementary positions with one another.
All three of these are equivalent statements, because there is no such
thing as "time-domain" vs "frequency-domain" processing - these are
words that we use to describe the perspective we take to analyze the
behavior of a physical system, not the behavior of the system itself.
Statements made in the time domain can be turned into equivalent
statements in the frequency domain, and vice-versa. It's like us
arguing about whether the brain is making use of an "object-oriented"
or a "procedural" algorithm to perform f0 estimation.

Anyway, assuming you understand that, the statement that spiker
waveforms = more buzz is the same thing as saying phase coherence =
more buzz, you're just taking a different frame on the whole thing.
This still doesn't propose a mechanism for why spikier waveforms would
produce more buzz than non-spikier waveforms. Filterbanks propose such
a mechanism. If you're more fond of thinking about things along
"time-domain" lines, I could reformulate my statements about the
cochlea from a time-domain perspective, although I think that would be
even more confusing.

> I found some time domain plots, though not next to
> matching gammatone plots with the same time scale
> (see below). Please, nobody will begrudge you if you
> take a breather for other activities!

You can rest assured that a breather is on its way, but for now I'm
trying to contribute as much as possible.

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> Now, if you're trying to make the argument that the brain adds
> up the time domain waveform and sees the original again,
> and then gets buzz from it

Yep, that's what I proposed (you clipped it).

> first off, the statement that the waveform has more spikes,
[snip]
> Anyway, assuming you understand that,

It looked interesting, but I'm afraid to say you lost me.

> This still doesn't propose a mechanism for why spikier
> waveforms would produce more buzz than non-spikier waveforms.
> Filterbanks propose such a mechanism.
(from another msg)
> That still wouldn't explain why there's buzz, no.

Why not? And if the output of the filterbank is
resynthesized, how does the filterbank supply such a
mechanism?

> This also suggests that this is not the case:
> /tuning/topicId_95634.html#95634

Again, it's not clear such signals are relevant the present
investigation. But they'd certainly be welcome with the
rest in a time-domain vs gammatone comparison.

-Carl