I'm saying nothing, but Daniel Levitin's excellent book This Is Your Brain On Music is well worth a read.

Nice. Another book for my neurology list. I'm getting really into nuerology of late. It's my latest science fad. It's overtaken practically everything else in my science reading.

The Levitin is excellent, both on the music and the science. Oliver Sachs' Musicophilia: Tales of Music and the Brain is quite good - more current, but I think the Levitin gets to grips with the science in a way the Sachs mostly skips through.

I really want to like Levitin's book. And I must admit I haven't read much of it. But from what I have read, there are problems on the music side of things. There is some sloppy terminology, but more important are mistakes of real substance. And these are curious: he will make a claim, and it will be perfectly good one. And he chooses an excellent example to illustrate it. But then he misuses the example, thus missing the point by a mile. It's very weird. At first I thought he had a bad editor. But the mistakes are deeper than those that would be within the purview of an editor.

As I said, I haven't read much of it. But what I did read left me doubting his understanding (and maybe knowledge) of music. Too, it causes me to wonder how accurate he is on the science (about which I can't judge, since I don't really know much about it).

Hmm. I found the musical side of the book quite acceptable. But then again, I am a Jazzer at heart.


EDIT: Just to add, the only area where I found the book fall short was on that of the big why-question. "Why music?" He does well explaining the process of perception, and the relationship between music and the synapses, music as cultural artefact, and music and emotion response, but when it comes to that big question he has about two paragraphs on evolutions and sexual selection, and leaves it at that. That, I must admit, did seem a bit lightweight.

Oh drat, and I said I was go to say nothing.

So I didn't get any inspirations on how to wrap my head around how all the harmonics of the notes of a chord were interacting so I thought maybe going the simplify route might shed some light on this. I used Animoog on my Ipad to create a 1 harmonic instrument, i.e. just the root of each note. Major and minor chords still had their familiar feel even without all the higher harmonic interactions. (Animoog may still add a bit of "English" to the notes because it still sounded pretty good for just a few sin waves.) And then I just played the root and the major or minor third and the feel was still there.

Animoog sort of shows the waveform that you're hearing and I noticed that one of the 2 note chords seemed to sync up in the display better than the other 2 note chord so I decided that it might be interesting to plot adding a major 3rd, or minor 3rd, 1 harmonic note to a 1 harmonic root note. The following graphs shows what that looks like. Each image has 3 graphs. The top graph is the root with a major third added. The middle graph is root with minor third. The bottom graph is root with 5th. Each image graphs a longer period of time.









To my eyes, it seemed pretty obvious that the major 3rd and 5th displayed a "pattern" of a type that wasn't nearly as prevalent with the minor 3rd. I wondered what would happen if the 3rd and 5th were a octave higher than the root. Here's what I got:





Same sort of patterns!

So this got me wondering what was going on, it looked to me like the major 3rd and 5th were creating a beat frequency that somehow fit better with the root. When you add two sine waves of different frequencies you get a beat frequency that is 2 times the difference of the frequencies of the sin waves. You can learn all about it here

So somehow it seems that the interference patterns of the major 3rd and 5th create a more regular pattern.

Looking at the frequencies involved:

A - 440
C - 523.25
C# - 554.37
E - 659.26

We can calculate the beat frequency of adding the minor 3rd, major 3rd, and 5th to the root.

A + C gives beat frequency of (523.25- 440) = 83.25
A+ C# gives a beat frequency of (554.37-440) = 114.37
A+ E gives a beat frequency of (659.26 -440) = 219.26

And then I noticed that the E, the 5th, which seems to sync real well with the root, interacts with the root to create a beat frequency which is very close to 1/2 the frequency of the root. And the major third interacts with the root to create a beat frequency that is near 1/4 the root. But the minor 3rd interacts with the root to create a beat frequency that doesn't seem as related.

Which leads to my theory of why a major chord sounds "happy" and a minor chord sounds "sad" (here comes the woo ..) The beat frequency created when a Major 3rd fundamental is added to the Root fundamental is more harmonious to the fundamental than the beat frequency created when a minor 3rd fundamental is added to the Root fundamental (which isn't really all that harmonious). And the mind maps the more harmonious beat frequency to happy and the dissonance of the minor 3rd beat frequency to sad. And these mappings occur because we get a similar feeling when we are happy and when we hear harmonious sounds and a similar feeling when we are sad and hear dissonant sounds. - And the same sort of things happens with all the harmonics we get with real musical instruments. In short, the basic phenomenon afoot is created by the actual ratio of the frequency of notes and how the beat frequencies created by their addition relate to the root of the chord.

(Edit 1 to fix beat frequencies)

John Platko wrote:So I didn't get any inspirations on how to wrap my head around how all the harmonics of the notes of a chord were interacting so I thought maybe going the simplify route might shed some light on this. I used Animoog on my Ipad to create a 1 harmonic instrument, i.e. just the root of each note. Major and minor chords still had their familiar feel even without all the higher harmonic interactions. (Animoog may still add a bit of "English" to the notes because it still sounded pretty good for just a few sin waves.) And then I just played the root and the major or minor third and the feel was still there.

[... omitted for brevity ...]

Which leads to my theory of why a major chord sounds "happy" and a minor chord sounds "sad" (here comes the woo ..) The beat frequency created when a Major 3rd fundamental is added to the Root fundamental is more harmonious to the fundamental than the beat frequency created when a minor 3rd fundamental is added to the Root fundamental (which isn't really all that harmonious). And the mind maps the more harmonious beat frequency to happy and the dissonance of the minor 3rd beat frequency to sad. And these mappings occur because we get a similar feeling when we are happy and when we hear harmonious sounds and a similar feeling when we are sad and hear dissonant sounds. - And the same sort of things happens with all the harmonics we get with real musical instruments. In short, the basic phenomenon afoot is created by the actual ratio of the frequency of notes and how the beat frequencies created by their addition relate to the root of the chord.

(Edit 1 to fix beat frequencies)

You might possibly gain more from looking at beat frequencies between overtones; for sine waves, the "critical bandwidth" for perceiving dissonance is close to a minor third (although wider in lower registers, which is why chords played entirely in the bassiest register of any instrument tend to sound like shit), if you add up the dissonances between successive overtones, the regular major chord is a pretty obvious local minimum as far as triads go. (The only better local minimums I can think of either are not real triads due to octaves, or are inversions of the major triad.)

Simple sine waves are not very good things to work with when reasoning about harmony, since so much of the harmony is a result from the interactions of overtones. A tritone played by pure sinewaves is not particularly dissonant (except in the lower registers - but in those, even a fifth is dissonant)

Zwaarddijk wrote:

John Platko wrote:So I didn't get any inspirations on how to wrap my head around how all the harmonics of the notes of a chord were interacting so I thought maybe going the simplify route might shed some light on this. I used Animoog on my Ipad to create a 1 harmonic instrument, i.e. just the root of each note. Major and minor chords still had their familiar feel even without all the higher harmonic interactions. (Animoog may still add a bit of "English" to the notes because it still sounded pretty good for just a few sin waves.) And then I just played the root and the major or minor third and the feel was still there.

[... omitted for brevity ...]

Which leads to my theory of why a major chord sounds "happy" and a minor chord sounds "sad" (here comes the woo ..) The beat frequency created when a Major 3rd fundamental is added to the Root fundamental is more harmonious to the fundamental than the beat frequency created when a minor 3rd fundamental is added to the Root fundamental (which isn't really all that harmonious). And the mind maps the more harmonious beat frequency to happy and the dissonance of the minor 3rd beat frequency to sad. And these mappings occur because we get a similar feeling when we are happy and when we hear harmonious sounds and a similar feeling when we are sad and hear dissonant sounds. - And the same sort of things happens with all the harmonics we get with real musical instruments. In short, the basic phenomenon afoot is created by the actual ratio of the frequency of notes and how the beat frequencies created by their addition relate to the root of the chord.

(Edit 1 to fix beat frequencies)

You might possibly gain more from looking at beat frequencies between overtones; for sine waves, the "critical bandwidth" for perceiving dissonance is close to a minor third (although wider in lower registers, which is why chords played entirely in the bassiest register of any instrument tend to sound like shit), if you add up the dissonances between successive overtones, the regular major chord is a pretty obvious local minimum as far as triads go. (The only better local minimums I can think of either are not real triads due to octaves, or are inversions of the major triad.)

I'm not sure we're on the same page here, you might be a few chapters ahead of me.

As I understand it, when we talk about harmony/dissonance there are at least three levels involved.

1) How the frequency of the waves are numerically related, i.e. are they harmonically related (multiple of integers) or not.

2) How physiology interacts with 1) - I think this is where your statement about "critical bandwidth" comes in. And although I don't know much about it, I assume this is on the path of the work started by Plomp and Levelt as seen here

3) The effect culture has on how we process what we get from 2)

So at this point I'm just dealing with 1) - is there something obviously different about the frequency relationship between a minor third and major third? It seems pretty obvious to me that the answer is yes. I don't seem to have to depend on the physiology of my ears to sense it, I can see it with my eyes- of course what that means is another matter. Never-the-less, there seems to be something obviously more harmonic and less dissident in the graphs of the major third as opposed to the minor third. And that makes me think that I must be talking about some other kind of dissidence than Plomp because, at least as far as I can understand it, his work seems to predict that minor third intervals are slightly less dissident than major third intervals. As seen here (this graph is also in the paper I reference above):



Is there a good modern reference that deals more completely with "critical bandwidth" theory that you recommend?

Simple sine waves are not very good things to work with when reasoning about harmony, since so much of the harmony is a result from the interactions of overtones. A tritone played by pure sinewaves is not particularly dissonant (except in the lower registers - but in those, even a fifth is dissonant)

Again I'm not sure I'm following this. Imagine I have a musical instrument, let's call it a "fundamental". The notes on a fundamental only sound the fundamental without any other harmonics. Are you saying that when playing a fundamental that major and minor chords would loose their majorness or minorness?

I don't think that's the case but perhaps I need to spend more time playing my fundamental to be sure. This is really the kind of thing I'm interested in looking at with as simple a situation as I can make work.

Again, sorry if I missed your point.

I find these graphs interesting. They show how the fundamental of a root interacts with the fundamental of every interval.



And zoomed in closer:

John Platko wrote:
Again I'm not sure I'm following this. Imagine I have a musical instrument, let's call it a "fundamental". The notes on a fundamental only sound the fundamental without any other harmonics. Are you saying that when playing a fundamental that major and minor chords would loose their majorness or minorness?

I don't think that's the case but perhaps I need to spend more time playing my fundamental to be sure. This is really the kind of thing I'm interested in looking at with as simple a situation as I can make work.

Again, sorry if I missed your point.

The fundamental would have much fewer rough chords - the little dissonance you'd have in a "straight" voicing of a dom7 chord would be from the minor third between the fifth and seventh, not from the tritone between the third and seventh; a minor chord would possibly be more dissonant than a major chord, but in both the very very slight amount of dissonance would be produced by the minor third in them. A dom7#9 or any extended kind of jazz chord would be rather smooth - most of the effects of consonance and dissonance would be smoothed out, except for those caused by seconds (and minor thirds). Beyond major and minor triads, a lot of the sound of our chords are basically different types of dissonance - the type of dissonance really helps us identify them as well. This gets lost with pure sine waves.

Zwaarddijk wrote:

John Platko wrote:
Again I'm not sure I'm following this. Imagine I have a musical instrument, let's call it a "fundamental". The notes on a fundamental only sound the fundamental without any other harmonics. Are you saying that when playing a fundamental that major and minor chords would loose their majorness or minorness?

I don't think that's the case but perhaps I need to spend more time playing my fundamental to be sure. This is really the kind of thing I'm interested in looking at with as simple a situation as I can make work.

Again, sorry if I missed your point.

The fundamental would have much fewer rough chords - the little dissonance you'd have in a "straight" voicing of a dom7 chord would be from the minor third between the fifth and seventh, not from the tritone between the third and seventh; a minor chord would possibly be more dissonant than a major chord, but in both the very very slight amount of dissonance would be produced by the minor third in them. A dom7#9 or any extended kind of jazz chord would be rather smooth - most of the effects of consonance and dissonance would be smoothed out, except for those caused by seconds (and minor thirds). Beyond major and minor triads, a lot of the sound of our chords are basically different types of dissonance - the type of dissonance really helps us identify them as well. This gets lost with pure sine waves.

I'm spending time trying to understand how the fundamentals of different chords interact with each other in the hope that it will provide a better understanding of how various harmonics of chords which are close in frequency will also interact. Besides the plots of two fundamentals at all intervals I've also made plots of the how the most common tri tone fundamentals interact which I can post if there's any interest in that sort of thing.

Now when we add the harmonics of each note to the story things get pretty complicated. I think I've finally wrapped my mind around a way to show the big picture of this. Here's a chart showing the harmonics of the notes of a major and minor A chord.



(edit: It's a bit hard to read that, here's a link to a pdf of the chart
https://drive.google.com/file/d/0B0wung ... 9QRTg/view)

It's better to try and illustrate it by sound. I should have a sample lying around. Oh, here it is:
http://yourlisten.com/miekko/tritonus-b ... es#comment

That's just for a tritonus though - it adds the overtones alternating between one tone and the other. Notice how the first three or so don't really contribute much in ways of dissonance. I could make a similar thing for a major and a minor chord if you want?

Zwaarddijk wrote:It's better to try and illustrate it by sound.

Well yes, listening to sound has its advantages. But I since I'm not all that musically gifted, I find it helps if I supplement what my ears are hearing with some visual ways to see what's going on. Then I can listen to aspects of it and I cant test if I really understand it by making predictions, synthesizing the sound and seeing if it really works the way I think it works. Of course, I wish I was just born with better ears ...

I should have a sample lying around. Oh, here it is:
http://yourlisten.com/miekko/tritonus-b ... es#comment

That's just for a tritonus though - it adds the overtones alternating between one tone and the other. Notice how the first three or so don't really contribute much in ways of dissonance.

I'm not quite following you. Could you say in a bit more detail exactly which overtones from which notes are being played?

I could make a similar thing for a major and a minor chord if you want?

It might help. It would also be helpful if you could point out exactly what overtones are contributing to dissonance and why.

John Platko wrote:

Zwaarddijk wrote:It's better to try and illustrate it by sound.

Well yes, listening to sound has its advantages. But I since I'm not all that musically gifted, I find it helps if I supplement what my ears are hearing with some visual ways to see what's going on. Then I can listen to aspects of it and I cant test if I really understand it by making predictions, synthesizing the sound and seeing if it really works the way I think it works. Of course, I wish I was just born with better ears ...

I should have a sample lying around. Oh, here it is:
http://yourlisten.com/miekko/tritonus-b ... es#comment

That's just for a tritonus though - it adds the overtones alternating between one tone and the other. Notice how the first three or so don't really contribute much in ways of dissonance.

I'm not quite following you. Could you say in a bit more detail exactly which overtones from which notes are being played?

It starts out with (IIRC, won't go and check which exact tones I use - the same applies for any pair of tritones anyway) the fundamental of C, then the fundamental of F#, then alternates between adding one to each of them as follows:
C F#
c f#
g c#'
c' f#'
e' a#'
g' c#''
bb' d#''
. .
. .
. .

It alternates between the two rows, C->F#->c->f#->g->c#'...
Now, I've retuned all the pitches to within 1/64 semitone of the actual pitch of the overtone (better precision than that is not available with the software I use). There's no dissonance in it, imho at least, until it hits g.

The dissonance appears because of the beats that are produced between g and f# - if you were to produce a recording with sine wave tones where you had C and f#, or c and f# or G and f# or anything like that, it would not be dissonant. This is because our hearing apparatus is weird and only really perceives beats between intervals that are narrower than about a minor third as dissonances.

A more detailed explanation can be found on wikipedia at http://en.wikipedia.org/wiki/Critical_band

Zwaarddijk wrote:

John Platko wrote:

Zwaarddijk wrote:It's better to try and illustrate it by sound.

Well yes, listening to sound has its advantages. But I since I'm not all that musically gifted, I find it helps if I supplement what my ears are hearing with some visual ways to see what's going on. Then I can listen to aspects of it and I cant test if I really understand it by making predictions, synthesizing the sound and seeing if it really works the way I think it works. Of course, I wish I was just born with better ears ...

I should have a sample lying around. Oh, here it is:
http://yourlisten.com/miekko/tritonus-b ... es#comment

That's just for a tritonus though - it adds the overtones alternating between one tone and the other. Notice how the first three or so don't really contribute much in ways of dissonance.

I'm not quite following you. Could you say in a bit more detail exactly which overtones from which notes are being played?

It starts out with (IIRC, won't go and check which exact tones I use - the same applies for any pair of tritones anyway) the fundamental of C, then the fundamental of F#, then alternates between adding one to each of them as follows:
C F#
c f#
g c#'
c' f#'
e' a#'
g' c#''
bb' d#''
. .
. .
. .

It alternates between the two rows, C->F#->c->f#->g->c#'...
Now, I've retuned all the pitches to within 1/64 semitone of the actual pitch of the overtone (better precision than that is not available with the software I use). There's no dissonance in it, imho at least, until it hits g.

OK, I understand what's going on in that clip now. It all happened a bit fast for me so I made a similar version in audacity where I could bring in each harmonic at a slower rate - although I didn't fine tune the frequencies like you did. The most obvious thing I hear is the F#-G and C-C# beating, or roughness as some call it, but there's something about the C-F# that doesn't sound as appealing as a C-G does to me.

The dissonance appears because of the beats that are produced between g and f# - if you were to produce a recording with sine wave tones where you had C and f#, or c and f# or G and f# or anything like that, it would not be dissonant. This is because our hearing apparatus is weird and only really perceives beats between intervals that are narrower than about a minor third as dissonances.

A more detailed explanation can be found on wikipedia at http://en.wikipedia.org/wiki/Critical_band

I found this paper that tried to sus out if the beating effect or the harmonic frequency relations are more important to how we perceive dissonance interesting:

http://www.pnas.org/content/109/48/19858.abstract

Evolution

Zwaarddijk wrote:

John Platko wrote:

Zwaarddijk wrote:It's better to try and illustrate it by sound.

Well yes, listening to sound has its advantages. But I since I'm not all that musically gifted, I find it helps if I supplement what my ears are hearing with some visual ways to see what's going on. Then I can listen to aspects of it and I cant test if I really understand it by making predictions, synthesizing the sound and seeing if it really works the way I think it works. Of course, I wish I was just born with better ears ...

I should have a sample lying around. Oh, here it is:
http://yourlisten.com/miekko/tritonus-b ... es#comment

That's just for a tritonus though - it adds the overtones alternating between one tone and the other. Notice how the first three or so don't really contribute much in ways of dissonance.

I'm not quite following you. Could you say in a bit more detail exactly which overtones from which notes are being played?

It starts out with (IIRC, won't go and check which exact tones I use - the same applies for any pair of tritones anyway) the fundamental of C, then the fundamental of F#, then alternates between adding one to each of them as follows:
C F#
c f#
g c#'
c' f#'
e' a#'
g' c#''
bb' d#''
. .
. .
. .

It alternates between the two rows, C->F#->c->f#->g->c#'...
Now, I've retuned all the pitches to within 1/64 semitone of the actual pitch of the overtone (better precision than that is not available with the software I use). There's no dissonance in it, imho at least, until it hits g.

The dissonance appears because of the beats that are produced between g and f# - if you were to produce a recording with sine wave tones where you had C and f#, or c and f# or G and f# or anything like that, it would not be dissonant. This is because our hearing apparatus is weird and only really perceives beats between intervals that are narrower than about a minor third as dissonances.

A more detailed explanation can be found on wikipedia at http://en.wikipedia.org/wiki/Critical_band

I've added some analysis to my chord harmonic grapher. It connects harmonics that are a min 2nd, Maj 2nd, or min 3rd apart with a line so you can get the big picture view of that type of dissonance. It's not obvious to me what role that type of distance plays in the difference between major and minor. Here's what it now looks like:

https://drive.google.com/open?id=0B0wun ... authuser=0

I think the main difference is that dissonance appears *earlier* up the overtone series of the root in the minor chord than it does in the major chord. The minor chord is pretty close to a local minimum for triadic dissonance, but the major chord is almost smack dab on a global minimum.

Zwaarddijk wrote:I think the main difference is that dissonance appears *earlier* up the overtone series of the root in the minor chord than it does in the major chord. The minor chord is pretty close to a local minimum for triadic dissonance, but the major chord is almost smack dab on a global minimum.

I too think that the earlier in the series that the dissonance appears the more it matters but I'm starting to believe that it's not just the kind of dissonance the lines on my chart represent but also the way the harmonics of the non root tones "fit in"/match the harmonics of the root. That, as that last paper I linked to suggests, may be the most important factor - as opposed to actual beating effects.

I updated my chart to add markers to make it easy to see where the root, Maj3r, and 5th of the root note appear in all the harmonics of the chord to help show how non root notes are related to the root of the chord.
https://docs.google.com/file/d/0B0wung_ ... FtWnM/edit

Here's a comparison of min7, Maj7, and 7 chords.

https://drive.google.com/file/d/0B0wung ... sp=sharing

I spent some time trying to digest the harmonic matches and beat lines that I plotted by applying a simple algorithm to a bunch of chords and seeing what I could see from that. The y axis is the percentage of harmonics in the chord that fall on a root, M3rd, or P5 of the root of the chord. The x axis is based on the "beat" lines, ie, the number of m2nds, M2nds, and m3rds between harmonics in a chord (with a weighting of 3,2,1) normalized to an ideal tritone of 3 root notes. (the altered chords 7_b5 and 7_#5 plotted over each other). i1 and i2 are inversions. RSSS = is a root, m2, M2, m3 chord.



What sticks out as an obvious problem to me is that this algorithm doesn't adequately distinguish between a 7th and Major7th chord or a 9th and Major 9th chord. I'm thinking where the m2nd, M2nd, and m3rd, dissonance is matters. I'll factor that in next and see what happens.