Beatings vs Intermodulation tones

I am a new member of this very long-lasting discussion group, so:
my greetings to all members, and apologies in case my contribution
(s) were something already well known or understood.

I have always loved music, together with physics and electronics. I
have been playing keyboards, guitar and bass guitar for a long while
but always "by ear" – that is, I have no classic education, I can't
read the staff well, and so on...

Recently – say, since last year – I have started to ask myself some
of the questions that I have also found in this discussion group. I
have worked on my curiosities with the background and the method of
an experimental physicist, so perhaps my view is strongly oriented
in such sense. However, I would like to share with you some of my
questions and answers, hoping that they could be of common interest,
and – in case – that someone of you could give me directions for a
more correct approach, and criticize my point of view.

I would like to start with a question recently found on this group:
why the 4:5:6 major triad sounds more consonant than the 10:12:15
minor triad – the "old chestnut" question. But before that, I
believe that a clarification about some physical concepts is needed.

Initially, following Von Helmholtz work, I have started regarding
consonance as due to good overlapping of musical harmonics. After
several considerations (not reported here and now) I tend now to
conclude that this explanation is far not sufficient. The main
argument seems to me that higher harmonics are too weak to account
for the strong dissonance of many of the possible triads. I have
tried to sample a number of musical instruments (piano, guitar,
violin) and for all of them the intensity of harmonics higher than
the third one is much (at least 10 dB) weaker than the fundamental
or second harmonic. If you plot the amplitude spectrum on a linear
scale, you can clearly see how weak such harmonics are compared to
the fundamental. To my ear, instead, heard dissonance is strong, and
concerns the fundamental, or the very first harmonics.

Then I turned to an explanation involving "difference tones." I
would like to stress that such tones can also be present when
starting tones are pure, i.e. with no higher harmonics. At this
point, from most of the discussion on the web and the literature
that I could find, I see that there are two (physically different)
concepts that are concerned, that is, beatings and intermodulation
tones. Both of them lead to some "difference tones," the origin and
nature of which is, however, very different. I will try to explain
this below; I apologize for some simple maths, that on the other
hand is necessary to understand the core of the question.

Beatings – It is the phenomenon occurring when one tries to tune a
string instrument like a guitar. By playing the same note on two
adjacent (or anyway different) strings, if the two are not perfectly
in tune one hears a "beating", that is, a long-period modulation of
LOUDNESS. [Note: in physics the word "modulation" means periodic
change of some quantity, e.g. amplitude modulation (or loudness
modulation in this case) is a periodic change of amplitude,
frequency modulation is a periodic change of frequency, like in
vibrato, and so on. In musical terminology, modulation is a change
of tonality, that happens once and has no periodicity!] Beatings
happen always between two tones, it is a process due to wave
interference and does not need any nonlinear phenomenon to happen.
Mathematically it is described by trigonometric reverse
prosthaphaeresis formulae, e.g.:

sin a + sin b = 2sin[½(a + b)]cos[½(a - b)]

You can find those at http://en.wikipedia.org/wiki/Prosthaphaeresis

Intermodulation tones – If two tones get "mixed" in some way, that
is, they are not just summed up like in the previous case, but are
also multiplied with each other, sum and difference frequencies are
generated. This process requires a nonlinear interaction between
tones, therefore is not always present. Moreover, nonlinear
processes need high intensities to become dominant compared to
linear ones, or the other way around, they tend to vanish when
intensity is decreased. Mathematically those are described by
trigonometric prosthaphaeresis formulae, e.g.:

sin a sin b = ½[cos(a - b) - cos(a + b)]

The difference with the previous one is the following: sum of two
tones is what happens normally in nature, that is, a linear process
with no dependency of intensity. Product of two terms arises when
intensity increases and a nonlinear interaction is introduced. The
nonlinear response (sum and difference tones) is usually small
compared to the original tones. This is why, for instance, Tartini's
third sound is weak. Instead, a beating is strong: if the intensity
of beating waves is the same, wave interference also causes periodic
nulling of loudness, that is, loudness oscillates between zero and a
maximum value. This is surely a strong effect.

The main difference between the two phenomena is exactly related to
the different nature of produced tones. Intermodulation tones are
two additional tones, the frequency of which is exactly the sum and
the difference of the two starting frequency, and the amplitude of
those will be much smaller of the two starting amplitudes (but will
depend on the overall loudness). You will hear four tones at the
same time: the two starting ones, plus the sum and difference tones.
In beatings instead, you will hear only two tones, that is, the
starting ones. However, the way they sum up is such that you hear
a "mean tone" (whose frequency is proportional to ½ (a+b), that is
the average of the two) with a loudness envelope at frequency
proportional to (a-b). [Note: Actually, what we hear is a beating at
frequency (a-b) and not ½ (a-b) because the loudness envelope is at
twice the frequency of the wave envelope, it needs a sketch to
understand this but please trust for the moment.] Therefore, in
guitar tuning the two tones are very close, then the "mean tone" is
about the same, and the beating is at very low frequency but it can
still be heard just because it is a loudness modulation – and not an
instantaneous pressure modulation, that would be beyond the audible
frequency range.

I would very much appreciate your feedback, and if you were
interested in discussing this issue, we could also talk about how
this could be relevant, in my opinion, for harmonic analysis.

Best wishes of a happy 2009 to everybody,

Max Labardi

--I would like to start with a question recently found on this group:
--why the 4:5:6 major triad sounds more consonant than the 10:12:15
---minor triad – the "old chestnut" question.

---To my ear, instead, heard dissonance is strong, and
---concerns the fundamental, or the very first harmonics.
   Sethares formula for dissonance pretty much says the same thing IE dissonance is scaled/multiplied by the amplitude of the tone (so those first few overtones matter the most in terms of determining consonance).  So it makes sense...

Major triad is composed of
root
5/4 * 2 = 5/2 (first overtone)
3/2 * 2 = 3 (first overtone)

root
6/5 * 2 = 12/5 (first overtone)
3/2 * 2 = 3 (first overtone)

     But, note, 5/2 is a perfect 5th on the next octave
(right smack intersecting with not just the scale, but the harmonic series itself)...while 12/5 (minor triad) is not.  Even that strong/loud second harmonic would begin to explain why the minor triad is more dissonant sounding.
*****************************************************************************************************
---Moreover, nonlinear
---processes need high intensities to become dominant compared to
---linear ones, or the other way around, they tend to vanish when
---intensity is decreased.
    So, again, the higher overtones matter less (and the non-linear beating they produces thus matters less)...if I am following what you are saying.
****************************************************************************************
---which is exactly the sum and
---the difference of the two starting frequency
AKA the second heard "tone" is produced by beating between the root and "1st"
tone....and the difference can be used to calculate the rate if beating...again, if I am still following you correctly.

---Therefore, in
---guitar tuning the two tones are very close, then the "mean tone" is
---about the same
     Hmm...I wonder how this figures in...you seem to be describing a circumstance that would produce very slow beating that could easily be seen as a gradual change in amplitude.  I do know that slow beating is generally considered pleasant, while fast beating is not.  Sethares' consonance formula points to that...it is the tones relatively close to the root frequency, and not those "virtually on it"...that cause the worst dissonance. 

   I do know, that when all the beating produced by several tones in a scale played together are the same distance away IE beat at the same rate against each other...the result is pleasant/consonant (such is the case with the
harmonic series which is root note, root note * 2, root note * 3, root note * 4....).

   If I were to try and sum it up I would say something like
"The major triad is more tonal because the overtones of the second harmonic of its second note intersect with the harmonically beating (and pleasant) harmonic series and the minor triads second note does not.  Plus, since the first and second tones have the most amplitude they have the most effect on consonance so there is little need to worry about whether or not higher octaves of a tone in each chord also intersect the harmonic series."

[ Attachment content not displayed ]

---then with sine waves.... not much difference between the two chords?
Exactly...it has mostly to do, it seems, with the second (and perhaps third) harmonic of the middle/second note in the chord.  The root tones IE "sine wave" should not sound different so far as consonance is concerned...in fact, in that case, the chords are essentially the same 2 intervals added in sequence, but in reverse (for the minor chord).

--- On Wed, 1/7/09, Chris Vaisvil <chrisvaisvil@...> wrote:

From: Chris Vaisvil <chrisvaisvil@gmail.com>
Subject: Re: [tuning] Beatings vs Intermodulation tones
To: tuning@yahoogroups.com
Date: Wednesday, January 7, 2009, 1:46 PM

then with sine waves.... not much difference between the two chords?

"The major triad is more tonal because the overtones of the second
harmonic of its second note intersect with the harmonically beating
(and pleasant) harmonic series and the minor triads second note does
not.  Plus, since the first and second tones have the most amplitude
they have the most effect on consonance so there is little need to
worry about whether or not higher octaves of a tone in each chord also
intersect the harmonic series."

Then that implies chord progressions are not functional with sine waves. There would be no interplay of tension and relaxation.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Michael Sheiman <djtrancendance@yahoo.com>

Date: Wed, 7 Jan 2009 14:14:44
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Beatings vs Intermodulation tones

---then with sine waves.... not much difference between the two chords?
Exactly...it has mostly to do, it seems, with the second (and perhaps third) harmonic of the middle/second note in the chord.  The root tones IE "sine wave" should not sound different so far as consonance is concerned...in fact, in that case, the chords are essentially the same 2 intervals added in sequence, but in reverse (for the minor chord).

--- On Wed, 1/7/09, Chris Vaisvil <chrisvaisvil@gmail.com> wrote:

From: Chris Vaisvil <chrisvaisvil@gmail.com>
Subject: Re: [tuning] Beatings vs Intermodulation tones
To: tuning@yahoogroups.com
Date: Wednesday, January 7, 2009, 1:46 PM

then with sine waves.... not much difference between the two chords?

"The major triad is more tonal because the overtones of the second
harmonic of its second note intersect with the harmonically beating
(and pleasant) harmonic series and the minor triads second note does
not.  Plus, since the first and second tones have the most amplitude
they have the most effect on consonance so there is little need to
worry about whether or not higher octaves of a tone in each chord also
intersect the harmonic series."

Welcome, Max!

> I would like to start with a question recently found on this group:
> why the 4:5:6 major triad sounds more consonant than the 10:12:15
> minor triad – the "old chestnut" question. But before that, I
> believe that a clarification about some physical concepts is needed.
>
> Initially, following Von Helmholtz work, I have started regarding
> consonance as due to good overlapping of musical harmonics. After
> several considerations (not reported here and now) I tend now to
> conclude that this explanation is far not sufficient.

On this we can certainly agree.

> Then I turned to an explanation involving "difference tones."
> I would like to stress that such tones can also be present when
> starting tones are pure, i.e. with no higher harmonics.

Indeed. However they seldom occur at volumes sufficient to
explain the range of musical phenomena sometimes attributed
to them.

> Beatings – ... modulation of LOUDNESS. ... e.g. amplitude
> modulation (or loudness modulation in this case) is a periodic
> change of amplitude, frequency modulation is a periodic change
> of frequency, like in vibrato, and so on. ... it is a process
> due to wave interference and does not need any nonlinear
> phenomenon to happen.

Again we agree.

> The nonlinear response (sum and difference tones) is usually
> small compared to the original tones.

Exactly.

-Carl

Chris wrote:
> then with sine waves.... not much difference between the two chords?

Why don't you try it? Better yet, try 4:5:6:7:9:11 and
1/(4:5:6:7:9:11).

-Carl

>>Then that implies chord progressions are not functional with sine
waves. There would be >>no interplay of tension and relaxation.
    Just because the major and minor triad have similar tension does NOT mean every chord will.   For example, the triad and a sus4 chord have different degrees of tension. 

   It would make sense that the effect of changing between major and minor changes the sense of "average tone"...just like transposing between keys changes the mood WITHOUT changing the level of consonance/tension.  Think about it...taking the average of the minor triad's frequencies yields a small number/"center" than the major triad.

--- On Wed, 1/7/09, chrisvaisvil@... <chrisvaisvil@...> wrote:

From: chrisvaisvil@... <chrisvaisvil@...>
Subject: Re: [tuning] Beatings vs Intermodulation tones
To: tuning@yahoogroups.com
Date: Wednesday, January 7, 2009, 4:01 PM

Then that implies chord progressions are not functional with sine waves. There would be no interplay of tension and relaxation. Sent via BlackBerry from T-MobileFrom: Michael Sheiman
Date: Wed, 7 Jan 2009 14:14:44 -0800 (PST)
To: <tuning@yahoogroups. com>
Subject: Re: [tuning] Beatings vs Intermodulation tones
---then with sine waves.... not much difference between the two chords?
Exactly...it has mostly to do, it seems, with the second (and perhaps third) harmonic of the middle/second note in the chord.  The root tones IE "sine wave" should not sound different so far as consonance is concerned... in fact, in that case, the chords are essentially the same 2 intervals added in sequence, but in reverse (for the minor chord).

--- On Wed, 1/7/09, Chris Vaisvil <chrisvaisvil@ gmail.com> wrote:

From: Chris Vaisvil <chrisvaisvil@ gmail.com>
Subject: Re: [tuning] Beatings vs Intermodulation tones
To: tuning@yahoogroups. com
Date: Wednesday, January 7, 2009, 1:46 PM

then with sine waves.... not much difference between the two chords?

"The major triad is more tonal because the overtones of the second harmonic of its second note intersect with the harmonically beating (and pleasant) harmonic series and the minor triads second note does not.  Plus, since the first and second tones have the most amplitude they have the most effect on consonance so there is little need to worry about whether or not higher octaves of a tone in each chord also intersect the harmonic series."

[ Attachment content not displayed ]

Chris wrote:

> what scale degrees does 4:5:6:7:9:11 equate to?

No scale... just a JI hexad. You can synthesize it in
cool edit at 400, 500, 600 etc. Hz. Or divide everything
by 2 to put it an octave lower.

> and... why the skip at the end?

You mean 8 & 10... octave copies of 4 and 5. 11:10
and such start getting into the semitone range, making
the chord more dissonant from roughness. This way it's
a stack of thirds.

> (I'm not at all used to the nomenclature. this stuff is
> more complicated than organic chemistry - so far anyway.)

"They use that as a weed-out course", my high school
chem teacher used to say.

Have you read Doty's JI Primer or Partch's book? Kindof
prerequisites for this list, I'm afraid.

1/(4:5:6:7:9:11) = 1/1 11/9 11/7 11/6 11/5 11/4. To
make the comparison as fair as possible, tune the otonal
chord 1/1 5/4 3/2 7/4 9/4 11/4, keeping 1/1 and 11/4 the
same pitch in both chords, so only the middle four notes
change.

-Carl

//No scale... just a JI hexad. You can synthesize it in
//cool edit at 400, 500, 600 etc.

     In that case,  the interval gaps seem to fit right in with the harmonic series IE 5/4, 6/5, 7/6,(skip 8/7 and 9/8 and multiply them to form 9/7, etc.)

  As for 1/1 11/9 11/7 11/6 11/5 11/4....that seems to be an exactly inverse IE dividing the base tone by a fairly sequential series instead of multiplying.  If I take, say, 500hz, I would get 500/6 (83.3333), 500/5 (100), 500/4 (125)...so it looks like a chain of similar intervals. 
  
   I would also guess both of these would sound about the same in consonance and pointing toward their root note as the harmonic series does IE the results would likely be very consonance.

  One terminology question...how do you get a "stack of thirds" when the interval between each note in the scale is constantly changing (IE the 5/4 gap is not = 6/5)?

-Michael 

--- On Wed, 1/7/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Beatings vs Intermodulation tones
To: tuning@yahoogroups.com
Date: Wednesday, January 7, 2009, 9:20 PM

Chris wrote:

> what scale degrees does 4:5:6:7:9:11 equate to?

No scale... just a JI hexad. You can synthesize it in

cool edit at 400, 500, 600 etc. Hz. Or divide everything

by 2 to put it an octave lower.

> and... why the skip at the end?

You mean 8 & 10... octave copies of 4 and 5. 11:10

and such start getting into the semitone range, making

the chord more dissonant from roughness. This way it's

a stack of thirds.

> (I'm not at all used to the nomenclature. this stuff is

> more complicated than organic chemistry - so far anyway.)

"They use that as a weed-out course", my high school

chem teacher used to say.

Have you read Doty's JI Primer or Partch's book? Kindof

prerequisites for this list, I'm afraid.

1/(4:5:6:7:9: 11) = 1/1 11/9 11/7 11/6 11/5 11/4. To

make the comparison as fair as possible, tune the otonal

chord 1/1 5/4 3/2 7/4 9/4 11/4, keeping 1/1 and 11/4 the

same pitch in both chords, so only the middle four notes

change.

-Carl

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
>    I would also guess both of these would sound about the same
> in consonance and pointing toward their root note as the
> harmonic series does IE the results would likely be very
> consonance.

You're in good company with that assessment, but, like I said...
try it!

>   One terminology question...how do you get a "stack of thirds"
> when the interval between each note in the scale is constantly
> changing (IE the 5/4 gap is not = 6/5)?

They're all roughly the size of diatonic thirds.

-Carl

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

>....try 4:5:6:7:9:11 and...

On the one hand:
that contains some
http://en.wikipedia.org/wiki/Overtone
s out of the
http://en.wikipedia.org/wiki/Harmonic_series_(music)
.
On the other hand: The reverse virtual series:
>... 1/(4:5:6:7:9:11).
consists in an choice of less frequent occuring
http://en.wikipedia.org/wiki/Subharmonic
pitches.
"Subharmonics can be produced by signal amplification through
loudspeakers. They are naturally produced by bells,
giving them their distinct sound."

http://www.ausbell.com/Harmonic/titleharm.html
"The lower virtual pitch percepts are subharmonics of 220 Hz and are
very weak, partly spurious results. The average pure tonalness of this
bell is 1.16 as compared to the european bell shown elsewhere at 0.81
and the 8/5 polytone at 0.46."

Historical review of concepts about that theroies in:
http://www.lamadeguido.com/artangles.pdf

But i do prefer to explain both, the
http://en.wikipedia.org/wiki/Major_and_minor
chords

'4':'5':'6' as Major C:E:G tonic, and
"10":"12":"15" as it's corresponding dominant-minor e:g:b counterpart

That two triads are barely
common subparts of the complete overtone series:

1:2:3:'4=C':'5=E':'6=G':7:8:9:"10=e":11:"12=g":13:14:"15=b":16:.....

in reference to theirs common fundamental root 1:1,
two octaves below C.

That view even answers Veroli's/Tom's quest in:
/tuning/topicId_79725.html#79725
"One hoary old unanswered question I bumped
into reading the first few dozen pages:
Why is 4:5:6 more consonant than 10:12:15?"

simply by Werckmeister's, Euler's & Helmholtz's argument
that combinations of lower partials out of the harmonic series
do sound more consonant than choices among higher components:
http://sonic-arts.org/monzo/euler/euler-en.htm

bye
A.S.

>>Why is 4:5:6 more consonant than 10:12:15?"
>simply by Werckmeister' s, Euler's & Helmholtz's argument

t>hat combinations of lower partials out of the harmonic series

>do sound more consonant than choices among higher components:

http://sonic- arts.org/ monzo/euler/ euler-en. htm

I am not quite following how/why that would work.
   The ratio 5/4 in the major triad's first interval is >equal< to the 15/12 in the minor triad.
   And...the 6/5 in the major triad's first interval is also equal to the 12/10 in the minor triad.
Hence these are EXACTLY the same intervals...just in the reverse order.  So you would think it would not matter so long as the instruments you use with the triads are just pure sine waves.

  The only reason I can think of that the order would matter has to do with overtones.  The middle note in both triads in different.  An octave above the middle note in the major triad (first overtone) produces 10/4 (2.5...a perfect 5th...right in line with a ratio very early in the harmonic series: 3/2).  And...an octave above the middle note in the second triad produced 24/10 = 12/5.  Note 12/5 occurs much further into the harmonic series IE (6/5*7/6*8/7*9/8*10/9*11/10*12/11) = 12/5...that may help explain why the overtones generated by the minor triad sound more tense when using non-sine waves as instruments.

-Michael

--- On Thu, 1/8/09, Andreas Sparschuh <a_sparschuh@...> wrote:

From: Andreas Sparschuh <a_sparschuh@...>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation
To: tuning@yahoogroups.com
Date: Thursday, January 8, 2009, 11:08 AM

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

>....try 4:5:6:7:9:11 and...

On the one hand:

that contains some

http://en.wikipedia .org/wiki/ Overtone

s out of the

http://en.wikipedia .org/wiki/ Harmonic_ series_(music)

.

On the other hand: The reverse virtual series:

>... 1/(4:5:6:7:9: 11).

consists in an choice of less frequent occuring

http://en.wikipedia .org/wiki/ Subharmonic

pitches.

"Subharmonics can be produced by signal amplification through

loudspeakers. They are naturally produced by bells,

giving them their distinct sound."

http://www.ausbell. com/Harmonic/ titleharm. html

"The lower virtual pitch percepts are subharmonics of 220 Hz and are

very weak, partly spurious results. The average pure tonalness of this

bell is 1.16 as compared to the european bell shown elsewhere at 0.81

and the 8/5 polytone at 0.46."

Historical review of concepts about that theroies in:

http://www.lamadegu ido.com/artangle s.pdf

But i do prefer to explain both, the

http://en.wikipedia .org/wiki/ Major_and_ minor

chords

'4':'5':'6' as Major C:E:G tonic, and

"10":"12":"15" as it's corresponding dominant-minor e:g:b counterpart

That two triads are barely

common subparts of the complete overtone series:

1:2:3:'4=C': '5=E':'6= G':7:8:9: "10=e":11: "12=g":13: 14:"15=b" :16:.....

in reference to theirs common fundamental root 1:1,

two octaves below C.

That view even answers Veroli's/Tom' s quest in:

http://launch. groups.yahoo. com/group/ tuning/message/ 79725

"One hoary old unanswered question I bumped

into reading the first few dozen pages:

Why is 4:5:6 more consonant than 10:12:15?"

simply by Werckmeister' s, Euler's & Helmholtz's argument

that combinations of lower partials out of the harmonic series

do sound more consonant than choices among higher components:

http://sonic- arts.org/ monzo/euler/ euler-en. htm

bye

A.S.

Thanks for the explanation Mr. Sparschuh.

Obviously the treatment of comparative consonances/dissonances of both
triads in my UT book is not fully complete/consistent. I intend to
rewrite that page in future versions.

Kind regards
Claudio

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
...

----- Original Message -----

From: Michael Sheiman

To: tuning@yahoogroups.com

Sent: Thursday, January 08, 2009 9:16 PM

Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Michael wrote:

> I am not quite following how/why that would work.
> The ratio 5/4 in the major triad's first interval is >equal< to the 15/12 in the minor triad.
> And...the 6/5 in the major triad's first interval is also equal to the 12/10 in the minor triad.
> Hence these are EXACTLY the same intervals...just in the reverse order. So you would think it would not matter so long as the instruments you use with the triads are just pure sine waves.

Sorry to be suddenly so open to you but I'm just surprised that you can say you would think it wouldn't matter. Personally, what I'm often hearing very clearly, while listening to JI intervals, is the common fundamental; and a bit less clearly I can hear the actual difference tones; and, even less clearly than that, second difference tones (meaning frequencies of 2x*y and 2y*x). If you také 400:500:600Hz, for example, then the common fundamental is 100Hz, the difference tone in the lower interval is also 100Hz, the difference tone in the upper interval is also 100Hz, the difference tone between the highest and the lowest tone is 200Hz, and the second difference tones are 200Hz, 300Hz, 400Hz, 600Hz, 700Hz, and 800Hz, which means that ALL of them are multiples of 100Hz and therefore the periodicity is very clearly audible and the frequency of 100Hz is very prominent, even though it's not actually being intentionally "emitted". On the other side, if you try 1000:1200:1500Hz, then the common fundamental is 100Hz, the difference tone in the lower interval is 200Hz, the difference tone in the upper interval is 300Hz, the difference tone of the highest and the lowest tone is 500Hz, and the second difference tones are 800Hz, 500Hz, 900Hz, 1400Hz, 2000Hz, and 1800Hz. Obviously, the fifth of 1000:1500Hz has a bit stronger difference tone than a third because its difference tone is equal to its fundamental and will be therefore a bit more "pronounced", giving more emphasis on itself rather than making all the intervals equally significant. Anyway, what this means is actually that while a major triad of C4-E4-G4 has an "acoustical fundamental" of C2, the "acoustical fundamental" of a minor triad of C4-Eb4-G4 is Ab0, which is certainly not the same as the harmonic fundamental, for one thing, and, for another thing, is more than 3 octaves lower than the sounding tones. The fact that classical harmony determines the fundamental tone according to the position of the fifth is just a proof of what I've said.

Petr

b" :16:.....

in reference to theirs common fundamental root 1:1,
two octaves below C.

That view even answers Veroli's/Tom' s quest in:
http://launch. groups.yahoo. com/group/ tuning/message/ 79725
"One hoary old unanswered question I bumped
into reading the first few dozen pages:
Why is 4:5:6 more consonant than 10:12:15?"

simply by Werckmeister' s, Euler's & Helmholtz's argument
that combinations of lower partials out of the harmonic series
do sound more consonant than choices among higher components:
http://sonic- arts.org/ monzo/euler/ euler-en. htm

bye
A.S.

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

> Sorry to be suddenly so open to you but I'm just surprised
>that you can say you would think it wouldn't matter. Personally,
>what I'm often hearing very clearly, while listening to JI
>intervals, is the common fundamental;

To what mechanism do you attribute this "common fundamental"?

> If you take 400:500:600Hz, for example, then the common
> fundamental is 100Hz,

Or is it 200 Hz?

>the difference tone in the lower interval is also 100Hz, the
>difference tone in the upper interval is also 100Hz, the
>difference tone between the highest and the lowest tone is 200Hz,
>and the second difference tones are 200Hz, 300Hz, 400Hz, 600Hz,
>700Hz, and 800Hz, which means that ALL of them are multiples of
>100Hz and therefore the periodicity is very clearly audible and
>the frequency of 100Hz is very prominent,

And what happens when you temper the chord?

> even though it's not actually being intentionally "emitted".
> On the other side, if you try 1000:1200:1500Hz, then the common
> fundamental is 100Hz,

Not 500 Hz?

-Carl

[ Attachment content not displayed ]

I'll be damned....

    In fact, I will agree 4:5:6 sounds better even with pure sines...10:12:15 seems to sound more "off center" in mood, though the amount of beating between the two sounds quite similar even in the triangle version. 

   It seems that, for some reason, putting the intervals in reverse has negative mood effects.  IE the consonance may be the same...but having the intervals grow smaller as they get higher sounds better.

     Come to think of it...I remember I read a long time ago that the critical band gets smaller (on the exponential scale we use for hearing) as the notes get higher IE more notes can fit per octave as you go higher up without registering to the brain as significantly beating/dissonance.  So, if that
holds here...it makes perfect sense why putting smaller intervals between notes at higher frequencies IE "further up the chord" sounds
better.

-Michael

--- On Thu, 1/8/09, Chris Vaisvil <chrisvaisvil@gmail.com> wrote:

From: Chris Vaisvil <chrisvaisvil@...>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation
To: tuning@yahoogroups.com
Date: Thursday, January 8, 2009, 7:51 PM

Ok, I put together a series that some of you may find interesting

http://soonlabel. com/cgi-bin/ yabb2/YaBB. pl?num=123147207 5

http://tinyurl. com/9hzzlt

I did what Carl suggested and played around with pure sines, and also with pure triangles.
I only had time to do this in 12 TET.

By using an oscilloscope program I captured the FFT spectrum of C major C minor and C sus 4th.

There is a music example as well.

I also did the same with a triangle wave.

From a screen shot of cool edit you can see the beating or roughness of the chord. However, the lack of harmonics do not make a difference to me in the perceived consonance. In fact is you load the sine example in cool edit it looks pretty much the same as the triangle version.

This is all I had the time to do - any suggestions for exploration?

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> wrote:
>
> Ok, I put together a series that some of you may find interesting
>
> http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1231472075
>
> http://tinyurl.com/9hzzlt
>
> I did what Carl suggested and played around with pure sines, and
> also with pure triangles.
> I only had time to do this in 12 TET.

Good work, Chris!

> However, the lack of harmonics do not make a difference to me
> in the perceived consonance.

From this we should be able to conclude that coincidences of
partials and/or combination tones (if you listen to the sine
example at soft to medium volume on good speakers or headphones)
do not account for the primary qualities of musical consonances.

Now Chris, if you get time to do this for the two 11-limit JI
chords I posted, we should further get a demonstration that:
A lack of critical band roughness is a necessary but not
sufficient condition for musical consonance.

-Carl

Michael wrote:

>    It seems that, for some reason, putting the intervals in
> reverse has negative mood effects.  IE the consonance may be the
> same...but having the intervals grow smaller as they get higher
> sounds better.

The consonance isn't the same, especially for higher-limit
otonal / utonal chords (hopefully Chris will oblige with that
demo shortly). The difference is subtle enough in the 5-limit
that it winds up sounding like a mood change.

>      Come to think of it...I remember I read a long time ago
> that the critical band gets smaller

Hopefully wasn't too long ago, since I've posted about it
here several times in the past couple months. Anyway, it has
no bearing on the chords in Chris' demonstration. The width
of the critical band changes gradually across the range of
human hearing, and both these chords are in almost the same
range. Anyway, both the major and minor chords in the demo
have the same amount of roughness, dictated by the 12-ET
error and the fact that they contain the same intervals, and
in complete agreement with roughness models like Sethares'.

> So, if that holds here...it makes perfect sense why putting
> smaller intervals between notes at higher frequencies IE
> "further up the chord" sounds better.

I'd like to suggest an alternative explanation: The human
hearing system evolved to do scene analysis on natural sounds,
and especially to do sophisticated signal processing and
pattern recognition on human speech sounds. In particular,
we use changing overtone structure (different resonant filters)
to communicate vowels. The ability to infer a single source
with changing qualities (like different vowels) from rapidly-
changing partial complexes produced in human vocalization is
paramount to human survival. Therefore it may make sense that
our hearing apparatus is programmed to dissect partial
complexes containing *harmonic relationships*, since those are
found in nature and especially in human speech. Subharmonic
relationships (like those created in minor or utonal chords),
on the other hand, are nowhere found in nature.

-Carl

Carl wrote:

> To what mechanism do you attribute this "common fundamental"?

To the unremovable synchronicity of the beats. If you play two tones at frequencies of 300Hz and 500Hz, even if the waves never get perfectly in phase, the phase shifts start repeating in exactly the same "stages" every 100th of a second.

> > If you take 400:500:600Hz, for example, then the common
> > fundamental is 100Hz,
>
> Or is it 200 Hz?

I thought someone in the long past had already defined the common fundamental frequency of a JI interval to be the GCD of the sounding frequencies, which is exactly why I was using the term like this. I've seen it labeled like this quite a few times so I'm doing the same. Moreover, it seems very logical to me to do it like that because it perfectly reflects the way I think about what I can hear.

> And what happens when you temper the chord?

Depends on how strong the tempering is. But for a 12-EDO triad, for example, the difference tones are still close enough to be understood as modifications of some harmonic relationships, albeit very rough ones. Similarly as the bands around the "split guide tone" (I'm saying "split" because getting away from JI splits the guide tone actually into two different frequencies very close to each other) start beating fast at that moment (which is what tuners count to get tempered intervals), the "split fundamental" also starts beating, but obviously at a much slower rate.

I should also point out that the fundamental frequency is very well audible even with sharp timbres full of loud overtones while the difference tones are better heard in timbres with soft overtones like flutes or pipe organs played with single pipe registers, for example. What's more, all of these "special" tones are best heard when there are only two sounding tones. The more tones you add, the more they get "hidden" by these because the sounding tones are obviously much louder.

> > On the other side, if you try 1000:1200:1500Hz, then the common
> > fundamental is 100Hz,
>
> Not 500 Hz?

That would be the case of the fifth in this example because it's the GCD of the highest and the lowest sounding frequency.

Petr

[ Attachment content not displayed ]

>From this we should be able to conclude that coincidences of

>partials and/or combination tones (if you listen to the sine

>example at soft to medium volume on good speakers or headphones)

>do not account for the primary qualities of musical consonances.
   Actually, I would not fully agree IE I believe they are a part of what makes musical consonance/"concordance"...but having partials align does not automatically solve the whole problem just part of it.  IE a major sounds better than a minor...but the minor by not means sounds "terribly dissonant". 
    For some reason though (I am interest to know exactly why)...minor triads appear to have both of their harmonic partials intersect very well and, as you have said, still lack the feeling of pure consonance that major intervals/triads have...I again suspect it has something to do with the the ascending interval closeness (as you go higher on a major triad) better mirrors the harmonic series and more obviously points the brain to the existence of a base tone (even without partials). 

   If you try playing 11TET with normal instruments with normal partials (which clash) and then compare it to Sethares "Blue Dabo Girl" done in 11TET with partials skewed to match...you'll notice a huge gain in musical consonance IE partial intersection is a large part of the problem...but certainly fixing it does not fix the whole thing or guarantee the mind will easily find a base tone and thus register the chords as "tonal".
   Yet, at the same time, you'll notice there is still a lot of consonance lacking between that and a piece with JI interval chords IE only part of the problem is solved.

--- On Fri, 1/9/09, Carl Lumma <carl@lumma.org> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation
To: tuning@yahoogroups.com
Date: Friday, January 9, 2009, 12:07 AM

--- In tuning@yahoogroups. com, "Chris Vaisvil" <chrisvaisvil@ ...> wrote:

>

> Ok, I put together a series that some of you may find interesting

>

> http://soonlabel. com/cgi-bin/ yabb2/YaBB. pl?num=123147207 5

>

> http://tinyurl. com/9hzzlt

>

> I did what Carl suggested and played around with pure sines, and

> also with pure triangles.

> I only had time to do this in 12 TET.

Good work, Chris!

> However, the lack of harmonics do not make a difference to me

> in the perceived consonance.

From this we should be able to conclude that coincidences of

partials and/or combination tones (if you listen to the sine

example at soft to medium volume on good speakers or headphones)

do not account for the primary qualities of musical consonances.

Now Chris, if you get time to do this for the two 11-limit JI

chords I posted, we should further get a demonstration that:

A lack of critical band roughness is a necessary but not

sufficient condition for musical consonance.

-Carl

>    It seems that, for some reason, putting the intervals in

> reverse has negative mood effects.  IE the consonance may be the

> same...but having the intervals grow smaller as they get higher

> sounds better.

--The consonance isn't the same, especially for higher-limit
--otonal / utonal chords

   I think here we are going back to our old consonance vs. concordance argument. :-D
I agree the concordance/"musical consonance" is less...but the mathematical beating/"consonance" appears to be the same.  Of course, musical consonance is the ultimate goal.

---Therefore it may make sense that
---our hearing apparatus is programmed to dissect partial
---complexes containing *harmonic relationships*
    So, again, for musical consonance...it seems we must obey the exact order the harmonic series evident in nature implies.  Thus, reversing it simply confuses the mind, even if the mathematical consonance makes sense on paper for minor/reversed intervals.

--- On Fri, 1/9/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation
To: tuning@yahoogroups.com
Date: Friday, January 9, 2009, 12:23 AM

Michael wrote:

>    It seems that, for some reason, putting the intervals in

> reverse has negative mood effects.  IE the consonance may be the

> same...but having the intervals grow smaller as they get higher

> sounds better.

The consonance isn't the same, especially for higher-limit

otonal / utonal chords (hopefully Chris will oblige with that

demo shortly). The difference is subtle enough in the 5-limit

that it winds up sounding like a mood change.

>      Come to think of it...I remember I read a long time ago

> that the critical band gets smaller

Hopefully wasn't too long ago, since I've posted about it

here several times in the past couple months. Anyway, it has

no bearing on the chords in Chris' demonstration. The width

of the critical band changes gradually across the range of

human hearing, and both these chords are in almost the same

range. Anyway, both the major and minor chords in the demo

have the same amount of roughness, dictated by the 12-ET

error and the fact that they contain the same intervals, and

in complete agreement with roughness models like Sethares'.

> So, if that holds here...it makes perfect sense why putting

> smaller intervals between notes at higher frequencies IE

> "further up the chord" sounds better.

I'd like to suggest an alternative explanation: The human

hearing system evolved to do scene analysis on natural sounds,

and especially to do sophisticated signal processing and

pattern recognition on human speech sounds. In particular,

we use changing overtone structure (different resonant filters)

to communicate vowels. The ability to infer a single source

with changing qualities (like different vowels) from rapidly-

changing partial complexes produced in human vocalization is

paramount to human survival. Therefore it may make sense that

our hearing apparatus is programmed to dissect partial

complexes containing *harmonic relationships* , since those are

found in nature and especially in human speech. Subharmonic

relationships (like those created in minor or utonal chords),

on the other hand, are nowhere found in nature.

-Carl

Hi Petr,

> > To what mechanism do you attribute this "common fundamental"?
>
> To the unremovable synchronicity of the beats. If you play two
> tones at frequencies of 300Hz and 500Hz, even if the waves never
> get perfectly in phase, the phase shifts start repeating in
> exactly the same "stages" every 100th of a second.

That won't be audible in the least. Audio equipment stomps
on phase (and so does the cochlea!) yet we don't have a
problem hearing consonance and dissonance through speakers.
Try playing with the phases in a good additive synth sometime.

> > > If you take 400:500:600Hz, for example, then the common
> > > fundamental is 100Hz,
> >
> > Or is it 200 Hz?
>
> I thought someone in the long past had already defined the
> common fundamental frequency of a JI interval to be the GCD
> of the sounding frequencies,

There is no such mathematical definition of the virtual
fundamental (which is the result of a stochastic process in
your brain... but you can measure the likelihood of it coming
out the same every time with harmonic entropy).

I don't know what a "common fundamental" is.

> it seems very logical to me to do it like that because it
> perfectly reflects the way I think about what I can hear.

And what fundamental do you hear for 400:500:600 Hz?

> > > On the other side, if you try 1000:1200:1500Hz, then the
> > > common fundamental is 100Hz,
> >
> > Not 500 Hz?
>
> That would be the case of the fifth in this example because
> it's the GCD of the highest and the lowest sounding frequency.

Put away your calculator and listen to it!

-Carl

Thanks to the whole alternate tuning list for the nice feedback. I
found Chris' test very interesting, perhaps because as an
experimental physicist I like making direct tests….. starting from
this, I'll tell you what I think about the issue of consonance of
triads, and ask Chris, if he likes and has time, to "explore" a bit
more....

I have tried to analyze consonance of triads by the following
hypothesis: loudness beatings (that I have been mentioning in my
previous post) are perceived by the ear, along with the mean tone.
This seems well assessed to me, since the widely accepted definition
of critical band (Plomp & Levelt) states that the beating note of a
dyadic chord should be out of a certain critical band of frequency
to sound not dissonant. In other words, when the beating frequency
is within a certain frequency range, it is perceived as "annoying."
This means that it can be assumed that loudness modulations (=
beatings) are perceived by the ear as sounds themselves. [Note: this
point of my analysis is, in my opinion, questionable. To the ear
there must be difference between a tone at frequency "a" and a tone
at frequency "b" much greater than "a," the loudness of which is
modulated at a rate "a." However, let me proceed on this route.]

Let us start from the fact that a dyadic chord gives always raise to
a beating (lower frequency, loudness modulation) and a mean tone
(higher frequency, sine wave tone). The fifth interval (1,3/2) (is
my notation acceptable?) has a beating at frequency ½ and a mean
tone at frequency 5/4. Both the beating frequency and mean tone
belong to the harmonic series etc etc. This can be also interpreted
in terms of "virtual pitch:" we can hear either the dyadic chord as
(1,3/2) or as (1/2,5/4) (bearing in mind that the tone ½ is actually
a loudness modulation and not a "real" tone!). How the brain choses
between the two possibilities could be an important part of the
issue, but I have absolutely no background to comment on that!

Let us now move to triads. The simplest case is a triad (1,x,3/2)
with the mediant (middle tone of the triad) to be determined. I have
looked for the most consonant mediants by asking for the following
requirements:

a. The tones of the triad should belong to the scale.
b. The beating tones of each of the dyads composing the triad
(that is: (1,x),(x,3/2), (1,3/2)) should be the same or multiples of
each other, with the smallest possible ratio.
c. The beating tone should belong to the triad, or at least to
the scale.
d. The mean tones should belong to the triad, or at least to
the scale.

This just means that, in any case, if such annoying beatings were to
exist, one could think that they could be less annoying
if "simplified." Perhaps my approach is too naïve, however let us
just use it for the natural scale and see what is the response to
the "old chestnut" question about different consonance of major and
minor chords. Let's see how it works.

In the case of the Just Intonation (natural) scale, one should
expect the most consonant chords. If we look for beating tones in
small ratios between each other, we require that:

(x-1)=n1(3/2-x) this means that the beating tone between mediant
and tonic is required to be n times the beating tone between the
fifth and the mediant; [Note: the more general form would use two
integers, like n1a(x-1)=n1b(3/2-x), but I prefer to use one index n1
that can assume integer or rational values]

(3/2-1)=n2(x-1) the same for the dyads fifth-tonic and mediant-tonic;

(3/2-1)=n3(3/2-x) the same for fifth-tonic and fifth-mediant.

I have tried with n = 1,2,3,1/3,1/2, 2/3 and 3/2, but of course one
can use higher (or more complicated) ratios. You can find that there
are three mediants x that provide solutions for all three equations
(with n1,n2 and n3 within the given ensamble):

x=5/4, for n1=1, n2=2, n3=2;
x=4/3, for n1=2, n2=3/2, n3=3;
x=7/6, for n1=1/2, n2=3, n3=3/2.

There are also solutions to just two or one of those, but I guess
they would provide worse consonance than the ones fulfilling all
three equations.

The far simplest one is the first, that is the standard major triad
(1,5/4,3/2), where the ratios n are the smallest (1 – 2 – 2). For
such triad, the beatings analysis fits well with the rules of
consonance that I have found in classic harmony texts, where
difference frequencies are calculated for each dyadic chord. For
instance, the interval (1,3/2) gives beating = ½, that is the tonic
1 octave below the fundamental (but remember: it is not a real tone,
but a loudness modulation). (1,5/4) gives ¼, that is the tonic 2
octaves below the fundamental. (5/4,3/2) gives ¼, that is again the
tonic 2 octaves below. Then we see that all three beatings are tones
belonging to the triad, and also to the scale.

Let us now determine the mean tones. They are also rather good:
(1,3/2) gives 5/4, that belongs to the triad; (1,5/4) gives 9/8,
that belongs to the scale; (5/4,3/2) gives 11/8, that in this case
does not belong to the scale.

The second triad is the one with x = 4/3. With the same procedure,
the "sus4" triad (1,4/3,3/2) is consonant with beating index (2 –
3/2 – 3). They are not as small and nice as in the first case, but
still acceptable. The tones corresponding to such beatings are ½,
the tonic 1 octave below, 1/3, the fourth 2 octave below, and 1/6,
the fourth 3 octaves below. In this case, the beatings again belong
to the triad, although only one of them is the tonic.As for the mean
tones, we have 5/4 (that belongs to the scale but no longer to the
triad), 7/6 and 17/12, that both do not belong to the scale (even
from the minor scale, if we consider the minor third being 6/5).

Finally, the third triad with x=7/6 is (1,7/6,3/2), I will call it a
minor triad. The tone 7/6 does not belong to the natural scale…. So
this shoud discourage us. However, the beating index is (1/2,3,3/2),
so comparably simple to the 4sus triad. Beating tones are ½, 1/6 and
1/3, but in this case the fourth does not belong to the triad. Mean
tones are 5/4, 13/12 (not belonging to the scale) and 4/3, that
belongs to the scale but not to the triad.

What about the minor triad 10:12:15? That would correspond to
(1,6/5,3/2). This triad is not a solution of our equations for the
simplest beating index. The beating index corresponding to this
case, would be (2/3 – 5/2 – 5/3). This really looks more complicated
than the previous ones! Let us calculate the difference tones: ½,
that is (as usual) in the triad; 1/5 (this introduces a minor sixth
3 octaves below) that is in the scale but not in the triad, and 3/10
(a minor third 2 octaves below) that is in the triad. Mean tones are
5/4 (that belongs to the scale but, this time, not to the triad),
11/10 and 27/10, that both do not belong to the scale.

To conclude, this kind of analysis tells us the following: if the
beatings coincidence were to rule the consonance of a triad, one
would expect the major triad to be the most consonant, followed by
the 4sus. Below is the minor triad with 7/6 minor third, although
such tone does not to the natural scale. After, the classic minor
triad with 6/5 minor third. One should judge which of those is more
consonant.

If, instead, also the absolute coincidences of beating tones and
mean tones with the tones of the triad and of the scale were
important, then there would be different combinations to take into
account, but from the overall appearance of the results, I would say
that the rank of consonance given above should be respected as well.

One could replace the fifth 3/2 with some different frequency, to
analyze harmonic behavior of different kinds of triads, e.g.
reversed) as well as of n-TETs more carefully. For instance, for 12-
TET with a fifth of 1.49831 you find out 1.2492 for the major triad,
1.3322 for the 4sus and 1.1661 for the minor triad (compare with the
frequencies of the 12-TET scale 1.2599, 1.3348 and 1.1892). For 17-
TET, with a fifth of 1.50341, you find 1.2517, 1.3356 and 1.1678
(compare with 1.2772, 1.3303, 1.1771). You see that for 12-TET there
is more difference for the minor triad (1.1661 vs 1.1892) than for
the major one (1.2492 vs 1.2599), that is, in 12-TET – according to
the above criteria of consonance of course! – the minor triad should
sound more dissonant than the major triad. For the 17-TET instead,
there is more difference for the major triad (1.2517 vs 1.2772) than
for the minor triad (1.1678 vs 1.1771), that is, in 17-TET the major
triad should sound more dissonant than the minor triad. Actually I
have to say that I consider more dissonant the minor triad in 12-TET
and the major triad in 17-TET. For sure I can say that, because I
have refretted a classic guitar with 17-TET and the minor triad
sounds nicer, in my opinion, than the standard one, while the major
triad is a bit annoying, especially in some positions on the
fretboard. Perhaps the different sound in different positions could
be due to the role of higher harmonics?…

Indeed, when we introduce the role of the higher harmonics,
everything becomes more complex. But all the considerations valid
for a given triad (combination of three fundamental tones) is also
valid for each of the higher harmonics (the triad of second
harmonics, the triad of third harmonics….). Additionally, there is
also overlapping of different harmonic orders that completes the
whole stuff, but in my opinion, at this point, it becomes secondary –
I mean that if I hear more or less consonance just on the basis of
the fundamental tones, to introduce higher harmonics won't change
the results too much, since they are much weaker after all. If you
have a look at Chris' spectra, you see how higher harmonics are at
least 12 dB lower than the fundamental. Of course this should be
measured on a "physical" instrument, not a generator, but I have
tried that and the result shows, as well known, that harmonics are
at least 10 dB weaker than the fundamental.

So, a proposal for exploration is to try the 7/6 minor triad with no
overtones and with them, as Chris already did (actually, all these
tests should be performed in Just Intonation). One should try to
realize whether the fact that 7/6 is not on the natural scale is
more important than the fact that beating index is a simple one and
that beating tones, as well as two of the three mean tones, belong
to the scale.

Max

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...>
wrote:
>
> Ok, I put together a series that some of you may find interesting
>
> http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1231472075
>
> http://tinyurl.com/9hzzlt
>
> I did what Carl suggested and played around with pure sines, and
also with
> pure triangles.
> I only had time to do this in 12 TET.
> *
> *By using an oscilloscope program I captured the FFT spectrum of C
major C
> minor and C sus 4th.
> There is a music example as well.
>
> I also did the same with a triangle wave.
>
> From a screen shot of cool edit you can see the beating or
roughness of the
> chord. However, the lack of harmonics do not make a difference to
me in the
> perceived consonance. In fact is you load the sine example in cool
edit it
> looks pretty much the same as the triangle version.
>
> This is all I had the time to do - any suggestions for exploration?
>

>> > > On the other side, if you try 1000:1200:1500Hz, then the
>> > > common fundamental is 100Hz,
>> >
>> > Not 500 Hz?
>>
>> That would be the case of the fifth in this example because
>> it's the GCD of the highest and the lowest sounding frequency.
>
> Put away your calculator and listen to it!

I played a few minor triads on my piano just to hear what fundamentals
jumped out at me. A few times it was an octave above what calculations
would dictate: A C minor triad would produce an Ab but an octave up
from where it would normally be. A few times I heard something in the
ballpark of a few octaves below the root of the triad, which
presumably indicates I'd heard the minor third as being in a 16:19
ratio... And a few times I heard some pretty interesting and
unexpected results. For an F minor triad, the dominant fundamental
that popped out was an A that was slightly sharp. Perhaps that
indicates I was hearing it as a subminor triad, but the sharp minor
third moved the fundamental down?

I think the volumes of each note in the chord contributes a lot to the
perceived fundamental. The calculated fundamentals of each dyad
analyzed separately will end up being in a harmonic series of the
fundamental of the whole chord, and when one overtone of the real VF
dominates, you might be inclined to hear that as the fundamental.

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> wrote:
>
> Carl,
>
> can I trouble you to post the chords again?
>
> please note the korg only has 4 note polyphony (a big flaw but
> the keyboard itself does more) so only 4 notes per chord please

Then use 1/1 5/4 3/2 7/4, and 1/1 7/6 7/5 7/4. I'll do
an 11-limit comparison in cool edit, but it'll have to wait
a couple weeks until I can get Vista off my damn machine!

-Carl

Hi Michael,

>    Actually, I would not fully agree IE I believe they are a
> part of what makes musical consonance/"concordance"...but having
> partials align does not automatically solve the whole problem
> just part of it.

No partials OR combination tones are audible with sine
tones at normal listening levels.

> IE a major sounds better than a minor...but
> the minor by not means sounds "terribly dissonant".

Just wait for the next demo.
 
> I again suspect it has something to do with the the ascending
> interval closeness (as you go higher on a major triad) better
> mirrors the harmonic series and more obviously points the brain
> to the existence of a base tone (even without partials). 

That makes sense.

>    If you try playing 11TET with normal instruments with normal
> partials (which clash) and then compare it to Sethares "Blue
> Dabo Girl" done in 11TET with partials skewed to match...you'll
> notice a huge gain in musical consonance IE partial intersection
> is a large part of the problem...

It's mainly the lack of partial near-intersection that results
in the improvement there, not the addition of exact intersection.

>    Yet, at the same time, you'll notice there is still a lot of
> consonance lacking between that and a piece with JI interval
> chords IE only part of the problem is solved.

Right, I hear it the same way.

-Carl

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> >> > > On the other side, if you try 1000:1200:1500Hz, then the
> >> > > common fundamental is 100Hz,
> >> >
> >> > Not 500 Hz?
> >>
> >> That would be the case of the fifth in this example because
> >> it's the GCD of the highest and the lowest sounding frequency.
> >
> > Put away your calculator and listen to it!
>
> I played a few minor triads on my piano just to hear what
> fundamentals jumped out at me. A few times it was an octave
> above what calculations would dictate: A C minor triad would
> produce an Ab but an octave up from where it would normally be.

Ab is one possible fundamental. Most people will hear an
octave below the C of the triad though (e.g. 500 Hz.).

-Carl

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...>
wrote:
>
> Ok, I put together a series that some of you may find interesting
>
> http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1231472075
>
> http://tinyurl.com/9hzzlt
>
> I did what Carl suggested and played around with pure sines, and
also with
> pure triangles.
> I only had time to do this in 12 TET.
> *
> *By using an oscilloscope program I captured the FFT spectrum of C
major C
> minor and C sus 4th.
> There is a music example as well.
>
> I also did the same with a triangle wave.
>
> From a screen shot of cool edit you can see the beating or
roughness of the
> chord. However, the lack of harmonics do not make a difference to
me in the
> perceived consonance. In fact is you load the sine example in cool
edit it
> looks pretty much the same as the triangle version.
>
> This is all I had the time to do - any suggestions for exploration?

Chris, this is great, as far as it goes, especially since there are
triangle waveforms that we can compare with the pure sine waves.
However, to compare the case of coincident vs. non-coincident
combinational tones the chords *must be in JI*, i.e., 4:5:6 vs.
10:12:15 vs. 6:8:9.

Since you said that you could only do four pitches at a time, then I
suggest that you compare these 7-limit chords (in JI):

4:5:6:7 vs. 60:70:84:105
6:7:9 vs. 14:18:21
5:6:7:9 vs. 70:90:105:126

An 11-limit comparison could be:
6:7:9:11 vs. 126:154:198:231

Each pair of chords consists of same intervals in reverse order.

--George

What trouble are you having with Vista? And if you are going to do that DL windows 7 from a torrent and try it before you install XP. The 7000 build beta is faster than XP - it even DL's file from the net faster. I think M$ did it right this time. Also I can cool edit as well if you want but I'm not sure I can route it into the analyzer. The korg is easy too.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Fri, 09 Jan 2009 20:32:12
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> wrote:
>
> Carl,
>
> can I trouble you to post the chords again?
>
> please note the korg only has 4 note polyphony (a big flaw but
> the keyboard itself does more) so only 4 notes per chord please

Then use 1/1 5/4 3/2 7/4, and 1/1 7/6 7/5 7/4. I'll do
an 11-limit comparison in cool edit, but it'll have to wait
a couple weeks until I can get Vista off my damn machine!

-Carl

>    Actually, I would not fully agree IE I believe they are a

> part of what makes musical consonance/" concordance" ...but having

> partials align does not automatically solve the whole problem

> just part of it.

--No partials OR combination tones are audible with sine
--tones at normal listening levels.
   My bad...I was talking about the triangle example, the one with overtones/partials, not the sine example.

> IE a major sounds better than a minor...but

> the minor by no means sounds "terribly dissonant".

----Just wait for the next demo.
   Aww man, am I going to have to start avoiding "reverse intervals" like the plague after you prove this one? :-D

>    If you try playing 11TET with normal instruments with normal

> partials (which clash) and then compare it to Sethares "Blue

> Dabo Girl" done in 11TET with partials skewed to match...you' ll

> notice a huge gain in musical consonance IE partial intersection

> is a large part of the problem...

---It's mainly the lack of partial near-intersection that results
---in the improvement there, not the addition of exact intersection.
   Near intersection meaning...when it gets near half the critical band IE the "harmonic entropy" point?  I think (although I am missing a few terms) we are talking about the same thing IE the fact that avoiding getting near that "maximum mathematical dissonance" point always helps to a decent extent.   And it does so even if it does not solve the "musical dissonance" problem (then we go back to observing the harmonic series order...which it seems clear we both agree means a lot toward the idea of "concordance/musical-consonance").
   Simple translation: Sethares has a very strong point and a fun musical option, but (as you said before), there are physiological factors related to the ear and the harmonic series and scales built to match it (including the series itself) which generally also observes its interval order (greater to smaller sizes between tones as you go up IE as a major triad does) will always give the most musically consonant results.

-Michael

--- On Fri, 1/9/09, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation
To: tuning@yahoogroups.com
Date: Friday, January 9, 2009, 12:36 PM

Hi Michael,

>    Actually, I would not fully agree IE I believe they are a

> part of what makes musical consonance/" concordance" ...but having

> partials align does not automatically solve the whole problem

> just part of it.

No partials OR combination tones are audible with sine

tones at normal listening levels.

> IE a major sounds better than a minor...but

> the minor by not means sounds "terribly dissonant".

Just wait for the next demo.

> I again suspect it has something to do with the the ascending

> interval closeness (as you go higher on a major triad) better

> mirrors the harmonic series and more obviously points the brain

> to the existence of a base tone (even without partials). 

That makes sense.

>    If you try playing 11TET with normal instruments with normal

> partials (which clash) and then compare it to Sethares "Blue

> Dabo Girl" done in 11TET with partials skewed to match...you' ll

> notice a huge gain in musical consonance IE partial intersection

> is a large part of the problem...

It's mainly the lack of partial near-intersection that results

in the improvement there, not the addition of exact intersection.

>    Yet, at the same time, you'll notice there is still a lot of

> consonance lacking between that and a piece with JI interval

> chords IE only part of the problem is solved.

Right, I hear it the same way.

-Carl

***Each pair of chords consists of same intervals in reverse order.

****--George

    Again this seems to be at the crux of the problem with "minor" scales...that chords with the intervals growing from larger to smaller on the exponential scale of human hearing (as you move higher up in frequency...just like the harmonic series does) sound more natural musically even though the "mathematical consonance" is the same.

-Michael

--- On Fri, 1/9/09, George D. Secor <gdsecor@...> wrote:

From: George D. Secor <gdsecor@...>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation
To: tuning@yahoogroups.com
Date: Friday, January 9, 2009, 12:45 PM

--- In tuning@yahoogroups. com, "Chris Vaisvil" <chrisvaisvil@ ...>

wrote:

>

> Ok, I put together a series that some of you may find interesting

>

> http://soonlabel. com/cgi-bin/ yabb2/YaBB. pl?num=123147207 5

>

> http://tinyurl. com/9hzzlt

>

> I did what Carl suggested and played around with pure sines, and

also with

> pure triangles.

> I only had time to do this in 12 TET.

> *

> *By using an oscilloscope program I captured the FFT spectrum of C

major C

> minor and C sus 4th.

> There is a music example as well.

>

> I also did the same with a triangle wave.

>

> From a screen shot of cool edit you can see the beating or

roughness of the

> chord. However, the lack of harmonics do not make a difference to

me in the

> perceived consonance. In fact is you load the sine example in cool

edit it

> looks pretty much the same as the triangle version.

>

> This is all I had the time to do - any suggestions for exploration?

Chris, this is great, as far as it goes, especially since there are

triangle waveforms that we can compare with the pure sine waves.

However, to compare the case of coincident vs. non-coincident

combinational tones the chords *must be in JI*, i.e., 4:5:6 vs.

10:12:15 vs. 6:8:9.

Since you said that you could only do four pitches at a time, then I

suggest that you compare these 7-limit chords (in JI):

4:5:6:7 vs. 60:70:84:105

6:7:9 vs. 14:18:21

5:6:7:9 vs. 70:90:105:126

An 11-limit comparison could be:

6:7:9:11 vs. 126:154:198: 231

Each pair of chords consists of same intervals in reverse order.

--George

Hi Max,

> In other words, when the beating frequency
> is within a certain frequency range, it is perceived as
> "annoying." This means that it can be assumed that loudness
> modulations (= beatings) are perceived by the ear as sounds
> themselves.

How do you conclude this? The mean tone is heard because
of the limited frequency resolution of the basilar membrane --
the two stimuli excite the same area, and their center
frequency is perceived, along with the AM. The AM isn't
necessarily the annoying part. The fact that there's a
typical beating rate that tends to be most dissonant can be
explained, for example, in terms of the width of the
excitation pattern on the basilar membrane.

> Let us start from the fact that a dyadic chord gives always
> raise to a beating (lower frequency, loudness modulation) and
> a mean tone (higher frequency, sine wave tone). The fifth
> interval (1,3/2) (is my notation acceptable?)

Generally just "3:2".

> has a beating at frequency

I don't hear any beats at all when I play a 3:2, nor do I hear
a mean tone.

-Carl

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> Chris, this is great, as far as it goes, especially since there are
> triangle waveforms that we can compare with the pure sine waves.
> However, to compare the case of coincident vs. non-coincident
> combinational tones the chords *must be in JI*, i.e., 4:5:6 vs.
> 10:12:15 vs. 6:8:9.

Why? I still hear the phenomenon. Is this more evidence that
coincident combination tones can't explain it?

-Carl

Carl wrote:

> That won't be audible in the least. Audio equipment stomps
> on phase (and so does the cochlea!) yet we don't have a
> problem hearing consonance and dissonance through speakers.
> Try playing with the phases in a good additive synth sometime.

I'm not sure if I've understood you now. If I listen to a chord of 300:500:700:900:1100Hz, I can very "brightly" hear 100Hz and, slightly less "brightly", also 200Hz -- I mean, as long as the overtones are rather soft. If they are loud, the frequency of 100Hz will sound much more "pronounced" to me than 200Hz. Similarly, when I hear 200:500:800:1100:1400Hz, I can clearly hear 100Hz there and less clearly I can hear 300Hz. In both cases, the first one is the GCD frequency and the second one is the difference frequency. I've asked a few other people about what tones thei were hearing when I played similar examples to them, and all of them chose the fundamental (i.e. the GCD) frequency as the most "prominent" one. Yes, it's true that all of them worked either in the field of acoustics or composing music, I've not tried so far to ask more than a few people.

> There is no such mathematical definition of the virtual
> fundamental (which is the result of a stochastic process in
> your brain... but you can measure the likelihood of it coming
> out the same every time with harmonic entropy).
>
> I don't know what a "common fundamental" is.

If there were no mathematical definition of the fundamental frequency or tone (I'm not talking about the VF right now), then I'd ask why so many people including Fokker and Partch gave so much significance to it. In order you knew what I'm talking about, I'll try to clarify: http://tonalsoft.com/enc/f/fundamental.aspx

> Put away your calculator and listen to it!

But Carl, it really IS what I can hear there, I'm not lead by the calculations, I'm getting to the calculations after listening carefully to the intervals. Whenever I hear some of the "combination tones" or whatever, I first listen to find out, as exactly as I can, in what intervals these are away from the sounding tones. And if I really DO care about the frequency relationships, sometimes I do a bit of quick maths to convert the exponential interval sizes to linear frequency relationships. For example, recently, I've experimented with some sine wave "peculiarities" and ran 1800Hz and 2500Hz through my headphones. When I played the sound pretty loud and then converted the intervals I could hear into linear ratios, it turned out I was actually able to hear 1100, 700, and 400Hz there. I can let you think of your own explanation for why I could hear these tones if you don't want me to saywhat I think.

Petr

I wrote:

> I was actually able to hear 1100, 700, and 400Hz there.

And as soon as the periods weren't sine waves, 100Hz appeared in a way which was just impossible to "overhear".

Petr

Hi George

is pythagorean JI? I'm just a beginner. my korg can be programed - not sure if its cents (I think so) or frequency off the top of my head. The ratios mean little to me at my level of tuning knowledge.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "George D. Secor" <gdsecor@yahoo.com>

Date: Fri, 09 Jan 2009 20:45:54
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...>
wrote:
>
> Ok, I put together a series that some of you may find interesting
>
> http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1231472075
>
> http://tinyurl.com/9hzzlt
>
> I did what Carl suggested and played around with pure sines, and
also with
> pure triangles.
> I only had time to do this in 12 TET.
> *
> *By using an oscilloscope program I captured the FFT spectrum of C
major C
> minor and C sus 4th.
> There is a music example as well.
>
> I also did the same with a triangle wave.
>
> From a screen shot of cool edit you can see the beating or
roughness of the
> chord. However, the lack of harmonics do not make a difference to
me in the
> perceived consonance. In fact is you load the sine example in cool
edit it
> looks pretty much the same as the triangle version.
>
> This is all I had the time to do - any suggestions for exploration?

Chris, this is great, as far as it goes, especially since there are
triangle waveforms that we can compare with the pure sine waves.
However, to compare the case of coincident vs. non-coincident
combinational tones the chords *must be in JI*, i.e., 4:5:6 vs.
10:12:15 vs. 6:8:9.

Since you said that you could only do four pitches at a time, then I
suggest that you compare these 7-limit chords (in JI):

4:5:6:7 vs. 60:70:84:105
6:7:9 vs. 14:18:21
5:6:7:9 vs. 70:90:105:126

An 11-limit comparison could be:
6:7:9:11 vs. 126:154:198:231

Each pair of chords consists of same intervals in reverse order.

--George

Carl, (and the physicist) are we *not* going to see the combination tones in the FFT spectrum?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Fri, 09 Jan 2009 20:38:57
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> >> > > On the other side, if you try 1000:1200:1500Hz, then the
> >> > > common fundamental is 100Hz,
> >> >
> >> > Not 500 Hz?
> >>
> >> That would be the case of the fifth in this example because
> >> it's the GCD of the highest and the lowest sounding frequency.
> >
> > Put away your calculator and listen to it!
>
> I played a few minor triads on my piano just to hear what
> fundamentals jumped out at me. A few times it was an octave
> above what calculations would dictate: A C minor triad would
> produce an Ab but an octave up from where it would normally be.

Ab is one possible fundamental. Most people will hear an
octave below the C of the triad though (e.g. 500 Hz.).

-Carl

You know ... It is dangerous to extrapolate with too little data. :-)
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Michael Sheiman <djtrancendance@yahoo.com>

Date: Fri, 9 Jan 2009 13:30:03
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

***Each pair of chords consists of same intervals in reverse order.

****--George

    Again this seems to be at the crux of the problem with "minor" scales...that chords with the intervals growing from larger to smaller on the exponential scale of human hearing (as you move higher up in frequency...just like the harmonic series does) sound more natural musically even though the "mathematical consonance" is the same.

-Michael

--- On Fri, 1/9/09, George D. Secor <gdsecor@yahoo.com> wrote:

From: George D. Secor <gdsecor@yahoo.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation
To: tuning@yahoogroups.com
Date: Friday, January 9, 2009, 12:45 PM

--- In tuning@yahoogroups. com, "Chris Vaisvil" <chrisvaisvil@ ...>

wrote:

>

> Ok, I put together a series that some of you may find interesting

>

> http://soonlabel. com/cgi-bin/ yabb2/YaBB. pl?num=123147207 5

>

> http://tinyurl. com/9hzzlt

>

> I did what Carl suggested and played around with pure sines, and

also with

> pure triangles.

> I only had time to do this in 12 TET.

> *

> *By using an oscilloscope program I captured the FFT spectrum of C

major C

> minor and C sus 4th.

> There is a music example as well.

>

> I also did the same with a triangle wave.

>

> From a screen shot of cool edit you can see the beating or

roughness of the

> chord. However, the lack of harmonics do not make a difference to

me in the

> perceived consonance. In fact is you load the sine example in cool

edit it

> looks pretty much the same as the triangle version.

>

> This is all I had the time to do - any suggestions for exploration?

Chris, this is great, as far as it goes, especially since there are

triangle waveforms that we can compare with the pure sine waves.

However, to compare the case of coincident vs. non-coincident

combinational tones the chords *must be in JI*, i.e., 4:5:6 vs.

10:12:15 vs. 6:8:9.

Since you said that you could only do four pitches at a time, then I

suggest that you compare these 7-limit chords (in JI):

4:5:6:7 vs. 60:70:84:105

6:7:9 vs. 14:18:21

5:6:7:9 vs. 70:90:105:126

An 11-limit comparison could be:

6:7:9:11 vs. 126:154:198: 231

Each pair of chords consists of same intervals in reverse order.

--George

Chris wrote:

> are we *not* going to see the combination tones in the FFT spectrum?

Probably not, if I understand the concept of FFT correctly. If you mix two sine waves of 50Hz and 60Hz and listen to it, you may also hear 10Hz, which is the difference frequency and also the fundamental (i.e. GCD) frequency, but in the case of a good-quality FFT algorithm, the sine waves of 50Hz and 60Hz will be „extracted“ from the actual waveform, leaving nothing else to extract.

Petr

Correct. The FFT/DFT of the signal will show only what frequencies are
present in the signal. The 10 Hz phantom fundamental that you might
hear is an "artificial" frequency produced in the brain. The concept
of "frequency" here is something different from and much more general
than pitch.

Sum, difference, and harmonic tones might be produced if the signal is
run through some kind of nonlinear system, such as a distortion, and
the spectrum of the resulting output signal would reflect those.
Perhaps the air itself has some kind of nonlinear effect, or perhaps
the ear does, or perhaps the brain does - most speakers certainly do.
If you could somehow perform some kind of Fourier analysis on that
signal, you'll see combination tones and such. But Fourier analysis of
the original signal will only give you what frequencies were only in
the signal.

-Mike

On Fri, Jan 9, 2009 at 5:57 PM, Petr Parízek <p.parizek@...> wrote:
> Chris wrote:
>
>> are we *not* going to see the combination tones in the FFT spectrum?
>
> Probably not, if I understand the concept of FFT correctly. If you mix two
> sine waves of 50Hz and 60Hz and listen to it, you may also hear 10Hz, which
> is the difference frequency and also the fundamental (i.e. GCD) frequency,
> but in the case of a good-quality FFT algorithm, the sine waves of 50Hz and
> 60Hz will be „extracted" from the actual waveform, leaving nothing else to
> extract.
>
> Petr
>
>
>
>
>
>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > That won't be audible in the least. Audio equipment stomps
> > on phase (and so does the cochlea!) yet we don't have a
> > problem hearing consonance and dissonance through speakers.
> > Try playing with the phases in a good additive synth sometime.
>
> I'm not sure if I've understood you now. If I listen to a chord
> of 300:500:700:900:1100Hz, I can very "brightly" hear 100Hz and,
> slightly less "brightly", also 200Hz -- I mean, as long as the
> overtones are rather soft.

That should be the VF, not difference tones.

> Similarly, when I hear 200:500:800:1100:1400Hz, I can clearly
> hear 100Hz there and less clearly I can hear 300Hz.

Do you hear 300 Hz. with sine tones here?

> In both cases, the first one is the GCD frequency and the second
> one is the difference frequency.

You've changed the examples to ones where the GCD is also
the most likely VF. What about 10:12:15?

> > There is no such mathematical definition of the virtual
> > fundamental (which is the result of a stochastic process in
> > your brain... but you can measure the likelihood of it coming
> > out the same every time with harmonic entropy).
> >
> > I don't know what a "common fundamental" is.
>
> If there were no mathematical definition of the fundamental
> frequency or tone (I'm not talking about the VF right now),

The VF is the fundamental. I don't know of anything else.

> In order you knew what I'm talking about, I'll try to clarify:
> http://tonalsoft.com/enc/f/fundamental.aspx

Both of these definitions refer to virtual pitch ("VF").

-Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Carl, (and the physicist) are we *not* going to see the
> combination tones in the FFT spectrum?

You're not -- they're typically generated in the ear itself.

-Carl

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Chris wrote:
>
> > are we *not* going to see the combination tones in the FFT
> > spectrum?
>
> Probably not, if I understand the concept of FFT correctly. If
> you mix two sine waves of 50Hz and 60Hz and listen to it, you may
> also hear 10Hz, which is the difference frequency and also the
> fundamental (i.e. GCD) frequency, but in the case of a good-
> quality FFT algorithm, the sine waves of 50Hz and 60Hz will be
> "extracted" from the actual waveform, leaving nothing else to
> extract.

Let's be clear where these things come from. Combination tones
are due to nonlinearities in the response in the ear (and
occasionally in guitar amplifiers). VFs are created by the
brain out of whole cloth. In most cases, combination tones are
inaudible in music, or audible only through very careful
analytical listening. They cannot explain consonance and
dissonance phenomena, and no psychoacoustic models I'm aware of
claim they do.

-Carl

Mike wrote:

> Sum, difference, and harmonic tones might be produced if the
> signal is run through some kind of nonlinear system, such as a
> distortion, and the spectrum of the resulting output signal
> would reflect those. Perhaps the air itself has some kind of
> nonlinear effect,

A room temperature gas is about as elastic as it gets.
Nevertheless, if you hit it with enough power, it does get
nonlinear. "Audio beam" technology uses this effect --
you create a signal such that the distortion products are
the desired output.

http://www.holosonics.com/technology.html

> or perhaps the ear does,

It does, but not very much.

> most speakers certainly do.

If we're talking about the 6x9s in the rear deck of your
'82 camero, then yes. Any modern loudspeaker in decent
shape, essentially none.

-Carl

Petre,

I don't think anybody with normal hearing can hear 10 Hz directly. We
can hear only resulting effect, like beating, or amplitude
modulation. This was probably what you wanted to say.

Daniel Forro

On 10 Jan 2009, at 7:57 AM, Petr Parízek wrote:

>
> Chris wrote:
>
> > are we *not* going to see the combination tones in the FFT spectrum?
>
> Probably not, if I understand the concept of FFT correctly. If you
> mix two sine waves of 50Hz and 60Hz and listen to it, you may also
> hear 10Hz, which is the difference frequency and also the
> fundamental (i.e. GCD) frequency, but in the case of a good-quality
> FFT algorithm, the sine waves of 50Hz and 60Hz will be „extracted“
> from the actual waveform, leaving nothing else to extract.
>
> Petr

But the beating - the amplitude modulation shows up in cool edit. Something to think about. Because sound is just periodic pressure waves. And amplitude modulate - beating - is derived from constructive and deconstructive interference - now the difference tones - say 400 and 500 Hz - giving ± 100 Hz - is from ring modulation. I can't remember exactly the result there.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Sat, 10 Jan 2009 00:26:45
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Carl, (and the physicist) are we *not* going to see the
> combination tones in the FFT spectrum?

You're not -- they're typically generated in the ear itself.

-Carl

I was confused about this too a while ago. Although the amplitude
envelope does appear in the signal with a certain frequency, no sine
wave of that frequency exists in the signal. If it did, the signal
would also be moving up and down on the graph at the same rate that
the envelope exists at. When we talk about frequency analysis or
taking the "FFT" of something, we're trying to figure out what
sinusoids exist in the signal, and at what frequency they exist, and
at what phase (starting point). Although it is true that the amplitude
of the signal is being modulated in a sinusoidal pattern with a
certain frequency, no actual sine wave with that frequency exists in
the signal.

-Mike

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Ring modulation is kind of amplitude modulation, but side bands are
on both sides (differential and sum tones), and original input
frequencies are suppressed. So in this case there will be 100 Hz and
900 Hz on the output only.

Daniel Forro

On 10 Jan 2009, at 12:18 PM, chrisvaisvil@... wrote:

> But the beating - the amplitude modulation shows up in cool edit.
> Something to think about. Because sound is just periodic pressure
> waves. And amplitude modulate - beating - is derived from
> constructive and deconstructive interference - now the difference
> tones - say 400 and 500 Hz - giving ± 100 Hz - is from ring
> modulation. I can't remember exactly the result there.

Daniel wrote:

> I don't think anybody with normal hearing can hear 10 Hz directly. We
> can hear only resulting effect, like beating, or amplitude
> modulation. This was probably what you wanted to say.

Of course, we can�t hear 10Hz for two reasons: A) our hearing filters this out, B) there actually isn�t any 10Hz period in the example I gave. We�ll hear, as you correctly pointed out, the amplitude modulation at a frequency of 10Hz. If you tak� a set of sine waves with frequencies of 40:50:60:70 ... up to 20000Hz, you�ll see there�s also no 10Hz sine wave but we can hear the �synchronicity� of all of the frequency components that get in the same phase every 10th of a second. so we�re not actually hearing any 10Hz tones but we�re hearing the 10Hz periodicity in the higher registers. Similarly, if you talked to me on the phone and let your vocal chords vibrate at 70Hz which the phone doesn�t transmit, I would only hear the 70Hz periodicity in the higher frequencies, yet this would make it clear to me your voice probably spoke with a 70Hz tone.

Petr

On 10 Jan 2009, at 6:18 PM, Petr Parízek wrote:

> Of course, we can’t hear 10Hz for two reasons: A) our hearing
> filters this
> out,

Petre,

I always learned you to express your thoughts exactly which you
usually do :-) So here you can't say "filtered out", because that
would mean it's recognized and then filtered. But such low frequency
is not recognized as a tone at all as you know well, so it can't be
filtered in the sense how filter works (not to allow to pass some
frequencies on input to the output). Just say "it's out of the range
of our hearing system".

And don't think please I'm too pedantic. We were given a speech and
writing to express clearly our thoughts, so let's do it. If all of us
will use language this way, the world would look much better than now...

Daniel F

Carl wrote:

> That should be the VF, not difference tones.

The definition of the fundamental frequency being the GCD of the sounding frequencies is not something I've made up in my mind, I've read that more times from various sources, some of which also had a definition for the "guide tone" to be the LCM of the sounding frequencies, like, for example, this one: www.xs4all.nl/~huygensf/doc/efg-e.html

> > Similarly, when I hear 200:500:800:1100:1400Hz, I can clearly
> > hear 100Hz there and less clearly I can hear 300Hz.
>
> Do you hear 300 Hz. with sine tones here?

Depends on how high the sounding frequencies are. If they were only 200:500:800Hz, then 300Hz is very poorly audible to me. But if they were 1400:1700:2000Hz, then I can hear that better. But what I can hear much clearer in both of these cases is 100Hz.

> You've changed the examples to ones where the GCD is also
> the most likely VF. What about 10:12:15?

I've tried once again, to see if I wasn't kidding, to mix 1000:1200:1500Hz and listen to it. What I could hear best of all was 100Hz, which is, as I've already said, a few octaves below any of the sounding tones. When I turned the volume a bit louder, I could hear, although only very softly, 200, 300, and 500Hz.

> Both of these definitions refer to virtual pitch ("VF").

Even if they do, they also explicitely specify the fundamental frequency to be the GCD of the frequencies in question.

Maybe I should give a particular example of what I'm thinking about. Suppose you mix a set of equally loud static sinusoidal waves (or, more precisely, cosine waves) which have frequencies of 3:7:11:15:19:23Hz and so on and so on, up to the highest frequency you can hear, or (if we work on a digital sampling level) the highest frequency which the actual sampling rate makes possible to transmit. What you get is a single full-volume impulse at the very begining, then after 0.25 seconds you get this impulse phase-shifted by -90 degrees, which, in the case of digital sampling, results in a DHT (see http://en.wikipedia.org/wiki/Hilbert_transform#Discrete_Hilbert_transforms), then after 0.5 seconds you get the impulse phase shifted by 180 degrees (which is essentially the same as inverting its polarity), and after 0.75 seconds you get the impulse phase-shifted by +90 degrees, which again, in the case of digital sampling, results in an inverted version of the DHT. This means that there are actually two kinds of periodicity, one of 1Hz and the other of 4Hz, even though neither of these frequencies is actually present. Now imagine all of this played 100 times faster. Don't know about you, but I can clearly hear 100Hz in such a sound and slightly less clearly also 400Hz. If you want, you can download an experiment of mine where I tried something similar with phase shifts of 0, -120, and +120 degrees. In order the impulses were possible to clearly separate, I left the general periodicity way below our hearing range. In this example, 4Hz is the lowest frequency present and 6Hz is the difference between consecutive frequencies, which results in an actual period of 2Hz. Here is the link: www.sendspace.com/file/r6cbzu

Petr

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Max,
>
> > In other words, when the beating frequency
> > is within a certain frequency range, it is perceived as
> > "annoying." This means that it can be assumed that loudness
> > modulations (= beatings) are perceived by the ear as sounds
> > themselves.
>
> How do you conclude this? The mean tone is heard because
> of the limited frequency resolution of the basilar membrane --
> the two stimuli excite the same area, and their center
> frequency is perceived, along with the AM. The AM isn't
> necessarily the annoying part. The fact that there's a
> typical beating rate that tends to be most dissonant can be
> explained, for example, in terms of the width of the
> excitation pattern on the basilar membrane.
>
> > Let us start from the fact that a dyadic chord gives always
> > raise to a beating (lower frequency, loudness modulation) and
> > a mean tone (higher frequency, sine wave tone). The fifth
> > interval (1,3/2) (is my notation acceptable?)
>
> Generally just "3:2".
>
> > has a beating at frequency
>
> I don't hear any beats at all when I play a 3:2, nor do I hear
> a mean tone.
>
> -Carl
>

Hi Carl, Ok I think I see your point.

But: when you tune your guitar, you hear beating and use it to tune.
When you start going out of tune, you hear fast beatings that are not
nice. At some point, when the interval is wide enough, you cannot
hear anymore the beatings, but intervals still are more or less
pleasant to hear. In Plomp&Levelt experiments, pure tones were used
to determine most dissonant intervals, and this was done, as far as I
know, with a continuum of frequency ratios. For some dissonant
intervals, you don't hear the beating for sure, but still it sounds
dissonant. So I am just saying: since beatings (at least the ones
fast enough) give an unpleasant sensation, perhaps faster beatings
(even if you don't realize they are there) do the same if within the
critical band.

Now I also say: prosthapheresis formulae show superposition of two
distinct tones is equivalent to a mean tone with an amplitude
modulation on it. So, since (in the case of a dissonant interval of
course) the mean tone is not annoying by itself, it must be made
annoying by its loudness modulation - even if it is fast and you
don't realize it.

I also understand (and I already arose this doubt in my previous
post) that the ear cannot feel the beating frequency just as a FFT
analyzer can't. If I have a resonator tuned at the beating frequency,
and I play my interval, I am sure it will not resonate. Nonetheless,
the waveform and mathematics itself show a periodicity, and it is
because of this periodicity that a couple of tones played together
sound dissonant while the frequency corresponding to their mean tone,
when played alone and steadily, is not. But you are also right, you
don't hear any mean tone when you play a fifth interval, and you
don't hear any beats as well.

Let's say that in the following way. The two (beating) tones when
inside the ear excite different parts of it. If frequencies are too
close, the ear gets excited in adjacent sections and is not able to
identify the tone, therefore the annoying sensation. But when the
tones are sufficiently far apart (let's say, more than a minor third)
there should be negligible superposition and the tones be well
splitted. Let's think about a major triad; then, if I go slightly out
of tune with the third, or the fifth, why does the triad get worse
so "quickly" (by means of so small change of tune)? The only reason I
could imagine was the conflict of beating frequencies of the three
dyads with each other. I don't know well the mechanism of hearing, so
I'm stuck at this point. And my curiosity is limited to the
fundamentals, no harmonics. What do you think?

To comment about some other points arosen:

I think we are not going to see combination tones in the FFT
spectrum. A Fourier transformation is, by definition, extracting the
amplitude and phase of single frequencies within a signal. One has to
bear in mind that a single frequency exists only if the signal has
infinite duration. In mathematical terms, you integrate the product
of your signal with a reference cosine wave over an infinite time,
and span all possible frequencies of the reference wave. The results
for each reference wave compose the Fourier spectrum.

The explanation of why the beating (or amplitude modulation) does not
show up in the Fourier spectrum is that it is not an oscillation
itself, but a change of loudness. If you Fourier-transform the "mean
tone" fa=(f1+f2)/2 with an amplitude modulation on it, you do not get
the same result than Fourier-transforming the mean tone with constant
amplitude. This last transform would give you exactly (f1+f2)/2,
while the first would give you the couple of frequencies f1 and f2
ONLY if the beating frequency is exactly fb=f1-f2 (by beating
frequency I mean the frequency of repetition of loudness modulation,
that is, the inverse of a period between two zeroes in amplitude). If
you change the beating frequency fb while keeping the mean tone fixed
fa, you will get two frequencies (different from f1 and f2) spaced by
a frequency difference fb.

Now, if cool edit shows frequency beats, it means it is not
performing Fourier transform, possibly something else. I have
downloaded it to make some tests (not yet done). If you use a real
spectrum analyzer, the beating should not show up.

Ring modulation, as far as I understand, is completely different. It
is due to the two oscillators physically coupled to each other.
However this is not necessarily a nonlinear effect, can be linear
coupling. Think of two independent springs, and join them by a third
spring. The system is physically modified, the resonance frequencies
of single springs are lost and new resonant modes of the coupled
system arise. For instance, if the two springs are identical,
coupling between them splits their common resonant frequencies into
two different ones. Depending on the geometry of the system, one of
the two "modes" can be much less damped than the original one and
provide therefore long-lasting tones. The tuning fork works on this
principle (this is why you do not use a single bar, but instead a
fork, to obtain the tuning tone). The amount of frequency separation
depends on the amount of coupling.

Max

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

> > That should be the VF, not difference tones.
>
> The definition of the fundamental frequency being the GCD of
> the sounding frequencies is not something I've made up in my
> mind, I've read that more times from various sources, some of
> which also had a definition for the "guide tone" to be the LCM
> of the sounding frequencies, like, for example, this one:
> www.xs4all.nl/~huygensf/doc/efg-e.html

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

> Maybe I should give a particular example of what I'm thinking
> about. Suppose you mix a set of equally loud static sinusoidal
> waves (or, more precisely, cosine waves) which have frequencies
> of 3:7:11:15:19:23Hz and so on and so on, up to the highest
> frequency you can hear, or (if we work on a digital sampling
> level) the highest frequency which the actual sampling rate
> makes possible to transmit. What you get is a single full-volume
> impulse at the very begining, then after 0.25 seconds you get
> this impulse phase-shifted by -90 degrees, [snip]

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

-Carl

Max wrote:

> > How do you conclude this? The mean tone is heard because
> > of the limited frequency resolution of the basilar membrane --
> > the two stimuli excite the same area, and their center
> > frequency is perceived, along with the AM. The AM isn't
> > necessarily the annoying part. The fact that there's a
> > typical beating rate that tends to be most dissonant can be
> > explained, for example, in terms of the width of the
> > excitation pattern on the basilar membrane.
//
> Hi Carl, Ok I think I see your point.
>
> But: when you tune your guitar, you hear beating and use it to
> tune. When you start going out of tune, you hear fast beatings
> that are not nice. At some point, when the interval is wide
> enough, you cannot hear anymore the beatings, but intervals still
> are more or less pleasant to hear. In Plomp&Levelt experiments,
> pure tones were used to determine most dissonant intervals, and
> this was done, as far as I know, with a continuum of frequency
> ratios. For some dissonant intervals, you don't hear the beating
> for sure, but still it sounds dissonant.

Plomp & Levelt used continuously-tunable sine tone generators.
As two sine tones diverge, we hear a gradually-increasing rate
of beating in the mean tone. At about half the critical
bandwidth, the beating goes away and we start to hear two
separate tones instead of the mean tone, but these tones are
"rough" and poorly resolved. The roughness diminishes until
two clear tones are heard with no beating. Beyond this, P&L
found no consonance distinctions with interval size. However,
if they had tested triads and tetrads, they would indeed have
seen consonance distinctions that cannot be explained by
critical band interactions, as Chris will hopefully demonstrate
soon, by rendering harmonic and subharmonic 7th chords with
sine tones.

> So I am just saying: since beatings (at least the ones fas
> enough) give an unpleasant sensation, perhaps faster beatings
> (even if you don't realize they are there) do the same if
> within the critical band.

The point of maximum critical band dissonance is right around
where the beating stops and the two separate tones begin.

> Let's say that in the following way. The two (beating) tones when
> inside the ear excite different parts of it. If frequencies are
> too close, the ear gets excited in adjacent sections and is not
> able to identify the tone, therefore the annoying sensation.

Yes, exactly. It can be seen as a failure of pitch discrimination.
Really, consonance is our level of success assigning a pitch or
pitches to what we hear.

> if I go slightly out of tune with the third, or the fifth,
> why does the triad get worse so "quickly" (by means of so small
> change of tune)? The only reason I could imagine was the
> conflict of beating frequencies of the three dyads with each
> other.

That may be the case. With chords such as 7:9:11 for example,
there is beating between the upper partials even when the chord
is tuned exactly, but it is not unpleasant. This must be partly
due to the fact that only the upper partials are affected, and
they are at much lower amplitudes than the 3rd and 2nd partials
that are affected in the case of a mistuned 2:3. But
additionally, one cannot help but report a pleasant synchronicity
to the beatings...

> Now, if cool edit shows frequency beats, it means it is not
> performing Fourier transform, possibly something else. I have
> downloaded it to make some tests (not yet done). If you use a
> real spectrum analyzer, the beating should not show up.

Correct. Cool edit, like most wave editors, by default shows
simply an amplitude-time plot, where the beatings appear just
fine. :)

It also has an FFT function that can show spectrograms, but it
is not enabled by default.

-Carl

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--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...>
wrote:
>
> Ok,
>
> Isn't ring modulation the multiplication of one input by the other?
>
> One can make a simple ring modulator out of 4 diodes and nothing
more, no
> oscillators needed.
>
> http://en.wikipedia.org/wiki/Ring_modulation
>
> *Ring modulation* is a signal-processing effect in electronics,
> related to amplitude
> modulation <http://en.wikipedia.org/wiki/Amplitude_modulation> or
frequency
> mixing <http://en.wikipedia.org/wiki/Frequency_mixer>, performed by
> multiplying two signals, where one is typically a
> sine-wave<http://en.wikipedia.org/wiki/Sine-wave>or another simple
> waveform. It is referred to as "ring" modulation because
> the analog circuit <http://en.wikipedia.org/wiki/Analog_circuit> of
> diodes<http://en.wikipedia.org/wiki/Diode>originally used to
implement
> this technique took the shape of a ring. This
> circuit is similar to a bridge
> rectifier<http://en.wikipedia.org/wiki/Bridge_rectifier>,
> except that instead of the diodes facing "left" or "right", they go
> "clockwise" or "anti-clockwise".
>

Hi Chris,

yes, we are talking about different topics. By ringing I just meant
the phenomenon by which energy goes back and forth between the two
coupled oscillators. It is not standard terminology though, I found
it somewhere once and liked it. However, it seems to me that the ring
modulation you mention is just one possible way to obtain some
frequency mixing (by means of the diode nonlinearity)?

Ciao

Max

ps by any chance, would you find a moment to try fundamental tones
combination 1 - 7/6 - 3/2 (or 6:7:9) the same way you did with 4:5:6
and 10:12:15?.......... Thanks

Better ring modulators use additionally two small transformers 1:1. There are also ring modulators using IC's.

It's one of the most interesting audio processing method heavily used in electronic music, even in pop and rock.

Daniel Forro

On 11 Jan 2009, at 8:33 AM, Chris Vaisvil wrote:

> Ok,
>
> Isn't ring modulation the multiplication of one input by the other?
>
> One can make a simple ring modulator out of 4 diodes and nothing > more, no oscillators needed.
>
> http://en.wikipedia.org/wiki/Ring_modulation
>
> Ring modulation is a signal-processing effect in electronics, > related toamplitude modulation or frequency mixing, performed by > multiplying two signals, where one is typically a sine-wave or > another simple waveform. It is referred to as "ring" modulation > because the analog circuit of diodes originally used to implement > this technique took the shape of a ring. This circuit is similar to > abridge rectifier, except that instead of t he diodes facing "left" > or "right", they go "clockwise" or "anti-clockwise".
>

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--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@> wrote:
>
> > > That should be the VF, not difference tones.
> >
> > The definition of the fundamental frequency being the GCD of
> > the sounding frequencies is not something I've made up in my
> > mind, I've read that more times from various sources, some of
> > which also had a definition for the "guide tone" to be the LCM
> > of the sounding frequencies, like, for example, this one:
> > www.xs4all.nl/~huygensf/doc/efg-e.html
>
> I'm not accusing you of making it up, I'm asking you why
> you believed it. As already mentioned, VF extraction is a
> stochastic, listener-dependent process and there's no
> simple formula that predicts it with certainty. That is why
> we model it with harmonic _entropy_. For chords with low
> entropy, something like the GCD may give quasi-reliable
> results, though the heard VF will often be an octave above
> or below the GCD tone.
>
> > Maybe I should give a particular example of what I'm thinking
> > about. Suppose you mix a set of equally loud static sinusoidal
> > waves (or, more precisely, cosine waves) which have frequencies
> > of 3:7:11:15:19:23Hz and so on and so on, up to the highest
> > frequency you can hear, or (if we work on a digital sampling
> > level) the highest frequency which the actual sampling rate
> > makes possible to transmit. What you get is a single full-volume
> > impulse at the very begining, then after 0.25 seconds you get
> > this impulse phase-shifted by -90 degrees, [snip]
>
> What does this have to do with anything? Of course one can
> construct any number of synthesized examples where phase
> cancelations occur in the signal generation domain. Such
> effects are almost never heard in musical contexts, and are
> not a significant source of anything remotely related to
> consonance or dissonance in normal musical settings. If you
> have evidence to the contrary, I would like to download it.
>
> -Carl
>
But why make the initial distinction b/w fundamental frequency as GCD
and the sounding frequencies in the first place? If we play two
periodic waves which share a GCD then the number of cycles per second
will equal the inverse of the GCD, which is a fact of nature. And
since this is also the very definition of frequency then we cannot
assume that the original frequencies did not also meet this condition
i.e. that they are already GCD to some other wave addition, that these
can be added to other waves, and so on ad infinitum. This GCD not only
seems to have the advantage of harmonic complexity, worthy of real
music, but it also seems to agree with tonality i.e. tonic = 8ve
equivalent of GCD (frequency),as opposed to combination tones which
don't produce the correct tonic and must enter into the picture long
after the fact.If a computer model did not produce this successfully,
then perhaps this is due to an oversight on behalf of the programmer
and not on some higher 'natural' principle.

-Rick

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@> wrote:
> >
> > > > That should be the VF, not difference tones.
> > >
> > > The definition of the fundamental frequency being the GCD of
> > > the sounding frequencies is not something I've made up in my
> > > mind, I've read that more times from various sources, some of
> > > which also had a definition for the "guide tone" to be the LCM
> > > of the sounding frequencies, like, for example, this one:
> > > www.xs4all.nl/~huygensf/doc/efg-e.html
> >
> > I'm not accusing you of making it up, I'm asking you why
> > you believed it. As already mentioned, VF extraction is a
> > stochastic, listener-dependent process and there's no
> > simple formula that predicts it with certainty. That is why
> > we model it with harmonic _entropy_. For chords with low
> > entropy, something like the GCD may give quasi-reliable
> > results, though the heard VF will often be an octave above
> > or below the GCD tone.
> >
> > > Maybe I should give a particular example of what I'm thinking
> > > about. Suppose you mix a set of equally loud static sinusoidal
> > > waves (or, more precisely, cosine waves) which have frequencies
> > > of 3:7:11:15:19:23Hz and so on and so on, up to the highest
> > > frequency you can hear, or (if we work on a digital sampling
> > > level) the highest frequency which the actual sampling rate
> > > makes possible to transmit. What you get is a single full-volume
> > > impulse at the very begining, then after 0.25 seconds you get
> > > this impulse phase-shifted by -90 degrees, [snip]
> >
> > What does this have to do with anything? Of course one can
> > construct any number of synthesized examples where phase
> > cancelations occur in the signal generation domain. Such
> > effects are almost never heard in musical contexts, and are
> > not a significant source of anything remotely related to
> > consonance or dissonance in normal musical settings. If you
> > have evidence to the contrary, I would like to download it.
> >
> > -Carl
> >
> But why make the initial distinction b/w fundamental frequency as GCD
> and the sounding frequencies in the first place? If we play two
> periodic waves which share a GCD then the number of cycles per second
> will equal the inverse of the GCD, which is a fact of nature. And
> since this is also the very definition of frequency then we cannot
> assume that the original frequencies did not also meet this condition
> i.e. that they are already GCD to some other wave addition, that these
> can be added to other waves, and so on ad infinitum. This GCD not only
> seems to have the advantage of harmonic complexity, worthy of real
> music, but it also seems to agree with tonality i.e. tonic = 8ve
> equivalent of GCD (frequency),as opposed to combination tones which
> don't produce the correct tonic and must enter into the picture long
> after the fact.If a computer model did not produce this successfully,
> then perhaps this is due to an oversight on behalf of the programmer
> and not on some higher 'natural' principle.
>
> -Rick
>
Sorry, correction. The number of cycles/periods per sec will equal the
GCD number, and the period value is its inverse.

-Rick

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Hi George
>
> is pythagorean JI?

No. JI consists of tones having frequencies related by simple-number
ratios.

> I'm just a beginner. my korg can be programed - not sure if its
cents (I think so) or frequency off the top of my head. The ratios
mean little to me at my level of tuning knowledge.

If it's cent-adjustments to 12-equal that you need, then, keeping C
as root tone (zero cents adjustment):
4:5:6 (just major triad) has E lowered 13.68c and G raised 1.96c
10:12:15 (just minor triad) has Eb raised 15.64c and G raised 1.96c
4:5:6:7 has E lowered 13.68c, G raised 1.96c, and Bb lowered 31.17c
60:70:84:105 has Eb lowered 33.13c, Gb lowered 17.49c, and Bb lowered
31.17c
6:7:9 has Eb lowered 33.13c and G raised 1.96c
14:18:21 has E raised 35.08c and G raised 1.96c
5:6:7:9 has Eb raised 15.64c, Gb lowered 17.49c, and Bb lowered 31.17c
70:90:105:126 has E raised 35.08c, G raised 1.96c, and Bb lowered
31.17c

--George

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > Chris, this is great, as far as it goes, especially since there
are
> > triangle waveforms that we can compare with the pure sine waves.
> > However, to compare the case of coincident vs. non-coincident
> > combinational tones the chords *must be in JI*, i.e., 4:5:6 vs.
> > 10:12:15 vs. 6:8:9.
>
> Why? I still hear the phenomenon. Is this more evidence that
> coincident combination tones can't explain it?
>
> -Carl

(Carl, I apologize for the delay in answering. There have been too
many messages to read in the limited amount of free time I've had
lately, and I still haven't gotten thru them all.)

What exactly is it that you're hearing?

What I hear in the 12-ET sine example is a relatively slow "beating"
(at a rate comparable to a vibrato) in the C major triad and a much
faster "beating" in the C minor triad. This "beating" was easily
heard at both a loud and moderate volume level.

To determine whether this perceived "beating" is due to differences
in the frequencies of first-order difference tones, I calculated the
following:

Difference between C-E and E-G difference tones in major triad: ~5.6
beats/second
Difference between C-Eb and Eb-G difference tones in minor triad:
~31.5 beats/second

These numbers are consistent with the beat rates that I heard.

Do you have another explanation for these observations?

--George

Hi George,

>>> However, to compare the case of coincident vs. non-coincident
>>> combinational tones the chords *must be in JI*, i.e., 4:5:6
>>> vs. 10:12:15 vs. 6:8:9.
>>
>> Why? I still hear the phenomenon. Is this more evidence that
>> coincident combination tones can't explain it?
>
> What exactly is it that you're hearing?

The thread started by someone observing that major and minor
triads are of a fundamentally different character. This
difference gets more extreme in the 7-limit, more extreme
still in the 11-limit, and so on. You proposed that
combination tones explain this, and I disagreed. Chris
synthesized the chords with sine tones, and I still heard the
difference when listening at low volume levels, when no
combination tones were present.

> What I hear in the 12-ET sine example is a relatively slow
> "beating"
//
> Do you have another explanation for these observations?

I didn't really hear any beating when I listened, but if
you do it could very well be from combination tones. How
good is your speaker system? Try turning down the volume?

But regardless of the beating, I assume you can still
recognize that one is a minor chord and one is a major
chord...

Hopefully we'll get a 7-limit demo soon.

-Carl

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

> The thread started by someone observing that major and minor
> triads are of a fundamentally different character. This
> difference gets more extreme in the 7-limit, more extreme
> still in the 11-limit, and so on. You proposed that
> combination tones explain this, and I disagreed. Chris
> synthesized the chords with sine tones, and I still heard the
> difference when listening at low volume levels, when no
> combination tones were present.
>
> > What I hear in the 12-ET sine example is a relatively slow
> > "beating"
> //
> > Do you have another explanation for these observations?
>
> I didn't really hear any beating when I listened, but if
> you do it could very well be from combination tones. How
> good is your speaker system? Try turning down the volume?
>
> But regardless of the beating, I assume you can still
> recognize that one is a minor chord and one is a major
> chord...
>
> Hopefully we'll get a 7-limit demo soon.
>
> -Carl
>

Hi,

I also hear, in the pure tone sample by Chris, the low-frequency
beating (loudness variation) in the major triad, and not on the minor
one, that however sounds more dissonant to me.

I guess the heard beating is also what Chris sees on "cool edit" as
an amplitude modulation in the waveform. As we have already agreed,
such beatings do not show up in Fourier spectra.

Since the beatings are also heard at low volume, they can't be due to
combination tones. Furthermore, combination tones would produce a
pitch, that could not be heard at least in the case of 12-ET major
triad, at 5 Hz or so.

But loudness modulations do exist, they are both heard, and visible
on the waveform plot. I have tried to understand why are they
generated. This goes along with my curiosity about why a major triad
becomes dissonant by a small variation of the third. All I say
concerns, as usual, pure tones only, no higher harmonics.

I have developed the sum of three cosine waves by prosthaphaeresis
formulae, by expressing

Cos(a)+Cos(b)+Cos(c) =

1/2(Cos(a)+Cos(b))+1/2(Cos(a)+Cos(c))+1/2(Cos(c)+Cos(b))=

Cos((a-b)/2)Cos((a+b)/2)+Cos((a-c)/2)Cos((a+c)/2)+Cos((c-b)/2)Cos
((c+b)/2).

I started with the simple case of a:b:c proportional to 4:5:6. Then I
shift b by a frequency deviation delta. I am not going into the maths
here, but as a result, I see that the loudness of the triad is
modulated by a frequency delta as well as by a frequency delta/2.

I have not extended the calculation for the case of minor triads but
I guess that the effect would be similar. I will check calculations
again but they seem reasonably simple and the results resemble what I
see on the waveform (I see the beatings with a "denser" band around
zero, that could be the beating at frequency delta, that should have
a smaller AM depth). Beatings at low frequency do not show up in
dyads, or let's say, their frequency is of the same order of
magnitude of the one of the mean tone or of each of the composing
tones; for instance, for 1:3/2 the beating would be at 1/2 and the
mean tone at 5/4 of course. But with triads, a kind of "beating
between beatings" can take place. If we move for instance the third
of the major triad as 5/4 + delta, the beating frequency
corresponding to the dyad 1:5/4+delta will be 1/4+delta, the one of
the dyad 5/4 + delta : 3/2 will be 1/4 - delta, the beating of the
dyad 1:3/2 is still 1/2. If we beat the first and the second, for
instance, we get a difference of 2 delta. This is just as a
simplification, the actual effect on the wave is seen from the
complete result of calculations.

In the case of Chris' samples made in equal temperament, the
difference of fifth and third from the just intonation ones accounts
for the heard low-frequency beat of the major triad. Perhaps, as from
my starting guess already discussed with Carl, in the case that
beatings become too fast and may be heard as a "strange" type of
tone, this could also account for the slight dissonance effect of the
minor triad; this explanation is not yet satisfying to me and I would
like to explore this point further.

However, I would like to go on. Of course, beatings cannot fully
account for consonance/dissonance, but may as well be part of it. As
for the example suggested by Carl, I have tested by generating the
waveforms by Mathemathica and playing them, so it is not reliable and
we are waiting for Chris' new sample. I had two different impressions
in the case of 4:5:6:7 and 1/(4:5:6:7) and in the case 4:5:6:7:9:11
and 1/(4:5:6:7:9:11), but to comment about this I would like to hear
good quality sounds. However, I guess that Carl would like to show us
that the difference tone coincidence (that one gets by taking equally-
frequency-spaced tones in the chord, like the ones obtainable from
the harmonic series) does not tell much, so in his example both
chords have "well synchronized difference tones" but nevertheless one
case is much more consonant than the other one, so "difference tones
(as well as coincidence of partials itself) are not all".

Finally, about the central topic of this discussion, just one
comment. It has been explained that the most consonant chords fit the
harmonic series at smallest terms. In this respect, the minor chord
10:12:15 has rather high indices. The minor triad 6:7:9, that I found
by optimizing beating ratios (see my previous post about Beatings and
Consonance of triads) is placed lower on the harmonic series and
maintains the fifth ratio; however, it is to be checked whether it
will result more or less consonant than the 10:12:15 minor triad
(indeed, I also asked Chris to play also such triad in pure tones,
just to check that).

I hope this will stimulate further discussion. Meanwhile, thanks a
lot to you and the rest of the list for the precious explanations and
debate.

Max

> I have developed the sum of three cosine waves by prosthaphaeresis
> formulae, by expressing
>
> Cos(a)+Cos(b)+Cos(c) =
>
> 1/2(Cos(a)+Cos(b))+1/2(Cos(a)+Cos(c))+1/2(Cos(c)+Cos(b))=
>
> Cos((a-b)/2)Cos((a+b)/2)+Cos((a-c)/2)Cos((a+c)/2)+Cos((c-b)/2)Cos
> ((c+b)/2).
>
> I started with the simple case of a:b:c proportional to 4:5:6. Then I
> shift b by a frequency deviation delta. I am not going into the maths
> here, but as a result, I see that the loudness of the triad is
> modulated by a frequency delta as well as by a frequency delta/2.

What was the final expression that you got when delta was thrown in?
I'm interested to see this. Previously the existence of beating was
written off as a failure of the basilar membrane to discern between
fine pitch differences, but I've long suspected there's more to it
than that. By that logic, if the basilar membrane had infinitely fine
frequency resolution, then we'd never hear any changes in the sound
over time - we'd perceive all of sound as one static, unchanging sound
throughout our lifetime! I suspect there is some neurological
mechanism to explain why the brain sometimes "flips" its perspective
on the incoming signal to hear it as a modulation envelope over a tone
vs. different tones - something beyond critical band effects.

> In the case of Chris' samples made in equal temperament, the
> difference of fifth and third from the just intonation ones accounts
> for the heard low-frequency beat of the major triad. Perhaps, as from
> my starting guess already discussed with Carl, in the case that
> beatings become too fast and may be heard as a "strange" type of
> tone, this could also account for the slight dissonance effect of the
> minor triad; this explanation is not yet satisfying to me and I would
> like to explore this point further.

I made this point earlier as well. If you take 5/4 and you widen it
slowly, you'll hear the beating start. Keep widening it until you get
to 9/7. At this point the 5/4 will sound uncomfortably sharp, and the
beating will have gotten too fast to hear, as you put it... But if you
stay on 9/7 for a second, you might experience a sudden perception
shift whereas you suddenly realize that the "beating" is in a regular
pattern, and then you'll suddenly become aware of the different
fundamental and hear it as 9/7.

> However, I would like to go on. Of course, beatings cannot fully
> account for consonance/dissonance, but may as well be part of it. As
> for the example suggested by Carl, I have tested by generating the
> waveforms by Mathemathica and playing them, so it is not reliable and
> we are waiting for Chris' new sample.

What sample are you guys waiting for? 7 and 11 limit major vs minor
chords? I have MATLAB going over here, I could whip something up.

-Mike

Max wrote:

> However, I would like to go on. Of course, beatings cannot fully
> account for consonance/dissonance, but may as well be part of it.
> As for the example suggested by Carl, I have tested by generating
> the waveforms by Mathemathica and playing them, so it is not
> reliable and we are waiting for Chris' new sample. I had two
> different impressions in the case of 4:5:6:7 and 1/(4:5:6:7) and
> in the case 4:5:6:7:9:11 and 1/(4:5:6:7:9:11), but to comment
> about this I would like to hear good quality sounds.

Strange, I would think mathematica would be the ideal tool
for creating these sounds, with the highest quality, as they
can be perfect to the sample and written directly to disk.

> Finally, about the central topic of this discussion, just one
> comment. It has been explained that the most consonant chords
> fit the harmonic series at smallest terms. In this respect,
> the minor chord 10:12:15 has rather high indices. The minor
> triad 6:7:9, that I found by optimizing beating ratios (see my
> previous post about Beatings and Consonance of triads) is placed
> lower on the harmonic series and maintains the fifth ratio;
> however, it is to be checked whether it will result more or
> less consonant than the 10:12:15 minor triad (indeed, I also
> asked Chris to play also such triad in pure tones, just to check
> that).

Having used 6:7:9 chords on several instruments over the years,
including several pianos, my own impression is that it is
indeed more consonant, in a sense, than either 10:12:15 or
16:19:24. However, there are two things against it:

1. 7/6 is getting close to the critical band, and even in the
middle of the piano, we hear some beating between its partials.
As mentioned before, this beating is nicely synchronous, but
it lends the interval to a somewhat 'tense' feeling. This can
be a good thing for minor chords however. :)

2. The chord is, in fact, low enough in the harmonic series
that the implied fundamental is likely to be 4 (or 2 or 1
depending on the voicing). Compare my analysis of 10:12:15,
where almost always 10 or 5 will be heard as the root, with
the 12 being dismissed by the hearing system as unrecoverable
(unless you listen to 15-chords all the time, as Mike seems
to do, when you may hear 4 or 8 for the root. :)

In summary, 6:7:9 is an excellent minor chord and can be
used even in performances of existing classical music. But
overall I think either 10:12:15 or 16:19:24 are to be
preferred, because of their greater tendency to evoke a
fundamental that is the same as the root (lowest tone, or
octave extension of the lowest tone), in the chord.

-Carl

> 2. The chord is, in fact, low enough in the harmonic series
> that the implied fundamental is likely to be 4 (or 2 or 1
> depending on the voicing). Compare my analysis of 10:12:15,
> where almost always 10 or 5 will be heard as the root, with
> the 12 being dismissed by the hearing system as unrecoverable
> (unless you listen to 15-chords all the time, as Mike seems
> to do, when you may hear 4 or 8 for the root. :)

I'm slowly getting tired of them. I've been taking Debussy's reverie
and retuning it to 72tet, which, while not JI, is close enough for me
to get the job done here. Not only are there comma pumps all over the
place (vi-ii-V-I is particularly nasty, as is iim7-V7-I), but the
10:12:15 minor chords don't always "fit" - sometimes it seems like
16:19:24 is better. Same with major chords - sometimes a 12-tet major
chord seems to sound better than 4:5:6, and I'm not sure why. Maybe if
it functions as the III chord of a 16:19:24 minor or something.

I gave 10:12:15 a better listen - that 4/8 fundamental is there, but
indeed 5/10 stands out as the "root". To me at least, I hear the root
as 5/10, but that 4/8 is indeed there, but lower - I sort of feel like
it's partially responsible for the "quality" of the chord. I don't
think it's completely dismissed by the hearing system as
unrecoverable, as 11/9 is if you play it low enough. That one I feel
is completely thrown out the window.

BTW, what listening examples are people still waiting for? 11-limit
major vs 11-limit minor in sines? I can whip that up in MATLAB here.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi George,
>
> >>> However, to compare the case of coincident vs. non-coincident
> >>> combinational tones the chords *must be in JI*, i.e., 4:5:6
> >>> vs. 10:12:15 vs. 6:8:9.
> >>
> >> Why? I still hear the phenomenon. Is this more evidence that
> >> coincident combination tones can't explain it?
> >
> > What exactly is it that you're hearing?
>
> The thread started by someone observing that major and minor
> triads are of a fundamentally different character. This
> difference gets more extreme in the 7-limit, more extreme
> still in the 11-limit, and so on. You proposed that
> combination tones explain this, and I disagreed. Chris
> synthesized the chords with sine tones, and I still heard the
> difference when listening at low volume levels, when no
> combination tones were present.
>
> > What I hear in the 12-ET sine example is a relatively slow
> > "beating"
> //
> > Do you have another explanation for these observations?
>
> I didn't really hear any beating when I listened, but if
> you do it could very well be from combination tones. How
> good is your speaker system? Try turning down the volume?
>
> But regardless of the beating, I assume you can still
> recognize that one is a minor chord and one is a major
> chord...
>
> Hopefully we'll get a 7-limit demo soon.
>
> -Carl

"Let's be absolutely clear on what we mean by "frequency" here: 1 Hz is
the frequency of the 3:7:11:15:19 waveform. However, no 1 Hz SINE WAVE
is present anywhere in that signal. When people refer to hearing a
certain "frequency", what they're really talking about is hearing a
sine wave with that frequency. The brain, in certain circumstances,
manages to artificially create that 1Hz frequency so that it DOES seem
like we hear a sine wave there."
I do understand that you're speaking about only a sine wave
corresponding to the fundamental. It is that assumption that "sine
wave = frequency = fundamental" that I find reductionist. However,
someone did answer my question that it can be heard if the harmonic
"circumstances" are in the range of human hearing, which is what I
wanted to know. But even more to my point, I suspect that that GCD
fundamental DOES make its presence felt in the very concept of KEY, so
that there is nothing "artificial" about it at all, and in fact it
might provide the vital clue to understanding musical harmony. The
danger is that we experiment with chords out of musical context, only
considering notes appearing simultaneously. How then do we "model"
apeggios or entire orchestral pieces of music? (Leonard Bernstein
makes the point that the reason why the teasing song "ner ner na ner
ner", you know from major 3 to 5th, is teasing is because we never get
to hear the tonic).
>EG: Take the overtones from one note from 2 upwards, say an E note as
5 from C = 1, giving 10, 15, etc... Then take the overtones from 2
upwards from a G note as 6 from C, giving 12,18, and so on. What is
interesting is that the pairs of 2nd, then 3rd etc... harmonics will
all give GCD of the overtones of the fundamental C. In other words,
each of these overtones is itself a GCD: 10 and 12 have 2, 15 and 18
have 3, etc...And each of these GCD are overtones of the "artificial"
frequency C = 1, which is itself the "greatest" GCD of them all. Now
you're trying to tell me that this C is not present because no sine
wave corresponds to it?
-Rick

[ Attachment content not displayed ]

Max wrote:

> Since the beatings are also heard at low volume, they can't be due to
> combination tones. Furthermore, combination tones would produce a
> pitch, that could not be heard at least in the case of 12-ET major
> triad, at 5 Hz or so.

If you are, as you say, using only pure sine or cosine waves, then I'm not sure what kind of "beating" you mean. Generally, what's most often called "beats" in tempered intervals is when some harmonics of one tone start clashing with different harmonics of another tone, which can't happen with sine waves that have no harmonics at all. For example, if I play two pitches simultaneously, one of 200Hz and another of 301Hz, then there will be 2 beats per second because the 3rd harmonic of the lower tone is 600Hz and the 2nd harmonic of the higher tone is 602Hz -- i.e. the guide tone (which would be 600Hz if the frequencies were 200 and 300) turned into two close frequencies, so close that we can hear beats.

> I have developed the sum of three cosine waves by prosthaphaeresis
> formulae, by expressing
>
> Cos(a)+Cos(b)+Cos(c) =
>
> 1/2(Cos(a)+Cos(b))+1/2(Cos(a)+Cos(c))+1/2(Cos(c)+Cos(b))=
>
> Cos((a-b)/2)Cos((a+b)/2)+Cos((a-c)/2)Cos((a+c)/2)+Cos((c-b)/2)Cos
> ((c+b)/2).

Okay, I think you're on a good track. But summing cosines is useful for things like filtering, phase shifting, or making amplitude envelopes for sounds that are to be interpolated to prevent pops and clicks. But to prove the trigonometric identity that you were describing, it's usually better to use sines which, for one thing, have their starting point at 0 and, for another thing, can make distinction between positive and negative frequencies. So I would probably do it like this:
sin(a)+sin(b)+sin(c) =
1/2(sin(a)+sin(b))+1/2(sin(a)+sin(c))+1/2(sin(c)+sin(b))=
sin((a+b)/2)*cos((a-b)/2) + sin((a+c)/2)*cos((a-c)/2) + sin((b+c)/2)*cos((b-c)/2)

> I started with the simple case of a:b:c proportional to 4:5:6. Then I
> shift b by a frequency deviation delta. I am not going into the maths
> here, but as a result, I see that the loudness of the triad is
> modulated by a frequency delta as well as by a frequency delta/2.

I'm afraid you couldn't argue with this. If you také two values of x and y, then the sum and difference values are x+y and x-y. If you make sum and difference values of that, you get 2x and 2y. AFAIK, this is true for any number.

If you are talking about the amplitude modulation itself, I don't think you could argue with that either, because the actual periodicity occurs at a frequency of "delta", or at a time of, let's say, "delta-t", and the length of the period by which the amplitude modulation is done to make this happen is just an integer multiple of this "delta-t" -- i.e. you could say, for example, that mixing two sine waves of 400Hz and 500Hz is the same as multipliing a 450Hz sine with a 50Hz cosine, but the phase patterns in such a sound repeat exactly the same every 10 miliseconds (i.e. every 100th of a second), and the time of 20 miliseconds (which is 1/50 of a second) is just an integer multiple of that, so I see no point in using that as a reference here.

> I have not extended the calculation for the case of minor triads but
> I guess that the effect would be similar. I will check calculations
> again but they seem reasonably simple and the results resemble what I
> see on the waveform (I see the beatings with a "denser" band around
> zero, that could be the beating at frequency delta, that should have
> a smaller AM depth). Beatings at low frequency do not show up in
> dyads, or let's say, their frequency is of the same order of
> magnitude of the one of the mean tone or of each of the composing
> tones; for instance, for 1:3/2 the beating would be at 1/2 and the
> mean tone at 5/4 of course.

First of all, why say "1:3/2" when you can simply say "3/2"? And then, you're probably not using the right terminology -- what you describe is not beating.

> But with triads, a kind of "beating
> between beatings" can take place. If we move for instance the third
> of the major triad as 5/4 + delta, the beating frequency
> corresponding to the dyad 1:5/4+delta will be 1/4+delta, the one of
> the dyad 5/4 + delta : 3/2 will be 1/4 - delta, the beating of the
> dyad 1:3/2 is still 1/2. If we beat the first and the second, for
> instance, we get a difference of 2 delta. This is just as a
> simplification, the actual effect on the wave is seen from the
> complete result of calculations.

These are difference tones, not beats.

Petr

Petr, with all due respect, why would the sine in a harmonic act differently than the sine in the fundamental? It seems to me that the physics would be the same. And I did see beating in chords made of pure sines.

Chris

If you are, as you say, using only pure sine or cosine waves, then I’m not sure what kind of „beating“ you mean. Generally, what’s most often called „beats“ in tempered intervals is when some harmonics of one tone start clashing with different harmonics of another tone, which can’t happen with sine waves that have no harmonics at all. For example, if I play two pitches simultaneously, one of 200Hz and another of 301Hz, then there will be 2 beats per second because the 3rd harmonic of the lower tone is 600Hz and the 2nd harmonic of the higher tone is 602Hz -- i.e. the guide tone (which would be 600Hz if the frequencies were 200 and 300) turned into two close frequencies, so close that we can hear beats
Sent via BlackBerry from T-Mobile

Chris wrote:

> Petr, with all due respect, why would the sine in a harmonic act > differently
> than the sine in the fundamental? It seems to me that the physics would > be the same.

I'm not sure what you mean by "the sine in the fundamental". Of course, if you mix sines of 600Hz and 601Hz, you CAN get some beats; but remember that this is a rather small interval. Generally, I've always seen the term "beats" used in situations where the intervals were smaller than about a quarter-tone or whatever. A minor second can have perceivable difference tones as well, but i wouldn't think of calling them "beats".

Petr

I think solution is more simple:

- difference tones - there is clearly written "tone" here, and tone
must be a periodic vibration we can hear (directly from the source,
or from high quality amplifier/speaker). For such low border usually
20 Hz is used, but I have found 16 Hz somewhere, too. Usually
frequency range of high quality PA or generally electronic audio system is written as 20 Hz - 20 kHz.

- everything lower then this border is percepted as beats as we can
hear it as individual peaks. Sometimes it's referred as infrasound.

Daniel Forro

On 14 Jan 2009, at 9:44 PM, Petr Parízek wrote:

> Chris wrote:
>
>> Petr, with all due respect, why would the sine in a harmonic act
>> differently
>> than the sine in the fundamental? It seems to me that the physics
>> would
>> be the same.
>
> I'm not sure what you mean by "the sine in the fundamental". Of
> course, if
> you mix sines of 600Hz and 601Hz, you CAN get some beats; but
> remember that
> this is a rather small interval. Generally, I've always seen the term
> "beats" used in situations where the intervals were smaller than
> about a
> quarter-tone or whatever. A minor second can have perceivable
> difference
> tones as well, but i wouldn't think of calling them "beats".
>
> Petr

I wrote:

> Generally, I've always seen the term
> "beats" used in situations where the intervals were smaller than about a
> quarter-tone or whatever.

Sometimes even that is too much. Take the example of 8/5 and compare 13/8 to that. The distance between these two is 65/64 (which you can hear on your own by mixing a relative frequency of 40 with either 65 or 64), which is just about 27 cents in size; yet you won't hear one as a variation of the other because the entire characteristics of each dyad will differ substantially. For example, take a 400Hz tone and add 640Hz to it in one version and 650Hz in the other. Don't know about you, but the first change I would realize in that case would be the GCD (or fundamental) frequency change from 80 in the first version to 50 in the second one. And if the tones were pure sines, next I would probably hear the difference tone change from 240 to 250 and the lower second difference tone (i.e. 2*lf - hf) change from 160 to 150. So you see, two completely different sounds -- the fundamental frequency has changed, the difference tones compared to each other make a perfect fifth and then a major sixth, ... Well, just a clear change.

And for another thing, to a large extent, I agree with DF that difference tones in the range of our hearing are generally not considered "beats".

Petr

What makes a "beat" different from a "difference tone"? with respect the only difference I see between the two is the frequency range in which they occur. Thanks Chris
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-----Original Message-----
From: Petr Parízek <p.parizek@chello.cz>

Date: Wed, 14 Jan 2009 13:44:57
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Chris wrote:

> Petr, with all due respect, why would the sine in a harmonic act
> differently
> than the sine in the fundamental? It seems to me that the physics would
> be the same.

I'm not sure what you mean by "the sine in the fundamental". Of course, if
you mix sines of 600Hz and 601Hz, you CAN get some beats; but remember that
this is a rather small interval. Generally, I've always seen the term
"beats" used in situations where the intervals were smaller than about a
quarter-tone or whatever. A minor second can have perceivable difference
tones as well, but i wouldn't think of calling them "beats".

Petr

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Chris wrote:

> What makes a "beat" different from a "difference tone"? with respect the > only difference I see between the two is the frequency range in which they > occur.

As Daniel has pointed out, sometimes it is the frequency of the difference tone itself, which we may perceive as beats if it lies outside our hearing range. In some other conditions, it may have to do with the actual (exponential) interval size and with the general characteristics of the intervals; and in some of these cases, even a deviation as small as 27 cents can remarcably change them (see my last post).

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Chris wrote:
>
> > What makes a "beat" different from a "difference tone"? with
respect the
> > only difference I see between the two is the frequency range in
which they
> > occur.
>
> As Daniel has pointed out, sometimes it is the frequency of the
difference
> tone itself, which we may perceive as beats if it lies outside our
hearing
> range. In some other conditions, it may have to do with the actual
> (exponential) interval size and with the general characteristics
of the
> intervals; and in some of these cases, even a deviation as small
as 27 cents
> can remarcably change them (see my last post).
>
> Petr
>

To my understanding, difference tone is a tone, i.e. periodical
changes of air pressure. For instance, combinational tones (due only
to some nonlinear effect involved in production, transmission or
perception of sound) of two frequencies "a" and "b" are frequencies
a-b and a+b, and they are tones as well. With combinational tones
you actually see the two new frequencies to appear in the spectrum,
a difference tone and a sum tone, and their intensity relative to
the source tones depends on volume.

Instead, beatings are periodical changes of the loudness of some
other (typically faster) tone. Beating is a linear effect due to
wave interference. It does not introduce new frequencies in the
signal, that is, if you make a Fourier spectrum you don't see
additional frequencies.

In my view, beatings are always beatings (linear wave interference)
regardless their frequency. The fact that the ear cannot follow too
rapid beatings does not change the nature of the stimulus. In other
words, can a modulation of the amplitude of a pressure wave (that
has itself a constant average pressure, that is the atmospheric
pressure) turn into a time-dependent average pressure at the
frequency of the beating? I think not, but this topic is actually
what I am trying to understand.

Let's put it in different terms. Take two ultrasonic waves with an
audible frequency difference (say 30000 Hz and 30100 Hz). We should
be able to hear 100 Hz (we know it also from Tartini's third sound).
But we hear that sound only because of nonlinear interaction inside
our ear. If you lower the overall volume such effect should
disappear.

So if you took a (hypothetical) guitar with two strings tuned at
30000 and 30001 Hz, you do not hear any beating at 1 Hz, because you
do not hear the frequency at 30000 Hz and therefore you do not hear
its amplitude modulation! You can say the same with a detuning of
100 Hz, that is in the audible range: if your guitar strings are
tuned at 30000 and 30100 Hz, you do not hear any beating (although
this time it is fast and within the audible range) just because you
can't hear the frequency at 30000 Hz. Or do you? (note: if the
volume is increased enough, as in the previous example of third
sound, combinational tones appear, but in this case we are not
talking about beatings, but of "real" combinational sounds, that you
can see in your Fourier spectra, regardless it is made by a spectrum
analyzer or by your brain)

So, I repute the distinction between pressure (instantaneous)
stimulus and loudness (amplitude of pressure wave) crucial for a
better understanding. I can hear very well 1 Hz beating of two
slightly detuned guitar strings while I try to tune it, but for sure
my ear is not able to hear any 1 Hz (sinewave, pressure change...)
tone.

Max

beating is a change of loudness, while the

Max wrote:

> To my understanding, difference tone is a tone, i.e. periodical
> changes of air pressure. For instance, combinational tones (due only
> to some nonlinear effect involved in production, transmission or
> perception of sound) of two frequencies "a" and "b" are frequencies
> a-b and a+b, and they are tones as well. With combinational tones
> you actually see the two new frequencies to appear in the spectrum,
> a difference tone and a sum tone, and their intensity relative to
> the source tones depends on volume.

Then you are not talking about combination tones. Combination tones do not appear themselves in the spectrum and you don't see them in FFT analysis. If they, for any reason, eventually do appear in the spectrum, then they are not combination tones anymore.

Petr

> Then you are not talking about combination tones. Combination
tones do not appear themselves in the spectrum and you don't see
them in FFT analysis. If they, for any reason, eventually do appear
in the spectrum, then they are not combination tones anymore.
>
> Petr
>

Exactly Petr. Here is the excerpt from my post about "beating
beats"....

"Since the beatings are also heard at low volume, they can't be due
to combination tones. Furthermore, combination tones would produce a
pitch, that could not be heard at least in the case of 12-ET major
triad, at 5 Hz or so."

Max

Sorry Petr, perhaps I misunderstood. I am talking about the FFT
spectrum of the resulting signal, after nonlinear combination. In
such a case combination tones appear in the spectrum.

Instead, I meant that FFT spectrum of a couple of pure tones (low
volume, no nonlinear interactions) is just made up of such two
tones, even if you hear a clear beating (loudness modulation) as in
the case of a guitar being tuned.

I understand this is a delicate point, I am doing also tests with
FFTs to clarify that.....

Max

--- In tuning@yahoogroups.com, "massimilianolabardi" <labardi@...>
wrote:
>
>
> > Then you are not talking about combination tones. Combination
> tones do not appear themselves in the spectrum and you don't see
> them in FFT analysis. If they, for any reason, eventually do
appear
> in the spectrum, then they are not combination tones anymore.
> >
> > Petr
> >
>
> Exactly Petr. Here is the excerpt from my post about "beating
> beats"....
>
> "Since the beatings are also heard at low volume, they can't be
due
> to combination tones. Furthermore, combination tones would produce
a
> pitch, that could not be heard at least in the case of 12-ET major
> triad, at 5 Hz or so."
>
> Max
>

Max wrote:

> Sorry Petr, perhaps I misunderstood. I am talking about the FFT
> spectrum of the resulting signal, after nonlinear combination. In
> such a case combination tones appear in the spectrum.

And from then on, they can no longer be called combination tones.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Max wrote:
>
> > Sorry Petr, perhaps I misunderstood. I am talking about the FFT
> > spectrum of the resulting signal, after nonlinear combination. In
> > such a case combination tones appear in the spectrum.
>
> And from then on, they can no longer be called combination tones.
>
> Petr
>

Ok. So let's call them "tones that arise from a nonlinear combination
of two pure tones", What is the correct terminology for those?

Thanks

Max

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> What makes a "beat" different from a "difference tone"? with
> respect the only difference I see between the two is the frequency
> range in which they occur. Thanks Chris

They're completely different phenomena. Beating is amplitude
modulation caused by interference, and difference tones are
nonlinear distortion products. As I thought everyone agreed
by now, beating frequencies are not tones, but difference
tones are. -Carl

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

> As Daniel has pointed out, sometimes it is the frequency of the
> difference tone itself, which we may perceive as beats if it
> lies outside our hearing range.

Incorrect! I'm not sure what else to say about this... -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@> wrote:
>
> > As Daniel has pointed out, sometimes it is the frequency of the
> > difference tone itself, which we may perceive as beats if it
> > lies outside our hearing range.
>
> Incorrect! I'm not sure what else to say about this... -Carl

I guess the simplest demonstration is to play a pair of
converging sine tones softly. There will be no difference
tones, but roughness and then beating will appear when
they get within the critical band.

-Carl

Max wrote:

> "Since the beatings are also heard at low volume, they can't be due
> to combination tones."

You read my mind! -C.

To play the devil's advocate - I thought what you are calling difference tones was the result of ring modulation. My point is pressure waves will or will not interact without regard to frequencies. However the ear's interpetation has nothing to do with that pont.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Wed, 14 Jan 2009 16:35:51
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> What makes a "beat" different from a "difference tone"? with
> respect the only difference I see between the two is the frequency
> range in which they occur. Thanks Chris

They're completely different phenomena. Beating is amplitude
modulation caused by interference, and difference tones are
nonlinear distortion products. As I thought everyone agreed
by now, beating frequencies are not tones, but difference
tones are. -Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> To play the devil's advocate - I thought what you are calling
> difference tones was the result of ring modulation.

Other way around - beating is the ring modulation one. -Carl

It doesn't seem to be what everyone is describing. On wikipedia ring modulation results in f1+f2 and f2-f1. These seem to be the difference tones as describe many times in this thread. I don't think that can happen without some sort of signal processing even if it is the ear or mind processing. Beats are flat out constructive and destructive interference. And I think that is all that can hjappen "in air" as it were.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@...>

Date: Wed, 14 Jan 2009 18:14:10
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> To play the devil's advocate - I thought what you are calling
> difference tones was the result of ring modulation.

Other way around - beating is the ring modulation one. -Carl

If you take a 440 Hz sine wave, and you ring modulate it with a
frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
whole thing. Fourier analysis of the resultant signal will show
frequencies at 440.5 Hz and 439.5 Hz.

-Mike

On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@...> wrote:
> It doesn't seem to be what everyone is describing. On wikipedia ring
> modulation results in f1+f2 and f2-f1. These seem to be the difference tones
> as describe many times in this thread. I don't think that can happen without
> some sort of signal processing even if it is the ear or mind processing.
> Beats are flat out constructive and destructive interference. And I think
> that is all that can hjappen "in air" as it were.
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: "Carl Lumma"
> Date: Wed, 14 Jan 2009 18:14:10 -0000
> To: <tuning@yahoogroups.com>
> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>
>> To play the devil's advocate - I thought what you are calling
>> difference tones was the result of ring modulation.
>
> Other way around - beating is the ring modulation one. -Carl
>
>

Or would it be at 441 Hz and 439 Hz? The convolution theorem would
seem to dictate that, but now I'm confused as to why the beating isn't
2 Hz then, but rather 1 Hz.
-Mike

On Wed, Jan 14, 2009 at 1:41 PM, Mike Battaglia <battaglia01@...> wrote:
> If you take a 440 Hz sine wave, and you ring modulate it with a
> frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
> whole thing. Fourier analysis of the resultant signal will show
> frequencies at 440.5 Hz and 439.5 Hz.
>
> -Mike
>
>
>
> On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@...> wrote:
>> It doesn't seem to be what everyone is describing. On wikipedia ring
>> modulation results in f1+f2 and f2-f1. These seem to be the difference tones
>> as describe many times in this thread. I don't think that can happen without
>> some sort of signal processing even if it is the ear or mind processing.
>> Beats are flat out constructive and destructive interference. And I think
>> that is all that can hjappen "in air" as it were.
>>
>> Sent via BlackBerry from T-Mobile
>>
>> ________________________________
>> From: "Carl Lumma"
>> Date: Wed, 14 Jan 2009 18:14:10 -0000
>> To: <tuning@yahoogroups.com>
>> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
>> triads,Re:Beatings vs Intermodulation
>>
>> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>>
>>> To play the devil's advocate - I thought what you are calling
>>> difference tones was the result of ring modulation.
>>
>> Other way around - beating is the ring modulation one. -Carl
>>
>>
>

OK, I see - it WOULD be 441 and 439 Hz. The "difference" of the two
there is 2 Hz. If you treat 1 "beat" as an journey from soft to high
volume and back to soft, then you do indeed hear two of those per
second, but every other beat here is phase inverted, so one whole
cycle from start to finish would incorporate two "beats".
-Mike

On Wed, Jan 14, 2009 at 1:49 PM, Mike Battaglia <battaglia01@...> wrote:
> Or would it be at 441 Hz and 439 Hz? The convolution theorem would
> seem to dictate that, but now I'm confused as to why the beating isn't
> 2 Hz then, but rather 1 Hz.
> -Mike
>
>
>
> On Wed, Jan 14, 2009 at 1:41 PM, Mike Battaglia <battaglia01@...> wrote:
>> If you take a 440 Hz sine wave, and you ring modulate it with a
>> frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
>> whole thing. Fourier analysis of the resultant signal will show
>> frequencies at 440.5 Hz and 439.5 Hz.
>>
>> -Mike
>>
>>
>>
>> On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@...> wrote:
>>> It doesn't seem to be what everyone is describing. On wikipedia ring
>>> modulation results in f1+f2 and f2-f1. These seem to be the difference tones
>>> as describe many times in this thread. I don't think that can happen without
>>> some sort of signal processing even if it is the ear or mind processing.
>>> Beats are flat out constructive and destructive interference. And I think
>>> that is all that can hjappen "in air" as it were.
>>>
>>> Sent via BlackBerry from T-Mobile
>>>
>>> ________________________________
>>> From: "Carl Lumma"
>>> Date: Wed, 14 Jan 2009 18:14:10 -0000
>>> To: <tuning@yahoogroups.com>
>>> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
>>> triads,Re:Beatings vs Intermodulation
>>>
>>> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>>>
>>>> To play the devil's advocate - I thought what you are calling
>>>> difference tones was the result of ring modulation.
>>>
>>> Other way around - beating is the ring modulation one. -Carl
>>>
>>>
>>
>

I respectfully disagree. The formula says the result would be 441 hz and 439 hz sine waves. This would not be beating.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Mike Battaglia" <battaglia01@gmail.com>

Date: Wed, 14 Jan 2009 13:41:28
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

If you take a 440 Hz sine wave, and you ring modulate it with a
frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
whole thing. Fourier analysis of the resultant signal will show
frequencies at 440.5 Hz and 439.5 Hz.

-Mike

On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@gmail.com> wrote:
> It doesn't seem to be what everyone is describing. On wikipedia ring
> modulation results in f1+f2 and f2-f1. These seem to be the difference tones
> as describe many times in this thread. I don't think that can happen without
> some sort of signal processing even if it is the ear or mind processing.
> Beats are flat out constructive and destructive interference. And I think
> that is all that can hjappen "in air" as it were.
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: "Carl Lumma"
> Date: Wed, 14 Jan 2009 18:14:10 -0000
> To: <tuning@yahoogroups.com>
> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>
>> To play the devil's advocate - I thought what you are calling
>> difference tones was the result of ring modulation.
>
> Other way around - beating is the ring modulation one. -Carl
>
>

Yeah, I screwed up on the math - see above. The point I was making is
that this would, in fact, be beating.

-Mike

On Wed, Jan 14, 2009 at 2:07 PM, <chrisvaisvil@...> wrote:
> I respectfully disagree. The formula says the result would be 441 hz and 439
> hz sine waves. This would not be beating.
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: "Mike Battaglia"
> Date: Wed, 14 Jan 2009 13:41:28 -0500
> To: <tuning@yahoogroups.com>
> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> If you take a 440 Hz sine wave, and you ring modulate it with a
> frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
> whole thing. Fourier analysis of the resultant signal will show
> frequencies at 440.5 Hz and 439.5 Hz.
>
> -Mike
>
> On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@...> wrote:
>> It doesn't seem to be what everyone is describing. On wikipedia ring
>> modulation results in f1+f2 and f2-f1. These seem to be the difference
>> tones
>> as describe many times in this thread. I don't think that can happen
>> without
>> some sort of signal processing even if it is the ear or mind processing.
>> Beats are flat out constructive and destructive interference. And I think
>> that is all that can hjappen "in air" as it were.
>>
>> Sent via BlackBerry from T-Mobile
>>
>> ________________________________
>> From: "Carl Lumma"
>> Date: Wed, 14 Jan 2009 18:14:10 -0000
>> To: <tuning@yahoogroups.com>
>> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
>> triads,Re:Beatings vs Intermodulation
>>
>> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>>
>>> To play the devil's advocate - I thought what you are calling
>>> difference tones was the result of ring modulation.
>>
>> Other way around - beating is the ring modulation one. -Carl
>>
>>
>
>

It is not beating, it is ring modulation. The constructive destructive interference that occurs "in air" upon the resulting two sine waves is incidentail and has nothing to do with ring modulation. What would happen if you ring modulated 100 and 200 hz sine waves? Would you get what you term beating?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Mike Battaglia" <battaglia01@gmail.com>

Date: Wed, 14 Jan 2009 14:08:29
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Yeah, I screwed up on the math - see above. The point I was making is
that this would, in fact, be beating.

-Mike

On Wed, Jan 14, 2009 at 2:07 PM, <chrisvaisvil@gmail.com> wrote:
> I respectfully disagree. The formula says the result would be 441 hz and 439
> hz sine waves. This would not be beating.
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: "Mike Battaglia"
> Date: Wed, 14 Jan 2009 13:41:28 -0500
> To: <tuning@yahoogroups.com>
> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> If you take a 440 Hz sine wave, and you ring modulate it with a
> frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
> whole thing. Fourier analysis of the resultant signal will show
> frequencies at 440.5 Hz and 439.5 Hz.
>
> -Mike
>
> On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@gmail.com> wrote:
>> It doesn't seem to be what everyone is describing. On wikipedia ring
>> modulation results in f1+f2 and f2-f1. These seem to be the difference
>> tones
>> as describe many times in this thread. I don't think that can happen
>> without
>> some sort of signal processing even if it is the ear or mind processing.
>> Beats are flat out constructive and destructive interference. And I think
>> that is all that can hjappen "in air" as it were.
>>
>> Sent via BlackBerry from T-Mobile
>>
>> ________________________________
>> From: "Carl Lumma"
>> Date: Wed, 14 Jan 2009 18:14:10 -0000
>> To: <tuning@yahoogroups.com>
>> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
>> triads,Re:Beatings vs Intermodulation
>>
>> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>>
>>> To play the devil's advocate - I thought what you are calling
>>> difference tones was the result of ring modulation.
>>
>> Other way around - beating is the ring modulation one. -Carl
>>
>>
>
>

Let me ask you a question: ring modulating 440 Hz by 1 Hz gives you
frequencies of 441 Hz and 439 Hz. As you might expect, the resultant
sound sounds like a 440 Hz tone whose volume is being modulated in a
sinusoidal pattern.

What happens if you just take 441 Hz and 439 Hz sine waves and add
them together? You hear a 440 Hz tone whose volume seems to modulate
in a sinusoidal pattern.

What's the difference? The frequency spectra are the same, the time
domain waveforms are the same... What is different between them other
than a choice of perspective?

The beating would become so fast that it would stop sounding like
beating and the whole thing would start sounding like a complex tone.
What would happen if you played 430 Hz and 450 Hz together? Would it
sound like fast beating? How about 435 Hz and
445 Hz?

-Mike

On Wed, Jan 14, 2009 at 2:13 PM, <chrisvaisvil@...> wrote:
> It is not beating, it is ring modulation. The constructive destructive
> interference that occurs "in air" upon the resulting two sine waves is
> incidentail and has nothing to do with ring modulation. What would happen if
> you ring modulated 100 and 200 hz sine waves? Would you get what you term
> beating?
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: "Mike Battaglia"
> Date: Wed, 14 Jan 2009 14:08:29 -0500
> To: <tuning@yahoogroups.com>
> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> Yeah, I screwed up on the math - see above. The point I was making is
> that this would, in fact, be beating.
>
> -Mike
>
> On Wed, Jan 14, 2009 at 2:07 PM, <chrisvaisvil@...> wrote:
>> I respectfully disagree. The formula says the result would be 441 hz and
>> 439
>> hz sine waves. This would not be beating.
>>
>> Sent via BlackBerry from T-Mobile
>>
>> ________________________________
>> From: "Mike Battaglia"
>> Date: Wed, 14 Jan 2009 13:41:28 -0500
>> To: <tuning@yahoogroups.com>
>> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
>> triads,Re:Beatings vs Intermodulation
>>
>> If you take a 440 Hz sine wave, and you ring modulate it with a
>> frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
>> whole thing. Fourier analysis of the resultant signal will show
>> frequencies at 440.5 Hz and 439.5 Hz.
>>
>> -Mike
>>
>> On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@...> wrote:
>>> It doesn't seem to be what everyone is describing. On wikipedia ring
>>> modulation results in f1+f2 and f2-f1. These seem to be the difference
>>> tones
>>> as describe many times in this thread. I don't think that can happen
>>> without
>>> some sort of signal processing even if it is the ear or mind processing.
>>> Beats are flat out constructive and destructive interference. And I think
>>> that is all that can hjappen "in air" as it were.
>>>
>>> Sent via BlackBerry from T-Mobile
>>>
>>> ________________________________
>>> From: "Carl Lumma"
>>> Date: Wed, 14 Jan 2009 18:14:10 -0000
>>> To: <tuning@yahoogroups.com>
>>> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
>>> triads,Re:Beatings vs Intermodulation
>>>
>>> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>>>
>>>> To play the devil's advocate - I thought what you are calling
>>>> difference tones was the result of ring modulation.
>>>
>>> Other way around - beating is the ring modulation one. -Carl
>>>
>>>
>>
>>
>
>

> The beating would become so fast that it would stop sounding like
> beating and the whole thing would start sounding like a complex tone.
> What would happen if you played 430 Hz and 450 Hz together? Would it
> sound like fast beating? How about 435 Hz and
> 445 Hz?
>
> -Mike

Sorry, forgot to quote - this last part was in response to your
question about the RM between 100 and 200 Hz.

The difference is you are mixing two processes. One is electronic and the other is acoustic. And it won't sound like modulated 440. It will sound like two tones almost at unison.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Mike Battaglia" <battaglia01@gmail.com>

Date: Wed, 14 Jan 2009 14:21:32
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Let me ask you a question: ring modulating 440 Hz by 1 Hz gives you
frequencies of 441 Hz and 439 Hz. As you might expect, the resultant
sound sounds like a 440 Hz tone whose volume is being modulated in a
sinusoidal pattern.

What happens if you just take 441 Hz and 439 Hz sine waves and add
them together? You hear a 440 Hz tone whose volume seems to modulate
in a sinusoidal pattern.

What's the difference? The frequency spectra are the same, the time
domain waveforms are the same... What is different between them other
than a choice of perspective?

The beating would become so fast that it would stop sounding like
beating and the whole thing would start sounding like a complex tone.
What would happen if you played 430 Hz and 450 Hz together? Would it
sound like fast beating? How about 435 Hz and
445 Hz?

-Mike

On Wed, Jan 14, 2009 at 2:13 PM, <chrisvaisvil@gmail.com> wrote:
> It is not beating, it is ring modulation. The constructive destructive
> interference that occurs "in air" upon the resulting two sine waves is
> incidentail and has nothing to do with ring modulation. What would happen if
> you ring modulated 100 and 200 hz sine waves? Would you get what you term
> beating?
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: "Mike Battaglia"
> Date: Wed, 14 Jan 2009 14:08:29 -0500
> To: <tuning@yahoogroups.com>
> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> Yeah, I screwed up on the math - see above. The point I was making is
> that this would, in fact, be beating.
>
> -Mike
>
> On Wed, Jan 14, 2009 at 2:07 PM, <chrisvaisvil@gmail.com> wrote:
>> I respectfully disagree. The formula says the result would be 441 hz and
>> 439
>> hz sine waves. This would not be beating.
>>
>> Sent via BlackBerry from T-Mobile
>>
>> ________________________________
>> From: "Mike Battaglia"
>> Date: Wed, 14 Jan 2009 13:41:28 -0500
>> To: <tuning@yahoogroups.com>
>> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
>> triads,Re:Beatings vs Intermodulation
>>
>> If you take a 440 Hz sine wave, and you ring modulate it with a
>> frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
>> whole thing. Fourier analysis of the resultant signal will show
>> frequencies at 440.5 Hz and 439.5 Hz.
>>
>> -Mike
>>
>> On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@gmail.com> wrote:
>>> It doesn't seem to be what everyone is describing. On wikipedia ring
>>> modulation results in f1+f2 and f2-f1. These seem to be the difference
>>> tones
>>> as describe many times in this thread. I don't think that can happen
>>> without
>>> some sort of signal processing even if it is the ear or mind processing.
>>> Beats are flat out constructive and destructive interference. And I think
>>> that is all that can hjappen "in air" as it were.
>>>
>>> Sent via BlackBerry from T-Mobile
>>>
>>> ________________________________
>>> From: "Carl Lumma"
>>> Date: Wed, 14 Jan 2009 18:14:10 -0000
>>> To: <tuning@yahoogroups.com>
>>> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
>>> triads,Re:Beatings vs Intermodulation
>>>
>>> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>>>
>>>> To play the devil's advocate - I thought what you are calling
>>>> difference tones was the result of ring modulation.
>>>
>>> Other way around - beating is the ring modulation one. -Carl
>>>
>>>
>>
>>
>
>

Both resultant waveforms will be identical and have the same spectrum.
How could they possibly sound different? The speaker putting the
sounds into the air will do the same exact thing for both of them.
-Mike

On Wed, Jan 14, 2009 at 3:24 PM, <chrisvaisvil@...> wrote:
> The difference is you are mixing two processes. One is electronic and the
> other is acoustic. And it won't sound like modulated 440. It will sound like
> two tones almost at unison.
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: "Mike Battaglia"
> Date: Wed, 14 Jan 2009 14:21:32 -0500
> To: <tuning@yahoogroups.com>
> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> Let me ask you a question: ring modulating 440 Hz by 1 Hz gives you
> frequencies of 441 Hz and 439 Hz. As you might expect, the resultant
> sound sounds like a 440 Hz tone whose volume is being modulated in a
> sinusoidal pattern.
>
> What happens if you just take 441 Hz and 439 Hz sine waves and add
> them together? You hear a 440 Hz tone whose volume seems to modulate
> in a sinusoidal pattern.
>
> What's the difference? The frequency spectra are the same, the time
> domain waveforms are the same... What is different between them other
> than a choice of perspective?
>
> The beating would become so fast that it would stop sounding like
> beating and the whole thing would start sounding like a complex tone.
> What would happen if you played 430 Hz and 450 Hz together? Would it
> sound like fast beating? How about 435 Hz and
> 445 Hz?
>
> -Mike
>
> On Wed, Jan 14, 2009 at 2:13 PM, <chrisvaisvil@...> wrote:
>> It is not beating, it is ring modulation. The constructive destructive
>> interference that occurs "in air" upon the resulting two sine waves is
>> incidentail and has nothing to do with ring modulation. What would happen
>> if
>> you ring modulated 100 and 200 hz sine waves? Would you get what you term
>> beating?
>>
>> Sent via BlackBerry from T-Mobile
>>
>> ________________________________
>> From: "Mike Battaglia"
>> Date: Wed, 14 Jan 2009 14:08:29 -0500
>> To: <tuning@yahoogroups.com>
>> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
>> triads,Re:Beatings vs Intermodulation
>>
>> Yeah, I screwed up on the math - see above. The point I was making is
>> that this would, in fact, be beating.
>>
>> -Mike
>>
>> On Wed, Jan 14, 2009 at 2:07 PM, <chrisvaisvil@...> wrote:
>>> I respectfully disagree. The formula says the result would be 441 hz and
>>> 439
>>> hz sine waves. This would not be beating.
>>>
>>> Sent via BlackBerry from T-Mobile
>>>
>>> ________________________________
>>> From: "Mike Battaglia"
>>> Date: Wed, 14 Jan 2009 13:41:28 -0500
>>> To: <tuning@yahoogroups.com>
>>> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
>>> triads,Re:Beatings vs Intermodulation
>>>
>>> If you take a 440 Hz sine wave, and you ring modulate it with a
>>> frequency of 1 Hz, you will hear 440 Hz with a 1 Hz "beating" over the
>>> whole thing. Fourier analysis of the resultant signal will show
>>> frequencies at 440.5 Hz and 439.5 Hz.
>>>
>>> -Mike
>>>
>>> On Wed, Jan 14, 2009 at 1:26 PM, <chrisvaisvil@...> wrote:
>>>> It doesn't seem to be what everyone is describing. On wikipedia ring
>>>> modulation results in f1+f2 and f2-f1. These seem to be the difference
>>>> tones
>>>> as describe many times in this thread. I don't think that can happen
>>>> without
>>>> some sort of signal processing even if it is the ear or mind processing.
>>>> Beats are flat out constructive and destructive interference. And I
>>>> think
>>>> that is all that can hjappen "in air" as it were.
>>>>
>>>> Sent via BlackBerry from T-Mobile
>>>>
>>>> ________________________________
>>>> From: "Carl Lumma"
>>>> Date: Wed, 14 Jan 2009 18:14:10 -0000
>>>> To: <tuning@yahoogroups.com>
>>>> Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15
>>>> triads,Re:Beatings vs Intermodulation
>>>>
>>>> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>>>>>
>>>>> To play the devil's advocate - I thought what you are calling
>>>>> difference tones was the result of ring modulation.
>>>>
>>>> Other way around - beating is the ring modulation one. -Carl
>>>>
>>>>
>>>
>>>
>>
>>
>
>

Chris wrote:

> The difference is you are mixing two processes. One is electronic and the other is acoustic.
> And it won't sound like modulated 440. It will sound like two tones almost at unison.

First of all, I don’t know what you mean by „acoustic“ because here we were discussing pure sine waves which never occur in nature and can only be made electronically. Second, as I’ve already said earlier, if you také two sine waves of 439Hz and 441Hz and mix them in equal loudnesses, you get EXACTLY the same as if you multiplied (i.e. ring-modulated) a 440Hz sine wave with a 1Hz cosine wave because there’s the trigonometric identity that sais „sin(a+b)+sin(a-b) = sin(a)*cos(b)*2“. So you couldn’t notice any difference in the two sounds because there actually wouldn’t be any at all.

Petr

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

> What was the final expression that you got when delta was thrown
in?

Hi Mike,

I have reviewed my calculations about beating of triads, they now
seem reliable. The basic idea works, but I had misinterpreted
something in the conclusions, actually what is found is a beating at
frequency 2 df where df is the frequency shift of the middle tone.

The starting point is the following wave:

cos(w1 t) + cos(w2 t) + cos (w3 t)

where w1 = 2 Pi f1, w2 = 2 Pi f2 and w3 = 2 Pi f3, w are angular
frequencies corresponding to the three tones composing thr triad
(let's assume that f1 is the tonic, f3 the fifth, and f2 the middle
note, but in general they can be any frequency, no approximations),
and f are usual frequencies in Hz. Being delta the angular frequency
shift of the middle tone, so that delat = 2 Pi df, one gets by
applying prosthaphaeresis formulae and collecting terms:

cos(delta t/2) [cos(da t/2)cos((sa+delta)t/2)+cos(db t/2)cos
((sb+delta)t/2)] +

+ sin(delta t/2) [-sin(da t/2)cos((sa+delta)t/2)+sin(db t/2)cos
((sb+delta)t/2)] +

+ cos(dg t/2)cos(sg t/2)

with da=w2-w1, db=w3-w2, dg=w3-w1 are the difference (angular)
frequencies of the single dyads (unshifted), sa=w2+w1, sb=w3+w2,
sg=w3+w1 are the corresponding sum frequencies.

If we assume df << f1 (now this is an approximation: small detuning,
but the formulae above are always valid) then we can say that the
resulting wave will be composed of three parts:

1) a "complicated" wave made up by the all the three frequencies,
but with an overall amplitude modulation proportional to cos(delta
t/2); all the oscillations within are "fast" with respect to the
modulant cosine;

2) a "complicated" wave made up by the all the three frequencies,
but with an overall amplitude modulation proportional to sin(delta
t/2); all the oscillations within are "fast" with respect to the
modulant sine;

3) a steady loudness (not amplitude modulated) wave ("fast" as well).

The overall amplitude of this wave is made up by the amplitude of 1)
(2|cos(delta t/2)|) plus the amplitude of 2) (2|cos(delta t/2)|)
plus the amplitude of 3) (1). Note: in my previous estimation I
disregarded the fact that amplitude is proportional to the modulus
of cos(delta t/2) etc. and not to just cos(delta t/2). Then when you
sum up the two amplitudes you have to sum up the moduli.

In conclusion, the overall amplitude modulation has the shape of a 1
+ 2|cos(delta t/2)| + 2|sin(delta t/2)|, therefore has frequency 2
delta (one factor 2 comes from the modulus, another factor 2 comes
from the sum of sine and cosine) and has spans from amplitude 3 to
amplitude 1+2Sqrt(2) that is about 3.828. This analysis is
straightforward only if delta is small enough, otherwise amplitudes
behave differently. I have checked all this with Mathematica, that
could also play the waveforms, in the case of the just major triad
with tonic at 220 Hz and with a detuning of the middle tone of 1 Hz.
I could then check that what I hear is actually what I say in this
case. I have also made other tests, all confirming that everything
should be ok.

> I'm interested to see this. Previously the existence of beating was
> written off as a failure of the basilar membrane to discern between
> fine pitch differences, but I've long suspected there's more to it
> than that. By that logic, if the basilar membrane had infinitely
fine
> frequency resolution, then we'd never hear any changes in the sound
> over time - we'd perceive all of sound as one static, unchanging
sound
> throughout our lifetime! I suspect there is some neurological
> mechanism to explain why the brain sometimes "flips" its
perspective
> on the incoming signal to hear it as a modulation envelope over a
tone
> vs. different tones - something beyond critical band effects.

Your analysis is interesting. I think that one should distinguish
between the real behaviour of sound, and what is created into the
ear or the brain. As a physicist I would first like to understand
what is already in the sound, and as my personal interest, I would
study in particular the issue of consonance/dissonance.

> > In the case of Chris' samples made in equal temperament, the
> > difference of fifth and third from the just intonation ones
accounts
> > for the heard low-frequency beat of the major triad. Perhaps, as
from
> > my starting guess already discussed with Carl, in the case that
> > beatings become too fast and may be heard as a "strange" type of
> > tone, this could also account for the slight dissonance effect
of the
> > minor triad; this explanation is not yet satisfying to me and I
would
> > like to explore this point further.
>
> I made this point earlier as well. If you take 5/4 and you widen it
> slowly, you'll hear the beating start. Keep widening it until you
get
> to 9/7. At this point the 5/4 will sound uncomfortably sharp, and
the
> beating will have gotten too fast to hear, as you put it... But if
you
> stay on 9/7 for a second, you might experience a sudden perception
> shift whereas you suddenly realize that the "beating" is in a
regular
> pattern, and then you'll suddenly become aware of the different
> fundamental and hear it as 9/7.

This is for sure a matter of interpretation of our brain
("perception shift"). To understand better this, I think it is
necessary to know very well how the sound is composed and "why" it
is so (in the sense of what are the factors influencing etc) I will
try what you propose, although I suspect the result depends on
how "good" and trained is one's ear...

> > However, I would like to go on. Of course, beatings cannot fully
> > account for consonance/dissonance, but may as well be part of
it. As
> > for the example suggested by Carl, I have tested by generating
the
> > waveforms by Mathemathica and playing them, so it is not
reliable and
> > we are waiting for Chris' new sample.
>
> What sample are you guys waiting for? 7 and 11 limit major vs minor
> chords? I have MATLAB going over here, I could whip something up.
>
> -Mike
>

Mathematica is fine for generating waveforms, I just was not sure on
how is the quality of the audio system of my PC. With an mp3 I can
play it everywhere. I just don't know if and how I can generate an
audio file with my version of Mathematica... perhaps it is possible
but I have not looked to it!

Cheers

Max

if you are interested in the Mathematica nb file I can email it to
you, I have no place in my webpage to upload it for you do
download...

You will not get interference when the sine waves are electrical. Ring modulation is a process that occurs with electrical signals. Beating as you describe will not occur with electrical signals. Furthermore I do not think you can multiply waveforms after they leave the electrical domain and enter the acoustic domain as pressure waves.

I think equating the two domains is clouding the discussion. I believe we are only interested in acoustic phenomena.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Petr Parízek <p.parizek@chello.cz>

Date: Thu, 15 Jan 2009 08:37:33
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Chris wrote:

> The difference is you are mixing two processes. One is electronic and the other is acoustic.
> And it won't sound like modulated 440. It will sound like two tones almost at unison.

First of all, I don’t know what you mean by „acoustic“ because here we were discussing pure sine waves which never occur in nature and can only be made electronically. Second, as I’ve already said earlier, if you také two sine waves of 439Hz and 441Hz and mix them in equal loudnesses, you get EXACTLY the same as if you multiplied (i.e. ring-modulated) a 440Hz sine wave with a 1Hz cosine wave because there’s the trigonometric identity that sais „sin(a+b)+sin(a-b) = sin(a)*cos(b)*2“. So you couldn’t notice any difference in the two sounds because there actually wouldn’t be any at all.

Petr

But oops. My experiment did show what is considered beating which is interference which is summation of the signal.

However I still hold that one cannot ring modulate in the acoustic domain.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Petr Parízek <p.parizek@chello.cz>

Date: Thu, 15 Jan 2009 08:37:33
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Chris wrote:

> The difference is you are mixing two processes. One is electronic and the other is acoustic.
> And it won't sound like modulated 440. It will sound like two tones almost at unison.

First of all, I don’t know what you mean by „acoustic“ because here we were discussing pure sine waves which never occur in nature and can only be made electronically. Second, as I’ve already said earlier, if you také two sine waves of 439Hz and 441Hz and mix them in equal loudnesses, you get EXACTLY the same as if you multiplied (i.e. ring-modulated) a 440Hz sine wave with a 1Hz cosine wave because there’s the trigonometric identity that sais „sin(a+b)+sin(a-b) = sin(a)*cos(b)*2“. So you couldn’t notice any difference in the two sounds because there actually wouldn’t be any at all.

Petr

> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@>
> wrote:
> >
>
> > What was the final expression that you got when delta was thrown
> in?
>
...
>
> if you are interested in the Mathematica nb file I can email it to
> you, I have no place in my webpage to upload it for you do
> download...
>

Hi Mike,

I have realized that I could upload files within this list,
therefore i have uploaded my Mathematica notebook
file "beating_of_triads.nb" in the new folder "Max"

If you can't read it just let me know, I'll produce a pdf file out
of it.

Max

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> But oops. My experiment did show what is considered beating
> which is interference which is summation of the signal.

I think your sine waves weren't pure.

> However I still hold that one cannot ring modulate in the
> acoustic domain.

Petr and Mike just showed you how you can.

-Carl

Why do you think the sine waves were not pure. 2 I am nopt convinced you can multipy sines waves acoustically which is what ring modulation does.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@...>

Date: Thu, 15 Jan 2009 17:27:01
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> But oops. My experiment did show what is considered beating
> which is interference which is summation of the signal.

I think your sine waves weren't pure.

> However I still hold that one cannot ring modulate in the
> acoustic domain.

Petr and Mike just showed you how you can.

-Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Why do you think the sine waves were not pure.

Because the chords were beating.

-Carl

Carl... Harmonics are nothing more than additional sinusoidal waves usually at significantly decreasing volume going up in frequency. I am home and have sometime for experiments. Does anyone know of a program to capture screen shot equivalent as a movie?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@...>

Date: Thu, 15 Jan 2009 17:58:34
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Why do you think the sine waves were not pure.

Because the chords were beating.

-Carl

I believe the summation of sines + cosines is fundamental to Fourier's theorm.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@...>

Date: Thu, 15 Jan 2009 17:58:34
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Why do you think the sine waves were not pure.

Because the chords were beating.

-Carl

Chris wrote:

> You will not get interference when the sine waves are electrical. Ring modulation is a
> process that occurs with electrical signals. Beating as you describe will not occur with
> electrical signals.

Either I don’t understand what you mean or I must just wonder where you got to this conclusion. I don’t know what should be the reason for „electrical signals“ containing no interference.

> Furthermore I do not think you can multiply waveforms after they leave
> the electrical domain and enter the acoustic domain as pressure waves.

I wasn’t saying you could realize that acoustically. I was saying that if you (electronically) make a sine wave with a frequency of A and a cosine wave with a frequency of B and multiply the two, you get just the same as if you (electronically) make one sine wave with a frequency of A+B and another sine wave with a frequency of A-B and mix them. And since they’re both the same (because of the trigonometric identity I mentioned earlier) and you were saying they were not, I just didn’t know why.

> I think equating the two domains is clouding the discussion. I believe we are only interested
> in acoustic phenomena.

Even if you can’t do strict multiplication of two acoustic sounds in the „acoustic enviro“ment", at least similar things can sometimes happen even there and we can’t just ignore the fact. For example, whistling has very soft overtones and therefore if two people try whistling at the same pitch, they will soon hear beating because such a thing is essentially impossible to do and therefore one person will then whistle a bit higher or lower than the other.

Petr

[ Attachment content not displayed ]

On Thu, Jan 15, 2009 at 8:53 AM, <chrisvaisvil@...> wrote:
> But oops. My experiment did show what is considered beating which is
> interference which is summation of the signal.
>
> However I still hold that one cannot ring modulate in the acoustic domain.

Sure you can. Say "ahhhhh... ahhhhh... ahhhhh... ahhhhh..." so that
the pitch stays the same. Try to make the amplitude change in as
sinusoidal a pattern as possible. The "aaahhhhhh" signal is being
modulated by the sinusoidal "envelope" of your choosing. If you were
to perform a fourier analysis of this, and if your "aaahhhhh" was a
perfect sine wave, you would see three frequencies: the center
frequency, the center frequency - the frequency of the amplitude
modulation, and the center frequency + the frequency of the amplitude
modulation. So if you sing at A 440 and the volume goes up and down in
a sinusoidal pattern with a complete repetition every second, you'll
see 439 Hz, 440 Hz, and 441 Hz.

This isn't strictly "ring modulation" since the modulator in this case
is "unipolar" (the volume is going from 1 to 0, not from 1 to -1) and
so the 440 Hz will still be in there as well. If you want to devise a
clever way to reverse the phase every other time, then that would be
strictly "ring modulation". This is more like ring modulation with the
original signal added back in.

Regardless, the point I'm making is something about how Fourier
analysis works. It takes the whole signal and tells you how to rebuild
that signal up from scratch by taking sine waves of different
"frequencies" that stretch from the beginning of time to an infinite
distance away from here. Envelope changes are dealt with by figuring
out what infinite series of sine wave will add up to somehow yield the
waveform anyway. It isn't REALLY doing everything that the "ear" is
doing when it converts a waveform in the air. A frequency present in a
Fourier transform doesn't necessarily translate into a pitch we hear,
and for more reasons than just human threshold of hearing limits and
such.

This is sort of a simplification, and for the DFT/FFT, the sine waves
don't stretch back from -Infinity to Infinity, but rather from the
beginning of the signal we're analyzing to the end. You can take the
DFT of a whole song and get a ton of frequencies present. You don't
hear all of those frequencies as static "pitches", do you? You didn't
suddenly hear three pitches jump into existence when you went
"aaaaahhhh.....aaaaahhhh...", right? But there are in fact three
"frequencies" in that signal.

The cause of all of this confusion is that while it is a useful
abstraction to say that what the ear/brain system is doing when a
signal comes in is some kind of "Fourier Analysis", it's a gross
oversimplification. After all, we hear changes in sound over time - we
hear 439 Hz, 440 Hz, and 441 Hz as being 440 Hz modulated by a 1 Hz
sine wave, and there's NO REASON to assume that that isn't just
fundamentally what is is, rather than thinking of it as three
different frequencies of infinite span. After all, perceiving the
envelope of a sound is just as fundamental as hearing its harmonic
content.

So, another analogy that's useful to think of is that perhaps the
auditory system performs some kind of overlapping FT with small
"windows" of sound that it evaluates chunk by chunk and such, which is
what they call a Short-Time Fourier Transform (STFT) -- but even
that's a bit nonsensical as the window size would have to be the same
for all frequencies, and so there'd be some pretty nasty artifacts
which are luckily absent from the end result of what we perceive.
Likely the brain is performing something more like a wavelet
transform, which utilizes different window sizes for different
frequencies, in a sense.

-Mike

> Even if you can't do strict multiplication of two acoustic sounds in the
> „acoustic enviro"ment", at least similar things can sometimes happen even
> there and we can't just ignore the fact. For example, whistling has very
> soft overtones and therefore if two people try whistling at the same pitch,
> they will soon hear beating because such a thing is essentially impossible
> to do and therefore one person will then whistle a bit higher or lower than
> the other.
>
> Petr

Whenever you say "uh-uh", you're multiplying "uhhhhhhhhh" by a square
wave on top of it whose values range from 1 to 0. Any time the volume
of a tone changes in real life, that tone's amplitude is being
"multiplied" by its envelope. Frequency and amplitude modulation CAN
happen in an acoustic environment, a la the Doppler effect.

> Hi Mike,
>
> I have realized that I could upload files within this list,
> therefore i have uploaded my Mathematica notebook
> file "beating_of_triads.nb" in the new folder "Max"
>
> If you can't read it just let me know, I'll produce a pdf file out
> of it.
>
> Max

Could you? I don't have Mathematica set up on my laptop.

Thanks,
Mike

Mike we've gone full circle. Someone claimed you won't see beats in FFT you are saying we will. Ring modulate IS different. When you expained the compromise you are saying its different. I propose experiments design to replace our opinions, claculations and assumption with facts. This freeware FFT program seems pretty powerful the more I play with. I'm still working on video capture because interactive examples will be conclusive for some.

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-----Original Message-----
From: Mike Battaglia <battaglia01@gmail.com>

Date: Thu, 15 Jan 2009 15:39:28
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
triads,Re:Beatings vs Intermodulation

On Thu, Jan 15, 2009 at 8:53 AM, <chrisvaisvil@gmail.com> wrote:
> But oops. My experiment did show what is considered beating which is
> interference which is summation of the signal.
>
> However I still hold that one cannot ring modulate in the acoustic domain.

Sure you can. Say "ahhhhh... ahhhhh... ahhhhh... ahhhhh..." so that
the pitch stays the same. Try to make the amplitude change in as
sinusoidal a pattern as possible. The "aaahhhhhh" signal is being
modulated by the sinusoidal "envelope" of your choosing. If you were
to perform a fourier analysis of this, and if your "aaahhhhh" was a
perfect sine wave, you would see three frequencies: the center
frequency, the center frequency - the frequency of the amplitude
modulation, and the center frequency + the frequency of the amplitude
modulation. So if you sing at A 440 and the volume goes up and down in
a sinusoidal pattern with a complete repetition every second, you'll
see 439 Hz, 440 Hz, and 441 Hz.

This isn't strictly "ring modulation" since the modulator in this case
is "unipolar" (the volume is going from 1 to 0, not from 1 to -1) and
so the 440 Hz will still be in there as well. If you want to devise a
clever way to reverse the phase every other time, then that would be
strictly "ring modulation". This is more like ring modulation with the
original signal added back in.

Regardless, the point I'm making is something about how Fourier
analysis works. It takes the whole signal and tells you how to rebuild
that signal up from scratch by taking sine waves of different
"frequencies" that stretch from the beginning of time to an infinite
distance away from here. Envelope changes are dealt with by figuring
out what infinite series of sine wave will add up to somehow yield the
waveform anyway. It isn't REALLY doing everything that the "ear" is
doing when it converts a waveform in the air. A frequency present in a
Fourier transform doesn't necessarily translate into a pitch we hear,
and for more reasons than just human threshold of hearing limits and
such.

This is sort of a simplification, and for the DFT/FFT, the sine waves
don't stretch back from -Infinity to Infinity, but rather from the
beginning of the signal we're analyzing to the end. You can take the
DFT of a whole song and get a ton of frequencies present. You don't
hear all of those frequencies as static "pitches", do you? You didn't
suddenly hear three pitches jump into existence when you went
"aaaaahhhh.....aaaaahhhh...", right? But there are in fact three
"frequencies" in that signal.

The cause of all of this confusion is that while it is a useful
abstraction to say that what the ear/brain system is doing when a
signal comes in is some kind of "Fourier Analysis", it's a gross
oversimplification. After all, we hear changes in sound over time - we
hear 439 Hz, 440 Hz, and 441 Hz as being 440 Hz modulated by a 1 Hz
sine wave, and there's NO REASON to assume that that isn't just
fundamentally what is is, rather than thinking of it as three
different frequencies of infinite span. After all, perceiving the
envelope of a sound is just as fundamental as hearing its harmonic
content.

So, another analogy that's useful to think of is that perhaps the
auditory system performs some kind of overlapping FT with small
"windows" of sound that it evaluates chunk by chunk and such, which is
what they call a Short-Time Fourier Transform (STFT) -- but even
that's a bit nonsensical as the window size would have to be the same
for all frequencies, and so there'd be some pretty nasty artifacts
which are luckily absent from the end result of what we perceive.
Likely the brain is performing something more like a wavelet
transform, which utilizes different window sizes for different
frequencies, in a sense.

-Mike

Isn't it additive? Not multiplicative ?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Mike Battaglia <battaglia01@gmail.com>

Date: Thu, 15 Jan 2009 15:43:13
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
triads,Re:Beatings vs Intermodulation

> Even if you can't do strict multiplication of two acoustic sounds in the
> „acoustic enviro"ment", at least similar things can sometimes happen even
> there and we can't just ignore the fact. For example, whistling has very
> soft overtones and therefore if two people try whistling at the same pitch,
> they will soon hear beating because such a thing is essentially impossible
> to do and therefore one person will then whistle a bit higher or lower than
> the other.
>
> Petr

Whenever you say "uh-uh", you're multiplying "uhhhhhhhhh" by a square
wave on top of it whose values range from 1 to 0. Any time the volume
of a tone changes in real life, that tone's amplitude is being
"multiplied" by its envelope. Frequency and amplitude modulation CAN
happen in an acoustic environment, a la the Doppler effect.

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On Thu, Jan 15, 2009 at 3:50 PM, <chrisvaisvil@...> wrote:
> Mike we've gone full circle. Someone claimed you won't see beats in FFT you
> are saying we will. Ring modulate IS different. When you expained the
> compromise you are saying its different. I propose experiments design to
> replace our opinions, claculations and assumption with facts. This freeware
> FFT program seems pretty powerful the more I play with. I'm still working on
> video capture because interactive examples will be conclusive for some.

I never said you'll see the beats in the FFT. If you add 439 Hz and
441 Hz together, you won't see 2 Hz or 1 Hz spontaneously appear in
the FFT. But if you take 440 Hz and you multiply that signal by a 1 Hz
sinusoid, you will see 439 and 441 Hz together.

Let me ask you a question: what's the difference in the FFT between a
sine wave at 440 Hz for the whole signal and between a 440 Hz sine
wave that is only on for half of the signal?

Are you saying amplitute modulate with a 440 Hz 50 percent duty cycle squarewave?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Mike Battaglia <battaglia01@gmail.com>

Date: Thu, 15 Jan 2009 16:03:09
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
triads,Re:Beatings vs Intermodulation

On Thu, Jan 15, 2009 at 3:50 PM, <chrisvaisvil@gmail.com> wrote:
> Mike we've gone full circle. Someone claimed you won't see beats in FFT you
> are saying we will. Ring modulate IS different. When you expained the
> compromise you are saying its different. I propose experiments design to
> replace our opinions, claculations and assumption with facts. This freeware
> FFT program seems pretty powerful the more I play with. I'm still working on
> video capture because interactive examples will be conclusive for some.

I never said you'll see the beats in the FFT. If you add 439 Hz and
441 Hz together, you won't see 2 Hz or 1 Hz spontaneously appear in
the FFT. But if you take 440 Hz and you multiply that signal by a 1 Hz
sinusoid, you will see 439 and 441 Hz together.

Let me ask you a question: what's the difference in the FFT between a
sine wave at 440 Hz for the whole signal and between a 440 Hz sine
wave that is only on for half of the signal?

On Thu, Jan 15, 2009 at 3:52 PM, <chrisvaisvil@...> wrote:
> Isn't it additive? Not multiplicative ?
> Sent via BlackBerry from T-Mobile

It's multiplicative. The amplitude of the sine wave is changing over
time. When it's at 0, that's like saying the sine wave is being
multiplied by 0. If it's ever twice as loud as normal, that's like
multiplying it by 2. Adding a 1 Hz sine wave to a 440 Hz sine wave
would sound to us like a 440 Hz sine wave, because the 1 Hz would be
largely filtered out by the acoustics of the ear canal.

To put it another way: You know you can't hear 1 Hz, but you can
definitely hear changes in sound in time over 1 second, right?
Maybe this picture will help explain things:

http://rabbit.eng.miami.edu/students/mbattaglia/sines.png

The top one is a 440 Hz sine wave multiplied by a 1 Hz envelope, and
the right is the FFT of it, zoomed in to show you that it's made up of
two frequencies right next to each other, which, incidentally, gives
you "beating," which is the same exact thing as taking a 440 Hz sine
wave and multiplying it by a 1 Hz sine wave, which is exactly what you
hear beating to be. The numbers on the bottom don't indicate what
frequency it is, but what the "bin number" of the frequency is. I just
put it there to give you a sense of the scale.

The bottom one is a 440 Hz sine wave added to a 1 Hz sine wave. The
FFT on the right shows two distinct frequencies, and only two: 440 Hz,
and 1 Hz.

-Mike

I have just posted a pdf file with Mathematica output about the
beating of triads.

Max

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Hi Mike,
> >
> > I have realized that I could upload files within this list,
> > therefore i have uploaded my Mathematica notebook
> > file "beating_of_triads.nb" in the new folder "Max"
> >
> > If you can't read it just let me know, I'll produce a pdf file out
> > of it.
> >
> > Max
>
> Could you? I don't have Mathematica set up on my laptop.
>
> Thanks,
> Mike
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi George,
>
> >>> However, to compare the case of coincident vs. non-coincident
> >>> combinational tones the chords *must be in JI*, i.e., 4:5:6
> >>> vs. 10:12:15 vs. 6:8:9.
> >>
> >> Why? I still hear the phenomenon. Is this more evidence that
> >> coincident combination tones can't explain it?
> >
> > What exactly is it that you're hearing?
>
> The thread started by someone observing that major and minor
> triads are of a fundamentally different character. This
> difference gets more extreme in the 7-limit, more extreme
> still in the 11-limit, and so on. You proposed that
> combination tones explain this, and I disagreed. Chris
> synthesized the chords with sine tones, and I still heard the
> difference when listening at low volume levels, when no
> combination tones were present.

Yes, I understand that. When I asked, "What exactly is it that
you're hearing?", I wanted you to be more specific. All you're
telling me is that you heard a difference. You're not giving me much
indication of what sort of difference you perceive. I'd say that the
minor triad sounds rougher, and I perceive that roughness as a
beating on the order of 31/second.

> > What I hear in the 12-ET sine example is a relatively slow
> > "beating"
> //
> > Do you have another explanation for these observations?
>
> I didn't really hear any beating when I listened, but if
> you do it could very well be from combination tones. How
> good is your speaker system? Try turning down the volume?

I used cheap headphones on a laptop computer. I also tried a desktop
computer, on which I heard the same result, but it doesn't have
expensive speakers. However, my hearing isn't really hi-fi anymore --
around 20 years ago I had a middle-ear infection in both ears,
severe enough to burst both eardrums. It took a couple of months for
my hearing to get back to anything resembling normal.

However, all of the foregoing may be irrelevant, because I noticed
something very interesting happening with the Winamp sound frequency-
display bars as I was playing the mp3 files. The triangle-wave file
activated several of the bars, but the sine-wave file activated only
one of the bars (which was to be expected, since the output
frequencies were all in a single octave). What surprised me was that
I actually saw the single bar go up and down several times a second
in exact synchronization with the beating I was hearing. So it's not
all in my head!

> But regardless of the beating, I assume you can still
> recognize that one is a minor chord and one is a major
> chord...

But of course. :-)

> Hopefully we'll get a 7-limit demo soon.

Yes. (I'm still far behind on my reading in this thread, so I'm
unaware of the latest developments. Hopefully I'll be able to catch
up soon.)

--George

Hi Mike,

> The cause of all of this confusion is that while it is a useful
> abstraction to say that what the ear/brain system is doing when a
> signal comes in is some kind of "Fourier Analysis", it's a gross
> oversimplification.

The best way to model the cochlea is with a filterbank, typically
of gammatone filters. Did you say you have Matlab? In that case,
I think you can create these filterbanks easily with
Malcolm Slaney's Auditory Toolbox, um, here

http://www.speech.cs.cmu.edu/comp.speech/Section1/HumanAudio/auditory.tlbx.html

I'm sure that technically, any time-domain transform can be
represented as a filterbank and vice versa, but it's very
natural to see the basilar membrane as a filterbank, so I
prefer to think of it that way. As far as STFT-based methods,
"time-frequency reassignment" is state of the art (last I
checked).

http://www.cerlsoundgroup.org/Kelly/timefrequency.html

Oh, and it looks like there's a Matlab toolbox for this too:

http://tftb.nongnu.org

As far as Melodyne DNA, the output of such methods is just
the beginning. Some sort of harmonic-template extraction
has to be used. I've suggested a greedy algorithm (based
on removing as much energy from the signal as possible at
each go) in the past...

-Carl

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> > The thread started by someone observing that major and minor
> > triads are of a fundamentally different character. This
> > difference gets more extreme in the 7-limit, more extreme
> > still in the 11-limit, and so on. You proposed that
> > combination tones explain this, and I disagreed. Chris
> > synthesized the chords with sine tones, and I still heard the
> > difference when listening at low volume levels, when no
> > combination tones were present.
>
> Yes, I understand that. When I asked, "What exactly is it that
> you're hearing?", I wanted you to be more specific. All you're
> telling me is that you heard a difference. You're not giving me
> much indication of what sort of difference you perceive.

The inherent difference between major and minor triads, which
you can hear on your piano, or any instrument you like.
No special equipment required. The difference persists even
in 12-ET, and is therefore fairly robust to whatever tuning
is used.

> I used cheap headphones on a laptop computer. I also tried a
> desktop computer, on which I heard the same result, but it
> doesn't have expensive speakers. However, my hearing isn't
> really hi-fi anymore

I was just wondering if some of the beating you heard could
have been caused by distortion products produced by your
equipment. But there's beating in the file too...

-Carl

Can I interject that small distortions will be swamped out by the major
components?

I found a video capture program that's free and works on 32 bit windows 7
Let me see if I can coax it into life in Vista 64

On Thu, Jan 15, 2009 at 6:26 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "George D.
> Secor" <gdsecor@...> wrote:
>
> > > The thread started by someone observing that major and minor
> > > triads are of a fundamentally different character. This
> > > difference gets more extreme in the 7-limit, more extreme
> > > still in the 11-limit, and so on. You proposed that
> > > combination tones explain this, and I disagreed. Chris
> > > synthesized the chords with sine tones, and I still heard the
> > > difference when listening at low volume levels, when no
> > > combination tones were present.
> >
> > Yes, I understand that. When I asked, "What exactly is it that
> > you're hearing?", I wanted you to be more specific. All you're
> > telling me is that you heard a difference. You're not giving me
> > much indication of what sort of difference you perceive.
>
> The inherent difference between major and minor triads, which
> you can hear on your piano, or any instrument you like.
> No special equipment required. The difference persists even
> in 12-ET, and is therefore fairly robust to whatever tuning
> is used.
>
> > I used cheap headphones on a laptop computer. I also tried a
> > desktop computer, on which I heard the same result, but it
> > doesn't have expensive speakers. However, my hearing isn't
> > really hi-fi anymore
>
> I was just wondering if some of the beating you heard could
> have been caused by distortion products produced by your
> equipment. But there's beating in the file too...
>
> -Carl
>
>
>

No such luck with vista 64 - but I cobbled together something with audacity,
any video converter, microsoft movie maker and easy video capture.

The capturing was on the Acer dual core Windows 7 system.

While the FFT is averaging and therefore sluggish the video and audio were
recorded with separate programs - so they are out of sync, and the clicks I
think are form audacity's recording because they were not there in the first
place.

The first couple minutes is a demo of being a sine, major 3rd, minor 3rd,
5th, major chord, minor chord, and then at about 2 minutes I amplitude
modulate the middle C sine with another sine and change the frequency. What
happens is a complex waveform is made....

http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1232067334/0#0

or

*http://tinyurl.com/microvideotest3am

if you look on the lower right will be an icon for full screen. that's
recommended.
*

> The best way to model the cochlea is with a filterbank, typically
> of gammatone filters. Did you say you have Matlab? In that case,
> I think you can create these filterbanks easily with
> Malcolm Slaney's Auditory Toolbox, um, here
>
> http://www.speech.cs.cmu.edu/comp.speech/Section1/HumanAudio/auditory.tlbx.html

I've been throwing the same idea around, but I'm not sure how to take
the information from each filterbank and do anything useful with it in
terms of sound manipulation. I'd like to design some kind of real-time
pitch-shifting algorithm so that I could do all kinds of useful things
:) But I'll give it another go.

> As far as STFT-based methods,
> "time-frequency reassignment" is state of the art (last I
> checked).
>
> http://www.cerlsoundgroup.org/Kelly/timefrequency.html

I don't know what to say about this other than that it's amazing. I
wish you'd shown me this a year ago :) For the last year I've been
struggling to understand wavelet analysis and trying to design fast
CWT algorithms to do precisely this, but the literature on wavelets
can be extremely difficult to understand, especially when the discrete
version is introduced (and I can't see what the point of the DWT is,
to be honest, it seems almost useless).

The thing I don't understand is how to define the difference between a
change in a frequency's amplitude and the existence of multiple
frequencies. It has to be based on psychoacoustic parameters rather
than rooted in some kind of unwavering theory. It certainly does make
intuitive sense to reframe 441 Hz + 439 Hz as simply being 440 Hz with
its amplitude changing in a 1 Hz fashion, but where does that cutoff
lie? Especially when you consider the gray area of "roughness" in
which some beating is present along with the emergence of two separate
tones.

This certainly gets a thousand times more complicated when envelopes
other than simple sinusoids are introduced.

> As far as Melodyne DNA, the output of such methods is just
> the beginning. Some sort of harmonic-template extraction
> has to be used. I've suggested a greedy algorithm (based
> on removing as much energy from the signal as possible at
> each go) in the past...

I agree, and I've been kicking around the same idea for a few months
now. In terms of wavelets, another thing i'd been considering is to
start with the low frequencies (let's simplify and say that each
wavelet is a single period of a sinewave), and for each low frequency,
detect how much energy from each harmonic of that frequency is present
as well with the same window size, and then remove that from the
signal, and start again. It's a bit difficult to really put on a firm
theoretical basis when you factor in bin spreading and such as well,
as well as the fact that plenty of timbres are fairly inharmonic.

Thanks for the resources though, the time-frequency reassignment stuff
looks extremely promising - and hopefully it'll yield better results
than wavelet analysis.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The best way to model the cochlea is with a filterbank,
> > typically of gammatone filters.
>
> I've been throwing the same idea around, but I'm not sure how
> to take the information from each filterbank

You mean each filter? There's only one bank per ear.

> and do anything useful with it in terms of sound manipulation.

Well, it's not for manipulation per se, but for measuring.
You can plot the output of the filters to get a spectrogram...

> > As far as STFT-based methods, "time-frequency reassignment"
> > is state of the art (last I checked).
> >
> > http://www.cerlsoundgroup.org/Kelly/timefrequency.html
>
> I don't know what to say about this other than that it's
> amazing. I wish you'd shown me this a year ago :)

Well, why didn't you ask? :)

> For the last year I've been struggling to understand wavelet
> analysis and trying to design fast CWT algorithms to do
> precisely this, but the literature on wavelets can be
> extremely difficult to understand,

Yeah. I understand they're like a standard moving-window
STFT, but with a specially shaped window. I should probably
look into it more, but everyone seems to agree they're no
panacea for sound analysis.

> The thing I don't understand is how to define the difference
> between a change in a frequency's amplitude and the existence
> of multiple frequencies. // It certainly does make intuitive
> sense to reframe 441 Hz + 439 Hz as simply being 440 Hz with
> its amplitude changing in a 1 Hz fashion, but where does that
> cutoff lie? Especially when you consider the gray area of
> "roughness" in which some beating is present along with the
> emergence of two separate tones.

Try this:
http://en.wikipedia.org/wiki/Equivalent_rectangular_bandwidth

Keep in mind it varies with loudness (the louder, the wider)
and between subjects. More:
http://ccrma.stanford.edu/~jos/bbt/Equivalent_Rectangular_Bandwidth.html

Oh, and apparently "gammachirp" is better than gammatone:
http://www.mrc-cbu.cam.ac.uk/~roy.patterson/RoyCV/pdfs/IP97.pdf

> This certainly gets a thousand times more complicated when
> envelopes other than simple sinusoids are introduced.

Envelopes? If the waveform is more complicated, it shouldn't
matter, since it's getting decomposed into sinusoids anyway.

-Carl

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:

> http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1232067334/0#0
>
> or
>
> http://tinyurl.com/microvideotest3am

Hi Chris- it's giving me "You are not allowed to access
this section." -Carl

Ouch - ok

I do apologize for that. I've changed the permissions and it should work.

On Fri, Jan 16, 2009 at 4:06 AM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
>
> > http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1232067334/0#0
> >
> > or
> >
> > http://tinyurl.com/microvideotest3am
>
> Hi Chris- it's giving me "You are not allowed to access
> this section." -Carl
>
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Mike,

Not to belabor the point too much:

If I play the open A and open D strings simultaneously on my classical
guitar I do not believe and multiplicative phenomena will occur.

If I have magic strings and continue to tune the A up (and a lot of
patience) by a cent at a time I also believe no multiplicative phenomena
will occur, even when the strings become very close, and then identical in
tuning.

My understanding is that the only phenomena that occurs is constructive and
destructive interference that results from adding the waveforms. It is the
same phenomenon that led to the wave theory of light:

http://en.wikipedia.org/wiki/File:Young_Diffraction.png

http://en.wikipedia.org/wiki/Light

This is the context I am talking about - not where we are involving
envelopes.

People here have been saying we should see difference tones.... and
difference tones appear, as far as I can tell, to be the same result as a
ring modulator, which is multiplicative. And that these tones mysteriously
do not show up in a FFT even though they can be heard.

What keeps knocking in my head is that a ring modulator is a great way to
make inharmonic sounds... and that just doesn't happen in most western
acoustic music......... where would Sethares' theory be if it did?

I figure I can get my classical guitar later today (which has a contact mic)
and run it through the analyzer.

On Thu, Jan 15, 2009 at 4:34 PM, Mike Battaglia <battaglia01@...>wrote:

> On Thu, Jan 15, 2009 at 3:52 PM, <chrisvaisvil@...<chrisvaisvil%40gmail.com>>
> wrote:
> > Isn't it additive? Not multiplicative ?
> > Sent via BlackBerry from T-Mobile
>
> It's multiplicative. The amplitude of the sine wave is changing over
> time. When it's at 0, that's like saying the sine wave is being
> multiplied by 0. If it's ever twice as loud as normal, that's like
> multiplying it by 2. Adding a 1 Hz sine wave to a 440 Hz sine wave
> would sound to us like a 440 Hz sine wave, because the 1 Hz would be
> largely filtered out by the acoustics of the ear canal.
>
> To put it another way: You know you can't hear 1 Hz, but you can
> definitely hear changes in sound in time over 1 second, right?
> Maybe this picture will help explain things:
>
> http://rabbit.eng.miami.edu/students/mbattaglia/sines.png
>
> The top one is a 440 Hz sine wave multiplied by a 1 Hz envelope, and
> the right is the FFT of it, zoomed in to show you that it's made up of
> two frequencies right next to each other, which, incidentally, gives
> you "beating," which is the same exact thing as taking a 440 Hz sine
> wave and multiplying it by a 1 Hz sine wave, which is exactly what you
> hear beating to be. The numbers on the bottom don't indicate what
> frequency it is, but what the "bin number" of the frequency is. I just
> put it there to give you a sense of the scale.
>
> The bottom one is a 440 Hz sine wave added to a 1 Hz sine wave. The
> FFT on the right shows two distinct frequencies, and only two: 440 Hz,
> and 1 Hz.
>
> -Mike
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
wrote:

>
> The first couple minutes is a demo of being a sine, major 3rd,
minor 3rd,
> 5th, major chord, minor chord, and then at about 2 minutes I
amplitude
> modulate the middle C sine with another sine and change the
frequency. What
> happens is a complex waveform is made....
>

Chris,

What is the range of modulation frequency you have used? And what
kind of amplitude modulation is performed in your system? Is it of
kind sin (a t) * cos (b t) (ring modulator) or of kind VCA (voltage
controlled amplifier) that changes the volume (amplitude) of the
generated wave on top of an average volume level, in such a way that
you still have the original tone with additional "sidebands"
(difference and sum frequencies), or else?

I see that there are many kinds of amplitude modulations, ring
modulating seems to be just a particular one of those, I have found
for instance a webpage (not wiki) that analyzes results of different
kinds of mixing in the way they are actually performed in synths:

http://www.soundonsound.com/sos/mar00/articles/synthsecrets.htm

(there are some wrong parentheses within the equations but one can
easily figure out that by doing the mathematical passages that are
not complicated)

Thanks

Max

Hi Max,

It is a VCA modulation with a sine wave. That is evident in the oscilloscope
portion, especially when the modulation rate is slow. I do not have a way,
off the top of my head, to determine the modulating signal's frequency
range.

Let me see if the Korg manual has a mathematical description though I think
they don't.

Chris

(very nice page in the link you gave!)

On Fri, Jan 16, 2009 at 6:52 AM, massimilianolabardi <labardi@...>wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...>
> wrote:
>
> >
> > The first couple minutes is a demo of being a sine, major 3rd,
> minor 3rd,
> > 5th, major chord, minor chord, and then at about 2 minutes I
> amplitude
> > modulate the middle C sine with another sine and change the
> frequency. What
> > happens is a complex waveform is made....
> >
>
> Chris,
>
> What is the range of modulation frequency you have used? And what
> kind of amplitude modulation is performed in your system? Is it of
> kind sin (a t) * cos (b t) (ring modulator) or of kind VCA (voltage
> controlled amplifier) that changes the volume (amplitude) of the
> generated wave on top of an average volume level, in such a way that
> you still have the original tone with additional "sidebands"
> (difference and sum frequencies), or else?
>
> I see that there are many kinds of amplitude modulations, ring
> modulating seems to be just a particular one of those, I have found
> for instance a webpage (not wiki) that analyzes results of different
> kinds of mixing in the way they are actually performed in synths:
>
> http://www.soundonsound.com/sos/mar00/articles/synthsecrets.htm
>
> (there are some wrong parentheses within the equations but one can
> easily figure out that by doing the mathematical passages that are
> not complicated)
>
> Thanks
>
> Max
>
>
>

On Fri, Jan 16, 2009 at 6:38 AM, Chris Vaisvil <chrisvaisvil@...> wrote:
> Mike,
>
> Not to belabor the point too much:
>
> If I play the open A and open D strings simultaneously on my classical
> guitar I do not believe and multiplicative phenomena will occur.
>
> If I have magic strings and continue to tune the A up (and a lot of
> patience) by a cent at a time I also believe no multiplicative phenomena
> will occur, even when the strings become very close, and then identical in
> tuning.
>

I never said they did. There would be no multiplicative phenomena
there. Look at the picture I posted.

> My understanding is that the only phenomena that occurs is constructive and
> destructive interference that results from adding the waveforms. It is the
> same phenomenon that led to the wave theory of light:

> This is the context I am talking about - not where we are involving
> envelopes.

When you bring two notes close together, somehow, mysteriously, it
starts to sound like one note with a sinusoidal envelope, and we call
that beating. How does that not involve envelopes?

> People here have been saying we should see difference tones.... and
> difference tones appear, as far as I can tell, to be the same result as a
> ring modulator, which is multiplicative. And that these tones mysteriously
> do not show up in a FFT even though they can be heard.

I don't think difference tones in this case are the same thing as a
ring modulator. And these tones don't show up in an FFT because they
aren't present in the original signal. When you play them out of a
speaker, the speaker might distort very faintly (which most speakers
do) which would cause the appearance of sum and difference tones. Try
playing one tone out of the left speaker, and one tone out of the
right speaker. If you still hear it, it's possible that nonlinear
effects (aka slight distortion, in a way) are occuring somewhere in
the ear, in the brain, or both for you to hear the difference tones.
But the point of the FFT isn't to model the behavior of that whole
system - just to tell you what frequencies are present in the original
signal. The difference tones are NOT present in the original signal -
some part of the signal chain between you playing it from the speakers
and you perceiving it in your brain is creating them. If you could
somehow model the whole process and get the end result in an audio
file and take the FFT of THAT, you'd see those tones existing.

> What keeps knocking in my head is that a ring modulator is a great way to
> make inharmonic sounds... and that just doesn't happen in most western
> acoustic music......... where would Sethares' theory be if it did?

Well, that's how a ring modulator is often used, but that isn't the
only way it works. A ring modulator multiplies two waveforms together.
Are you intuitively familiar with what that means? Don't get hung up
on the term "ring modulator" - just because it's often used to make
crazy bell-like tones from the future doesn't mean that that is
fundamentally what it IS.

Let's put it this way: If I take one sine wave and I artificially make
it "beat" at a low frequency, that's the same thing as "multiplying"
it by another sine wave with a really low frequency. When the
modulating sine wave's value is 1, the original sine wave will be
operating at full blast. When its value is 0, the sine wave will be
off. And so the original sine wave will go off and on in some kind of
a regular, repeating, "sinusoidal" pattern, which is the exact same
that happens when you have two frequencies close together. If you have
439 Hz and 441 Hz, you'll hear 440 Hz with a sinusoidal envelope.
Isn't that what beating is?

Hi Mike,

" The difference tones are NOT present in the original signal -
some part of the signal chain between you playing it from the speakers
and you perceiving it in your brain is creating them. If you could
somehow model the whole process and get the end result in an audio
file and take the FFT of THAT, you'd see those tones existing."

Perhaps I'm getting hung up on what some people are saying they hear and
what actually is there.

For me at least, I'd like to make and keep that distinction.

For the 440 with 1 hz amplitude modulation vs. 439 + 441 hz simultaneous
sine example:

On an oscilloscope they may very well look the same... and therefore sound
the same.
But I don't think they would look the same in an FFT of the signal.

I can't do anything about brain perception - right now no one (except,
maybe, the military) has the technology to hear as the brain hears.
However.... using a microphone and an acoustic source (speaker, guitar) one
should get really close to what actually is there in the signal.

Any distortion of the input by the brain should go into the psycho-acoustics
bin and be discussed there - not as a physical phenomenon.

=>I have resorted to talking in generalities as to things stated on this
list because I've totally lost track of all the participants :-)

"When you bring two notes close together, somehow, mysteriously, it
starts to sound like one note with a sinusoidal envelope, and we call
that beating. How does that not involve envelopes?"

The cause here is interference... not the imposition of an envelope. In this
particular case you may get to the same place but it is not same phenomena
causing the output.

On Fri, Jan 16, 2009 at 9:31 AM, Mike Battaglia <battaglia01@...>wrote:

> On Fri, Jan 16, 2009 at 6:38 AM, Chris Vaisvil <chrisvaisvil@...<chrisvaisvil%40gmail.com>>
> wrote:
> > Mike,
> >
> > Not to belabor the point too much:
> >
> > If I play the open A and open D strings simultaneously on my classical
> > guitar I do not believe and multiplicative phenomena will occur.
> >
> > If I have magic strings and continue to tune the A up (and a lot of
> > patience) by a cent at a time I also believe no multiplicative phenomena
> > will occur, even when the strings become very close, and then identical
> in
> > tuning.
> >
>
> I never said they did. There would be no multiplicative phenomena
> there. Look at the picture I posted.
>
> > My understanding is that the only phenomena that occurs is constructive
> and
> > destructive interference that results from adding the waveforms. It is
> the
> > same phenomenon that led to the wave theory of light:
>
> > This is the context I am talking about - not where we are involving
> > envelopes.
>
> When you bring two notes close together, somehow, mysteriously, it
> starts to sound like one note with a sinusoidal envelope, and we call
> that beating. How does that not involve envelopes?
>
> > People here have been saying we should see difference tones.... and
> > difference tones appear, as far as I can tell, to be the same result as a
> > ring modulator, which is multiplicative. And that these tones
> mysteriously
> > do not show up in a FFT even though they can be heard.
>
> I don't think difference tones in this case are the same thing as a
> ring modulator. And these tones don't show up in an FFT because they
> aren't present in the original signal. When you play them out of a
> speaker, the speaker might distort very faintly (which most speakers
> do) which would cause the appearance of sum and difference tones. Try
> playing one tone out of the left speaker, and one tone out of the
> right speaker. If you still hear it, it's possible that nonlinear
> effects (aka slight distortion, in a way) are occuring somewhere in
> the ear, in the brain, or both for you to hear the difference tones.
> But the point of the FFT isn't to model the behavior of that whole
> system - just to tell you what frequencies are present in the original
> signal. The difference tones are NOT present in the original signal -
> some part of the signal chain between you playing it from the speakers
> and you perceiving it in your brain is creating them. If you could
> somehow model the whole process and get the end result in an audio
> file and take the FFT of THAT, you'd see those tones existing.
>
> > What keeps knocking in my head is that a ring modulator is a great way to
> > make inharmonic sounds... and that just doesn't happen in most western
> > acoustic music......... where would Sethares' theory be if it did?
>
> Well, that's how a ring modulator is often used, but that isn't the
> only way it works. A ring modulator multiplies two waveforms together.
> Are you intuitively familiar with what that means? Don't get hung up
> on the term "ring modulator" - just because it's often used to make
> crazy bell-like tones from the future doesn't mean that that is
> fundamentally what it IS.
>
> Let's put it this way: If I take one sine wave and I artificially make
> it "beat" at a low frequency, that's the same thing as "multiplying"
> it by another sine wave with a really low frequency. When the
> modulating sine wave's value is 1, the original sine wave will be
> operating at full blast. When its value is 0, the sine wave will be
> off. And so the original sine wave will go off and on in some kind of
> a regular, repeating, "sinusoidal" pattern, which is the exact same
> that happens when you have two frequencies close together. If you have
> 439 Hz and 441 Hz, you'll hear 440 Hz with a sinusoidal envelope.
> Isn't that what beating is?
>
>

On Fri, Jan 16, 2009 at 10:00 AM, Chris Vaisvil <chrisvaisvil@...> wrote:
> Hi Mike,
>
> " The difference tones are NOT present in the original signal -
> some part of the signal chain between you playing it from the speakers
> and you perceiving it in your brain is creating them. If you could
> somehow model the whole process and get the end result in an audio
> file and take the FFT of THAT, you'd see those tones existing."
>
> Perhaps I'm getting hung up on what some people are saying they hear and
> what actually is there.
>
> For me at least, I'd like to make and keep that distinction.

You hear 439 Hz + 441 Hz as 440 Hz with a sinusoidal envelope. Do you not?

> For the 440 with 1 hz amplitude modulation vs. 439 + 441 hz simultaneous
> sine example:
>
> On an oscilloscope they may very well look the same... and therefore sound
> the same.
> But I don't think they would look the same in an FFT of the signal.

They would, in fact, look and be exactly the same. If the little
squiggly waveform would look the same, then the FFT operating on it
would also look the same.

> "When you bring two notes close together, somehow, mysteriously, it
> starts to sound like one note with a sinusoidal envelope, and we call
> that beating. How does that not involve envelopes?"
>
> The cause here is interference... not the imposition of an envelope. In this
> particular case you may get to the same place but it is not same phenomena
> causing the output.

Well I'm not saying that when you add two sine waves together that you
should suddenly lose your memory and think that your tone generation
mechanism was one sine wave and an envelope. The point I'm making is
one of signal processing. No matter what your tone generation
mechanism is, from a Fourier analysis standpoint, the resultant signal
from both of those methods is going to be identical both in terms of
the time-domain waveform and the FFT spectrum. In other words, the two
methods will yield the same signal. This interesting little factoid is
not one of mere coincidence, but reflects something deep and more
fundamental about Fourier analysis and about how we hear sound in
general.

The interference causes a perceived envelope. That's what we call
beating. Similarly, it works in reverse. manually creating this
envelope will miraculously cause two frequencies to exist close to
each other where previously there was one. That is to say, taking a
440 Hz single sine wave and making it "beat" manually will cause 439
Hz and 441 Hz to appear, and 440 Hz to disappear.

-Mike

"That is to say, taking a
440 Hz single sine wave and making it "beat" manually will cause 439
Hz and 441 Hz to appear, and 440 Hz to disappear."

I need to test this because this is, in fact, the thing that makes ring
modulation different. The original signals are not in the output of a ring
modulator.

Chris wrote:

> For the 440 with 1 hz amplitude modulation vs. 439 + 441 hz simultaneous sine example:
> On an oscilloscope they may very well look the same... and therefore sound the same.
> But I don't think they would look the same in an FFT of the signal.

Not only would they look the same there but they would both be exactly the same -- I mean, 100% mathematically identical. If you think they wouldn't, then you're probably using a different effect than the one called "ring modulation".

As to your statement that amplitude modulation results in inharmonic sounds, bear in mind that this is true only for some cases and not for others. Let me show you an excerpt from my "Linear music, version 2". In order you heard precisely what I mean, I'll let you first listen to the recording without any amplitude modulation, without any reverb added, without any noise reduction or any other effects (but I DID use them in the final recording, of course): www.sendspace.com/file/enurtx
And now, here's what comes out after I "ring-modulate" it with a periodic pure sine wave of 180Hz: www.sendspace.com/file/fl6wt6

Petr

Hi Chris,

so if it is a VCA, for sure it is not doing sin (at) * cos (bt).
That could be a part of the total. In particular, it is not possible
to eliminate the tone frequency (the one being modulated in
amplitude). It will always be there. So you cannot compare anymore
the effect of beating between two tones as the appearance of a
middle tone modulated in amplitude (in the sense of sin * cos) and
the effect of amplitude modulation that you are performing by your
synth. If you had a ring modulator - a true sin * cos operation, you
would see only difference and sum frequency.

Cosine product could be easily performed by digital signal
processing too. Something similar is what one can do by matlab,
mathematica etc. I have tried such products with Mathematica and it
worked fine. If you wish you can download the Mathematica notebook
file from a folder Max here within Tuning list.

I hope this helps to understand better this issue of "what is the
difference between: beatings, intermodulation tones, difference
tones, combination tones... Perhaps we should write down a
vocabulaty with the exact meaning of all such terms, and the
relation/differences among them, just to simplify future discussions!

Max

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> Hi Max,
>
> It is a VCA modulation with a sine wave. That is evident in the
oscilloscope
> portion, especially when the modulation rate is slow. I do not
have a way,
> off the top of my head, to determine the modulating signal's
frequency
> range.
>
> Let me see if the Korg manual has a mathematical description
though I think
> they don't.
>
> Chris
>
> (very nice page in the link you gave!)
>
> On Fri, Jan 16, 2009 at 6:52 AM, massimilianolabardi
<labardi@...>wrote:
>
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>,
Chris Vaisvil
> > <chrisvaisvil@>
> > wrote:
> >
> > >
> > > The first couple minutes is a demo of being a sine, major 3rd,
> > minor 3rd,
> > > 5th, major chord, minor chord, and then at about 2 minutes I
> > amplitude
> > > modulate the middle C sine with another sine and change the
> > frequency. What
> > > happens is a complex waveform is made....
> > >
> >
> > Chris,
> >
> > What is the range of modulation frequency you have used? And what
> > kind of amplitude modulation is performed in your system? Is it
of
> > kind sin (a t) * cos (b t) (ring modulator) or of kind VCA
(voltage
> > controlled amplifier) that changes the volume (amplitude) of the
> > generated wave on top of an average volume level, in such a way
that
> > you still have the original tone with additional "sidebands"
> > (difference and sum frequencies), or else?
> >
> > I see that there are many kinds of amplitude modulations, ring
> > modulating seems to be just a particular one of those, I have
found
> > for instance a webpage (not wiki) that analyzes results of
different
> > kinds of mixing in the way they are actually performed in synths:
> >
> > http://www.soundonsound.com/sos/mar00/articles/synthsecrets.htm
> >
> > (there are some wrong parentheses within the equations but one
can
> > easily figure out that by doing the mathematical passages that
are
> > not complicated)
> >
> > Thanks
> >
> > Max
> >
> >
> >
>

Hi Max

A vocabulary is probably a good idea. The korg MS2000 does have a ring modulator. Just haven't got there yet in these tests. Figuring out how to capture video was a time consumer. I might have to resort to my daughters xp machine to do it right. Unless someone knows of one for ubuntu.

Chris
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "massimilianolabardi" <labardi@df.unipi.it>

Date: Fri, 16 Jan 2009 14:20:10
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Hi Chris,

so if it is a VCA, for sure it is not doing sin (at) * cos (bt).
That could be a part of the total. In particular, it is not possible
to eliminate the tone frequency (the one being modulated in
amplitude). It will always be there. So you cannot compare anymore
the effect of beating between two tones as the appearance of a
middle tone modulated in amplitude (in the sense of sin * cos) and
the effect of amplitude modulation that you are performing by your
synth. If you had a ring modulator - a true sin * cos operation, you
would see only difference and sum frequency.

Cosine product could be easily performed by digital signal
processing too. Something similar is what one can do by matlab,
mathematica etc. I have tried such products with Mathematica and it
worked fine. If you wish you can download the Mathematica notebook
file from a folder Max here within Tuning list.

I hope this helps to understand better this issue of "what is the
difference between: beatings, intermodulation tones, difference
tones, combination tones... Perhaps we should write down a
vocabulaty with the exact meaning of all such terms, and the
relation/differences among them, just to simplify future discussions!

Max

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> Hi Max,
>
> It is a VCA modulation with a sine wave. That is evident in the
oscilloscope
> portion, especially when the modulation rate is slow. I do not
have a way,
> off the top of my head, to determine the modulating signal's
frequency
> range.
>
> Let me see if the Korg manual has a mathematical description
though I think
> they don't.
>
> Chris
>
> (very nice page in the link you gave!)
>
> On Fri, Jan 16, 2009 at 6:52 AM, massimilianolabardi
<labardi@...>wrote:
>
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>,
Chris Vaisvil
> > <chrisvaisvil@>
> > wrote:
> >
> > >
> > > The first couple minutes is a demo of being a sine, major 3rd,
> > minor 3rd,
> > > 5th, major chord, minor chord, and then at about 2 minutes I
> > amplitude
> > > modulate the middle C sine with another sine and change the
> > frequency. What
> > > happens is a complex waveform is made....
> > >
> >
> > Chris,
> >
> > What is the range of modulation frequency you have used? And what
> > kind of amplitude modulation is performed in your system? Is it
of
> > kind sin (a t) * cos (b t) (ring modulator) or of kind VCA
(voltage
> > controlled amplifier) that changes the volume (amplitude) of the
> > generated wave on top of an average volume level, in such a way
that
> > you still have the original tone with additional "sidebands"
> > (difference and sum frequencies), or else?
> >
> > I see that there are many kinds of amplitude modulations, ring
> > modulating seems to be just a particular one of those, I have
found
> > for instance a webpage (not wiki) that analyzes results of
different
> > kinds of mixing in the way they are actually performed in synths:
> >
> > http://www.soundonsound.com/sos/mar00/articles/synthsecrets.htm
> >
> > (there are some wrong parentheses within the equations but one
can
> > easily figure out that by doing the mathematical passages that
are
> > not complicated)
> >
> > Thanks
> >
> > Max
> >
> >
> >
>

Max wrote:

> I hope this helps to understand better this issue of "what is the
> difference between: beatings, intermodulation tones, difference
> tones, combination tones... Perhaps we should write down a
> vocabulaty with the exact meaning of all such terms, and the
> relation/differences among them, just to simplify future discussions!

I would very much welcome such an effort. I could at least
add it to my growing collection of material for a FAQ.

-Carl

Let me explain (as another happy MS2000 owner) that MS2000 is virtual analog machine, where all sound generating and processing is done digitally, on software basis. As I suppose, including ring modulation. Nevertheless the result is the same like with real analog ring modulator.

Older Korg analog synthesizers (like MS20) had RM built from NAND gate IC's as far as I know. I can find somewhere electronic scheme...

Daniel Forro

On 17 Jan 2009, at 7:28 AM, chrisvaisvil@... wrote:

> Hi Max
>
> A vocabulary is probably a good idea. The korg MS2000 does have a > ring modulator. Just haven't got there yet in these tests. Figuring > out how to capture video was a time consumer. I might have to > resort to my daughters xp machine to do it right. Unless someone > knows of one for ubuntu.
>
> Chris

Happy yes!

On Fri, Jan 16, 2009 at 6:53 PM, Daniel Forro <dan.for@tiscali.cz> wrote:

> Let me explain (as another happy MS2000 owner) that MS2000 is virtual
> analog machine, where all sound generating and processing is done
> digitally, on software basis. As I suppose, including ring
> modulation. Nevertheless the result is the same like with real analog
> ring modulator.
>
> Older Korg analog synthesizers (like MS20) had RM built from NAND
> gate IC's as far as I know. I can find somewhere electronic scheme...
>
> Daniel Forro
>
>
> On 17 Jan 2009, at 7:28 AM, chrisvaisvil@...<chrisvaisvil%40gmail.com>wrote:
>
> > Hi Max
> >
> > A vocabulary is probably a good idea. The korg MS2000 does have a
> > ring modulator. Just haven't got there yet in these tests. Figuring
> > out how to capture video was a time consumer. I might have to
> > resort to my daughters xp machine to do it right. Unless someone
> > knows of one for ubuntu.
> >
> > Chris
>
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I would very much welcome such an effort. I could at least
> add it to my growing collection of material for a FAQ.
>

Hi Carl,

I could start to write down definitions for a few terms, that of course
would reflect my point of view. If you and/or others of the list could
then improve or add differences and similarities, probably the final
result could be useful to others and included in FAQs. Perhaps we could
then also contribute to some entry in Wikipedia. I have never done
that, but sooner or later one should start...

Max

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, chrisvaisvil@ wrote:
> >
> > Why do you think the sine waves were not pure.
>
> Because the chords were beating.
>
> -Carl
>

Hi Carl,

you can see that there may be beatings in chords also using pure
tones. They seem to come out from a combination of beating
frequencies of the composing dyads with each other. If you like, have
a look to files in folder "Max" in this list. They show the waveform
and envelope of a triad made up of pure tones, in the case of the
middle tone of a perfect major third is displaced by a detuning
frequency df. Beatings (periodic amplitude variations) are coming out
at frequency 2 df.

Cheers

Max

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

> > I would very much welcome such an effort. I could at least
> > add it to my growing collection of material for a FAQ.
>
> Hi Carl,
>
> I could start to write down definitions for a few terms, that
> of course would reflect my point of view. If you and/or others
> of the list could then improve or add differences and
> similarities, probably the final result could be useful to
> others and included in FAQs. Perhaps we could then also
> contribute to some entry in Wikipedia. I have never done that,
> but sooner or later one should start...
>
> Max

There is

http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Tunings,_Temperaments,_and_Scales

But the scope doesn't really include psychoacoustics.
The psychoacoustics content on wikipedia is badly in need
of help, though.

-Carl

--- In tuning@yahoogroups.com, "massimilianolabardi" <labardi@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > --- In tuning@yahoogroups.com, chrisvaisvil@ wrote:
> > >
> > > Why do you think the sine waves were not pure.
> >
> > Because the chords were beating.
> >
> > -Carl
> >
>
> Hi Carl,
>
> you can see that there may be beatings in chords also using pure
> tones. They seem to come out from a combination of beating
> frequencies of the composing dyads with each other.

You'll have to convince me.

> If you like, have a look to files in folder "Max" in this list.
> They show the waveform and envelope of a triad made up of pure
> tones, in the case of the middle tone of a perfect major third
> is displaced by a detuning frequency df. Beatings (periodic
> amplitude variations) are coming out at frequency 2 df.

I can't view PDFs on this computer at the moment. I guess I
should just fire up my old computer and synthesize some
beatless triads (as well as the other demos I suggested)
before the thread dies.

-Carl

Let me try the Korg Pure Major scale with sines

and.... yes I got off track on the video, but honestly I've been busy.
Wife, kid, job and all that... I imagine I'm not alone in that respect.

I also have to learn how to use the tools to accomplish this as well.

On Sat, Jan 17, 2009 at 3:40 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>,
> "massimilianolabardi" <labardi@...> wrote:
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Carl Lumma"
> <carl@> wrote:
> > >
> > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>,
> chrisvaisvil@ wrote:
> > > >
> > > > Why do you think the sine waves were not pure.
> > >
> > > Because the chords were beating.
> > >
> > > -Carl
> > >
> >
> > Hi Carl,
> >
> > you can see that there may be beatings in chords also using pure
> > tones. They seem to come out from a combination of beating
> > frequencies of the composing dyads with each other.
>
> You'll have to convince me.
>
> > If you like, have a look to files in folder "Max" in this list.
> > They show the waveform and envelope of a triad made up of pure
> > tones, in the case of the middle tone of a perfect major third
> > is displaced by a detuning frequency df. Beatings (periodic
> > amplitude variations) are coming out at frequency 2 df.
>
> I can't view PDFs on this computer at the moment. I guess I
> should just fire up my old computer and synthesize some
> beatless triads (as well as the other demos I suggested)
> before the thread dies.
>
> -Carl
>
>
>

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

> > Hi Carl,
> >
> > you can see that there may be beatings in chords also using pure
> > tones. They seem to come out from a combination of beating
> > frequencies of the composing dyads with each other.
>
> You'll have to convince me.
>

Hi Carl,

I have uploaded a zipped folder containing an HTML output of the
Mathematica notebook. You should be able to see it with any web browser.

About the maths please have a look to one of my previous posts:

/tuning/topicId_79751.html#79976

However, if you plot cos(at)+cos(bt)+cos(ct) you will see exactly the
beatings. Modulation depth is not full (it is something like 27% if
amplitude of the three tones are equal). What is their origin is
another issue: by mathematical development we can have a hint. At least
for a major triad with detuning of the middle tone, it seems to happen
that the 5-1 dyad gives a mean tone whose amplitude modulation
is "fast", that is, at the difference frequency of the dyad, plus
entangled combinations of 3-1 and 5-3 dyads that are amplitude-
modulated at the double of the detuning frequency. This is what comes
out from both mathematics... and speakers (even at low volume). You can
also play Mathematica's waves and you can clearly hear the envelopes
seen in the waveforms. The sound is pretty the same than the one of
Chris' synth samples.

Max

http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1232242785
*http://tinyurl.com/test2tuning* <http://tinyurl.com/test2tuning>

ok, there is again noise that wasn't present when monitoring. I think the
problem is a difference in sampling rate between the thinkpad input and
video capture... but since this is on an XP machine everything was captured
real time.

The pure major chord did beat.

of interest is when I played C4 and C#4 - the peaks split - 4 signals were
present. When I went back and did osc 1 and osc 2 added together with osc 2
freely tuned there is an area where the peaks split - at osc 2 pitches about
and below (but still above C4) the peaks do not spilt.

C5 and C#5 together do not split....

So.... this is an interesting result.

On Sat, Jan 17, 2009 at 6:40 PM, massimilianolabardi <labardi@....it>wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Carl Lumma"
> <carl@...> wrote:
>
> > > Hi Carl,
> > >
> > > you can see that there may be beatings in chords also using pure
> > > tones. They seem to come out from a combination of beating
> > > frequencies of the composing dyads with each other.
> >
> > You'll have to convince me.
> >
>
> Hi Carl,
>
> I have uploaded a zipped folder containing an HTML output of the
> Mathematica notebook. You should be able to see it with any web browser.
>
> About the maths please have a look to one of my previous posts:
>
> /tuning/topicId_79751.html#79976
>
> However, if you plot cos(at)+cos(bt)+cos(ct) you will see exactly the
> beatings. Modulation depth is not full (it is something like 27% if
> amplitude of the three tones are equal). What is their origin is
> another issue: by mathematical development we can have a hint. At least
> for a major triad with detuning of the middle tone, it seems to happen
> that the 5-1 dyad gives a mean tone whose amplitude modulation
> is "fast", that is, at the difference frequency of the dyad, plus
> entangled combinations of 3-1 and 5-3 dyads that are amplitude-
> modulated at the double of the detuning frequency. This is what comes
> out from both mathematics... and speakers (even at low volume). You can
> also play Mathematica's waves and you can clearly hear the envelopes
> seen in the waveforms. The sound is pretty the same than the one of
> Chris' synth samples.
>
> Max
>
>
>
>

> I guess I should just fire up my old computer and synthesize
> some beatless triads (as well as the other demos I suggested)
> before the thread dies.

I did so:

http://lumma.org/stuff/majmin.zip

~ 2.7 MB.

Unfortunately, I can't synthesize 6-voice chords with
Cool Edit. But I did make otonal and utonal versions of
4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
Each file is 8 seconds long, and begins with 2 seconds of
the outer dyad alone, followed by 6 seconds of the complete
chord. All chords are rooted on C = 264 Hz. Files are
rendered directly at -15dB. Combination tones didn't much
effect the comparison, but I found I could indeed make them
disappear on my speakers by playing them at moderate volume.
With in-ear monitors, combination tones were present at
almost any volume unless I pulled the phones out of my ear
canals and only rested them on my conchae
http://en.wikipedia.org/wiki/Pinna_(anatomy)

I hear no beating in any of the samples, but I do hear that
the otonal chords are more consonant. The contrast becomes
complete in the 11-limit. The 9-limit utonal pentad only
sounds as good as it does because of the 4:6:9 backbone of
fifths that it shares with the otonal pentad. Otherwise,
the otonal/utonal contrast steadily increases with odd limit.

-Carl

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

> I hear no beating in any of the samples, but I do hear that
> the otonal chords are more consonant. The contrast becomes
> complete in the 11-limit. The 9-limit utonal pentad only
> sounds as good as it does because of the 4:6:9 backbone of
> fifths that it shares with the otonal pentad. Otherwise,
> the otonal/utonal contrast steadily increases with odd limit.

Hi Carl,

there is no doubt that just intonation chords should not beat.
Beating chords are, up to now, the ones played by piano (12-TET) as
well as the ones played by Chris by pure tones in 12-TET. The beating
we are talking about is the one due to slight deviation of equal
temperament from just intonation.

Thanks for the samples, they are very explanatory. Actually otonal
chords are much more consonant than utonal ones.

Max

But, with all due respect, you can see the beating, ie, the constructive /
destructive interference.

That is plainly visible when you look a the wave with sufficient zoom.

For what it is worth my last video used the "pure major" scale .

On Sun, Jan 18, 2009 at 4:50 AM, Carl Lumma <carl@...> wrote:

> > I guess I should just fire up my old computer and synthesize
> > some beatless triads (as well as the other demos I suggested)
> > before the thread dies.
>
> I did so:
>
> http://lumma.org/stuff/majmin.zip
>
> ~ 2.7 MB.
>
> Unfortunately, I can't synthesize 6-voice chords with
> Cool Edit. But I did make otonal and utonal versions of
> 4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
> Each file is 8 seconds long, and begins with 2 seconds of
> the outer dyad alone, followed by 6 seconds of the complete
> chord. All chords are rooted on C = 264 Hz. Files are
> rendered directly at -15dB. Combination tones didn't much
> effect the comparison, but I found I could indeed make them
> disappear on my speakers by playing them at moderate volume.
> With in-ear monitors, combination tones were present at
> almost any volume unless I pulled the phones out of my ear
> canals and only rested them on my conchae
> http://en.wikipedia.org/wiki/Pinna_(anatomy)
>
> I hear no beating in any of the samples, but I do hear that
> the otonal chords are more consonant. The contrast becomes
> complete in the 11-limit. The 9-limit utonal pentad only
> sounds as good as it does because of the 4:6:9 backbone of
> fifths that it shares with the otonal pentad. Otherwise,
> the otonal/utonal contrast steadily increases with odd limit.
>
> -Carl
>
>
>

http://clones.soonlabel.com/tis/tones.jpg

On Sun, Jan 18, 2009 at 11:14 AM, Chris Vaisvil <chrisvaisvil@...>wrote:

> But, with all due respect, you can see the beating, ie, the constructive /
> destructive interference.
>
> That is plainly visible when you look a the wave with sufficient zoom.
>
> For what it is worth my last video used the "pure major" scale .
>
>
>
> On Sun, Jan 18, 2009 at 4:50 AM, Carl Lumma <carl@...> wrote:
>
>> > I guess I should just fire up my old computer and synthesize
>> > some beatless triads (as well as the other demos I suggested)
>> > before the thread dies.
>>
>> I did so:
>>
>> http://lumma.org/stuff/majmin.zip
>>
>> ~ 2.7 MB.
>>
>> Unfortunately, I can't synthesize 6-voice chords with
>> Cool Edit. But I did make otonal and utonal versions of
>> 4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
>> Each file is 8 seconds long, and begins with 2 seconds of
>> the outer dyad alone, followed by 6 seconds of the complete
>> chord. All chords are rooted on C = 264 Hz. Files are
>> rendered directly at -15dB. Combination tones didn't much
>> effect the comparison, but I found I could indeed make them
>> disappear on my speakers by playing them at moderate volume.
>> With in-ear monitors, combination tones were present at
>> almost any volume unless I pulled the phones out of my ear
>> canals and only rested them on my conchae
>> http://en.wikipedia.org/wiki/Pinna_(anatomy)
>>
>> I hear no beating in any of the samples, but I do hear that
>> the otonal chords are more consonant. The contrast becomes
>> complete in the 11-limit. The 9-limit utonal pentad only
>> sounds as good as it does because of the 4:6:9 backbone of
>> fifths that it shares with the otonal pentad. Otherwise,
>> the otonal/utonal contrast steadily increases with odd limit.
>>
>> -Carl
>>
>>
>>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> But, with all due respect, you can see the beating, ie, the
constructive /
> destructive interference.
>
> That is plainly visible when you look a the wave with sufficient zoom.
>
> For what it is worth my last video used the "pure major" scale .
>

Hi Chris,

in your video there are many tests all together, and frankly I could
clearly follow just a few of them. Especially, it seems that FFT is
calculated on a rather long time window so that when you change a tone,
you see the old one slowly disappearing from the spectrum while you see
the newer one slowly appearing. Sometimes the tones are changed too
quickly, so that the "steady state" spectrum does never show up.

Anyway, it would be nice to see the waveform on a longer timescale,
e.g. 1 or 2 seconds, so that if you have slow beatings (e.g. 1 Hz or
even 5 Hz) you can see the envelope of the waveform.

With just intonation there should be no "slow" envelope when you play
the major triad with pure tones. There should be some "fast" beating
(or what I call fast beating), that is an amplitude envelope at a
frequency equal to the difference frequency of the single dyads, that
on the other hand is not much different from the frequency of the tones
themselves (e.g. 55 Hz (and 110 Hz) for a triad made up by 220 Hz A,
275 Hz C# and 330 Hz E), and for sure that the ear cannot follow--- or
let'say it better: when the ear can follow the amplitude beating, it is
not able to hear distinct tones (we hear one mean tone with amplitude
envelope), while when the ear cannot follow the amplitude beating
anymore (because it becomes too fast), then it becomes able to "split"
the distinct tones of the dyad or chord.

What is intriguing me is: is there a physical mechanism (I am not
looking for psychoacoustic explanations, because I have no scientific
basis to understand those) lying below this phenomenon of passage from
one mean tone with amplitude beating to split tones with no beating -
and all the intermediate situations of course, that seemingly cause the
sensation of dissonance - occurs in the ear? I mean, are there physical
reasons for that, or it is completely or mainly due to psychoacoustics?

After the interesting discussion going on here on the tuning list, the
focus of my curiosity definitely moves to the ear's (physical) working
principle. That is: why a detector (ear) is capable to record slow
amplitude variations, to hear a range of acoustic frequencies fron 20
Hz to 20kHz or so, but not to hear an amplitude variation faster than
20-30 Hz (or, better, is capable to hear it but it renders such beating
as the presence of two tones)? There must be something tricky in this,
since at the threshold between such two behaviors an "annoying"
sensation comes out (dissonance)... The explanation could be that
everything is due to a psychoacoustical effect, of course, but I would
love to rule out possible purely-physical effects before giving the
easier explanation that "brain is doing everything"!

Max

Hi Max,

I agree about the flaws of the video. (and the rest of your points too -
especially fast beating)

The FFT is averaged to eliminate noise and causes it to be really sluggish.

If no one has a higher priority I think it would be most interesting to zoom
in on the beating/spliting/beating morfing area. I want to nail down the
frequencies and see if I can reproduce it in cool edit or audacity like
Carl's examples.

I can pause the video while doing this and set up the next example - or I'll
do stills.

Chris

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> But, with all due respect, you can see the beating, ie, the
> constructive / destructive interference.
>
> That is plainly visible when you look a the wave with sufficient
> zoom.

No. You're either talking about the shape of the waveform
itself, or the way wave editors truncate it when they are
displaying it at less than full scale.

-Carl

Max wrote:
> What is intriguing me is: is there a physical mechanism (I am
> not looking for psychoacoustic explanations, because I have no
> scientific basis to understand those) lying below this
> phenomenon of passage from one mean tone with amplitude
> beating to split tones with no beating - and all the
> intermediate situations of course, that seemingly cause the
> sensation of dissonance - occurs in the ear? I mean, are
> there physical reasons for that, or it is completely or
> mainly due to psychoacoustics?

It's completely due to the spectral resolution of the
cochlea. If you want to call that psychoacoustics, it's
fine with me. When the two tones are closer than the
cochlea can resolve, we hear a single tone.

Basilar membrane with two clear tones heard:

_____/^^\___/^^\______

Basilar membrane with two clear tones heard:

______/^^\_/^^\_______

Basilar membrane with one rough tone heard:

_______/^^^^\_________

Basilar membrane with one clear tone heard:

________/^^\__________

> but not to hear an amplitude variation faster than
> 20-30 Hz (or, better, is capable to hear it but it
> renders such beating as the presence of two tones)?

It has nothing to do with the rate of beating. The
rate at which the two tones finally emerge will depend
on the width of the critical band (specral resolution
of the cochlea). Since the critical bandwidth changes
with respect to frequency, the beat rate, f1-f2, at
1/2CB or whatever point you consider the two tones to
be distinct, will also be different.

-Carl

Look at the supplied jpeg. Its pretty clear that there is a longer periodicity.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@...>

Date: Sun, 18 Jan 2009 22:30:52
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> But, with all due respect, you can see the beating, ie, the
> constructive / destructive interference.
>
> That is plainly visible when you look a the wave with sufficient
> zoom.

No. You're either talking about the shape of the waveform
itself, or the way wave editors truncate it when they are
displaying it at less than full scale.

-Carl

Is this the waveform with beating? I don't see any beating anywhere.
-Mike

On Sun, Jan 18, 2009 at 11:36 AM, Chris Vaisvil <chrisvaisvil@...> wrote:
> http://clones.soonlabel.com/tis/tones.jpg
>
>
> On Sun, Jan 18, 2009 at 11:14 AM, Chris Vaisvil <chrisvaisvil@...>
> wrote:
>>
>> But, with all due respect, you can see the beating, ie, the constructive /
>> destructive interference.
>>
>> That is plainly visible when you look a the wave with sufficient zoom.
>>
>> For what it is worth my last video used the "pure major" scale .
>>
>>
>> On Sun, Jan 18, 2009 at 4:50 AM, Carl Lumma <carl@...> wrote:
>>>
>>> > I guess I should just fire up my old computer and synthesize
>>> > some beatless triads (as well as the other demos I suggested)
>>> > before the thread dies.
>>>
>>> I did so:
>>>
>>> http://lumma.org/stuff/majmin.zip
>>>
>>> ~ 2.7 MB.
>>>
>>> Unfortunately, I can't synthesize 6-voice chords with
>>> Cool Edit. But I did make otonal and utonal versions of
>>> 4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
>>> Each file is 8 seconds long, and begins with 2 seconds of
>>> the outer dyad alone, followed by 6 seconds of the complete
>>> chord. All chords are rooted on C = 264 Hz. Files are
>>> rendered directly at -15dB. Combination tones didn't much
>>> effect the comparison, but I found I could indeed make them
>>> disappear on my speakers by playing them at moderate volume.
>>> With in-ear monitors, combination tones were present at
>>> almost any volume unless I pulled the phones out of my ear
>>> canals and only rested them on my conchae
>>> http://en.wikipedia.org/wiki/Pinna_(anatomy)
>>>
>>> I hear no beating in any of the samples, but I do hear that
>>> the otonal chords are more consonant. The contrast becomes
>>> complete in the 11-limit. The 9-limit utonal pentad only
>>> sounds as good as it does because of the 4:6:9 backbone of
>>> fifths that it shares with the otonal pentad. Otherwise,
>>> the otonal/utonal contrast steadily increases with odd limit.
>>>
>>> -Carl
>>>
>>
>
>

I would appreciate if you and Carl (and anyone else how cares) could come up with a solid definition of beating based on physics. To me what I read here is inconsistant.

On another note this phenomenon of spliting that did occur is quite interesting. I want to determine if it is a hardware limitation or real. My point of view did not anticipate this. If it is real then there is a frequency ratio range that is different under the condition of the test I ran.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Mike Battaglia <battaglia01@gmail.com>

Date: Sun, 18 Jan 2009 18:51:55
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
triads,Re:Beatings vs Intermodulation

Is this the waveform with beating? I don't see any beating anywhere.
-Mike

On Sun, Jan 18, 2009 at 11:36 AM, Chris Vaisvil <chrisvaisvil@gmail.com> wrote:
> http://clones.soonlabel.com/tis/tones.jpg
>
>
> On Sun, Jan 18, 2009 at 11:14 AM, Chris Vaisvil <chrisvaisvil@gmail.com>
> wrote:
>>
>> But, with all due respect, you can see the beating, ie, the constructive /
>> destructive interference.
>>
>> That is plainly visible when you look a the wave with sufficient zoom.
>>
>> For what it is worth my last video used the "pure major" scale .
>>
>>
>> On Sun, Jan 18, 2009 at 4:50 AM, Carl Lumma <carl@lumma.org> wrote:
>>>
>>> > I guess I should just fire up my old computer and synthesize
>>> > some beatless triads (as well as the other demos I suggested)
>>> > before the thread dies.
>>>
>>> I did so:
>>>
>>> http://lumma.org/stuff/majmin.zip
>>>
>>> ~ 2.7 MB.
>>>
>>> Unfortunately, I can't synthesize 6-voice chords with
>>> Cool Edit. But I did make otonal and utonal versions of
>>> 4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
>>> Each file is 8 seconds long, and begins with 2 seconds of
>>> the outer dyad alone, followed by 6 seconds of the complete
>>> chord. All chords are rooted on C = 264 Hz. Files are
>>> rendered directly at -15dB. Combination tones didn't much
>>> effect the comparison, but I found I could indeed make them
>>> disappear on my speakers by playing them at moderate volume.
>>> With in-ear monitors, combination tones were present at
>>> almost any volume unless I pulled the phones out of my ear
>>> canals and only rested them on my conchae
>>> http://en.wikipedia.org/wiki/Pinna_(anatomy)
>>>
>>> I hear no beating in any of the samples, but I do hear that
>>> the otonal chords are more consonant. The contrast becomes
>>> complete in the 11-limit. The 9-limit utonal pentad only
>>> sounds as good as it does because of the 4:6:9 backbone of
>>> fifths that it shares with the otonal pentad. Otherwise,
>>> the otonal/utonal contrast steadily increases with odd limit.
>>>
>>> -Carl
>>>
>>
>
>

Chris,

Constructive and destructive interference is only part of the reason
that beating exists. If you took an entire song and you got the FFT of
the whole song, it would give you a bunch of sine waves of different
frequencies that add together and magically interfere constructively
and destructively to create the whole song. Frequencies don't
necessarily translate to pitches.

Another thing: a sawtooth wave has the particular shape it does
because of constructive and destructive interference between the
tones.

A little thought exercise for you: Let's say you have a sine wave at
A440 that lasts for 2 seconds. Then let's say you have a sine wave at
A440 that lasts for 1 second and is then off for one second. How will
the FFT's of those two signals differ? The FFT of the first one will
contain a single tone. The FFT of the second one will contain a whole
bunch of tones that miraculously add together to create a sine wave
for the first half and silence for the second half.

440 Hz and 880 Hz will add together and constructively and
destructively interfere with one another, nonetheless, you don't
really hear beating there.

-Mike

On Sun, Jan 18, 2009 at 7:03 PM, <chrisvaisvil@...> wrote:
> I would appreciate if you and Carl (and anyone else how cares) could come up
> with a solid definition of beating based on physics. To me what I read here
> is inconsistant.
>
> On another note this phenomenon of spliting that did occur is quite
> interesting. I want to determine if it is a hardware limitation or real. My
> point of view did not anticipate this. If it is real then there is a
> frequency ratio range that is different under the condition of the test I
> ran.
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: Mike Battaglia
> Date: Sun, 18 Jan 2009 18:51:55 -0500
> To: <tuning@yahoogroups.com>
> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> Is this the waveform with beating? I don't see any beating anywhere.
> -Mike
>
> On Sun, Jan 18, 2009 at 11:36 AM, Chris Vaisvil <chrisvaisvil@...>
> wrote:
>> http://clones.soonlabel.com/tis/tones.jpg
>>
>>
>> On Sun, Jan 18, 2009 at 11:14 AM, Chris Vaisvil <chrisvaisvil@...>
>> wrote:
>>>
>>> But, with all due respect, you can see the beating, ie, the constructive
>>> /
>>> destructive interference.
>>>
>>> That is plainly visible when you look a the wave with sufficient zoom.
>>>
>>> For what it is worth my last video used the "pure major" scale .
>>>
>>>
>>> On Sun, Jan 18, 2009 at 4:50 AM, Carl Lumma <carl@...> wrote:
>>>>
>>>> > I guess I should just fire up my old computer and synthesize
>>>> > some beatless triads (as well as the other demos I suggested)
>>>> > before the thread dies.
>>>>
>>>> I did so:
>>>>
>>>> http://lumma.org/stuff/majmin.zip
>>>>
>>>> ~ 2.7 MB.
>>>>
>>>> Unfortunately, I can't synthesize 6-voice chords with
>>>> Cool Edit. But I did make otonal and utonal versions of
>>>> 4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
>>>> Each file is 8 seconds long, and begins with 2 seconds of
>>>> the outer dyad alone, followed by 6 seconds of the complete
>>>> chord. All chords are rooted on C = 264 Hz. Files are
>>>> rendered directly at -15dB. Combination tones didn't much
>>>> effect the comparison, but I found I could indeed make them
>>>> disappear on my speakers by playing them at moderate volume.
>>>> With in-ear monitors, combination tones were present at
>>>> almost any volume unless I pulled the phones out of my ear
>>>> canals and only rested them on my conchae
>>>> http://en.wikipedia.org/wiki/Pinna_(anatomy)
>>>>
>>>> I hear no beating in any of the samples, but I do hear that
>>>> the otonal chords are more consonant. The contrast becomes
>>>> complete in the 11-limit. The 9-limit utonal pentad only
>>>> sounds as good as it does because of the 4:6:9 backbone of
>>>> fifths that it shares with the otonal pentad. Otherwise,
>>>> the otonal/utonal contrast steadily increases with odd limit.
>>>>
>>>> -Carl
>>>>
>>>
>>
>>
>
>

>>> But, with all due respect, you can see the beating, ie, the
>>> constructive / destructive interference.
>>>
>>> That is plainly visible when you look a the wave with sufficient
>>> zoom.
>>
>> No. You're either talking about the shape of the waveform
>> itself, or the way wave editors truncate it when they are
>> displaying it at less than full scale.
>
> Look at the supplied jpeg. Its pretty clear that there is a
> longer periodicity.

I'm looking at
http://clones.soonlabel.com/tis/tones.jpg
not sure what you mean.

Utonal chords will have a longer periodicity than otonal chords,
but that's got nothing to do with beating.

-Carl

> Hi Carl,
>
> I have uploaded a zipped folder containing an HTML output of the
> Mathematica notebook. You should be able to see it with any web
> browser.
>

Thanks Max (and Chris), there does seem to be something going
on here. I must say I was not aware of this effect. Interesting!

-Carl

Mike in 1978 I took a course in electronic music as part of my musuic degree. I am in the process of scanning a couple pages for you and Carl. Essentially interference is the only reason for beating and combination tones and the two phenomena differ only in frequency.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Mike Battaglia <battaglia01@gmail.com>

Date: Sun, 18 Jan 2009 19:45:10
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
triads,Re:Beatings vs Intermodulation

Chris,

Constructive and destructive interference is only part of the reason
that beating exists. If you took an entire song and you got the FFT of
the whole song, it would give you a bunch of sine waves of different
frequencies that add together and magically interfere constructively
and destructively to create the whole song. Frequencies don't
necessarily translate to pitches.

Another thing: a sawtooth wave has the particular shape it does
because of constructive and destructive interference between the
tones.

A little thought exercise for you: Let's say you have a sine wave at
A440 that lasts for 2 seconds. Then let's say you have a sine wave at
A440 that lasts for 1 second and is then off for one second. How will
the FFT's of those two signals differ? The FFT of the first one will
contain a single tone. The FFT of the second one will contain a whole
bunch of tones that miraculously add together to create a sine wave
for the first half and silence for the second half.

440 Hz and 880 Hz will add together and constructively and
destructively interfere with one another, nonetheless, you don't
really hear beating there.

-Mike

On Sun, Jan 18, 2009 at 7:03 PM, <chrisvaisvil@gmail.com> wrote:
> I would appreciate if you and Carl (and anyone else how cares) could come up
> with a solid definition of beating based on physics. To me what I read here
> is inconsistant.
>
> On another note this phenomenon of spliting that did occur is quite
> interesting. I want to determine if it is a hardware limitation or real. My
> point of view did not anticipate this. If it is real then there is a
> frequency ratio range that is different under the condition of the test I
> ran.
>
> Sent via BlackBerry from T-Mobile
>
> ________________________________
> From: Mike Battaglia
> Date: Sun, 18 Jan 2009 18:51:55 -0500
> To: <tuning@yahoogroups.com>
> Subject: Re: [tuning] Explaining major 4:5:6 and minor 10:12:15
> triads,Re:Beatings vs Intermodulation
>
> Is this the waveform with beating? I don't see any beating anywhere.
> -Mike
>
> On Sun, Jan 18, 2009 at 11:36 AM, Chris Vaisvil <chrisvaisvil@gmail.com>
> wrote:
>> http://clones.soonlabel.com/tis/tones.jpg
>>
>>
>> On Sun, Jan 18, 2009 at 11:14 AM, Chris Vaisvil <chrisvaisvil@gmail.com>
>> wrote:
>>>
>>> But, with all due respect, you can see the beating, ie, the constructive
>>> /
>>> destructive interference.
>>>
>>> That is plainly visible when you look a the wave with sufficient zoom.
>>>
>>> For what it is worth my last video used the "pure major" scale .
>>>
>>>
>>> On Sun, Jan 18, 2009 at 4:50 AM, Carl Lumma <carl@lumma.org> wrote:
>>>>
>>>> > I guess I should just fire up my old computer and synthesize
>>>> > some beatless triads (as well as the other demos I suggested)
>>>> > before the thread dies.
>>>>
>>>> I did so:
>>>>
>>>> http://lumma.org/stuff/majmin.zip
>>>>
>>>> ~ 2.7 MB.
>>>>
>>>> Unfortunately, I can't synthesize 6-voice chords with
>>>> Cool Edit. But I did make otonal and utonal versions of
>>>> 4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
>>>> Each file is 8 seconds long, and begins with 2 seconds of
>>>> the outer dyad alone, followed by 6 seconds of the complete
>>>> chord. All chords are rooted on C = 264 Hz. Files are
>>>> rendered directly at -15dB. Combination tones didn't much
>>>> effect the comparison, but I found I could indeed make them
>>>> disappear on my speakers by playing them at moderate volume.
>>>> With in-ear monitors, combination tones were present at
>>>> almost any volume unless I pulled the phones out of my ear
>>>> canals and only rested them on my conchae
>>>> http://en.wikipedia.org/wiki/Pinna_(anatomy)
>>>>
>>>> I hear no beating in any of the samples, but I do hear that
>>>> the otonal chords are more consonant. The contrast becomes
>>>> complete in the 11-limit. The 9-limit utonal pentad only
>>>> sounds as good as it does because of the 4:6:9 backbone of
>>>> fifths that it shares with the otonal pentad. Otherwise,
>>>> the otonal/utonal contrast steadily increases with odd limit.
>>>>
>>>> -Carl
>>>>
>>>
>>
>>
>
>

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Mike in 1978 I took a course in electronic music as part of my
> music degree. I am in the process of scanning a couple pages for
> you and Carl. Essentially interference is the only reason for
> beating and combination tones and the two phenomena differ only
> in frequency.

Mike's right Chris, but sure, I'd be happy to look at some
scans. In the meantime, maybe you could try some research
of your own on the WWW.

-Carl

I am most certainly not wrong.

Please refer to:

http://clones.soonlabel.com/tis/scan10001.jpg
http:/clones.soonlabel.com/tis/scan10002.jpg

On Sun, Jan 18, 2009 at 5:30 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > But, with all due respect, you can see the beating, ie, the
> > constructive / destructive interference.
> >
> > That is plainly visible when you look a the wave with sufficient
> > zoom.
>
> No. You're either talking about the shape of the waveform
> itself, or the way wave editors truncate it when they are
> displaying it at less than full scale.
>
> -Carl
>
>
>

Carl.... the phenomena is the same.

Otherwise point me to a reference that describes the physics of sound that
differs from the reference I have provided which comes from:

Electronic Music Synthesis
Concepts, Facilities, Techniques

Author Hubert S. Howe, Jr
Associate Professor of Music, queens College of the City University of New
York
1975

I honestly think the definitions around here have not been consistent and it
causes much confusion.

On Sun, Jan 18, 2009 at 11:24 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, chrisvaisvil@...
> wrote:
> >
> > Mike in 1978 I took a course in electronic music as part of my
> > music degree. I am in the process of scanning a couple pages for
> > you and Carl. Essentially interference is the only reason for
> > beating and combination tones and the two phenomena differ only
> > in frequency.
>
> Mike's right Chris, but sure, I'd be happy to look at some
> scans. In the meantime, maybe you could try some research
> of your own on the WWW.
>
> -Carl
>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I am most certainly not wrong.
>
> Please refer to:
>
> http://clones.soonlabel.com/tis/scan10001.jpg
> http:/clones.soonlabel.com/tis/scan10002.jpg

You are wrong, Chris, and so is this textbook. It's not
uncommon.

-Carl

This textbook has information in it that simply is wrong. Adding two
sine waves together will not generate another sine wave of a different
frequency. The beat frequency never -becomes- audible - when do you
ever hear a third note pop out when you add two together? And even
more misleading is the fact that what they referred to as the "beat
frequency" there is really twice what it REALLY is, as every other
"beat" is phase inverted. 439 Hz and 441 Hz is equal to a 440 Hz sine
wave modified sinusoidally by 1 Hz. However, we don't hear any
difference between the two "humps" in the beating, even though every
other one is phase inverted, and so it is often said that the beat
frequency here is 2 Hz.

Here, let's put it this way. If you add 440 Hz and 410 Hz together,
does a low bass 30 Hz note suddenly appear? No. Does a high 850 Hz
combination appear? No way. And even if you do faintly perceive some
of these tones due to some nonlinear effect of the ear, the speaker,
the brain, or anything, that certainly doesn't mean that it is present
in the original signal from a Fourier Analysis standpoint. In fact, by
DEFINITION, 440 Hz and 410 Hz waves added together will yield only 440
Hz and 410 Hz in the resulting spectrum when a Fourier Transform is
done on it. That's the whole point of Fourier Analysis - to tell you
what sine waves to add together to give you back the original signal.
The notion that the combination tone is physically created as a sine
wave is equivalent to saying that 440 Hz + 410 Hz = 440 Hz + 410 Hz +
30 Hz, which is simply a false statement.

Now, I'm a bit rusty on the exact picture of the inner ear, but this
should give you an idea of the general picture of what causes beating:

The usual stated cause for beating when two sinusoids are close in
frequency is due somewhat to the physics of the ear. In the cochlea,
along the basilar membrane, lie tons of little hairs (or nerve fibers,
as I've sometimes heard them cited to be) that resonate at different
frequencies. They are arranged logarithmically in order from high to
low frequencies (or is it low to high?), and when those hairs are
stimulated, they trigger nerve endings under the hears, and resultant
signal is sent straight through the nervous system to the brain, where
more stuff happens.

Now each of these hairs doesn't resonate at only one frequency: rather
a general "range" of frequencies centered around one frequency,
something like a bandpass filter. The filters of the different hairs
overlap in such a way that the entire hearing spectrum is covered.
When you put in 440 Hz and 6000 Hz, the 440 Hz tone is ideally
captured by one of the bandpass filters and the 6000 Hz tone is
captured by another one, and the two are sent along separate pathways
to the brain with no constructive or destructive interference.

When you put in 440 Hz and 441 Hz, both of them are snatched up by the
same bandpass filter and the combined signal is sent to the brain, so
constructive and destructive interference actually takes place, and we
hear beating. These bandpass filters are commonly referred to as
"critical bands" in literature about this stuff. The theory goes that
the neural "channels" that critical band dumps information to can only
operate on one tone at a time, and so when 440 Hz and 441 Hz is sent,
you only do hear one tone. If you listen carefully, it will sound like
440.5 Hz, but as the tones are widened, it starts to sound
increasingly like the pitch is "wobbling" a bit, and eventually it
widens into two tones.

Now, even this picture is incomplete, as it assumes a few things about
how the brain makes a time-frequency map of a signal that are a bit
arbitrary. A certain amount of an incoming signal needs to be looked
at in RETROSPECT to get any kind of idea about what "frequencies" are
present in the signal. There needs to be a "window" of time in which
we look at the signal to get an idea about its frequency content.
There is no reason that this time window needs to be the same for all
frequencies or that it has to be a static constant that doesn't
change. The length of this window will also radically alter the
frequency content of the signal.

I suspect there is some kind of "perspective shift" that occurs in the
brain whereby different frequencies added together end up becoming
other frequencies being modulated by some kind of envelope. This is
something distinct from the critical band effects I mentioned above -
I'm not sure if this corresponds to a "changing" of the window size or
what's going on there. Max's recent posts validate my suspicions on
this.

Nonetheless: wherever this new train of thought about beating and
other band effects ends up leading, it has NOTHING to do with anything
labeled "difference tones". Difference tones and combination tones are
terms that refer to some kind of intermodulation distortion, like when
you play two notes simultaneously on a distorted guitar and it sounds
"noisy". And constructive and destructive interference alone is only
half of the story, since the signal is SPLIT UP within the inner ear
into separate channels. Within -one- critical band, there can be
constructive and destructive interference that could partially account
for beating, even if the question of how the brain manages to perceive
changing frequency content at all is ignored.

That's the full picture, in all of its complicated glory. If you are
interested in this stuff, and you want to understand more about
Fourier Analysis in general, then you should probably first learn
about the mathematical operation of convolution and the "convolution
theorem" as it applies to Fourier Analysis. Despite the scary name,
they aren't that complicated if explained the right way. You should
also understand the distinction between linear and non-linear systems
(which also isn't that complicated), as these concepts are hard to
explain without referring to them with that terminology. You should
also learn about some of the basic properties of linear,
time-invariant systems. Fourier analysis is an extremely deep subject
with a lot of twists and turns, and it's essential to be able to
navigate it to discuss the causes of more complicated, emergent
phenomena such as beating.

-Mike

To Both you and Carl,

Either combination tones AND beats are a physical phenomenon (and not a
perception phenomenon) OR it is not a physical phenomenon and lies in the
realm of perception.

I guess I could go back and dredge through all the posts and show you where
people were saying things like "beating was a real physical phenomenon but
will never show in Fourier analysis" and beating and combination tones were
different phenomena. I really don't care too much who is right - as long as
we ALL agree to somethine in order to stop wasting time on basic
definitions. But mixing physical and psycho-acoustics at this point in this
discussion clouds the issue. The two are separate.

As for my familiarity with Fourier analysis - I've programmed a computer to
do Fourier analysis. (Have you?) To be sure it was not an FFT... I could not
do an FFT because the Ti-99/4A had no way to manipulate bits in BASIC. But I
could do a discrete transform. To be sure... it took several hours to run.

I don't understand why there is an assumption of people being uneducated or
stupid on this list. Disagrrements occur all the time.

I have a music and a chemistry degree - I've had college level physics and
math through differential equations. So I'm not brilliant - but I have
-some- basic understanding...

On Mon, Jan 19, 2009 at 1:07 AM, Mike Battaglia <battaglia01@...>wrote:

> This textbook has information in it that simply is wrong. Adding two
> sine waves together will not generate another sine wave of a different
> frequency. The beat frequency never -becomes- audible - when do you
> ever hear a third note pop out when you add two together? And even
> more misleading is the fact that what they referred to as the "beat
> frequency" there is really twice what it REALLY is, as every other
> "beat" is phase inverted. 439 Hz and 441 Hz is equal to a 440 Hz sine
> wave modified sinusoidally by 1 Hz. However, we don't hear any
> difference between the two "humps" in the beating, even though every
> other one is phase inverted, and so it is often said that the beat
> frequency here is 2 Hz.
>
> Here, let's put it this way. If you add 440 Hz and 410 Hz together,
> does a low bass 30 Hz note suddenly appear? No. Does a high 850 Hz
> combination appear? No way. And even if you do faintly perceive some
> of these tones due to some nonlinear effect of the ear, the speaker,
> the brain, or anything, that certainly doesn't mean that it is present
> in the original signal from a Fourier Analysis standpoint. In fact, by
> DEFINITION, 440 Hz and 410 Hz waves added together will yield only 440
> Hz and 410 Hz in the resulting spectrum when a Fourier Transform is
> done on it. That's the whole point of Fourier Analysis - to tell you
> what sine waves to add together to give you back the original signal.
> The notion that the combination tone is physically created as a sine
> wave is equivalent to saying that 440 Hz + 410 Hz = 440 Hz + 410 Hz +
> 30 Hz, which is simply a false statement.
>
> Now, I'm a bit rusty on the exact picture of the inner ear, but this
> should give you an idea of the general picture of what causes beating:
>
> The usual stated cause for beating when two sinusoids are close in
> frequency is due somewhat to the physics of the ear. In the cochlea,
> along the basilar membrane, lie tons of little hairs (or nerve fibers,
> as I've sometimes heard them cited to be) that resonate at different
> frequencies. They are arranged logarithmically in order from high to
> low frequencies (or is it low to high?), and when those hairs are
> stimulated, they trigger nerve endings under the hears, and resultant
> signal is sent straight through the nervous system to the brain, where
> more stuff happens.
>
> Now each of these hairs doesn't resonate at only one frequency: rather
> a general "range" of frequencies centered around one frequency,
> something like a bandpass filter. The filters of the different hairs
> overlap in such a way that the entire hearing spectrum is covered.
> When you put in 440 Hz and 6000 Hz, the 440 Hz tone is ideally
> captured by one of the bandpass filters and the 6000 Hz tone is
> captured by another one, and the two are sent along separate pathways
> to the brain with no constructive or destructive interference.
>
> When you put in 440 Hz and 441 Hz, both of them are snatched up by the
> same bandpass filter and the combined signal is sent to the brain, so
> constructive and destructive interference actually takes place, and we
> hear beating. These bandpass filters are commonly referred to as
> "critical bands" in literature about this stuff. The theory goes that
> the neural "channels" that critical band dumps information to can only
> operate on one tone at a time, and so when 440 Hz and 441 Hz is sent,
> you only do hear one tone. If you listen carefully, it will sound like
> 440.5 Hz, but as the tones are widened, it starts to sound
> increasingly like the pitch is "wobbling" a bit, and eventually it
> widens into two tones.
>
> Now, even this picture is incomplete, as it assumes a few things about
> how the brain makes a time-frequency map of a signal that are a bit
> arbitrary. A certain amount of an incoming signal needs to be looked
> at in RETROSPECT to get any kind of idea about what "frequencies" are
> present in the signal. There needs to be a "window" of time in which
> we look at the signal to get an idea about its frequency content.
> There is no reason that this time window needs to be the same for all
> frequencies or that it has to be a static constant that doesn't
> change. The length of this window will also radically alter the
> frequency content of the signal.
>
> I suspect there is some kind of "perspective shift" that occurs in the
> brain whereby different frequencies added together end up becoming
> other frequencies being modulated by some kind of envelope. This is
> something distinct from the critical band effects I mentioned above -
> I'm not sure if this corresponds to a "changing" of the window size or
> what's going on there. Max's recent posts validate my suspicions on
> this.
>
> Nonetheless: wherever this new train of thought about beating and
> other band effects ends up leading, it has NOTHING to do with anything
> labeled "difference tones". Difference tones and combination tones are
> terms that refer to some kind of intermodulation distortion, like when
> you play two notes simultaneously on a distorted guitar and it sounds
> "noisy". And constructive and destructive interference alone is only
> half of the story, since the signal is SPLIT UP within the inner ear
> into separate channels. Within -one- critical band, there can be
> constructive and destructive interference that could partially account
> for beating, even if the question of how the brain manages to perceive
> changing frequency content at all is ignored.
>
> That's the full picture, in all of its complicated glory. If you are
> interested in this stuff, and you want to understand more about
> Fourier Analysis in general, then you should probably first learn
> about the mathematical operation of convolution and the "convolution
> theorem" as it applies to Fourier Analysis. Despite the scary name,
> they aren't that complicated if explained the right way. You should
> also understand the distinction between linear and non-linear systems
> (which also isn't that complicated), as these concepts are hard to
> explain without referring to them with that terminology. You should
> also learn about some of the basic properties of linear,
> time-invariant systems. Fourier analysis is an extremely deep subject
> with a lot of twists and turns, and it's essential to be able to
> navigate it to discuss the causes of more complicated, emergent
> phenomena such as beating.
>
> -Mike
>
>

Actually I hear about 4 beats per second in the utonal 4:5:6:7:9, for
some reason. But the difference in consonance is quite clear anyway.

I am still slightly puzzled by what you meant by saying (if I remember
correctly) minor chords aren't musically viable beyond 9-limit:

"minor chords beyond the 9-limit are essentially unusable."

- What would such a minor chord beyond the 9-limit consist of?

It may not be quite on-topic, but how does 16:19:24 compare with
10:12:15 aurally as minor triad?
~~~T~~~

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

> http://lumma.org/stuff/majmin.zip
>
> ~ 2.7 MB.
>
> Unfortunately, I can't synthesize 6-voice chords with
> Cool Edit. But I did make otonal and utonal versions of
> 4:5:6, 4:5:6:7, and 4:5:6:7:9 chords, using perfect sines.
> Each file is 8 seconds long, and begins with 2 seconds of
> the outer dyad alone, followed by 6 seconds of the complete
> chord. All chords are rooted on C = 264 Hz. Files are
> rendered directly at -15dB. Combination tones didn't much
> effect the comparison, but I found I could indeed make them
> disappear on my speakers by playing them at moderate volume.
> With in-ear monitors, combination tones were present at
> almost any volume unless I pulled the phones out of my ear
> canals and only rested them on my conchae
> http://en.wikipedia.org/wiki/Pinna_(anatomy)
>
> I hear no beating in any of the samples, but I do hear that
> the otonal chords are more consonant. The contrast becomes
> complete in the 11-limit. The 9-limit utonal pentad only
> sounds as good as it does because of the 4:6:9 backbone of
> fifths that it shares with the otonal pentad. Otherwise,
> the otonal/utonal contrast steadily increases with odd limit.
>
> -Carl
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Max wrote:
> > What is intriguing me is: is there a physical mechanism (I am
> > not looking for psychoacoustic explanations, because I have no
> > scientific basis to understand those) lying below this
> > phenomenon of passage from one mean tone with amplitude
> > beating to split tones with no beating - and all the
> > intermediate situations of course, that seemingly cause the
> > sensation of dissonance - occurs in the ear? I mean, are
> > there physical reasons for that, or it is completely or
> > mainly due to psychoacoustics?
>
> It's completely due to the spectral resolution of the
> cochlea. If you want to call that psychoacoustics, it's
> fine with me. When the two tones are closer than the
> cochlea can resolve, we hear a single tone.
>
> (...)
>
> > but not to hear an amplitude variation faster than
> > 20-30 Hz (or, better, is capable to hear it but it
> > renders such beating as the presence of two tones)?
>
> It has nothing to do with the rate of beating. The
> rate at which the two tones finally emerge will depend
> on the width of the critical band (specral resolution
> of the cochlea). Since the critical bandwidth changes
> with respect to frequency, the beat rate, f1-f2, at
> 1/2CB or whatever point you consider the two tones to
> be distinct, will also be different.
>
> -Carl

Indeed, with relatively low frequencies it is possible to hear a not
so very fast beating *and* to experience two distinct pitches.

eg bass A has frequency 110Hz, a diatonic semitone below it is G#
103Hz, the beating is 7Hz but the pitches are very much distinct.

By the way it is perhaps better to speak of *power* variation with
respect to beating.

Of course the power of an oscillating travelling wave such as sound is
continually varying round each oscillation, so what we mean is the
power *averaged* over some timescale. Obviously there is a shortest
timescale over which the ear/brain doesn't register power variation as
such, so the beat frequency should be slower than this if we are to
hear it.

In the case of slow beats where the averaging timescale is much
shorter than the beating, the variation in power is very clear.
~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
>
> Indeed, with relatively low frequencies it is possible to hear a
not
> so very fast beating *and* to experience two distinct pitches.
>
> eg bass A has frequency 110Hz, a diatonic semitone below it is G#
> 103Hz, the beating is 7Hz but the pitches are very much distinct.
>
> By the way it is perhaps better to speak of *power* variation with
> respect to beating.
>
> Of course the power of an oscillating travelling wave such as
sound is
> continually varying round each oscillation, so what we mean is the
> power *averaged* over some timescale. Obviously there is a shortest
> timescale over which the ear/brain doesn't register power
variation as
> such, so the beat frequency should be slower than this if we are to
> hear it.
>
> In the case of slow beats where the averaging timescale is much
> shorter than the beating, the variation in power is very clear.
> ~~~T~~~
>

This is interesting to me. It is ok to speak of acoustic power, that
should be the square of amplitude, and as amplitude of a complicated
wavelet should be defined on a given time window, so also power
should.

So my curiosity turns to: is there something physical to determine
the shortest averaging timescale at which ear is able to respond and
so on? I really hope to catch some glimpse of what could be at the
basis of this without resorting to psychoacoustics, up to now I have
only small clues.

Max

Tom,

"Of course the power of an oscillating travelling wave such as sound is
continually varying round each oscillation, so what we mean is the
power *averaged* over some timescale."

I wonder of power can be a measure of consonance?

Observations:

In phase unison should be more powerful than out of phase

slightly out of tune unison (beats) should be less powerful than in tune
unison (no beats)

a "perfect" 5th should be more powerful than an 12 - edo 5th.

And I'm supposing that a major chord is more poerful than a minor... etc.

Chris wrote:
> I don't understand why there is an assumption of people being
> uneducated or stupid on this list. Disagrrements occur all
> the time.

I certainly don't assume that -- I tend to assume anyone who's
on this list is smarter than average. But I don't know what
to say about difference tones at this point that hasn't been
said already. You don't seem receptive to what's been said.
Let me try once more. Difference tones are physical phenomena
alright -- they can be recorded coming out of the ears using
a microphone! They are, by definition, musical tones created
in the ear. Beating is AM. As with difference tones, the
frequency of beating is f1-f2, but the thing at that frequency
in the case of beating is not a _tone_, but rather a variation
in the volume of a tone. Both can occur at the same time.
You could hear a wah-wah-wah at f1-f2 Hz, and a pitch at
f1-f2 Hz. Or you could turn down the volume until the difference
tone became inaudible and you'd still hear the beating.

-Carl

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> Actually I hear about 4 beats per second in the utonal
> 4:5:6:7:9, for some reason. But the difference in consonance
> is quite clear anyway.
>
> I am still slightly puzzled by what you meant by saying (if I
> remember correctly) minor chords aren't musically viable
> beyond 9-limit:
>
> "minor chords beyond the 9-limit are essentially unusable."
>
> - What would such a minor chord beyond the 9-limit consist of?

Sorry, I use "minor" and "utonal" as synonyms sometimes.
I believe this use is justified, but I suppose it is
debatable.

> It may not be quite on-topic, but how does 16:19:24 compare with
> 10:12:15 aurally as minor triad?

You should listen to them. And I've copied a recent description
of mine below.

-Carl

>When presented with an auditory stimulus, the brain will attempt
>to assign it a pitch. It does this by extracting harmonically-
>related signals in the spectrum analysis of the stimulus. The
>stimulus need not be periodic for this to work, but even if it
>is, the brain is only so good at:
> 1. performing spectrum analysis -- failure here results in
>roughness
> 2. detecting periodicity -- failure here results in high
>harmonic entropy
>For clean stimuli with periodicities in the range the brain is
>good at detecting (such as 5-limit major triads), the extracted
>pitch will fairly unambiguous (though it will sometimes get
>confused by octaves). For other kinds of stimuli, the stochastic
>nature of processes in the brain becomes evident -- populations
>of thousands or millions or tens of millions of neurons are used
>to accomplish signal-processing tasks. There will be several
>pitches competing for the 'answer', and harmonic entropy can be
>used to model this uncertainty.
>
>In the case of 10:12:15, the probability distribution for the
>fundamental will be mostly covered by the pitches 5, 8, and 10.
>If the tones of the chord are complex tones, the timbre will
>influence the relative likelihoods of 5, 8, and 10 being heard
>as the fundamental. In the case that 8 is heard, the brain is
>interpreting 10:12:15 literally, as a segment of harmonics
>relatively high in the series. If 10 or 5 are heard, the brain
>is saying it considers 10:15 = 3:2, and is dismissing the 12 as
>an artifact.
>
>In the case of 16:19:24, the same tradeoff exists, but now the
>complete series-segment interpretation is even less likely
>because it is even higher in the series. However in this case,
>both interpretations lead to the same fundamental, namely 8 or 16
>(or 4 if the chord is registered in the soprano). Therefore
>entropy associated with the probability distribution for the
>fundamental tends to be as lower or lower than that for 10:12:15,
>with the result that 16:19:24 is slighly more stable, more
>"major", more "happy" if you will, than 10:12:15.
>
>Having used 6:7:9 chords on several instruments over the years,
>including several pianos, my own impression is that it is indeed
>more consonant, in a sense, than either 10:12:15 or 16:19:24.
>However, there are two things against it:
> 1. 7/6 is getting close to the critical band, and even in the
>middle of the piano, we hear some beating between its partials.
>As mentioned before, this beating is nicely synchronous, but it
>lends the interval to a somewhat 'tense' feeling. This can be a
>good thing for minor chords however. :)
> 2. The chord is, in fact, low enough in the harmonic series that
>the implied fundamental is likely to be 4 (or 2 or 1 depending on
>the voicing). Compare my analysis of 10:12:15, where almost
>always 10 or 5 will be heard as the root, with the 12 being
>dismissed by the hearing system as unrecoverable (unless you
>listen to 15-chords all the time, as Mike seems to do, when you
>may hear 4 or 8 for the root. :)
>
>In summary, 6:7:9 is an excellent minor chord and can be used
>even in performances of existing classical music. But overall I
>think either 10:12:15 or 16:19:24 are to be preferred, because of
>their greater tendency to evoke a fundamental that is the same as
>the root (lowest tone or octave extension of the lowest tone) of
>the chord.

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I wonder of power can be a measure of consonance?

So I can make things more consonant by turning up the volume?

-Carl

I believe I gave examples that clarified my meaning. In that context your comment makes little sense.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@...>

Date: Mon, 19 Jan 2009 17:57:25
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I wonder of power can be a measure of consonance?

So I can make things more consonant by turning up the volume?

-Carl

First order of business is to seperate the physics from the biological. And in that vein I fail to see where my one guitar string has the innate ability to AM the other. Could you defend your AM point with physics in the context of tuning a guitar?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Mon, 19 Jan 2009 17:37:15
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

Chris wrote:
> I don't understand why there is an assumption of people being
> uneducated or stupid on this list. Disagrrements occur all
> the time.

I certainly don't assume that -- I tend to assume anyone who's
on this list is smarter than average. But I don't know what
to say about difference tones at this point that hasn't been
said already. You don't seem receptive to what's been said.
Let me try once more. Difference tones are physical phenomena
alright -- they can be recorded coming out of the ears using
a microphone! They are, by definition, musical tones created
in the ear. Beating is AM. As with difference tones, the
frequency of beating is f1-f2, but the thing at that frequency
in the case of beating is not a_tone_, but rather a variation
in the volume of a tone. Both can occur at the same time.
You could hear a wah-wah-wah at f1-f2 Hz, and a pitch at
f1-f2 Hz. Or you could turn down the volume until the difference
tone became inaudible and you'd still hear the beating.

-Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> First order of business is to seperate the physics from the
> biological. And in that vein I fail to see where my one guitar
> string has the innate ability to AM the other. Could you defend
> your AM point with physics in the context of tuning a guitar?

Huh??

-Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> I believe I gave examples that clarified my meaning.

They did not. Frankly, there's very little meaning of any
kind in the five sentence fragments you posted. Here they
are:

>I wonder of power can be a measure of consonance?

That's not a question, but you've put a question mark after
it. It also doesn't make any sense.

>In phase unison should be more powerful than out of phase

Assuming we know what you mean by "out" of phase, yes.

>slightly out of tune unison (beats) should be less powerful
>than in tune unison (no beats)

It will vary from being more powerful, to be less powerful.
The average power is the same.

>a "perfect" 5th should be more powerful than an 12 - edo 5th.

??

-Carl

Please explain how one guitar string amplitude modulates the other. My position is that the phenomena is constructive and destructive interference. I have cited a source, and found serveral more on the web this morning, to support my position.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@...>

Date: Mon, 19 Jan 2009 19:15:50
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> First order of business is to seperate the physics from the
> biological. And in that vein I fail to see where my one guitar
> string has the innate ability to AM the other. Could you defend
> your AM point with physics in the context of tuning a guitar?

Huh??

-Carl

Carl

Picture if you can one second each of two sine waves of the same frequency, one pure, the other beating at 1 hrz.

For the sake of our discussion ignore the portion of each sine that is below zero. Now integrate the area above zero for the 1 second for each example. The result will be the pure sine will have a larger area.

Take this example and apply it to a pure JI 5th and a 12 TET 5th - I believe the result, though closer numerically, would rank the same.

I am thinking this will be true for all dyands triads etc. I'm not a math guy so I don't know how to formally prove this but bet someone already has.

Is this clear?

Chris.

Is this any clearer?
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Mon, 19 Jan 2009 19:20:23
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> I believe I gave examples that clarified my meaning.

They did not. Frankly, there's very little meaning of any
kind in the five sentence fragments you posted. Here they
are:

>I wonder of power can be a measure of consonance?

That's not a question, but you've put a question mark after
it. It also doesn't make any sense.

>In phase unison should be more powerful than out of phase

Assuming we know what you mean by "out" of phase, yes.

>slightly out of tune unison (beats) should be less powerful
>than in tune unison (no beats)

It will vary from being more powerful, to be less powerful.
The average power is the same.

>a "perfect" 5th should be more powerful than an 12 - edo 5th.

??

-Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Please explain how one guitar string amplitude modulates the
> other. My position is that the phenomena is constructive and
> destructive interference.

Chris, I'm happy to discuss this further, but let's take it
offlist. I'm sure people here are tired of reading about it
by now. :)

-Carl

Take it off-list if you wish, but tired of it?

Never!

caleb

On Jan 19, 2009, at 3:44 PM, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
> >
> > Please explain how one guitar string amplitude modulates the
> > other. My position is that the phenomena is constructive and
> > destructive interference.
>
> Chris, I'm happy to discuss this further, but let's take it
> offlist. I'm sure people here are tired of reading about it
> by now. :)
>
> -Carl
>
>
>

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> Take it off-list if you wish, but tired of it?
>
> Never!
>
> caleb

In the past we've had serious complaints about the message
volume here, and I personally know several wonderful musicians
and technical people who stopped using the list because of
the message volume. Speaking for myself, I think frequent
two-sentence replies from a mobile device is excessive.

-Carl

Well, I made some replies off list but nothing certainty has been decided
except Carl says that beating is from constructive/destructive interference
which results in amplitude modulation. I certainly agree with that.

I am thinking, that in the case of the summation of pure sines in dyads,
triads, etc. that the resulting waveform is a measure of the consonance and
that properly integrated over time we should be able to quantify said
consonance. The principal behind this is the observation that a pure sine
wave will integrate to a larger number over a set period of time than a sine
wave that is modulated by beating (or if you wish amplitude modulation). To
me it makes sense - if the modulated wave goes to zero for part of its
sampled time where the non-modulated since is positive of course the
integral will be larger. (This assumes that you discard the negative portion
of a waveform symetrical around zero).

Chris

On Mon, Jan 19, 2009 at 4:40 PM, caleb morgan <calebmrgn@...> wrote:

> Take it off-list if you wish, but tired of it?
>
> Never!
>
> caleb
>
>
> On Jan 19, 2009, at 3:44 PM, Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, chrisvaisvil@...
> wrote:
> >
> > Please explain how one guitar string amplitude modulates the
> > other. My position is that the phenomena is constructive and
> > destructive interference.
>
> Chris, I'm happy to discuss this further, but let's take it
> offlist. I'm sure people here are tired of reading about it
> by now. :)
>
> -Carl
>
>
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>

> To me it makes sense - if the modulated wave goes to zero for
> part of its sampled time where the non-modulated since is
> positive of course the integral will be larger. (This assumes
> that you discard the negative portion of a waveform symetrical
> around zero).

My advice: try working out some test cases and present your
results.

-Carl

OK,

I have to figure out a way to do it. I do not have a program besides Excel
to do math and my calculus is some 10 years old. Unless someone wishes to
enter the discussion who has skill in this area.

On Mon, Jan 19, 2009 at 5:12 PM, Carl Lumma <carl@...> wrote:

> > To me it makes sense - if the modulated wave goes to zero for
> > part of its sampled time where the non-modulated since is
> > positive of course the integral will be larger. (This assumes
> > that you discard the negative portion of a waveform symetrical
> > around zero).
>
> My advice: try working out some test cases and present your
> results.
>
> -Carl
>
>
>

On Mon, Jan 19, 2009 at 6:16 AM, Chris Vaisvil <chrisvaisvil@...> wrote:
> To Both you and Carl,
>
> Either combination tones AND beats are a physical phenomenon (and not a
> perception phenomenon) OR it is not a physical phenomenon and lies in the
> realm of perception.
>
> I guess I could go back and dredge through all the posts and show you where
> people were saying things like "beating was a real physical phenomenon but
> will never show in Fourier analysis" and beating and combination tones were
> different phenomena. I really don't care too much who is right - as long as
> we ALL agree to somethine in order to stop wasting time on basic
> definitions. But mixing physical and psycho-acoustics at this point in this
> discussion clouds the issue. The two are separate.

"Combination tones" and "difference tones" refer to tones resulting
from the intermodulation distortion aspect of a nonlinear system.
"Beating" is ABSOLUTELY NOT a difference tone. Even though it seems
like the frequency of beating is equal to the difference of the two
sine waves, it is NOT. It is actually HALF of what you'd estimate by
that method, since each "beat" is only half of a sine wave, and it
takes two full beats to be a sine wave (every other beat is phase
inverted).

Sum and difference tones are what happens when you take two notes and
you run them through some kind of distortion pedal, or play them
through a speaker that distorts slightly, or something like that. They
have nothing to do with beating. If you play 440 Hz and 441 Hz
together, you'll hear 440.5 Hz with a perceived 1 Hz beating (but
really 0.5 Hz). If you play the two together through some hypothetical
distortion unit that doesn't add harmonics, then you'll still hear
440.5 Hz with a perceived 1 Hz beating. You will also hear 881 Hz (the
sum tone) and 1 Hz (the difference tone). The 1 Hz signal tone will be
filtered out by the ear, and you won't hear it.

None of the above has anything to do with psychoacoustics.

It's as simple as this: a 440 Hz sine wave beating at "1 Hz" will NOT
show 1 Hz in the FFT of signal. Try it and see for yourself. You'll
get 439.5 Hz and 440.5 Hz. Look up the "convolution theorem" of
Fourier analysis for more information.

> As for my familiarity with Fourier analysis - I've programmed a computer to
> do Fourier analysis. (Have you?) To be sure it was not an FFT... I could not
> do an FFT because the Ti-99/4A had no way to manipulate bits in BASIC. But I
> could do a discrete transform. To be sure... it took several hours to run.

This is unnecessary, and there is no need for you to be defensive. I
understand how different FFT algorithms work, although I haven't
bothered to implement one myself. There are implementations already
available for every language that I work with. None of that means that
I don't understand Fourier analysis, nor does it mean that beating has
anything to do with difference tones, nor does it mean that the
envelope of the wave has anything to do with difference tones.

> I don't understand why there is an assumption of people being uneducated or
> stupid on this list. Disagrrements occur all the time.
>
> I have a music and a chemistry degree - I've had college level physics and
> math through differential equations. So I'm not brilliant - but I have
> -some- basic understanding...

Nobody's assuming you're stupid or uneducated. I am actually trying to
help you understand everything by posting so many times about it. In
order for that to happen, you have to accept that there are things
about Fourier Analysis that you don't know. Why are you being so
defensive about it?

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1232242785
> *http://tinyurl.com/test2tuning* <http://tinyurl.com/test2tuning>

> of interest is when I played C4 and C#4 - the peaks split - 4
signals were
> present. When I went back and did osc 1 and osc 2 added together
with osc 2
> freely tuned there is an area where the peaks split - at osc 2
pitches about
> and below (but still above C4) the peaks do not spilt.
>
> C5 and C#5 together do not split....
>
> So.... this is an interesting result.

Hi Chris,

I am trying to understand why, when using pure sinusoidal tones,
just added up, you get some spurious frequency peaks in your Fourier
spectrum (in the case C4 - C#4 that you mention). Some questions:
where you live, the mains frequency is 50 Hz or 60 Hz? Is C4 - C#4
the only case when you get the additional peaks, or you get those
also for other couples of tones?

Thanks

Max

Mike wrote:

> Even though it seems
> like the frequency of beating is equal to the difference of the two
> sine waves, it is NOT. It is actually HALF of what you'd estimate by
> that method, since each "beat" is only half of a sine wave, and it
> takes two full beats to be a sine wave (every other beat is phase
> inverted).

I've already explained once that you can't argue with this. The phase inversion is related to the modulated signal, not to the resulting periods. If you také a two seconds long sine wave of 440.5Hz and multiply it with a two seconds long cosine wave of 0.5Hz, the second half of the resulting sound will be identical to the first one because in both cases the second half of the sound is the exact 180-degree phase shift of the first half, which means that you have "plus times plus" in the first half of the result and "minus times minus" in the second half -- i.e. both are the same. What this means is that if you mix a 440Hz sine wave with a 441Hz sine wave, then the period of repetition is actually 1 second, not 2 seconds.

Petr

Hi Max,

I live in the usa midwest. The mains are 60 hz or so. I'm thinking it is a problem with doing discrete math on an analog problem now. I need to re-visit it. It certainly makes little sense in the context of the book I quoted. At first I thought it was a critical band property but reality is I'm not going to discover something new and when I re-read my text I realized what I saw can't be the actual physical result and must be an anomaly from the limitation of the tools. However good science requires experimentation (challenges) without bias and I need to revisit it as I said.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "massimilianolabardi" <labardi@df.unipi.it>

Date: Tue, 20 Jan 2009 08:30:36
To: <tuning@yahoogroups.com>
Subject: [tuning] Explaining major 4:5:6 and minor 10:12:15 triads,Re:Beatings vs Intermodulation

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> http://soonlabel.com/cgi-bin/yabb2/YaBB.pl?num=1232242785
> *http://tinyurl.com/test2tuning* <http://tinyurl.com/test2tuning>

> of interest is when I played C4 and C#4 - the peaks split - 4
signals were
> present. When I went back and did osc 1 and osc 2 added together
with osc 2
> freely tuned there is an area where the peaks split - at osc 2
pitches about
> and below (but still above C4) the peaks do not spilt.
>
> C5 and C#5 together do not split....
>
> So.... this is an interesting result.

Hi Chris,

I am trying to understand why, when using pure sinusoidal tones,
just added up, you get some spurious frequency peaks in your Fourier
spectrum (in the case C4 - C#4 that you mention). Some questions:
where you live, the mains frequency is 50 Hz or 60 Hz? Is C4 - C#4
the only case when you get the additional peaks, or you get those
also for other couples of tones?

Thanks

Max

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Hi Max,
>
> I live in the usa midwest. The mains are 60 hz or so. I'm thinking
it is a problem with doing discrete math on an analog problem now. I
need to re-visit it. It certainly makes little sense in the context
of the book I quoted. At first I thought it was a critical band
property but reality is I'm not going to discover something new and
when I re-read my text I realized what I saw can't be the actual
physical result and must be an anomaly from the limitation of the
tools. However good science requires experimentation (challenges)
without bias and I need to revisit it as I said.
> Sent via BlackBerry from T-Mobile
>

Hi Chris,

I suspected you lived in a place with 60 Hz mains frequency, and
this is because I noted that the difference frequency between C4 and
C#4 is about 30 Hz. I wonder whether, if for some reason you get
such 60 Hz "mixed" (in a way I don't know, of course....) to the
beat of your sine waves, you could get some artifact in your
spectrum (of a kind that I don't know, of course...). Can you try
with C3 - D3 (that has the same beating frequency of 30 Hz) and e.g.
with C3 - C#3 (that, based on my assumption, should not give the
same effect)?

Thanks, Max

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Max wrote:
>
> > I hope this helps to understand better this issue of "what is
the
> > difference between: beatings, intermodulation tones, difference
> > tones, combination tones... Perhaps we should write down a
> > vocabulaty with the exact meaning of all such terms, and the
> > relation/differences among them, just to simplify future
discussions!
>
> I would very much welcome such an effort. I could at least
> add it to my growing collection of material for a FAQ.
>
> -Carl
>

Hi,

the following could be just a start of a vocabulary. I have written
it with no reference to wikipedia or other sources, just with what
is in my mind, and quickly, so it could be wrong somewhere. Please
contribute... also excuse my non-native english.

Perhaps we could revise each time one entry, proposing the new one
with highlighting changes and explaining why, and one of us could
keep this updated in a file (perhaps Carl in his FAQ file?...) Or
propose some other procedure to revise entries or add new ones....
for instance, I see now that one should define ENVELOPE....

Max

WAVE - solution of wave equation. Typically a stationary solution is
a sine or cosine wave.

TONE - a steady, sinusoidal wave at frequency f.

ADDITION - The algebraic summation of two signals, that is a purely
linear process. Addition of two tones at frequency f1 and f2 is cos
(2 Pi f1 t) + cos(2 Pi f2 t). Pressure waves generated by two
closeby sources and heard from a long distance appear generally as
summed together.

BEATING (of two tones) - modulation of the amplitude (or power) due
to addition of two tones. If two tones have frequency f1 and f2, it
is seen by the trigonometric formula cos(2 Pi f1 t) + cos(2 Pi f2 t)
= 2 cos(2 Pi (f2 - f1) t/2) cos(2 Pi (f2 + f1) t/2) that the
enveloping cosine is at frequency (f1 - f2)/2, so that amplitude
oscillations are at frequency f2-f1.

MIXING - Multiplicative combinations of two signals, generated as a
consequence of the nonlinear interaction between waves. If such
nonlinear interaction is of quadratic type, difference and sum
frequencies are generated (second-order mixing terms). In this case,
mixing of two tones at frequency f1 and f2 is cos(2 Pi f1 t) * cos(2
Pi f2 t). This produces additional tones, that are a difference tone
at frequency f2 - f1 and a sum tone at frequency f2 + f1. If higher
nonlinearities are present, higher order mixing terms may also be
generated. Sometimes the word mixing is used instead of "addition"
of two signals, like e.g. in the case of audio "mixers", but such
terminology creates confusion and it is preferable to speak more
properly of "addition" in that case.

COMBINATION TONE - Additional tone generated as a consequence of the
nonlinear interaction of sound waves, that cause mixing of the
source waves.

INTERMODULATION TONE - synonimous of combination tone?

RING MODULATOR - Electronic device in which mixing of two input
signals is realized in such a way that the output generated by two
tones at frequency f1 and f2 is composed only by their combination
tones at frequency f2 - f1 and f2 + f1. [as far as I understand!....
from the circuits exploiting diodes I see no reason why higher order
nonlinearities should not come out]

FREQUENCY MIXER - Electronic device in which mixing of two input
signals is realized in such a way that the output generated by two
tones at frequency f1 and f2 is added by combination tones at
frequency f2 - f1 and f2 + f1.

Thanks ... I'm afraid I didn't have the time to trawl through all the
contributions about 'fundamentals' when they came up a few days ago.
Remarks below.

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

> >When presented with an auditory stimulus, the brain will attempt
> >to assign it a pitch. It does this by extracting harmonically-
> >related signals in the spectrum analysis of the stimulus. The
> (...)
> >There will be several
> >pitches competing for the 'answer', and harmonic entropy can be
> >used to model this uncertainty.
>
This sounds somewhat like Terhardt, if we take his model of
subharmonic virtual pitches as a means to detect/determine 'harmonic
relations'. But...

> >In the case of 10:12:15, the probability distribution for the
> >fundamental will be mostly covered by the pitches 5, 8, and 10.

This is slightly confusing immediately, because 8 is not a subharmonic
of any of these. Even if we just take 10:12, it is 2 octaves above
their proper fundamental in terms of shared subharmonic (or
'periodicity pitch'). Now I can perfectly live with a fudging of one
octave either way (2:1 being the simplest interval absolutely beyond
the unison) but 2 or 3 seems a stretch for the explanation.

And isn't it an additional assumption that brains fudge fundamentals
by octaves, and not some other interval? Shouldn't a good theory
explain this (limited) octave equivalence of 'fundamentals' rather
than assuming it?

> >If the tones of the chord are complex tones, the timbre will
> >influence the relative likelihoods of 5, 8, and 10 being heard
> >as the fundamental.

OK, certainly with more complex timbres 8 will be a subharmonic of
more of the frequencies present

> >In the case that 8 is heard, the brain is
> >interpreting 10:12:15 literally, as a segment of harmonics
> >relatively high in the series.

so the question is: why 8 rather than 4 or 2 or 1? Or 6 or 3? What is
psychologically special about octaves above '1', compared to other
simple intervals?

> >In the case of 16:19:24, the same tradeoff exists, (...)
> >However in this case,
> >both interpretations lead to the same fundamental, namely 8 or 16
> >(or 4 if the chord is registered in the soprano).

again, assumes the brain is happy constructing a perceived fundamental
by triple or quadruple octaves above '1'...

OK, with 4:5:6(:7:9...) we are quite happy saying that 4 is the
'fundamental', but that is a rather special case, where it is also the
lowest out of three or more contiguous and nicely spaced harmonics.
What about 3:4:5(:6...)? Logically, 3 ought to be able to be the
fundamental here, unless the powers of 2 have some really special
psychoacoustic property.

> > two things against it (6:7:9):
> (...)
> > 2. The chord is, in fact, low enough in the harmonic series that
> >the implied fundamental is likely to be 4 (or 2 or 1 depending on
> >the voicing). Compare my analysis of 10:12:15, where almost
> >always 10 or 5 will be heard as the root, with the 12 being
> >dismissed by the hearing system as unrecoverable (...)