Yahoo Tuning Groups Ultimate Backup tuning Defend your rights with the GCD

Hi everyone,

Aaron K.J's kindly been trying to teach me Csound. So far I've been a very bad student and feel like I understand about 2% of what I'm supposed to. However, I've managed to get my act together enough to do a crude experiment. Using a basic sine wave generator (oscil opcode) and GEN10 sub-routine, I first listened to an A440Hz fundamental. Next I gave the first 19 harmonics equal amplitude (at 1) but cut out 1,2,4,8,and 16 harmonics (i.e. = 0). It looks like this in the .sco file:

f1 0 65536 10 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1

where the nth harmonic is the nth row after the 10. As I always suspected from studying waves, the note was still heard as an A440Hz the only difference being in the timbre (which admittedly sounded awful, but that's beside the point at this stage). I don't know how to do it yet, but I'm sure that if we tweeked each harmonic according to Erlich/Carl's entropy model so that each was slightly out-of-tune, then the result would be pretty much the same.

IOW, what this proves is that frequency = GCD = tonality. And mathematically speaking this applies to all waves without exception. ALL that is required is that two waves exist at the same place and time for this to apply, regardless as to how those waves were generated. That is, independently of 'boundary condition'. Yet if we do a Google search on 'GCD', 'Euclidean Algorithm' etc, NOT A SINGLE MENTION is made of Pythagoras or musical harmony. If we look at a textbook on differential equations, he might get a tiny mention in the back pages under the title 'boundary conditions', this despite the fact that the entire book is unwittingly speaking of wave motion. It is as if the whole world has forgotten that Euclid lived centuries after Pythagoras, that the discovery b/w string lengths and frequency in whole numbers combined the hitherto separate worlds of 'pure' maths, geometry and music, that the proportionality b/w string length and period allowed time to be represented by geometrical lengths in space i.e. the so-called 'fourth dimension'. In short, I would say that the very raison d'etre of the Euclidean Algorithm is that it is a model of musical harmony. It's divorce from music into 'pure' maths was likely the mistake of a group celibate monks during the dark ages. Judging from the standard educational curriculum, it is little wonder that we musicians are not getting paid and left to squabble over the remains! So I suggest that instead of wasting our time fighting over $200, we wage an internet war on all fronts. (Does anyone know how to write in wiki?)

Here endeth the lecture

Rick

🔗Kraig Grady <kraiggrady@...>

🔗Aaron Krister Johnson <aaron@...>

🔗Petr Parízek <p.parizek@...>

🔗Aaron Krister Johnson <aaron@...>

🔗Petr Parízek <p.parizek@...>

🔗Michael <djtrancendance@...>

Rick> "the note was still heard as an A440Hz the only difference being in the
timbre (which admittedly sounded awful, but that's beside the point at
this stage)."

I am a bit confused by your experiment: why are you pulling only the 2^x even harmonics out in your example and how is that related to GCD?

Except...that any two numbers in 2^x apparently have a GCD of 2 (the lowest possible beside a single tone), in which case I agree that keeping harmonics with the lowest GCD positively influences tonality.

   If that is indeed what you're implying, though...I guess my other questions are
1) What happens to the 5th as the 3/2 ratio disappears in your example (especially when there seems nothing wrong on the surface with how a 5th sounds)?
2) On the surface, it looks like your GCD system enforces scales either based on stacked octaves or tritaves with no notes in-between...wouldn't that restriction be a bit too fierce to allow many possible notes and/or degrees of musical expression (or even something as complex as a simple pentatonic scale)?   AKA is there any way to expand this theory to allow for more diverse combinations of ratios?

--- On Thu, 7/16/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@...>
Subject: [tuning] Defend your rights with the GCD
To: tuning@yahoogroups.com
Date: Thursday, July 16, 2009, 11:12 PM

Hi everyone,

f1 0 65536 10 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1

where the nth harmonic is the nth row after the 10. As I always suspected from studying waves, the note was still heard as an A440Hz the only difference being in the timbre (which admittedly sounded awful, but that's beside the point at this stage). I don't know how to do it yet, but I'm sure that if we tweeked each harmonic according to Erlich/Carl' s entropy model so that each was slightly out-of-tune, then the result would be pretty much the same.

IOW, what this proves is that frequency = GCD = tonality. And mathematically speaking this applies to all waves without exception. ALL that is required is that two waves exist at the same place and time for this to apply, regardless as to how those waves were generated. That is, independently of 'boundary condition'. Yet if we do a Google search on 'GCD', 'Euclidean Algorithm' etc, NOT A SINGLE MENTION is made of Pythagoras or musical harmony. If we look at a textbook on differential equations, he might get a tiny mention in the back pages under the title 'boundary conditions', this despite the fact that the entire book is unwittingly speaking of wave motion. It is as if the whole world has forgotten that Euclid lived centuries after Pythagoras, that the discovery b/w string lengths and frequency in whole numbers combined the hitherto separate worlds of 'pure' maths, geometry and music, that the proportionality b/w string length and period allowed time
to be represented by geometrical lengths in space i.e. the so-called 'fourth dimension'. In short, I would say that the very raison d'etre of the Euclidean Algorithm is that it is a model of musical harmony. It's divorce from music into 'pure' maths was likely the mistake of a group celibate monks during the dark ages. Judging from the standard educational curriculum, it is little wonder that we musicians are not getting paid and left to squabble over the remains! So I suggest that instead of wasting our time fighting over $200, we wage an internet war on all fronts. (Does anyone know how to write in wiki?)

Here endeth the lecture

Rick

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

Let's say it did have to do entirely with difference tones. So you
have pure tones consisting of 200, 300, 400, 500, 600, 700, and 800
Hz. The difference tone is 100 Hz. So you would then hear 100, 200,
300, 400, 500, 600, 700, and 800 Hz. This still does nothing to
explain why you hear those eight sinusoids as combining to make one
note of a certain complex timbre... If it really had to do entirely
with difference tones, why wouldn't we still hear those notes as being
8 separate sinusoids?

The short answer is that something else is happening as well. There is
a mechanism in your brain grouping frequencies in harmonic or
near-harmonic series together. We all know the usual explanation is
that you hear a "virtual pitch" which corresponds to the brain's best
estimate of the GCD of all these frequencies, but that might be a bit
misleading. The important thing occurring here is that a
psychoacoustic "perspective shift" is taking place; you go from
hearing separate frequencies to a combination of them all with a
certain timbre. This can happen even if the fundamental is absent, or
if the harmonics don't line up perfectly with a harmonic series.

-Mike

On Fri, Jul 17, 2009 at 10:35 AM, Aaron Krister
Johnson<aaron@...> wrote:
>
>
> Intersting question, Kraig. Instincts tell me that GCD=diff_tones but I
> haven't thought it through rigorously enough to 'prove' it. Wish Gene were
> here to answer.
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>>
>> Hi Rick~
>> why the preference as defining this as The GCD as opposed to the result
>> of difference tones. or is there any way to separate the phenomenon of
>> one from the other.
>> --
>>
>>
>> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
>> Mesotonal Music from:
>> _'''''''_ ^North/Western Hemisphere:
>> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>>
>> _'''''''_ ^South/Eastern Hemisphere:
>> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>>
>> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>>
>> a momentary antenna as i turn to water
>> this evaporates - an island once again
>>
>
>

🔗Carl Lumma <carl@...>

🔗Aaron Krister Johnson <aaron@...>

🔗Kraig Grady <kraiggrady@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Rick> "the note was still heard as an A440Hz the only difference being in the
> timbre (which admittedly sounded awful, but that's beside the point at
> this stage)."
>
>     I am a bit confused by your experiment: why are you pulling only the 2^x even harmonics out in your example and how is that related to GCD?
>
>    Except...that any two numbers in 2^x apparently have a GCD of 2 (the lowest possible beside a single tone), in which case I agree that keeping harmonics with the lowest GCD positively influences tonality.
>
>    If that is indeed what you're implying, though...I guess my other questions are
> 1) What happens to the 5th as the 3/2 ratio disappears in your example (especially when there seems nothing wrong on the surface with how a 5th sounds)?
> 2) On the surface, it looks like your GCD system enforces scales either based on stacked octaves or tritaves with no notes in-between...wouldn't that restriction be a bit too fierce to allow many possible notes and/or degrees of musical expression (or even something as complex as a simple pentatonic scale)?   AKA is there any way to expand this theory to allow for more diverse combinations of ratios?
>
>
>
> --- On Thu, 7/16/09, rick_ballan <rick_ballan@...> wrote:
>
> From: rick_ballan <rick_ballan@...>
> Subject: [tuning] Defend your rights with the GCD
> To: tuning@yahoogroups.com
> Date: Thursday, July 16, 2009, 11:12 PM
>
> Hi Mike,

I pulled out any frequency that could be heard as an A note, which of course are 2^x. Just to make sure that it was not a difference tone, I did another test which is in my reply to Petr.

In answer to your question what happens to the fifth 3/2, well I'm not saying that we're not allowed to have the fifth. It's just an experiment to show that the concept of frequency comes from the GCD and nothing else. To chuck the 1, 2, etc back in doesn't change the fact that the GCD is still A440Hz.

To the second question I'll say that the GCD is not restrictive but expansive. The older idea that we can only obtain the harmonic series via boundary conditions (such as the length of a string or pipe) is also a psychological boundary. The GCD applies also between boundary conditions themselves and when none exist at all. For eg, to the A440 note given in my example, add another frequency E660Hz with the same harmonics. The ratio is 3/2 and the GCD is A220Hz which will be the freq of the new wave. But again no sine wave will have this frequency.

Rick
>
>
>
>
>
>
>
>
>
>
> Hi everyone,
>
>
>
> Aaron K.J's kindly been trying to teach me Csound. So far I've been a very bad student and feel like I understand about 2% of what I'm supposed to. However, I've managed to get my act together enough to do a crude experiment. Using a basic sine wave generator (oscil opcode) and GEN10 sub-routine, I first listened to an A440Hz fundamental. Next I gave the first 19 harmonics equal amplitude (at 1) but cut out 1,2,4,8,and 16 harmonics (i.e. = 0). It looks like this in the .sco file:
>
>
>
> f1 0 65536 10 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1
>
>
>
> where the nth harmonic is the nth row after the 10. As I always suspected from studying waves, the note was still heard as an A440Hz the only difference being in the timbre (which admittedly sounded awful, but that's beside the point at this stage). I don't know how to do it yet, but I'm sure that if we tweeked each harmonic according to Erlich/Carl' s entropy model so that each was slightly out-of-tune, then the result would be pretty much the same.
>
>
>
> IOW, what this proves is that frequency = GCD = tonality. And mathematically speaking this applies to all waves without exception. ALL that is required is that two waves exist at the same place and time for this to apply, regardless as to how those waves were generated. That is, independently of 'boundary condition'. Yet if we do a Google search on 'GCD', 'Euclidean Algorithm' etc, NOT A SINGLE MENTION is made of Pythagoras or musical harmony. If we look at a textbook on differential equations, he might get a tiny mention in the back pages under the title 'boundary conditions', this despite the fact that the entire book is unwittingly speaking of wave motion. It is as if the whole world has forgotten that Euclid lived centuries after Pythagoras, that the discovery b/w string lengths and frequency in whole numbers combined the hitherto separate worlds of 'pure' maths, geometry and music, that the proportionality b/w string length and period allowed time
> to be represented by geometrical lengths in space i.e. the so-called 'fourth dimension'. In short, I would say that the very raison d'etre of the Euclidean Algorithm is that it is a model of musical harmony. It's divorce from music into 'pure' maths was likely the mistake of a group celibate monks during the dark ages. Judging from the standard educational curriculum, it is little wonder that we musicians are not getting paid and left to squabble over the remains! So I suggest that instead of wasting our time fighting over $200, we wage an internet war on all fronts. (Does anyone know how to write in wiki?)
>
>
>
> Here endeth the lecture
>
>
>
> Rick
>

🔗rick_ballan <rick_ballan@...>

🔗Marcel de Velde <m.develde@...>

🔗Carl Lumma <carl@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Let's say it did have to do entirely with difference tones. So you
> have pure tones consisting of 200, 300, 400, 500, 600, 700, and 800
> Hz. The difference tone is 100 Hz. So you would then hear 100, 200,
> 300, 400, 500, 600, 700, and 800 Hz. This still does nothing to
> explain why you hear those eight sinusoids as combining to make one
> note of a certain complex timbre... If it really had to do entirely
> with difference tones, why wouldn't we still hear those notes as being
> 8 separate sinusoids?
>
> The short answer is that something else is happening as well. There is
> a mechanism in your brain grouping frequencies in harmonic or
> near-harmonic series together. We all know the usual explanation is
> that you hear a "virtual pitch" which corresponds to the brain's best
> estimate of the GCD of all these frequencies, but that might be a bit
> misleading. The important thing occurring here is that a
> psychoacoustic "perspective shift" is taking place; you go from
> hearing separate frequencies to a combination of them all with a
> certain timbre. This can happen even if the fundamental is absent, or
> if the harmonics don't line up perfectly with a harmonic series.
>
> -Mike
>
>Hi Mike,

Nicely said. Just to add to your point, I have a tendency to think of it more as an objective reality since the wave composite is what's 'out there' if you know what I mean. The number of cps hitting the ear drum per second will be the GCD. But its interesting how our brain can still 'tune' these detuned harmonics (I call it the Mick Jagger syndrome). We can even see this if we change one wave slightly (There's a sine wave applet on Google). If we draw 60 and 90 say to get a wave of 30 and change the 90 to 91, 92, etc...we see that it takes a while for the wave to change its shape.

Rick
>
> On Fri, Jul 17, 2009 at 10:35 AM, Aaron Krister
> Johnson<aaron@...> wrote:
> >
> >
> > Intersting question, Kraig. Instincts tell me that GCD=diff_tones but I
> > haven't thought it through rigorously enough to 'prove' it. Wish Gene were
> > here to answer.
> >
> > --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@> wrote:
> >>
> >> Hi Rick~
> >> why the preference as defining this as The GCD as opposed to the result
> >> of difference tones. or is there any way to separate the phenomenon of
> >> one from the other.
> >> --
> >>
> >>
> >> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> >> Mesotonal Music from:
> >> _'''''''_ ^North/Western Hemisphere:
> >> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> >>
> >> _'''''''_ ^South/Eastern Hemisphere:
> >> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
> >>
> >> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >>
> >> a momentary antenna as i turn to water
> >> this evaporates - an island once again
> >>
> >
> >
>

🔗Michael <djtrancendance@...>

>It could be that the brain interprets this chord as a 16 19 24 (1/1
19/16 3/2) which makes it >1 as the tonic again. (infact in some music
1/1 6/5 3/2 the minor third sounds very much >out of tune to me, where
1/1 19/16 3/2 for instance does not)

A bizarre note: 1/1 19/16 3/2 has 16 as the least common denominator while 1/1 6/5 3/2 has 10. And, I'll agree: 16 (at least to my ears) sounds better than 10. But, meanwhile, too many root tones at x/16 (or x/any constant denominator) causes so much harmonic distortion that it becomes robotic sounding (to my ears)... It's my personal opinion that trying to summarize all optimum chords as one huge x/16 chord is by FAR an overly simplified method (though I assume it would be a convenient way of quickly programming an adaptive JI MIDI toolkit: rounding each note in the chord to the nearest x/16 from the root tone).

I've also noticed that being able to combine pattern in different ways (IE relatively conform to many patterns at once or make many things near but not at one pattern).

I believe someone had a theory of consonance that rated consonance as "how much complexity something has / how much effort it take to listen to" so the easier it is to listen to despite complexity the more consonance it has. Anyone recall who said that?

-Michael

--- On Sat, 7/18/09, Marcel de Velde <m.develde@...> wrote:

From: Marcel de Velde <m.develde@...>
Subject: Re: [tuning] Re: Defend your rights with the GCD
To: tuning@yahoogroups.com
Date: Saturday, July 18, 2009, 9:15 AM

Well yes but then there's the doubt again if when you play 10 12 15 (1/1 6/5 3/2) if this is what your brain is really interpreting it to be.It could be that the brain interprets this chord as a 16 19 24 (1/1 19/16 3/2) which makes it 1 as the tonic again. (infact in some music 1/1 6/5 3/2 the minor third sounds very much out of tune to me, where 1/1 19/16 3/2 for instance does not)
Yet I also think often 1/1 32/27 3/2 could be the correct tuning.These things have been mentioned many times before but there's no clear answer yet it seems. I'm still working on this myself.

2009/7/18 Kraig Grady <kraiggrady@anaphori a.com>

if i play a 10 12 15 chord i doubt if i would hear 1 as the tonic

🔗Marcel de Velde <m.develde@...>

🔗Mike Battaglia <battaglia01@...>

> I believe someone had a theory of consonance that rated consonance as "how
> much complexity something has / how much effort it take to listen to" so the
> easier it is to listen to despite complexity the more consonance it has.
> Anyone recall who said that?
>
> -Michael

I doubt I'm the only one who's said this, but a while ago I advanced
the theory that the brain gets accustomed to high-entropy intervals
the more it is exposed to them, which is why I see the dualistic
concept of "consonance/dissonance" as more of a single notion of
"complexity." Intervals that you would call "dissonant" are simply
more complex than "consonant" ones, and can be experienced as "complex
consonances" rather than "dissonances" under the right circumstances
(especially the timbre of the instruments used!)

You should also take note that people take will a lot of different
perceptual characteristics and lump them together into the
"consonance/dissonance" terminology. I've heard it used to refer to

1) Critical band effects
2) High-entropy intervals or chords
3) Intervals or chords that are out of place within the harmonic
environment of the piece
4) Intervals, chords, or modes that sound "dark" in character, perhaps
due to cultural or psychological associations
5) Intervals, chords, or modes that the listener simply just does not like

There are plenty of others too. And as you can see, some of these
criteria are more scientific than others. Let's keep in mind exactly
what perceptual quality it is we're trying to uncover when we talk
about "dissonance" and "consonance," then. The notion of "complexity"
I posted above refers only to the second one.

Another interesting idea I've been throwing around - is "consonance"
and "dissonance" really a linear thing when you get into triads and
such? Perhaps it's more like color: we could describe how "colorful" a
certain shade of color is, with gray being the least colorful and any
highly saturated shade being the most colorful, but we would be losing
information on precisely which way the shade is "colored" (red, green,
turquoise). Perhaps there are different "flavors" of consonance when
you get to triads and beyond, and this would more fully describe a
given chord beyond whether as a whole it's "consonant" or "dissonant"
or how "complex" or what not. In fact, this must be true on some
level, since the whole notion of "chord quality" exists.

That might be a way in which the "accordance" of certain intervals
might lead to the higher-level multidimensional experience of "chord
quality." Something to think about.

-Mike

🔗Carl Lumma <carl@...>

🔗Daniel Forró <dan.for@...>

Consonance and dissonance in music are dependent on culture (not every music culture found chords), musical style, tuning, and context. Besides in our Western culture on the music education and listening experience of a listener. I doubt there are objective criteria valid for every brain in the same way. Or maybe yes but a person is not aware about it as he/she has no knowledge about it and missing vocabulary to describe it. You can't expect sophisticated dialogue about Bach's enharmonic-chromatic modulations or using of extra-tonal diminished chord after Napolitan chord with an Amazonia, New Guinea or Australia aborigine (unless he/she has Ph.D. or M.A. in music :-) ). Even not with an average common music consumer.

Daniel Forro

On 19 Jul 2009, at 1:44 PM, Mike Battaglia wrote:

>
> > I believe someone had a theory of consonance that rated > consonance as "how
> > much complexity something has / how much effort it take to listen > to" so the
> > easier it is to listen to despite complexity the more consonance > it has.
> > Anyone recall who said that?
> >
> > -Michael
>
> I doubt I'm the only one who's said this, but a while ago I advanced
> the theory that the brain gets accustomed to high-entropy intervals
> the more it is exposed to them, which is why I see the dualistic
> concept of "consonance/dissonance" as more of a single notion of
> "complexity." Intervals that you would call "dissonant" are simply
> more complex than "consonant" ones, and can be experienced as "complex
> consonances" rather than "dissonances" under the right circumstances
> (especially the timbre of the instruments used!)
>
> You should also take note that people take will a lot of different
> perceptual characteristics and lump them together into the
> "consonance/dissonance" terminology. I've heard it used to refer to
>
> 1) Critical band effects
> 2) High-entropy intervals or chords
> 3) Intervals or chords that are out of place within the harmonic
> environment of the piece
> 4) Intervals, chords, or modes that sound "dark" in character, perhaps
> due to cultural or psychological associations
> 5) Intervals, chords, or modes that the listener simply just does > not like
>
> There are plenty of others too. And as you can see, some of these
> criteria are more scientific than others. Let's keep in mind exactly
> what perceptual quality it is we're trying to uncover when we talk
> about "dissonance" and "consonance," then. The notion of "complexity"
> I posted above refers only to the second one.
>
> Another interesting idea I've been throwing around - is "consonance"
> and "dissonance" really a linear thing when you get into triads and
> such? Perhaps it's more like color: we could describe how "colorful" a
> certain shade of color is, with gray being the least colorful and any
> highly saturated shade being the most colorful, but we would be losing
> information on precisely which way the shade is "colored" (red, green,
> turquoise). Perhaps there are different "flavors" of consonance when
> you get to triads and beyond, and this would more fully describe a
> given chord beyond whether as a whole it's "consonant" or "dissonant"
> or how "complex" or what not. In fact, this must be true on some
> level, since the whole notion of "chord quality" exists.
>
> That might be a way in which the "accordance" of certain intervals
> might lead to the higher-level multidimensional experience of "chord
> quality." Something to think about.
>
> -Mike

🔗Carl Lumma <carl@...>

🔗Daniel Forró <dan.for@...>

🔗Mike Battaglia <battaglia01@...>

🔗Petr Parízek <p.parizek@...>

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

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

> > > Consonance and dissonance in music are dependent on culture
> > > (not every music culture found chords),
> >
> > Yes, but not concordance and discordance, which are
> > psychoacoustic universals. Exhaustive crosscultural studies
> > have yet to be done, but the evidence so far points to this
> > conclusion.
>
> Concordance and discordance were not mentioned in the original
> message. My reaction reflected only what was mentioned -
> consonance and dissonance.

Understood.

> > In real music, these universal perceptions take on musical
> > functions, partly from the grammar in the music itself and
> > partly from the previous experience (culture) of the
> > listeners.
>
> As far as I know, if we talk about music, there's no universal
> grammar common to all music cultures. It's a language with many
> dialects.

I don't really think it has anything to do with culture.
I think it has to do with learning the grammar through
repeated listening. This may or may not be easy for people
to do. I like jazz now, but I didn't when I was a kid.
It took some listening. Other people may get it 'instantly' --
by some chance they were more receptive the first time they
heard it.

These days, I also like circumcision festival music from
Cameroon. What culture tells us is that we can probably
expect an indigenous teenager from Cameroon to have already
learned that grammar, whereas a teenager from the U.S. may
not have (even though both may be circumcised!). So any
grammar can be learned, but as a practical matter there are
some barriers. Even though I love maqam music, I don't have
enough practice with it yet to appreciate it on the same
level than a native Turk might, for example.

> And if your "universal perceptions" concept depends on the
> previous experience (culture) of the listeners,

It doesn't, that's the point. Psychoacoustic experiments
have shown that people from different parts of the world
agree on consonance judgments. As you can tell above, I
also think the grammar part is universal, though each
grammar must be learned by listening to music based on it,
whereas the consonance judgments are learned at a very young
age through exposure to human speech (and the timbre of
human speech changes little across cultures).

-Carl

🔗Mike Battaglia <battaglia01@...>

> It depends on the timbre and on the pitches used. I actually HAVE heard the
> relative frequency of 1 many times. I think there are actually more things
> playing a part here, wto of which I find especially important. One is the
> fact that we somehow try to subconsciously compare chords with the harmonic
> series, which means that when I hear a 10:12:15, the nearest set of „simple“
> numbers is 4:5:6 and the 4 and 6 are actually preserved here, only the 5 is
> lowered by 25/24. Another thing is something which I can’t generazile until
> someone else says if he/she can or can’t hear the same: When listening to
> intervals played using timbres with soft overtones, I can clearly hear
> (let’s call the lower frequency X and the higher one Y) not onlyY-X but also
> 2X-Y. In the case of a perfect fifth, the two are the same, which makes the
> difference tone significantly stronger than in other JI intervals.
>
> Petr

I do think that second and even-order difference tones in general are
likely going to play a bigger role in this than first-order difference
tones, with the exception of the VF. If you have any sound examples,
I'd be happy to tell you what I hear, as I've been paying particular
attention to combination tones within chords lately. I'm curious to
see if your 2X-Y explanation is putting your finger on a particular
perceptual phenomenon I've been noticing for a while.

I've heard the 10:12:15 being heard as an altered 4:5:6 chord
explanation, but I'm not sure if that's the whole story -- if the 4:5
is instead replaced with a neutral 9:11 chord, that one is much
harsher for me to listen to than 10:12:15, even though it's closer to
4:5:6. 10:12:15 perceptually also seems to "lock in" differently for
me than 4:5:6 on a basic level, which I would expect is the common
perception of it.

I've wondered if what's going on to explain the difference between
4:5:6 and 10:12:15 is that the brain might have some degree of
recursion when it places the fundamental. For example, take all of the
dyads in 4:5:6:
4:6 - 2
4:5 - 1
5:6 - 1

And so the fundamentals from each dyad are in a 2:1 ratio, which
itself is an easy dyad to recursively place. How about all of the
dyads in 10:12:15?

10:15 - 5
10:12 - 2
12:15 - 3

2:3:5 should be pretty easy to place, but it certainly isn't as easy
as 2:1. And this chord has a multistable perception whereby the
perceived root is usually either 5, or 3, or sometimes 1 or 2.
12:15:20 is even more ubiquitous.

Just a thought. And it would explain why major 6 chords are usually
perceived with the root they have.

-Mike

🔗Mike Battaglia <battaglia01@...>

> Another thing is something which I can’t generazile until
> someone else says if he/she can or can’t hear the same: When listening to
> intervals played using timbres with soft overtones, I can clearly hear
> (let’s call the lower frequency X and the higher one Y) not onlyY-X but also
> 2X-Y. In the case of a perfect fifth, the two are the same, which makes the
> difference tone significantly stronger than in other JI intervals.
>
> Petr

Also note here, from http://www.silcom.com/~aludwig/Nonlinear.htm:

"A intriguing question is: if there are two frequencies above the
limit of human hearing, say at 23 kHz and 24 kHz, would the
non-linearity cause audible intermodulation products below 20 kHz?
Everest states (page 55) that a difference tone of 1 kHz can be heard
in this case. I did several tests to see if I could hear such a
product, and I could not. Hartmann also states (page 514), without
mentioning the frequencies, that a difference tone can be heard for
tone levels greater than 50 dB SPL. I also could not hear any
difference tones listening to 4 kHz and 5 kHz tones. I thought this
indicated that the non-linearities of the ear were virtually
inaudible. Not true! It turns out that a difference tone of 2f1-f2 is
quite audible, for certain choices of frequencies.

The ear apparently also produces harmonic distortion. This is
difficult to detect directly, but there is convincing indirect
evidence from tests on the audibility of phase differences between a
tone and its second harmonic (see section on audibility of phase)"

-------------

You might be onto something after all. I don't like how the article
separates harmonic distortion from intermodulation distortion, being
as they're really the same thing, but oh well.

-Mike

🔗Petr Parízek <p.parizek@...>

Mike wrote:

> I do think that second and even-order difference tones in general are
> likely going to play a bigger role in this than first-order difference
> tones, with the exception of the VF. If you have any sound examples,
> I'd be happy to tell you what I hear, as I've been paying particular
> attention to combination tones within chords lately. I'm curious to
> see if your 2X-Y explanation is putting your finger on a particular
> perceptual phenomenon I've been noticing for a while.

I think you can try it yourself if you have some ï¿½meansï¿½ to make, for example, a sound of about 20 seconds in length where two sine waves are sounding together, one of which has a stable frequency of 1000Hz and the other gradually rises from 1400Hz to 1600Hz.

> I've heard the 10:12:15 being heard as an altered 4:5:6 chord
> explanation, but I'm not sure if that's the whole story -- if the 4:5
> is instead replaced with a neutral 9:11 chord, that one is much
> harsher for me to listen to than 10:12:15, even though it's closer to
> 4:5:6. 10:12:15 perceptually also seems to "lock in" differently for
> me than 4:5:6 on a basic level, which I would expect is the common
> perception of it.

For me there is the problem that if I change the 5/4 to 11/9, then the top dyad is 27/22, which is such a complex factor that I simply canï¿½t hear the ï¿½momentï¿½ when the interval is the right size, regardless of the fact that itï¿½s close to 9/11. Itï¿½s a bit similar to comparing 4:6:9 to 16:20:25. While 4:6:9 sounds okay to me, 16:20:25 is just like chaos because 25/16 simply canï¿½t be treated as a ï¿½relaxedï¿½ sound.

>I've wondered if what's going on to explain the difference between
> 4:5:6 and 10:12:15 is that the brain might have some degree of
> recursion when it places the fundamental.

Some months ago, there was a very ï¿½livelyï¿½ discussion here about this topic and I remember myself talking about the fact that when I listen to 10:12:15 in some high octaves, I can often clearly hear these tones (especially if the overtones are soft enough), which is why I sometimes view a high-pitched 10:12:15 as a kind of major 7th chord with a missing 8.

> Just a thought. And it would explain why major 6 chords are usually
> perceived with the root they have.

Why particularly major 6 chords?

Petr

🔗Klaus Schmirler <KSchmir@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Indeed. I wonder if it has to do with the general Fourier uncertainty
> principle whereby any signal can have multiple time-frequency
> representations... If it takes 3 seconds to hear an audible difference
> between x:y and x:y+0.0001, then how does that manifest itself in
> terms of the brain's time-frequency representation of the signal?
>
> -Mike
>
That's interesting. I hadn't thought of using time from a psychoacoustic standpoint. On a related note, I asked a similar question from a more mathematical stand point (I think we already spoke about this a while ago?): since frequency is the inverse of period, which is itself a length of time, then the idea of 'instantaneous frequency' must be nonsense. IOW we can't apply calculus to frequencies. But then how do we model changing freq over time, for eg a constantly increasing pitch? The only answer I could think of was that the period is still the time required for one cycle so that the frequency lasts for the length of its own period. See "Chirp function".

Rick

> >>Hi Mike,
> >
> > Nicely said. Just to add to your point, I have a tendency to think of it
> > more as an objective reality since the wave composite is what's 'out there'
> > if you know what I mean. The number of cps hitting the ear drum per second
> > will be the GCD. But its interesting how our brain can still 'tune' these
> > detuned harmonics (I call it the Mick Jagger syndrome). We can even see this
> > if we change one wave slightly (There's a sine wave applet on Google). If we
> > draw 60 and 90 say to get a wave of 30 and change the 90 to 91, 92, etc...we
> > see that it takes a while for the wave to change its shape.
> >
> > Rick
>

🔗rick_ballan <rick_ballan@...>

🔗Carl Lumma <carl@...>

🔗Daniel Forró <dan.for@...>

On 19 Jul 2009, at 6:05 PM, Carl Lumma wrote:

>
> --- In tuning@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
>
> > As far as I know, if we talk about music, there's no universal
> > grammar common to all music cultures. It's a language with many
> > dialects.
>
> I don't really think it has anything to do with culture.
> I think it has to do with learning the grammar through
> repeated listening. This may or may not be easy for people
> to do. I like jazz now, but I didn't when I was a kid.
> It took some listening. Other people may get it 'instantly' --
> by some chance they were more receptive the first time they
> heard it.
>

Very well said, I think the same - education can change culture. When
I talk about culture here, I mean some "local music grammar" which is
imprinted in the brain thanks to the ambient where a person lives.
But anyway last 50 years or so such pure type of culture disappeared,
as globalisation and media "polluted" and destroyed this, and there
are no isolated regions with preserved traditional culture. Anybody
has chance to hear any music anywhere anytime.

I don't want to say it's bad or good, just that such process started
and can't be stopped. Nobody can judge what it will bring for future
of music.

Extreme example of this is Hawaiian music. It was heavily influenced
by European music since the beginning of the 19th century, and lot of
the songs were composed by a German musician Berger...

Of course a lot of cross styles and fusions are interesting and we
take them many years after they were created like one style...

I have the same story concerning jazz as you :-)

> These days, I also like circumcision festival music from
> Cameroon. What culture tells us is that we can probably
> expect an indigenous teenager from Cameroon to have already
> learned that grammar, whereas a teenager from the U.S. may
> not have (even though both may be circumcised!). So any
> grammar can be learned, but as a practical matter there are
> some barriers. Even though I love maqam music, I don't have
> enough practice with it yet to appreciate it on the same
> level than a native Turk might, for example.
>

Exactly this I call "culture", something what's imprinted in genes,
or at least learned by repeated listening since the very early
childhood. Still there's a practical problem - not every mother sings
Schönberg twelve tone rows, or Haba quarter tone arias as
lullabies :-), and there's more possibility to hear some mainstream
world pop from local radio station than Debussy...

An average teenager here in Japan has very little experience with
Japanese traditional music, as there's not much chances to listen it
without some effort in finding it... Then it's difficult to talk
about local culture at all.

> > And if your "universal perceptions" concept depends on the
> > previous experience (culture) of the listeners,
>
> It doesn't, that's the point. Psychoacoustic experiments
> have shown that people from different parts of the world
> agree on consonance judgments. As you can tell above, I
> also think the grammar part is universal, though each
> grammar must be learned by listening to music based on it,
> whereas the consonance judgments are learned at a very young
> age through exposure to human speech (and the timbre of
> human speech changes little across cultures).
>
> -Carl
>

That's interesting. Could you recommend some links about such
research? I would say there's some knowledge about linguistics at my
side, and sound of some languages is far from consonance. Then there
must be differences in understanding what's consonant.

Daniel Forro

🔗Mike Battaglia <battaglia01@...>

> That's interesting. I hadn't thought of using time from a psychoacoustic
> standpoint. On a related note, I asked a similar question from a more
> mathematical stand point (I think we already spoke about this a while ago?):
> since frequency is the inverse of period, which is itself a length of time,
> then the idea of 'instantaneous frequency' must be nonsense. IOW we can't
> apply calculus to frequencies. But then how do we model changing freq over
> time, for eg a constantly increasing pitch? The only answer I could think of
> was that the period is still the time required for one cycle so that the
> frequency lasts for the length of its own period. See "Chirp function".

There are a few times in which the idea of "instantaneous frequency"
is defined - namely as the derivative of "instantaneous phase", which
might apply in the chirp scenario. However, the ear doesn't hear the
whole signal as having one "instantaneous frequency" at each second,
but as having multiple frequencies going on at any given time. If you
ran the signal through a filterbank like the one in the ear (of which
the filters change constantly!) and then got the instantaneous
frequency and amplitude of THAT output, then you might have something
closer to what we hear. I haven't actually experimented with that
method, so I can't say how accurate it is.

However, note that running the signal through a filterbank at all will
immediately produce a mixed time-frequency representation, and the
amount of time-frequency representations out there are infinite. The
question can be reformulated as "how far back in time is the brain
looking when it performs frequency analysis" (and is it even the same
for all frequencies)?

Your idea, however, of finding the "smallest unit" of a frequency is
the basic idea behind the wavelet transform. Although, if you take one
period of a sine wave, followed by an inverted period of another sine
wave, and loop that -- you're going to have complete "periods" of sine
waves in there, and I guarantee you it won't sound anything like a
sine wave. This is more or less off topic, so see
http://en.wikipedia.org/wiki/Continuous_wavelet_transform for more
about all of that.

> Instead of thinking of the x-axis (frequencies) as sine waves we think of
> them as gcd's. AND/OR we apply harmonic entropy to the upper harmonics in
> any combination and define frequency as approximating gcd's.

It's a good idea, and I've thought of it too. You're basically trying
to use sawtooth waves as basis functions rather than sine waves. But
we would then have to find some way to encode the actual timbre of
each wave, as well as account for inharmonic effects and distortions
in timbre and such. I think this would only work really well for
time-frequency representations. If you want to work out the math
involved, I'd be happy to discuss it with you offlist. Also check out
http://en.wikipedia.org/wiki/Harmonic_wavelet_transform.

-Mike

🔗William Gard <billygard@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > It depends on the timbre and on the pitches used. I actually HAVE heard the
> > relative frequency of 1 many times. I think there are actually more things
> > playing a part here, wto of which I find especially important. One is the
> > fact that we somehow try to subconsciously compare chords with the harmonic
> > series, which means that when I hear a 10:12:15, the nearest set of simple"
> > numbers is 4:5:6 and the 4 and 6 are actually preserved here, only the 5 is
> > lowered by 25/24. Another thing is something which I can't generazile until
> > someone else says if he/she can or can't hear the same: When listening to
> > intervals played using timbres with soft overtones, I can clearly hear
> > (let's call the lower frequency X and the higher one Y) not onlyY-X but also
> > 2X-Y. In the case of a perfect fifth, the two are the same, which makes the
> > difference tone significantly stronger than in other JI intervals.
> >
> > Petr
>
> I do think that second and even-order difference tones in general are
> likely going to play a bigger role in this than first-order difference
> tones, with the exception of the VF. If you have any sound examples,
> I'd be happy to tell you what I hear, as I've been paying particular
> attention to combination tones within chords lately. I'm curious to
> see if your 2X-Y explanation is putting your finger on a particular
> perceptual phenomenon I've been noticing for a while.
>
> I've heard the 10:12:15 being heard as an altered 4:5:6 chord
> explanation, but I'm not sure if that's the whole story -- if the 4:5
> is instead replaced with a neutral 9:11 chord, that one is much
> harsher for me to listen to than 10:12:15, even though it's closer to
> 4:5:6. 10:12:15 perceptually also seems to "lock in" differently for
> me than 4:5:6 on a basic level, which I would expect is the common
> perception of it.
>
> I've wondered if what's going on to explain the difference between
> 4:5:6 and 10:12:15 is that the brain might have some degree of
> recursion when it places the fundamental. For example, take all of the
> dyads in 4:5:6:
> 4:6 - 2
> 4:5 - 1
> 5:6 - 1
>
> And so the fundamentals from each dyad are in a 2:1 ratio, which
> itself is an easy dyad to recursively place. How about all of the
> dyads in 10:12:15?
>
> 10:15 - 5
> 10:12 - 2
> 12:15 - 3
>
> 2:3:5 should be pretty easy to place, but it certainly isn't as easy
> as 2:1. And this chord has a multistable perception whereby the
> perceived root is usually either 5, or 3, or sometimes 1 or 2.
> 12:15:20 is even more ubiquitous.
>
> Just a thought. And it would explain why major 6 chords are usually
> perceived with the root they have.
>
> -Mike
>
Hi Mike and Petr,

I've said it before but strictly speaking 10:12:15 is the second relative minor in a major key i.e. Emin triad in key of Cmaj. A possibility for a minor key might be something like 32:38:48:57 which still has two perfect fifths as the surrounding interval for both maj and min triads and is therefore 'simple' enough for the ear to recognise. Here the maj 3 48/38 = 24/19 is sharp, but it might sound good in the minor key (eg Ebmaj in key of C min, relative major)??

While I'm aware of the tests done on entropy etc, I still suspect that our instinctive avoidance of higher harmonics comes from the fact that if we concentrate on individual instruments then (as you know) the amplitude of upper partials decrease the higher you go. This gives the impression that the upper partials have less strength than the lower and therefore less meaning. But the gcd of course lifts this unrealistic restriction. Besides, if the brain seems to reduce them to simple whole numbers anyway, then what's the problem?

At any rate, I simply don't think difference tones have the correct properties to model music (IMO a useful model should try to uncover simple general principles, not bog us down with every conceivable detail which can only be attended to in writing music anyway. Why spend our time writing an algorithm for "Ode to Joy" when we can write the actual music?). Sure, difference tones might play a part, but only a mitigating one. At the end of the day what we are really looking for in diff tones is what ratios they form with the tonic. And this is modelled by the fact that if 'freq = gcd' then we are really looking for diff's b/w two gcd's to see what ratio it forms with a third.

Rick

🔗Mike Battaglia <battaglia01@...>

> I've said it before but strictly speaking 10:12:15 is the second relative
> minor in a major key i.e. Emin triad in key of Cmaj. A possibility for a
> minor key might be something like 32:38:48:57 which still has two perfect
> fifths as the surrounding interval for both maj and min triads and is
> therefore 'simple' enough for the ear to recognise. Here the maj 3 48/38 =
> 24/19 is sharp, but it might sound good in the minor key (eg Ebmaj in key of
> C min, relative major)??

I think there are times in which a minor triad is going to be heard as
an upper extension of an overall major tonic, which might indicate
that it's being perceived as 16:19:24. I've often wondered if the
equal tempered m3, when played over a Mixolydian modality, is somehow
processed as being 16:19. It certainly does have a completely
different sound in that context than the usual "minor" sonority that
we're used to. I've also heard that "the blues b3" interval is 7:6,
but given that there are a thousand different b3's used in different
contexts in the blues (including 11:9), I can't see why 16:19 wouldn't
be in there as well.

But even with stuff like that though, it's hard to tell if the
different chord quality percept really does have to do with the
perceptual quality of experiencing an interval as being 16:19 vs 5:6
or anything like that.

> While I'm aware of the tests done on entropy etc, I still suspect that our
> instinctive avoidance of higher harmonics comes from the fact that if we
> concentrate on individual instruments then (as you know) the amplitude of
> upper partials decrease the higher you go. This gives the impression that
> the upper partials have less strength than the lower and therefore less
> meaning. But the gcd of course lifts this unrealistic restriction. Besides,
> if the brain seems to reduce them to simple whole numbers anyway, then
> what's the problem?

Because sometimes the gcd of an individual dyad in the chord or an
upper structure triad in the chord is so strong that it tends to be
heard as the "root" of the chord itself. Take a major 6 chord. C E G
A. Technically, the GCD of that is going to be an F a few octaves
down, but I'll be extremely surprised if you hear the root of that
chord as being anything besides a C. When I hear a chord like that (or
10:12:15, for that matter), I do hear another faint note "A" at the
bottom, perhaps from the E-A dyad, which doesn't quite mix with the
4:5:6 so evenly. But it's a real stretch for me to then go all the way
down and place that A and the C from the C-E-G and hear the F at the
bottom.

If you play the notes "A" and "C", you might hear that as the upper
partials of an F major triad for sure. If you play the note "A" an
octave below the "C", it might a bit more difficult to hear it that
way. If you play the note "A" an octave below the "C" and play the
note "A" at 1/4 the volume of the "C", it's unlikely that you're going
to hear it that way at all. That's why the simple straightforward GCD
approach might not work in all cases.

Further complicating things is that the notes you imagine in your head
are going to change the way you hear the chord as well. I did an
experiment once where I played "happy birthday" on my keyboard on a
timbre with just harmonics 1 and 3 in. I slowly lowered the level of
the first harmonic until only 3 was left. While playing like that, I
could still hear the VF corresponding to what the note was before the
1 dropped out, if I concentrated on what the note was "supposed" to be
and followed myself mentally like that. The perception was fragile,
and often I'd revert to hearing that isolated 3 as its own 1, but I
got better at it with practice. Without the 1 even being there, I got
it to sound like a 1:3 timbre just being run through an extreme high
pass filter.

So the individual's own mental energy can, to a certain extent, push
the perception of an interval into a certain direction. This is pretty
easy to test yourself just by listening to a 350-cent third and trying
to hear it as a minor third, and then as a major third, and keep
flipping your perception of it like that. Not that this invalidates
any theory you'd make, but it is a point worth noting. It is, of
course, much more difficult to hear a major third as a fourth or
anything like that.

-Mike

🔗Mike Battaglia <battaglia01@...>

> Because sometimes the gcd of an individual dyad in the chord or an
> upper structure triad in the chord is so strong that it tends to be
> heard as the "root" of the chord itself. Take a major 6 chord. C E G
> A. Technically, the GCD of that is going to be an F a few octaves
> down, but I'll be extremely surprised if you hear the root of that
> chord as being anything besides a C. When I hear a chord like that (or
> 10:12:15, for that matter), I do hear another faint note "A" at the
> bottom, perhaps from the E-A dyad, which doesn't quite mix with the
> 4:5:6 so evenly. But it's a real stretch for me to then go all the way
> down and place that A and the C from the C-E-G and hear the F at the
> bottom.

One more thing about this is that you can also hear the root of that
chord as being an A, as if it were an Am7 chord in inversion. You can
mentally "flip back and forth" between the two perceived roots if you
want. I think this corresponds to a figure-ground switch whereby you
hear the low C from the C-E-G, combined with the C-E-G itself, as
being dominant, and the synchronicity between the "G-A" on top as part
of the "background noise" of the environment; or hearing the low "A"
as dominant and the low C as part of the background noise.
Simultaneously grasping the two of them as being the upper extensions
of a third, much lower note is of course also possible. Playing the
corresponding root that you want in the bass only weakens the other
one.

But these perceptual things are important (and extremely interesting),
and it's just a good thing to keep them in mind, IMO.

-Mike

🔗Carl Lumma <carl@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I've said it before but strictly speaking 10:12:15 is the second relative
> > minor in a major key i.e. Emin triad in key of Cmaj. A possibility for a
> > minor key might be something like 32:38:48:57 which still has two perfect
> > fifths as the surrounding interval for both maj and min triads and is
> > therefore 'simple' enough for the ear to recognise. Here the maj 3 48/38 =
> > 24/19 is sharp, but it might sound good in the minor key (eg Ebmaj in key of
> > C min, relative major)??
>
> I think there are times in which a minor triad is going to be heard as
> an upper extension of an overall major tonic, which might indicate
> that it's being perceived as 16:19:24. I've often wondered if the
> equal tempered m3, when played over a Mixolydian modality, is somehow
> processed as being 16:19. It certainly does have a completely
> different sound in that context than the usual "minor" sonority that
> we're used to. I've also heard that "the blues b3" interval is 7:6,
> but given that there are a thousand different b3's used in different
> contexts in the blues (including 11:9), I can't see why 16:19 wouldn't
> be in there as well.
>
> But even with stuff like that though, it's hard to tell if the
> different chord quality percept really does have to do with the
> perceptual quality of experiencing an interval as being 16:19 vs 5:6
> or anything like that.
>
Hi Mike. In fact 16:19 is closer to the tempered m3 than 5:6. On this point I've always argued the possibility that the ET minor has created a new "phenomenon" which has a logic outside historical JI i.e. it evolved as a new consequence of 12 tet not JI but could be approx rational numbers anyway.

> > While I'm aware of the tests done on entropy etc, I still suspect that our
> > instinctive avoidance of higher harmonics comes from the fact that if we
> > concentrate on individual instruments then (as you know) the amplitude of
> > upper partials decrease the higher you go. This gives the impression that
> > the upper partials have less strength than the lower and therefore less
> > meaning. But the gcd of course lifts this unrealistic restriction. Besides,
> > if the brain seems to reduce them to simple whole numbers anyway, then
> > what's the problem?
>
> Because sometimes the gcd of an individual dyad in the chord or an
> upper structure triad in the chord is so strong that it tends to be
> heard as the "root" of the chord itself. Take a major 6 chord. C E G
> A. Technically, the GCD of that is going to be an F a few octaves
> down, but I'll be extremely surprised if you hear the root of that
> chord as being anything besides a C. When I hear a chord like that (or
> 10:12:15, for that matter), I do hear another faint note "A" at the
> bottom, perhaps from the E-A dyad, which doesn't quite mix with the
> 4:5:6 so evenly. But it's a real stretch for me to then go all the way
> down and place that A and the C from the C-E-G and hear the F at the
> bottom.

But actually Mike I play 6 chords all the time and there is no ambiguity. First, the strength of the major triad clearly outweighs the presence of the maj 6 (which is generally regarded as one of the neutral 'colour' notes). Secondly, A can belong to the HS of C tonic without F. For eg, the open Pyth 27/16 is a distinct possibility, nice odd number over an 8ve. While thirdly, it is also possible that the gcd is an A since C6 = Am is just another inversion of Am i.e. relative major.
>
> If you play the notes "A" and "C", you might hear that as the upper
> partials of an F major triad for sure. If you play the note "A" an
> octave below the "C", it might a bit more difficult to hear it that
> way. If you play the note "A" an octave below the "C" and play the
> note "A" at 1/4 the volume of the "C", it's unlikely that you're going
> to hear it that way at all. That's why the simple straightforward GCD
> approach might not work in all cases.

Sorry Mike, have to disagree with you there. If you play A and C without anything else, I bet that most people like myself would immediately hear it as A minor every time. This is because thirds from the tonic give a much stronger tonality than thirds from the third to the fifth. If I was asked to write a basic tune for two horns alone, I would always choose standard 3rds, 10ths and 6ths. If asked to write two horns with accompaniment, I'd usually choose the guide tones b/w 3rds and 7ths. And remember that I'm talking about rules of harmony here, an idealistic set of useful guidelines independent of volume and timbre, not musical composition where all factors are taken into account (the periodicity of a wave is independent of amplitude and has many choice of harmonics). I'm not for instance disagreeing with what you tell me about psychoacoustics which are very interesting but just categorising them under 'composition'.

Much of the difficulty with simple first principles is that they appear obvious and therefore trivial. In fact they usually represent the solution to a much bigger problem and it is only when we put them back into action in more complex examples that their breadth once again becomes apparent. For eg, I said that 10:12:15 is an Em in the key of C maj, as opposed to an Em 16:19:24 in its own key. Or the Pyth minor 27:32 could be seen as an A min in the key of C maj. 54:64:81 for the A min triad now has the Pyth maj 3rd as well as the latter interval. Now here we find our two basic minor chord substitutions for a C major. By extending these three we obtain all possibilities. If we chuck an F in the mix to obtain D min (and strictly speaking a new A min), then this is basically a change of key (placing the C in the role of 5th). What other model has the ability to distinguish between root note and tonic? But besides this, the gcd is what happens when we add waves. It is a universal principle.
>
> Further complicating things is that the notes you imagine in your head
> are going to change the way you hear the chord as well. I did an
> experiment once where I played "happy birthday" on my keyboard on a
> timbre with just harmonics 1 and 3 in. I slowly lowered the level of
> the first harmonic until only 3 was left. While playing like that, I
> could still hear the VF corresponding to what the note was before the
> 1 dropped out, if I concentrated on what the note was "supposed" to be
> and followed myself mentally like that. The perception was fragile,
> and often I'd revert to hearing that isolated 3 as its own 1, but I
> got better at it with practice. Without the 1 even being there, I got
> it to sound like a 1:3 timbre just being run through an extreme high
> pass filter.
>
> So the individual's own mental energy can, to a certain extent, push
> the perception of an interval into a certain direction. This is pretty
> easy to test yourself just by listening to a 350-cent third and trying
> to hear it as a minor third, and then as a major third, and keep
> flipping your perception of it like that. Not that this invalidates
> any theory you'd make, but it is a point worth noting. It is, of
> course, much more difficult to hear a major third as a fourth or
> anything like that.
>
> -Mike
>
That's interesting Mike. I remember Leonard Bernstein pointing out that the "ner ner na ner ner song" (5th, maj 3rd and maj 6th) is teasing because we always expect to hear the tonic but never do. A smart arse friend said that we could hear the A min, but I said it was likely a passing note. In any case, I know exactly what you mean. Sometimes I get so lazy at a gig I'll play only the first few notes of a known melody and let the listener imagine the rest.

Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Because sometimes the gcd of an individual dyad in the chord or an
> > upper structure triad in the chord is so strong that it tends to be
> > heard as the "root" of the chord itself. Take a major 6 chord. C E G
> > A. Technically, the GCD of that is going to be an F a few octaves
> > down, but I'll be extremely surprised if you hear the root of that
> > chord as being anything besides a C. When I hear a chord like that (or
> > 10:12:15, for that matter), I do hear another faint note "A" at the
> > bottom, perhaps from the E-A dyad, which doesn't quite mix with the
> > 4:5:6 so evenly. But it's a real stretch for me to then go all the way
> > down and place that A and the C from the C-E-G and hear the F at the
> > bottom.
>
> One more thing about this is that you can also hear the root of that
> chord as being an A, as if it were an Am7 chord in inversion. You can
> mentally "flip back and forth" between the two perceived roots if you
> want. I think this corresponds to a figure-ground switch whereby you
> hear the low C from the C-E-G, combined with the C-E-G itself, as
> being dominant, and the synchronicity between the "G-A" on top as part
> of the "background noise" of the environment; or hearing the low "A"
> as dominant and the low C as part of the background noise.
> Simultaneously grasping the two of them as being the upper extensions
> of a third, much lower note is of course also possible. Playing the
> corresponding root that you want in the bass only weakens the other
> one.
>
> But these perceptual things are important (and extremely interesting),
> and it's just a good thing to keep them in mind, IMO.
>
> -Mike
>
Ah I spoke too soon. Yes I agree with you totally. In one of the Bach violin concerto's he plays a violin on one note over and over while the rest modulate. He then resolves to that note and you think its finished, but he keeps on going modulating further.

PS: On the topic of 'instantaneous frequency" I'll send you a doc I wrote years ago.

Cheers mate

Rick

🔗Mike Battaglia <battaglia01@...>

> Hi Mike. In fact 16:19 is closer to the tempered m3 than 5:6. On this point
> I've always argued the possibility that the ET minor has created a new
> "phenomenon" which has a logic outside historical JI i.e. it evolved as a
> new consequence of 12 tet not JI but could be approx rational numbers
> anyway.

And 19:24 is closer to the equal tempered major 3 than 5:4, but I
doubt you're going to hear it that way most times.

> But actually Mike I play 6 chords all the time and there is no ambiguity.
> First, the strength of the major triad clearly outweighs the presence of the
> maj 6 (which is generally regarded as one of the neutral 'colour' notes).
> Secondly, A can belong to the HS of C tonic without F. For eg, the open Pyth
> 27/16 is a distinct possibility, nice odd number over an 8ve. While thirdly,
> it is also possible that the gcd is an A since C6 = Am is just another
> inversion of Am i.e. relative major.

Putting 27/16 would work in theory, but putting 5/3 in sounds much
less dissonant and resonates much more. And I'm specifically talking
about the 5/3 version of that chord.

And as you said yourself, if the major 6 chord can be heard equally as
being a maj6 chord and an m7 chord in inversion... That's the
ambiguity I'm talking about.

> Sorry Mike, have to disagree with you there. If you play A and C without
> anything else, I bet that most people like myself would immediately hear it
> as A minor every time. This is because thirds from the tonic give a much
> stronger tonality than thirds from the third to the fifth. If I was asked to
> write a basic tune for two horns alone, I would always choose standard 3rds,
> 10ths and 6ths. If asked to write two horns with accompaniment, I'd usually
> choose the guide tones b/w 3rds and 7ths. And remember that I'm talking
> about rules of harmony here, an idealistic set of useful guidelines
> independent of volume and timbre, not musical composition where all factors
> are taken into account (the periodicity of a wave is independent of
> amplitude and has many choice of harmonics). I'm not for instance
> disagreeing with what you tell me about psychoacoustics which are very
> interesting but just categorising them under 'composition'.

You might be right there. I guess my example went a bit too far. But
it does still seem to apply to difference tones, or possible choices
of GCD's, generated when you play a minor chord.

> Much of the difficulty with simple first principles is that they appear
> obvious and therefore trivial. In fact they usually represent the solution
> to a much bigger problem and it is only when we put them back into action in
> more complex examples that their breadth once again becomes apparent. For
> eg, I said that 10:12:15 is an Em in the key of C maj, as opposed to an Em
> 16:19:24 in its own key. Or the Pyth minor 27:32 could be seen as an A min
> in the key of C maj. 54:64:81 for the A min triad now has the Pyth maj 3rd
> as well as the latter interval. Now here we find our two basic minor chord
> substitutions for a C major. By extending these three we obtain all
> possibilities. If we chuck an F in the mix to obtain D min (and strictly
> speaking a new A min), then this is basically a change of key (placing the C
> in the role of 5th). What other model has the ability to distinguish between
> root note and tonic? But besides this, the gcd is what happens when we add
> waves. It is a universal principle.

That doesn't change the fact that the perceived root of 10:12:15
usually isn't its GCD. And just because 16:19 is closer to 300 cents
than 5:6 doesn't mean that every minor third we hear is perceived that
way. The perceived musical root of a chord has a lot to do with the
listener's imagination.

> That's interesting Mike. I remember Leonard Bernstein pointing out that the
> "ner ner na ner ner song" (5th, maj 3rd and maj 6th) is teasing because we
> always expect to hear the tonic but never do. A smart arse friend said that
> we could hear the A min, but I said it was likely a passing note. In any
> case, I know exactly what you mean. Sometimes I get so lazy at a gig I'll
> play only the first few notes of a known melody and let the listener imagine
> the rest.

Hence the "ghost note" technique commonly used among bebop players.

-Mike

🔗Carl Lumma <carl@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
Just one thing Mike, you said

"That doesn't change the fact that the perceived root of 10:12:15
> usually isn't its GCD. And just because 16:19 is closer to 300 cents
> than 5:6 doesn't mean that every minor third we hear is perceived that
> way. The perceived musical root of a chord has a lot to do with the
> listener's imagination."

I fully agree that context, conditioning and imagination, loudness and emphasis will all not only play a vital role but will override the gcd everytime. For eg, in a tempered system there would be no distinction between an E min in the key of C maj and an E min in its own key. Both are physically exactly the same chord. Having said that, the problem with this is that it doesn't give us any indication as to how to make that distinction. And when a musician learns harmony, he/she knows that the "same" note is not the same at all if you know what I mean. In JI, they would probably have different frequencies. It is here in teaching or forming rules of harmony that the gcd might come into its own. It behaves like music.

Rick

> > Hi Mike. In fact 16:19 is closer to the tempered m3 than 5:6. On this point
> > I've always argued the possibility that the ET minor has created a new
> > "phenomenon" which has a logic outside historical JI i.e. it evolved as a
> > new consequence of 12 tet not JI but could be approx rational numbers
> > anyway.
>
> And 19:24 is closer to the equal tempered major 3 than 5:4, but I
> doubt you're going to hear it that way most times.
>
>
> > But actually Mike I play 6 chords all the time and there is no ambiguity.
> > First, the strength of the major triad clearly outweighs the presence of the
> > maj 6 (which is generally regarded as one of the neutral 'colour' notes).
> > Secondly, A can belong to the HS of C tonic without F. For eg, the open Pyth
> > 27/16 is a distinct possibility, nice odd number over an 8ve. While thirdly,
> > it is also possible that the gcd is an A since C6 = Am is just another
> > inversion of Am i.e. relative major.
>
> Putting 27/16 would work in theory, but putting 5/3 in sounds much
> less dissonant and resonates much more. And I'm specifically talking
> about the 5/3 version of that chord.
>
> And as you said yourself, if the major 6 chord can be heard equally as
> being a maj6 chord and an m7 chord in inversion... That's the
> ambiguity I'm talking about.
>
> > Sorry Mike, have to disagree with you there. If you play A and C without
> > anything else, I bet that most people like myself would immediately hear it
> > as A minor every time. This is because thirds from the tonic give a much
> > stronger tonality than thirds from the third to the fifth. If I was asked to
> > write a basic tune for two horns alone, I would always choose standard 3rds,
> > 10ths and 6ths. If asked to write two horns with accompaniment, I'd usually
> > choose the guide tones b/w 3rds and 7ths. And remember that I'm talking
> > about rules of harmony here, an idealistic set of useful guidelines
> > independent of volume and timbre, not musical composition where all factors
> > are taken into account (the periodicity of a wave is independent of
> > amplitude and has many choice of harmonics). I'm not for instance
> > disagreeing with what you tell me about psychoacoustics which are very
> > interesting but just categorising them under 'composition'.
>
> You might be right there. I guess my example went a bit too far. But
> it does still seem to apply to difference tones, or possible choices
> of GCD's, generated when you play a minor chord.
>
> > Much of the difficulty with simple first principles is that they appear
> > obvious and therefore trivial. In fact they usually represent the solution
> > to a much bigger problem and it is only when we put them back into action in
> > more complex examples that their breadth once again becomes apparent. For
> > eg, I said that 10:12:15 is an Em in the key of C maj, as opposed to an Em
> > 16:19:24 in its own key. Or the Pyth minor 27:32 could be seen as an A min
> > in the key of C maj. 54:64:81 for the A min triad now has the Pyth maj 3rd
> > as well as the latter interval. Now here we find our two basic minor chord
> > substitutions for a C major. By extending these three we obtain all
> > possibilities. If we chuck an F in the mix to obtain D min (and strictly
> > speaking a new A min), then this is basically a change of key (placing the C
> > in the role of 5th). What other model has the ability to distinguish between
> > root note and tonic? But besides this, the gcd is what happens when we add
> > waves. It is a universal principle.
>
> That doesn't change the fact that the perceived root of 10:12:15
> usually isn't its GCD. And just because 16:19 is closer to 300 cents
> than 5:6 doesn't mean that every minor third we hear is perceived that
> way. The perceived musical root of a chord has a lot to do with the
> listener's imagination.
>
> > That's interesting Mike. I remember Leonard Bernstein pointing out that the
> > "ner ner na ner ner song" (5th, maj 3rd and maj 6th) is teasing because we
> > always expect to hear the tonic but never do. A smart arse friend said that
> > we could hear the A min, but I said it was likely a passing note. In any
> > case, I know exactly what you mean. Sometimes I get so lazy at a gig I'll
> > play only the first few notes of a known melody and let the listener imagine
> > the rest.
>
> Hence the "ghost note" technique commonly used among bebop players.
>
> -Mike
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> > > > Instead of thinking of the x-axis (frequencies) as sine waves we
> > > > think of them as gcd's. AND/OR we apply harmonic entropy to the
> > > > upper harmonics in any combination and define frequency as
> > > > approximating gcd's.
> > > >
> > > > Rick
> > >
> > > Can you give an example?
> >
> > I'm not good enough yet in Csound to design a proper experiment.
>
> I meant, can you work out an example on paper, of either or
> both of the suggestions you are making above? Because I don't
> understand either suggestion above, and an example would help
> make it clear for me.
>
> -Carl
>
Well as you know the HE is lowest for simple whole-numbered intervals like 1:2, 2:3. For intervals close to these the HE is still low and decreases the further we get from them. So there is a certain 'give' around each interval. We can choose ones that are close (within the Gaussian bell) and the HE will still remain relatively low.

Now, the tests were done with simple sine waves or notes with harmonic spectra (which I can assume means harmonic series starting from the fundamental). However, the concepts of 'intervals' and 'notes' is not all that separate. Intervals within the harmonic spectra are 'part' of the note while notes go into making intervals and are 'part' of them too. So its a bit chicken and the egg. (This is why I had trouble distinguishing frequencies from intervals if you remember). And theoretically speaking, the gcd gives 'one note at a time': if we play 'two notes' simultaneously which form a ratio, then a third (single) note is produced which is the gcd. I'm guessing that if we made new tests with for example the note f1 0 65536 10 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1, that is with the fundamental and its 8ve's missing, the results would be the same.

But what if we applied HE to the harmonics themselves since these too are intervals? I also suspect that the 'give' around each interval would still apply and give values close to the GCD for the pure originals. IOW an 'almost' GCD frequency.

Does that make more sense?

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> You pointed out that 101, 202, and 301? Hz would have differences 100Hz but gcd 1Hz.
> But since 1 is small compared with 100 etc, I suggested that 101 approx equals 100
> by HE and that the approx gcd will be closer to 100.

You've hit just the same question I was trying to clarify a few years ago. I still have to play around with it many times to see whether I can eventually reach a meaningful conclusion. I think the explanation of approximate GCDs here is actually much much more difficult than approximate LCMs. Now let's stop thinking about how humans hear sounds for a while and just "observe" the behavior (or the actual properties) of the sounds themselves. If you také two periods of 201Hz and 301Hz whose overtones are loud enough, then the 2nd harmonic of the higher tone is 602Hz and the 3rd harmonic of the lower tone is 603Hz, which means that this is the lowest "band" where you get beating. But recall the trigonometric identity which tells you that "sin(a+b( + sin(a-b) = sin(a( * cos(b) * 2". This means that the 602Hz and 603Hz beats can also be understood as a 602.5Hz signal which is "amplitude-modulated" by another signal whose actual frequency is 0.5Hz and whose "half-wave" frequency is 1Hz (I know this term is nonsense, I just meant something like "when I don't care about plus or minus"). Dividing 602.5 by 2 and 3 makes 301.25 and ~200.83333, whose GCD is ~100.41667. This leads me to the "merely speculative" idea for finding the approximate fundamental frequency this way because if you divide 301 by 3 or 201 by 2, you get ~100.33333 and 100.5 and the arithmetic mean of these is also ~100.41667, similarly as 602.5 was the arithmetic mean of 602 and 603. But I'm not sure how well that could work.

Petr

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> > You pointed out that 101, 202, and 301? Hz would have
> > differences 100Hz but gcd 1Hz. But since 1 is small
> > compared with 100 etc, I suggested that 101 approx
> > equals 100 by HE and that the approx gcd will be closer
> > to 100.

> let's stop thinking about how humans hear sounds for a while
> and just "observe" the behavior (or the actual properties) of
> the sounds themselves. If you take two periods of 201Hz and
> 301Hz whose overtones are loud enough,

In this example they were to be pure tones, but OK for
the sake of a new argument...

> then the 2nd harmonic of the higher tone is 602Hz and the
> 3rd harmonic of the lower tone is 603Hz, which means that
> this is the lowest "band" where you get beating. But recall
> the trigonometric identity which tells you that
> "sin(a+b( + sin(a-b) = sin(a( * cos(b) * 2". This means
> that the 602Hz and 603Hz beats can also be understood as
> a 602.5Hz signal which is "amplitude-modulated" by another
> signal whose actual frequency is 0.5Hz and whose "half-wave"
> frequency is 1Hz (I know this term is nonsense, I just meant
> something like "when I don't care about plus or minus").
> Dividing 602.5 by 2 and 3 makes 301.25 and ~200.83333, whose
> GCD is ~100.41667.

Do you get the same fundamental by repeating this procedure
for other pairs of beating partials in this example?

-Carl

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:

> > > > > Instead of thinking of the x-axis (frequencies) as
> > > > > sine waves we think of them as gcd's. AND/OR we apply
> > > > > harmonic entropy to the upper harmonics in any
> > > > > combination and define frequency as approximating gcd's.

> > > > Can you give an example?

> Well as you know the HE is lowest for simple whole-numbered
> intervals like 1:2, 2:3. For intervals close to these the HE
> is still low and decreases the further we get from them.

Increases, yes.

> So there is a certain 'give' around each interval. We can
> choose ones that are close (within the Gaussian bell) and
> the HE will still remain relatively low.

HE minima are not Gaussians on the HE curve, but OK, for
any low-HE interval, I can find another low-HE interval for
you nearby.

> Now, the tests were done with simple sine waves or notes
> with harmonic spectra (which I can assume means harmonic
> series starting from the fundamental). However, the concepts
> of 'intervals' and 'notes' is not all that separate.
> Intervals within the harmonic spectra are 'part' of the
> note while notes go into making intervals and are 'part'
> of them too.

Only if we're talking about JI intervals.

> And theoretically speaking, the gcd gives 'one note at a
> time': if we play 'two notes' simultaneously which form a
> ratio, then a third (single) note is produced which is
> the gcd. I'm guessing that if we made new tests with for
> example the note f1 0 65536 10 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1,
> that is with the fundamental and its 8ve's missing, the results
> would be the same.

You lost me here.

> But what if we applied HE to the harmonics themselves since
> these too are intervals? I also suspect that the 'give' around
> each interval would still apply and give values close to the
> GCD for the pure originals. IOW an 'almost' GCD frequency.

HE as it's currently formulated can only be applied to pitches
at a time. If we have two arbitrary pitches, how can HE, or
the GCD, tell us the perceived fundamental?

-Carl

🔗Petr Parízek <p.parizek@...>

🔗Charles Lucy <lucy@...>

In JI they have different frequencies, because the interval sizes are
determined by the position of the intervals in the scale.
The size of the Large and small intervals is inconsistent.
This may well be the result of assuming that "harmonics" should only
be at integer frequency ratios, so that beating is zero or minimised.

Some people still judge the "validity"/"goodness" (?) of a tuning by
how closely it approximates to JI intervals.

<quote> "It takes all sorts to make a galaxy; some like you; some like
me" </quote> - Steve Hammond song "Galaxy" from "Flash Fearless and
the Zorg Women".

Don't ask me to defend this distorted perception, but if you wantexplore it further, check this page:

http://www.lucytune.com/tuning/just_intonation.html

BTW more info. on scales and scalecoding can be found here:

http://www.lucytune.com/scales/

On 22 Jul 2009, at 06:50, Petr Parízek wrote:

>
> Rick wrote:
>
> > I fully agree that context, conditioning and imagination, loudness
> and emphasis
> > will all not only play a vital role but will override the gcd
> everytime. For eg,
> > in a tempered system there would be no distinction between an E
> min in the key of C maj
> > and an E min in its own key. Both are physically exactly the same
> chord. Having said that,
> > the problem with this is that it doesn't give us any indication as
> to how to make that
> > distinction. And when a musician learns harmony, he/she knows that
> the "same" note is not
> > the same at all if you know what I mean. In JI, they would
> probably have different
> > frequencies.
>
> Why should they have different frequencies?
>
> Petr
>
>
>

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Petr Parízek <p.parizek@...>

🔗rick_ballan <rick_ballan@...>

🔗Petr Parízek <p.parizek@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > You pointed out that 101, 202, and 301? Hz would have differences 100Hz but gcd 1Hz.
> > But since 1 is small compared with 100 etc, I suggested that 101 approx equals 100
> > by HE and that the approx gcd will be closer to 100.
>
> You've hit just the same question I was trying to clarify a few years ago. I still have to play around with it many times to see whether I can eventually reach a meaningful conclusion. I think the explanation of approximate GCDs here is actually much much more difficult than approximate LCMs. Now let's stop thinking about how humans hear sounds for a while and just "observe" the behavior (or the actual properties) of the sounds themselves. If you také two periods of 201Hz and 301Hz whose overtones are loud enough, then the 2nd harmonic of the higher tone is 602Hz and the 3rd harmonic of the lower tone is 603Hz, which means that this is the lowest "band" where you get beating. But recall the trigonometric identity which tells you that "sin(a+b( + sin(a-b) = sin(a( * cos(b) * 2". This means that the 602Hz and 603Hz beats can also be understood as a 602.5Hz signal which is "amplitude-modulated" by another signal whose actual frequency is 0.5Hz and whose "half-wave" frequency is 1Hz (I know this term is nonsense, I just meant something like "when I don't care about plus or minus"). Dividing 602.5 by 2 and 3 makes 301.25 and ~200.83333, whose GCD is ~100.41667. This leads me to the "merely speculative" idea for finding the approximate fundamental frequency this way because if you divide 301 by 3 or 201 by 2, you get ~100.33333 and 100.5 and the arithmetic mean of these is also ~100.41667, similarly as 602.5 was the arithmetic mean of 602 and 603. But I'm not sure how well that could work.
>
> Petr
>
Hi Petr,

I had more of a simple-minded approach in mind where we use approximation signs ~ instead of equals = to get the approx gcd, 301/201 ~ 3/2, 301/3 ~ 100, that type of thing. Then I've been trying to think of a more rigorous approach based on the HE model. But I think your approach of beats and averages is a very interesting possibility. What did you do for the lcm's?

Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> > > > > > Instead of thinking of the x-axis (frequencies) as
> > > > > > sine waves we think of them as gcd's. AND/OR we apply
> > > > > > harmonic entropy to the upper harmonics in any
> > > > > > combination and define frequency as approximating gcd's.
>
> > > > > Can you give an example?
>
> > Well as you know the HE is lowest for simple whole-numbered
> > intervals like 1:2, 2:3. For intervals close to these the HE
> > is still low and decreases the further we get from them.
>
> Increases, yes.

Sorry, increases.
>
> > So there is a certain 'give' around each interval. We can
> > choose ones that are close (within the Gaussian bell) and
> > the HE will still remain relatively low.
>
> HE minima are not Gaussians on the HE curve, but OK, for
> any low-HE interval, I can find another low-HE interval for
> you nearby.

Oh, I thought you said they were Gaussians. IAC I trust your model.
>
> > Now, the tests were done with simple sine waves or notes
> > with harmonic spectra (which I can assume means harmonic
> > series starting from the fundamental). However, the concepts
> > of 'intervals' and 'notes' is not all that separate.
> > Intervals within the harmonic spectra are 'part' of the
> > note while notes go into making intervals and are 'part'
> > of them too.
>
> Only if we're talking about JI intervals.

Of course since gcd is the topic.
>
> > And theoretically speaking, the gcd gives 'one note at a
> > time': if we play 'two notes' simultaneously which form a
> > ratio, then a third (single) note is produced which is
> > the gcd. I'm guessing that if we made new tests with for
> > example the note f1 0 65536 10 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1,
> > that is with the fundamental and its 8ve's missing, the results
> > would be the same.
>
> You lost me here.

A single sine wave can be a note. A complex wave is made up of intervals between these harmonics, and can itself be used in intervals with other such waves to form yet another 'note', and so on. The gcd covers them all. The Csound example has only relative primes and no difference tones equal to the fundamental so it proves that it is the gcd at work. So I'm asking if the HE interval tests would be the same for these frequencies i.e. gcd's themselves, not only intervals b/w sine waves.
>
> > But what if we applied HE to the harmonics themselves since
> > these too are intervals? I also suspect that the 'give' around
> > each interval would still apply and give values close to the
> > GCD for the pure originals. IOW an 'almost' GCD frequency.
>
> HE as it's currently formulated can only be applied to pitches
> at a time. If we have two arbitrary pitches, how can HE, or
> the GCD, tell us the perceived fundamental?

Now you've lost me. The gcd IS the perceived fundamental.
>
> -Carl
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> In JI they have different frequencies, because the interval sizes are
> determined by the position of the intervals in the scale.
> The size of the Large and small intervals is inconsistent.
> This may well be the result of assuming that "harmonics" should only
> be at integer frequency ratios, so that beating is zero or minimised.
>
> Some people still judge the "validity"/"goodness" (?) of a tuning by
> how closely it approximates to JI intervals.

Hi Charles,

where have you been? Any case, nice to hear from you. True, the problem with 'perfection' is that its usually not so perfect. I was surprised when I got a microtuner (no one told me it was free) and could finally test 4:5 to find that it sounded very flat to my ears. But what we're (or at least I'm) talking about is that "harmonics" might not be so close to the integers as we previously thought.

Rick
>
> <quote> "It takes all sorts to make a galaxy; some like you; some like
> me" </quote> - Steve Hammond song "Galaxy" from "Flash Fearless and
> the Zorg Women".
>
> Don't ask me to defend this distorted perception, but if you want
> explore it further, check this page:
>
> http://www.lucytune.com/tuning/just_intonation.html
>
> BTW more info. on scales and scalecoding can be found here:
>
> http://www.lucytune.com/scales/
>
>
>
> On 22 Jul 2009, at 06:50, Petr Parízek wrote:
>
> >
> > Rick wrote:
> >
> > > I fully agree that context, conditioning and imagination, loudness
> > and emphasis
> > > will all not only play a vital role but will override the gcd
> > everytime. For eg,
> > > in a tempered system there would be no distinction between an E
> > min in the key of C maj
> > > and an E min in its own key. Both are physically exactly the same
> > chord. Having said that,
> > > the problem with this is that it doesn't give us any indication as
> > to how to make that
> > > distinction. And when a musician learns harmony, he/she knows that
> > the "same" note is not
> > > the same at all if you know what I mean. In JI, they would
> > probably have different
> > > frequencies.
> >
> > Why should they have different frequencies?
> >
> > Petr
> >
> >
> >
>
> Charles Lucy
> lucy@...
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>

🔗Carl Lumma <carl@...>

🔗Petr Parízek <p.parizek@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> > > HE minima are not Gaussians on the HE curve, but OK, for
> > > any low-HE interval, I can find another low-HE interval for
> > > you nearby.
> >
> > Oh, I thought you said they were Gaussians. IAC I trust your model.
>
> You may be thinking of Gaussian probability distributions
> in interval space, which are used to compute HE.
>
> > > > Now, the tests were done with simple sine waves or notes
> > > > with harmonic spectra (which I can assume means harmonic
> > > > series starting from the fundamental). However, the concepts
> > > > of 'intervals' and 'notes' is not all that separate.
> > > > Intervals within the harmonic spectra are 'part' of the
> > > > note while notes go into making intervals and are 'part'
> > > > of them too.
> > >
> > > Only if we're talking about JI intervals.
> >
> > Of course since gcd is the topic.
>
> Well, that's a pretty limited theory of virtual fundamentals
> wouldn't you say, since we still hear them when chords are
> tempered?

But that's exactly my point. Theoretically gcd's should only exist for frequencies in whole-numbered relations, right? Since we can sometimes hear them for tempered intervals you jump to the conclusion that the definition must be a more "limited" version of a more "general" theory of "virtual fundamentals", that it is the difference tone after all, for example. But this is true only when the difference happens to equal or approximate the gcd. I say it over and over again that the "frequency" of a wave IS its gcd, irrespective of whether it is subsonic or not, or whatever loose and unexamined terms scientists have applied in the past. There is nothing "virtual" about them; whenever two or more waves occupy the same place at the same time, regardless of how they are generated, if their freq's are relatively prime then a third single wave will be produced with a freq corresponding to the gcd. And since this is the more general term for "frequency", then nothing stops us from saying that each of the original two or more "frequencies" were themselves gcd's. Or to say it the other way, we can take this "third" gcd wave, add it to another gcd, and if these are relatively prime then this will create a new wave with that new gcd. And this proof can be repeated ad infinitum. Therefore, there is nothing "limited" about them either. When you say below "how can...the GCD tell us the perceived fundamental?", the sentence reads "how can the fundamental tell us about the perceived fundamental?". And since the difference tone is the difference between two frequencies, then it is the difference between two gcd's. The case of simple sine waves or harmonic spectra starting from 1 is but a small limited case e.g. series 1: 9Hz, 18Hz, 27...series 2: 12Hz, 24, 36...gcd between respective harmonics: 3Hz, 6, 9...(observe that the gcd of harmonics = harmonics of the gcd. Much easier than the Euclidean algorithm once we know the first). Difference tones: 3, 6, 9...However, new series 2: 15Hz, 30, 45...gcd series: 3Hz, 6Hz, 9Hz...differences: 6Hz,12,18...which we see is not necessarily the same.

So now you see the importance of trying to find how HE might apply to get approximate gcd's? For instance by HE we might apply 9Hz and 15.013Hz giving 15.013/9 ~ 5/3 and ~3Hz as the gcd. And/or we apply HE to the 3Hz and it applies to the harmonics. Are these two approaches equivalent or will they give different results? etc...This would be a far more realistic model for explaining tempered "virtual" fundamentals than applying the wrong theory of difference tones and then compounding the uncertainty by applying HE to them.

Sorry for lecturing Carl but I think this is important.

-Rick
>
> > > HE as it's currently formulated can only be applied to pitches
> > > at a time. If we have two arbitrary pitches, how can HE, or
> > > the GCD, tell us the perceived fundamental?
> >
> > Now you've lost me. The gcd IS the perceived fundamental.
>
> What if the GCD is subsonic?
>
> -Carl
>

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> But that's exactly my point. Theoretically gcd's should only exist for frequencies
> in whole-numbered relations, right? Since we can sometimes hear them for tempered
> intervals you jump to the conclusion that the definition must be a more "limited" version
> of a more "general" theory of "virtual fundamentals",

The problem here is that the term "virtual fundamental" is often linked directly to the question how humans hear sounds. However, the term "fundamental frequency" is, at least in the cases I've witnessed, used without any reference to our hearing, simply in the meaning of the general frequency of a complex periodic waveform, no matter if someone is actually hearing the sound in question or not. So while the VF is probably a result of our individual interpretation of a sound we hear, the "fundamental frequency" is just an "acoustical property" of any complex waveform whose frequencies can be expressed in integer ratios and is in no way trying to answer the question how we hear it.

> I say it over and over again that the "frequency" of a wave IS its gcd,
> irrespective of whether it is subsonic or not, or whatever loose and unexamined terms
> scientists have applied in the past.

Yes, but the important question is: Are you trying to answer the question what you can actually hear when listening to the sound? If you are, then the "fundamental frequency" (or, more precisely, the general frequency of a complex waveform) is not always the best starting point. If you mix frequencies of 150-190-227Hz, can you hear the general frequency of 1Hz? I can't -- I mean, I really can't hear any 1Hz periodicity in such a sound even though it IS there.

Petr

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

Rick wrote:
> > Well, that's a pretty limited theory of virtual fundamentals
> > wouldn't you say, since we still hear them when chords are
> > tempered?
>
> But that's exactly my point. Theoretically gcd's should only
> exist for frequencies in whole-numbered relations, right?
> Since we can sometimes hear them for tempered intervals

We don't "hear" GCDs. We hear pitches.

> you jump to the conclusion that the definition must be a more
> "limited" version of a more "general" theory of "virtual
> fundamentals", that it is the difference tone after all, for
> example.

Where did I mention difference tones?

>I say it over and over again that the "frequency" of a wave
>IS its gcd, irrespective of whether it is subsonic or not, or
>whatever loose and unexamined terms scientists have applied
>in the past.

How are you defining "frequency"? It seems like it just
means whatever you want it to mean. Back in the real world,
frequency has a precise definition, and complex tones or
chords of complex tones are not characterized by any single
frequency. Complex tones are often characterized by a
single pitch, but pitches are never subsonic.

>And since this is the more general term for "frequency", then
>nothing stops us from saying that each of the original two or
>more "frequencies" were themselves gcd's. Or to say it the
>other way, we can take this "third" gcd wave, add it to another
>gcd, and if these are relatively prime then this will create
>a new wave with that new gcd. And this proof can be repeated
>ad infinitum. Therefore, there is nothing "limited" about them
>either.

You lost me, but frankly I don't think you're making any sense.

> Sorry for lecturing Carl but I think this is important.

It is important, but until you think things through more
carefully, you're not going to make any progress.

-Carl

🔗Carl Lumma <carl@...>

🔗Petr Parízek <p.parizek@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > frequency has a precise definition, and complex tones or
> > chords of complex tones are not characterized by any single
> > frequency.
>
> Depends on what you mean by "complex tones". If you mix sine waves of 200-500-800-1100Hz and so on up to around 20kHz, then you get a different timbre whose length of the general period (i.e. the shortest time until the waveform starts repeating) is 10 ms or 1/100 of a second. Even though there isn't any 100Hz sine wave in such a sound, the general frequency of 100Hz is audible. IIRC, this is what we were discussing in the 10:12:15 thread some time ago; someone else had pointed this out and I thought you had replied you agreed. At that time, I was making the "Inharmonic spectra" recordings to try this with ratios of 3:7:11:15:19 and so on. The only situation when this "fundamental frequency" is inaudible is if it's so low that we can't hear it as a tone and at the same time it's masked by something else. For example, if you mix sine waves of 3Hz, 7Hz, 11Hz, 15Hz, and so on and so on, then, for one thing, there's no way you can recognize the GCD of 1Hz -- and, for another thing, it's perfectly masked by the common difference frequency of 4Hz which can be heard not as a tone but rather as some periodic clicks. And because these clicks don't differ from each other in anything else than phase shifts, they sound to us just like if all of them were the same.
>
> Petr
>
Thanks Petr,

You said it better than I. As I replied to Carl's last message to me (which will come out in the next 'issue'), 1. while we might imagine that we hear two or more different notes simultaneously, the air pressure on the ear drum always arrives 'one wave at a time' (principle of superposition), and 2. traditional instruments usually come with a ready made 'fundamental = 1st harmonic' so that if we play two or more together to form a gcd it is likely to be sub-audible. However, we do seem to recognise 'the tonic' when we land on something close to one of its 8ve equivalents within the audible range. Perhaps we can somehow detect that it is a 8ve to the fundamental? I don't know. In any case, the advent of computers seems to have freed us from this traditional limitation and, for once, practice can catch up with theory. The (albeit forgivable) mistake that fundamental necessarily equals the 1st harmonic is now only serving to unnaturally keep the gcd in the subsonic range. There are in fact the SAME NUMBER of GCD frequencies within the audible range as there are sine waves, namely aleph-naught (countable infinity).
(eg, in Csound, there are just as many GEN10 0 0 1 0 1...that is, only the relative primes, as GEN10 1 1 1 1 1...where all the harmonics have the same amplitude).

Rick

🔗Mike Battaglia <battaglia01@...>

> You said it better than I. As I replied to Carl's last message to me (which
> will come out in the next 'issue'), 1. while we might imagine that we hear
> two or more different notes simultaneously, the air pressure on the ear drum
> always arrives 'one wave at a time' (principle of superposition),

That's not true. How would the principle of superposition ever be able
to say something like that?

And by "wave" here do you mean monocomponent sinusoid?

> and 2. traditional instruments usually come with a ready made 'fundamental = 1st
> harmonic' so that if we play two or more together to form a gcd it is likely
> to be sub-audible. However, we do seem to recognise 'the tonic' when we land
> on something close to one of its 8ve equivalents within the audible range.
> Perhaps we can somehow detect that it is a 8ve to the fundamental? I don't
> know.

If your goal is to come up with an algorithm that can take a signal
and get a time-frequency plot in which frequencies are then
subsequently "grouped" as belonging to a "complex waveform" of a
certain fundamental frequency, you could get a pretty decent initial
run with the STFT and then just doing the math to get it done. I'm not
sure how you would encode the timbre of each GCD, but you sure could
somehow. You could detect if a certain note is an octave equivalent to
the fundamental of the GCD by checking to see if the frequency is a
power of two of the fundamental. If the two notes an octave apart are
played at the same time you're going to have to do some clever math to
figure out whether the two notes aren't in fact just one more complex
timbre. This is going to be a "best guess" process, as it's an
auditory illusion that human beings experience all the time.

If, on the other hand, your goal is to take a signal and then somehow
do a musical analysis on it whereby you can see how human beings would
"musically" perceive the signal over time, then go for it! Even if you
made the crude assumption that the only perceived fundamental that we
hear is the GCD of the wave and limit yourself to JI only, it would
still be miles above anything I've seen.

> In any case, the advent of computers seems to have freed us from this
> traditional limitation and, for once, practice can catch up with theory. The
> (albeit forgivable) mistake that fundamental necessarily equals the 1st
> harmonic is now only serving to unnaturally keep the gcd in the subsonic
> range. There are in fact the SAME NUMBER of GCD frequencies within the
> audible range as there are sine waves, namely aleph-naught (countable
> infinity).

Are you using the term "harmonic" here to mean that the 1st is the
fundamental, the 2nd is the octave, the 3rd is the octave + fifth,
etc?

-Mike

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > You said it better than I. As I replied to Carl's last message to me (which
> > will come out in the next 'issue'), 1. while we might imagine that we hear
> > two or more different notes simultaneously, the air pressure on the ear drum
> > always arrives 'one wave at a time' (principle of superposition),
>
> That's not true. How would the principle of superposition ever be able
> to say something like that?

But Mike, by 'one wave' here I don't mean 'one sine wave'. If we add just 2 or 1000 waves, there will only ever be one amount of air pressure, one displacement of the string or loud speaker etc, at any given place and time. This is what the superposition principle is. As you say yourself, why we hear 'different' waves might come down to psychoacoustic properties.
>
> And by "wave" here do you mean monocomponent sinusoid?
>
> > and 2. traditional instruments usually come with a ready made 'fundamental = 1st
> > harmonic' so that if we play two or more together to form a gcd it is likely
> > to be sub-audible. However, we do seem to recognise 'the tonic' when we land
> > on something close to one of its 8ve equivalents within the audible range.
> > Perhaps we can somehow detect that it is a 8ve to the fundamental? I don't
> > know.
>
> If your goal is to come up with an algorithm that can take a signal
> and get a time-frequency plot in which frequencies are then
> subsequently "grouped" as belonging to a "complex waveform" of a
> certain fundamental frequency, you could get a pretty decent initial
> run with the STFT and then just doing the math to get it done. I'm not
> sure how you would encode the timbre of each GCD, but you sure could
> somehow. You could detect if a certain note is an octave equivalent to
> the fundamental of the GCD by checking to see if the frequency is a
> power of two of the fundamental. If the two notes an octave apart are
> played at the same time you're going to have to do some clever math to
> figure out whether the two notes aren't in fact just one more complex
> timbre. This is going to be a "best guess" process, as it's an
> auditory illusion that human beings experience all the time.
>
> If, on the other hand, your goal is to take a signal and then somehow
> do a musical analysis on it whereby you can see how human beings would
> "musically" perceive the signal over time, then go for it! Even if you
> made the crude assumption that the only perceived fundamental that we
> hear is the GCD of the wave and limit yourself to JI only, it would
> still be miles above anything I've seen.

Mike, you know very well by our recent personal correspondence that there is much more to it than that, that my "crude assumption" is actually an elementary first principle, which among other things reminds us that the Fourier series and all of its variants are subject to the principle of superposition, not the other way around. Remember that Fourier's first paper just asked the question; lets assume that any arbitrary periodic function f(t) is equal to an infinite trigonometric series (i.e. superposition of whole-numbered sine waves, otherwise known as the harmonic series). Then what are its coefficients? Therefore superposition was assumed to begin with and the rest followed from this one basic insight. The fact that the transform deals with non-periodic functions doesn't change the fact that we are still dealing with superimposed sine waves one iota. (And as I've shown you, the 'time' domain in the STFT can always and in principle be written as a function of the cycles and period of the waves themselves according to t = nT, so that signal processing models are designed on this logic, but that's getting too complicated for the moment).

At any rate, it is very easy to prove that a Fourier Series f(t) corresponds to the GCD and not on the first coefficient which can be zero. And my question to the group is; is this what we hear as the tonic? So where is the conflict? Both of us are just trying to figure out one more part of the puzzle from different angles. I've never once said or implied that yours or anyone else's work or opinions are not valid and see no reason for you to go on the defensive.

>
> > In any case, the advent of computers seems to have freed us from this
> > traditional limitation and, for once, practice can catch up with theory. The
> > (albeit forgivable) mistake that fundamental necessarily equals the 1st
> > harmonic is now only serving to unnaturally keep the gcd in the subsonic
> > range. There are in fact the SAME NUMBER of GCD frequencies within the
> > audible range as there are sine waves, namely aleph-naught (countable
> > infinity).
>
> Are you using the term "harmonic" here to mean that the 1st is the
> fundamental, the 2nd is the octave, the 3rd is the octave + fifth,
> etc?
>
Yep. I recall that some people like to call N > 1 the overtone series and from 1 etc the harmonic series. Is that correct?

Rick

> -Mike
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Carl wrote:

> How are you defining "frequency"? It seems like it just
> means whatever you want it to mean. Back in the real world,
> frequency has a precise definition, and complex tones or
> chords of complex tones are not characterized by any single
> frequency. Complex tones are often characterized by a
> single pitch, but pitches are never subsonic.

Correct me if I'm wrong Carl but frequency is defined equivalently as a) the number of cycles occurring per second, and b) the inverse of the time period required for the wave to pass through one cycle. But one cycle is determined by the GCD and not necessarily on the period of a sine wave. And 'back in the real world' the principle of superposition says that any complex tone or chord will only have a single string displacement or air pressure etc... at any given place and time. The term 'complex' comes from the fact it can be Fourier Analysed into separate sine wave components. Therefore the frequency of this wave will always be its GCD and the fact that it sometimes corresponds to the sine wave of the fundamental is incidental.
>
Rick said:
> >And since this is the more general term for "frequency", then
> >nothing stops us from saying that each of the original two or
> >more "frequencies" were themselves gcd's. Or to say it the
> >other way, we can take this "third" gcd wave, add it to another
> >gcd, and if these are relatively prime then this will create
> >a new wave with that new gcd. And this proof can be repeated
> >ad infinitum. Therefore, there is nothing "limited" about them
> >either.
>
> Carl: You lost me, but frankly I don't think you're making any sense.
>
Of course I was making sense. Given two sine waves of frequency p and q, p > q, if their ratio is p/q = a/b where a and b are relatively prime, then the frequency of the resultant wave is p/a = q/b, IOW their greatest common divisor. The proof of this is that if we add or subtract whole-numbered multiples N of this period T = a/p = b/q then the wave function remains the same:
cos2pi*p*(t + NT) =
cos2pi*p*(t + Na/p) =
cos2pi*(pt + Na) =
cos2pi*pt since Na is whole and adding NA*2pi doesn't change the function.

My point above was that this proof works also when p and q are assumed to be GCD's rather than sine waves so that there is no 'bottoming out' to the process. And despite how it might otherwise appear, a little thought will reveal that this too is the basis of all periodic Fourier Analysis i.e. not all infinite trig series will have first harmonic.

> > Sorry for lecturing Carl but I think this is important.
>
> It is important, but until you think things through more
> carefully, you're not going to make any progress.
>
> -Carl
>
So now I hope you see that I have in fact thought things out carefully and it is you who needs to read my posts with more precision and less assumption.

Regards

Rick

🔗Mike Battaglia <battaglia01@...>

> But Mike, by 'one wave' here I don't mean 'one sine wave'. If we add just 2
> or 1000 waves, there will only ever be one amount of air pressure, one
> displacement of the string or loud speaker etc, at any given place and time.
> This is what the superposition principle is. As you say yourself, why we
> hear 'different' waves might come down to psychoacoustic properties.

That's a slightly different take on the superposition principle than
what I know. And the reason we hear 'different' waves is that the
inner ear basically consists of a bunch of hair cells that only
resonate at certain frequencies, and this splits the combined wave
form out into a bunch of different waves, producing a bunch of
different signals.

> Mike, you know very well by our recent personal correspondence that there is
> much more to it than that, that my "crude assumption" is actually an
> elementary first principle, which among other things reminds us that the
> Fourier series and all of its variants are subject to the principle of
> superposition, not the other way around. Remember that Fourier's first paper
> just asked the question; lets assume that any arbitrary periodic function
> f(t) is equal to an infinite trigonometric series (i.e. superposition of
> whole-numbered sine waves, otherwise known as the harmonic series). Then
> what are its coefficients? Therefore superposition was assumed to begin with
> and the rest followed from this one basic insight. The fact that the
> transform deals with non-periodic functions doesn't change the fact that we
> are still dealing with superimposed sine waves one iota. (And as I've shown
> you, the 'time' domain in the STFT can always and in principle be written as
> a function of the cycles and period of the waves themselves according to t =
> nT, so that signal processing models are designed on this logic, but that's
> getting too complicated for the moment).

I don't think you're ever going to hear me say that Fourier analysis
has nothing to do with superimposing sinusoids on top of each other to
recreate the original signal. But if you're not dealing with a
periodic signal, then there IS no GCD. And most real world signals, as
you will see, are usually aperiodic. And when the signal IS periodic,
if you claim that you hear the perceived fundamental of every possible
chord as being the GCD, then that claim simply isn't true.

> At any rate, it is very easy to prove that a Fourier Series f(t) corresponds
> to the GCD and not on the first coefficient which can be zero. And my
> question to the group is; is this what we hear as the tonic? So where is the
> conflict? Both of us are just trying to figure out one more part of the
> puzzle from different angles. I've never once said or implied that yours or
> anyone else's work or opinions are not valid and see no reason for you to go
> on the defensive.

I don't understand this. The first coefficient of a Fourier Series is
going to be the DC term, the second is going to be the fundamental. If
what you mean is the second coefficient, or the first harmonic, then
that would indeed be the fundamental, even if it is zero, and its
period length would equal the length of the window you use.

And furthermore, your approach here has a fatal flaw, which is the
same flaw I've been talking about all along: the GCD of any Fourier
Series is going to equal the frequency corresponding to the window
size you use, by definition. So if you evaluate the Fourier Series of
a periodic signal with a window size equaling exactly one period of
the signal, everything will work out fine. Otherwise, you're going to
have a few issues to deal with.

How are you going to figure out what one period of the signal is when
you don't know the signal in question? What if the signal is
aperiodic? What if the signal is periodic with a volume envelope? That
will add its own frequencies into the signal.

Now, as I've said, you COULD perform some clever mathematical analysis
on the signal to somehow figure out what the optimum window size is to
use at any given moment, or even do something more complicated
involving splitting the signal into different signals and use
different window sizes for each, or you could go right to wavelets and
get it over with. And if you have some idea of how to do this in a
mathematically "pure" way, using some insight that you have, then go
for it! If we're all just misunderstanding what you're saying, don't
let us get you down, just make your idea!

> Yep. I recall that some people like to call N > 1 the overtone series and
> from 1 etc the harmonic series. Is that correct?
>
> Rick

Yeah. The terminology is pretty loose as far as I can tell.

-Mike

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
And just to reconfirm once and for all that the GCD is the frequency of a Fourier Series we need only consider that a square wave is composed only of odd harmonics. QED

Rick

> > You said it better than I. As I replied to Carl's last message to me (which
> > will come out in the next 'issue'), 1. while we might imagine that we hear
> > two or more different notes simultaneously, the air pressure on the ear drum
> > always arrives 'one wave at a time' (principle of superposition),
>
> That's not true. How would the principle of superposition ever be able
> to say something like that?
>
> And by "wave" here do you mean monocomponent sinusoid?
>
> > and 2. traditional instruments usually come with a ready made 'fundamental = 1st
> > harmonic' so that if we play two or more together to form a gcd it is likely
> > to be sub-audible. However, we do seem to recognise 'the tonic' when we land
> > on something close to one of its 8ve equivalents within the audible range.
> > Perhaps we can somehow detect that it is a 8ve to the fundamental? I don't
> > know.
>
> If your goal is to come up with an algorithm that can take a signal
> and get a time-frequency plot in which frequencies are then
> subsequently "grouped" as belonging to a "complex waveform" of a
> certain fundamental frequency, you could get a pretty decent initial
> run with the STFT and then just doing the math to get it done. I'm not
> sure how you would encode the timbre of each GCD, but you sure could
> somehow. You could detect if a certain note is an octave equivalent to
> the fundamental of the GCD by checking to see if the frequency is a
> power of two of the fundamental. If the two notes an octave apart are
> played at the same time you're going to have to do some clever math to
> figure out whether the two notes aren't in fact just one more complex
> timbre. This is going to be a "best guess" process, as it's an
> auditory illusion that human beings experience all the time.
>
> If, on the other hand, your goal is to take a signal and then somehow
> do a musical analysis on it whereby you can see how human beings would
> "musically" perceive the signal over time, then go for it! Even if you
> made the crude assumption that the only perceived fundamental that we
> hear is the GCD of the wave and limit yourself to JI only, it would
> still be miles above anything I've seen.
>
> > In any case, the advent of computers seems to have freed us from this
> > traditional limitation and, for once, practice can catch up with theory. The
> > (albeit forgivable) mistake that fundamental necessarily equals the 1st
> > harmonic is now only serving to unnaturally keep the gcd in the subsonic
> > range. There are in fact the SAME NUMBER of GCD frequencies within the
> > audible range as there are sine waves, namely aleph-naught (countable
> > infinity).
>
> Are you using the term "harmonic" here to mean that the 1st is the
> fundamental, the 2nd is the octave, the 3rd is the octave + fifth,
> etc?
>
> -Mike
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
Oops, I meant another function which only has even harmonics. My apologies (I can't think of it but it does exist). UNQED

Rick

🔗Mike Battaglia <battaglia01@...>

That would be a sawtooth wave an octave up - 2,4,6,8 is just going to
be 1,2,3,4 * 2.

-Mike

On Tue, Jul 28, 2009 at 4:39 AM, rick_ballan<rick_ballan@...> wrote:
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>>
> Oops, I meant another function which only has even harmonics. My apologies
> (I can't think of it but it does exist). UNQED
>
> Rick
>
>> > You said it better than I. As I replied to Carl's last message to me
>> > (which
>> > will come out in the next 'issue'), 1. while we might imagine that we
>> > hear
>> > two or more different notes simultaneously, the air pressure on the ear
>> > drum
>> > always arrives 'one wave at a time' (principle of superposition),
>>
>> That's not true. How would the principle of superposition ever be able
>> to say something like that?
>>
>> And by "wave" here do you mean monocomponent sinusoid?
>>
>> > and 2. traditional instruments usually come with a ready made
>> > 'fundamental = 1st
>> > harmonic' so that if we play two or more together to form a gcd it is
>> > likely
>> > to be sub-audible. However, we do seem to recognise 'the tonic' when we
>> > land
>> > on something close to one of its 8ve equivalents within the audible
>> > range.
>> > Perhaps we can somehow detect that it is a 8ve to the fundamental? I
>> > don't
>> > know.
>>
>> If your goal is to come up with an algorithm that can take a signal
>> and get a time-frequency plot in which frequencies are then
>> subsequently "grouped" as belonging to a "complex waveform" of a
>> certain fundamental frequency, you could get a pretty decent initial
>> run with the STFT and then just doing the math to get it done. I'm not
>> sure how you would encode the timbre of each GCD, but you sure could
>> somehow. You could detect if a certain note is an octave equivalent to
>> the fundamental of the GCD by checking to see if the frequency is a
>> power of two of the fundamental. If the two notes an octave apart are
>> played at the same time you're going to have to do some clever math to
>> figure out whether the two notes aren't in fact just one more complex
>> timbre. This is going to be a "best guess" process, as it's an
>> auditory illusion that human beings experience all the time.
>>
>> If, on the other hand, your goal is to take a signal and then somehow
>> do a musical analysis on it whereby you can see how human beings would
>> "musically" perceive the signal over time, then go for it! Even if you
>> made the crude assumption that the only perceived fundamental that we
>> hear is the GCD of the wave and limit yourself to JI only, it would
>> still be miles above anything I've seen.
>>
>> > In any case, the advent of computers seems to have freed us from this
>> > traditional limitation and, for once, practice can catch up with theory.
>> > The
>> > (albeit forgivable) mistake that fundamental necessarily equals the 1st
>> > harmonic is now only serving to unnaturally keep the gcd in the subsonic
>> > range. There are in fact the SAME NUMBER of GCD frequencies within the
>> > audible range as there are sine waves, namely aleph-naught (countable
>> > infinity).
>>
>> Are you using the term "harmonic" here to mean that the 1st is the
>> fundamental, the 2nd is the octave, the 3rd is the octave + fifth,
>> etc?
>>
>> -Mike
>>
>
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > But Mike, by 'one wave' here I don't mean 'one sine wave'. If we add just 2
> > or 1000 waves, there will only ever be one amount of air pressure, one
> > displacement of the string or loud speaker etc, at any given place and time.
> > This is what the superposition principle is. As you say yourself, why we
> > hear 'different' waves might come down to psychoacoustic properties.
>
> That's a slightly different take on the superposition principle than
> what I know. And the reason we hear 'different' waves is that the
> inner ear basically consists of a bunch of hair cells that only
> resonate at certain frequencies, and this splits the combined wave
> form out into a bunch of different waves, producing a bunch of
> different signals.

Well that's interesting Mike. I know little or nothing about this end of things. And since my mind tells me that the 'real' wave out there is a single entity but my ears tell me quite another, then I knew there had to be more to it.
>
> > Mike, you know very well by our recent personal correspondence that there is
> > much more to it than that, that my "crude assumption" is actually an
> > elementary first principle, which among other things reminds us that the
> > Fourier series and all of its variants are subject to the principle of
> > superposition, not the other way around. Remember that Fourier's first paper
> > just asked the question; lets assume that any arbitrary periodic function
> > f(t) is equal to an infinite trigonometric series (i.e. superposition of
> > whole-numbered sine waves, otherwise known as the harmonic series). Then
> > what are its coefficients? Therefore superposition was assumed to begin with
> > and the rest followed from this one basic insight. The fact that the
> > transform deals with non-periodic functions doesn't change the fact that we
> > are still dealing with superimposed sine waves one iota. (And as I've shown
> > you, the 'time' domain in the STFT can always and in principle be written as
> > a function of the cycles and period of the waves themselves according to t =
> > nT, so that signal processing models are designed on this logic, but that's
> > getting too complicated for the moment).
>
> I don't think you're ever going to hear me say that Fourier analysis
> has nothing to do with superimposing sinusoids on top of each other to
> recreate the original signal. But if you're not dealing with a
> periodic signal, then there IS no GCD. And most real world signals, as
> you will see, are usually aperiodic. And when the signal IS periodic,
> if you claim that you hear the perceived fundamental of every possible
> chord as being the GCD, then that claim simply isn't true.

I was simply pointing out that Fourier analysis, both historically and logically, came from the principle of superposition rather than vice-versa, not saying that you disagreed but counting on the fact that you did agree. And it would be absurd if you truly believed I didn't know that waves must be periodic for the GCD to exist, since I've been saying that they are one and the same thing all along. For example, given any periodic f(t) = f(t + T), if we set this equal to an infinite trig series, then we are ALREADY assuming that some GCD wave exists which will approximate the original f(t) with the same period T. IOW it is quite possible for the resultant summation wave to be periodic while missing its 1st harmonic (I said coefficient before). So I'm not talking about 'real world' signals such as hand claps and firecrackers which require a Fourier Transform with an infinite number of frequencies in a finite range (the discreet FT being for digitalisation I believe?). Rather I'm talking about periodic waves and there association to tradition musical harmony, and mathematically about all periodic Fourier Series in general. Nothing here contradicts FA in any way.
>
>
> I don't understand this. The first coefficient of a Fourier Series is going to be the DC term, the second is going to be the fundamental. If what you mean is the second coefficient, or the first harmonic, then that would indeed be the fundamental, even if it is zero, and its period length would equal the length of the window you use.

Yes I was getting confused by the fact that some texts start with the A(0) coefficient separately while others include it in A(N). I meant first harmonic.
>
> And furthermore, your approach here has a fatal flaw, which is the
> same flaw I've been talking about all along: the GCD of any Fourier
> Series is going to equal the frequency corresponding to the window
> size you use, by definition. So if you evaluate the Fourier Series of a periodic signal with a window size equalling exactly one period of the signal, everything will work out fine. Otherwise, you're going to have a few issues to deal with.

Yet I still don't see how we're in disagreement. By 'window' do you mean something a 'window in time' of period T? (It sounds more like a finite Transform rather than a series since the latter is theoretically infinite in time. Am I missing something here?). But if you are talking about choosing your window to equal T, then I can't see how you would do otherwise.
>
> How are you going to figure out what one period of the signal is when you don't know the signal in question? What if the signal is aperiodic? What if the signal is periodic with a volume envelope? That will add its own frequencies into the signal.

Ah, but that's a completely different question. I think I see what's going on here. You're talking about practicalities of signal processing whereas I'm making a more general point about periodic waves, for these ARE what are usually used in musical harmony. Signal processing is not the only area of physics in which Fourier Analysis applies and its 'rules' are not always applicable across the board. FA is also used to figure out the shape of a plucked string or periodic air pressure in a pipe for example. But neither of these require a 'window' to do that analysis. Now admittedly I don't know much about Signal processing and have much to learn there, but I suspect that it is only when we try to record or recreate analogue waves in a digital environment, or transmit and receive them as radio 'signals' that we mightn't know what the period is and need to do a Fourier Transform to retrieve the information.

In any case, what I'm talking about here is how superposition can be used to explain musical harmony between periodic waves. I'm also making the point that this same rule will apply to any wave which is periodic, by definition. And finally, I've been pointing out that, mathematically speaking at least, the variable t appearing on the left side of a Fourier equation can always be written as a function of the class of periods appearing on the right. This is because time 'behaves' like a tempo, which has been (unconsciously?) encoded in its representation by numbers. Therefore, trying to find the 'optimum window' in your example below is reflected in this fact; for Fourier series, it is easy since there's only one choice. For aperiodic Transforms it is much more difficult. There's more work to do in this, but it's there.

Rick
>
> Now, as I've said, you COULD perform some clever mathematical analysis
> on the signal to somehow figure out what the optimum window size is to
> use at any given moment, or even do something more complicated
> involving splitting the signal into different signals and use
> different window sizes for each, or you could go right to wavelets and
> get it over with. And if you have some idea of how to do this in a
> mathematically "pure" way, using some insight that you have, then go
> for it! If we're all just misunderstanding what you're saying, don't
> let us get you down, just make your idea!
>
> > Yep. I recall that some people like to call N > 1 the overtone series and
> > from 1 etc the harmonic series. Is that correct?
> >
> > Rick
>
> Yeah. The terminology is pretty loose as far as I can tell.
>
> -Mike
>

🔗rick_ballan <rick_ballan@...>

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

Have you ever heard the story about Alexander the Great sitting around the palace playing the Lyre and his father Phillip of Macedon walks past and says "Stop! You're getting too good at that"? It was not the place of the king-to-be to appear like a mere tradesman, especially a musician which was seen as being little more than a prostitute. Aspects of this Ancient Greek prejudice that musikae is "effeminate" and geometria is "manly" is still with us today. Although Pythagoras unified the two and two centuries later Euclid worked out the inner logic of harmonia (his algorithm), we must also remember that both Plato and Aristotle, who didn't like this idea, came from the Athenian aristocracy and influenced the course of Western education for the next 1500 years. Most notable is Aristotle's 'criticism' of Pythagoras' claim that "everything is ratio!", which he took in the geometria 'counting number' sense and then 'proved' that it couldn't possibly be true. But Plato and Aristotle were wrong! It HAS been discovered that elementary 'particles' like electrons and photons are in fact waves and that the frequency-energy of these waves is quantised in whole-numbers. And an analysis of the equations reveals that, contrary to the interpretation that was originally given to them, the Euclidean algorithm still does actually hold. It has to and it does. The essential flaw of the original interpretation was that it used a space-time model that was designed for discreet particles, not waves.

Now, these waves are continuous. Look through any textbook and you'll see that they are described by complex Euler waves of the form
e^i(wt - kx). They cannot be otherwise since they are based on probabilities of localising the energy which must be continuous. Therefore, any conclusions we can draw from these equations must also be true, in and for itself. In contrast, the whole area of the discreet Fourier Transform and (as you've recently taught me) wavelets are designed for MODELLING these continuous waves in a discontinuous digital environment. From what I can gather, the fact that wavelets use different sized 'windows' for different parts of a wave has more to do with sampling rates, which in itself is a simulated form of frequency. At any rate, saying these waves are 'real' should be taken in the sense that the dragons in Lord of the Rings look 'realistic'. This is not to say that they are not valid - both science and art transform raw materials to suit human needs - only that they can no longer hold the moral high ground, especially if they then want to then claim that they are representing untarnished objective reality. If we listen to a fugue by J.S. Bach on the radio, the radio waves are no more or less valid than the recorded waves being transmitted. This is what I meant when I gave this post its original heading.

Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

Hi Carl,

Ive been reading over Erlich's post on Harmonic Entropy from 97' and here's what he says:
"There is a very strong propensity for the ear to try to fit what it hears into one or a small number of harmonic series, and the fundamentals of these series, even if not physically present, are either heard outright, or provide a more subtle sense of overall pitch known to musicians as the "root". As a component of consonance, the ease with which the ear/brain system can resolve the fundamental is known as "tonalness." I have proposed a concept called "relative harmonic entropy" to model this component of dissonance. The harmonic entropy is based on the concept that the critical band represents a certain degree of uncertainty in the perception of pitch, and for any "true" interval, the auditory system will perceive a range of intervals spanning a number of simple-integer ratios. Simple-integer ratios come into the picture because if the heard tones are to be understood as harmonic overtones of some missing fundamental or root, they must form a simple-integer ratio with one another. The range is a sort of probability distribution, and a certain amount of probability is associated with each of the simple-integer ratios."

Now many years ago when I was trying to explain musical tonality, (how do we hear a tonic when its not present?), I discovered independently that all waves have one basic characteristic property; one cycle always occurs for an amount of time equal to the inverse of the highest common factor between component sine waves, not on the fundamental. In other words, the concept of 'frequency' is no longer confined to that of the first harmonic. I then saw that this gives a possible alternate explanation of HE which, on review, it does.

Earlier you said that "We don't hear GCD's. We hear pitches". I too thought this until I did an experiment. I thought that they were sub-audible and that we somehow recognise when we hit one of its 8ve's in the audible range. However, my very first post on this strand proved the complete opposite! I first listened to an A440Hz sine wave. Next I added the first 19 harmonics all with equal amplitude and still heard A440Hz as expected. All that was different was the tone. Next I took out all harmonics except the relative primes. That is, all that remained was 3,5,7,11,13,17 and 19. And what note did I hear? An A440Hz. The only difference was in the tonal quality which sounded weak and trebly.Observe also that none of the difference tones equals the tonic.

So in answer to your statement, we do hear GCD's because they ARE pitches. If we take the set of all pitches in the audible range from 20Hz upward, then these can be sine waves, GCD's and everything in between. Our possibilities are greatly expanded, not limited, and we also arrive at an objective theory which possibly explains "virtual fundamentals" by showing that they are not that "virtual" at all but are the actual frequency of the incoming wave.

Looking over Erlich's work it is clear to me now that, since his experiments were done with pure sine waves, then HE must apply to the harmonic series itself. We could take approximate values for the relative primes 3,5,7,...above and we should still hear something closer to A440Hz. Or to use your previous example,although 201Hz, 301Hz, 401Hz will have a GCD of 1Hz, since 1 is small compared to the component frequencies, then by HE what we will actually hear is something closer to 100Hz. This explains why we can still hear the GCD in tempered systems. Besides, I've said many times that the tempered intervals are not as irrational as previously thought for this would make them aperiodic to infinity. More realistically we are only able to tune to within a few decimal places. The flat-fifth or sqrt 2 is not 1.41421356...all the way to infinity but 1.414 which is a high rational number.

-Rick

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:

> Earlier you said that "We don't hear GCD's. We hear pitches".
> I too thought this until I did an experiment. I thought that
> they were sub-audible and that we somehow recognise when we hit
> one of its 8ve's in the audible range. However, my very first
> post on this strand proved the complete opposite! I first
> listened to an A440Hz sine wave. Next I added the first
> 19 harmonics all with equal amplitude and still heard A440Hz
> as expected. All that was different was the tone. Next I took
> out all harmonics except the relative primes. That is, all that
> remained was 3,5,7,11,13,17 and 19. And what note did I hear?
> An A440Hz. The only difference was in the tonal quality which
> sounded weak and trebly.

So??????

>Observe also that none of the difference tones equals the tonic.

Right.

> So in answer to your statement, we do hear GCD's because they
> ARE pitches.

Incorrect.

> and we also arrive at an objective theory which possibly
> explains "virtual fundamentals" by showing that they are not
> that "virtual" at all but are the actual frequency of the
> incoming wave.

Do it with irrational intervals and you'll get my attention.

> Looking over Erlich's work it is clear to me now that, since
> his experiments were done with pure sine waves,

They weren't.

> then HE must
> apply to the harmonic series itself.

> We could take approximate values for the relative primes
> 3,5,7,...above and we should still hear something closer
> to A440Hz. Or to use your previous example, although 201Hz,
> 301Hz, 401Hz will have a GCD of 1Hz, since 1 is smal
> l compared to the component frequencies, then by HE what
> we will actually hear is something closer to 100Hz. This
> explains why we can still hear the GCD in tempered systems.

How you can conclude that we are still hearing the "GCD"
when the GCD is 1 is beyond me.

>Besides, I've said many times that the tempered intervals are
>not as irrational as previously thought for this would make
>them aperiodic to infinity. More realistically we are only
>able to tune to within a few decimal places.

The limited accuracy of human hearing steps in long before
the accuracy of your synthesizer runs out.

> The flat-fifth or sqrt 2 is not 1.41421356...all the way
> to infinity but 1.414 which is a high rational number.

And do we hear the GCD of that high rational? No.

-Carl

🔗rick_ballan <rick_ballan@...>

🔗Carl Lumma <carl@...>

🔗Graham Breed <gbreed@...>

🔗Carl Lumma <carl@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> > Earlier you said that "We don't hear GCD's. We hear pitches".
> > I too thought this until I did an experiment. I thought that
> > they were sub-audible and that we somehow recognise when we hit
> > one of its 8ve's in the audible range. However, my very first
> > post on this strand proved the complete opposite! I first
> > listened to an A440Hz sine wave. Next I added the first
> > 19 harmonics all with equal amplitude and still heard A440Hz
> > as expected. All that was different was the tone. Next I took
> > out all harmonics except the relative primes. That is, all that
> > remained was 3,5,7,11,13,17 and 19. And what note did I hear?
> > An A440Hz. The only difference was in the tonal quality which
> > sounded weak and trebly.
>
> So??????

So the audible frequency A440Hz here (oh sorry "pitch") is the GCD, not the 1st harmonic, not the difference tones, and since it is no more or less "virtual" than when the 1st harmonic was included, then it is not the "virtual fundamental" either (sorry Graham). Since all options are excluded except the GCD then the result speaks for itself. I've given the mathematical proof more than once; The time period required for a wave to pass through one cycle is equal to its inverse GCD. The number of cycles passed per second (Hertz) is equal to the GCD, both of which represent the very definition of frequency. The frequency only corresponds to the 1st harmonic when they happen to be equal.
>
> >Observe also that none of the difference tones equals the tonic.
>
> Right.
>
> > So in answer to your statement, we do hear GCD's because they
> > ARE pitches.
>
> Incorrect.

I can't see that Carl. I just showed that it is correct both mathematically and experimentally. Given a frequency of A440Hz, then the pitch will be the absolute value of this GCD (which is the air pressure arriving at the ear drum). ALL were heard distinctly as A440Hz. And when we extend this via harmonic entropy, a whole new horizon of possibilities might open up.
>
> > and we also arrive at an objective theory which possibly
> > explains "virtual fundamentals" by showing that they are not
> > that "virtual" at all but are the actual frequency of the
> > incoming wave.
>
> Do it with irrational intervals and you'll get my attention.

That's what I've been trying to draw your attention to all along. Obviously GCD's by definition can't apply to irrationals (Recall for eg Pythagoras' proof that sqrt2 is not a rational number). Since many tempered systems are based on irrationals and yet are tonal, then we must conclude that the GCD is not sufficiently general. However, this is where HE steps in (as I said below about 301Hz/201Hz approx 3/2 so that approx GCD is 301/3 approx 201/2). Transitive law: applying HE to the harmonic intervals we are automatically applying it to the GCD. It was this that attracted me to HE in the first place and why I asked for your help.
>
> > Looking over Erlich's work it is clear to me now that, since
> > his experiments were done with pure sine waves,
>
> They weren't.

Well that's much of a muchness now. It doesn't effect what I'm saying one way or the other.
>
> > then HE must
> > apply to the harmonic series itself.
>
> ?

Look at it another way. In the real world a piano string is not perfectly flexible so the harmonics 2, 3,...are not that exact. HE would apply to them.
>
> > We could take approximate values for the relative primes
> > 3,5,7,...above and we should still hear something closer
> > to A440Hz. Or to use your previous example, although 201Hz,
> > 301Hz, 401Hz will have a GCD of 1Hz, since 1 is smal
> > l compared to the component frequencies, then by HE what
> > we will actually hear is something closer to 100Hz. This
> > explains why we can still hear the GCD in tempered systems.
>
> How you can conclude that we are still hearing the "GCD"
> when the GCD is 1 is beyond me.

This confused me too. Yet if we look at the actual waves of these two examples, we see that the wave shape actually changes very little. IOW the resultant wave will be much closer to 100Hz than 1Hz.

> >Besides, I've said many times that the tempered intervals are
> >not as irrational as previously thought for this would make
> >them aperiodic to infinity. More realistically we are only
> >able to tune to within a few decimal places.
>
> The limited accuracy of human hearing steps in long before
> the accuracy of your synthesizer runs out.
>
> > The flat-fifth or sqrt 2 is not 1.41421356...all the way
> > to infinity but 1.414 which is a high rational number.
>
> And do we hear the GCD of that high rational? No.

No of course not. But I've chosen here a "purposefully irrational" interval which takes a long time before we find a rational number approximation. And by HE it is an interval with high entropy. OTOH, 2^1/3 for instance has approximations far earlier in the rationals. Besides, you saying that this is not 'close' to 5/4 or 81/64? I agree, the brain does seem to group these into one 'package'.
>
> -Carl
>

🔗Graham Breed <gbreed@...>

2009/8/3 rick_ballan <rick_ballan@...>:

"""
> So the audible frequency A440Hz here (oh sorry "pitch") is the GCD, not the 1st harmonic, not the difference tones, and since it is no more or less "virtual" than when the 1st harmonic was included, then it is not the "virtual fundamental" either (sorry Graham). Since all options are excluded except the GCD then the result speaks for itself. I've given the mathematical proof more than once; The time period required for a wave to pass through one cycle is equal to its inverse GCD. The number of cycles passed per second (Hertz) is equal to the GCD, both of which represent the very definition of frequency. The frequency only corresponds to the 1st harmonic when they happen to be equal.
"""

The term "virtual fundamental" may be my own coinage, so pay it no
heed. "Virtual pitch" is an established term. According to
Wikipedia, it comes from Ernst Terhardt. And, whaddayaknow, Terhardt
has a web page explaining it:

http://www.mmk.ei.tum.de/persons/ter/top/virtualp.html

However much you play with the words, what you're talking about looks
very much like virtual pitch. It's been around for a long time and is
not something you're likely to have discovered a critical but
previously overlooked flaw in.

When did you exclude other options? I missed that. In particular I
missed your consideration of subharmonic matching and windowed
autocorrelations.

You'll notice that Terhardt talks about bells. They're one common
timbre that can be written as tones with integer frequency
relationships, but where the GCD gives you the wrong pitch. That's
been known for as long as I've been alive, and the GCD theory has been
in the doghouse as a result. Now you're bringing it out into the open
again, you must have found something new. How do you deal with bells?

Graham

🔗Petr Pařízek <p.parizek@...>

Graham wrote:

> You'll notice that Terhardt talks about bells. They're one common
> timbre that can be written as tones with integer frequency
> relationships, but where the GCD gives you the wrong pitch. That's
> been known for as long as I've been alive, and the GCD theory has been
> in the doghouse as a result. Now you're bringing it out into the open
> again, you must have found something new. How do you deal with bells?

I thought Rick was talking about sounds which are approximating linearly „regular“ spectra in some way, wasn’t he? And while I’m able to easily find linear retularity in spectra whose consecutive overtones have a common difference frequency (like the aforementioned 500:800:1100:1400...), I’m unable to find any approximation to linear regularity in sounds like bells. To be honest, I’ve actually heard some people estimating the „fundamentals“ of church bells from a few overtones which they found close enough to a part of the harmonic series. Don’t know about other countries, but as far as I’ve listened to bells around the Czech Republic, the most „civilized“ bells I’ve heard have an audible minor tenth from the lowest tone, which is quite interesting. And if I should make up two imaginary spectra resembling church bell sounds rounded to, let’s say, 36-equal, it might be something like this (these are only a few tones, I would have to slow down recordings of real bells to have a good image in my mind)
======
Example 1:
C4 C5 Eb5 C6 G6 C7 F-7 A7
Example 2:
C4 Bb-4 D5 B5 F#6 B+6 E-7 Ab+8
======
In the second example, some people would argue that three of the tones are close enough to B5-F#6-B6, which they think is reasonable for claiming that the fundamental is B4, although there isn’t any B4 in the actual set of overtones.

Petr

PS: Anyone who could suggest a website with bell recordings?

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:

> > > I first
> > > listened to an A440Hz sine wave. Next I added the first
> > > 19 harmonics all with equal amplitude and still heard A440Hz
> > > as expected. All that was different was the tone. Next I
> > > took out all harmonics except the relative primes. That is,
> > > all that remained was 3,5,7,11,13,17 and 19. And what note
> > > did I hear? An A440Hz. The only difference was in the tonal
> > > quality which sounded weak and trebly.
> >
> > So??????
>
> So the audible frequency A440Hz here (oh sorry "pitch") is
> the GCD, not the 1st harmonic, not the difference tones, and
> since it is no more or less "virtual" than when the 1st harmonic
> was included, then it is not the "virtual fundamental" either
> (sorry Graham).

It is the "missing fundamental" or virtual pitch as Graham
stated. This phenomenon has been known for at least 50 years,
so I'm not sure why you're rehashing it here.

>Since all options are excluded except the GCD

You excluded all other options? That's pretty impressive.

> > > So in answer to your statement, we do hear GCD's because they
> > > ARE pitches.
> >
> > Incorrect.
>
> I can't see that Carl. I just showed that it is correct both
> mathematically and experimentally. Given a frequency of A440Hz,
> then the pitch will be the absolute value of this GCD (which
> is the air pressure arriving at the ear drum). ALL were heard
> distinctly as A440Hz.

You've been asked several times about the pitch of things
like piano tones and bells. Your method often gives a
subsonic "pitch" for such tones. Why you keep ignoring
this point is beyond me.

>And when we extend this via harmonic entropy, a whole new
>horizon of possibilities might open up.

Again, I've asked for an example of this but you haven't
provided one. You'll need to understand harmonic entropy
to create one.

> > Do it with irrational intervals and you'll get my attention.
>
> That's what I've been trying to draw your attention to all
> along. Obviously GCD's by definition can't apply to irrationals

I believe I was the one who pointed out that it can't apply
to irrationals.

>However, this is where HE steps in (as I said below about
>301Hz/201Hz approx 3/2 so that approx GCD is 301/3 approx 201/2).
>Transitive law: applying HE to the harmonic intervals we are
>automatically applying it to the GCD. It was this that attracted
>me to HE in the first place and why I asked for your help.

Here's where things go haywire. As Mike B. said, this line
of inquiry is certainly interesting, but it's a long way
from producing anything tangible. All of us would be more
than interested if you were to present something.

> > > then HE must apply to the harmonic series itself.
> >
> > ?
>
> Look at it another way. In the real world a piano string is
> not perfectly flexible so the harmonics 2, 3,...are not that
> exact. HE would apply to them.

Harmonic entropy measures the uncertainty of a dyad's
virtual pitch. That's all. Implicitly, the dyad can be
two sine tones or two pitched complex tones.

> > > We could take approximate values for the relative primes
> > > 3,5,7,...above and we should still hear something closer
> > > to A440Hz. Or to use your previous example, although 201Hz,
> > > 301Hz, 401Hz will have a GCD of 1Hz, since 1 is smal
> > > l compared to the component frequencies, then by HE what
> > > we will actually hear is something closer to 100Hz. This
> > > explains why we can still hear the GCD in tempered systems.
> >
> > How you can conclude that we are still hearing the "GCD"
> > when the GCD is 1 is beyond me.
>
> This confused me too. Yet if we look at the actual waves of
> these two examples, we see that the wave shape actually changes
> very little. IOW the resultant wave will be much closer to
> 100Hz than 1Hz.

Right. And the two conclusions I draw from this are:
1. It would be useful to quantify it and 2. The GCD can't
quantify it. Fortunately, it's a problem that's mostly
solved (for monophonic sources), and I've already posted a
link to a paper that tells you how to solve it. And the
source code is available at tartini.net. And precompiled
binaries are available there too, so you can see it works.

> 2^1/3 for instance has approximations far earlier in the
> rationals. Besides, you saying that this is not 'close'
> to 5/4 or 81/64? I agree, the brain does seem to group these
> into one 'package'.

Right, and here's where some quantitative capability comes
in handy. The nearest HE local minimum to 600 cents is 7/5.
The nearest to 400 cents is 5/4. So, if you like, that is
"earlier in the rationals" but how do you trade complexity
of the rational and the distance from the irrational?
Harmonic entropy does this in a precise way, and therefore
you can predict what the "packages" will be.

-Carl

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> 2009/8/3 rick_ballan <rick_ballan@...>:

> > So the audible frequency A440Hz here (oh sorry "pitch") is the GCD, not the 1st harmonic, not the difference tones, and since it is no more or less Â "virtual" than when the 1st harmonic was included, then it is not the "virtual fundamental" either (sorry Graham). Since all options are excluded except the GCD then the result speaks for itself. I've given the mathematical proof more than once; The time period required for a wave to pass through one cycle is equal to its inverse GCD. The number of cycles passed per second (Hertz) is equal to the GCD, both of which represent the very definition of frequency. The frequency only corresponds to the 1st harmonic when they happen to be equal.
> """
>
> The term "virtual fundamental" may be my own coinage, so pay it no
> heed. "Virtual pitch" is an established term. According to
> Wikipedia, it comes from Ernst Terhardt. And, whaddayaknow, Terhardt
> has a web page explaining it:
>
> http://www.mmk.ei.tum.de/persons/ter/top/virtualp.html
>
> However much you play with the words, what you're talking about looks
> very much like virtual pitch. It's been around for a long time and is
> not something you're likely to have discovered a critical but
> previously overlooked flaw in.

Thanks for the article Graham.
>
> When did you exclude other options? I missed that. In particular I
> missed your consideration of subharmonic matching and windowed
> autocorrelations.

My point was that it was not difference tones and the GCD was no less "virtual" than the original sine wave. Concerning subharmonics I've already shown in previous posts that they still have a GCD and are therefore just 'harmonics'. And as for autocorrelations, they were not a factor in my original 'test' which used the simple oscil in Csound and therefore need not be called as a witness for the defence. (The relationship between the sound produced and my ears did not need an autocorrelation program). IOW in this instance what we are hearing is not the virtual pitch but simply "the pitch", which is true both in definition and in fact. And since it's true in at least one case, then it's something that needs to be taken into account. If we take a simple sine wave y(t) = Acos2pi440t, it gives a nice rounded curve of period (1/440)sec. Add a few harmonics and the shape of the wave changes but the period remains the same. This is perceived by the ear as a change in tone. Finally, take out the original sine wave A440Hz and, provided that at least two of the remaining harmonics are relatively prime, then the shape of the wave changes but the period does not. And of course this is perceived as a change in tone.

Rick
>
> You'll notice that Terhardt talks about bells. They're one common
> timbre that can be written as tones with integer frequency
> relationships, but where the GCD gives you the wrong pitch. That's
> been known for as long as I've been alive, and the GCD theory has been
> in the doghouse as a result. Now you're bringing it out into the open
> again, you must have found something new. How do you deal with bells?
>
>
> Graham
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
>
> > Erlich: "There is a very strong propensity for the ear to try
> > to fit what it hears into one or a small number of harmonic series,
> > and the fundamentals of these series, even if not physically
> > present, are either heard outright, or provide a more subtle sense
> > of overall pitch known to musicians as the "root"."
> >
> > So when Erlich say's "even if not physically present", a study of
> > the waves does in fact show that they are physically present.
> > They are the frequency of the incoming waveform.
>
> Paul's talking about virtual pitch here, which is a well-accepted
> phenomenon in psychoacoustics. You haven't answered what happens
> when the "frequency" is subsonic. -Carl
>
I have but its just that you're not applying what you already know to the extended definition of frequency. The question of audibility/inaudibility is the same as it always has been for sine waves. It depends on choice of frequency range (above 20Hz) and relative amplitudes as usual, but with the additional clause of proximity of the partials to the GCD. For example, since we are assuming that the partials are in the audible range, if we choose small numbers such as 3 and 5 then these are close to the GCD 1 and it too will likely lie within the audible range. However, this begins to break down for higher harmonics as the relative position of the GCD moves farther away. It will also be inaudible if the 3 and 5 are too close to 20Hz.

To refer back the GEN 10 example I sent with frequency A440Hz, taking harmonics 0 0 1 0 1, that is the 3rd and 5th harmonics with unit amplitude of 1, produces a very clear A440Hz:

<CsScore>
; Table #1: a simple sine wave (using GEN10).
f 1 0 16384 10 0 0 1 0 1

This is as clear as if we chose the 1st harmonic:

<CsScore>
; Table #1: a simple sine wave (using GEN10).
f 1 0 16384 10 1

To test the results, try the 3rd and 5th harmonics on their own where we'll hear a distinct E1320Hz and C#2200Hz note:

f 1 0 16384 10 0 0 1

and

f 1 0 16384 10 0 0 0 0 1.

I've tested it up to the 19th harmonic using only two consecutive relative primes and the A440Hz can still be distinctly heard, though it begins to sounds very thin and piercing. Here's 17 and 19:

f 1 0 16384 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1.

To conclude on this point, we wouldn't say that all sine waves are not real because some of them happen to lie within the subsonic range. The same applies to the GCD. However, two other possibilities now present themselves. The first is that in the case where the waveform frequency (now synonymous with GCD) is subsonic, the ear still recognises the tonic when we play one of its 8ve equivalents that do lie within the audible range. The second possibility is that there is a greater association between tempo and harmony than was previously thought. If we treat tempo just like a subsonic frequency - that is, we treat note durations like quarter notes and eighth notes like periods - then the number of cycles represents the interval between the two frequencies. For example, C256Hz played for 2 seconds gives 512 cycles which is also the ratio between 256Hz and 0.5Hz. It says that the audible C is the "512th harmonic" of the tempo C. The vertical 'harmonic' and horizontal 'temporal' aspects of a musical score are now unified much like a cycles-time graph, and frequency is the 'slope of the line'. This is useful for mathematical wave analysis if not for musical composition (for example, it is the 'vt' part of a sine wave cos2pivt).

Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
>
>Okay Carl, below you stated "I believe I was the one who pointed out that it [the GCD] can't apply to irrationals". How you would think I wouldn't know this is truly beyond me. Lets' just say I've 'known' about it since the early nineties. And now I see that you don't fully appreciate what we're really dealing with if waves are truly irrational.

Since all periodic waves require a GCD between rationals then it follows automatically that all irrationals must by definition be aperiodic. Mathematically speaking, we could have a nice periodic infinite trigonometric series and throw just one truly irrational number in to make the entire wave completely aperiodic. This would mean that for an infinite amount of time no two consecutive waves would repeat. Or in other words, we could never pin-down an exact frequency. Scientists have had a computer running since, I think the early eighties, calculating PI to all its decimal places and, as far as I know, it's still going strong. And this is just for ONE irrational number. Let us now consider ALL irrational numbers.

According to Georg Cantor, the number of irrational numbers (aleph-one or uncountable infinity) is so vastly superior when compared to the number of rational numbers (aleph-naught or countable infinity) that if we were to throw a dart at random onto a number line then the chances of hitting a rational is zero. Now let us combine two relative irrationals, then three and so on up to aleph-one. Each one will give a new aperiodic wave completely different from the next. And each of these will be aperiodic to infinity.

Now I ask you; is THIS how our tempered tunings behave? No. As I've said all along they are not irrationals at all but are rationals high up in the series. We 'say' sqrt2 but in the real world create something closer to 1.414. It is for this reason I kept on asking you whether Erlich believed the frequencies along the axis were truly continuous. Because this would mean that if we chose an irrational interval arbitrarily close to the small whole-numbered ratio that the wave would become completely aperiodic, and I'm sure that this wasn't what he intended. In fact in a post to Graham a few months back I proposed the hypothesis that much of what we categorise as 'noise' might be such irrational waves. Of course I never heard back.

In contrast, the nice whole numbers seem like an oasis in a chaotic desert. Harmony is something to be nurtured and valued. As Kronecker, Cantor's maths teacher and eventual nemesis, said, "God created the natural numbers, mankind created the rest". Perhaps he had a point?

Rick

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > > > I first
> > > > listened to an A440Hz sine wave. Next I added the first
> > > > 19 harmonics all with equal amplitude and still heard A440Hz
> > > > as expected. All that was different was the tone. Next I
> > > > took out all harmonics except the relative primes. That is,
> > > > all that remained was 3,5,7,11,13,17 and 19. And what note
> > > > did I hear? An A440Hz. The only difference was in the tonal
> > > > quality which sounded weak and trebly.
> > >
> > > So??????
> >
> > So the audible frequency A440Hz here (oh sorry "pitch") is
> > the GCD, not the 1st harmonic, not the difference tones, and
> > since it is no more or less "virtual" than when the 1st harmonic
> > was included, then it is not the "virtual fundamental" either
> > (sorry Graham).
>
> It is the "missing fundamental" or virtual pitch as Graham
> stated. This phenomenon has been known for at least 50 years,
> so I'm not sure why you're rehashing it here.
>
> >Since all options are excluded except the GCD
>
> You excluded all other options? That's pretty impressive.
>
> > > > So in answer to your statement, we do hear GCD's because they
> > > > ARE pitches.
> > >
> > > Incorrect.
> >
> > I can't see that Carl. I just showed that it is correct both
> > mathematically and experimentally. Given a frequency of A440Hz,
> > then the pitch will be the absolute value of this GCD (which
> > is the air pressure arriving at the ear drum). ALL were heard
> > distinctly as A440Hz.
>
> You've been asked several times about the pitch of things
> like piano tones and bells. Your method often gives a
> subsonic "pitch" for such tones. Why you keep ignoring
> this point is beyond me.
>
> >And when we extend this via harmonic entropy, a whole new
> >horizon of possibilities might open up.
>
> Again, I've asked for an example of this but you haven't
> provided one. You'll need to understand harmonic entropy
> to create one.
>
> > > Do it with irrational intervals and you'll get my attention.
> >
> > That's what I've been trying to draw your attention to all
> > along. Obviously GCD's by definition can't apply to irrationals
>
> I believe I was the one who pointed out that it can't apply
> to irrationals.
>
> >However, this is where HE steps in (as I said below about
> >301Hz/201Hz approx 3/2 so that approx GCD is 301/3 approx 201/2).
> >Transitive law: applying HE to the harmonic intervals we are
> >automatically applying it to the GCD. It was this that attracted
> >me to HE in the first place and why I asked for your help.
>
> Here's where things go haywire. As Mike B. said, this line
> of inquiry is certainly interesting, but it's a long way
> from producing anything tangible. All of us would be more
> than interested if you were to present something.
>
> > > > then HE must apply to the harmonic series itself.
> > >
> > > ?
> >
> > Look at it another way. In the real world a piano string is
> > not perfectly flexible so the harmonics 2, 3,...are not that
> > exact. HE would apply to them.
>
> Harmonic entropy measures the uncertainty of a dyad's
> virtual pitch. That's all. Implicitly, the dyad can be
> two sine tones or two pitched complex tones.
>
> > > > We could take approximate values for the relative primes
> > > > 3,5,7,...above and we should still hear something closer
> > > > to A440Hz. Or to use your previous example, although 201Hz,
> > > > 301Hz, 401Hz will have a GCD of 1Hz, since 1 is smal
> > > > l compared to the component frequencies, then by HE what
> > > > we will actually hear is something closer to 100Hz. This
> > > > explains why we can still hear the GCD in tempered systems.
> > >
> > > How you can conclude that we are still hearing the "GCD"
> > > when the GCD is 1 is beyond me.
> >
> > This confused me too. Yet if we look at the actual waves of
> > these two examples, we see that the wave shape actually changes
> > very little. IOW the resultant wave will be much closer to
> > 100Hz than 1Hz.
>
> Right. And the two conclusions I draw from this are:
> 1. It would be useful to quantify it and 2. The GCD can't
> quantify it. Fortunately, it's a problem that's mostly
> solved (for monophonic sources), and I've already posted a
> link to a paper that tells you how to solve it. And the
> source code is available at tartini.net. And precompiled
> binaries are available there too, so you can see it works.
>
> > 2^1/3 for instance has approximations far earlier in the
> > rationals. Besides, you saying that this is not 'close'
> > to 5/4 or 81/64? I agree, the brain does seem to group these
> > into one 'package'.
>
> Right, and here's where some quantitative capability comes
> in handy. The nearest HE local minimum to 600 cents is 7/5.
> The nearest to 400 cents is 5/4. So, if you like, that is
> "earlier in the rationals" but how do you trade complexity
> of the rational and the distance from the irrational?
> Harmonic entropy does this in a precise way, and therefore
> you can predict what the "packages" will be.
>
> -Carl
>

🔗martinsj013 <martinsj@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > It is the "missing fundamental" or virtual pitch as Graham
> > stated. This phenomenon has been known for at least 50 years,
> > so I'm not sure why you're rehashing it here.
> ...
> Thanks Carl, I have been trying to follow this thread, and have some thoughts, but I realise there are many blanks in my knowledge, and this has filled some of them. The position seems to be that difference tones, subharmonics and GCD can satisfactorily explain the virtual pitch in some, perhaps many, cases but what we are looking for is a theory that can explain it in all cases.
>
> Group members may be interested in this (found via Google of course):
> http://www.chameleongroup.org.uk/research/link_virtual_analysis.html
>
> Steve M.
>
Hi Steve,

The over-riding problem in this thread was not to explain the virtual pitch. It began with my initial Csound experiment which showed once and for all that, as predicted by wave theory, we do actually hear the GCD as the pitch-frequency; provided it's in the audible range and not covered over by other factors, then we hear it just as plain as any old sine wave. Now others are trying to convince me that this is just the "virtual pitch" that has been well-known for 50 years. However, I don't believe this to be true.

The only difference between a pure sine wave of A440Hz and a GCD complex wave of the same frequency is the shape of the wave which alters the tone. The question of its amplitude i.e. loudness, and audible range is independent from questions of periodicity just as it is with sine waves. Therefore, it's no more or less "virtual" than the notion of pitch-frequency itself. And IMO it begins to explain musical tonality, the ability to hear the tonic, far more realistically than the virtual pitch theory. After all, it is the actual frequency of the wave!

Now, since the GCD has been proven to be synonymous with 'frequency' itself, it follows that both sub-harmonics and difference tones are also subject to the GCD. Given any finite collection of sub-harmonics 1/2, 1/3, ...there will still be a GCD (equal to the lcm of their denominators eg 1/2 and 1/3 have gcd 1/6). And difference tones, being the difference between two frequencies, now means the difference between two GCD's. Because the maths is quite simple, the gaps in your knowledge are probably not as wide as the some would have you believe.

Rick

🔗Michael <djtrancendance@...>

>The question of its amplitude i.e. loudness, and audible range is
independent from questions of periodicity just as it >is with sine
waves.
One experiment: try playing even a loud drum IE a kick tuned to C and another to F....then try the same thing with C and E. Note that the wave formed by the fundamental frequencies of C and F is more periodic than C and E and the amplitude formed by mixing the same two drums at the same volume is lower (look at a graph of the resulting wave form in any audio editor like Audacity or Soundforge).
When music production magazine talk about tuning drums for clarity and loudness (esp. getting loudness with little to no compression)...there seems to be a strong tie with periodicity.

I'm not saying your method is wrong, but rather that periodicity can be used to enhance music production and avoid things like major-label style massive compression that "roboticize" the feeling of music and still get excellent loudness and clarity.

>And difference tones, being the difference between two frequencies, now means the difference between two >GCD's.
So, if I have it right...if you have 200hz and 300hz how would you calculate the difference tones (in an example using these numbers)? It seems to me like your method who give 1/2 vs. 1/3 = 1/6 GCD as the answer but I'm pretty sure I'm wrong.

-Michael

________________________________
From: rick_ballan <rick_ballan@...>
To: tuning@yahoogroups.com
Sent: Wednesday, August 5, 2009 10:13:39 AM
Subject: [tuning] Re:Defend your rights with the GCD

--- In tuning@yahoogroups. com, "martinsj013" <martinsj@.. .> wrote:
>
> --- In tuning@yahoogroups. com, "Carl Lumma" <carl@> wrote:
> > It is the "missing fundamental" or virtual pitch as Graham
> > stated. This phenomenon has been known for at least 50 years,
> > so I'm not sure why you're rehashing it here.
> ...
> Thanks Carl, I have been trying to follow this thread, and have some thoughts, but I realise there are many blanks in my knowledge, and this has filled some of them. The position seems to be that difference tones, subharmonics and GCD can satisfactorily explain the virtual pitch in some, perhaps many, cases but what we are looking for is a theory that can explain it in all cases.
>
> Group members may be interested in this (found via Google of course):
> http://www.chameleo ngroup.org. uk/research/ link_virtual_ analysis. html
>
> Steve M.
>
Hi Steve,

Rick

🔗Mike Battaglia <battaglia01@...>

> Hi Steve,
>
> The over-riding problem in this thread was not to explain the virtual pitch.
> It began with my initial Csound experiment which showed once and for all
> that, as predicted by wave theory, we do actually hear the GCD as the
> pitch-frequency; provided it's in the audible range and not covered over by
> other factors, then we hear it just as plain as any old sine wave. Now
> others are trying to convince me that this is just the "virtual pitch" that
> has been well-known for 50 years. However, I don't believe this to be true.

It IS the "missing fundamental" phenomenon. "Missing fundamental",
"virtual pitch", and all of that mean the same exact thing you're
talking about. People have stopped responding because you've stopped
listening.

> The only difference between a pure sine wave of A440Hz and a GCD complex
> wave of the same frequency is the shape of the wave which alters the tone.
> The question of its amplitude i.e. loudness, and audible range is
> independent from questions of periodicity just as it is with sine waves.
> Therefore, it's no more or less "virtual" than the notion of pitch-frequency
> itself. And IMO it begins to explain musical tonality, the ability to hear
> the tonic, far more realistically than the virtual pitch theory. After all,
> it is the actual frequency of the wave!

No.

A time domain signal is going to be split up in the ear into several
time domain signals. If you take sinusoids of frequency 2, 3, 4, 5, 6,
7, and 8, each with decreasing amplitude, and add them together,
you'll "hear" frequency 1 in the signal even though an actual SINE
WAVE with that frequency isn't present. This is what is referred to as
the "missing fundamental" or "virtual fundamental" or "phantom
fundamental" or "virtual pitch" or anything like that. Even though "1"
isn't present, you perceive it anyway. This is because the brain is
doing some type of analysis on those waves and arriving at "1" for the
best fit for that series.

A friend of mine studying neurobiology told me it had something to do
with cells in the ear that track repeating patterns in the original
signal (or maybe it was cells in the brain). This means that some type
of autocorrelation might be utilized in this analysis.

And everything you're saying IS the virtual pitch theory! You haven't
done anything except propose that we use the word "frequency" to refer
to the fundamental of a periodic wave. It's referred to as "virtual"
because again, no actual SINE WAVE of that frequency is present in the
signal, and so the hair cells in the ear that would resonate at a sine
wave of that frequency won't resonate (except perhaps for a little bit
due to nonlinearities present somewhere). Nonetheless, your brain is
going to perceive that pitch anyway. And, this is a process that you
have some control over - you can "imagine" a root or fundamental
underneath a given chord and your brain will attempt to place the
chord information accordingly in the harmonic series resulting from
your imagined root and the chord.

> Now, since the GCD has been proven to be synonymous with 'frequency' itself,
> it follows that both sub-harmonics and difference tones are also subject to
> the GCD. Given any finite collection of sub-harmonics 1/2, 1/3, ...there
> will still be a GCD (equal to the lcm of their denominators eg 1/2 and 1/3
> have gcd 1/6). And difference tones, being the difference between two
> frequencies, now means the difference between two GCD's. Because the maths
> is quite simple, the gaps in your knowledge are probably not as wide as the
> some would have you believe.

The GCD hasn't been "proven" to be anything. You have decided that you
want to use the word "frequency" to refer to the inverse of the period
of a periodic wave. That's fine, and sometimes it's used that way. And
keep in mind that if you have a chord with frequencies of 4:5.001:6,
you're going to perceive it as 4:5:6 in all cases, perhaps with a
slowly changing phase shift or some kind of beating in the 5 over
time. The fact that the brain chooses to take this perspective instead
of actually locking the chord in at 4000:5001:6000 is fairly arbitrary
and reflects nothing mathematically fundamental about the nature of
"waves" or anything like that. After all, nature doesn't give a damn
about the scaling we use on the time axis. And if you take a chord
with cents 0-350-702, it's so ambiguous that you can flip your
perspective around to hear it as 4:5:6 or 10:12:15 or 16:19:24 or
whatever you'd like. The GCD doesn't seem to have a monopoly on your
perception there.

-Mike

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

> As discussed, Rick's experiment showed nothing new at all.
> Also as discussed, we do NOT hear the GCD, though sometimes
> the pitch we hear is the same as we would hear when listening
> to a sine wave at the GCD frequency, if the stimulus meets a
> the conditions needed to make Rick's ideas match reality.
> That means the GCD can't be subsonic, the stimulus must have
> a Fourier spectrum with most of its energy in the first 16
> harmonics (and preferrably in the first 8), and spectra
> where most of the energy is in, say, harmonics 2 4 6 8 but
> with a touch of 5 (where we hear 2 as the pitch) are excluded.
> Spectra where two or more of the loudest partials fall within
> a critical bandwidth may wind up being unpitched, though the
> GCD would remain unchanged.

Why couldn't we hear subsonic GCDs, if the range of human hearing has
primarily to do with the frequency response of the ear? I saw a demo
at the last ASA convention where they had an extremely high powered
subwoofer playing a tone you could hear at something like 5 Hz - the
idea was that the ear doesn't have a sharp cutoff at 20 Hz, but rather
that it would require a extremely high volume to hear a 5 Hz tone.

Here's a link to the website: http://www.eminent-tech.com/RWbrochure.htm

Rather, I think a better explanation might have to do with critical
band effects: Isn't 14 Hz about the width at which two sinusoids will
beat? So when a complex tone drops low enough, all of its partials
will start to interfere with themselves via critical band effects.

This would infer that it's possible to retain the "pitchedness" of a
square wave at lower frequencies than a sawtooth wave - that would be
an interesting experiment.

-Mike

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > It is the "missing fundamental" or virtual pitch as Graham
> > stated. This phenomenon has been known for at least 50 years,
> > so I'm not sure why you're rehashing it here.
> ...
> Thanks Carl, I have been trying to follow this thread, and
> have some thoughts, but I realise there are many blanks in my
> knowledge, and this has filled some of them. The position
> seems to be that difference tones, subharmonics and GCD can
> satisfactorily explain the virtual pitch in some, perhaps many,
> cases but what we are looking for is a theory that can explain
> it in all cases.
>
> Group members may be interested in this (found via Google
> of course):
> http://www.chameleongroup.org.uk/research/link_virtual_analysis.html
>
> Steve M.

That's an excellent article Steve; thanks for posting it.
It's not clear to me who the author is, but it's well-
researched and it bridges the gap between music analysis
and psychoacoustics nicely.

Hofmann-Engl's model is quite reasonable. It is a more
ad hoc version of Van Eck's approach and harmonic entropy.
Which is to say, Van Eck & Erlich have a much more direct
and principled way to get the same sorts of answers.

It's also worth pointing out that in the analysis of the
tempered chords, it's not clear whether the subharmonics
are being generated in JI or in 12-ET (apparently the
latter). With Van Eck's approach, one doesn't have to
make this choice. When dealing with tempered intervals,
it takes the degree of temperament error in each interval
fully into account. And the issue of where to stop
considering partials (e.g. Rameau 9, Leman 15, etc) also
goes away (Erlich showed the results of the model converge
as the limit approaches infinity).

Speaking of which, I have finally obtained a copy of
Van Eck's book, "J.S. Bach's Crtique of Pure Music",
which is mostly an analysis of Bach's music. But in
Appendix II (pp. 133-144), the model just mentioned is
described. I never realized he had gotten all the way to
the probability distribution -- only the entropy part of
harmonic entropy was missing. Simply amazing, that this
chemist from the Netherlands would publish a book in his
spare time, containing a model of virtual pitch which is
superior to all similar models I know of in the academic
literature, buried in an appendix and now out-of-print,
and virtually unknown to the world! We can all thank
Paul Erlich for finding it and realizing its potential.

He cites both Fokker and our own Claudio Di Veroli, among
others. Claudio- are you still here?

-Carl

🔗Carl Lumma <carl@...>

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

> Why couldn't we hear subsonic GCDs, if the range of human hearing
> has primarily to do with the frequency response of the ear?

If you have lazy eye, you often will go blind in that eye,
even if the eye is working perfectly.

> I saw a demo at the last ASA convention where they had an
> extremely high powered subwoofer playing a tone you could hear
> at something like 5 Hz - the idea was that the ear doesn't have
> a sharp cutoff at 20 Hz, but rather that it would require a
> extremely high volume to hear a 5 Hz tone.
>
> Here's a link to the website: http://www.eminent-tech.com
> /RWbrochure.htm

Yep, I know about it. Haven't heard it though. Which
AES were they at? And what kinda stuff were they playing
through it? I doubt it's very accurate except when driven
by a steady sine wave. I'm also 1000% sure you don't hear
a pitch at 5Hz. It's true the ear has an amplitude-frequency
response, not just a frequency response. But it's very hard
to hear pitches below 20 Hz or above 4KHz. In the case of
music, we already know that the root of chords does not go
this low -- in fact, it's often an octave multiple of the
GCD even when the GCD itself is perfectly audible.

> Rather, I think a better explanation might have to do with
> critical band effects: Isn't 14 Hz about the width at which
> two sinusoids will beat? So when a complex tone drops low
> enough, all of its partials will start to interfere with
> themselves via critical band effects.

I'm sure nobody knows what the critical band is at 14Hz,
since it's never been measured. It's probably outside the
linear response of the cochlea.

-Carl

🔗Mike Battaglia <battaglia01@...>

> Yep, I know about it. Haven't heard it though. Which
> AES were they at? And what kinda stuff were they playing
> through it? I doubt it's very accurate except when driven
> by a steady sine wave. I'm also 1000% sure you don't hear
> a pitch at 5Hz. It's true the ear has an amplitude-frequency
> response, not just a frequency response. But it's very hard
> to hear pitches below 20 Hz or above 4KHz. In the case of
> music, we already know that the root of chords does not go
> this low -- in fact, it's often an octave multiple of the
> GCD even when the GCD itself is perfectly audible.

They were at the ASA convention this past year in Doral, FL - not the
San Fran AES convention in 09. I only saw their poster, I didn't get
to see the live demonstration. A bunch of people in my group went to
it and said he played some extremely low subsonic tone that you could
actually feel, as the amplitude was probably ridiculously loud. I
can't answer about the "pitchedness" of the tone, as I wasn't there,
but I would imagine that it would be extremely hard to hear it as
having a pitch being that low.

> I'm sure nobody knows what the critical band is at 14Hz,
> since it's never been measured. It's probably outside the
> linear response of the cochlea.

Well, whatever it is, whenever the partials get into hearing range,
they'd going to interfere with each other either way. I just screwed
around with squares vs sawtooths at low frequencies and the squares
are considerably smoother, although I'm not sure if this has anything
to do with critical band effects or not.

-Mike

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Yep, I know about it. Haven't heard it though. Which
> > AES were they at? And what kinda stuff were they playing
> > through it? I doubt it's very accurate except when driven
> > by a steady sine wave. I'm also 1000% sure you don't hear
> > a pitch at 5Hz. It's true the ear has an amplitude-frequency
> > response, not just a frequency response. But it's very hard
> > to hear pitches below 20 Hz or above 4KHz. In the case of
> > music, we already know that the root of chords does not go
> > this low -- in fact, it's often an octave multiple of the
> > GCD even when the GCD itself is perfectly audible.
>
> They were at the ASA convention this past year in Doral, FL -
> not the San Fran AES convention in 09.

Ah. I haven't been to AES since 2005, which is also the
same year I found that webpage.

> I only saw their poster, I didn't get to see the live
> demonstration. A bunch of people in my group went to it and
> said he played some extremely low subsonic tone that you
> could actually feel, as the amplitude was probably
> ridiculously loud. I can't answer about the "pitchedness" of
> the tone, as I wasn't there, but I would imagine that it
> would be extremely hard to hear it as having a pitch being
> that low.

Right, you can feel it. There are two pipe organs in the
world with acoustic 64' stops that can get down to 8 Hz,
but people say it's more a sensation than a pitch.

-Carl

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Graham wrote:
> > pitch detection
> > algorithms, which attempt to do whatever it is your ear does.
>
> Several pitch detection algorithms are based on autocorrelation.
> The best pitch detection algorithm out there seems
> to be:
> http://miracle.otago.ac.nz/postgrads/tartini/papers/A_Smarter_Way_to_Find_Pitch.pdf
>
> -Carl

>Hi Carl,

Both yourself and Graham have said that "virtual pitch" has been a well-known phenomenon for 50 years. But only in the psychoacoustics department! Indeed, you've forgotten that psychoacoustics is not the "big picture". Theoretical physics is. The pecking order is: wave (quantum) mechanics - chemistry - biochemistry - biology. And it was this loss of perspective, caused from over specialisation and lack of communication between 'fields', that made psychoacoustics look for and 'find' tonality in the wrong place (as Charles' "chimps prefer consonant music" post showed, where scientists try to perpetuate the myth that music must have a biological basis). It was this loss of perspective which made Graham decide that what I was saying was true but "unimportant". And lets face it Carl, with all due respect, you kept on insisting that I seem to have my "own definition of frequency" which merely confessed a lack of wave theoretical knowledge on your part. So please reciprocate that respect.

Now I've already proven, both theoretically and experimentally, that the pitch detection algorithm is the Euclidean algorithm gone unrecognised for almost 2000 years (whatever else may be true, it's case closed on that point). The fact that the maths is 'known' to any high school student is beside the point. Because mathematicians from Euclid onwards didn't have the necessary background knowledge in wave theory to interpret it correctly and therefore decided for the rest of us that it must be a "thing in itself". Looking over the article you sent I see the same old unquestioned assumptions going on. The placement of the microphone relative to the speakers measured in feet rather than wavelengths (or perhaps only as an afterthought when it happens to be convenient), the fact that all algorithms are based in numbers which grew out of wave theory and not vice-versa, and of course the assumption that the equations which were originally designed to emulate real analogue waves in a digital environment are now also taken as a "thing in itself". Therefore I already know in advance that IF the pitch detection algorithms you sent work then they are unwittingly modelling the real-wave Euclidean algorithm in one way or another. And as I said, Paul's original assumption that the test subjects in HE were hearing a "virtual pitch" now need to be reconsidered, so the HE experiments were really showing the extent to which we can detune the GCD. Scratch beneath the surface and this is what's really going on.

Regards

Rick

🔗Carl Lumma <carl@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >The question of its amplitude i.e. loudness, and audible range is
> independent from questions of periodicity just as it >is with sine
> waves.
> One experiment: try playing even a loud drum IE a kick tuned to C and another to F....then try the same thing with C and E. Note that the wave formed by the fundamental frequencies of C and F is more periodic than C and E and the amplitude formed by mixing the same two drums at the same volume is lower (look at a graph of the resulting wave form in any audio editor like Audacity or Soundforge).
> When music production magazine talk about tuning drums for clarity and loudness (esp. getting loudness with little to no compression)...there seems to be a strong tie with periodicity.
>
> I'm not saying your method is wrong, but rather that periodicity can be used to enhance music production and avoid things like major-label style massive compression that "roboticize" the feeling of music and still get excellent loudness and clarity.

Sure Mike, when it comes to real composition a whole lot of factors come into play. I'm just trying to get across a model that explains something more like traditional harmony, you know, C major scales and all that stuff.
>
> >And difference tones, being the difference between two frequencies, now means the difference between two >GCD's.
> So, if I have it right...if you have 200hz and 300hz how would you calculate the difference tones (in an example using these numbers)? It seems to me like your method who give 1/2 vs. 1/3 = 1/6 GCD as the answer but I'm pretty sure I'm wrong.

The difference tone is just the same as usual i.e. 100Hz. Its just that it is possible to substitute GCD's for the 200Hz and 300Hz. For example, if we play sine waves of 600Hz and 1000Hz together, the 3rd and 5th harmonics of 200Hz, then the GCD is 200Hz which is the frequency. We could do the same for 300Hz. But the GCD between these is also 100Hz which makes all the sine waves involved upper harmonics of this new tonic i.e. the 3rd harmonic of 200Hz, 600Hz, is now the 6th harmonic of the new fundamental and so on.
>
> -Michael
>
>
>
>
>
>
> ________________________________
> From: rick_ballan <rick_ballan@...>
> To: tuning@yahoogroups.com
> Sent: Wednesday, August 5, 2009 10:13:39 AM
> Subject: [tuning] Re:Defend your rights with the GCD
>
>
> --- In tuning@yahoogroups. com, "martinsj013" <martinsj@ .> wrote:
> >
> > --- In tuning@yahoogroups. com, "Carl Lumma" <carl@> wrote:
> > > It is the "missing fundamental" or virtual pitch as Graham
> > > stated. This phenomenon has been known for at least 50 years,
> > > so I'm not sure why you're rehashing it here.
> > ...
> > Thanks Carl, I have been trying to follow this thread, and have some thoughts, but I realise there are many blanks in my knowledge, and this has filled some of them. The position seems to be that difference tones, subharmonics and GCD can satisfactorily explain the virtual pitch in some, perhaps many, cases but what we are looking for is a theory that can explain it in all cases.
> >
> > Group members may be interested in this (found via Google of course):
> > http://www.chameleo ngroup.org. uk/research/ link_virtual_ analysis. html
> >
> > Steve M.
> >
> Hi Steve,
>
> The over-riding problem in this thread was not to explain the virtual pitch. It began with my initial Csound experiment which showed once and for all that, as predicted by wave theory, we do actually hear the GCD as the pitch-frequency; provided it's in the audible range and not covered over by other factors, then we hear it just as plain as any old sine wave. Now others are trying to convince me that this is just the "virtual pitch" that has been well-known for 50 years. However, I don't believe this to be true.
>
> The only difference between a pure sine wave of A440Hz and a GCD complex wave of the same frequency is the shape of the wave which alters the tone. The question of its amplitude i.e. loudness, and audible range is independent from questions of periodicity just as it is with sine waves. Therefore, it's no more or less "virtual" than the notion of pitch-frequency itself. And IMO it begins to explain musical tonality, the ability to hear the tonic, far more realistically than the virtual pitch theory. After all, it is the actual frequency of the wave!
>
> Now, since the GCD has been proven to be synonymous with 'frequency' itself, it follows that both sub-harmonics and difference tones are also subject to the GCD. Given any finite collection of sub-harmonics 1/2, 1/3, ...there will still be a GCD (equal to the lcm of their denominators eg 1/2 and 1/3 have gcd 1/6). And difference tones, being the difference between two frequencies, now means the difference between two GCD's. Because the maths is quite simple, the gaps in your knowledge are probably not as wide as the some would have you believe.
>
> Rick
>

🔗Michael <djtrancendance@...>

>Newsflash: theoretical physics is off-topic for this list.
>Not that you're guilty of doing any.
Hmm....I get a strong impression what Rick was saying is wave mechanics is the top of the order not theoretical physics...and it just happens the wave mechanics hint at theoretical physics. The only thing I see being approached about GCD is wave mechanics, which do related directly to how the ear interprets music (although whether it relates much or less than psychoacoustics and things like varied results of tests to see the critical band is up to opinion). The point again though seems obvious; both can be used to help music and, on the side, both seem to point strongly toward periodicity as a basis, though not the only basis, of consonance (I think at least we can agree on that).

________________________________
From: Carl Lumma <carl@...>
To: tuning@yahoogroups.com
Sent: Thursday, August 6, 2009 3:10:45 AM
Subject: [tuning] Re:Defend your rights with the GCD

--- In tuning@yahoogroups. com, "rick_ballan" <rick_ballan@ ...> wrote:

> Both yourself and Graham have said that "virtual pitch" has
> been a well-known phenomenon for 50 years. But only in the
> psychoacoustics department! Indeed, you've forgotten that
> psychoacoustics is not the "big picture". Theoretical physics
> is. The pecking order is: wave (quantum) mechanics - chemistry -
> biochemistry - biology.

Newsflash: theoretical physics is off-topic for this list.
Not that you're guilty of doing any.

-Carl

🔗Michael <djtrancendance@...>

> Indeed, you've forgotten that psychoacoustics is not the "big picture". Theoretical physics is. And lets face it Carl, with all due respect, you kept on >insisting that I seem to have my "own definition of frequency" which merely confessed a lack of wave >theoretical knowledge on your part. So please >reciprocate that respect.

I don't think either of you are right or wrong...but I will say that both signal processing and psycho-acoustics have contributed immensely to our understanding of music and don't see any reason why one should be ignored for the sake of the other. That kind of thing comes across like saying all music across should be either based on 12-tone or Shrutis because only one can be right and I don't see how that attitude could help anyone.

Still I'm a bit confused about the GCD/"Euclidean" method vs. HE: again, given a 300 350, 360, and 362 hz wave what would be the root tone implied by each method (Harmonic Entropy vs. GCD) and why (mathematically)? And, better yet, also include a simple sound test where you hear A) the sine waves at the above tones B) the sine wave representing what SHOULD be the root tone according to each methods...so we listeners can compare and say "this root tone matches the sense of tone created by the overtone best and I prefer the method which deduced it."

I'm not taking sides on this issue...but I do feel there's a lack of tangible evidence on both sides of the argument (or, at least, that's directly visible on this list without a whole lot of post searching). Also, for the record, I think people on both sides (HE and GCD) are taking a lot of effort to explain why they think how they do about the issue...for sure effort is not the problem.

-Michael

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > As discussed, Rick's experiment showed nothing new at all.
> > Also as discussed, we do NOT hear the GCD, though sometimes
> > the pitch we hear is the same as we would hear when listening
> > to a sine wave at the GCD frequency, if the stimulus meets a
> > the conditions needed to make Rick's ideas match reality.
> > That means the GCD can't be subsonic, the stimulus must have
> > a Fourier spectrum with most of its energy in the first 16
> > harmonics (and preferrably in the first 8), and spectra
> > where most of the energy is in, say, harmonics 2 4 6 8 but
> > with a touch of 5 (where we hear 2 as the pitch) are excluded.
> > Spectra where two or more of the loudest partials fall within
> > a critical bandwidth may wind up being unpitched, though the
> > GCD would remain unchanged.
>
> Why couldn't we hear subsonic GCDs, if the range of human hearing has
> primarily to do with the frequency response of the ear? I saw a demo
> at the last ASA convention where they had an extremely high powered
> subwoofer playing a tone you could hear at something like 5 Hz - the
> idea was that the ear doesn't have a sharp cutoff at 20 Hz, but rather
> that it would require a extremely high volume to hear a 5 Hz tone.
>
> Here's a link to the website: http://www.eminent-tech.com/RWbrochure.htm
>
> Rather, I think a better explanation might have to do with critical
> band effects: Isn't 14 Hz about the width at which two sinusoids will
> beat? So when a complex tone drops low enough, all of its partials
> will start to interfere with themselves via critical band effects.
>
> This would infer that it's possible to retain the "pitchedness" of a
> square wave at lower frequencies than a sawtooth wave - that would be
> an interesting experiment.
>
> -Mike
>
And just to add, we can hear the GCD if it's harmonics are within the audible range and close to "1". I explained this already. I've heard the GCD myself many times and have not yet read any tangible proof that its not the real wave I'm hearing, only other theories.

Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> >
> > Okay Carl, below you stated "I believe I was the one who pointed
> > out that it [the GCD] can't apply to irrationals". How you would
> > think I wouldn't know this is truly beyond me.
>
> I'm sorry Rick, but I can't go on replying to your replies when
> you don't seem to address any of the points being made. It's
> like you're not even reading the posts you're replying to.
>
> -Carl
>
Of course I'm reading the posts. Don't imagine that I don't get what you're saying. What I'm trying to get across is giving something that was thought to be "well-known" a new perspective. The technique is called mathematical philosophy founded by Russell, Whitehead and Frege. I was in the middle of working out an answer to the paper you sent.

It is believed that the sine wave is the elementary starting point of Fourier analysis. However, given y(t) = Acos2pivt, since vt = n, where n is number of cycles, we have:
y(t) = Acos2pivt = Acos2pin.
The derivative becomes dy/dt = (dy/dn).(dn/dt) = (dy/dn).v, since frequency is the derivative of cycles with respect to time. The second derivative gives v^2, the third v^3 and so on.
Applying the same operation backwards for the integral requires multiplying by the period T = 1/v, so that the second integral requires T^2 etc...
Notice that the variable 't' has completely disappeared from our equation! Could it be that waves bring their own measure of time in the form of period/frequency?
Now, let us assume that our original sine wave represents one of the component sine waves in a Fourier series f(t) = f(t + T). It is usually believed that the periodicity of f(t) is given in advance and is independent of the component waves. Yet if we substitute this new n-dependent integral form of the wave into the Fourier coefficients, we find that this is not the case at all. Try it! Since the periodicity of the components on the right side of the equation require the GCD, then so does the original function. Hence, Fourier series depends upon the GCD and not the other way around. And it is not difficult to apply the same method to the Fourier transform and all of its variants (but of course they are no longer periodic).

So you see that what I'm saying doesn't actually disagree with the results of what you say, only the interpretation. To say "Rick hasn't found anything new, end of discussion" is not exactly true and doesn't get on my good side.

Regards

Rick

🔗Michael <djtrancendance@...>

🔗Mike Battaglia <battaglia01@...>

> It is believed that the sine wave is the elementary starting point of
> Fourier analysis. However, given y(t) = Acos2pivt, since vt = n, where n is
> number of cycles, we have:
> y(t) = Acos2pivt = Acos2pin.
> The derivative becomes dy/dt = (dy/dn).(dn/dt) = (dy/dn).v, since frequency
> is the derivative of cycles with respect to time. The second derivative
> gives v^2, the third v^3 and so on.
> Applying the same operation backwards for the integral requires multiplying
> by the period T = 1/v, so that the second integral requires T^2 etc...
> Notice that the variable 't' has completely disappeared from our equation!
> Could it be that waves bring their own measure of time in the form of
> period/frequency?

> Both yourself and Graham have said that "virtual pitch" has been a
> well-known phenomenon for 50 years. But only in the psychoacoustics
> department! Indeed, you've forgotten that psychoacoustics is not the "big
> picture". Theoretical physics is. The pecking order is: wave (quantum)
> mechanics - chemistry - biochemistry - biology.

No offense here, and not to speak for Carl, but this has been
discussed over, and over, and over again. In fact, it's been discussed
so much that I can foresee only three possible outcomes to this whole
conversation:

1) You realize that your reframing of the virtual pitch phenomenon is
equivalent to your choosing to use different terminology than what is
common
2) We all stop responding because we're tired of a pointless argument
that is clearly going nowhere
3) The debate continues until various list members start posting about
how their inboxes are flooded and how they're leaving the tuning list
forever.

I am by no means a veteran on this list, having been here for a year,
but this seems to be the general pattern. Shall we attempt to avoid
number three?

-Mike

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> > > Okay Carl, below you stated "I believe I was the one who
> > > pointed out that it [the GCD] can't apply to irrationals".
> > > How you would think I wouldn't know this is truly beyond me.
> >
> > I'm sorry Rick, but I can't go on replying to your replies when
> > you don't seem to address any of the points being made. It's
> > like you're not even reading the posts you're replying to.
>
> Of course I'm reading the posts.

Then why don't you respond to the points made in them?
Here are some for easy reference:

* We all agree 301/201 sounds like a 3/2, so you're not
bringing anything new there. What would get our attention,
and Mike B. and I have both asked, is to quantify the
phenomenon. You claim to be able to do this by 'combining
the GCD with harmonic entropy' but you haven't shown how
it might work.

* What happens when the GCD is subsonic?

* Can you predict the pitch of metallophones using your
method? (Asked by Graham)

* Howabout a spectrum with most of its energy is in
harmonics 2 4 6 8 but with a touch of 5?

>What I'm trying to get across is giving something that was
>thought to be "well-known" a new perspective.

It would help if you studied the literature you intend to
give a new perspective to, first.

>I was in the middle of working out an answer to the paper you sent.

Which one? McLeod & Wyvill? What's your response, then?

-Carl

🔗Michael <djtrancendance@...>

>And just to add, we can hear the GCD if it's harmonics are within the audible range and close to "1". I >explained this already.
Interesting, wouldn't this imply that we want to keep GCD to a fairly high number (IE at least 100-200hz) to keep it clearly within the audible range?

I admit...I haven't heard any counter-theories either, only alternative explanations such as the theory of Virtual Pitch (which is, if I have it right, simply that you will hear the base frequency of who's harmonic overtones best align with the frequencies given). (Other list members) please feel free to correct me if I'm wrong but, if that definition is correct and virtual pitch is indeed the only widespread accepted theory...Rick is indeed adding something new to the theory (which could, I imagine, be used to calculate different assumed root tones for certain instruments with odd harmonic structures).

-Michael

________________________________
From: rick_ballan <rick_ballan@...>
To: tuning@yahoogroups.com
Sent: Thursday, August 6, 2009 3:21:14 PM
Subject: [tuning] Re:Defend your rights with the GCD

--- In tuning@yahoogroups. com, Mike Battaglia <battaglia01@ ...> wrote:
>
> > As discussed, Rick's experiment showed nothing new at all.
> > Also as discussed, we do NOT hear the GCD, though sometimes
> > the pitch we hear is the same as we would hear when listening
> > to a sine wave at the GCD frequency, if the stimulus meets a
> > the conditions needed to make Rick's ideas match reality.
> > That means the GCD can't be subsonic, the stimulus must have
> > a Fourier spectrum with most of its energy in the first 16
> > harmonics (and preferrably in the first 8), and spectra
> > where most of the energy is in, say, harmonics 2 4 6 8 but
> > with a touch of 5 (where we hear 2 as the pitch) are excluded.
> > Spectra where two or more of the loudest partials fall within
> > a critical bandwidth may wind up being unpitched, though the
> > GCD would remain unchanged.
>
> Why couldn't we hear subsonic GCDs, if the range of human hearing has
> primarily to do with the frequency response of the ear? I saw a demo
> at the last ASA convention where they had an extremely high powered
> subwoofer playing a tone you could hear at something like 5 Hz - the
> idea was that the ear doesn't have a sharp cutoff at 20 Hz, but rather
> that it would require a extremely high volume to hear a 5 Hz tone.
>
> Here's a link to the website: http://www.eminent- tech.com/ RWbrochure. htm
>
> Rather, I think a better explanation might have to do with critical
> band effects: Isn't 14 Hz about the width at which two sinusoids will
> beat? So when a complex tone drops low enough, all of its partials
> will start to interfere with themselves via critical band effects.
>
> This would infer that it's possible to retain the "pitchedness" of a
> square wave at lower frequencies than a sawtooth wave - that would be
> an interesting experiment.
>
> -Mike
>
And just to add, we can hear the GCD if it's harmonics are within the audible range and close to "1". I explained this already. I've heard the GCD myself many times and have not yet read any tangible proof that its not the real wave I'm hearing, only other theories.

Rick

🔗Carl Lumma <carl@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > It is believed that the sine wave is the elementary starting point of
> > Fourier analysis. However, given y(t) = Acos2pivt, since vt = n, where n is
> > number of cycles, we have:
> > y(t) = Acos2pivt = Acos2pin.
> > The derivative becomes dy/dt = (dy/dn).(dn/dt) = (dy/dn).v, since frequency
> > is the derivative of cycles with respect to time. The second derivative
> > gives v^2, the third v^3 and so on.
> > Applying the same operation backwards for the integral requires multiplying
> > by the period T = 1/v, so that the second integral requires T^2 etc...
> > Notice that the variable 't' has completely disappeared from our equation!
> > Could it be that waves bring their own measure of time in the form of
> > period/frequency?
>
> > Both yourself and Graham have said that "virtual pitch" has been a
> > well-known phenomenon for 50 years. But only in the psychoacoustics
> > department! Indeed, you've forgotten that psychoacoustics is not the "big
> > picture". Theoretical physics is. The pecking order is: wave (quantum)
> > mechanics - chemistry - biochemistry - biology.
>
> No offense here, and not to speak for Carl, but this has been
> discussed over, and over, and over again. In fact, it's been discussed
> so much that I can foresee only three possible outcomes to this whole
> conversation:
>
> 1) You realize that your reframing of the virtual pitch phenomenon is
> equivalent to your choosing to use different terminology than what is
> common
> 2) We all stop responding because we're tired of a pointless argument
> that is clearly going nowhere
> 3) The debate continues until various list members start posting about
> how their inboxes are flooded and how they're leaving the tuning list
> forever.
>
> I am by no means a veteran on this list, having been here for a year,
> but this seems to be the general pattern. Shall we attempt to avoid
> number three?
>
> -Mike
>
Hi Mike,

I do wish to avoid 3) and apologise if I have offended. I do get a little frustrated sometimes because I'm just not getting the 4th possibility across, that it really isn't the virtual pitch at work here at all but something quite different. Yet at the same time this doesn't disprove virtual pitch in the slightest. IOW it's not 1) and therefore we don't need 2).

Referring to the original experiment, it would be described by the sine waves like y(t) = sin2pi440Hz*t, y(t) = sin2pi1320Hz*t + sin2pi2200Hz*t, that is, the 3rd and 5th partials of A440Hz, then the 5th and 7th, 7th and 11th, and so on through the relative primes. All waves have the same maximum unit amplitude and a period of T = 440Hz^-1. Hence, they all vibrate at the GCD frequency A440Hz. If we plot a graph of y-t for these waves, it is seen that the shape of each wave changes but have the same period. Of course a change in shape means a change in tone. Listening to the waves, all have a frequency of A440Hz and the tones thin out the higher up we go in the series, all as expected.

Now, the energy arriving at the ear drum is defined by the intensity, which is energy in joules per sq meter. It's equation is
I = Z[dy/dt]^2 where Z is a coefficient for air and depends on pressure and temperature. As we see, the GCD waves do not necessarily contain less energy than the sine wave of A440Hz, at least not enough in this instance to effect their audibility. And if we wish to calculate average energy, then this involves taking the definite integral of energy with respect to time over one period, which is now of course the GCD T = 440Hz^-1, and multiplying frequency, again the GCD. Therefore, knowledge of this must be presumed before an energy calculation can take place. A calculation of these energies would show that they all lie within the same range.

Finally, in order to prove that it is the virtual pitch and not the GCD we are hearing, we would have to find a way to recreate the same set of circumstances without the GCD being present, which is clearly impossible. By Occam's Razor, the simplest explanation is that we are hearing the GCD. This does not disprove virtual pitch and I have no doubt that this too plays a vital role in harmonic analysis.

Regards

Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
>
> >I admit...I haven't heard any counter-theories either, only
> >alternative explanations such as the theory of Virtual Pitch
> >(which is, if I have it right, simply that you will hear the
> >base frequency of who's harmonic overtones best align with the
> >frequencies given). (Other list members) please feel free to
> >correct me if I'm wrong but, if that definition is correct and
> >virtual pitch is indeed the only widespread accepted theory...
> >Rick is indeed adding something new to the theory (which could,
> >I imagine, be used to calculate different assumed root tones
> >for certain instruments with odd harmonic structures).
>
> You can do your own literature search for "virtual pitch" and
> reach your own conclusions about its acceptance. A good place
> to start would be the article Steve just posted.
> As Mike B. states, the phenomenon Rick is discussing *is* that
> of virtual pitch. It's not clear what calculations he's
> suggesting we do, but the notion that the GCD is somehow
> involved goes back to Rameau
>
> http://books.google.com/books?id=TdUtrQ1cJLgC&pg=PA54&lpg=PA54
>
> However, the fact that the GCD alone fails to account for many
> psychoacoustic experiments has led researchers to abandon it.
> Harmonic entropy is one way of solving the 301/201 problem.
> We're still waiting to see if Rick has a solution.
>
> -Carl
>
Thanks Carl,

What I had in mind initially was not so much that GCD's emulate any particular individual instrument (although they could in principle) but that it also applies between fundamentals of two or more instruments. For example, I can conceive that two violins, say, each with a well defined harmonic series starting from 1, could both play upper harmonics to some other fundamental and that the resultant wave the listener hears is the GCD. One could play the Fourier series starting from E1320Hz and the other from C#2200Hz and the resultant wave is A440Hz. This, as you know, might give us a clue as to what musicians call "the key". The music is in "the key" of A. It is common that musicians tend to recognise the key even when the note has not been played, and the summation of all these combinations in a musical composition will all accumulate in the listeners mind to build that sense of "A". Of course many will claim that in this case it is the virtual tonic at work and they might be correct, or partly correct. My only defence is that something about it "feels" right, that part of me who is a musician I mean. It is not sufficient, for example, to study the wave harmony of an entire orchestra by focusing in on one instrument (and I do think that classical mechanics is guilty of this).

Concerning 301/201, if we momentarily accept for the sake of argument that the GCD model is correct and that what we are hearing in ET's are approximations to these, then my intuition tells me that we would hear this as around 100Hz. Looking at a wave graph, if we begin with the idealistic wave 300/200 = 3/2, which of course would repeat every T = 100Hz^-1, and begin to slightly detune one of the notes,it is seen that the overall wave shape changes very little. To me this suggests the possibility of an "almost GCD" model. But of course trying to pin this down in some rigorous way is like asking "how near is almost, exactly?". How far can I detune my A string before it sounds out of tune, or before it becomes another note completely? Difficult questions. The only thought that comes to mind at the moment is that Erlich's HE applies to both harmonics and the resultant GCD simultaneously and that what he called virtual pitch might be wholly or in part this waveform frequency. This of course is not a problem; theories are always being modified in light of new evidence or possibilities and still remain largely intact. If anyone has a better idea I'd love to hear it.

Rick

🔗gooeyfruitbat69 <zhang@...>

This topic seems to come up quite often on this list (and other similar forums). The following is one of the best essays on this subject of cultural affects of music.

~ "The Dark Side" from _Harmonic Experience_ by W.A. Mathieu:

The Dark Side

The ear is deep; it is difficult to overestimate the collective profundity of the ear. And the depth of the ear's musical response is cumulatively enhanced and nurtured by culture, passed on from one person to the next, from generation to generation.

I am alive in the first generation of ears that can listen to the greatest master musicians of virtually every culture and subculture. For a few dollars I can buy recordings of the best musicians in the world, some of whom are long gone from us.

The more I listen the more I can hear the wit, pathos, and intellectual complexity that connects the Australian aborigine¹s didjeridoo and the European conductor¹s baton, and
the more I realize there is only one pair of ears.

What continually amazes me is the fierce love of harmonic resonance that is evident everywhere. Not all music has the complex tonal system or the counterpoint of the European legacy, or the modal variety of India, or the rhythmic complexity of Ghana, but everywhere one listens one hears the same passionate intelligence. The most subtle conceivable nuances of intonation are shared worldwide. It becomes apparent that our one pair of ears recognizes how wide harmonic territories can be traversed by small melodic intervals.

Even in music that does not require commas, micro-intervals that
radically affect the harmonic flow show up again and again. This is true for the best musicians regardless of culture or style - the hip pop singer in the recording booth no less than the muezzin in the tower or cellist in the concert hall.

Worldwide listening reveals that master musicians sing and play the gamut running from the dead-in-tuneness of low-prime norms, through harmonic complexities of every sort, all the way to momentary, deliberate out-of-tuneness.

Playing with simple pure harmonies is like playing with light, and the fascination with that cusp between complexity and out-of-tuneness is like playing with the dark side.

The flirtation between the light and the dark is a kind of confrontation with destiny itself, a reenactment of the battle between survival and annihilation, harmony and noise. We cherish this in our music.

As long as we are singing about life and death we know we are still alive. The various commas, whether they are used as expressive shading or as a functional way of zapping through harmonic territory, far from being the pesky problems that many theorists have assumed them to be, are in fact windows to the affective dark side of the psyche, and they enable music to fill up to the brim with the full range of human feeling.

Even the commas implicit in twelve-tone equal temperament imbue
equal-tempered music with a tremendous range of human response. We need to elevate the status of the equal-tempered comma from that of a pun, where a single thing stands for two incidentally related things, to that of matchmaker, where complementary energies are conjoined and synthesized into a higher meaning.

--- In tuning@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
>
> Consonance and dissonance in music are dependent on culture (not
> every music culture found chords), musical style, tuning, and
> context. Besides in our Western culture on the music education and
> listening experience of a listener. I doubt there are objective
> criteria valid for every brain in the same way. Or maybe yes but a
> person is not aware about it as he/she has no knowledge about it and
> missing vocabulary to describe it. You can't expect sophisticated
> dialogue about Bach's enharmonic-chromatic modulations or using of
> extra-tonal diminished chord after Napolitan chord with an Amazonia,
> New Guinea or Australia aborigine (unless he/she has Ph.D. or M.A. in
> music :-) ). Even not with an average common music consumer.
>
> Daniel Forro
>
>
>
> On 19 Jul 2009, at 1:44 PM, Mike Battaglia wrote:
>
> >
> > > I believe someone had a theory of consonance that rated
> > consonance as "how
> > > much complexity something has / how much effort it take to listen
> > to" so the
> > > easier it is to listen to despite complexity the more consonance
> > it has.
> > > Anyone recall who said that?
> > >
> > > -Michael
> >
> > I doubt I'm the only one who's said this, but a while ago I advanced
> > the theory that the brain gets accustomed to high-entropy intervals
> > the more it is exposed to them, which is why I see the dualistic
> > concept of "consonance/dissonance" as more of a single notion of
> > "complexity." Intervals that you would call "dissonant" are simply
> > more complex than "consonant" ones, and can be experienced as "complex
> > consonances" rather than "dissonances" under the right circumstances
> > (especially the timbre of the instruments used!)
> >
> > You should also take note that people take will a lot of different
> > perceptual characteristics and lump them together into the
> > "consonance/dissonance" terminology. I've heard it used to refer to
> >
> > 1) Critical band effects
> > 2) High-entropy intervals or chords
> > 3) Intervals or chords that are out of place within the harmonic
> > environment of the piece
> > 4) Intervals, chords, or modes that sound "dark" in character, perhaps
> > due to cultural or psychological associations
> > 5) Intervals, chords, or modes that the listener simply just does
> > not like
> >
> > There are plenty of others too. And as you can see, some of these
> > criteria are more scientific than others. Let's keep in mind exactly
> > what perceptual quality it is we're trying to uncover when we talk
> > about "dissonance" and "consonance," then. The notion of "complexity"
> > I posted above refers only to the second one.
> >
> > Another interesting idea I've been throwing around - is "consonance"
> > and "dissonance" really a linear thing when you get into triads and
> > such? Perhaps it's more like color: we could describe how "colorful" a
> > certain shade of color is, with gray being the least colorful and any
> > highly saturated shade being the most colorful, but we would be losing
> > information on precisely which way the shade is "colored" (red, green,
> > turquoise). Perhaps there are different "flavors" of consonance when
> > you get to triads and beyond, and this would more fully describe a
> > given chord beyond whether as a whole it's "consonant" or "dissonant"
> > or how "complex" or what not. In fact, this must be true on some
> > level, since the whole notion of "chord quality" exists.
> >
> > That might be a way in which the "accordance" of certain intervals
> > might lead to the higher-level multidimensional experience of "chord
> > quality." Something to think about.
> >
> > -Mike
>