Yahoo Tuning Groups Ultimate Backup tuning Some new? stuff on wikipedia

NICE!

Indeed, if you did it using all possible intervals with 3 sine waves, you would likely find the ideal areas for triads (assuming instruments have a timbre that matches the harmonic series).
Now I wonder....what audio program are they using to do that? If I knew...perhaps I could try....

Once thing I found myself is that the harmonic series partials about about x/9 (IE 10/9 11/9, etc.) are about the smallest set you can put next to each other before dissonance starts really creeping in. Note most of my latest scales use little more than a 9 denominator IE "x/9" format for the fractions. The fractions obtained seemed to confirm this: the highest denominator that appears in the graph is a 9 (13/9, 17/9). I think just-intonation diatonic actually goes astray because when you look at

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 you notice the smallest gap between tones is 15/14 (and 14 is a lot more than 9).

One thing to consider....a lot can go wrong soon as you try a chord without 1/1 as the root. Also the triad DFA, for example, is 9/8 4/3 5/3...which can only be summarized at 27/24 32/24 40/24 (24 is obviously far more than the 9 denominator "goal"). Taking ratios directly from the graph the scale I can form

7:4 = 1.75
5:3 = 1.6666666666666
3:2 = 1.5
4:3 = 1.3333333333333
5:4 = 1.25
7:6 = 1.166666666
1:1

.......seems like a much better alternative as everything reduces to x/12 and not x/24. My guess is around that set of notes would be what you got if you used, say, 7 sine waves instead of 2 for the test. One problem though (IMVHO)...7:6, 5:4, and 4:3 (plus 7:4 and 5:3) are too close to use consecutively in chords...so IMVHO it still isn't quite ideal because it does not deal well with the roughness issue for those notes. Sure seems to give a lot of (according to JI) possible perfect triads, though....

One thing I've found is that making a scale with either a lowest common denominator of 2 or 3 between fractions (IE 3/2 4/3 15/8) works well to maximize the number of just triads possible...but anything with two tones closer than about 12/11 from each other (roughness) starts to sound bad. Sure a note at15/8 sounds good next to 8/8...but put the octave as a third note in the chord and it sounds very rough (minor second interval).

________________________________
From: Chris Vaisvil <chrisvaisvil@...>
To: tuning@yahoogroups.com
Sent: Wed, January 13, 2010 3:42:24 PM
Subject: [tuning] Some new? stuff on wikipedia

Have you seen this audio spectrum?

http://upload. wikimedia. org/wikipedia/ commons/f/ f9/Dissonance- a220-a440- notated.jpg

in article

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

It is a graph that shows the spectra created when two sine waves are diverged from the unison to the octave.
Major intervals are notated.

I'd love to see the same with 3 tones.

Chris

🔗sevishmusic <sevish@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Mike Battaglia <battaglia01@...>

Furthermore, the overtones present at the end that magically pop up when the
two sine waves enter into a 2:1 ratio indicate that there was some
fundamental flaw in the rendering process. Two sine waves added together
will not create the presence of a third sinusoid, unless they are clipped or
in some way distorted.

My best guess is that you have happened upon yet another of the millions of
errors that exist on Wikipedia. Most likely, when this was rendered, the two
sine waves being added together were full scale sinusoids, so that two of
them added together would lead to clipping. Since clipping is a particularly
harsh form of distortion, it would explain the overtone series present at
the end.

In fact, if you look at the very first tone, you will see that the first,
third, and fifth overtones are present -- but not the second or fourth. Look
at the spacing of the overtones to confirm this for yourself. This indicates
that some kind of symmetrical waveform is present -- perhaps a square wave,
which would be precisely what results when a sine wave is clipped.

-Mike

On Wed, Jan 13, 2010 at 8:19 PM, Mike Battaglia <battaglia01@...>wrote:

> Not to burst your bubble, but all of these lines and patterns popping up --
> other than the original two sine waves -- are artefacts and byproducts of
> the analyzing process. They reflect fundamental limitations in the STFT (the
> mathematical transform behind the "spectrogram"), not laws of nature that
> happen whenever two sine waves are played together.
>
> -Mike
>
>
>
> On Wed, Jan 13, 2010 at 4:42 PM, Chris Vaisvil <chrisvaisvil@...>wrote:
>
>>
>>
>> Have you seen this audio spectrum?
>>
>>
>> http://upload.wikimedia.org/wikipedia/commons/f/f9/Dissonance-a220-a440-notated.jpg
>>
>> in article
>>
>> http://en.wikipedia.org/wiki/Consonance_and_dissonance
>>
>> It is a graph that shows the spectra created when two sine waves are
>> diverged from the unison to the octave.
>> Major intervals are notated.
>>
>> I'd love to see the same with 3 tones.
>>
>> Chris
>>
>>
>
>

🔗Mike Battaglia <battaglia01@...>

A last response to really drive the point home, here is an excerpt from the
article:

"The physical basis for Pythagoras's observation (when using
harmonic<http://en.wikipedia.org/wiki/Harmonic_series_%28music%29>
timbres <http://en.wikipedia.org/wiki/Timbre> as Pythagoras did) can be seen
in the spectral analysis above and heard in the accompanying sound file. At
the points where the ratios of the frequencies of the tones are simpler
(indicated by arrows near the top of the graph), the overtones as observed
in the spectral analysis are more ordered and simple. Most listeners
perceive the tone of the interval at these points to be more "pure" or
"harmonious"."

Another excerpt:

"However, such simple frequency ratios are relevant only to instruments
(including the human voice) that produce
harmonic<http://en.wikipedia.org/wiki/Harmonic_series_%28music%29>
timbres <http://en.wikipedia.org/wiki/Timbre>, because the intervals between
the strongest partials in such timbres follow such simple ratios."

As you can see, there are lots of references to "overtones" in the above
example. This is clearly absurd, because there are no overtones present in a
pure sine wave - by definition.

However, as you might notice, the graph clearly has some kind of
relationship to real world consonance/dissonance. So what does it mean? What
it means is, if you plug your guitar into an amp and crank the distortion up
to 11 -- and you make your guitar strings output sinusoidal tones instead of
complex tones (turn that tone knob all the way down!) -- the graph posted
corresponds roughly to the intervals which would sound most or least
"coherent."

As you might have noticed, this doesn't quite correspond perfectly to real
life - a maj7 chord sounds beautiful on a piano, but with guitar distortion,
it's a total mess. The same applies to a bare maj7 interval as well as tons
of other ones you are no doubt familiar with.

I'll post on the discussion page about this error. You have just witnessed
firsthand the problem with Wikipedia: that the musical experts editing these
pages aren't necessarily experts in acoustics and DSP, and vice versa.

-Mike

On Wed, Jan 13, 2010 at 8:23 PM, Mike Battaglia <battaglia01@...>wrote:

> Furthermore, the overtones present at the end that magically pop up when
> the two sine waves enter into a 2:1 ratio indicate that there was some
> fundamental flaw in the rendering process. Two sine waves added together
> will not create the presence of a third sinusoid, unless they are clipped or
> in some way distorted.
>
> My best guess is that you have happened upon yet another of the millions of
> errors that exist on Wikipedia. Most likely, when this was rendered, the two
> sine waves being added together were full scale sinusoids, so that two of
> them added together would lead to clipping. Since clipping is a particularly
> harsh form of distortion, it would explain the overtone series present at
> the end.
>
> In fact, if you look at the very first tone, you will see that the first,
> third, and fifth overtones are present -- but not the second or fourth. Look
> at the spacing of the overtones to confirm this for yourself. This indicates
> that some kind of symmetrical waveform is present -- perhaps a square wave,
> which would be precisely what results when a sine wave is clipped.
>
> -Mike
>
>
>
> On Wed, Jan 13, 2010 at 8:19 PM, Mike Battaglia <battaglia01@...>wrote:
>
>> Not to burst your bubble, but all of these lines and patterns popping up
>> -- other than the original two sine waves -- are artefacts and byproducts of
>> the analyzing process. They reflect fundamental limitations in the STFT (the
>> mathematical transform behind the "spectrogram"), not laws of nature that
>> happen whenever two sine waves are played together.
>>
>> -Mike
>>
>>
>>
>> On Wed, Jan 13, 2010 at 4:42 PM, Chris Vaisvil <chrisvaisvil@gmail.com>wrote:
>>
>>>
>>>
>>> Have you seen this audio spectrum?
>>>
>>>
>>> http://upload.wikimedia.org/wikipedia/commons/f/f9/Dissonance-a220-a440-notated.jpg
>>>
>>> in article
>>>
>>> http://en.wikipedia.org/wiki/Consonance_and_dissonance
>>>
>>> It is a graph that shows the spectra created when two sine waves are
>>> diverged from the unison to the octave.
>>> Major intervals are notated.
>>>
>>> I'd love to see the same with 3 tones.
>>>
>>> Chris
>>>
>>>
>>
>>
>

🔗Chris Vaisvil <chrisvaisvil@...>

With all due respect it says sinusoidal, not sines. So they are not
necessarily pure.

We had an argument about digital representation of audio data previously
(many times).

To a degree these representation do have to be trusted. Otherwise there are
a lot of
scientific experimets that would be in error because almost everything is
digital now.

I am not saying you are wrong - I'm just saying you may not be right.

Also - one can't assume on wikipedia that the same author wrote everything,
right?
If you are a member you might know this is the case though...

Chris

On Wed, Jan 13, 2010 at 8:34 PM, Mike Battaglia <battaglia01@...>wrote:

>
>
> A last response to really drive the point home, here is an excerpt from the
> article:
>
> "The physical basis for Pythagoras's observation (when using harmonic<http://en.wikipedia.org/wiki/Harmonic_series_%28music%29>
> timbres <http://en.wikipedia.org/wiki/Timbre> as Pythagoras did) can be
> seen in the spectral analysis above and heard in the accompanying sound
> file. At the points where the ratios of the frequencies of the tones are
> simpler (indicated by arrows near the top of the graph), the overtones as
> observed in the spectral analysis are more ordered and simple. Most
> listeners perceive the tone of the interval at these points to be more
> "pure" or "harmonious"."
>
>

🔗Mike Battaglia <battaglia01@...>

> With all due respect it says sinusoidal, not sines. So they are not necessarily pure.
>
> We had an argument about digital representation of audio data previously (many times).
>
> To a degree these representation do have to be trusted. Otherwise there are a lot of
> scientific experimets that would be in error because almost everything is digital now.
>
> I am not saying you are wrong - I'm just saying you may not be right.
>
> Also - one can't assume on wikipedia that the same author wrote everything, right?
> If you are a member you might know this is the case though...
>
> Chris

OK Chris, I just posted not one, not two, but three messages
containing technical explanations, explained in a way that a
non-engineer should be able to understand, of why this is
theoretically absurd. I did this out of good will and because I see
precisely where this diagram you posted went wrong, having studied
this subject for four years now. Given that, let's look at your
response:

> With all due respect it says sinusoidal, not sines. So they are not necessarily pure.

Except that that's what "sinusoidal" means. A sinusoidal signal is a
sine wave, or a cosine wave, or a sine wave of whatever phase. That is
what the word means as far as everyone who has ever even half studied
theory behind this stuff is concerned. If you would like to instead
argue that a "sinusoidal signal" is instead a signal made of any
number of sine waves added together, then here's a news flash for you:
every signal in the entire universe is a "sinusoidal signal."

But most importantly, after I said what I said, why was your initial
response to start searching for absurd flaws with what I said - the
obvious agenda being to "defend" this image you found - and present
them as an argument? There is no argument here: what I am saying is a
mathematical fact, not opinion, and not interpretation. Should you
disagree, you unfortunately are wrong.

I can't comprehend why someone who is trying to understand this stuff
would choose to learn twisted fragments of the basics, and then assume
knowledge enough to argue when someone knowledgeable on the subject is
out of good will trying to explain it. After spending a lot of time on
this list trying to help explain the basics of signal processing and
acoustics to those who have not let yearned - I can't help but feel at
this point that there is no point.

-Mike

🔗Chris Vaisvil <chrisvaisvil@...>

Mike,

I'm an old analytical chemist so I'm used to estimating from graphs on
paper. (Ask me some day about using cut and weight to integrate peaks in GPC
and HPLC.)

Ok - it appear that the tones hold for 2 seconds and then start to migrate.
At 30 seconds they reach the octave. So....
28 seconds to travel 220 Hz. The factor then is 220/28 or... ~7.857 Hz per
second.

I check about 75% of the annotated features. All of them were spot on within
the limits of my ability to estimate (+/- 0.05 sec) - agreement within a Hz
or two. The ones which fell on an X axis time stamp agreed to a high degree
- agreement to less than a Hz.

I check the ratios against the actual frequency ratios I calculated. It is
again spot on.

So... I don't know what the author did. But it really does appear to be
holding water. It *could* all be coincidence.

I implore you, being a wikipedia member, to contact this author and find
out what is actually going on here. For all these numbers to work out is ...
an unusual coincidence. PLEASE I am not defending this graphic. I am mearly
suggesting that the graphic *could* be accurate. If it really is would be
very useful to know how it was generated - even if the author used the wrong
terms - or didn't know what in hell they were doing. Because it looks right.

Chris.

On Wed, Jan 13, 2010 at 9:30 PM, Chris Vaisvil <chrisvaisvil@...>wrote:

>
> I really hate when conversations on this list devolve to religion.
>
> From what I see you are making assumptions that I am uncomfortable with -
> like tying the image to statements in the text. Unless you know that they
> come form the same author it is an assumption that can be wrong.
>
> I have NO vested interest in this image. I didn't make it. It just looks
> really interesting which is why I brought it up here.
>
> All I suggested was that there may be some alternate explanations.
>
> However, I wait for you to contact the author of the image because that is
> what I understood you to say you were going to do.
>
> In the meantime I intend to calculate, roughly obviously, the pitches at
> some of those points. Artifact or not, if it works it could be useful.
>
> Chris
>
>

🔗Mike Battaglia <battaglia01@...>

On Wed, Jan 13, 2010 at 9:30 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
> I really hate when conversations on this list devolve to religion.

Sigh...

> From what I see you are making assumptions that I am uncomfortable with - like tying the image to statements in the text. Unless you know that they come form the same author it is an assumption that can be wrong.

I assumed that the statement "The physical basis for Pythagoras's
observation (when using harmonic timbres as Pythagoras did) can be
seen in the spectral analysis above and heard in the accompanying
sound file." was by the same person who made the picture. But
honestly, even if not, that doesn't change my point, which is that the
picture does not accurately reflect the spectral content of the
divergence of two sinusoids.

In fact, something I missed on the side:

"The sinusoidal-like tones are filtered triangle waves, which have
strong fundamentals, no even overtones (at 2x the fundamental
frequency, 4x, 6x, and so on) and highly attenuated odd overtones (at
3x, 5x, 7x, and so on)."

This is extremely important information to have left out of the
picture. In fact, it is partially responsible for the pattern in which
you see. The other element clearly present in the spectrogram is the
presence of some kind of nonlinearity (again, think guitar
distortion).

> All I suggested was that there may be some alternate explanations.

For what?

> However, I wait for you to contact the author of the image because that is what I understood you to say you were going to do.

I'm not going to contact the author of the image. I wrote an entry on
the discussion page for the article. Perhaps the author will come
forward. If not, I'll edit the article myself -- sometime.

> In the meantime I intend to calculate, roughly obviously, the pitches at some of those points. Artifact or not, if it works it could be useful.

Go for it, but know what you're calculating.

-Mike

🔗Mike Battaglia <battaglia01@...>

> Ok - it appear that the tones hold for 2 seconds and then start to migrate. At 30 seconds they reach the octave. So....
> 28 seconds to travel 220 Hz. The factor then is 220/28 or... ~7.857 Hz per second.
>
> I check about 75% of the annotated features. All of them were spot on within the limits of my ability to estimate (+/- 0.05 sec) - agreement within a Hz or two. The ones which fell on an X axis time stamp agreed to a high degree - agreement to less than a Hz.
>
> I check the ratios against the actual frequency ratios I calculated. It is again spot on.
>
> So... I don't know what the author did. But it really does appear to be holding water. It *could* all be coincidence.
>
> I implore you, being a wikipedia member, to contact this author and find out what is actually going on here. For all these numbers to work out is ... an unusual coincidence. PLEASE I am not defending this graphic. I am mearly suggesting that the graphic *could* be accurate. If it really is would be very useful to know how it was generated - even if the author used the wrong terms - or didn't know what in hell they were doing. Because it looks right.
>
> Chris.

What numbers exactly are working out? That the frequencies listed in
Hz coincide with the labels? I never said they wouldn't.

What is going on here that is not quite explained is the peculiar
pattern that emerges, which is the very thing you have noticed in your
message, in which intervals such as 5:3 are marked by very clear
vertical "fields" on the graph, intervals such as 13:9 marked by
smaller, more complex fields, and intervals corresponding to no clear
JI ratio marked by complete chaos.

That pattern isn't just something that magically happens when two
sinusoids are added together and diverge. In fact, it isn't something
that magically happens when two filtered triangle waves are added
together and diverge either. It is something that happens when two
filtered triangle waves are added together and run through some kind
of nonlinearity - such as a distortion pedal, or saturation on the
synth this was recorded with, or perhaps the signal clipped somewhat
when processed.

It could also have to do with the windowing function used to perform
the STFT - the picture's annotation says that a Blackman window was
used, the frequency response of which can be found at
http://en.wikipedia.org/wiki/File:Window_function_%28blackman%29.png.
Each frequency component on the spectrogram will be "blurred"
vertically in a way that corresponds with the frequency response of
that filter. I don't however see how that would create the pattern.

🔗Carl Lumma <carl@...>

🔗Michael <djtrancendance@...>

>"They reflect fundamental limitations in the STFT (the mathematical
transform behind the "spectrogram"), not laws of nature that happen
whenever two sine waves are played together."

Bleh....so, apparently, it's likely just FFT bin distortion (IE for those waves that are not periodic within the length of the FFT). FFTs are annoying that way...each of, say, the 1024 bins each represents 20 or so frequencies...so of course the phasing and amplitude information is just an average for each set of 20 and thus skews the results.

________________________________
From: Mike Battaglia <battaglia01@...>
To: tuning@yahoogroups.com
Sent: Wed, January 13, 2010 7:19:15 PM
Subject: Re: [tuning] Some new? stuff on wikipedia

Not to burst your bubble, but all of these lines and patterns popping up -- other than the original two sine waves -- are artefacts and byproducts of the analyzing process. They reflect fundamental limitations in the STFT (the mathematical transform behind the "spectrogram"), not laws of nature that happen whenever two sine waves are played together.

-Mike

On Wed, Jan 13, 2010 at 4:42 PM, Chris Vaisvil <chrisvaisvil@ gmail.com> wrote:

>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> >
>
>>
>
>>
>
>Have you seen this audio spectrum?
>
>http://upload. wikimedia. org/wikipedia/ commons/f/ f9/Dissonance- a220-a440- notated.jpg
>
>in article
>
>http://en.wikipedia.org/wiki/Consonance_and_dissonance
>
>It is a graph that shows the spectra created when two sine waves are diverged from the unison to the octave.
>>
>
>Major intervals are notated.
>
>I'd love to see the same with 3 tones.
>
>Chris
>

🔗Michael <djtrancendance@...>

>"Two sine waves added together will not create the presence of a third
sinusoid, unless they are clipped or in some way distorted."

Ugh...exactly, so the smooth tips of the sine wave got flattened against the top into a square-ish wave with extra harmonics. Funny, because this could be used for studies of how/if "over-driving" certain waveforms can lead to definition of pure harmonics.

________________________________
From: Mike Battaglia <battaglia01@...>
To: tuning@yahoogroups.com
Sent: Wed, January 13, 2010 7:23:42 PM
Subject: Re: [tuning] Some new? stuff on wikipedia

Furthermore, the overtones present at the end that magically pop up when the two sine waves enter into a 2:1 ratio indicate that there was some fundamental flaw in the rendering process. Two sine waves added together will not create the presence of a third sinusoid, unless they are clipped or in some way distorted.

My best guess is that you have happened upon yet another of the millions of errors that exist on Wikipedia. Most likely, when this was rendered, the two sine waves being added together were full scale sinusoids, so that two of them added together would lead to clipping. Since clipping is a particularly harsh form of distortion, it would explain the overtone series present at the end.

In fact, if you look at the very first tone, you will see that the first, third, and fifth overtones are present -- but not the second or fourth. Look at the spacing of the overtones to confirm this for yourself. This indicates that some kind of symmetrical waveform is present -- perhaps a square wave, which would be precisely what results when a sine wave is clipped.

-Mike

On Wed, Jan 13, 2010 at 8:19 PM, Mike Battaglia <battaglia01@ gmail.com> wrote:

>
>Not to burst your bubble, but all of these lines and patterns popping up -- other than the original two sine waves -- are artefacts and byproducts of the analyzing process. They reflect fundamental limitations in the STFT (the mathematical transform behind the "spectrogram"), not laws of nature that happen whenever two sine waves are played together.
>
>-Mike
>
>
>
>
>On Wed, Jan 13, 2010 at 4:42 PM, Chris Vaisvil <chrisvaisvil@ gmail.com> wrote:
>
>>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> >>
>>
>>>>
>>
>>>>
>>
>>Have you seen this audio spectrum?
>>
>>http://upload. wikimedia. org/wikipedia/ commons/f/ f9/Dissonance- a220-a440- notated.jpg
>>
>>in article
>>
>>http://en.wikipedia.org/wiki/Consonance_and_dissonance
>>
>>It is a graph that shows the spectra created when two sine waves are diverged from the unison to the octave.
>>>>
>>
>>
>>Major intervals are notated.
>>
>>I'd love to see the same with 3 tones.
>>
>>Chris
>>
>

🔗Carl Lumma <carl@...>

🔗Michael <djtrancendance@...>

Chris>"or didn't know what in hell they were doing. Because it looks right."
I actually agree with both of you.

Mike, I agree on your evidence that is looks to author(s) mistakenly over-drove two sine waves (IE added them together and forgot that they can both be pointing the same direction and add up to greater than 32768 and distort forming extra never-in-sine-wave harmonics.

Chris, I agree that regardless of the fact whoever wrote this paper apparently not aware of his bad DSP testing technique, the results seem quite useful.

Personally I recently applied maximization (which introduces some distortion) to a 7-tone-per-octave chord (played with sine waves( in order to reduce beating by flattening the resulting waveform a bit.
And, guess what; it also produces rational ratios that were within about 2-6 cents of the original tones of the notes.

And when I tried to make a scale with those new frequencies (from scratch with pure sine waves and no maximization) guess what: the result beated significantly less and sounded better: it retained the advantages of the maximization without the distortion!

I think this coincidence points strongly to a tie between the supposedly somewhat dis-jointed theories of periodicity and roughness.

-Michael

🔗Michael <djtrancendance@...>

Correction to one part: I just read the short blurb about the use of a triangle wave for the test/graph. So apparently the author didn't make a "mistake" in the experiment...just in the fact he didn't put that information in a spot where it's clear to the listener...because, of course, the fact the wave used is a triangle and not a sine wave is extremely important (sine has no overtones while a triangle is loaded with odd overtones).

________________________________
From: Michael <djtrancendance@...>
To: tuning@yahoogroups.com
Sent: Wed, January 13, 2010 10:03:08 PM
Subject: [tuning] A possible Conclusion: Some new? stuff on wikipedia

Chris>"or didn't know what in hell they were doing. Because it looks right."
I actually agree with both of you.

Chris, I agree that regardless of the fact whoever wrote this paper apparently not aware of his bad DSP testing technique, the results seem quite useful.

I think this coincidence points strongly to a tie between the supposedly somewhat dis-jointed theories of periodicity and roughness.

-Michael

🔗Mike Battaglia <battaglia01@...>

There's more to it than that. Two triangle waves being summed together
wouldn't create the interference pattern that you see (where would we be
seeing subtones from?) There's some kind of nonlinearity at work.

In fact, to put the matter to rest, look at the discussion page:
http://en.wikipedia.org/wiki/Talk:Consonance_and_dissonance

I wasn't the first to notice this, and someone did a comparison of two
actual triangle waves diverging for comparison:
http://en.wikipedia.org/wiki/File:Dissonance_A220-A440_bandlimited_sawtooths_dBV2.png

As you can see, the result is quite different.

-Mike

On Wed, Jan 13, 2010 at 10:46 PM, Carl Lumma <carl@...> wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
>
> > I implore you, being a wikipedia member, to contact this author
> > and find out what is actually going on here. For all these numbers
> > to work out is ... an unusual coincidence. PLEASE I am not
> > defending this graphic. I am mearly suggesting that the graphic
> > *could* be accurate.
>
> It is accurate, for two complex tones. It's mislabeled. This
> is the real 'problem of wikipedia' - that people don't read it
> critically. Of course printed texts have just as many errors,
> if not more. People just don't get to discuss them in the open
> with others like this.
>
> -Carl
>
>
>

🔗Mike Battaglia <battaglia01@...>

An alternate explanation, from the site I linked:

"I've restored the deleted description on
Commons:File:Dissonance-a220-a440-notated.jpg The captions and quality
of this image are pretty poor, and I think the subject matter is
rather misleading. If there are overtones, then these are not (quite)
sinusoids. The fact that you've got lines going up and down at the
same time means that there is significant aliasing, too. This is
because the tones were generated in Cool Edit, which does a "naive"
waveform generation instead of a bandlimited waveform. This seems to
be the reason for the pronounced effect at consonances, not the
consonance itself. I'm not sure if these bands are actually even due
to consonance, or if they're due to the waveform frequency being a
submultiple of the sampling frequency at those points."

He thinks it's due to aliasing. I disagree, and think it has to do
more with harmonic distortion. As far as I can tell, aliasing (being a
type of nonlinearity in and of itself) wouldn't cause such a neat
pattern in which we see maximal coherence at low integer JI intervals
and maximal incoherence at spaces between them. Rather it would be
most coherent when the upper sinusoid happened to divide perfectly
into the Nyquist frequency.

It's possible I suppose that aliasing could somehow produce that
pattern, but I'm not seeing it. It looks like the hallmark of simple
even-order harmonic distortion to me.

Alternatively, the explanation might be much more simple:

"Then you've got the lossy nature of both the ogg vorbis and the
.jpg... — Omegatron (talk) 02:34, 31 July 2009 (UTC)'

The image's caption says that it was made in Cool Edit Pro. Perhaps he
ran the spectrogram on the file after it was compressed, meaning that
perhaps the lossy OGG compression algorithm added a bit of THD or EOD
on its own.

-Mike

On Wed, Jan 13, 2010 at 11:11 PM, Mike Battaglia <battaglia01@...> wrote:
>
> Sorry, those are actually diverging sawtooths, but the concept remains the same nonetheless. THAT is what the spectrogram would look like of two diverging triangle waves had everything been done properly, except with no even-order tones and a steeper rolloff.
>
> -Mike
>
>
> On Wed, Jan 13, 2010 at 11:10 PM, Mike Battaglia <battaglia01@...> wrote:
>>
>> There's more to it than that. Two triangle waves being summed together wouldn't create the interference pattern that you see (where would we be seeing subtones from?) There's some kind of nonlinearity at work.
>>
>> In fact, to put the matter to rest, look at the discussion page: http://en.wikipedia.org/wiki/Talk:Consonance_and_dissonance
>>
>> I wasn't the first to notice this, and someone did a comparison of two actual triangle waves diverging for comparison: http://en.wikipedia.org/wiki/File:Dissonance_A220-A440_bandlimited_sawtooths_dBV2.png
>>
>> As you can see, the result is quite different.
>>
>> -Mike
>>
>>
>> On Wed, Jan 13, 2010 at 10:46 PM, Carl Lumma <carl@...> wrote:
>>>
>>>
>>>
>>> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>>>
>>> > I implore you, being a wikipedia member, to contact this author
>>> > and find out what is actually going on here. For all these numbers
>>> > to work out is ... an unusual coincidence. PLEASE I am not
>>> > defending this graphic. I am mearly suggesting that the graphic
>>> > *could* be accurate.
>>>
>>> It is accurate, for two complex tones. It's mislabeled. This
>>> is the real 'problem of wikipedia' - that people don't read it
>>> critically. Of course printed texts have just as many errors,
>>> if not more. People just don't get to discuss them in the open
>>> with others like this.
>>>
>>> -Carl
>>>
>>>
>

🔗Mike Battaglia <battaglia01@...>

Sorry, those are actually diverging sawtooths, but the concept remains the
same nonetheless. THAT is what the spectrogram would look like of two
diverging triangle waves had everything been done properly, except with no
even-order tones and a steeper rolloff.

-Mike

On Wed, Jan 13, 2010 at 11:10 PM, Mike Battaglia <battaglia01@...>wrote:

> There's more to it than that. Two triangle waves being summed together
> wouldn't create the interference pattern that you see (where would we be
> seeing subtones from?) There's some kind of nonlinearity at work.
>
> In fact, to put the matter to rest, look at the discussion page:
> http://en.wikipedia.org/wiki/Talk:Consonance_and_dissonance
>
> I wasn't the first to notice this, and someone did a comparison of two
> actual triangle waves diverging for comparison:
> http://en.wikipedia.org/wiki/File:Dissonance_A220-A440_bandlimited_sawtooths_dBV2.png
>
> As you can see, the result is quite different.
>
> -Mike
>
>
>
> On Wed, Jan 13, 2010 at 10:46 PM, Carl Lumma <carl@...> wrote:
>
>>
>>
>> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
>> <chrisvaisvil@...> wrote:
>>
>> > I implore you, being a wikipedia member, to contact this author
>> > and find out what is actually going on here. For all these numbers
>> > to work out is ... an unusual coincidence. PLEASE I am not
>> > defending this graphic. I am mearly suggesting that the graphic
>> > *could* be accurate.
>>
>> It is accurate, for two complex tones. It's mislabeled. This
>> is the real 'problem of wikipedia' - that people don't read it
>> critically. Of course printed texts have just as many errors,
>> if not more. People just don't get to discuss them in the open
>> with others like this.
>>
>> -Carl
>>
>>
>>
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Mike Battaglia <battaglia01@...>

Chris,

The fact that that graphic coincides with small-integer JI ratios is why
people at one point thought nonlinearities were responsible for our
perception of consonance.

This concept that harmonic distortion (perhaps occuring in the ear) are at
the root of consonance has been disproven for years and years now. See
http://en.wikipedia.org/wiki/Missing_fundamental#Explanation for more
information. Licklider's experiment pretty much singlehandedly shot down the
notion.

-Mike

On Wed, Jan 13, 2010 at 11:50 PM, Chris Vaisvil <chrisvaisvil@...>wrote:

>
>
> Mike B.
>
> With all due respect - I don't care *how* the graphic was arrived at.
>
> I wish you'd consider the fact that the data itself is consistent.
> I don't care if it was derived from sampling two cars crashing on I-80.
> And what I mean consistent is that the vertical artifacts do occur where JI
> ratios occur between the two source frequencies.
>
> If it takes a certain FFT size and certain graph parameters and certain
> program and certain waveforms - that is all OK.
> As long as the technique is consistent in usage and reliably gives the
> same result for the same input it *could* be useful.
>
>
> Chris
>
>
> On Wed, Jan 13, 2010 at 11:19 PM, Mike Battaglia <battaglia01@...>wrote:
>
>> An alternate explanation, from the site I linked:
>>
>> "I've restored the deleted description on
>>
>
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

Mike B.

I'm not sure how that relates to the graph we are discussing per se.

But it did give me this thought:

Does this imply then - since the Sethares "custom timbre for a tuning
works", that the "autocorrelation" occurs for non-octave based harmonic
spectra as well?

I am understanding your position as "we don't know the why of dissonance /
consonance. " Am I correct?

Its late for me - thanks for the link - see you all tomorrow.

Chris

On Wed, Jan 13, 2010 at 11:58 PM, Mike Battaglia <battaglia01@...>wrote:

>
>
> Chris,
>
> The fact that that graphic coincides with small-integer JI ratios is why
> people at one point thought nonlinearities were responsible for our
> perception of consonance.
>
> This concept that harmonic distortion (perhaps occuring in the ear) are at
> the root of consonance has been disproven for years and years now. See
> http://en.wikipedia.org/wiki/Missing_fundamental#Explanation for more
> information. Licklider's experiment pretty much singlehandedly shot down the
> notion.
>
> -Mike
>
>
>
> On Wed, Jan 13, 2010 at 11:50 PM, Chris Vaisvil <chrisvaisvil@...>wrote:
>
>>
>>
>> Mike B.
>>
>> With all due respect - I don't care *how* the graphic was arrived at.
>>
>> I wish you'd consider the fact that the data itself is consistent.
>> I don't care if it was derived from sampling two cars crashing on I-80.
>> And what I mean consistent is that the vertical artifacts do occur where
>> JI ratios occur between the two source frequencies.
>>
>> If it takes a certain FFT size and certain graph parameters and certain
>> program and certain waveforms - that is all OK.
>> As long as the technique is consistent in usage and reliably gives the
>> same result for the same input it *could* be useful.
>>
>>
>> Chris
>>
>>
>> On Wed, Jan 13, 2010 at 11:19 PM, Mike Battaglia <battaglia01@...>wrote:
>>
>>> An alternate explanation, from the site I linked:
>>>
>>> "I've restored the deleted description on
>>>
>>
>>
>
>

🔗Mike Battaglia <battaglia01@...>

It relates to the graph we are discussing because if there were something
happening in the ear that actually took the incoming frequencies and
distorted them like that and that was all there was to consonance, that
graph would basically be THE picture of consonance and the entire problem
would be solved with there being nothing else to it.

This is not the case, because even if you take a signal containing two
sinusoids and add some THD to the signal -- there is still a far more
complicated and much deeper level of the brain which makes sense of it and
the resulting spectra. This happens whether your signal consists of two
sinusoids, two sinusoids that have been run through a distortion pedal, or
clipped, or had something special done to them in the ear, or what not. It
is on this level that something much closer to actual "consonance" resides.

For the record, there are, in certain circumstances, nonlinearities that can
occur in the ear -- but as I said above, all they do is simply add more
sinusoids to the equation. It still takes your brain to make sense of
everything, split the signal into multiple sources, and perform a scene
analysis on everything coming in.

As for this:

> Does this imply then - since the Sethares "custom timbre for a tuning
works", that the "autocorrelation" occurs for non-octave based harmonic
spectra as well?

> I am understanding your position as "we don't know the why of dissonance /
consonance. " Am I correct?

Right. We aren't quite sure exactly HOW this periodicity-finder in the brain
works, but the most modern theory AFAIK is that some kind of autocorrelation
is being done within the auditory nerve (so that it originates in the ear).

From a signal processing standpoint, I'm not sure exactly HOW this would
answer the entire question, since autocorrelation is a linear process (read:
NOT nonlinear, produces NO distortion) and so would never produce a "phantom
fundamental" or anything like that.

I attribute this to a lack of understanding on my part, however, as I am not
well read on the most recent research into autocorrelation. Perhaps the use
of the term "autocorrelation" is just a metaphor. But whatever this
mysterious autocorrelation mechanism is, it can indeed detect
pseudo-harmonic relationships as well as perfect harmonic relationships. In
the literature I believe that they are referred to as
"pseudo-periodicities," and I was reading an interesting study a while ago
about how certain types of periodic noise can fool the ear into thinking the
fundamental is an octave higher or lower (I don't remember which) than it
really is, since this mechanism is apparently easily fooled into thinking
that the pseudo-periodicities are actually real periodicities.

-Mike

On Thu, Jan 14, 2010 at 12:16 AM, Chris Vaisvil <chrisvaisvil@...>wrote:

>
>
> Mike B.
>
> I'm not sure how that relates to the graph we are discussing per se.
>
> But it did give me this thought:
>
> Does this imply then - since the Sethares "custom timbre for a tuning
> works", that the "autocorrelation" occurs for non-octave based harmonic
> spectra as well?
>
> I am understanding your position as "we don't know the why of dissonance /
> consonance. " Am I correct?
>
> Its late for me - thanks for the link - see you all tomorrow.
>
> Chris
>
>
> On Wed, Jan 13, 2010 at 11:58 PM, Mike Battaglia <battaglia01@gmail.com>wrote:
>
>>
>>
>> Chris,
>>
>> The fact that that graphic coincides with small-integer JI ratios is why
>> people at one point thought nonlinearities were responsible for our
>> perception of consonance.
>>
>> This concept that harmonic distortion (perhaps occuring in the ear) are at
>> the root of consonance has been disproven for years and years now. See
>> http://en.wikipedia.org/wiki/Missing_fundamental#Explanation for more
>> information. Licklider's experiment pretty much singlehandedly shot down the
>> notion.
>>
>> -Mike
>>
>>
>>
>> On Wed, Jan 13, 2010 at 11:50 PM, Chris Vaisvil <chrisvaisvil@...>wrote:
>>
>>>
>>>
>>> Mike B.
>>>
>>> With all due respect - I don't care *how* the graphic was arrived at.
>>>
>>> I wish you'd consider the fact that the data itself is consistent.
>>> I don't care if it was derived from sampling two cars crashing on I-80.
>>> And what I mean consistent is that the vertical artifacts do occur where
>>> JI ratios occur between the two source frequencies.
>>>
>>> If it takes a certain FFT size and certain graph parameters and certain
>>> program and certain waveforms - that is all OK.
>>> As long as the technique is consistent in usage and reliably gives the
>>> same result for the same input it *could* be useful.
>>>
>>>
>>> Chris
>>>
>>>
>>> On Wed, Jan 13, 2010 at 11:19 PM, Mike Battaglia <battaglia01@...>wrote:
>>>
>>>> An alternate explanation, from the site I linked:
>>>>
>>>> "I've restored the deleted description on
>>>>
>>>
>>>
>>
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Mike B.

You seem to be saying as far as I can tell that if a tool doesn't explain
everything then it explains nothing. I suggest that this is not the case.
Your explanation suggests that the phenomenon of hearing and discriminating
/ analyzing pitch is a very complex process. By virtue of this argument it
stands to reason that one tool or view or series of data will NOT
necessarily suffice to fully represent the process of human hearing and
perception of pitch interaction.

What *seems* to be apparent in the graph is that is a simple, easy to
understand representation of the ratio between two pitches. I am at this
point very curious as to the identity of some of the other "resonance
points" that seem apparent in the graph. If these are random unknown points
then, yes, the technique is probably worthless - no cherry picking allowed.
However if they do indicate even more known relationships - I'm even more
curious.

While the relationship between two pitches is certainly no mystery in our
community, if this could be extended to 3 or more pitches it could have
value. It could show places to look for triads or represent some knowledge
that has been ascertained to this point empirically. To my knowledge no one
has been able to represent a series of triads (or more pitches) in an easy
to understand graphical representation of the interaction of the constituent
pitches. This seems to me to be a goal worth a bit of effort to realize.

Chris

On Thu, Jan 14, 2010 at 12:29 AM, Mike Battaglia <battaglia01@...>wrote:

>
>
> It relates to the graph we are discussing because if there were something
> happening in the ear that actually took the incoming frequencies and
> distorted them like that and that was all there was to consonance, that
> graph would basically be THE picture of consonance and the entire problem
> would be solved with there being nothing else to it.
>
> This is not the case, because even if you take a signal containing two
> sinusoids and add some THD to the signal -- there is still a far more
> complicated and much deeper level of the brain which makes sense of it and
> the resulting spectra. This happens whether your signal consists of two
> sinusoids, two sinusoids that have been run through a distortion pedal, or
> clipped, or had something special done to them in the ear, or what not. It
> is on this level that something much closer to actual "consonance" resides.
>
>

🔗Michael <djtrancendance@...>

Those aren't triangle waves, those are sawtooth waves: as an analog synth programmer I know; big difference!

________________________________
From: Mike Battaglia <battaglia01@...>
To: tuning@yahoogroups.com
Sent: Wed, January 13, 2010 10:10:22 PM
Subject: Re: [tuning] Re: Some new? stuff on wikipedia

There's more to it than that. Two triangle waves being summed together wouldn't create the interference pattern that you see (where would we be seeing subtones from?) There's some kind of nonlinearity at work.

In fact, to put the matter to rest, look at the discussion page: http://en.wikipedia.org/wiki/Talk:Consonance_and_dissonance

I wasn't the first to notice this, and someone did a comparison of two actual triangle waves diverging for comparison: http://en.wikipedia.org/wiki/File:Dissonance_A220-A440_bandlimited_sawtooths_dBV2.png

As you can see, the result is quite different.

-Mike

On Wed, Jan 13, 2010 at 10:46 PM, Carl Lumma <carl@lumma.org> wrote:

>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> >
>
>>
>
>>
>
>--- In tuning@yahoogroups. com, Chris Vaisvil <chrisvaisvil@ ...> wrote:
>
>
>>> I implore you, being a wikipedia member, to contact this author
>>> and find out what is actually going on here. For all these numbers
>>> to work out is ... an unusual coincidence. PLEASE I am not
>>> defending this graphic. I am mearly suggesting that the graphic
>>> *could* be accurate.
>
>
>It is accurate, for two complex tones. It's mislabeled. This
>>is the real 'problem of wikipedia' - that people don't read it
>>critically. Of course printed texts have just as many errors,
>>if not more. People just don't get to discuss them in the open
>>with others like this.
>
>>-Carl
>
>

🔗Michael <djtrancendance@...>

>"Perhaps the use of the term "autocorrelation" is just a metaphor. But
whatever this mysterious autocorrelation mechanism is, it can indeed
detect pseudo-harmonic relationships as well as perfect harmonic
relationships."

If that's true, doesn't that a small amount of tempering will not influence the sense of tonality (implying that "near perfect" ratios may essentially be equally good to the brain as perfect ones)?

________________________________
From: Mike Battaglia <battaglia01@...>
To: tuning@yahoogroups.com
Sent: Wed, January 13, 2010 11:29:44 PM
Subject: Re: [tuning] Re: Some new? stuff on wikipedia

It relates to the graph we are discussing because if there were something happening in the ear that actually took the incoming frequencies and distorted them like that and that was all there was to consonance, that graph would basically be THE picture of consonance and the entire problem would be solved with there being nothing else to it.

This is not the case, because even if you take a signal containing two sinusoids and add some THD to the signal -- there is still a far more complicated and much deeper level of the brain which makes sense of it and the resulting spectra. This happens whether your signal consists of two sinusoids, two sinusoids that have been run through a distortion pedal, or clipped, or had something special done to them in the ear, or what not. It is on this level that something much closer to actual "consonance" resides.

For the record, there are, in certain circumstances, nonlinearities that can occur in the ear -- but as I said above, all they do is simply add more sinusoids to the equation. It still takes your brain to make sense of everything, split the signal into multiple sources, and perform a scene analysis on everything coming in.

As for this:

> Does this imply then - since the Sethares "custom timbre for a
tuning works", that the "autocorrelation" occurs for non-octave based
harmonic spectra as well?

> I am understanding your position as "we don't know the why of dissonance / consonance. " Am I correct?

Right. We aren't quite sure exactly HOW this periodicity- finder in the brain works, but the most modern theory AFAIK is that some kind of autocorrelation is being done within the auditory nerve (so that it originates in the ear).

From a signal processing standpoint, I'm not sure exactly HOW this would answer the entire question, since autocorrelation is a linear process (read: NOT nonlinear, produces NO distortion) and so would never produce a "phantom fundamental" or anything like that.

I attribute this to a lack of understanding on my part, however, as I am not well read on the most recent research into autocorrelation. Perhaps the use of the term "autocorrelation" is just a metaphor. But whatever this mysterious autocorrelation mechanism is, it can indeed detect pseudo-harmonic relationships as well as perfect harmonic relationships. In the literature I believe that they are referred to as "pseudo-periodicitie s," and I was reading an interesting study a while ago about how certain types of periodic noise can fool the ear into thinking the fundamental is an octave higher or lower (I don't remember which) than it really is, since this mechanism is apparently easily fooled into thinking that the pseudo-periodicitie s are actually real periodicities.

-Mike

On Thu, Jan 14, 2010 at 12:16 AM, Chris Vaisvil <chrisvaisvil@ gmail.com> wrote:

>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> >
>
>>
>
>>
>
>Mike B.
>
>I'm not sure how that relates to the graph we are discussing per se.
>
>But it did give me this thought:
>
>Does this imply then - since the Sethares "custom timbre for a tuning works", that the "autocorrelation" occurs for non-octave based harmonic spectra as well?
>
>I am understanding your position as "we don't know the why of dissonance / consonance. " Am I correct?
>
>Its late for me - thanks for the link - see you all tomorrow.
>
>Chris
>>
>
>
>>On Wed, Jan 13, 2010 at 11:58 PM, Mike Battaglia <battaglia01@ gmail.com> wrote:
>
>>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> >>
>>
>>>>
>>
>>>>
>>
>>Chris,
>>
>>The fact that that graphic coincides with small-integer JI ratios is
>>why people at one point thought nonlinearities were responsible for our
>>perception of consonance.
>>
>>This concept that harmonic distortion (perhaps occuring in the ear) are at the root of consonance has been disproven for years and years now. See http://en.wikipedia.org/wiki/Missing_fundamental#Explanation for more information. Licklider's experiment pretty much singlehandedly shot down the notion.
>>
>>-Mike
>>
>>
>>
>>
>>On Wed, Jan 13, 2010 at 11:50 PM, Chris Vaisvil <chrisvaisvil@ gmail.com> wrote:
>>
>>>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> >>>
>>>
>>>>>>
>>>
>>>>>>
>>>
>>>Mike B.
>>>
>>>With all due respect - I don't care *how* the graphic was arrived at.
>>>
>>>I wish you'd consider the fact that the data itself is consistent.
>>>I don't care if it was derived from sampling two cars crashing on I-80.
>>>>>>
>>>
>>>
>>>
>>>
>>>And what I mean consistent is that the vertical artifacts do occur where JI ratios occur between the two source frequencies.
>>>
>>>If it takes a certain FFT size and certain graph parameters and certain program and certain waveforms - that is all OK.
>>>>>>
>>>
>>>
>>>
>>>
>>>As long as the technique is consistent in usage and reliably gives the same result for the same input it *could* be useful.
>>>
>>>
>>>Chris
>>>
>>>
>>>
>>>On Wed, Jan 13, 2010 at 11:19 PM, Mike Battaglia <battaglia01@ gmail.com> wrote:
>>>
>>>An alternate explanation, from the site I linked:
>>>>
>>>>>>>>"I've restored the deleted description on
>>>>
>>>
>>
>

🔗Michael <djtrancendance@...>

Chris> "Hi Mike B.
You seem to be saying as far as I can tell that if a
tool doesn't explain everything then it explains nothing. I suggest
that this is not the case. Your explanation suggests that the
phenomenon of hearing and discriminating / analyzing pitch is a very
complex process. By virtue of this argument it stands to reason that
one tool or view or series of data will NOT necessarily suffice to
fully represent the process of human hearing and perception of pitch
interaction."

Here is something dramatic to think about; what if
A) The human ear can calculate tonality essentially just as well using not just pure intervals but, say, 1-8 cents of so off-pure intervals (and harmonic distortion simply makes fundamental frequency location more obvious, but neither it nor pure intervals are the only requirement for having the brain locate it).
B) The distortion itself introduces harmonics that are loud/easily-heard during short periods where other ones are near silent and caught in phase-cancellation due to beating (IE the type that takes place with 2 sine/triangle/sawtooth/etc. waves a 5/4 or so or less ratio apart). Thus while distortion may not create the missing fundamental, it may clarify it.
C) What if the brain is perfectly OK with more than one missing fundamental for each chord...so long as those fundamentals have a low-integer-fraction relationship to each other?
D) What if the brain is perfectly OK with more than one missing fundamental for each chord...so long as those fundamentals are symmetrical to each other (IE 3 where the third splits the other two in the middle where one half of the area is 1.618 times the size of the other, 2.414 times the size of the other (Silver Ratio), etc.

IMVHO, the brain may simply look for patterns to help it categorize...and the harmonic series (and certainly not only 100% perfect ratios from the series) are almost certainly not the only cue the brain can use to organize sounds.

Mike B, I'd be interested to see you explain this (if it turns out exact periodicity is by and far the only way the brain can categorize tones):

The following scale has a boat-load of by-harmonic-series perfect triads
1/1
7/6
5/4
4/3
3/2
5/3
7/4
2/1

This next one is a result of a distorted waveform of a scale with far less perfect triads possible

1
1.12626262
1.25252525
1.37878787
1.50505050
1.6717171
1.83838383
2

Now...try playing each with sine waves and creating a bunch of chords (esp. ones with 5+ notes per octave).
Which scale sounds better to you?

🔗Mike Battaglia <battaglia01@...>

Chris,

> You seem to be saying as far as I can tell that if a tool doesn't explain everything then it explains nothing. I suggest that this is not the case. Your explanation suggests that the phenomenon of hearing and discriminating / analyzing pitch is a very complex process. By virtue of this argument it stands to reason that one tool or view or series of data will NOT necessarily suffice to fully represent the process of human hearing and perception of pitch interaction.

That isn't what I'm saying at all. I've sent 11 messages in this
thread of technical explanations that explain exactly what the tool
is, what it's doing, what it explains and what it doesn't. At this
point I'm giving up.

My final word on the subject: that graph corresponds roughly to what
will result if you play two guitar strings at once with the tone knob
all the way down and the distortion cranked to 11. A perfect fifth
will sound resonant, a just maj3 will sound resonant, and an equal
tempered tritone will not.

> What *seems* to be apparent in the graph is that is a simple, easy to understand representation of the ratio between two pitches. I am at this point very curious as to the identity of some of the other "resonance points" that seem apparent in the graph. If these are random unknown points then, yes, the technique is probably worthless - no cherry picking allowed. However if they do indicate even more known relationships - I'm even more curious.

They indicate what spectra will result when two triangle waves are run
through a harmonic distortion unit.

> While the relationship between two pitches is certainly no mystery in our community, if this could be extended to 3 or more pitches it could have value. It could show places to look for triads or represent some knowledge that has been ascertained to this point empirically. To my knowledge no one has been able to represent a series of triads (or more pitches) in an easy to understand graphical representation of the interaction of the constituent pitches. This seems to me to be a goal worth a bit of effort to realize.
>
> Chris

Expect to see that an equal-tempered minor triad would look horrific.

-Mike

🔗christopherv <chrisvaisvil@...>

Mike,

For what it is worth, I can't seem to reproduce the original graphic with cool edit or audition.

Actually I'm getting the theoretical collection of harmonics w/o interference as discussed some year ago. (i.e. just a bunch of lines representing the harmonic series as appropriate and no interference. )

I have no idea how the original graphic was made.

Chris

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Not to burst your bubble, but all of these lines and patterns popping up --
> other than the original two sine waves -- are artefacts and byproducts of
> the analyzing process. They reflect fundamental limitations in the STFT (the
> mathematical transform behind the "spectrogram"), not laws of nature that
> happen whenever two sine waves are played together.
>
> -Mike
>
>
> On Wed, Jan 13, 2010 at 4:42 PM, Chris Vaisvil <chrisvaisvil@...>wrote:
>
> >
> >
> > Have you seen this audio spectrum?
> >
> >
> > http://upload.wikimedia.org/wikipedia/commons/f/f9/Dissonance-a220-a440-notated.jpg
> >
> > in article
> >
> > http://en.wikipedia.org/wiki/Consonance_and_dissonance
> >
> > It is a graph that shows the spectra created when two sine waves are
> > diverged from the unison to the octave.
> > Major intervals are notated.
> >
> > I'd love to see the same with 3 tones.
> >
> > Chris
> >
> >
>