Yahoo Tuning Groups Ultimate Backup tuning Porcupine-8 tonal theory; sum/difference tones and consonance

Hello all,

For those that don't know me from MMM, my name's Igliashon Jones and
I've been studying microtonal music for about 10 months now. I've
got 3 microtonal guitars (22- and 31-edo plus a Catler Ultra Plus)
and a 14-out-of-22-edo tubulong that is almost done, and I've
familiarized myself with nearly every microtonal resource on the
internet. I'm here because I want to talk a little theory and MMM
ain't quite the place for that.

I have two questions:

1) Has anyone yet written any papers/articles involving tonal theory
based on the 8-note Porcupine scale? If not, I've been doing a lot
of work in this system and have been thinking about writing an
essay/article on what I've discovered and put together, similar in
style to Paul Erlich's "Tuning, Tonality, and 22-tone Equal
Temperament". I may not be a veteran of the field, but then again
this paper I propose would be more aimed at microtonal neophytes
looking for new systematic tonal possibilities within a microtonal
framework. Any thoughts?

2) I haven't yet had time to read the massive catalogue of literature
on theories of consonance, but I've been noticing some rather bizarre
phenomena involving sum/difference tones in higher-prime-limit
intervals that have given me an idea that these sum/difference tones
may have something to do with consonance. Can anyone suggest any
reading on the subject that might address this idea? I have only the
most rudimentary knowledge of acoustical physics (I'm a junior
Philosophy major at San Fran State...physics was never really my
subject), so I'm sure I can't be the first to have thought about it.

Many thanks,

-Igliashon

In response to your 2nd question, a quick summary:

Summation tones, on a level, have EVERYTHING to do with "consonance." A
clear example is to think about the method a piano tuner uses -- to set
and tune a perfect fifth (the 3:2), he will need to hear the 6th partial
(the sum of the 2nd and 3rd partials) -- so to tune a C--G fifth, listen
for the G an octave above the struck "G." This can be applied, obviously,
to any ratio. The 4:3 would produce the 12th partial (I should say
PRIMARILY produce, for unless you are using a pure sine-wave for are other
less-strong partials) -- so in the case of a C--F (where "C" is the 3 and
"F" is the 4) you would want to listen for the "C" two octaves above (the
12th partial).

But what happens if you don't hit the 3:2 dead on (i.e. you aren't
"consonant") -- let's say you tune it as a 3.1:2 -- this gives rise to a
summation tone at 6.2 (or about 759 cents) -- so what you will be hearing
will be a "G" which is 59 cents sharps -- obviously not meshing with the
struck "G" an octave below.

That's the long and short of it

Paul

___________________________________________
Paul Greenhaw
Music Specialist II
The New York Public Library for the Performing Arts
40 Lincoln Center Plaza
New York, NY 10023
(212) 870-1892
__________________________________________

"Igliashon Jones" <igliashon@sbcglobal.net>
02/05/2005 12:00 AM
Please respond to tuning

To: tuning@yahoogroups.com
cc:
Subject: [tuning] Porcupine-8 tonal theory; sum/difference tones and consonance

Many thanks,

-Igliashon

> Summation tones, on a level, have EVERYTHING to do
> with "consonance."

Okay, good. Does this explain why you can't judge an interval's
consonance solely by its simplicity? Can you recommend any readings
on the subject?

-Igliashon

Hi Igs,

>I have two questions:
>
>1) Has anyone yet written any papers/articles involving tonal theory
>based on the 8-note Porcupine scale? If not, I've been doing a lot
>of work in this system and have been thinking about writing an
>essay/article on what I've discovered and put together, similar in
>style to Paul Erlich's "Tuning, Tonality, and 22-tone Equal
>Temperament". I may not be a veteran of the field, but then again
>this paper I propose would be more aimed at microtonal neophytes
>looking for new systematic tonal possibilities within a microtonal
>framework. Any thoughts?

I'm not familiar with any such work. Go for it!

>2) I haven't yet had time to read the massive catalogue of literature
>on theories of consonance, but I've been noticing some rather bizarre
>phenomena involving sum/difference tones in higher-prime-limit
>intervals that have given me an idea that these sum/difference tones
>may have something to do with consonance. Can anyone suggest any
>reading on the subject that might address this idea? I have only the
>most rudimentary knowledge of acoustical physics (I'm a junior
>Philosophy major at San Fran State...physics was never really my
>subject), so I'm sure I can't be the first to have thought about it.

I agree with Paul E. that combination tones are not of primary
importance in consonance.

Where to start on this subject? Maybe here...

http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937

Then here...

http://www.amazon.com/exec/obidos/tg/detail/-/1852337974/

Then here...

/harmonic_entropy

-Carl

>Summation tones, on a level, have EVERYTHING to do with "consonance."
>A clear example is to think about the method a piano tuner uses -- to
>set and tune a perfect fifth (the 3:2), he will need to hear the 6th
>partial (the sum of the 2nd and 3rd partials) -- so to tune a C--G
>fifth, listen for the G an octave above the struck "G."

Hi Paul G.,

I'm afraid I don't follow you here. One can put C-G in a pure 3:2
by eliminating the beats between the 3rd partial of the C and the
2nd partial of the G. . . not sure what this has to do with
combination (sum and difference) tones, which are a different
phenomenon than "analytical listening" (hearing partials in a
complex tone).

-Carl

>> Summation tones, on a level, have EVERYTHING to do
>> with "consonance."
>
>Okay, good. Does this explain why you can't judge an interval's
>consonance solely by its simplicity? Can you recommend any
>readings on the subject?

Hi Igs,

Combination tones can be prominent sometimes, but are all but
inaudible in most musical situations for most listeners. Even
when they are audible, in my experience they are heard as a
separate sensation on top of whatever consonance (or lack
thereof) would otherwise be there.

Consonance has traditionally been explained by critical-band
roughness -- a feature of the spectral analysis done in the
cochlea. But that is not the whole story. Consonance also
has a lot to do with how the information from the cochlea is
interpreted by the brain.

By "simplicity", do you mean the size of the ratio describing an
interval? This is in fact an excellent predictor of consonance
among dyads whose ratios have numerators and denominators less
than about 10-15. This doesn't work for ratios like 30001:20001
due to the phenomenon of 'approximation' -- the ear is very
forgiving in its search for justly-intoned intervals.

See my earlier post for recommended readings.

-Carl

> I'm not familiar with any such work. Go for it!

Cool. It'll be a while coming, though...I don't want to write too
much on a system I've barely composed in.

> Where to start on this subject? Maybe here...
>
> http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937
>
> Then here...
>
> http://www.amazon.com/exec/obidos/tg/detail/-/1852337974/
>
> Then here...
>
> /harmonic_entropy

Tanks! I'll check 'em out.

-Igs

Hi IJ!
The first person to come up with a theory of diff/sum tones being related to Consonance was Helmholtz which you can find in his book. Following along these lines i did quite a bit of music based on different inversions of the same chord ( usually 4 factors) arranged in consonance and dissonance curves. The system i found worked the best for me was taking the sum of all the difference tones and the generating intervals and omitting the duplicates. This seem to work better than just a straight addition and worked better that more involved math. An empirical way to evaluate this system is the arrangement which goes from most consonant to most dissonant. which can be seen and tested with this piece.
http://anaphoria.com/lullaby.html
the most trouble it has is with the first line or two since it is hard to measure a dissonant of a single note :) yet it all fails perfectly into place. My girlfriend at the time i could predict what her threshold would be before she would get uncomfortable, usually within a few numbers from this list. These pieces of which this was only one enabled one to teach players only a few tones and their octaves. their being more reluctance to such things at the time
Regardless more test should possibly be done yet find this method more fruitful than any of the "lattice " distance methods because it takes into consideration that different inversions are different chords.
I have since abandoned research and composition along these lines . While it greatly broaden my palette to different inversions ( there are always those we just don't think of) i found trying to incorporate it all the time into actual music became more burdensome than not and found , one can easily increase tension at will by so many different factors , that having to charts ones course before taking a step was just too big a price to pay.
Still it exposed my ear and imagination to what can be done with such methods , so once one does it, we hopefully absorb it intuitively.

>
>Message: 1 >
> From: "Igliashon Jones" <igliashon@sbcglobal.net>
>Subject: Porcupine-8 tonal theory; sum/difference tones and consonance
>
>
>
>Hello all,
>
>For those that don't know me from MMM, my name's Igliashon Jones and >I've been studying microtonal music for about 10 months now. I've >got 3 microtonal guitars (22- and 31-edo plus a Catler Ultra Plus) >and a 14-out-of-22-edo tubulong that is almost done, and I've >familiarized myself with nearly every microtonal resource on the >internet. I'm here because I want to talk a little theory and MMM >ain't quite the place for that.
>
>I have two questions:
>
>1) Has anyone yet written any papers/articles involving tonal theory >based on the 8-note Porcupine scale? If not, I've been doing a lot >of work in this system and have been thinking about writing an >essay/article on what I've discovered and put together, similar in >style to Paul Erlich's "Tuning, Tonality, and 22-tone Equal >Temperament". I may not be a veteran of the field, but then again >this paper I propose would be more aimed at microtonal neophytes >looking for new systematic tonal possibilities within a microtonal >framework. Any thoughts?
>
>2) I haven't yet had time to read the massive catalogue of literature >on theories of consonance, but I've been noticing some rather bizarre >phenomena involving sum/difference tones in higher-prime-limit >intervals that have given me an idea that these sum/difference tones >may have something to do with consonance. Can anyone suggest any >reading on the subject that might address this idea? I have only the >most rudimentary knowledge of acoustical physics (I'm a junior >Philosophy major at San Fran State...physics was never really my >subject), so I'm sure I can't be the first to have thought about it.
>
>Many thanks,
>
>-Igliashon
>
>
>
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 2 > Date: Sat, 5 Feb 2005 11:54:08 -0500
> From: pgreenhaw@nypl.org
>Subject: Re: Porcupine-8 tonal theory; sum/difference tones and consonance
>
>In response to your 2nd question, a quick summary:
>
>Summation tones, on a level, have EVERYTHING to do with "consonance." A >clear example is to think about the method a piano tuner uses -- to set >and tune a perfect fifth (the 3:2), he will need to hear the 6th partial >(the sum of the 2nd and 3rd partials) -- so to tune a C--G fifth, listen >for the G an octave above the struck "G." This can be applied, obviously, >to any ratio. The 4:3 would produce the 12th partial (I should say >PRIMARILY produce, for unless you are using a pure sine-wave for are other >less-strong partials) -- so in the case of a C--F (where "C" is the 3 and >"F" is the 4) you would want to listen for the "C" two octaves above (the >12th partial).
>
>But what happens if you don't hit the 3:2 dead on (i.e. you aren't >"consonant") -- let's say you tune it as a 3.1:2 -- this gives rise to a >summation tone at 6.2 (or about 759 cents) -- so what you will be hearing >will be a "G" which is 59 cents sharps -- obviously not meshing with the >struck "G" an octave below.
>
>That's the long and short of it
>
>Paul
>
>
>
>___________________________________________
>Paul Greenhaw
>Music Specialist II
>The New York Public Library for the Performing Arts
>40 Lincoln Center Plaza
>New York, NY 10023
>(212) 870-1892
>__________________________________________
>
>
>
>
>
>"Igliashon Jones" <igliashon@sbcglobal.net>
>02/05/2005 12:00 AM
>Please respond to tuning
>
> > To: tuning@yahoogroups.com
> cc: > Subject: [tuning] Porcupine-8 tonal theory; sum/difference tones and consonance
>
>
>
>
>2) I haven't yet had time to read the massive catalogue of literature >on theories of consonance, but I've been noticing some rather bizarre >phenomena involving sum/difference tones in higher-prime-limit >intervals that have given me an idea that these sum/difference tones >may have something to do with consonance. Can anyone suggest any >reading on the subject that might address this idea? I have only the >most rudimentary knowledge of acoustical physics (I'm a junior >Philosophy major at San Fran State...physics was never really my >subject), so I'm sure I can't be the first to have thought about it.
>
>Many thanks,
>
>-Igliashon
>
>
>
>
>
>
>
>
>
>[This message contained attachments]
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 3 > Date: Sat, 05 Feb 2005 20:48:02 -0000
> From: "Igliashon Jones" <igliashon@sbcglobal.net>
>Subject: Re: Porcupine-8 tonal theory; sum/difference tones and consonance
>
>
>
> >
>>Summation tones, on a level, have EVERYTHING to do >>with "consonance." >> >>
>
>Okay, good. Does this explain why you can't judge an interval's >consonance solely by its simplicity? Can you recommend any readings >on the subject?
>
>-Igliashon
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 4 > Date: Sat, 05 Feb 2005 21:14:30 -0000
> From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
>Subject: Announcement of radio programme
>
>
>Monday 7 Feb, Concertzender, 20-22 h. CET
>Music on the Russian ANS synthesizer.
>1. Edward Artemiev. Mosaic.
>2. Sofia Goebaidoelina. Vivente-Non Vivente.
>3. Edison Denisov. Birds Singing.
>4. Alfred Schnittke. Steam.
>5. Stanislav Kreitchi. Ruins in the Waste.
>6. Stanislav Kreitchi / Edward Artemiev. Filmmuziek voor `Cosmos'
>7. Edward Artemiev.Filmmuziek voor Solaris: `Ill' en `Ocean'.
>8. Coil. Coilans.
>See also
>http://www.concertzender.nl/cgi/site/index.php?id=10
>For streaming audio click on "luister live".
>
>Manuel
>
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 5 > Date: Sat, 05 Feb 2005 15:49:39 -0800
> From: Carl Lumma <ekin@lumma.org>
>Subject: Re: Porcupine-8 tonal theory; sum/difference tones and consonance
>
>Hi Igs,
>
> >
>>I have two questions:
>>
>>1) Has anyone yet written any papers/articles involving tonal theory >>based on the 8-note Porcupine scale? If not, I've been doing a lot >>of work in this system and have been thinking about writing an >>essay/article on what I've discovered and put together, similar in >>style to Paul Erlich's "Tuning, Tonality, and 22-tone Equal >>Temperament". I may not be a veteran of the field, but then again >>this paper I propose would be more aimed at microtonal neophytes >>looking for new systematic tonal possibilities within a microtonal >>framework. Any thoughts?
>> >>
>
>I'm not familiar with any such work. Go for it!
>
> >
>>2) I haven't yet had time to read the massive catalogue of literature >>on theories of consonance, but I've been noticing some rather bizarre >>phenomena involving sum/difference tones in higher-prime-limit >>intervals that have given me an idea that these sum/difference tones >>may have something to do with consonance. Can anyone suggest any >>reading on the subject that might address this idea? I have only the >>most rudimentary knowledge of acoustical physics (I'm a junior >>Philosophy major at San Fran State...physics was never really my >>subject), so I'm sure I can't be the first to have thought about it.
>> >>
>
>I agree with Paul E. that combination tones are not of primary
>importance in consonance.
>
>Where to start on this subject? Maybe here...
>
>http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937
>
>Then here...
>
>http://www.amazon.com/exec/obidos/tg/detail/-/1852337974/
>
>Then here...
>
>/harmonic_entropy
>
>-Carl
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 6 > Date: Sat, 05 Feb 2005 15:55:18 -0800
> From: Carl Lumma <ekin@lumma.org>
>Subject: Re: Porcupine-8 tonal theory; sum/difference tones and consonance
>
> >
>>Summation tones, on a level, have EVERYTHING to do with "consonance."
>>A clear example is to think about the method a piano tuner uses -- to
>>set and tune a perfect fifth (the 3:2), he will need to hear the 6th
>>partial (the sum of the 2nd and 3rd partials) -- so to tune a C--G
>>fifth, listen for the G an octave above the struck "G."
>> >>
>
>Hi Paul G.,
>
>I'm afraid I don't follow you here. One can put C-G in a pure 3:2
>by eliminating the beats between the 3rd partial of the C and the
>2nd partial of the G. . . not sure what this has to do with
>combination (sum and difference) tones, which are a different
>phenomenon than "analytical listening" (hearing partials in a
>complex tone).
>
>-Carl
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 7 > Date: Sat, 05 Feb 2005 16:06:25 -0800
> From: Carl Lumma <ekin@lumma.org>
>Subject: Re: Re: Porcupine-8 tonal theory; sum/difference tones and consonance
>
> >
>>>Summation tones, on a level, have EVERYTHING to do >>>with "consonance." >>> >>>
>>Okay, good. Does this explain why you can't judge an interval's >>consonance solely by its simplicity? Can you recommend any
>>readings on the subject?
>> >>
>
>Hi Igs,
>
>Combination tones can be prominent sometimes, but are all but
>inaudible in most musical situations for most listeners. Even
>when they are audible, in my experience they are heard as a
>separate sensation on top of whatever consonance (or lack
>thereof) would otherwise be there.
>
>Consonance has traditionally been explained by critical-band
>roughness -- a feature of the spectral analysis done in the
>cochlea. But that is not the whole story. Consonance also
>has a lot to do with how the information from the cochlea is
>interpreted by the brain.
>
>By "simplicity", do you mean the size of the ratio describing an
>interval? This is in fact an excellent predictor of consonance
>among dyads whose ratios have numerators and denominators less
>than about 10-15. This doesn't work for ratios like 30001:20001
>due to the phenomenon of 'approximation' -- the ear is very
>forgiving in its search for justly-intoned intervals.
>
>See my earlier post for recommended readings.
>
>-Carl
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 8 > Date: Sun, 06 Feb 2005 09:12:58 -0000
> From: "Igliashon Jones" <igliashon@sbcglobal.net>
>Subject: Re: Porcupine-8 tonal theory; sum/difference tones and consonance
>
>
>
> >
>>I'm not familiar with any such work. Go for it!
>> >>
>
>Cool. It'll be a while coming, though...I don't want to write too >much on a system I've barely composed in.
>
>
> >
>>Where to start on this subject? Maybe here...
>>
>>http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937
>>
>>Then here...
>>
>>http://www.amazon.com/exec/obidos/tg/detail/-/1852337974/
>>
>>Then here...
>>
>>/harmonic_entropy
>> >>
>
>
>Tanks! I'll check 'em out.
>
>-Igs
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>
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--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> Regardless more test should possibly be done yet find this method
> more fruitful than any of the "lattice " distance methods

What lattice distance methods are there for chords of more than two notes?
This could be an interesting topic.

And for two notes, what are the major differences compared with your
method?

>because it
> takes into consideration that different inversions are different chords.

In the paper I just sent you, Kraig, the lattice does show different inversions as
different chords; they have different shapes.

Best,
Paul

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Combination tones can be prominent sometimes, but are all but
> inaudible in most musical situations for most listeners.

If you're playing electric guitar with distortion, they're most audible (even the
neighbors can hear them)!

> Even
> when they are audible, in my experience they are heard as a
> separate sensation on top of whatever consonance (or lack
> thereof) would otherwise be there.

I, on the other hand, find that combinational tones can beat against one
another, against sounding tones, and possibly against virtual pitches, in each
casing becoming an important determinant of consonance.

--- In tuning@yahoogroups.com, pgreenhaw@n... wrote:
> In response to your 2nd question, a quick summary:
>
> Summation tones, on a level, have EVERYTHING to do with "consonance."

Bill Sethares's book _Tuning, Timbre, Spectrum, Scale" claims they have just
about nothing to do with consonance. Though I give them more credit than
that, I'm afraid your calculations below don't support the case. A great
explanation of nonlinear combinational tones, including summation tones, is
found in the Feynman Lectures on Physics.

>A
> clear example is to think about the method a piano tuner uses -- to set
> and tune a perfect fifth (the 3:2), he will need to hear the 6th partial
> (the sum of the 2nd and 3rd partials) -- so to tune a C--G fifth, listen
> for the G an octave above the struck "G." This can be applied, obviously,
> to any ratio. The 4:3 would produce the 12th partial (I should say
> PRIMARILY produce, for unless you are using a pure sine-wave for are
other
> less-strong partials) -- so in the case of a C--F (where "C" is the 3 and
> "F" is the 4) you would want to listen for the "C" two octaves above (the
> 12th partial).
>
> But what happens if you don't hit the 3:2 dead on (i.e. you aren't
> "consonant") -- let's say you tune it as a 3.1:2 -- this gives rise to a
> summation tone at 6.2 (or about 759 cents)

The summation tone would be 5.1, not 6.2. They're called "summatiion tones"
because you calculate the sum: 3.1 + 2 = 5.1

> -- so what you will be hearing
> will be a "G" which is 59 cents sharps -- obviously not meshing with the
> struck "G" an octave below.

6.2 is in fact the frequency of the second harmonic of the 3.1 note. So your
explanation is in fact invoking harmonics, and not summation tones at all.

Best,
Paul

If I well remember Hindemith wrote that some organ builders have used difference tones to produce deeper basses. This means that sometimes they are audible.

Lorenzo

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: Sunday, February 06, 2005 10:09 PM
Subject: [tuning] Re: Porcupine-8 tonal theory; sum/difference tones and consonance

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Combination tones can be prominent sometimes, but are all but
> inaudible in most musical situations for most listeners.

If you're playing electric guitar with distortion, they're most audible (even the
neighbors can hear them)!

> Even
> when they are audible, in my experience they are heard as a
> separate sensation on top of whatever consonance (or lack
> thereof) would otherwise be there.

You can configure your subscription by sending an empty email to one
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Lorenzo,

This is true! Notice that we are talking about REALLY powerful instruments
here -
your average municipal or cathedral pipe organ uses as much power to pump
air as
you do to heat your home in winter ...

This fact leads to an interesting anomaly on synthesiser keyboards -
typically the
pipe organ sounds do not seem to be low enough when you play at less than
default
volume (100 out of 127). For example, I usually set my master volume to 80,
out of
consideration for other family members, or even less very late at night. At
this
level, any note below about F3 seems very thin and weak, the second partial
tone,
or first octave above, tending to dominate the first partial, or fundamental
tone.
It doesn't begin to match the real pipe organs I've played - in fact, even
our old
parlour harmonium (12 stops) used to sound better. For satisfactory full
organ
sound on the electronic keyboard, I need to crank the volume way up - and
sit
back! :-)

Regards,
Yahya

-----Original Message-----
From: Lorenzo Frizzera
Sent: Monday 7 February 2005 10:17 am
To: tuning
Subject: Re: [tuning] Re: Porcupine-8 tonal theory; sum/difference tones
and consonance

If I well remember Hindemith wrote that some organ builders have used
difference tones to produce deeper basses. This means that sometimes they
are audible.

Lorenzo

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Combination tones can be prominent sometimes, but are all but
> inaudible in most musical situations for most listeners.

If you're playing electric guitar with distortion, they're most audible
(even the
neighbors can hear them)!

> Even
> when they are audible, in my experience they are heard as a
> separate sensation on top of whatever consonance (or lack
> thereof) would otherwise be there.

I, on the other hand, find that combinational tones can beat against one
another, against sounding tones, and possibly against virtual pitches,
in each
casing becoming an important determinant of consonance.

--
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>If I well remember Hindemith wrote that some organ builders have used
>difference tones to produce deeper basses. This means that sometimes
>they are audible.

Oh for sure. That's a tone-color thing, and they are maybe frequently
important (to varying degrees of subtly) there. But, like I just got
back from a string quartet concert, and no musical effect was made of
any difference tones -- I heard none out. Much of Kraig Grady's music,
La Monte Young's Dream House, are two examples I've heard of loud
combination tones being used to good affect. Claiming they have
everything to do with musical consonance is a very different breed of
cat.

-Carl

Yes, they are sometimes audible. But Hindemith lived before the
phenomenon of virtual pitch was well-understood. Today we know that
any near-harmonic series with a missing fundamental will have the
fundamental supplied by the brain as a "virtual pitch", and that this
is measurably different in pitch from the difference tones in the
case of systematically slightly inharmonic stimuli. At quiet volume
levels, this virtual pitch phenomenon is far more important than
difference tones in explaining the heard "deep bass" below low 2:3s
(the organ builders' trick).

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@c...> wrote:
> If I well remember Hindemith wrote that some organ builders have
used difference tones to produce deeper basses. This means that
sometimes they are audible.
>
> Lorenzo
>
>
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: Sunday, February 06, 2005 10:09 PM
> Subject: [tuning] Re: Porcupine-8 tonal theory; sum/difference
tones and consonance
>
>
>
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > Combination tones can be prominent sometimes, but are all but
> > inaudible in most musical situations for most listeners.
>
> If you're playing electric guitar with distortion, they're most
audible (even the
> neighbors can hear them)!
>
> > Even
> > when they are audible, in my experience they are heard as a
> > separate sensation on top of whatever consonance (or lack
> > thereof) would otherwise be there.
>
> I, on the other hand, find that combinational tones can beat
against one
> another, against sounding tones, and possibly against virtual
pitches, in each
> casing becoming an important determinant of consonance.
>
>
>
>
>
>
>
>
> You can configure your subscription by sending an empty email to
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Service.

There were other mistakes in this post, which I thought I'd point out:

>A
> clear example is to think about the method a piano tuner uses -- to
set
> and tune a perfect fifth (the 3:2), he will need to hear the 6th
partial
> (the sum of the 2nd and 3rd partials)

It's not the sum in any sense. And one doesn't need to hear the 6th
partial of either note in order to do the tuning.

> -- so to tune a C--G fifth, listen
> for the G an octave above the struck "G."

Yes, this is the 2nd partial of the G, and the 3rd parial of the C.
They need to coincide for the tuning to be just -- the tuner
eliminates the beating at this higher G if a just C-G fifth is
desired. All that is required is for the 2nd partial of the G and the
3rd partial of the C to be audible.

> This can be applied, obviously,
> to any ratio. The 4:3 would produce the 12th partial

The 12th partial is over 3-and-a-half-octaves above the fundamental --
it's really not necessary for tuning a 4:3.

> (I should say
> PRIMARILY produce, for unless you are using a pure sine-wave for
are other
> less-strong partials)

Sine waves have no partials whatsoever, and it's most difficult to
tune such intervals by ear using sine waves. As Aaron Johnston
recently said to me, almost any tuning can sound right with sine-wave
timbres.

> -- so in the case of a C--F (where "C" is the 3 and
> "F" is the 4) you would want to listen for the "C" two octaves
above (the
> 12th partial).

The C two octaves above is the 4th partial of the struck "C" and the
3rd partial of the struck "F". For the tuner to get the C-F to be a
just perfect fourth, they must make these two partials coincide --
typically done by eliminating the beating between them. The presence
or absence of the 12th partial in the timbres is largely irrelevant.

> But what happens if you don't hit the 3:2 dead on (i.e. you aren't
> "consonant") -- let's say you tune it as a 3.1:2 -- this gives rise
to a
> summation tone at 6.2 (or about 759 cents)

The actual summation tone, being the sum 3.1 + 2 = 5.1, would be an E
(a major tenth above the C), 21 cents sharp of the equal-tempered E,
or 1621 cents above the struck C. If summation tones are audible,
this is the note you would hear. Let me know if you can hear it.

>-- so what you will be hearing
> will be a "G" which is 59 cents sharps -- obviously not meshing
with the
> struck "G" an octave below.

The 3rd partial of the C will be 59 cents off the 2nd partial of the
C in this case. So once again, beating will be the tuner's guide and
by eliminating the beating between these partials, the tuner can
approach a just 2:3 between the C and the G. The summation tone,
which is in the vicinity of E in this case, will not be of any help
to the tuner, even if he or she somehow manages to hear it.

I think I see why you said "6th partial" and "12th partial" above.
It's true that if you have a simple integer ratio *in lowest terms*
p:q, then the lowest common harmonic the notes share is p*q in the
same units, such that the two notes and their lowest common harmonic
form the triad p:q:p*q. This lowest common harmonic (p*q) has been
referred to as a "guide tone" among other things, but it's completely
different from the summation tone. Moreover, for the example 3.1:2,
the guide tone is *not* 6.2 because 3.1:2 is not a simple integer
ratio in lowest terms. If this ratio qualifies for a "guide tone" at
all, it's by virtue of its being 31:20 in lowest terms, in which case
the guide tone would be 31*20=620; the chord 20:31:620 containing the
intervals 759 cents between the first two notes and 5945 cents
(almost 5 octaves) between the lowest and highest notes.

Can you make some short example of how virtual pitch works?
I' think I've understood the concept but I would like know, for example, if virtual pitch provides a "tonic" for each interval (dyad).

Lorenzo

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: Monday, February 07, 2005 8:40 PM
Subject: [tuning] Re: Porcupine-8 tonal theory; sum/difference tones and consonance

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--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@c...> wrote:

> Can you make some short example of how virtual pitch works?

There are some short examples of virtual pitch in Bill Sethares's
book _Tuning, Timbre, Spectrum, Scale_, which comes with a CD of
sound examples. If you don't have this book and don't feel like
purchasing it, I'll make some similar examples for you myself using
Matlab -- let me know.

> I' think I've understood the concept but I would like know, for
>example, if virtual pitch provides a "tonic" for each interval >
(dyad).

When more than two notes are playing, it may be the case that the
brain tried to provide a virtual pitch for each dyad -- I don't know
of any firm evidence one way or the other, though harmonic practice
is suggestive. But I think it normally takes three or more pitches
(or partials), approximating a harmonic series, to evoke a strong
virtual pitch, which is the best-fit fundamental of those pitches (or
partials). And what is clear from the evidence is that this virtual
pitch is different, in general, from the difference tone or any of
the higher-order combinational tones. Inharmonic partials which
differ from a harmonic series by a constant Hz offset will produce
the same difference tones as the harmonic series without the offset,
and yet the virtual pitch moves slightly in the direction of the
offset.

Heya Paul,

>When more than two notes are playing, it may be the case that the
>brain tried to provide a virtual pitch for each dyad -- I don't know
>of any firm evidence one way or the other, though harmonic practice
>is suggestive. But I think it normally takes three or more pitches
>(or partials), approximating a harmonic series, to evoke a strong
>virtual pitch, which is the best-fit fundamental of those pitches (or
>partials). And what is clear from the evidence is that this virtual
>pitch is different, in general, from the difference tone or any of
>the higher-order combinational tones. Inharmonic partials which
>differ from a harmonic series by a constant Hz offset will produce
>the same difference tones as the harmonic series without the offset,
>and yet the virtual pitch moves slightly in the direction of the
>offset.

Do you know of a model for finding virtual pitch of inharmonic
timbres? Did Terhardt take a stab at this?

I have the following in my notes...

>The pitch shifts of inharmonic complex tones were observed already
>in the 1950's by de Boer, who studied the so called first pitch-shift
>effect, the change of pitch as a function of center frequency for
>complex tones with a fixed and equal frequency spacing (de Boer 1976).
>He modelled the effect mathematically by a sawtooth function. A
>pitch-shift was also detected, when the frequency spacing of a complex
>tone with fixed center frequency was varied. Vassilakis (1998) related
>this to the first pitch-shift effect and explained both by a single
>model. The explanations of the pitch-shift effects as well as the
>pitch ambiguity are based on the detection of periodicity in the
>signal waveform (Cariani and Delgutte 1996), (Schneider 2000).
>Compared to harmonic tones, the waveform of inharmonic tones is less
>uniform, and the detection of periodicity becomes harder.
>
>de Boer, E. 1976.
>On the residue and auditory pitch perception.
>Handbook of sensory physiology. Keidel, W.D. and Neff, W.D (eds).
>Berlin: Springer-Verlag.
>
>Cariani, P.A. and Delgutte, B. 1996.
>Neural correlates of the pitch of complex tones, I: Pitch and pitch
>saliance, II: Pitch shift, pitch ambiguity, phase invariance, pitch
>circularity, rate pitch, and the dominanceregion of pitch.
>J. Neurophysiology, 76, 1698-1716, 1717-1734.
>
>Cohen, E.A. 1984.
>Some effects of inharmonic partials on interval perception.
>Music Perception 1, 323-349.
>
>Moore, B.C., Peters, R.V., and Glasberg, B.R. 1986.
>Thresholds for hearing mistuned partials as separate tones in
>harmonic complexes.
>J. Acoust. Soc. Am., 80(2), 479-483.

...but I don't have the text for any of these papers.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Heya Paul,
>
> >When more than two notes are playing, it may be the case that the
> >brain tried to provide a virtual pitch for each dyad -- I don't
know
> >of any firm evidence one way or the other, though harmonic
practice
> >is suggestive. But I think it normally takes three or more pitches
> >(or partials), approximating a harmonic series, to evoke a strong
> >virtual pitch, which is the best-fit fundamental of those pitches
(or
> >partials). And what is clear from the evidence is that this
virtual
> >pitch is different, in general, from the difference tone or any of
> >the higher-order combinational tones. Inharmonic partials which
> >differ from a harmonic series by a constant Hz offset will produce
> >the same difference tones as the harmonic series without the
offset,
> >and yet the virtual pitch moves slightly in the direction of the
> >offset.
>
> Do you know of a model for finding virtual pitch of inharmonic
> timbres?

A very simplistic model is presented in Sethares's book. If you don't
have the book, perhaps Bill would allow me to state it here . . .
What's interesting is that the same 'timbre' can be heard having two
or more different virtual pitches (corresponding to interpretations
of the physically present partials in terms of a different set of
harmonic numbers), and often the context will determine which virtual
pitch is more salient. Bill's book and CD contain an example where a
single inharmonic timbre is heard as having a certain pitch in one
context, and a different pitch in another context.

Essentially, harmonic entropy calculates the probability for various
different virtual pitches and calculates the degree of confusion
coming from the conflict between the various possibilities. The more
the input resembles a clear, low section of the harmonic series, the
lower the calculated harmonic entropy will be.

> Did Terhardt take a stab at this?
>
> I have the following in my notes...
>
> >The pitch shifts of inharmonic complex tones were observed already
> >in the 1950's by de Boer, who studied the so called first pitch-
shift
> >effect, the change of pitch as a function of center frequency for
> >complex tones with a fixed and equal frequency spacing (de Boer
1976).
> >He modelled the effect mathematically by a sawtooth function.

The big discontinuities in the sawtooth would correspond to where the
brain "flipped" between an interpretation in terms of one set of
consecutive harmonic numbers to an interpretation involving another
set of consecutive harmonic numbers, offset from the original set by
1. As partials having a fixed and equal frequency spacing will form a
clear harmonic series every time the center partial is a small
multiple of this frequency spacing, the sawtooth function would have
a period (on the graph, where the x-axis is Hz) equal to the fixed
frequency spacing itself.

> A
> >pitch-shift was also detected, when the frequency spacing of a
complex
> >tone with fixed center frequency was varied. Vassilakis (1998)
related
> >this to the first pitch-shift effect and explained both by a single
> >model.

I'd be interested to see it!

> >The explanations of the pitch-shift effects as well as the
> >pitch ambiguity are based on the detection of periodicity in the
> >signal waveform (Cariani and Delgutte 1996), (Schneider 2000).
> >Compared to harmonic tones, the waveform of inharmonic tones is
less
> >uniform, and the detection of periodicity becomes harder.
> >
> >de Boer, E. 1976.
> >On the residue and auditory pitch perception.
> >Handbook of sensory physiology. Keidel, W.D. and Neff, W.D (eds).
> >Berlin: Springer-Verlag.
> >
> >Cariani, P.A. and Delgutte, B. 1996.
> >Neural correlates of the pitch of complex tones, I: Pitch and pitch
> >saliance, II: Pitch shift, pitch ambiguity, phase invariance, pitch
> >circularity, rate pitch, and the dominanceregion of pitch.
> >J. Neurophysiology, 76, 1698-1716, 1717-1734.
> >
> >Cohen, E.A. 1984.
> >Some effects of inharmonic partials on interval perception.
> >Music Perception 1, 323-349.
> >
> >Moore, B.C., Peters, R.V., and Glasberg, B.R. 1986.
> >Thresholds for hearing mistuned partials as separate tones in
> >harmonic complexes.
> >J. Acoust. Soc. Am., 80(2), 479-483.
>
> ...but I don't have the text for any of these papers.
>
> -Carl

I found this:

http://epl.meei.harvard.edu/~bard/Papers/CedolinISH2003.pdf

There's a lot of Cariani stuff online too, accessible from his
website. Gotta run!

>> >Inharmonic partials which
>> >differ from a harmonic series by a constant Hz offset will produce
>> >the same difference tones as the harmonic series without the
>> >offset, and yet the virtual pitch moves slightly in the direction
>> >of the offset.
>>
>> Do you know of a model for finding virtual pitch of inharmonic
>> timbres?
>
>A very simplistic model is presented in Sethares's book. If you don't
>have the book, perhaps Bill would allow me to state it here . . .

I have the 1st edition in storage, and the 2nd edition in my
massive Amazon cart, which I hope to realize in a few weeks.

I read the 1st edition and listened to the CD in 1999, but that was
a while ago...

>What's interesting is that the same 'timbre' can be heard having two
>or more different virtual pitches (corresponding to interpretations
>of the physically present partials in terms of a different set of
>harmonic numbers), and often the context will determine which virtual
>pitch is more salient.

Yes, I know this from experience.

>Essentially, harmonic entropy calculates the probability for various
>different virtual pitches and calculates the degree of confusion
>coming from the conflict between the various possibilities. The more
>the input resembles a clear, low section of the harmonic series, the
>lower the calculated harmonic entropy will be.

Right, but it isn't helpful in, say, finding the true pitch of a
low piano string. Or is it?

>> I have the following in my notes...
>>
>>> The pitch shifts of inharmonic complex tones were observed already
>>> in the 1950's by de Boer, who studied the so called first pitch-
>>> shift effect, the change of pitch as a function of center frequency
>>> for complex tones with a fixed and equal frequency spacing (de Boer
>>> 1976). He modelled the effect mathematically by a sawtooth function.
>
>The big discontinuities in the sawtooth would correspond to where the
>brain "flipped" between an interpretation in terms of one set of
>consecutive harmonic numbers to an interpretation involving another
>set of consecutive harmonic numbers, offset from the original set by
>1. As partials having a fixed and equal frequency spacing will form a
>clear harmonic series every time the center partial is a small
>multiple of this frequency spacing,

What's the center partial?

>the sawtooth function would have a period (on the graph, where the
>x-axis is Hz) equal to the fixed frequency spacing itself.

Ok.

>>>Vassilakis (1998) related this to the first pitch-shift effect
>>>and explained both by a single model.
>
>I'd be interested to see it!

Don't know when I'll have time to dig these up.

>I found this:
>
> http://epl.meei.harvard.edu/~bard/Papers/CedolinISH2003.pdf

Downloaded and on the stack.

>There's a lot of Cariani stuff online too, accessible from his
>website. Gotta run!

Yeah, I should go over there and grab them.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Right, but it isn't helpful in, say, finding the true pitch of a
> low piano string. Or is it?

Not yet, at any rate . . . of course the place for this discussion is:

/harmonic_entropy/

> >The big discontinuities in the sawtooth would correspond to where
the
> >brain "flipped" between an interpretation in terms of one set of
> >consecutive harmonic numbers to an interpretation involving
another
> >set of consecutive harmonic numbers, offset from the original set
by
> >1. As partials having a fixed and equal frequency spacing will
form a
> >clear harmonic series every time the center partial is a small
> >multiple of this frequency spacing,
>
> What's the center partial?

Whichever one is in the center -- of course I could have
said "lowest" or "highest" instead, it would amount to the same thing
in this case.

> >the sawtooth function would have a period (on the graph, where the
> >x-axis is Hz) equal to the fixed frequency spacing itself.
>
> Ok.

I take it you follow, then . . .

> >There's a lot of Cariani stuff online too, accessible from his
> >website. Gotta run!
>
> Yeah, I should go over there and grab them.

http://www.cariani.com/

>> >the sawtooth function would have a period (on the graph, where the
>> >x-axis is Hz) equal to the fixed frequency spacing itself.
>>
>> Ok.
>
>I take it you follow, then . . .

Yes, I think so...

-Carl

Porcupine-8 tonal theory; sum/difference tones and consonance

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