Harmonics

[peter]

(I thought I was starting a new topic here Guys entitled Harmonics but I obiousely hit the reply Button by mistake - Hell; I'll ask the question any way!) There is something weird here I'm not getting. Sounds that sound 'harmonious' when played together do so, we are told, because their wavelengths/frequencies are divisable by whole number integers such that the 'nodes' will always lie directly 'on top' of each other .....ish (do you get what I'm trying to say?) Hence a chord played on a piano can only produce a harmonious sound if the rules of this are obeyed. If I walk up to a piano and plunk a fist full of any old notes it will sound disharmonious because the wave lenghts of the individual noted will not be divisible exactly but will invove fractions of intigers. (I think this is how it goes).

My question is why should this be. Why should the human ear have evolved such that it only hears harmony when a set of circumstances that it cannot possibly have been aware of (ie the whole number divisabilities of the wavelengths of harmonious notes) is satisfied. Why should we have evolved to 'like' the harmony and 'dislike' the disharmony. There is something here (hear?) I am missing. How could we know (obviousely we couldn't) that the sounds we were evolving to like were the onbes where the conditions of divisability were being satisfied. What is going on?

[Vraith]

Umm....without getting into the psycho-social-cultural stuff, neurol-stuff, lots of other complicated stuff...[cuz what you say, in a way, isn't true...music without dissonance isn't "pleasing," for long...what it is is boring]
In the simplest, most general sense, people like patterns and notes sound more harmonious when they share more frequencies, [harmonics] generating regular periodic "beats."
But no one really uses "perfect" mathematical scales. [just intonation] cuz it causes tuning problems with various instruments...and if you transpose to a different key serious ugliness ensues...and certain intervals become useless cuz of dissonance. So we compromise, putting up with a little bit of barely noticeable dissonance in lots of places to eliminate big nastiness in some places.
[the ordinary western scale now uses the 12th root of 2 IIRC...I'm aware of others that, also IIRC, are something like the 19th root, 24th, 31st? I think anyway]

[peter]

Just re-read the above Vraith and I think (until the end bit) I get it. The pattern thing makes sense to me at least!

[Obi-Wan Nihilo]

Perhaps people like music because in our evolutionary past we had to learn pitch in order to imitate sounds properly.

On the freezing water problem, I'm reminded a bit of the supercooling phenomenon, which requires a very sedate environment to achieve. Perhaps the cool water supports a a very sedate environment while the warmer water creates one that is much more dynamic (internal and external convection, etc.), as several others have already suggested in their own way. So perhaps the cool jug is actually freezing at a lower temperature achieved later than the hot jug which freezes earlier at a higher temperature -- something that could be true in view of the fact that the potential for heat transfer attenuates as the temperature gradient diminishes. Maybe the warm jug is freezing right at 32 while the cool jug is able to get down well into the 20s before also freezing, but takes additional time to get there.

[Zarathustra]

Why should the human ear have evolved such that it only hears harmony when a set of circumstances that it cannot possibly have been aware of (ie the whole number divisabilities of the wavelengths of harmonious notes) is satisfied. ... How could we know (obviousely we couldn't) that the sounds we were evolving to like were the onbes where the conditions of divisability were being satisfied.

You're conflating two different "levels." There is the phenomenon, which we can sense. And then there's the mathematical description of it, which we discover after the fact. We don't need to know about the mathematical description in order to be aware of the phenomenon, no more than we need to know the wavelength of red to see red. The "nodes matching up" as you say is a physical effect that we can sense. It hits the eardrum in a physically different way than if the nodes were out of synch. So the question should be: why should the human ear have evolved in such a way that it hears waves of air molecules "matching up"? It seems to me that this particular kind of hearing is nothing more than a temporal awareness .... hearing two nodes in synch or out of synch. You might as well ask why we are aware of time. It doesn't seem problematic to suppose--for creatures that have evolved hearing in the first place--that there would be survival advantages to temporally organizing this sensory input. Knowing that the frequencies of a predator's scream are happening in a certain linear, temporal sequence would help you locate it through time. The more accurate that temporal awareness, the more accurate the spatial awareness.

Why should we have evolved to 'like' the harmony and 'dislike' the disharmony.

Why do we like order instead of chaos? Why is ripe, symmetrical fruit "pretty" but rotting food is "ugly?" Obviously, the question answers itself: survival advantage. Life is itself order out of chaos. But there are also dangerous forms of life. Again, the predator's scream is a very dissonant. So are many sounds that indicate danger. Rockslides, earthquakes, thunder, stampedes, volcanoes, etc. Perhaps for this reason species often choose "musical" calls to attract mates ... useful, helpful information that rises above the noise of nature. Calls that have high signal-to-noise ratio, so to speak.

[peter]

Yes but Zarathustra - there are many (in fact an infinite number) of positions where the nodes are out of synch and only a very limited number (within the range of human hearing) where they are. I think Vraith is saying that the 'pattern' of sound that occurs when some degree of symetry exists between various frequencies simultaneously heard is more pleasing by virtue of this symetry and relates to deep seated need for symetry/pattern in the human psyche. We percieve the pattern subconciously even when we are unaware of it at a concious level and the 'liking' of it is but a side effect of this.

[Hashi Lebwohl]

I concur--mixtures of sounds that are "pleasant" to us are ones where the frequencies blend in particular ways to us. This is wired into our brains; there is no "why" for this, as far as I can guess. A list of musical notes, their associated frequencies, and their wavelengths is listed here. What you will note here is that each note has a frequency exactly 1.059454545 times the frequency of the previous note, which is awfully close to 2^(1/12), the exact value (12 steps = an octave).

The sonic weapons used to create headaches broadcast 16,000 Hz and 16,002 Hz, combining to give 32,000 Hz and 2 Hz.

[Zarathustra]

Peter, I'm not seeing the need for the "but." Did you feel that my answers didn't address your questions? I agree with everything in your last post, but I don't see how it contradicts or otherwise pokes holes in my own post.

[Hashi Lebwohl]

Perhaps what peter is trying to express is why F-major sounds "pleasing" but D-minor sounds "creepy".

[Vraith]

Peter, I'm not seeing the need for the "but." Did you feel that my answers didn't address your questions? I agree with everything in your last post, but I don't see how it contradicts or otherwise pokes holes in my own post.

I agree...I think you were just on a slightly different point/direction. OTOH, I also think we're mixing some definitions. I THINK you were addressing things I intentionally left out...perception/psychology/meaning in environment...and that what you call "dissonance" of predator for instance isn't that in the musical/harmony sense. I think what you meant by dissonance transferred to the area I was speaking to would actually be speaking to the timbre...the shape of the sound wave, not the agreement [or lack] in frequencies.

And peter...yea, that's the simplest, most basic direction of my point...though things like habit, culture, familiarity...lots of crap...complicates it.
The simplest physical/visualizable/demonstrable example I can think of:
Say you have a whole bunch of different length strings tied up [or a piano, perhaps]...overtones are the basis of harmony, and it's physical. If you bang on a really low A, for example...the string half that length [A an octave higher] will ALSO vibrate...Also the string a perfect 5th above that higher A [E] and higher the 4th [D] etc. All of this because the original low A is generating these sub-vibrations [which peeps with a really good ear can consciously tease out...most of us it just makes it sound "rich" or "full" if it's good strings, or thin/flat if crappy materials...and the shape and material of the instrument enhances [or ruins] these.
So, in harmony...the perfect harmony is "unison"...two different sources of exactly the same pitch. If I play an A and an E at the same time...they sound "harmonious" because they have family relationships. The E already "belongs" to the A.
And that's the simple stuff...our brains read inversions, too...and lots of fractionals generate regularities/overlaps.

[Avatar]

I'm tone deaf.

--A

[Iolanthe]

If you bang on a really low A, for example...the string half that length [A an octave higher] will ALSO vibrate...

One of my pupils had a direction in one of her examination pieces to play a low note silently and hold it down whilst playing in the upper register with the right hand, and very effective it was too. An etherial quality, altogether more subtle than using the sustaining pedal.

I've very carefully read the posts above, and have understood hardly a word (attracted by the subject "Harmonics"), yet I have been playing by ear since I was 4 (so my parents told me) and by 16 had Grade 8 (but only a pass, it took me another 2 trys to get a distinction). I can play anything I know, with the correct harmonies - not difficult classical stuff, but songs from shows, "pop" songs etc. I've never thought how I do it, it just comes naturally. I taught piano for 25 years, so had to know the theory, but "science" never came into it.

Oh, and Vraith, do not " bang" your piano keys - it's not what they were made for.

And Peter

If I walk up to a piano and plunk a fist full of any old notes it will sound disharmonious

- do not!!

[peter]

Yes - sorry about that Z. Once again I seem to have dipped my toe into the waters of questions whose answers are deeper than my ability to swim!

Hashi's examples are great in that the Bach does have that 'feel' of disharmony about it that could so easily in the hands of a lesser man been unpleasant on the ear. It just seems to me that here, in the physics of sound, in the pattern of frequencies and on what 'works' for our ear and what doesn't, we come slap up against the starting point of the aesthetic (in a manner that no other art form allows). I just find it amazing (and wonderful) that this should occur where somekind of internal symetry exists that I had not thought we could be aware of.

Almost as soon as I had finished the original posting however I thought of an example that totally went against what I had suggested, namely the music of Chinese and japanese opera. Clearly these combinations of sound have to resonate as deeply with the 'aesthetic' sensibilities of oriental (and no doubt western) afficienados as do the harmonies of our music to us. But to me these forms of music seem to be composed of the very kinds of asymetric sound that i could produce by a random fistfull of notes played on a piano (much in modern classical music seems to be of the same cut to me as well). Clearly there is a deep well of knowledge to be drunk from on my part before enlightenment is to be found .

Just a turn of phrase Iolanthe - promise!

[DoctorGamgee]

Hmmm...

Harmonious is a relative term, but within the west there are simple mathematical reasons why some notes sound more pleasing. Overtones are part of the answer. And those come naturally in the True-tuning model before the Baroque period.

Before Equal Temperment (each of the 12 1/2 steps seeming equal) we used true tuning. Back when the church monks sang a note (whatever note that may be) the properties of that note were the fundamental pitch (A-440 for example). We hear A440 clearly. Our ear also hears the overtones (which separate the sound of a Flute from a Soprano, Piano and Train Whistle). When I took Music Theory as an undergraduate (15 years before getting my Doctorate in Music and 27 years ago now...oy do I feel old!) they played for us these four instruments all playing A440 on an analog recording, and you could distinguish their sound (timbre). They then removed all overtones: they were indistinguishable.

I bore you with all of this, because it is the overtone series which put in place the current system of Harmony which we now use. If you play Middle C (C4) on the piano, you will hear it as normal. If, while you are doing that, you press C5, G5, and C6 above it without striking the strings, you can hear the three strongest overtones. And that is why the first types of harmonies which were favored in Western Music were Doubling the octave (men/women) same melody, different octaves, and Organum which would have one voice singing C and the second singing G (a fifth above...it fit the harmonic pattern and since the C3 hit C4 and G4, singing G3 augmented that 5th relationship).

From there, harmonies were favored which were pleasing (mostly because they fit the overtone form, but also because that is what was considered pleasing to some). We like octaves, 3rds, fifths and 6ths (inversion of the third). We find 2nds, and 7ths dissonant, and 4ths if they occur above the bass note. That is why Bach organ music ending in C woud have an F step down to E -- Create tension and release it. But also because if your piano is well tuned, when you play C2 and press E4 you will clearly hear E5 ringing. Which is also why many composers of music in minor keys ended on Major chords (as the disonance between the Minor Chord and the Major 3rd overtone was displeasing...)

With the Equal-tempered system we use now, we lose a little of that "Ring" when we perform, but we gain more choices and diversity in the process. If you listen to acapella choral music, especially really good stuff, they are able to maintain the true tuning sparkle unaccompanied as they are. Accompaniment forces them to adjust.

Not that anybody cares, but for what it is worth.

BTW: It should also be noted that not everyone uses our scales. Eastern music uses quarter-tones as the smallest step, and I have performed choral music which did that to great (and disturbing) results. You could watch the audience react as the pitch wavered between quarter tones...our ears are not used to it.

Doc

[Hashi Lebwohl]

Almost as soon as I had finished the original posting however I thought of an example that totally went against what I had suggested, namely the music of Chinese and japanese opera. Clearly these combinations of sound have to resonate as deeply with the 'aesthetic' sensibilities of oriental (and no doubt western) afficienados as do the harmonies of our music to us. But to me these forms of music seem to be composed of the very kinds of asymetric sound that i could produce by a random fistfull of notes played on a piano (much in modern classical music seems to be of the same cut to me as well). Clearly there is a deep well of knowledge to be drunk from on my part before enlightenment is to be found .

When I get bored with Western music I turn my radio to a regional station here and listen to Asian music. The different tonalities and qualities are quite refreshing.

Iolanthe, I keep my fingernails long so they click when I play the piano. Not really, but that is the title of a real song.

Doc, I helped the mezzo with her research when she took Music Theory in her graduate degree program about 20 years ago. (has it really been that long? yes, it has) Anyway...I used to think that Differential Equations and Topolgy were hard, but they were nowhere near as difficult as Music Theory.

[Zarathustra]

Some of this has to be cultural. A major chord doesn't necessarily have to seem happy, nor a minor chord seem creepy. We've come to associate these sounds with certain feelings due to a shared culture and musical "language." As others have noted, music from different cultures sounds bizarre to outsiders. That's because we haven't grown up with the social cues which inform those sounds.

We could also ask why red is associated with passion, anger, "hot," etc., while blue is associated with calm, peace, "cool," etc. These feelings aren't inherent to the colors, they are formed by analogy to objects which often display those colors. Peaceful sky, calming waters, vs bloody violence, or flushed skin.

[aliantha]

Doc, your post is fascinating! Thanks for the harmonics lesson. When I'd sing along to the radio as a teenager, I would sometimes challenge myself to sing harmony. I realized not long ago that on at least one song, my accustomed pitch is an octave above the highest note I now hear on the recording. Maybe I was picking up the harmonic? (Or maybe my ears are shot! )

Wasn't some of the perceived dissonance in Bach's work due to the limitations of the instruments at the time? I seem to remember something about technical advances allowing a truer reproduction of chromatics. Altho that doesn't explain some of the weird (to us) intervals in medieval choral music, I guess.

[Fist and Faith]

Centuries ago, there were several other musical scales, or "modes", used in western Europe. Lydian, Dorian, yadda yadda. And the combinations of notes used (intervals and chords), and the progressions of these intervals and chords, were not the same as we use today.

Over time, western music settled into what we usually hear. Major and minor scales/keys. The composers back in the day simply started using those two modes more and more, and the others less and less. And they started making their music end in the same key that it began in. It's a fascinating thing to watch these, and other, aspects of music change in Monteverdi's (1567-1643) eight books of madrigals. (Uh, fascinating to a nerd, perhaps? )

Today, for most people in our culture, it all seems so natural. Time was, it was not. The question is, COULD it have evolved in a different way? COULD the Dorian mode have become the top dog, if the composers had simply decided to ode it more? Or is the human ear, possibly the human psyche, more attuned to what we ended up with and it GUIDED music's evolution? Well, since other cultures use different musical systems, it seems obvious that it could have gone in other directions.

Familiarity is probably the biggest factor in music preference. Many, many people have thought there's no reason at all that music can't follow other rules. Debussy, Ives, Bartok, Coltrane... Different ways of establishing tonal centers, different groups of notes "allowed", different types of objects producing the sound... The more you hear it, the more likely you are to like it, even if you prefer one persons compositions in a given tonality over another's. (I love the Beatles, not so crazy about Motley Crue.)

Problem is, so many want to invent their own, unique system of music. So there's not as much opportunity to become familiar with it, and you can't come to like one composer's style over another's. We have to start from scratch with each composer.

[aliantha]

The "Sammy key"? That's what happens when you post from a glorified phone!

I like Mixolydian mode. Mostly because it's fun to say.

But seriously, folks, I prefer listening to medieval and Renaissance music over most of the Romantics. Yes, I know I'm weird.

[Hashi Lebwohl]

But seriously, folks, I prefer listening to medieval and Renaissance music over most of the Romantics. Yes, I know I'm weird.

I have introduced the kids to Medeival and Renaissance music from time to time, letting them hear it performed. They may not develop a lifelong interest in it but at least they have heard it.