10:12:15 and the fundamental again

Carl wrote:

> I'm not accusing you of making it up, I'm asking you why
> you believed it. As already mentioned, VF extraction is a
> stochastic, listener-dependent process and there's no
> simple formula that predicts it with certainty. That is why
> we model it with harmonic _entropy_. For chords with low
> entropy, something like the GCD may give quasi-reliable
> results, though the heard VF will often be an octave above
> or below the GCD tone.

I see, then we probably aren�t both speaking about the same thing. I was
talking about the thing which is also reflected in the �shape� of the sound,
no matter if someone is actually listening to that sound at that moment or
not. Nevertheless, it IS well audible and it does have a significant meaning
when we judge if one chord is more or less �concordant� or �harmonious� than
another. And because of that, it is one of the primary reasons why, for
example, European classical harmony developed the way it did. Even people
like Zarlino had known about that and were using that knowledge to describe
what �perfect harmony� should sound like � meaning, for example, that a
major triad of C4-E4-G4 is in �perfect harmony� if there are no beats heard
and the fundamental of C2 is audible (because all of these tones are
harmonics of the C2, which is what he considered a perfect reference for
explaining what �in tune� means).

> What does this have to do with anything? Of course one can
> construct any number of synthesized examples where phase
> cancelations occur in the signal generation domain. Such
> effects are almost never heard in musical contexts, and are
> not a significant source of anything remotely related to
> consonance or dissonance in normal musical settings. If you
> have evidence to the contrary, I would like to download it.

First of all, this is not about phase cancellation. Second, even if they
shouldn�t have a great deal to do with the �musical� view of con/dissonance,
they have a lot to do with the fundamental frequency and with the multiple
periodicity phenomenon, which I was trying to illustrate there. Should I try
to say it in a less complicated way, basically, what I meant was that if you
mix frequencies of 3:7:11:15:19Hz and so on, up to as high as you can hear,
you�ll get, as a result, a sound which exploits two kinds of periodicity,
one of 1Hz and another of 4Hz, even though neither of these frequencies
themselves is actually present in it. If you had a program for finding
frequencies of periodic sounds and loaded a sound like this into it, it
would say �1Hz�. If you then listened to such a sound, instead of hearing
proper tones, you would hear beats of 4Hz. And if, as they clearly can,
these things can be heard at frequencies that low in the form of beats, I
see no reason why they couldn't be heard in our hearing range in the form of
tones.

If you want to listen on your own, I�ve made a Rar archive containing a few
wave files demonstrating this. The number contained in each filename tells
the frequency whose multiples there are of 3, 7, 11, and so on. So, the
number 30 sais that the sound contains 90Hz, 210Hz, 330Hz and so on. The
�5over4� stands for 1.25. The names marked with �x� mean that all of the
frequencies are equally loud, the names marked with �y� mean that changes in
amplitude are inversely proportional to changes in frequency so that higher
tones sound softer than the lower ones. Here�s the link:
www.sendspace.com/file/i2z76s

Petr

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
>
> Nevertheless, it IS well audible and it does have a significant
> meaning when we judge if one chord is more or less "concordant"
> or "harmonious" than another.

First of all, if you really mean GCD, then all JI chords
have a GCD of 1, so I don't see how it helps us compare them.

Secondly, how do you apply it when the chords are tempered?

> Should I try to say it in a less complicated way, basically,
> what I meant was that if you mix frequencies of 3:7:11:15:19Hz
> and so on, up to as high as you can hear, you'll get, as a
> result, a sound which exploits two kinds of periodicity,
> one of 1Hz and another of 4Hz, even though neither of these
> frequencies themselves is actually present in it. If you had a
> program for finding frequencies of periodic sounds and loaded
> a sound like this into it, it would say "1Hz".

OK.

> If you then listened to such a sound, instead of hearing
> proper tones, you would hear beats of 4Hz.

I would?

> I see no reason why they couldn't be heard in our hearing
> range in the form of tones.

I'm not sure what you're saying, but beats are not ever
heard as tones, and vice versa.

> If you want to listen on your own, I've made a Rar archive
> containing a few wave files demonstrating this. The number
> contained in each filename tells the frequency whose multiples
> there are of 3, 7, 11, and so on. So, the number 30 sais that
> the sound contains 90Hz, 210Hz, 330Hz and so on. The "5over4"
> stands for 1.25. The names marked with "x" mean that all of
> the frequencies are equally loud, the names marked with "y"
> mean that changes in amplitude are inversely proportional to
> changes in frequency so that higher tones sound softer than
> the lower ones. Here's the link:
> www.sendspace.com/file/i2z76s

How exactly did you make these?

-Carl

Carl wrote:

> First of all, if you really mean GCD, then all JI chords
> have a GCD of 1, so I don't see how it helps us compare them.

Once again, if I play 400:500:600Hz, the GCD is equal to the difference frequency of both the top and the lower interval and therefore the "synchronicity" is quite obvious. OTOH, for 1000:1200:1500, the GCD of the entire chord is 100 and the difference tones in the top and bottom intervals are 300 and 200 (and their GCDs as well); BUT ... Because the GCD for the highest and the lowest tone is 500 and it's also equal not only to their proper difference tone but also to their lower second difference tone (i.e. 2*lf - hf) -- which can happen only in the case of a perfect fifth, its a bit more "pronounced" than the other difference tones.

> Secondly, how do you apply it when the chords are tempered?

As I've already said, similarly as the "guide tone" turns into two close frequencies which start beating against each other, so does also the fundamental frequency turn into two close frequencies that start beating against each other -- much slower, of course. The mathematical ways of proving these things are very similar one to the other, so if you are interested, I can give particular examples.

> > If you then listened to such a sound, instead of hearing
> > proper tones, you would hear beats of 4Hz.
>
> I would?

Ahum ... I'm sure.

> I'm not sure what you're saying, but beats are not ever
> heard as tones, and vice versa.

Okay, I didn't say that -- and I fully agree with you.

> How exactly did you make these?

These particular recordings were made by strict additive synthesis, others did I make by repeating phase-shifted impulses; and the results were generally indistinguishable.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> > First of all, if you really mean GCD, then all JI chords
> > have a GCD of 1, so I don't see how it helps us compare them.
>
> Once again, if I play 400:500:600Hz, the GCD is equal to the
> difference frequency of both the top and the lower interval and
> therefore the "synchronicity" is quite obvious. OTOH, for
> 1000:1200:1500, the GCD of the entire chord is 100 and the
> difference tones in the top and bottom intervals are 300 and 200
> (and their GCDs as well); BUT ... Because the GCD for the highest
> and the lowest tone is 500 and it's also equal not only to their
> proper difference tone but also to their lower second difference
> tone (i.e. 2*lf - hf) -- which can happen only in the case of a
> perfect fifth, its a bit more "pronounced" than the other
> difference tones.

That's some very baroque reasoning. But it doesn't begin to
answer my question. You said GCD let's us compare chords.
Now you're talking about difference tones again. But you still
haven't answered whether you think consonance phenomena
disappear when difference tones disappear.

> > Secondly, how do you apply it when the chords are tempered?
>
> As I've already said, similarly as the "guide tone" turns into
> two close frequencies which start beating against each other,
> so does also the fundamental frequency turn into two close
> frequencies that start beating against each other

How can the fundamental turn into two frequencies??

> The mathematical ways of proving these things are very similar
> one to the other, so if you are interested, I can give particular
> examples.

To be honest, I'm afraid to accept your offer. I must say
I'm quite confused as to where or how you have arrived at
these strange conclusions regarding psychoacoustics, and why
you seem to be so sure of them.

-Carl

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > I'm not accusing you of making it up, I'm asking you why
> > you believed it. As already mentioned, VF extraction is a
> > stochastic, listener-dependent process and there's no
> > simple formula that predicts it with certainty. That is why
> > we model it with harmonic _entropy_. For chords with low
> > entropy, something like the GCD may give quasi-reliable
> > results, though the heard VF will often be an octave above
> > or below the GCD tone.
>
> I see, then we probably aren't both speaking about the same thing. I was
> talking about the thing which is also reflected in the „shape" of
the sound,
> no matter if someone is actually listening to that sound at that
moment or
> not. Nevertheless, it IS well audible and it does have a significant
meaning
> when we judge if one chord is more or less „concordant" or
„harmonious" than
> another. And because of that, it is one of the primary reasons why, for
> example, European classical harmony developed the way it did. Even
people
> like Zarlino had known about that and were using that knowledge to
describe
> what „perfect harmony" should sound like – meaning, for example, that a
> major triad of C4-E4-G4 is in „perfect harmony" if there are no
beats heard
> and the fundamental of C2 is audible (because all of these tones are
> harmonics of the C2, which is what he considered a perfect reference for
> explaining what „in tune" means).
>
> > What does this have to do with anything? Of course one can
> > construct any number of synthesized examples where phase
> > cancelations occur in the signal generation domain. Such
> > effects are almost never heard in musical contexts, and are
> > not a significant source of anything remotely related to
> > consonance or dissonance in normal musical settings. If you
> > have evidence to the contrary, I would like to download it.
>
> First of all, this is not about phase cancellation. Second, even if they
> shouldn't have a great deal to do with the „musical" view of
con/dissonance,
> they have a lot to do with the fundamental frequency and with the
multiple
> periodicity phenomenon, which I was trying to illustrate there.
Should I try
> to say it in a less complicated way, basically, what I meant was
that if you
> mix frequencies of 3:7:11:15:19Hz and so on, up to as high as you
can hear,
> you'll get, as a result, a sound which exploits two kinds of
periodicity,
> one of 1Hz and another of 4Hz, even though neither of these frequencies
> themselves is actually present in it. If you had a program for finding
> frequencies of periodic sounds and loaded a sound like this into it, it
> would say „1Hz". If you then listened to such a sound, instead of
hearing
> proper tones, you would hear beats of 4Hz. And if, as they clearly can,
> these things can be heard at frequencies that low in the form of
beats, I
> see no reason why they couldn't be heard in our hearing range in the
form of
> tones.
>
> If you want to listen on your own, I've made a Rar archive
containing a few
> wave files demonstrating this. The number contained in each filename
tells
> the frequency whose multiples there are of 3, 7, 11, and so on. So, the
> number 30 sais that the sound contains 90Hz, 210Hz, 330Hz and so on. The
> „5over4" stands for 1.25. The names marked with „x" mean that all of the
> frequencies are equally loud, the names marked with „y" mean that
changes in
> amplitude are inversely proportional to changes in frequency so that
higher
> tones sound softer than the lower ones. Here's the link:
> www.sendspace.com/file/i2z76s
>
> Petr
>
Just to stick my head in between your argument for a second; "if you
mix frequencies of 3:7:11:15:19Hz and so on, up to as high as you can
hear,
you'll get, as a result, a sound which exploits two kinds of periodicity,
one of 1Hz and another of 4Hz, even though neither of these frequencies
themselves is actually present in it". But 1Hz IS the frequency in
this case. What I would like answered is whether we can hear it, and
if not, then why. Or perhaps we sense it in the harmonic relations.
This might seem like a simple question but it is important for a whole
range of questions which I can't go into here. Thanks

- Rick

Rick wrote:

> Just to stick my head in between your argument for a second; "if you
> mix frequencies of 3:7:11:15:19Hz and so on, up to as high as you can
> hear,
> you'll get, as a result, a sound which exploits two kinds of periodicity,
> one of 1Hz and another of 4Hz, even though neither of these frequencies
> themselves is actually present in it". But 1Hz IS the frequency in
> this case. What I would like answered is whether we can hear it, and
> if not, then why. Or perhaps we sense it in the harmonic relations.
> This might seem like a simple question but it is important for a whole
> range of questions which I can't go into here. Thanks

What I'm saying is that if this "general" frequency is in the range of our hearing, then I can hear it clearly and lots of other people whom I've asked told me they could as well. This can be easily verified with the "Y50" example in my archive, where there are frequencies used like 3*50, 7*50, 11*50, 15*50, and so on, all of them sounding together, and where the higher frequencies sound softer than the lower ones in order the entire thing didn't sound too "treblish".

Petr

Carl wrote:

> That's some very baroque reasoning. But it doesn't begin to
> answer my question. You said GCD let's us compare chords.
> Now you're talking about difference tones again. But you still
> haven't answered whether you think consonance phenomena
> disappear when difference tones disappear.

I'm not sure what such a "disappearing" should sound like -- and I'm actually not sure if I really know what you mean by that.I usually try not to make too quick decisions when talking about these things because the views on con/dissonance are affected not only by how we hear but also by our experience and "attitude" (like the fact that, for example, Javanese gamelan may sound terribly out of tune for people in Europe but perfectly okay for those in Java). Personally, what I find most challenging is that some people use the term for completely different things as they often call some intervals "dissonant" not because of acoustical dissonance but because of harmonic instability, which has, however, almost nothing to do with the sound of the intervals and is just a matter of how we view them from a classical harmony perspective (for example, ending a two-voiced piece with an augmented fourth would be usually considered a bit strange in that context, even if it were an acoustically consonant 7/5).

> To be honest, I'm afraid to accept your offer. I must say
> I'm quite confused as to where or how you have arrived at
> these strange conclusions regarding psychoacoustics, and why
> you seem to be so sure of them.

Now that you've said "psychoacoustics", it seems each of us is again talking about something different. I wasn't actually trying to talk about what we hear or how we hear it, I was trying to find an answer for what happens to the sound itself in that case, no matter if it's being listened to or not. To be honest, I'm losing the idea what the question was we're trying to sort out here.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > That's some very baroque reasoning. But it doesn't begin to
> > answer my question. You said GCD let's us compare chords.
> > Now you're talking about difference tones again. But you still
> > haven't answered whether you think consonance phenomena
> > disappear when difference tones disappear.
>
> I'm not sure what such a "disappearing" should sound like --

There will be no difference tones when the ear is driven at
energies where it responds linearly. At low volumes, there
are no difference tones, yet the distinction between major
and minor remains. That was one of the points we were chasing
in this thread.

>Personally, what I find most challenging is that some people
>use the term for completely different things as they often call
>some intervals "dissonant" not because of acoustical dissonance
>but because of harmonic instability, which has, however, almost
>nothing to do with the sound of the intervals and is just a
>matter of how we view them from a classical harmony
>perspective

Yes, and in fact that meaning even has precedence over the
psychoacoustic one. Which is why we really should be saying
"discordance" in this thread. OK, I'll start doing that.
I did it for a while but it just seemed to confuse people.

> > To be honest, I'm afraid to accept your offer. I must say
> > I'm quite confused as to where or how you have arrived at
> > these strange conclusions regarding psychoacoustics, and
> > why you seem to be so sure of them.
>
> Now that you've said "psychoacoustics", it seems each of us
> is again talking about something different. I wasn't actually
> trying to talk about what we hear or how we hear it, I was
> trying to find an answer for what happens to the sound itself
> in that case, no matter if it's being listened to or not.

There are 3 levels of distinction:

acoustics - elementary qualities of sound waves in the
domain of signal processing (fourier theory, etc.)

psychoacoustics - the interaction of acoustics with the
human hearing apparatus, yielding concordance/discordance,
and to some extent, auditory scene analysis

music theory - yielding consonance/dissonance, from an
interaction between musical grammar and psycoacoustics

> To be honest, I'm losing the idea what the question was we're
> trying to sort out here.

I think the original question was why minor chords are less
consonant than major chords.

-Carl

On Mon, Jan 12, 2009 at 7:05 PM, rick_ballan <rick_ballan@...> wrote:
> Just to stick my head in between your argument for a second; "if you
> mix frequencies of 3:7:11:15:19Hz and so on, up to as high as you can
> hear,
> you'll get, as a result, a sound which exploits two kinds of periodicity,
> one of 1Hz and another of 4Hz, even though neither of these frequencies
> themselves is actually present in it". But 1Hz IS the frequency in
> this case. What I would like answered is whether we can hear it, and
> if not, then why. Or perhaps we sense it in the harmonic relations.
> This might seem like a simple question but it is important for a whole
> range of questions which I can't go into here. Thanks
>
> - Rick

Let's be absolutely clear on what we mean by "frequency" here: 1 Hz is
the frequency of the 3:7:11:15:19 waveform. However, no 1 Hz SINE WAVE
is present anywhere in that signal. When people refer to hearing a
certain "frequency", what they're really talking about is hearing a
sine wave with that frequency. The brain, in certain circumstances,
manages to artificially create that 1Hz frequency so that it DOES seem
like we hear a sine wave there.

If you take 3 Hz, 7 Hz, 11 Hz, 15 Hz, etc, and you add them all
together, the resultant waveform will have a period of 1 second, and
thus a frequency of 1 Hz. I was explaining that this frequency of
"repeated information" within the waveform is not equivalent to having
a sine wave with that frequency present within the waveform, and vice
versa.

-Mike

On Tue, Jan 13, 2009 at 1:23 PM, Carl Lumma <carl@...> wrote:
> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
>> Let's be absolutely clear on what we mean by "frequency"
>> here: 1 Hz is the frequency of the 3:7:11:15:19 waveform.
>
> ???
>
> -Carl
>
>

>>Personally, what I find most challenging is that some people
>>use the term for completely different things as they often call
>>some intervals "dissonant" not because of acoustical dissonance
>>but because of harmonic instability, which has, however, almost
>>nothing to do with the sound of the intervals and is just a
>>matter of how we view them from a classical harmony
>>perspective
>
> Yes, and in fact that meaning even has precedence over the
> psychoacoustic one. Which is why we really should be saying
> "discordance" in this thread. OK, I'll start doing that.
> I did it for a while but it just seemed to confuse people.

I want to point out here that most musicians outside of classical
circles refer to "consonance" and "dissonance" closer to how people
refer to "concordance" and "discordance" here. For example, if you
listen to Art Tatum's recordings, he uses "dissonances", usually b9
intervals, as harmonic colors. I sort of have to think way back to my
high school AP theory days when the teacher would rant on about how a
major 7th was a "dissonance" that "needs to resolve outward" to get to
that other meaning of dissonance. And I always thought that that
version of "dissonance" had no foundation in anything anyway - it
always seemed completely arbitrary, and so I don't think it's all that
real.

Nonetheless, in a lot of Classical-era music, sticking a major seventh
in the middle of a piece where it's been notably absent might sound a
bit "unexpected". Labelling such as chord as "good" vs. "bad" is
obviously due to the preferences of the listener, cultural or
otherwise, but it is a worthwhile question whether or not there is
something fundamental going on in the scene analysis layer that is
responsible for the unexpectedness.

That is, of all of the auditory phenomena to suddenly appear midway
through a song, one of the most jarring is a completely unexpected
chord change, and for more than just cultural reasons. A new
instrument appearing midway that's playing the same chords doesn't
have to be all of that jarring at all, but if the song modulates up by
a half step, that's usually pretty surprising and colorful. Perhaps
there is some fundamental reason for this, other than one's cultural
preferences?

> There are 3 levels of distinction:
>
> acoustics - elementary qualities of sound waves in the
> domain of signal processing (fourier theory, etc.)
>
> psychoacoustics - the interaction of acoustics with the
> human hearing apparatus, yielding concordance/discordance,
> and to some extent, auditory scene analysis
>
> music theory - yielding consonance/dissonance, from an
> interaction between musical grammar and psycoacoustics

You define "musical grammar" as the level on which certain auditory
phenomena come to "symbolize" higher-order concepts? I.E., that
Gershwin's Rhapsody in Blue is often interpreted as "sounding like
jazz" rather than the way it would be evaluated by someone who'd never
heard jazz before?

Mike wrote:

> If you take 3 Hz, 7 Hz, 11 Hz, 15 Hz, etc, and you add them all
> together, the resultant waveform will have a period of 1 second, and
> thus a frequency of 1 Hz. I was explaining that this frequency of
> "repeated information" within the waveform is not equivalent to having
> a sine wave with that frequency present within the waveform, and vice
> versa.

Couldn't say that better myself. That's like taking the words out of my mind.

Petr

Yeah, sorry. I deleted my original but I guess you caught it. -C.

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> If you take 3 Hz, 7 Hz, 11 Hz, 15 Hz, etc,
>
> >> Let's be absolutely clear on what we mean by "frequency"
> >> here: 1 Hz is the frequency of the 3:7:11:15:19 waveform.
> >
> > ???
> >
> > -Carl
> >

Carl wrote:

> There will be no difference tones when the ear is driven at
> energies where it responds linearly. At low volumes, there
> are no difference tones, yet the distinction between major
> and minor remains. That was one of the points we were chasing
> in this thread.

Don't know about other people's hearing, but if the sounding tones are rich in overtones (as is the case of many musical instruments including bowed strings or, let's say,saxophones) and a chord is close enough to JI, even if the overall sound should be "supersoft", I still think I could hear the "false fundamental" there somewhere in the background -- at least in a form of some buzzing, if not proper tones (I say "false" because a chord of 0-312-696 cents doesn't have any mathematically exact fundamental frequency but it's so close to 10:12:15 that I can easily perceive it as a modified version of that and, to some extent, sense the appearing and disappearing non-existent relative frequency of 1).

> I think the original question was why minor chords are less
> consonant than major chords.

From a musical point of view, I'm really not sure. From an acoustical point of view, I still think it has something to do with the fact that major chords are direct parts of the harmonic series, which guarantees them the "synchronicity" I was talking about earlier, and that the fundamental of a minor chord, in most cases, results in an undesirable tone in the case of 10:12:15 or is too low in the case of 16:19:24. I'm thinking of those days when I heard some guitarists playing around with the overdriven sounds and feeling very uncomfortable about hearing a C1 when they tried to get an E minor triad (I mean, only the simple three-voiced version) as much "in tune" as they could. Later, I found them making jokes when they realized that, when they pulled the G about a quarter-tone lower, instead of a C1, suddenly an A1 came out. :-D

Petr

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> >>Personally, what I find most challenging is that some people
> >>use the term for completely different things as they often call
> >>some intervals "dissonant" not because of acoustical dissonance
> >>but because of harmonic instability, which has, however, almost
> >>nothing to do with the sound of the intervals and is just a
> >>matter of how we view them from a classical harmony
> >>perspective
> >
> > Yes, and in fact that meaning even has precedence over the
> > psychoacoustic one. Which is why we really should be saying
> > "discordance" in this thread. OK, I'll start doing that.
> > I did it for a while but it just seemed to confuse people.
>
> I want to point out here that most musicians outside of classical
> circles refer to "consonance" and "dissonance" closer to how people
> refer to "concordance" and "discordance" here.

That's probably true.

> > music theory - yielding consonance/dissonance, from an
> > interaction between musical grammar and psycoacoustics
>
> You define "musical grammar" as the level on which certain
> auditory phenomena come to "symbolize" higher-order concepts?

Don't get your hopes up, I just meant it in the
"functional harmony" sense.

-Carl

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:

> Don't know about other people's hearing, but if the sounding
> tones are rich in overtones I still think I could hear the
> "false fundamental" there somewhere in the background

The tones do not have to be rich in overtones, which is why
I suggested you perform the experiment with sine tones.

I'm not disputing the existence of a false fundamental.

> at least in a form of some buzzing,

Buzzing is not a pitch. A fundamental must be a pitch.

> I say "false" because a chord of 0-312-696 cents doesn't
> have any mathematically exact fundamental frequency

So you say! But *no* chord has a "mathematically exact
fundamental frequency" in the way you imagine, as I have
been trying to explain.

> > I think the original question was why minor chords are
> > less consonant than major chords.
>
> From a musical point of view, I'm really not sure. From an
> acoustical point of view, I still think it has something to
> do with the fact that major chords are direct parts of the
> harmonic series, which guarantees them the "synchronicity"
> I was talking about earlier,

The *only* way to explain the phenomenon is with some sort
of 'harmonic template' model, such as the one I sketched.
Difference tones, sum tones, critical band roughness, and all
other spectral arguments fail. You're on the right track
with the idea of periodicity, but the example of 0-312-696
shows it can't be as simple as you imagine.

> the fundamental of a minor chord, in most cases, results in
> an undesirable tone in the case of 10:12:15 or is too low in
> the case of 16:19:24.

In the case of 10:12:15, the probability distribution for
the fundamental will be mostly covered by the pitches 5, 8,
and 10. If the tones of the chord are complex tones, the
timbre will influence the relative likelihoods of 5, 8, and
10 being heard as the fundamental. In the case that 8 is
heard, the brain is interpreting 10:12:15 literally, as a
segment of harmonics relatively high in the series. If 10
or 5 are heard, the brain is saying it considers 10:15 = 3:2,
and is dismissing the 12 as an artifact.

In the case of 16:19:24, the same tradeoff exists, but now
the complete series-segment interpretation is even less
likely because it is even higher in the series. However,
in this case, both interpretations lead to the same
fundamental, namely 8 or 16 (or 4 if the chord is registered
in the soprano). Therefore entropy associated with the
probability distribution for the fundamental tends to be
as lower or lower than that for 10:12:15, with the result
that 16:19:24 is slighly more stable, more "major", more
"happy" if you will, than 10:12:15.

-Carl

On Tue, Jan 13, 2009 at 1:56 PM, Petr Parízek <p.parizek@...> wrote:
> Carl wrote:
>
>> There will be no difference tones when the ear is driven at
>> energies where it responds linearly. At low volumes, there
>> are no difference tones, yet the distinction between major
>> and minor remains. That was one of the points we were chasing
>> in this thread.
>
> Don't know about other people's hearing, but if the sounding tones are rich
> in overtones (as is the case of many musical instruments including bowed
> strings or, let's say,saxophones) and a chord is close enough to JI, even if
> the overall sound should be „supersoft", I still think I could hear the
> „false fundamental" there somewhere in the background -- at least in a form
> of some buzzing, if not proper tones (I say „false" because a chord of
> 0-312-696 cents doesn't have any mathematically exact fundamental frequency
> but it's so close to 10:12:15 that I can easily perceive it as a modified
> version of that and, to some extent, sense the appearing and disappearing
> non-existent relative frequency of 1).

I think that any time you hear a chord as existing as some kind of
harmonic entity in and of itself, and not as a bunch ofrandom
cacophonous notes, that you have succeeded in getting that phantom
fundamental, although it might be covered up by other notes or just
perceptually ignored. I think that cognitive factors can "bias" you
towards a certain fundamental if it's really ambiguous and push you
"over the edge" to hearing it a certain way.

Play a 12-tet minor triad and see if you can "trick" yourself into
hearing the low fundamental as if it were a subminor triad, then as if
it were 16:19:24, and then again as though it were 10:12:15. You can
succeed to some extent, and when I do it, I actually notice subtle
variations in the perceived pitch of the minor third. But if you can't
get your perception to flip entirely by willpower, try playing the
desired VF along with it, and then play just the minor triad again. I
think this is a lesser version of the same phenomenon by which you can
hear a neutral triad and "flip it mentally" back and forth between
major and minor - you are somehow influencing the VF that you hear,
which would support my hypothesis put forth months ago that cognitive
processes can influence the periodicity mechanism in the brain.

>> I think the original question was why minor chords are less
>> consonant than major chords.
>
> From a musical point of view, I'm really not sure. From an acoustical point
> of view, I still think it has something to do with the fact that major
> chords are direct parts of the harmonic series, which guarantees them the
> „synchronicity" I was talking about earlier, and that the fundamental of a
> minor chord, in most cases, results in an undesirable tone in the case of
> 10:12:15 or is too low in the case of 16:19:24. I'm thinking of those days
> when I heard some guitarists playing around with the overdriven sounds and
> feeling very uncomfortable about hearing a C1 when they tried to get an E
> minor triad (I mean, only the simple three-voiced version) as much „in tune"
> as they could. Later, I found them making jokes when they realized that,
> when they pulled the G about a quarter-tone lower, instead of a C1, suddenly
> an A1 came out. :-D

I agree with this. I've always noticed that chords with different
fundamentals than an octave-equivalent of the root were more dissonant
than chords that do have it. The fundamental of a major chord seems to
reinforce and stabilize the chord, and the fundamental of a minor
chord destabilizes it and adds to the "color" of it, I think.

But there has to be something else at work here too - think about a
5/3 major sixth chord. The GCD of a C E- G A- would be F -- and yet
it still seems to be a "strong, resonant" chord, albeit slightly less
strong than C major. But what about the chord C E- F G? That's much,
much less stable than C E- G A-, even though they have the same
fundamental. And I think there's more to it than just the minor second
over the E- is more dissonant than the major second over the G -- the
first one sounds like some kind of "C" chord, and the second one
sounds ambiguous as to whether it's a "C" chord or an "F" chord in
inversion.

Maybe we don't get the VF of the whole chord at once, but rather in
parts, so that the VF of the C major chord sounds like C, and then the
A- on top is some slightly "colorful" note that technically produces a
different VF, but it's much softer and so it's less obtrusive?

Carl wrote:

> The tones do not have to be rich in overtones, which is why
> I suggested you perform the experiment with sine tones.

Did you? That was the very first experiment I tried when we began discussing the fundamental frequency to make sure I was really able to hear it.

> Buzzing is not a pitch. A fundamental must be a pitch.

Okay, here comes the fact that English is not my native language -- I thought buzzing could be periodic as well.

> So you say! But *no* chord has a "mathematically exact
> fundamental frequency" in the way you imagine, as I have
> been trying to explain.

No chord has an exact VF, that's right. But JI chords do have an exact ... not VF, but the one that many people call fundamental frequency -- the relative frequency of 1.

> You're on the right track
> with the idea of periodicity, but the example of 0-312-696
> shows it can't be as simple as you imagine.

I don't know exactly what you mean but 0-312-696 cents is, at least for me, close enough to 10:12:15 so that I can, with some "concentration", hear a pitch close to the relative frequency of what would be 1 if it were 10:12:15.

> In the case of 10:12:15, the probability distribution for
> the fundamental will be mostly covered by the pitches 5, 8,
> and 10. If the tones of the chord are complex tones, the
> timbre will influence the relative likelihoods of 5, 8, and
> 10 being heard as the fundamental. In the case that 8 is
> heard, the brain is interpreting 10:12:15 literally, as a
> segment of harmonics relatively high in the series. If 10
> or 5 are heard, the brain is saying it considers 10:15 = 3:2,
> and is dismissing the 12 as an artifact.

Probably. I'm just not sure that I understand the first sentence.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> > Buzzing is not a pitch. A fundamental must be a pitch.
>
> Okay, here comes the fact that English is not my native
> language -- I thought buzzing could be periodic as well.

Infrared light can be periodic, but we do not hear a pitch.
Audio amplitude modulation can be periodic, but we do not
hear a pitch.
When longitudinal compression waves in the air are periodic,
we sometimes do hear a pitch, but it is not always the same
pitch that we would hear for a pure tone with the same period
as the longitudinal compression waves under consideration!

> > So you say! But *no* chord has a "mathematically exact
> > fundamental frequency" in the way you imagine, as I have
> > been trying to explain.
>
> No chord has an exact VF, that's right. But JI chords do have
> an exact ... not VF, but the one that many people call
> fundamental frequency -- the relative frequency of 1.

I've not heard many people refer to fundamentals of chords.
I've heard them refer to roots of chords, by which usually
the VF is meant.

> > You're on the right track with the idea of periodicity,
> > but the example of 0-312-696 shows it can't be as simple
> > as you imagine.
>
> I don't know exactly what you mean but 0-312-696 cents is,
> at least for me, close enough to 10:12:15

Sure... but how do you quantify this? Hearing doesn't stop
at integers so your model can't either.

> > In the case of 10:12:15, the probability distribution for
> > the fundamental will be mostly covered by the pitches 5, 8,
> > and 10. If the tones of the chord are complex tones, the
> > timbre will influence the relative likelihoods of 5, 8, and
> > 10 being heard as the fundamental. In the case that 8 is
> > heard, the brain is interpreting 10:12:15 literally, as a
> > segment of harmonics relatively high in the series. If 10
> > or 5 are heard, the brain is saying it considers 10:15 = 3:2,
> > and is dismissing the 12 as an artifact.
>
> Probably. I'm just not sure that I understand the first
> sentence.

When presented with an auditory stimulus, the brain will
attempt to assign it a pitch. It does this by extracting
harmonically-related signals in the spectrum analysis of the
stimulus. The stimulus need not be periodic for this to
work, but even if it is, the brain is only so good at
1. performing spectrum analysis -- failure here results
in roughness
2. detecting periodicity -- failure here results in high
harmonic entropy
For clean stimuli with periodicities in the range the brain
is good at detecting (such as 5-limit major triads), the
extracted pitch will fairly unambiguous (though it will
sometimes get confused by octaves). For other kinds of
stimuli, the stochastic nature of processes in the brain
becomes evident -- populations of thousands or millions
or tens of millions of neurons are used to accomplish
signal-processing tasks. There will be several pitches
competing for the 'answer', and harmonic entropy can be
used to model this uncertainty.

-Carl

Carl wrote:

> I've not heard many people refer to fundamentals of chords.
> I've heard them refer to roots of chords, by which usually
> the VF is meant.

When I was attending classical harmony lessons, some of which I was taught at an age of 11 (Daniel Forró may confirm), the term "fundamental tone" was always used specifically for the one tone in a chord from which a series of rising thirds spelled all of its tones -- i.e. a chord of F-G-B-D has a fundamental tone of G because G-B-D-F is the way to describe the chord as a series of thirds. There was the thing which I could try to translate perhaps as major or minor "tonal gender", which meant that the chord contained a rising major or minor third from the fundamental tone. But none of that appeared to have to do anything with the harmonic series. OTOH, when I later started my first web search about microtonal music and the harmonic series, many times I found a term of "fundamental frequency" referring specifically to the relative frequency of 1 in JI chords and some articles were even stressing that distinction should be made between a "fundamental tone" in classical harmony and a "fundamental tone" of the harmonic series, the latter meaning the lowest tone in the series.

> Sure... but how do you quantify this? Hearing doesn't stop
> at integers so your model can't either.

Depends on how high the tones are, I was trying to play various sliding intervals on my synth right today. If the tones are high enough, I can hear something as complex as a 17/13. If they are low, sometimes I can't even notice a simple 9/7.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> > I've not heard many people refer to fundamentals of chords.
> > I've heard them refer to roots of chords, by which usually
> > the VF is meant.
>
> When I was attending classical harmony lessons, some of which
> I was taught at an age of 11 (Daniel Forró may confirm), the
> term "fundamental tone" was always used specifically for the
> one tone in a chord from which a series of rising thirds
> spelled all of its tones -- i.e. a chord of F-G-B-D has a
> fundamental tone of G because G-B-D-F is the way to describe
> the chord as a series of thirds.

OK, fair enough.

> But none of that appeared to have to do anything with the
> harmonic series.

Right, everything's indirect in classical theory. We have
to adapt it to reality. Hilarity ensues. :)

> > Sure... but how do you quantify this? Hearing doesn't stop
> > at integers so your model can't either.
>
> Depends on how high the tones are, I was trying to play
> various sliding intervals on my synth right today. If the
> tones are high enough, I can hear something as complex as
> a 17/13. If they are low, sometimes I can't even notice
> a simple 9/7.

OK. What about irrational intervals?

-Carl

Carl wrote:

> OK. What about irrational intervals?

Well, it would have to be something heavily "out of tune" in order I didn't try to intentionally find a rational approximation for it. What I could think of in such a case is perhaps something like 7-EDO whose thirds and sixths are so neutral that I really don't try to find any other approximation while listening to them, or tunings like 11-EDO or 13-EDO which sound to me, in most cases, like chaos -- with the exception of one piece by Bill Sethares which I was listening to just a few hours ago. :-D Even temperaments like mavila have such a characteristic sound that always makes me think of very vague approximations of some rational chords or intervals to theirs, even though fifths in mavila can be almost 30 cents flat than pure quite happily. But it's not because of the fifths, its because of the other intervals surrounding them.

Petr

>
>
> I think the original question was why minor chords are less
> consonant than major chords.
>
> -Carl
>
Are they really? I didn't mention. Besides it depends on absolute frequency, in low registers everything sound dissonant.

Then who knows? Maybe because smaller interval is lower one from two :-) Chords imitating harmonic series (where intervals are smaller and smaller in the direction up) sound generally better.

Daniel Forro

>
> I think the original question was why minor chords are less
> consonant than major chords.
>
> -Carl
>
>
I add more to this:

Who cares? For me they both are part of one class of consonant sounding chords based on the chain of thirds (or sixths). I used them like this, and as a contrast against much more dissonant chords constructed on the other principles.

What's more interesting to me as a composer is a borderline between consonance and dissonance, that means making the consonant chords to sound dissonantly and dissonant chords to sound consonantly. There are many chords which don't exist in classical harmony theory, even it's impossible to describe their chord structure, or harmonic function, despite all this they sound beautifully. Opposite - even major triad can sound very ugly. During my studies I liked to provoke my teachers with similar things, unfortunately there were no answers from their sides.

Another interesting question are chord inversions, if somebody here can say something to it. From the point of music theory C-E-G is the same chord as E-G-C and G-C-E, even in harmony theory, be it modal or functional, but acoustically they are completely different beasts. Best sounding is of course C-E-G, then G-C-E (both are parts of harmonic series, both have smaller interval up), E-G-C sounds the worse (because smaller interval down?). Also different voicings (narrow, wide), instrumentation and absolute position in frequency range has big effect on the resulting feeling.

Daniel Forro

I agree with the thrust of what you write, but I adore the sound of "first inversion"--

E, G, C

The 5th partial of the E beating deliciously with the G, the 3rd partial of the E bickering and fighting with the C.

If the E is low enough, it's more like a minor chord with a dissonant b6, or something.

wow, I can't keep up, all the microencephalites chiming in...it's a party!

caleb

On Jan 13, 2009, at 7:54 PM, Daniel Forro wrote:

> >
> > I think the original question was why minor chords are less
> > consonant than major chords.
> >
> > -Carl
> >
> >
> I add more to this:
>
> Who cares? For me they both are part of one class of consonant
> sounding chords based on the chain of thirds (or sixths). I used them
> like this, and as a contrast against much more dissonant chords
> constructed on the other principles.
>
> What's more interesting to me as a composer is a borderline between
> consonance and dissonance, that means making the consonant chords to
> sound dissonantly and dissonant chords to sound consonantly. There
> are many chords which don't exist in classical harmony theory, even
> it's impossible to describe their chord structure, or harmonic
> function, despite all this they sound beautifully. Opposite - even
> major triad can sound very ugly. During my studies I liked to provoke
> my teachers with similar things, unfortunately there were no answers
> from their sides.
>
> Another interesting question are chord inversions, if somebody here
> can say something to it. From the point of music theory C-E-G is the
> same chord as E-G-C and G-C-E, even in harmony theory, be it modal or
> functional, but acoustically they are completely different beasts.
> Best sounding is of course C-E-G, then G-C-E (both are parts of
> harmonic series, both have smaller interval up), E-G-C sounds the
> worse (because smaller interval down?). Also different voicings
> (narrow, wide), instrumentation and absolute position in frequency
> range has big effect on the resulting feeling.
>
> Daniel Forro
>
>

> What's more interesting to me as a composer is a borderline between
> consonance and dissonance, that means making the consonant chords to
> sound dissonantly and dissonant chords to sound consonantly. There
> are many chords which don't exist in classical harmony theory, even
> it's impossible to describe their chord structure, or harmonic
> function, despite all this they sound beautifully. Opposite - even
> major triad can sound very ugly. During my studies I liked to provoke
> my teachers with similar things, unfortunately there were no answers
> from their sides.

Especially the latter. Every time I find a way to make some
"dissonance" work in a consonant way, I have a damn near heart attack.
It's awesome.

I haven't heard anything you've written - do you have any copies of
works online?

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@> wrote:
> >
> > > I've not heard many people refer to fundamentals of chords.
> > > I've heard them refer to roots of chords, by which usually
> > > the VF is meant.
> >
> > When I was attending classical harmony lessons, some of which
> > I was taught at an age of 11 (Daniel Forró may confirm), the
> > term "fundamental tone" was always used specifically for the
> > one tone in a chord from which a series of rising thirds
> > spelled all of its tones -- i.e. a chord of F-G-B-D has a
> > fundamental tone of G because G-B-D-F is the way to describe
> > the chord as a series of thirds.
>
> OK, fair enough.
>
> > But none of that appeared to have to do anything with the
> > harmonic series.
>
> Right, everything's indirect in classical theory. We have
> to adapt it to reality. Hilarity ensues. :)
>
> > > Sure... but how do you quantify this? Hearing doesn't stop
> > > at integers so your model can't either.
> >
> > Depends on how high the tones are, I was trying to play
> > various sliding intervals on my synth right today. If the
> > tones are high enough, I can hear something as complex as
> > a 17/13. If they are low, sometimes I can't even notice
> > a simple 9/7.
>
> OK. What about irrational intervals?
>
> -Carl

"OK. What about irrational intervals?" Well there's the rub! On the
one hand we have a rational model which seems to explain musical
harmony to some extent. On the other, we have a 12-t system which is
based on irrationals and therefore by definition cannot have a GCD.
Yet it has the advantage of equalizing all notes such that each tone
is placed in the same harmonic role as every other.

What I suspect - and it is just a theory - is that the 12-t notes in
the real world are not irrational at all, but are actually very high
overtones in the series. Why? First, irrationals are supposed to have
an infinite number of digits eg Flat 5 as sq root of 2 is 1.414...to
infinity. I don't think this would be even possible to realize on,
say, a piano. Second, we have limits of human hearing to consider,
like the schisma. Hindemith makes the point that we can still
recognise a melody when it is out of tune. So maybe each tempered
interval is a "cluster" of possible higher overtones, and that we have
a type of "uncertainty principle" at work? For instance, the harmonics
640 through to 645, taken over 512, all approximate what our brain
registers as a maj 3. The first 640/512 = 5/4, while the last 645/512
gives a very close approximation to the tempered maj 3 as 2 to the
power of 1/3. In fact, I seem to recall that the difference b/w them
is less than the schisma (but I haven't got my calculator). Maybe
playing notes together "pushes" these harmonics out or something to
that effect?

-Rick
>

If you are interested, some small part of my works of all kind is on www.soundclick.com/forrotronics, recently I have added my profile and MP3's here: netnewmusic.ning.com/profile/DanielForro
Still much more is waiting for uploading... Time consuming work, it goes slowly to process old records or make new ones.

Please enjoy!

Daniel Forro

On 14 Jan 2009, at 12:45 PM, Mike Battaglia wrote:

> > What's more interesting to me as a composer is a borderline between
> > consonance and dissonance, that means making the consonant chords to
> > sound dissonantly and dissonant chords to sound consonantly. There
> > are many chords which don't exist in classical harmony theory, even
> > it's impossible to describe their chord structure, or harmonic
> > function, despite all this they sound beautifully. Opposite - even
> > major triad can sound very ugly. During my studies I liked to > provoke
> > my teachers with similar things, unfortunately there were no answers
> > from their sides.
>
> Especially the latter. Every time I find a way to make some
> "dissonance" work in a consonant way, I have a damn near heart attack.
> It's awesome.
>
> I haven't heard anything you've written - do you have any copies of
> works online?
>
> -Mike

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:

> > I think the original question was why minor chords are less
> > consonant than major chords.
>
> Are they really?

Yes, very much so. In fact the effect is so extreme, minor
chords beyond the 9-limit are essentially unusable.

> Who cares?

It's quite important in designing instruments and so on.

> For me they both are part of one class of consonant
> sounding chords based on the chain of thirds (or sixths).

Even in the 5-limit, one may speculate why Bach favored
the minor keys. I would go so far as to say that counterpoint
works better in minor keys, partly because the consonance
of the tonic chord is weaker, and therefore the voices
are more free and not constantly heard as harmonics of it.

> From the point of music theory C-E-G is the same chord as
> E-G-C and G-C-E,

That's true if we accept the notion of octave-equivalence
100%. However, anyone who has ever scored for orchestra
or played jazz piano knows, it is not true 100%. Beyond
the 5-limit, it is even less true. The inversions of the
7-limit otonality are very different in character (try it!).

-Carl

On 14 Jan 2009, at 10:18 AM, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
>
> > > I think the original question was why minor chords are less
> > > consonant than major chords.
> >
> > Are they really?
>
> Yes, very much so. In fact the effect is so extreme, minor
> chords beyond the 9-limit are essentially unusable.
>
>
But depending on the voicing and absolute frequency location there are always some harmonics fighting, in major chord as well as in minor chord. Major: B-C, G-G#, Bb-B-C... Minor: D-Eb-E, Bb-B-C. From this point of view grade of percepted dissonance looks similar.
> > For me they both are part of one class of consonant
> > sounding chords based on the chain of thirds (or sixths).
>
> Even in the 5-limit, one may speculate why Bach favored
> the minor keys. I would go so far as to say that counterpoint
> works better in minor keys, partly because the consonance
> of the tonic chord is weaker, and therefore the voices
> are more free and not constantly heard as harmonics of it.
>
>
Maybe. I think that minor key generally offers more chromatism in functional harmony, more extra-key chords, more harmonic progressions, more rich harmonies in comparison with major... Chord progressions which sound boringly in major sound quite well when emulated in minor tonality, they can be even improved not to sound so commonly. For example:

Major:
Melody tone: G A A B B C....
Chord: C F D7 G E7 Am....
Function: T S (D7) D (D7) VI....

Minor:
Melody tone: G Ab A Bb B C....
Chord: Cm Fm Adim/Eb Bb/D Db7/Ab C7/G...
Function : T S (VII7) VII bII7 (D7)....

(tabs will be probably ignored after sending, please repair the table)
>
> From the point of music theory C-E-G is the same chord as
> > E-G-C and G-C-E,
>
> That's true if we accept the notion of octave-equivalence
> 100%. However, anyone who has ever scored for orchestra
> or played jazz piano knows, it is not true 100%. Beyond
> the 5-limit, it is even less true. The inversions of the
> 7-limit otonality are very different in character (try it!).
>
> -Carl
>
I was referring just about theoretical and abstract 12 tone ET, not about microintervals :-)

Daniel Forro

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:

> > minor chords beyond the 9-limit are essentially unusable.
>
> But depending on the voicing and absolute frequency location there
> are always some harmonics fighting, in major chord as well as in
> minor chord. Major: B-C, G-G#, Bb-B-C... Minor: D-Eb-E, Bb-B-C.
> From this point of view grade of percepted dissonance looks
> similar.

That is exactly why this point of view of dissonance is
incomplete.

> Chord progressions which sound boringly in major sound quite
> well when emulated in minor tonality, they can be even improved
> not to sound so commonly. For example:
>
>Major:
>Melody tone: G A A B B C....
>Chord: C F D7 G E7 Am....
>Function: T S (D7) D (D7) VI....
>
>Minor:
>Melody tone: G Ab A Bb B C....
>Chord: Cm Fm Adim/Eb Bb/D Db7/Ab C7/G...
>Function: T S (VII7) VII bII7 (D7)....

Sure, that's reasonable, and there may be any number of
reasonable explanations. Note I did say, "_partly_ because
the consonance of the tonic chord is weaker". :)

> > > From the point of music theory C-E-G is the same chord as
> > > E-G-C and G-C-E,
> >
> > That's true if we accept the notion of octave-equivalence
> > 100%. However, anyone who has ever scored for orchestra
> > or played jazz piano knows, it is not true 100%. Beyond
> > the 5-limit, it is even less true. The inversions of the
> > 7-limit otonality are very different in character (try it!).
>
> I was referring just about theoretical and abstract 12 tone ET,
> not about microintervals :-)

Call me crazy, but I think any reasonable music theory should
be able to handle both. :)

-Carl