Yahoo Tuning Groups Ultimate Backup tuning chords

Sorry if I use wrong descriptors/words; english is not my native language. I'dlike to solve a following issue.

When using multi-tone/sine setup, we can create intervals or chords (3-tone or more). Now - I'm trying to find some formulas, that allow to calculate a vibrations/beating, that happens in chords and intervals. There is a lot of data in the internet about ratios and tunings in general, but I can't find anything useful on that slight harmonic beating (not "binaural beating", when two freqs are very close together). Some parts I have figured out experimental, but still don't know if there is a way to generalize it. Harmonic series and "fractions" are involved.

On an example.

1) Let say, that you have an interval of two tones. 100Hz and 204Hz. If you see the waveform of that mix, you will notice, that there is 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first one).

2) Let say that you have a CEG chord of three tones. To start with - 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if you shift the mid tone, let say to 248 or 252Hz, then you get an audible vibration of 2Hz (250Hz seems to be the harmonic zero-point so to speak).

My question - how to calculate this stuff more general?
And also for more than 3 tones, and around various harmonic zero-points (various depths of vibration)?

🔗Chris Vaisvil <chrisvaisvil@...>

I think Paul Erlich and his Harmonic Entropy data + plots is a start

http://xenharmonic.wikispaces.com/harmonic+entropy

On Fri, May 3, 2013 at 4:06 PM, ayamahambho <ayamahambho@...> wrote:

> **
>
>
> Sorry if I use wrong descriptors/words; english is not my native language.
> I'dlike to solve a following issue.
>
> When using multi-tone/sine setup, we can create intervals or chords
> (3-tone or more). Now - I'm trying to find some formulas, that allow to
> calculate a vibrations/beating, that happens in chords and intervals. There
> is a lot of data in the internet about ratios and tunings in general, but I
> can't find anything useful on that slight harmonic beating (not "binaural
> beating", when two freqs are very close together). Some parts I have
> figured out experimental, but still don't know if there is a way to
> generalize it. Harmonic series and "fractions" are involved.
>
> On an example.
>
> 1) Let say, that you have an interval of two tones. 100Hz and 204Hz. If
> you see the waveform of that mix, you will notice, that there is 4Hz
> beating (second tone differs by 4Hz from H2 harmonics of the first one).
>
> 2) Let say that you have a CEG chord of three tones. To start with -
> 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" of CEG
> (1/1.25/1.5 ratios) because it rather does not vibrates. But if you shift
> the mid tone, let say to 248 or 252Hz, then you get an audible vibration of
> 2Hz (250Hz seems to be the harmonic zero-point so to speak).
>
> My question - how to calculate this stuff more general?
> And also for more than 3 tones, and around various harmonic zero-points
> (various depths of vibration)?
>
>
>

🔗Marcel de Velde <marcel@...>

First of all, all tones that are not perfect unison "beat" in a way.
Here is a definition of beating: http://en.wikipedia.org/wiki/Beat_(acoustics) <http://en.wikipedia.org/wiki/Beat_%28acoustics%29>
Even when we combine 100 and 200 Hertz we get a 100 Hertz beat. That is, a constructive and deconstructive interference occurring 100 times per second.
Since this falls exactly at the same frequency as the lower tone we don't usually call it beating though.
But for 200 and 300 Hertz, the beating frequency is 100 Hertz, and where there is no tone present one could call this beating.
If one uses timbres with overtones then the beating occurs between all the overtones.
For equal and perfect harmonic timbres the beating frequencies will be an exact copy of the timbre, though usually only the lowest beating frequency is meant.

Some people seem to feel beating means a slow phasing of the timbre when two tones sound together. One can describe this by creating a cutoff frequency above which one no longer calls the interference frequency "beating".

Furthermore, since we're talking about music and tuning, we can add that beating is something relative to a desired rational intonation system.
What I mean is if your aim is 5/4 for instance, yet you play 81/64 you may say that this 81/64 is beating with a certain frequency relative to 5/4 (which is actually beating itself as well if we take the strictest definition of beating).
But if your aim is 81/64 then if you play it correctly you may say you do not see it as beating, and choose to only call phasing relative to this 81/64 beating.
I personally think this is the most musical approach, and that this has some merits to how our brains/ears may perceive this beating.

-Marcel

> Sorry if I use wrong descriptors/words; english is not my native > language. I'dlike to solve a following issue.
>
> When using multi-tone/sine setup, we can create intervals or chords > (3-tone or more). Now - I'm trying to find some formulas, that allow > to calculate a vibrations/beating, that happens in chords and > intervals. There is a lot of data in the internet about ratios and > tunings in general, but I can't find anything useful on that slight > harmonic beating (not "binaural beating", when two freqs are very > close together). Some parts I have figured out experimental, but still > don't know if there is a way to generalize it. Harmonic series and > "fractions" are involved.
>
> On an example.
>
> 1) Let say, that you have an interval of two tones. 100Hz and 204Hz. > If you see the waveform of that mix, you will notice, that there is > 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first > one).
>
> 2) Let say that you have a CEG chord of three tones. To start with - > 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" > of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if > you shift the mid tone, let say to 248 or 252Hz, then you get an > audible vibration of 2Hz (250Hz seems to be the harmonic zero-point so > to speak).
>
> My question - how to calculate this stuff more general?
> And also for more than 3 tones, and around various harmonic > zero-points (various depths of vibration)?
>
>

🔗ayamahambho <ayamahambho@...>

Checking, thanks!

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I think Paul Erlich and his Harmonic Entropy data + plots is a start
>
> http://xenharmonic.wikispaces.com/harmonic+entropy
>
>
> On Fri, May 3, 2013 at 4:06 PM, ayamahambho <ayamahambho@...> wrote:
>
> > **
> >
> >
> > Sorry if I use wrong descriptors/words; english is not my native language.
> > I'dlike to solve a following issue.
> >
> > When using multi-tone/sine setup, we can create intervals or chords
> > (3-tone or more). Now - I'm trying to find some formulas, that allow to
> > calculate a vibrations/beating, that happens in chords and intervals. There
> > is a lot of data in the internet about ratios and tunings in general, but I
> > can't find anything useful on that slight harmonic beating (not "binaural
> > beating", when two freqs are very close together). Some parts I have
> > figured out experimental, but still don't know if there is a way to
> > generalize it. Harmonic series and "fractions" are involved.
> >
> > On an example.
> >
> > 1) Let say, that you have an interval of two tones. 100Hz and 204Hz. If
> > you see the waveform of that mix, you will notice, that there is 4Hz
> > beating (second tone differs by 4Hz from H2 harmonics of the first one).
> >
> > 2) Let say that you have a CEG chord of three tones. To start with -
> > 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" of CEG
> > (1/1.25/1.5 ratios) because it rather does not vibrates. But if you shift
> > the mid tone, let say to 248 or 252Hz, then you get an audible vibration of
> > 2Hz (250Hz seems to be the harmonic zero-point so to speak).
> >
> > My question - how to calculate this stuff more general?
> > And also for more than 3 tones, and around various harmonic zero-points
> > (various depths of vibration)?
> >
> >
> >
>

🔗ayamahambho <ayamahambho@...>

I don't speak about binaural beats. Play a CEG (ratio) chord (or use some app to generate 3 sine tones: let say 200, 300 and variations around 250 Hz) and you will notice a vibration, that is not binaural beat. I look for algorithms to calculate that sort of relationships between pure (sine) tones, but also within reasonable (audible) range (and amount of tones on board). From what I see and hear, these vibrations can be calculated/predicted in Hz. But there seem to be too much guessing. I'd like to implement such automated calculator to my software.

--- In tuning@yahoogroups.com, Marcel de Velde <marcel@...> wrote:
>
> First of all, all tones that are not perfect unison "beat" in a way.
> Here is a definition of beating:
> http://en.wikipedia.org/wiki/Beat_(acoustics)
> <http://en.wikipedia.org/wiki/Beat_%28acoustics%29>
> Even when we combine 100 and 200 Hertz we get a 100 Hertz beat. That is,
> a constructive and deconstructive interference occurring 100 times per
> second.
> Since this falls exactly at the same frequency as the lower tone we
> don't usually call it beating though.
> But for 200 and 300 Hertz, the beating frequency is 100 Hertz, and where
> there is no tone present one could call this beating.
> If one uses timbres with overtones then the beating occurs between all
> the overtones.
> For equal and perfect harmonic timbres the beating frequencies will be
> an exact copy of the timbre, though usually only the lowest beating
> frequency is meant.
>
> Some people seem to feel beating means a slow phasing of the timbre when
> two tones sound together. One can describe this by creating a cutoff
> frequency above which one no longer calls the interference frequency
> "beating".
>
> Furthermore, since we're talking about music and tuning, we can add that
> beating is something relative to a desired rational intonation system.
> What I mean is if your aim is 5/4 for instance, yet you play 81/64 you
> may say that this 81/64 is beating with a certain frequency relative to
> 5/4 (which is actually beating itself as well if we take the strictest
> definition of beating).
> But if your aim is 81/64 then if you play it correctly you may say you
> do not see it as beating, and choose to only call phasing relative to
> this 81/64 beating.
> I personally think this is the most musical approach, and that this has
> some merits to how our brains/ears may perceive this beating.
>
> -Marcel
>
>
>
> > Sorry if I use wrong descriptors/words; english is not my native
> > language. I'dlike to solve a following issue.
> >
> > When using multi-tone/sine setup, we can create intervals or chords
> > (3-tone or more). Now - I'm trying to find some formulas, that allow
> > to calculate a vibrations/beating, that happens in chords and
> > intervals. There is a lot of data in the internet about ratios and
> > tunings in general, but I can't find anything useful on that slight
> > harmonic beating (not "binaural beating", when two freqs are very
> > close together). Some parts I have figured out experimental, but still
> > don't know if there is a way to generalize it. Harmonic series and
> > "fractions" are involved.
> >
> > On an example.
> >
> > 1) Let say, that you have an interval of two tones. 100Hz and 204Hz.
> > If you see the waveform of that mix, you will notice, that there is
> > 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first
> > one).
> >
> > 2) Let say that you have a CEG chord of three tones. To start with -
> > 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord"
> > of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if
> > you shift the mid tone, let say to 248 or 252Hz, then you get an
> > audible vibration of 2Hz (250Hz seems to be the harmonic zero-point so
> > to speak).
> >
> > My question - how to calculate this stuff more general?
> > And also for more than 3 tones, and around various harmonic
> > zero-points (various depths of vibration)?
> >
> >
>

🔗ayamahambho <ayamahambho@...>

Okay, I think I have figured it out. Was too blind on something :-)

There seem to be two things that interfere on that level of my interest. First is something that internet calls "displacement frequencies" (at small difference - pitch centers), and second - interval points from harmonic series.

As for displacement freqs, I found that name following a math formula that emerged from practical experiments. They show something like this. If you have two sine sounds, f1 and f2, then you have a (f1+f2)/2 frequency point, that will vibrate with third f3 sound, close to that point. So if you make f1 = 1000Hz and f2 1200Hz, then mid freq is 1100Hz. If you play a third tone let say f3 = 1101Hz, then you get a 1Hz vibration.

As for intervals - If you have two sounds, f1 and f2, then f2 will produce such vibration at N*f1 points (harmonics; have to figure out to which N it is useful). But... Both, f1 and f2 can be considered as Nth harmonics of something lower else, so now I have to check to which degree it is usefu. I suspect, that 4 harmonics down (which produces 0.25 fractional harmonic steps) will be enough.

These two combined with each other, if they cross, they probably produce even stronger effect, or they change the nature of effect (consonance or something like that). Did I miss or over-complicated something?

Anyway - now it should be easier to create something that calculates these harmonic flows.

--- In tuning@yahoogroups.com, "ayamahambho" <ayamahambho@...> wrote:
>
> I don't speak about binaural beats. Play a CEG (ratio) chord (or use some app to generate 3 sine tones: let say 200, 300 and variations around 250 Hz) and you will notice a vibration, that is not binaural beat. I look for algorithms to calculate that sort of relationships between pure (sine) tones, but also within reasonable (audible) range (and amount of tones on board). From what I see and hear, these vibrations can be calculated/predicted in Hz. But there seem to be too much guessing. I'd like to implement such automated calculator to my software.
>
> --- In tuning@yahoogroups.com, Marcel de Velde <marcel@> wrote:
> >
> > First of all, all tones that are not perfect unison "beat" in a way.
> > Here is a definition of beating:
> > http://en.wikipedia.org/wiki/Beat_(acoustics)
> > <http://en.wikipedia.org/wiki/Beat_%28acoustics%29>
> > Even when we combine 100 and 200 Hertz we get a 100 Hertz beat. That is,
> > a constructive and deconstructive interference occurring 100 times per
> > second.
> > Since this falls exactly at the same frequency as the lower tone we
> > don't usually call it beating though.
> > But for 200 and 300 Hertz, the beating frequency is 100 Hertz, and where
> > there is no tone present one could call this beating.
> > If one uses timbres with overtones then the beating occurs between all
> > the overtones.
> > For equal and perfect harmonic timbres the beating frequencies will be
> > an exact copy of the timbre, though usually only the lowest beating
> > frequency is meant.
> >
> > Some people seem to feel beating means a slow phasing of the timbre when
> > two tones sound together. One can describe this by creating a cutoff
> > frequency above which one no longer calls the interference frequency
> > "beating".
> >
> > Furthermore, since we're talking about music and tuning, we can add that
> > beating is something relative to a desired rational intonation system.
> > What I mean is if your aim is 5/4 for instance, yet you play 81/64 you
> > may say that this 81/64 is beating with a certain frequency relative to
> > 5/4 (which is actually beating itself as well if we take the strictest
> > definition of beating).
> > But if your aim is 81/64 then if you play it correctly you may say you
> > do not see it as beating, and choose to only call phasing relative to
> > this 81/64 beating.
> > I personally think this is the most musical approach, and that this has
> > some merits to how our brains/ears may perceive this beating.
> >
> > -Marcel
> >
> >
> >
> > > Sorry if I use wrong descriptors/words; english is not my native
> > > language. I'dlike to solve a following issue.
> > >
> > > When using multi-tone/sine setup, we can create intervals or chords
> > > (3-tone or more). Now - I'm trying to find some formulas, that allow
> > > to calculate a vibrations/beating, that happens in chords and
> > > intervals. There is a lot of data in the internet about ratios and
> > > tunings in general, but I can't find anything useful on that slight
> > > harmonic beating (not "binaural beating", when two freqs are very
> > > close together). Some parts I have figured out experimental, but still
> > > don't know if there is a way to generalize it. Harmonic series and
> > > "fractions" are involved.
> > >
> > > On an example.
> > >
> > > 1) Let say, that you have an interval of two tones. 100Hz and 204Hz.
> > > If you see the waveform of that mix, you will notice, that there is
> > > 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first
> > > one).
> > >
> > > 2) Let say that you have a CEG chord of three tones. To start with -
> > > 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord"
> > > of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if
> > > you shift the mid tone, let say to 248 or 252Hz, then you get an
> > > audible vibration of 2Hz (250Hz seems to be the harmonic zero-point so
> > > to speak).
> > >
> > > My question - how to calculate this stuff more general?
> > > And also for more than 3 tones, and around various harmonic
> > > zero-points (various depths of vibration)?
> > >
> > >
> >
>

🔗kraiggrady <kraiggrady@...>

🔗Mike Battaglia <battaglia01@...>

On Fri, May 3, 2013 at 4:06 PM, ayamahambho <ayamahambho@...> wrote:
>
> Sorry if I use wrong descriptors/words; english is not my native language. I'dlike to solve a following issue.
>
> When using multi-tone/sine setup, we can create intervals or chords (3-tone or more). Now - I'm trying to find some formulas, that allow to calculate a vibrations/beating, that happens in chords and intervals. There is a lot of data in the internet about ratios and tunings in general, but I can't find anything useful on that slight harmonic beating (not "binaural beating", when two freqs are very close together). Some parts I have figured out experimental, but still don't know if there is a way to generalize it. Harmonic series and "fractions" are involved.

You're talking about the same thing that I investigated in this
thread: /tuning/topicId_95699.html#95700

We've been calling it "periodicity buzz." For harmonic timbres, I
found that here are the things that generally cause "maximum buzz":

1) Tempered chords will buzz more as they get closer to JI. For
instance, a JI 4:5:6 will buzz more than the 12-EDO version.
2) Assuming JI chords, "isoharmonic" chords, such as 7:9:11:13, will
buzz more than non-isoharmonic ones, like 7:8:11:14. (Isoharmonic
means that the chord is of the form a:a+b:a+2b:etc, so that the
differences between consecutive notes in the chord are the same (in
this case, they're equal to "b").)
3) Assuming you have two isoharmonic chords with equal "spacing"
between the notes, chords with more notes buzz more than chords with
less notes. For instance, 7:9:11:13:15:17 will buzz more than 7:9:11.
4) Assuming isoharmonic chords of the same number of notes, chords
with (logarithmically) smaller consecutive dyads will buzz more than
chords with larger ones. For instance, 8:9:10:11:12 will buzz more
than 1:3:5:7:9.

These are simplified "rules of thumb" and aren't hard and fast exact
rules, but they generally do the trick. There are some additional
factors as well, such as the phase of the notes involved, the total
amount of distortion present in the signal pathway as the sound leaves
the speaker and makes its way to your brain, and the timbre involved.
For instance, if the timbre is a sine wave, and there isn't a lot of
distortion, generally *any* chord whose frequencies form a subset of
an arithmetic series (e.g. there's a constant difference in Hz between
consecutive notes) will buzz, even if it's not close to JI at all. I
give some examples of "stretched JI chords" which still buzz in the
thread I linked to above. But, for harmonic timbres, I've found that
the four rules of thumb above generally encapsulate the relative
behavior to a decent enough degree that you can predict, correctly,
whats's going on. Note the importance of #4, which you might find
counterintuitive at first.

If you're interested in the psychoacoustics behind this phenomenon,
you can click on the thread above to see some more discussion on it.

Mike

🔗ayamahambho <ayamahambho@...>

Thanks.

I'm playing with this little fellow (windows only):
http://planetaziemia.net/explorers/hse/beeone-smod-workflow.png
http://www.planetaziemia.net/counter/click.php?id=66
http://planetaziemia.net/hemi-synths.htm

and there are few things left to implement. Generally the (f1+f2)/2 formula for interference midpoints (plus harmonic indications for them) - probably will cover the job. I just need to spend some hours to figure out how to easy re-matrix these calculations. :-) In sound production, I'm using such vibrations by purpose (producing "phantom tones"), and sometimes must avoid them.

J.K.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, May 3, 2013 at 4:06 PM, ayamahambho <ayamahambho@...> wrote:
> >
> > Sorry if I use wrong descriptors/words; english is not my native language. I'dlike to solve a following issue.
> >
> > When using multi-tone/sine setup, we can create intervals or chords (3-tone or more). Now - I'm trying to find some formulas, that allow to calculate a vibrations/beating, that happens in chords and intervals. There is a lot of data in the internet about ratios and tunings in general, but I can't find anything useful on that slight harmonic beating (not "binaural beating", when two freqs are very close together). Some parts I have figured out experimental, but still don't know if there is a way to generalize it. Harmonic series and "fractions" are involved.
>
> You're talking about the same thing that I investigated in this
> thread: /tuning/topicId_95699.html#95700
>
> We've been calling it "periodicity buzz." For harmonic timbres, I
> found that here are the things that generally cause "maximum buzz":
>
> 1) Tempered chords will buzz more as they get closer to JI. For
> instance, a JI 4:5:6 will buzz more than the 12-EDO version.
> 2) Assuming JI chords, "isoharmonic" chords, such as 7:9:11:13, will
> buzz more than non-isoharmonic ones, like 7:8:11:14. (Isoharmonic
> means that the chord is of the form a:a+b:a+2b:etc, so that the
> differences between consecutive notes in the chord are the same (in
> this case, they're equal to "b").)
> 3) Assuming you have two isoharmonic chords with equal "spacing"
> between the notes, chords with more notes buzz more than chords with
> less notes. For instance, 7:9:11:13:15:17 will buzz more than 7:9:11.
> 4) Assuming isoharmonic chords of the same number of notes, chords
> with (logarithmically) smaller consecutive dyads will buzz more than
> chords with larger ones. For instance, 8:9:10:11:12 will buzz more
> than 1:3:5:7:9.
>
> These are simplified "rules of thumb" and aren't hard and fast exact
> rules, but they generally do the trick. There are some additional
> factors as well, such as the phase of the notes involved, the total
> amount of distortion present in the signal pathway as the sound leaves
> the speaker and makes its way to your brain, and the timbre involved.
> For instance, if the timbre is a sine wave, and there isn't a lot of
> distortion, generally *any* chord whose frequencies form a subset of
> an arithmetic series (e.g. there's a constant difference in Hz between
> consecutive notes) will buzz, even if it's not close to JI at all. I
> give some examples of "stretched JI chords" which still buzz in the
> thread I linked to above. But, for harmonic timbres, I've found that
> the four rules of thumb above generally encapsulate the relative
> behavior to a decent enough degree that you can predict, correctly,
> whats's going on. Note the importance of #4, which you might find
> counterintuitive at first.
>
> If you're interested in the psychoacoustics behind this phenomenon,
> you can click on the thread above to see some more discussion on it.
>
> Mike
>

🔗bigAndrewM <bigandrewm@...>

It may also be useful to make a distinction between what this source calls first-degree and second-degree beats, because they're caused by different physical phenomena. Phenomenas? Phenomenae? Whatever the plural of that word is.

http://www.sfu.ca/sonic-studio/handbook/Beats.html

Andrew

--- In tuning@yahoogroups.com, "ayamahambho" <ayamahambho@...> wrote:
>
> Sorry if I use wrong descriptors/words; english is not my native language. I'dlike to solve a following issue.
>
> When using multi-tone/sine setup, we can create intervals or chords (3-tone or more). Now - I'm trying to find some formulas, that allow to calculate a vibrations/beating, that happens in chords and intervals. There is a lot of data in the internet about ratios and tunings in general, but I can't find anything useful on that slight harmonic beating (not "binaural beating", when two freqs are very close together). Some parts I have figured out experimental, but still don't know if there is a way to generalize it. Harmonic series and "fractions" are involved.
>
> On an example.
>
> 1) Let say, that you have an interval of two tones. 100Hz and 204Hz. If you see the waveform of that mix, you will notice, that there is 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first one).
>
> 2) Let say that you have a CEG chord of three tones. To start with - 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if you shift the mid tone, let say to 248 or 252Hz, then you get an audible vibration of 2Hz (250Hz seems to be the harmonic zero-point so to speak).
>
> My question - how to calculate this stuff more general?
> And also for more than 3 tones, and around various harmonic zero-points (various depths of vibration)?
>

🔗Cornell III, Howard M <howard.m.cornell.iii@...>

"Phenomena" IS the plural. The singular is "phenomenon."

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of bigAndrewM
Sent: Monday, May 06, 2013 3:57 PM
To: tuning@yahoogroups.com
Subject: EXTERNAL: [tuning] Re: chords - how to calculate their...

http://www.sfu.ca/sonic-studio/handbook/Beats.html

Andrew

--- In tuning@yahoogroups.com<mailto:tuning%40yahoogroups.com>, "ayamahambho" <ayamahambho@...> wrote:
>
> Sorry if I use wrong descriptors/words; english is not my native language. I'dlike to solve a following issue.
>
> When using multi-tone/sine setup, we can create intervals or chords (3-tone or more). Now - I'm trying to find some formulas, that allow to calculate a vibrations/beating, that happens in chords and intervals. There is a lot of data in the internet about ratios and tunings in general, but I can't find anything useful on that slight harmonic beating (not "binaural beating", when two freqs are very close together). Some parts I have figured out experimental, but still don't know if there is a way to generalize it. Harmonic series and "fractions" are involved.
>
> On an example.
>
> 1) Let say, that you have an interval of two tones. 100Hz and 204Hz. If you see the waveform of that mix, you will notice, that there is 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first one).
>
> 2) Let say that you have a CEG chord of three tones. To start with - 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if you shift the mid tone, let say to 248 or 252Hz, then you get an audible vibration of 2Hz (250Hz seems to be the harmonic zero-point so to speak).
>
> My question - how to calculate this stuff more general?
> And also for more than 3 tones, and around various harmonic zero-points (various depths of vibration)?
>

🔗bigAndrewM <bigandrewm@...>

🔗ayamahambho <ayamahambho@...>

I check how it sounds on Bee, then we will see. Thanks. I must limit the accuracy of audible depth/recognition to some usable degree.

J.K.

--- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@...> wrote:
>
> It may also be useful to make a distinction between what this source calls first-degree and second-degree beats, because they're caused by different physical phenomena. Phenomenas? Phenomenae? Whatever the plural of that word is.
>
> http://www.sfu.ca/sonic-studio/handbook/Beats.html
>
> Andrew
>
>
>
> --- In tuning@yahoogroups.com, "ayamahambho" <ayamahambho@> wrote:
> >
> > Sorry if I use wrong descriptors/words; english is not my native language. I'dlike to solve a following issue.
> >
> > When using multi-tone/sine setup, we can create intervals or chords (3-tone or more). Now - I'm trying to find some formulas, that allow to calculate a vibrations/beating, that happens in chords and intervals. There is a lot of data in the internet about ratios and tunings in general, but I can't find anything useful on that slight harmonic beating (not "binaural beating", when two freqs are very close together). Some parts I have figured out experimental, but still don't know if there is a way to generalize it. Harmonic series and "fractions" are involved.
> >
> > On an example.
> >
> > 1) Let say, that you have an interval of two tones. 100Hz and 204Hz. If you see the waveform of that mix, you will notice, that there is 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first one).
> >
> > 2) Let say that you have a CEG chord of three tones. To start with - 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if you shift the mid tone, let say to 248 or 252Hz, then you get an audible vibration of 2Hz (250Hz seems to be the harmonic zero-point so to speak).
> >
> > My question - how to calculate this stuff more general?
> > And also for more than 3 tones, and around various harmonic zero-points (various depths of vibration)?
> >
>

🔗ayamahambho <ayamahambho@...>

Okay, I think I finished the first part, which refers to interference with midpoints ("f3 ~ (f1+f2)/2" formula).

Now it's time for the second part - interval based. From what I see, points of audible/usable interference will go like this:

1f -- 2f -- 3f -- 4f ==> "f" is a first in the series
1/2f -- f -- 3/2f -- 2f ==> "f" is second in the series
1/3f -- 2/3f -- f -- 4/3f ==> "f" is third in the series
1/4f -- 1/2f -- 3/4f -- f ==> "f" is fourth in the series

"f" is a base, everything else are virtual harmonic points of interference with the second tone.

Elliminating all duplicates, we will end with a list of 11 items, from which one will contain physical differential beats. All other will represent "h-points" around which vibration will happen.

Am I correct?

J.K.