Unraveling the Mystery of How Music Evokes Emotions

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Music evokes emotions in the human mind through a complex interplay of sound waves and brain interpretation, with ongoing scientific exploration into the underlying mechanisms. While some argue that emotional responses are tied to cultural conditioning—such as associations with specific chords or rhythms—others suggest that the experience of music is more visceral and instinctual, transcending linguistic boundaries. The expectation theory posits that satisfaction arises when musical patterns meet our anticipations, which can explain why certain pieces grow on listeners over time. The discussion also touches on the subjective nature of music, where personal taste and emotional connection play significant roles. Instrumental music, devoid of lyrics, can still deeply affect listeners, highlighting the intrinsic power of sound itself. Ultimately, the relationship between music and emotion remains a rich area for exploration, blending neuroscience, psychology, and cultural studies.
  • #31
AlephZero said:
You don't need to look further that Country and Western to disprove the theory that major chords sound happy.

How about an example?
 
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  • #32
AlephZero said:
I wouldn't attempt to define it, beyond "music is whatever a particular group of people, at a particular time, call music"

lol, I'm more than happy to accept that as a definition, totally agree with that actually.

AlephZero said:
it leads to nonsense conclusions, like the fact that if you marked J S Bach by the standards of Cherubini's 19th century "rules for writing fugues", which was one of the classic texts for teaching composition in every music conservatiore in Europe, he would have failed the course.

Good example of how trying objectify something inherently subjective is an inane task!

AlephZero said:
Is it "music", or 45 minutes of random noise made by somebody horsing around in an organ loft?

I say it was somebody horsing around in an organ loft making music lol :)
 
  • #33
zoobyshoe said:
When someone speaks, there are two things going on: the words, and how they say the words. You can vastly change the meaning of an utterance by changing the tone of voice, rhythm, word emphasis, etc. Imagine removing the words, replacing them with non-significant gibberish, and being left only with tone of voice, rhythm, emotional emphasis. In the absence of words, what is communicated? Huge amounts about the mood, attitude, and personality texture of the speaker. What you'd be hearing, in the absence of understandable words, is that person's personal music.

Ever notice that you just love the sound of a certain person's voice? Math Is Hard once said she loved Morgan Freeman's voice so much she could sit and listen to him read the phone book. The opposite's also true: some people's personal music is quite ugly, and you can't stand the very sound of their voice. There's everything in between and more gradients along all other axes.

BenG549 said:
I'm not sure what your point is here. If it is that information content (in the sense of communicating speech) is not an important part of music, I totally agree. Some of my favourite music is instrumental, but again in the context of this discussion I'm not sure of the point. When a said "It just happens to be useful for communicating information as well" I was mealy saying that we can use our vocals for both music and communication of ideas, or both.

Let me just respond to this. The fact you didn't see my point here must be why you didn't see it elsewhere. The point is that people's speaking voices have musical properties. I went on to say that the reason we respond to music is because music captures and emphasizes the musical properties of speech.

I didn't say it, but what you might infer from that is that sound which does not ultimately reference the non-verbal aspects of human speech is not music, and that that which is music is so because it references the non-verbal aspects of human speech.

With that I'm answering the question "how/why music causes emotion", and also putting parameters around music.
 
  • #34
AlephZero said:
I wouldn't attempt to define it, beyond "music is whatever a particular group of people, at a particular time, call music"
Forget that you might be criticized by someone who disagrees with your definition and define what you personally respond to as being "music". (Like: If you know something is pornography, you don't have to pretend it's art just because that label could be upheld in court with enough insistence and recourse to legal technicality here.)
 
  • #35
zoobyshoe said:
I didn't say it, but what you might infer from that is that sound which does not ultimately reference the non-verbal aspects of human speech is not music, and that that which is music is so because it references the non-verbal aspects of human speech.

Oh right lol! OK I think I understand where you're coming from now, sorry about that.

So are you saying that the timbre etc. of a persons voice is musical, however the "information content" i.e. the dialogue, is out side of what you would describe as music?

If so, its a reasonable point, but I'd still disagree, there are plenty of musical forms that directly "reference the verbal aspects of speech" any rap, hip hop or grime for instance is primarily focused on lyrical content over the "non-verbal" aspects. People still relate to it emotionally, and to pick up on AlephZero's point, people call it music, despite if there is any other "musical accompaniment".
 
  • #36
zoobyshoe said:
Forget that you might be criticized by someone who disagrees with your definition and define what you personally respond to as being "music". (Like: If you know something is pornography, you don't have to pretend it's art just because that label could be upheld in court with enough insistence and recourse to legal technicality here.)

To be fair, if we are trying to establish how music causes emotion, we don't really need to get too bogged down in a discussion over the personal definitions of what music is, beyond the one we have i.e. what ever someone might conceivably describe as music; we know it's subjective so isn't it a bit of a side issue? I guess we're basically asking what mechanism is responsible for invoking emotion given audible stimuli (after all however we define it, that is essentially what music is), and what are the reasons for it i.e. is there any evolutionary basis for how we react to complex sounds? ... personally I'm not a neuroscientist or an evolutionary biologist though, so any thoughts on that might be interesting... for me anyway.
 
  • #37
BenG549 said:
So are you saying that the timbre etc. of a persons voice is musical, however the "information content" i.e. the dialogue, is out side of what you would describe as music?
Information as information is not music. For example, this is not music:

Newton said:
Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

It's a piece of text composed such that the purely informational aspect of the words should completely dominate how it is received.

When someone is speaking we can abstract some element of what they are saying as purely informational, and what's left will be the music: the tell tales that let us know their mood, how they feel about what they are saying, and that also tell us about the texture of their personality, etc.

This non-verbal side of speech has a name:

The catch here is that you have to be speaking to be speaking in paralanguage, so it's rarely separate from words. Music is, I think, a medium in which we can directly communicate paralanguage without words.

Jerry Lewis doesn't say a word in that clip, but he speaks volumes. We know everything about the type of bossy man-in-charge he's rendered into a cartoon there because the music takes the place of the words and speaks man's paralanguage.

If so, its a reasonable point, but I'd still disagree, there are plenty of musical forms that directly "reference the verbal aspects of speech" any rap, hip hop or grime for instance is primarily focused on lyrical content over the "non-verbal" aspects. People still relate to it emotionally, and to pick up on AlephZero's point, people call it music, despite if there is any other "musical accompaniment".
Hip hop and all that is low on melody but rich in rhythm, and rhythm is an essential component of music. In a sense these forms (rap, et al) are verbal percussion more than songs or poems. The lyrics are usually words that make you feel you're being beaten with a stick or stone (or at least threatened with them). Words heavily laden with paralanguage. There's a closer tie to ritual war music than anything else in rap.

You won't understand a word of this clip, but I bet you can figure out what they're saying:

https://www.youtube.com/watch?v=gFUf9dP1sWY
 
  • #38
BenG549 said:
To be fair, if we are trying to establish how music causes emotion, we don't really need to get too bogged down in a discussion over the personal definitions of what music is, beyond the one we have i.e. what ever someone might conceivably describe as music; we know it's subjective so isn't it a bit of a side issue? I guess we're basically asking what mechanism is responsible for invoking emotion given audible stimuli (after all however we define it, that is essentially what music is), and what are the reasons for it i.e. is there any evolutionary basis for how we react to complex sounds? ... personally I'm not a neuroscientist or an evolutionary biologist though, so any thoughts on that might be interesting... for me anyway.
The answer would be that what makes us respond to music is the same thing that makes us respond to the paralinguistic aspects of speech, by my take.

We've been bullied into accepting a lot of junk noise as music by the 20th Century avant guardists: 12 tone, John Cage, etc. But that time is past and we no longer have to pretend we love The Emperor's New Music.

I, personally, hate most Country-Western music, and I'm not very fond of Mariachi, either, but I don't claim they're not music. 12 tone, though, was never really music, and neither was John Cage. I don't accept that I have to accept as music whatever someone else presents as music. I think we can distill a good definition of music from what everyone agrees is music, (provided people don't exclude what they recognize to be music but don't enjoy).
 
  • #39
Here is an old file I found in my PDF library from College. Maybe it applies to this question of music.
 

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  • #40
AbbyLayne said:
Here is an old file I found in my PDF library from College. Maybe it applies to this question of music.
It's a very long paper, but I read the first few pages and I like it. It echos a lot of what Sacks says in Musicophilia, especially the point that music is very much more basic and important than it's often given credit for.
 
  • #41
zoobyshoe said:
It's a very long paper, but I read the first few pages and I like it. It echos a lot of what Sacks says in Musicophilia, especially the point that music is very much more basic and important than it's often given credit for.

Yea, I only understood about 50% of the thing, but it sounded like they were talking about how the music makes people feel emotions, so I thought it might be relevant to the thread :)
 
  • #42
zoobyshoe said:
When someone is speaking we can abstract some element of what they are saying as purely informational, and what's left will be the music: the tell tales that let us know their mood, how they feel about what they are saying, and that also tell us about the texture of their personality, etc.

Yeah OK that makes a lot of sense, just to play devils advocate, do you not think that information can be artistic? after all a lot of the time is it created (in a not breaking the 2nd law of thermodynamic sort of way) you produce the information and if that information is verbal could you not describe it as music... I don't know, maybe not. Anyway like I said I do see your point, and I agree that the tonal and temporal features of speech are the more 'musical elements' of what we hear

zoobyshoe said:
The catch here is that you have to be speaking to be speaking in paralanguage, so it's rarely separate from words. Music is, I think, a medium in which we can directly communicate paralanguage without words.

Yeah this was going to be my point, they are very intertwined, and the created by the same mechanisms, so it is hard to really distinguish them, or discuss them as separate things (in my mind anyway lol)

zoobyshoe said:
Jerry Lewis doesn't say a word in that clip, but he speaks volumes. We know everything about the type of bossy man-in-charge he's rendered into a cartoon there because the music takes the place of the words and speaks man's paralanguage.

I'd say it was more to do with his body language, which is a form of information. But I get the point this time!

zoobyshoe said:
Hip hop and all that is low on melody but rich in rhythm, and rhythm is an essential component of music.

Yeah that is very true, to a certain extent. I do feel that the information content goes some way to invoking emotion though, it allows us it empathise, when we hear a song about love most of us understand, or have had comparable feelings, that allow us to relate to the song, that's purely about information content and our inherent desire to feel attached or connected to people. That is information invoking emotion and in that context I would say it was musical, or at least part or the musical experience.

Although just to add to that I wouldn't argue that a sense of rhythm is an essential part of music.
 
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  • #43
BenG549 said:
...just to play devils advocate, do you not think that information can be artistic?
It's conceivable, but I didn't quite follow what you said next. Do you have an example of information you feel is artistic?
Yeah this was going to be my point, they are very intertwined, and the created by the same mechanisms, so it is hard to really distinguish them, or discuss them as separate things (in my mind anyway lol)
I'd say it was more to do with his body language, which is a form of information. But I get the point this time!
Speaking of intertwined, body language is very hard to separate from the information and the paralanguage. In the case of that clip we know the music preceeded the body language. There's no telling what Count Basie had in mind exactly, but Lewis heard a distinct, vivid paralanguage and supplied the body language to support what he heard so he could share it with the audience. Once you take the informational aspects away you have a more basic, primal thing that every individual hearing it can fill out according to his own confirmation bias.
I do feel that the information content goes some way to invoking emotion though, it allows us it empathise, when we hear a song about love most of us understand, or have had comparable feelings, that allow us to relate to the song, that's purely about information content and our inherent desire to feel attached or connected to people. That is information invoking emotion and in that context I would say it was musical, or at least part or the musical experience.
Adding lyrics is a way for the composer to prompt the listener to have a much more specific reaction to the paralanguage. It still ends up accommodating a huge variety of interpretations. What I like about the Lewis clip is that he clearly understood the music to be a voice speaking with a lot of attitude. The exact place he took it was just one of a multitude of potential places where a voice speaks with a lot of attitude. I could see it done as a dialog with two people going at each other with attitude, just as well.

Although just to add to that I wouldn't argue that a sense of rhythm is an essential part of music.
You must be some kinda crazy person, then. Hehe.
 
  • #44
zoobyshoe said:
It's conceivable, but I didn't quite follow what you said next. Do you have an example of information you feel is artistic?


I would probably say that the way we use speech is personally exclusive enough to argue there is some artistic quality do it. Creative use of mathematics may be considered artistic by some,... actually for fear of getting into a debate about the definition of art I think I'll just retract that question, it was not really worth answering anyway lol, bit off the point.


zoobyshoe said:
You must be some kinda crazy person, then. Hehe.

lol, generally speaking I'm inclined to agree with what you've been saying, but you could write music that has no consistent rhythmical structure

The last movement of Schoernberg's second string quartet, Opus 10, has no time signature; Gregorian chant use free rhythm; Performances of Carnatic music (South Indian classical music) frequently begin with a type of improvisation called alapana (melodic exposition) in free rhythm without percussion; Steve Reich's Tehillim, a musical setting of four psalms in Hebrew, is composed in free rhythm.

This article* details the fact that, although the term 'free rhythm' is not specifically defined, unmetred music is common in many cultures including some western examples... I'm not the only crazy person out there lol!

*http://oro.open.ac.uk/17650/1/FreeRhythm.pdf
 
  • #45
Just as any other sensory input, sound can stimulate pathways in the brain associated with pleasure and/or pain. Music is sequences of sounds that are studied to stimulate either of these particular responses. A great example of this is music used in movies, where sound can be used to create an ambience of tension and fear as much as one of romance and affection, or hatred and so forth, depending on the particular scene of the movie. There are people whose neural connections are a bit stranger than the norm and who mix up sensory information, thus 'seeing' sounds or hearing colors. These people probably have an even more interesting experience when hearing music.
 
  • #46
vappole said:
Just as any other sensory input, sound can stimulate pathways in the brain associated with pleasure and/or pain.

This might be a bit of a side issue, if so feel free to ignore this post, but is the threshold of pain w.r.t noise, not down to mechanical rather than neurological reasons... i.e. the reason you experience pain is not to do with your perception of the sound pre se, but because your ear drum is being deflected beyond what is comfortable or excessive vibration on the basilar membrane, I know there are muscles in the inner ear that react to loud impulses, effecting the position of the ossicles and reducing the transmission though the bones, but I'm not sure what actually causes the pain... Although I've studied acoustics we didn't do a lot on aural physiology or noise induced pain and hearing loss... Any knowledge would be appreciated.
 
  • #47
BenG549 said:
Yeah OK that makes a lot of sense, just to play devils advocate, do you not think that information can be artistic?

zoobyshoe said:
It's conceivable, but I didn't quite follow what you said next. Do you have an example of information you feel is artistic?

Some attempts are described on Dubnov's 18th slide. I believe the perspective is related to Huron's essay mentioned earlier by 256bits in post #9. http://musicweb.ucsd.edu/~sdubnov/SixthFun1.htm
 
  • #48
...you could write music that has no consistent rhythmical structure

CRAZY_by_zoobyshoe.jpg
 
  • #49
zoobyshoe said:
CRAZY_by_zoobyshoe.jpg

Hahaha nice touch... although I did provide examples of how that was true.
 
  • #50
zoobyshoe said:
CRAZY_by_zoobyshoe.jpg

https://www.youtube.com/watch?v=DGovCafPQAE

The text is key. The setting of the words is exquisite.
 
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  • #51
atyy said:
https://www.youtube.com/watch?v=DGovCafPQAE

The text is key. The setting of the words is exquisite.
OK, I get it. This fits the criteria of having no consistent rhythmic structure.

I am asking myself why, though, it doesn't at all suggest a lack of rhythmic sense. I think it is because all the little pieces of different rhythm (which have their own rhythmic integrity) are arranged in sequence with a definite eye (ear) to creating an overall structure that is actually quite satisfying. There's a good balance of slow rhythm, rapid rhythm, and silence. I feel like the composer had good instincts about varying that which is similar with that which is novel such that it comes off as deliberate and "composed".

I wouldn't call this music, but I would call it art. Maybe: "Rhythm Collage."

I couldn't follow the text at all, so I stopped trying. I think if text is the key, it's up to the composer to make sure it's easily accessed. I guess I'm an a hole that way.
 
  • #52
BenG549 said:
Hahaha nice touch... although I did provide examples of how that was true.
I looked for the Schoenberg but YouTube didn't have the piece you mentioned, so I was done with that quest. If the piece Atty posted is representative of what you're talking about I'd have to say this kind of thing is in the realm of "experimental" art and is an anomalous side eddy occurring at one particularly strange build up of old logs and rocks on one bank of the larger river that you can't use to characterize the main flow. I think a sense of rhythm is vital to music. The first non-vocal instruments must have been percussion, don't you think? Like you said, two rocks. (But more likely stick on log.)
 
  • #53
zoobyshoe said:
I looked for the Schoenberg but YouTube didn't have the piece you mentioned, so I was done with that quest. If the piece Atty posted is representative of what you're talking about I'd have to say this kind of thing is in the realm of "experimental" art and is an anomalous side eddy occurring at one particularly strange build up of old logs and rocks on one bank of the larger river that you can't use to characterize the main flow. I think a sense of rhythm is vital to music.

the final movement of Schoernberg's second string quartet "Opus 10" (not sure if you thought I meant Schoenberg's sting quartet and Opus 10 as different things) but yeah this is written with no time signature.



zoobyshoe said:
The first non-vocal instruments must have been percussion, don't you think? Like you said, two rocks. (But more likely stick on log.)

lol yeah, based on the complexity of whacking something against the complexity of developing or using resonant cavities attached to stings or even blow holes makes that assertion likely. Not sure it's totally relevant though. Just because it happened to be a big part of the way we used to do things, does that mean it has to be an integral part of how we do things now?

zoobyshoe said:
I wouldn't call this music, but I would call it art.

Could 'audible art' not qualify as a definition for music? On face value it seems rather fitting as a definition actually.
 
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  • #54
zoobyshoe said:
OK, I get it. This fits the criteria of having no consistent rhythmic structure.

I am asking myself why, though, it doesn't at all suggest a lack of rhythmic sense.

I'm not sure if BenG549 would agree, but let me try to explain why it's no big deal not to have a "consistent rhythmical structure". I think it's like the "rhythm" of prose or of free verse. While both are unmetred, the best authors clearly care about the rhythm of their prose or free verse. In the sense that music is heightened speech or narrative, then it need not have the "consistent rhythmical structure" that BenG549 mentioned. Many old forms such as Gregorian chant and the Baroque recitative very naturally have no "consistent rhythmical structure". I think it is also interesting to consider speech as a form of movement. Some movements such a jump for joy or changing bed sheets have no obvious repeated structure, but many such as heart beats, breathing, walking, running and ballroom dancing do. So we would expect all of these "rhyhms" to feel natural.

http://www.lphrc.org/Chant/
http://en.wikipedia.org/wiki/Recitative

Incidentally, the text for Berio's sequenza III is by Markus Kutter.

Give me a few words for a woman
to sing a truth allowing us
to build a house without worrying before night comes
 
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  • #55
I also think it should have some consistent thytmical structure. I wouldn't say Gregorian chant had NO consistent rhythmical structure, just very little (as it says in your link).

But Gregorian chant is right on the cusp between no music and music... (it's often considered the first music) so you might argue it's pseudo music.. underdeveloped music.
 
  • #56
How could Gregorian chant be the first music? Didn't the Sumerians have music? Isn't dance mentioned in the Old Testament?
 
  • #57
Yeah, I must have missed some qualifier from my humanities class. Maybe oldest western, or oldest western written music. I didn't realize Gregorian was so recent in human history. Didn't even know it was Christian music.

Anyway, remove the confusion of specific instances, same argument. Music developed from stuff that didn't have rhythm or integer ratios. Certainly the more integer your ratios, the more people are bound to like it (i.e. pop music with the I-IV-V, a small excerpt of the circle of fifths).
 
  • #58
atyy said:
I think it's like the "rhythm" of prose or of free verse. While both are unmetred, the best authors clearly care about the rhythm of their prose or free verse. In the sense that music is heightened speech or narrative, then it need not have the "consistent rhythmical structure" that BenG549 mentioned. Many old forms such as Gregorian chant and the Baroque recitative very naturally have no "consistent rhythmical structure". I think it is also interesting to consider speech as a form of movement. Some movements such a jump for joy or changing bed sheets have no obvious repeated structure, but many such as heart beats, breathing, walking, running and ballroom dancing do. So we would expect all of these "rhyhms" to feel natural.

That's quite an interesting point, I certainly wouldn't disagree.

Pythagorean said:
Yeah, I must have missed some qualifier from my humanities class. Maybe oldest western, or oldest western written music. I didn't realize Gregorian was so recent in human history. Didn't even know it was Christian music.

lol, you should probably at least run a quick google search to clarify your arguments before posting them. Other wise you are pretty much just making stuff up ;) .

Pythagorean said:
Certainly the more integer your ratios, the more people are bound to like it (i.e. pop music with the I-IV-V, a small excerpt of the circle of fifths).

In western culture maybe, the musical concepts we are 'used to' are not ubiquitous across all cultures, some have very different ideas of what constitutes 'pleasing' music... there are examples posted above. Gamalan is one of the first that springs to mind*, and this article** is one i posted earlier about the cultural differences in rhythmical structure in music as an example of how musical 'ideas and methods' are cultural... it's not necessarily that their music is 'less evolved' than ours.

* http://oro.open.ac.uk/17650/1/FreeRhythm.pdf

(no nice II-V-Is or I-IV-V chord structures in this, but things like this are written to be emotion invoking aids in Indonesian theatre)

**http://oro.open.ac.uk/17650/1/FreeRhythm.pdf
 
  • #59
BenG549 said:
lol, you should probably at least run a quick google search to clarify your arguments before posting them. Other wise you are pretty much just making stuff up ;) .

That's great advice, but for me it's more about impulse control. I know what I *should* do, but often don't realize until after the action. That's a much more general problem of mine. Probably not something any amount of information will change, just practice (and old age, maybe). Anyway, impulsive people have their place in society. There will always be the donkey jumping off the cliff and the elephant too stubborn to move.

In western culture maybe, the musical concepts we are 'used to' are not ubiquitous across all cultures, some have very different ideas of what constitutes 'pleasing' music... there are examples posted above. Gamalan is one of the first that springs to mind*, and this article** is one i posted earlier about the cultural differences in rhythmical structure in music as an example of how musical 'ideas and methods' are cultural... it's not necessarily that their music is 'less evolved' than ours.

* http://oro.open.ac.uk/17650/1/FreeRhythm.pdf

(no nice II-V-Is or I-IV-V chord structures in this, but things like this are written to be emotion invoking aids in Indonesian theatre)

**http://oro.open.ac.uk/17650/1/FreeRhythm.pdf
Western-influenced pop music is the most popular music world wide. You might argue this is just because they (we) developed a stronger media faster, but you have to admit that it's a strange coincidence that it's so popular and also has such perfect ratios... and humans are known for their love of symmetry.

Nobody's making an argument for less evolved, btw. I don't think fish are less evolved than humans. However, I will note that our common ancestor looked a lot more like fish than humans :)
 
  • #60
Pythagorean said:
Anyway, impulsive people have their place in society. There will always be the donkey jumping off the cliff and the elephant too stubborn to move.

lol, I totally agree. Pretty impulsive myself, nice analogy.

Pythagorean said:
Western-influenced pop music is the most popular music world wide. You might argue this is just because they (we) developed a stronger media faster, but you have to admit that it's a strange coincidence that it's so popular and also has such perfect ratios.

Yes that is exactly what I would argue. It's not a strange coincidence at all. For example there has never been any Korean pop music in the charts in the UK. You might conclude that people in the UK don't like Korean pop, it's not popular, but after major radio stations like Radio 1 played PSY's Gangman style tune for a few weeks it was number one and won a freaking MTV Award. People just buy what's fed to them, marketing sells.

And in this society we have incrementally developed a "liking" for certain musical intervals etc. because that's what's sold to us, but that's not to say that music as we know it couldn't have developed differently. People widely enjoy and accept different genres and types of music around the world, you can't really say they're enjoyed less just because they aren't as "widely popular", they might just not get the same marketing and radio coverage, it's not feed to the masses so to speak.

Pythagorean said:
Nobody's making an argument for less evolved, btw. I don't think fish are less evolved than humans. However, I will note that our common ancestor looked a lot more like fish than humans :)

Sorry I didn't mean "evolved from" in a human evolutionary sense... I was referring to this comment you made:

Pythagorean said:
Music developed from stuff that didn't have rhythm or integer ratios.

Should have made that clearer, sorry.
 

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