The Saddest Thing I Know about the Integers
December 1, 2014 12:52 PM   Subscribe

The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.

I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
posted by jenkinsEar (93 comments total) 51 users marked this as a favorite
 
The views expressed are those of the author and are not necessarily those of Scientific American.

Scientific American believes the saddest thing about integers is when seven ate nine.
posted by InfidelZombie at 1:01 PM on December 1, 2014 [52 favorites]


But would Lie Bot agree?
posted by tommasz at 1:03 PM on December 1, 2014 [2 favorites]


The 12TET perfect fifth is close enough for my tastes. I guess that's the one that a string player would focus on, but it's the thirds that I've always found tragically diminished (no pun intended) by their readjustment. Justly-intoned thirds are just amazing and vital things. Equally-tempered, well, they're fine for establishing chord quality or the qualia of the currently-sounding harmony, but they just don't have much in and of themselves that makes them compelling.
posted by invitapriore at 1:05 PM on December 1, 2014 [6 favorites]


Doesn't this article neglect the fact that sounds cannot be distinguished beyond a certain precision. The frequency of a note does not need to go beyond a fixed number of decimal places.
posted by humanfont at 1:06 PM on December 1, 2014


I know it's probably irrational but the more I read about music theory the more angry I get. Between this and a post a while back with "accessible" music theory lessons I just get the overwhelming notion that music theory is a big fancy framework built solely on human perception, like a scaffold built on Jell-O. Sure, it's fun to talk about fifths and thirds and octaves, but that's all based on what sounds nice to the human ear, which of course doesn't much care about how exactly perfect a given frequency is. So while it's useful to know what combinations of notes sound good and how to manipulate them, applying the sort of math you see in the hard sciences is pointless. Oh, and while we're at it, all this again but this time color theory.
posted by Mr.Encyclopedia at 1:07 PM on December 1, 2014 [19 favorites]


humanfont: Doesn't this article neglect the fact that sounds cannot be distinguished beyond a certain precision. The frequency of a note does not need to go beyond a fixed number of decimal places.
You're talking quantization error, which is a heavily-studied idea, and of course my ears have a wholly different quantization level from those of someone with perfect pitch.

Short answer: the quantization level of the typical, musically-educated human listener is lower than the errors in estimating perfect fifths while maintaining octave doubling. Oops, that's not really short.

Shorter answer: we can hear the problem, so the digits matter.
posted by IAmBroom at 1:13 PM on December 1, 2014 [5 favorites]


... where "we" does not include me, but does include many people I know.
posted by IAmBroom at 1:14 PM on December 1, 2014 [1 favorite]


Yes this is an absolutely ridiculous statement to make. Before seeking to apply math to music one might be better advised to apply physics to music, because that is the reason you can apply math to music. And in physics you quickly learn about things like error flags and precision, because you need them to make sense of even the simplest experiments.
posted by localroger at 1:15 PM on December 1, 2014 [2 favorites]


BTW, this is one reason why unaccompanied choirs tend to run flat. If you tune all your fifths and thirds perfectly, in the standard western music cadences, the adjustments tend to pull you flat over time. The usual adjustment is to try and pull all your passing tones slightly sharp to compensate. (If you want to hear that done masterfully, check out the Tallis Scholars' performance of William Byrd's Mass in Four Voices.) It's very frustrating! And yes, you can totally hear the difference.
posted by KathrynT at 1:18 PM on December 1, 2014 [14 favorites]


Mr.Encyclopedia: I know it's probably irrational but the more I read about music theory the more angry I get

What's interesting is that what makes you angry about music is exactly why this topic makes it so fascinating for me. If you're really into the mathematics of music, one of my favorite topics is the concept of alternate scales, such as the Bohlen-Pierce Scale which have 13 steps to a tritave instead of the usual 12 to an octave. Learning about these and other aspect theories give you an idea why the human ear interprets certain sounds as "pleasing" while others sound more "jarring". It's not really as arbitrary as one could be led to believe by only taking a superficial study of the matter.
posted by surazal at 1:18 PM on December 1, 2014 [3 favorites]


You're talking quantization error

No he is not; he is talking measurement error. Quantization is a form of measurement but not all measurement is quantization, and all measurement is subject to error.

The error in your own hearing cannot meaningfully be called quantization error. The signal is actually a combination analog responses by fibers tuned to respond to various frequencies, which are then blended by a processing schema that is not completely understood but which discards quite a bit of information before reporting to the brain. People with perfect pitch are able to discern which absolute fibres are being stimulated. This information is collected by everyone's ears but in people without perfect pitch it is discarded at some point in the processing chain which results in perception.
posted by localroger at 1:20 PM on December 1, 2014 [4 favorites]


Sure, it's fun to talk about fifths and thirds and octaves, but that's all based on what sounds nice to the human ear, which of course doesn't much care about how exactly perfect a given frequency is.

The ear does care, within limits. The fact that the piano can't get tuning perfect doesn't mean that it can't be done more closely. String players and choral singers who've worked in a tight ensemble -- or even attentive listeners -- have a feel for the difference between the tonal pallete that's available when you can dial in your chords more closely than equal temperament allows.

It can be a subtle difference, and whether it's "better" is arguably an aesthetic judgment, and it's also somewhat subject to the listener's acuity, attentiveness, and training, but it's there.
posted by weston at 1:25 PM on December 1, 2014 [3 favorites]


localroger, I'll defer somewhat to most of your arguments there (I still think it's a sort of quantization error, but the difference may be in context), but do you have a citation for this claim?

localroger: People with perfect pitch are able to discern which absolute fibres are being stimulated.
posted by IAmBroom at 1:25 PM on December 1, 2014


I know it's probably irrational but the more I read about music theory the more angry I get.

In my case, I know my anger is irrational. I keep trying to understand music theory, and keep utterly failing, and I'm completely unable to put my finger on what goes wrong in my mind when I try to understand it. It's like I'm trying to answer questions that I can't quite express, and the answers I see almost, but don't quite fit the questions I'm trying to form.

Almost everything else I've ever tried to learn I've been able to grasp (even if I quit trying before reaching full understanding, which is most of the time), but music completely baffles me. I like music, but don't understand it at all, and it drives me crazy.
posted by Ickster at 1:29 PM on December 1, 2014 [3 favorites]


I keep trying to understand music theory, and keep utterly failing, and I'm completely unable to put my finger on what goes wrong in my mind when I try to understand it.

You basically can only understand it by making music. You have to get a synthesizer out or something and play around with notes and chords and so on.

Imagine trying to understand chess strategy without ever playing it.
posted by empath at 1:32 PM on December 1, 2014 [5 favorites]


Oh, that's the saddest thing you know about the integers? Not that fact that so many of them are racists?
posted by PlusDistance at 1:36 PM on December 1, 2014 [7 favorites]


The book that made sense of music theory for me was The Music Instinct, which I recommend to anyone interested in learning more. It talks about the issue of temperament and also touches on completely different alternative scales, like the ones used by gamelan orchestras.

It's helpful if you think of music theory not as The Theory of Music, but as A Theory of Music. It's one possible semi-mathematical construct that's genuinely useful in analyzing music and guiding composition. You can't get it to be a completely hard science, because it's a field where how we perceive a thing is more important than how it actually is.
posted by echo target at 1:37 PM on December 1, 2014 [3 favorites]


This post is awesome. It reminds me of a fun and frustrating night in college when three friends and I were up until sunrise trying to map the relationship between different tunings (Pythagorean, equal temper, some others) as functions.
posted by doteatop at 1:38 PM on December 1, 2014 [1 favorite]


Then there are those of us who make music on the regular, and have a strong innate understanding of what works by ear, and who still don't understand a lick of theory...
posted by stenseng at 1:38 PM on December 1, 2014 [3 favorites]


Honestly, I question the value of being perfectly in tune. Most musicians warp notes constantly, either live or during production. One of the more interesting things about music theory is that what makes songs good is often how they break the rules.
posted by empath at 1:38 PM on December 1, 2014 [1 favorite]


I just get the overwhelming notion that music theory is a big fancy framework built solely on human perception

It's tempting to think of Music as some platonic-realm thing - It might be the art most open to formal description with mathematics, but that doesn't mean the source of musical beauty, creativity or enjoyment is inherent to the mathematics itself. It's still an interaction of perception, culture, education, and biology, like the other arts, at least so far as I can tell.
posted by Jon Mitchell at 1:43 PM on December 1, 2014 [4 favorites]


I vastly prefer the older Chinese approach (prominent before the immersion in Western musical influence) where they addressed the problem not by increasing the number of notes but minimizing the variety of ratios (as European music convention is fond to do), but rather arranged the scale to have as much variety of ratio available in a small number of notes.
posted by idiopath at 1:46 PM on December 1, 2014


localroger: Yes this is an absolutely ridiculous statement to make. Before seeking to apply math to music one might be better advised to apply physics to music, because that is the reason you can apply math to music. And in physics you quickly learn about things like error flags and precision, because you need them to make sense of even the simplest experiments.

In physics, you also learn about the obvious audible effects that are produced when two sounds have slightly different frequencies. And when you go around the circle of fifths, you're glad that they keyboard is tuned in 2^(1/12) intervals so that you end up right back where you started, instead of 1% off.
posted by clawsoon at 1:49 PM on December 1, 2014 [2 favorites]


Then there are those of us who make music on the regular, and have a strong innate understanding of what works by ear, and who still don't understand a lick of theory...

You have to put concerted effort into matching up the terminology with what you're doing, I think. It's not like you can read a book on music theory, then play some music and you'll get it. You have to read a chapter, play around on a piano, read another one, play some music, then maybe go back to a previous chapter, maybe take a class. It's not at all a natural thing, it's just a lot of studying and practice.

Based on every conversation I've had with a profressional musician, though, formal music theory is fairly useless for actually making music, though it does make it easier to talk about making music.
posted by empath at 1:50 PM on December 1, 2014 [2 favorites]


You basically can only understand it by making music.

I took a year of piano lessons a little while back; I enjoyed playing, but it did little for my understanding of the theory. I had some moments where I thought I understood things, but it always fell apart.

I'm not sure what I want to understand, which is a big part of the problem. All of my mental grasping at music feels like looking at a long series of letters which might be gibberish or might be a scrambled message, and I keep seeing patterns which suggest what the message might be, but am never able to solve it.

Hard to explain better than that.
posted by Ickster at 1:51 PM on December 1, 2014 [1 favorite]


Music theory is just a way of abstracting patterns that commonly occur in music. It's like math in that way; at some point someone realized that two apples and two oranges had something in common, and that commonality was "two".

Music theory starts to happen when you realized that, huh, I can sing part of Don't Cry when I play the chords for I Will Survive. If you go nuts about identifying those kinds of similarities, you'll discover that - again, just like math - a whole language has been created to describe and talk about those abstract similarities. That language is music theory.
posted by clawsoon at 2:03 PM on December 1, 2014 [8 favorites]


huh, I can sing part of Don't Cry when I play the chords for I Will Survive.

Or when you realize while idly listening to the classical music station on your drive home that the several-hundred-year-old Folia Ground has the same chords as Britney Spears' Oops I Did It Again. Or that the Pachelbel Canon is everything.
posted by KathrynT at 2:17 PM on December 1, 2014 [6 favorites]


So that's why I could never tune my guitar without an electronic thingy!
posted by infinitewindow at 2:34 PM on December 1, 2014


I'll confess to having set somebody's favourite Christmas poem to a tune that sat atop Pachelbel's chords for a Christmas pageant way back in high school. I have added to Rob Paravonian's paranoid fantasies.
posted by clawsoon at 2:37 PM on December 1, 2014 [3 favorites]


KathrynT: About La Folia, and its staggering number of appearances/variations...
posted by seyirci at 2:38 PM on December 1, 2014 [2 favorites]


empath: "Based on every conversation I've had with a profressional musician, though, formal music theory is fairly useless for actually making music, though it does make it easier to talk about making music."

I think it's actually very useful for making music, or at least during a certain stage of that creation for a certain kind of music (specifically, music written in a novel or idiosyncratic idiom). For me, the work involved in writing that type of music is basically a cycle that looks like:

1. Generate a seed of an idea intuitively.
2. Analyze that seed for organizing principles.
3. Develop the seed intuitively, but with those principles in mind.
4. Go to (1).

Step three may sound like a contradiction in terms, but it's really not any more self-contradictory than the notion of improvising over chords. Anyway, music theory really comes in handy at step two, where for instance I might discover that a passage I wrote is entirely drawn from the octatonic scale. That doesn't bind me to continue in that vein, but knowing of the existence of that structure and knowing that it underlies what I've already written helps me keep control over the extent to which I give off the sense that what I'm writing derives from a consistent musical idiom, even when it's not an idiom that I'm already familiar with. Trying to enforce that consistency entirely through intuition is exhausting and untenable when you don't have some comparatively long-ranging tradition to draw from,* as you might if you were writing a rock song or a hip-hop beat.

* I think there's some expectation, some vestigial Romantic thought-appendage, that a composer be sufficiently visionary so as for this to not be the case, and for sure I think if you've spent a long time fashioning your own idiom it probably stops being the case, but one of my biggest moments in learning to write music was when a teacher of mine, a really good and original composer, found out that I was writing so slowly because of the fact that I was trying to work out the whole piece from intuition. She told me the above, basically, and it really opened the floodgates.
posted by invitapriore at 2:51 PM on December 1, 2014 [8 favorites]


I know it's probably irrational...
Of course it is, when you're dealing with 2(1/12)
posted by MtDewd at 3:00 PM on December 1, 2014 [13 favorites]


Music is in fact a framework that emerges around perception. We have 16,000 to 20,000 cilia but they are not like tuning forks; a pure frequency will stimulate a group in a distribution pattern, with a maximum that can be estimated more closely than the individual cilia sensitive frequencies. That's why the ear is so sensitive to pitch and obvious to anyone who's designed instrumentation.

The entire hearing apparatus is very sensitive to harmonic relationships, to the point where if we generate an artificial sound containing all the harmonics but not the fundamental frequency we will hear the missing fundamental as the pitch of the sound. It's not known whether this sensitivity is due to the physical shape of the cochlea or hard-wired neurology or learned, but it makes a lot of sense because the harmonic structure of a sound is very informative as to its source, so it's a sensible thing to examine and be attentive to.

Actual music appears to involve patterns of sound simultaneously stimulating several areas of the brain connected with pleasure, but part of this process is arbitrary and learned because there are different tonal scales and different styles of music which sound preferentially "musical" to someone who is used to them. This is why dwelling on the math is kind of silly, because there is no math (except possibly a basic sensitivity to harmonic relationships) being exercised in our musical perception.

Western culture has had the technology for quite a while to explore very fine shades of musical nuance, and it turns out that there are some relationships which can separately be perceived as "perfect" but which cannot be experienced at the same time because they aren't quite compatible. This is why a particular tuning scheme works for some note combinations and not others, and it has less to do with math than with a collection of sensors designed to do different things all being brought to bear at the same time.
posted by localroger at 3:19 PM on December 1, 2014 [4 favorites]


My big question with music theory has always been whether there is some intrinsic perceptual reason to prefer e.g., octaves and perfect fifths, or whether our preference for these is instead the product of familiarity with these conventions. And it's irritating because it seems that none of the texts will go into this question, but rather assume that this is some obvious fact that is beyond discussion.
posted by Pyry at 3:20 PM on December 1, 2014 [2 favorites]


I guess that's the one that a string player would focus on, but it's the thirds that I've always found tragically diminished (no pun intended) by their readjustment. Justly-intoned thirds are just amazing and vital things.

In my experience non-fretted stringed instruments do care about thirds a lot, especially the higher members of the family. Chances are pretty good that any big cadence a string ensemble is playing isn't going to be played in equal temperament, and so the higher voices have to know which way to nudge their thirds.

Based on every conversation I've had with a profressional musician, though, formal music theory is fairly useless for actually making music, though it does make it easier to talk about making music.

Not really, take the example I just gave, in order to tune the chord it helps if each member of the ensemble knows if they're playing a root, third, or fifth, and then knows enough to figure out how to adjust as a result. On top of that, there are different tweaks for a half cadence than a full cadence. So they need to know that as well. I mean, yeah at some point players just do this by ear, but knowing WHY they do this helps them do it at the right point. Also, having a strong knowledge of harmony and forms and such helps figure out more effective phrasing, gives you a good idea of your role within the ensemble, and actually really helps take the pressure off of figuring out what notes you're playing and informs how you play them.

You can be a great musician without knowing a lick of theory, but theory's a great tool for figuring out a piece of music you want to play.
posted by Gygesringtone at 3:26 PM on December 1, 2014 [1 favorite]


Why is it when I listen to the various versions of The Well Tempered Clavier via this Wikipedia page about tuning systems, that I find "equal temperament" much easier to listen to than the other three examples cited on that page? Even if there's no such thing as a perfect 12-note scale, that tuning system seems to sound "right" to me.
posted by not_on_display at 3:31 PM on December 1, 2014


Why is it when I listen to the various versions of The Well Tempered Clavier via this Wikipedia page about tuning systems, that I find "equal temperament" much easier to listen to than the other three examples cited on that page?

Bach wrote the Well Tempered Clavier to show off what sort of neat musical tricks he could do in a system where every key sounds good. It's specifically written to sound good in a tuning system like equal temperament.
posted by Gygesringtone at 3:35 PM on December 1, 2014 [4 favorites]


I've read that, as an empirical or psychoacoustic fact, people (acculturated to Western music) find the sound of thirds out of tune more unpleasant than they do the sound of fifths or octaves out of tune. We are more forgiving of little differences in an octave. So, piano tuners created a practice of "borrowing" pitches from octaves or fifths to sweeten the thirds. In doing this, they deviate from what theoretical equal temperament would be. I think all this is fascinating, and I now kind of want to learn to tune pianos. Another issue is that the harmonics of the very low strings influence our perception of pitch more than the fundamental, and some of the harmonic series is sharp to the fundamental. So, they have to tune the low strings flat. Or maybe it's the other way around. I think there is a similar issue with the very high strings too, because, as they get very thin, they deviate from what an ideal, spherical cow, vibrating string does, just like the very heavy bass strings.
posted by thelonius at 3:59 PM on December 1, 2014


By the way, the deal with 5ths and 3rds being non-commensurable is why it is actually wrong wrong wrong to tune a guitar by using the harmonics at the 5th and 7th fret, as many people like to do. If you do this perfectly, then play a big ol' E chord, it will sound horribly out of tune, because the fretted G# will be way out.
posted by thelonius at 4:11 PM on December 1, 2014 [3 favorites]


It's specifically written to sound good in a tuning system like equal temperament.

Yeah, you wouldn't play a piece of music like that using Pythagorean or just tuning. I'm surprised they don't have an example in a "well" temperament, but not_on_display, you can hear a comparison between equal and well temperament here - it's more subtle. (Well is unequal but fudged in a sort of ad hoc way, so there are lots of different well temperaments.)
posted by en forme de poire at 4:26 PM on December 1, 2014 [1 favorite]


We have 16,000 to 20,000 cilia but they are not like tuning forks; a pure frequency will stimulate a group in a distribution pattern, with a maximum that can be estimated more closely than the individual cilia sensitive frequencies.

Are you referencing the stereocilia of the hair cells in the cochlea? If that is the case, we have many more than that. Outer hair cells seem to be the cochlear cells that are frequency sensitive and tonotopically organized. We have ~12,000 of them, each with any number of stereocilia that rest on top, acting as the organelles which are bent based on the movement of the basilar membrane, pulling tip links that connect the stereocilia, which open the ion channels that create the action potential to the brain which is perceived as sound.

Inner hair cells aren't really thought to have so much of a frequency-specific role. They have a much greater role in innervation and possibly some other things.

It's true that each hair cell is not tuned for some specific frequency. The cochlea is more like a set of band pass filters, like 1,500 of them. Some pure sine wave will excite a place on the cochlea, but not some exact number of hair cells. Really the band pass filter is a good analogy, because there is a sort of compressive, dropping off nature to the tonotopic mapping of the cochlea.

You're talking quantization error, which is a heavily-studied idea, and of course my ears have a wholly different quantization level from those of someone with perfect pitch.

Mmm, not exactly. This goes back to the cochlea thing. We do indeed experience a certain 'quantization,' but there's two aspects to it. Physically all of our ears are relatively similar, so on the ear level (peripheral hearing level), any person with healthy hearing will have similar quantization (as you call it), or, more accurately, frequency resolution. The average person can resolve about 1500 pitches from 20 Hz to 20,000 Hz. In music terms, a normal hearing person cannot resolve a pitch smaller than about 1/10th of a step. Now, that's a pretty damn small interval, so the slightly off tuning of the piano is certainly an issue.

Now as to whether someone less trained would notice a difference? Maybe not - but that's a brain thing, at the cortex level. Your auditory cortex is tonotopically organized as well (to a degree), and you can certainly learn to be better at perceiving small frequency discrepancies. Some people are sort of born better at it, but you can be trained.

My big question with music theory has always been whether there is some intrinsic perceptual reason to prefer e.g., octaves and perfect fifths, or whether our preference for these is instead the product of familiarity with these conventions.


The short answer is that it's learned.

The thing about octaves is that they are really the only interval that seems to be somehow more innately pleasing and intuitive than any other interval. There is something about that 2:1 ratio that is universal. Every single music, for example, across the entire world has scales that are based on the octave. All the other intervals are up for grabs, and are thought to be basically learned; that is, if you find the third and the fifth pleasing, that's more to do with the culture you were brought up in rather than any inherent trait to those intervals per se.

There was a time when there was a lot of focus in the West on trying to figure out a hierarchy of intervals, to come up with psychological theories about why certain intervals were pleasing and others weren't. But of course this was a bit backwards - in the West we are inundated with music that follows a very narrow structure from the time we are born, so it was a bit post facto to say, hmmm, why do we like those sounds? The reason is really that those are the sounds that formed deep networks in our auditory cortex when our brains were very plastic and it sticks with us. None of the music perception people who were working on this problem some 40 years ago stopped to think about why it might be that people in non-Western cultures enjoyed entirely different harmonic structures.

Based on every conversation I've had with a profressional musician, though, formal music theory is fairly useless for actually making music, though it does make it easier to talk about making music.

I once had a composition teacher who was fond of saying that when composing you never use theory until you get stuck - then it's like a little toolbox of what you might do next.

I mean, music theory is a huge field with lots of approaches, so it's hard to group it all together. Lots of theoretical approaches look at spectral content and such, and go far beyond the sort of chord change theory many are familiar with. So, eh. But generally, most theory is just that - one theory of trying to understand how some piece of music is put together. A lot of the theoretical concepts that have stuck over the past few hundred years are just conventions that, for one reason or another, have been deemed useful. Lest us never forget that, for the Greeks, the notes we think of as high were called low and vice versa.
posted by Lutoslawski at 4:31 PM on December 1, 2014 [13 favorites]


Wait, *that's* what this person think is the saddest thing about integers? How about the fact that Hippasus of Metapontum was brutally murdered for daring to speak the truth about integers?

Integers kill. They've killed before.

Oh, but your piano uses the equal-tempered scale? Sure, yeah. I weep for you.
posted by kyrademon at 4:35 PM on December 1, 2014 [2 favorites]


Integers don't kill people, crazed Pythagorean fanatics armed with integers kill people.
posted by localroger at 4:39 PM on December 1, 2014 [2 favorites]


I was once a music (piano performance!) student at Gordon College, a Christian liberal arts school located in Wenham, Massachusetts, which is perhaps the least urbanized of the towns that make up Boston's North Shore.

It was a bad time in my life, as I discovered that I lacked the discipline needed to be a good pianist, or even a good student, and concurrently learned that I was not, in fact, a Christian. I found that I possessed little innate capacity for spirituality once removed from the social influences of my childhood and adolescence.

Just before I failed out of college for the final time, I entered the auditorium of Phillips Music Hall to find that one of the college's prized full grand pianos, which had been tuned and used for a performance the evening before, remained unlocked. I tarried there for a bit, held by a desire to play just a few notes on this new instrument1, but at the same time painfully aware that I did not deserve to do so.

A fellow student at the college, named (if memory serves) Andre, who was a brilliant and committed pianist2, came up next to me and suggested I play a few bars. When I demurred, noting that it was obviously not there for me to play, he offered to serve as look-out.

This is all prelude to the moment I sat down, put my fingers on the keys, and played a D-Major root triad. I shivered, and nearly cried, at how good it sounded. I had never made music so beautiful as I did in that moment, and I expect that I never will again.

After I finished playing Andre told me that he could hear the despondence in my improvisation, and asked me what was up. It was, I think, the first time I admitted to anyone, including myself, that I simply wasn't cut out for college.
---
  1. It's nearly a compulsion for me to attempt to play any unlocked (and not conspicuously marked "do not play") piano, although my repertoire has shifted over the years. These days, if muscle memory serves, I'm likely to attempt everything up to the B♭-Major section of the FFVI Piano Collections version of Terra's Theme, or The Mamas and the Papas' California Dreamin'.
  2. ...and also as good or better a poor-man's basso profundo in the college choir as I was, although both of us were reduced to sotto voce below the low E♭.
posted by The Confessor at 4:39 PM on December 1, 2014 [17 favorites]


Lutoslawski: "The thing about octaves is that they are really the only interval that seems to be somehow more innately pleasing and intuitive than any other interval."

The logarithmic scaling of pitch perception is undeniable, but in truth this has much less to do with intervals per se than it does with the relative affinities between an interval and the spectral characteristics of the device with which it was sounded. I suspect a culture which by some turn of luck ended up with a bunch of sound-producing devices that produced dissonant octaves would not regard even this venerable interval as especially central to their conception of consonance.
posted by invitapriore at 4:43 PM on December 1, 2014 [3 favorites]


The thing about octaves is that they are really the only interval that seems to be somehow more innately pleasing and intuitive than any other interval. There is something about that 2:1 ratio that is universal. Every single music, for example, across the entire world has scales that are based on the octave. All the other intervals are up for grabs, and are thought to be basically learned; that is, if you find the third and the fifth pleasing, that's more to do with the culture you were brought up in rather than any inherent trait to those intervals per se.

Well the thing about octaves of course isn't just that they are a "pleasing" interval but that the octave is where the cyclical equivalence of pitch - which does seem to have some basis in the organization of the inner ear though I forget exactly how it's thought to work - happens. I was under the impression that the significance of 5ths is pretty close to universal too, though.
posted by atoxyl at 4:46 PM on December 1, 2014


Well the thing about octaves of course isn't just that they are a "pleasing" interval but that the octave is where the cyclical equivalence of pitch - which does seem to have some basis in the organization of the inner ear though I forget exactly how it's thought to work - happens.

Mmm, yeah, but there's not really a great intrinsic reason why a doubling of a frequency should equate to the perception of a similarity of pitch, that's the rub. It's a perceptual phenomenon that is somehow innately cognitive, but no one knows why (yet). It has nothing to do with the inner ear, however.
posted by Lutoslawski at 4:51 PM on December 1, 2014 [1 favorite]


I suspect a culture which by some turn of luck ended up with a bunch of sound-producing devices that produced dissonant octaves would not regard even this venerable interval as especially central to their conception of consonance.

That's a very interesting little thought experiment which I will have to think on a bit. Too bad we can't do a study where we have a group of children raised only in an environment with a microtonal Moog.
posted by Lutoslawski at 4:53 PM on December 1, 2014


Lutoslawski, does the perception of consonance really have nothing to do with the harmonic series, and that the first three overtones of a note are octaves and fifths (with the third being #4)?
posted by en forme de poire at 5:02 PM on December 1, 2014 [1 favorite]


Wow this thread is amazing.
posted by humanfont at 5:06 PM on December 1, 2014 [4 favorites]


I should rtfa but something doesn't make sense to me: if the octave interval is all doubling, then how is it that we can't get a piano "in tune"? Is it just that the exact frequency becomes too difficult to accurately implement in the instrument (like 440 and 880 are ok, but 934.3 repeating is tough to nail)?

Also I had a band recently who asked me to tune my piano (err KORG organ) to A442 was it? Or 430? Anyway, it made for an interesting first 30 seconds when I forgot to change the tuning in that first song...
posted by joecacti at 5:10 PM on December 1, 2014


I suspect a culture which by some turn of luck ended up with a bunch of sound-producing devices that produced dissonant octaves would not regard even this venerable interval as especially central to their conception of consonance.

Yeah, but the first sound-producing device any culture has is the voice, and humans seem to be wired to recognize the kind of octaves we've produced. I don't think that humans could actually produce a culture that produces a dissonant octave.
posted by Gygesringtone at 5:13 PM on December 1, 2014


It is not possible to make a piano "perfectly in tune" because it is not mathematically possible to reconcile having the octave as a perfect 1/2 ratio with a fifth as a perfect 2/3 ratio. If you run through the circle of fifths with perfect fifths, you miss your octave stop by a good margin when you get to the end.
posted by NMcCoy at 5:23 PM on December 1, 2014 [3 favorites]


The emphasis on doubling could very well be learned because nearly all sounds are rich in harmonics, so multiples are nearly always heard in some pattern or other along with any base frequency. One reason music seems to be so affecting is that sound is associated with brain areas where learning can occur very quickly, which is useful because transient sounds can be very important in nature.
posted by localroger at 5:23 PM on December 1, 2014


I should rtfa but something doesn't make sense to me: if the octave interval is all doubling, then how is it that we can't get a piano "in tune"?

You can tune octaves no problem. The problem is that a perfect fifth is a 3:2 ratio, and when you stack 12 of them on top of each other, you don't arrive back where you started from.
posted by KathrynT at 5:24 PM on December 1, 2014 [3 favorites]


And 12 fifths =/= 2x isn't a problem with math, and certainly not with irrational numbers; it's a problem because we have two pleasure-eliciting relationships that we are trying to use at the same time. But our fondness for fifths is very likely learned and arbitrary and our fondness for octaves might even be learned, although almost inevitable in any normal setting of background sounds.
posted by localroger at 5:27 PM on December 1, 2014 [2 favorites]


Mmm, yeah, but there's not really a great intrinsic reason why a doubling of a frequency should equate to the perception of a similarity of pitch, that's the rub. It's a perceptual phenomenon that is somehow innately cognitive, but no one knows why (yet). It has nothing to do with the inner ear, however.

I really meant auditory cortex and I though I had read that there was a partial understanding of how pitch classes map in the brain but I can't find anything very good at all on the subject right now so maybe not.

The logarithmic scaling of pitch perception is undeniable, but in truth this has much less to do with intervals per se than it does with the relative affinities between an interval and the spectral characteristics of the device with which it was sounded. I suspect a culture which by some turn of luck ended up with a bunch of sound-producing devices that produced dissonant octaves would not regard even this venerable interval as especially central to their conception of consonance.

Doesn't this ultimately come down to the intervals that are present in the combined spectrum of the base note and the octave up played together on this "instrument?" Is there another framework for predicting whether things will be consonant or dissonant? Am I missing something about how this example was constructed?
posted by atoxyl at 5:42 PM on December 1, 2014


And here I thought it was just that One was the Loneliest Number.
posted by chicobangs at 6:06 PM on December 1, 2014


Lutoslawski: Mmm, yeah, but there's not really a great intrinsic reason why a doubling of a frequency should equate to the perception of a similarity of pitch...

This seems flat-out wrong to me. Doesn't pretty much any instrument outside of a pure sine wave generator produce a tone an octave up as one of its overtones? So an octave gives the impression of similarity of pitch because it's amplifying a sound already present in the original tone.

That seems pretty intrinsic to me. Same thing with fifths: The second overtone is a natural fifth, so when you add a fifth, you're emphasizing a sound that's already in the original tone.

We might not know why we find that similarity pleasing, but we definitely know why they sound similar.
posted by clawsoon at 6:22 PM on December 1, 2014


thanks for sending me down a k-hole of alternate musical scales. The 31-equal-temperament death metal guitar shredding may have been the best.
posted by jepler at 7:57 PM on December 1, 2014 [2 favorites]


Doesn't pretty much any instrument outside of a pure sine wave generator produce a tone an octave up as one of its overtones? So an octave gives the impression of similarity of pitch because it's amplifying a sound already present in the original tone.

Well, yeah it's true that the octave is almost always a prominent overtone when you have some periodic wave in a tube or on a string attached to some resonant thing. But that doesn't have anything to do with why we perceive octaves as being particularly consonant sounding, or phenomenally the 'same pitch' as their fundamental. I mean, consider a pure sine wave. You're absolutely still going to perceive a 1000 Hz sine wave as having something inherently the same about it as a 2000 Hz sine wave that has nothing to do with an overtone series. The phenomenon isn't limited to instruments at all.

Or consider a situation in which you don't have the octave played together. You aren't 'amplifying' anything. You still have that same perception. Or even consider something that isn't so predictably resonant. Consider the voice - depending on the vowel, you aren't going to get as clear octave overtones as you will with, say, a cello (outside of the first formant). You're still going to get that quality of sameness. Or consider narrow band noise. Imagine you have all frequencies from 800-1000 Hz, evenly distributed. I guarantee you're going to experience the octave phenomenon if you then hear narrow band noise at 1600-2000 Hz.

I mean, frequency and pitch are related but distinct things. Pitch is the psychological correlate. You have a frequency and 2x that frequency and they are perceived as fundamentally related pitches. I mean, it's very weird. But why do we have this effect with 2x and not 3x? Or 3.5x? Or whatever? It isn't so clear. I get that it seems intuitive that it ought to be this way on some level, but I think we are just so used to thinking about octaves that we forget that there isn't really some inherent thing about them that should make the octave phenomenon such a universal thing.

And this leads a bit to: Lutoslawski, does the perception of consonance really have nothing to do with the harmonic series, and that the first three overtones of a note are octaves and fifths (with the third being #4)?

No, I don't think it does. There was definitely a time in music cognition circles where there was a lot of emphasis on trying to use the harmonic series as some sort of reasoning for why we perceive certain things as pleasing or consonant. But...it breaks down pretty quickly, considering that 1) you don't have to go very far in the harmonic series to get some pretty dissonant intervals, 2) the intervals we have sensed as being consonant have changed over time (namely the fifth), 3) the perception of consonance among intervals, outside of the octave, is not universal across cultures. I don't know many who argue now that there is a sort of fundamental basis for harmony perception rooted in overtones. I think the consensus seems to be that overtones are really timbre and little more (I say little as if timbre is not like maybe the most important factor in music).

but it makes total sense to me that our brains lump a doubling of frequency together because it helps us perceive the sounds that physical objects make as being a single sound, not multiple sounds.


The way the central auditory process groups sounds together is actually extremely, extremely complex, and actually not super well understood. In fact, it seems that frequency has much less to do with it and timing (namely phase-locking) and directionality are actually the key bits of information our brain uses in (what is technically called) Auditory Scene Analysis, or the phenomenon of hearing all these different complex signals and being able to say, oh, these are coming from that thing, and these are coming from this other thing.*

I actually think this is going to be an area of cognitive science/neuroscience/audiology/etc. that may break open in the next ten or twenty years, as our imaging and such gets better and better. We know surprisingly little about why we perceive sounds the way we do - we have taken a lot for granted over the years it turns out.

* Interestingly enough, there's mounting evidence that not all people are equally adept at doing this kind of auditory scene analysis, and it may turn out actually that some people who have language or learning disabilities may actually have deficits in their ability to do this kind of aural processing.
posted by Lutoslawski at 8:01 PM on December 1, 2014 [6 favorites]


the intervals we have sensed as being consonant have changed over time (namely the fifth)

Can you go into detail about this? I was taught that the fourth and fifth have been considered consonant for a very long time in Western music but thirds only (relatively) more recently.
posted by atoxyl at 8:08 PM on December 1, 2014


It's not just the generation of overtones but the ability to resonate at f, 2f and so on. Those cilia in your ear that react most strongly at a particular frequency will also react strongly at a variety of integral multiples of half the wavelength. That's purely mechanical, so the brain will be able to spot which components of the audio landscape are probably related - and are more likely to come from the same source. This improves the information we can get - processing gain for a particular channel amid competing noise - and it's not surprising that we like to pay attention.
posted by Devonian at 8:14 PM on December 1, 2014 [1 favorite]


I guess it depends on if you mean harmonizing with the fourth and the fifth or moving to them in relation to the tonic in some key. Certainly the fourth was a popular harmonizing interval in very early music, but you rarely hear it as a harmonizing interval by the baroque period. Same with the fifth outside of its triadic use. Certainly you would never hear fifths moving together in counterpoint, and rarely would you even land on a fifth. Depending on who you talk to, the fifth as a sort of destination point has been the major theme throughout Western music, but it's role, especially in like a discrete sense, has evolved quite a bit.

Those cilia in your ear that react most strongly at a particular frequency will also react strongly at a variety of integral multiples of half the wavelength.

This isn't true. Outer hair cells (I'm assuming you mean the stereocilia on the outer hair cells) are literally arranged as filters from high frequencies up the cochlea to low. Perception is cognitive, not mechanical, and frequency has little to do with placing sounds.
posted by Lutoslawski at 8:19 PM on December 1, 2014


I guess I should further qualify the 'fourth rarely heard as a harmonizing interval by the baroque period' by saying that rarely might have been strong, and of course you hear it as a harmonizing interval at times in a fugue, and not just in passing, but I think that's a special case.
posted by Lutoslawski at 8:21 PM on December 1, 2014


In fact, it seems that frequency has much less to do with it and timing (namely phase-locking) and directionality are actually the key bits of information our brain uses in (what is technically called) Auditory Scene Analysis, or the phenomenon of hearing all these different complex signals and being able to say, oh, these are coming from that thing, and these are coming from this other thing.*

In a lot of electronic music production, placing sounds in some kind of realistic psycho-acoustic space is super important, actually. So much time is spent on reverb, delays, panning, and so on, to make the song sound like it's coming from a real space, even if the sounds themselves are completely unreal. I've had a few producers tell me that they want songs to sound like they're coming from the jungle, or some other natural space. Melody in stuff like house music or techno is a factor, but it's mostly about rhythm and timbre, not melody.
posted by empath at 8:45 PM on December 1, 2014


The average person can resolve about 1500 pitches from 20 Hz to 20,000 Hz. In music terms, a normal hearing person cannot resolve a pitch smaller than about 1/10th of a step. Now, that's a pretty damn small interval, so the slightly off tuning of the piano is certainly an issue.

Another fact, though, is that if you play two tones of nearly equal frequency, you will hear 'beats' caused by their difference in frequency. For instance, if you play the a note with frequency 440 and another with frequence 441 you will hear a beat frequency of 1 per second.

It's very easy to hear, say, beats at the rate of 2 or 3 per second (say, 440 vs 442 hertz or 440 vs 443 hertz) but it's also quite possible to hear beats at the rate of 1 per second or even 1 every two seconds, 3 seconds, and so on.

My point is, by using this different means of detecting differences in pitch, we can detect pitch differences about an order of magnitude more fine than what you outlined above. (1/10th of a step at 440 hz is about the difference between 440 hz and 442.5 hz; using the 'beat difference' method it's pretty easy to hear that 440 and 440.25 are different tones.)

Part of the fun of music is any system of music theory or practice (like for instance a tuning system) is in essence a way of taking all these different physical and perceptual 'facts' about musical tones etc. that all sort of work together into a harmonious system but also have these strange little discrepancies, and one grand unified system that works at the practical level.

There are a lot of these little discrepancies, and a lot of different ways of resolving them, and that is why there are a lot of different systems for making music in the world . . .
posted by flug at 9:01 PM on December 1, 2014 [4 favorites]


Lutoslawski: Mmm, yeah, but there's not really a great intrinsic reason why a doubling of a frequency should equate to the perception of a similarity of pitch, that's the rub. ... I mean, consider a pure sine wave. You're absolutely still going to perceive a 1000 Hz sine wave as having something inherently the same about it as a 2000 Hz sine wave that has nothing to do with an overtone series. The phenomenon isn't limited to instruments at all.

I know absolutely nothing at all about this topic first-hand, but Dave Benson's book Music: A Mathematical Offering disagrees with you completely:
For pure sine waves, the ear detects nothing special about a pair of signals exactly an octave apart, and a mistuned octave does not sound unpleasant. Interval recognition among trained musicians is a factor being deliberately ignored here. On the other hand, a pair of pure sine waves whose frequencies only differ slightly give rise to an unpleasant sound. Moreover, it is possible to synthesize musical sounding tones for which the exact octave sounds unpleasant, while an interval of slightly more than an octave sounds pleasant. This is done by stretching the spectrum from what would be produced by a natural instrument. These experiments are described in Chapter 4.

The origin of the consonance of the octave turns out to be the instruments we play. Stringed and wind instruments naturally produce a sound that consists of exact integer multiples of a fundamental frequency. If our instruments were different, our musical scale would no longer be appropriate. For example, in the Indonesian gamelan, the instruments are all percussive. Percussive instruments do not produce exact integer multiples of a fundamental, for reasons explained in Chapter 3. So the western scale is inappropriate, and indeed not used, for gamelan music.
What do you think about this line of argument?
posted by narain at 9:16 PM on December 1, 2014 [5 favorites]


Octaves are perfectly natural. Its double the frequency. If you have two strobe lights, one flashing at 5 hz and the other flashing at 10 hz it won't be nearly as disorienting as having one set at 5 hz and the other at 9 hz. You would be able to clearly distinguish whether the second strobe is at 10 hz or not. The same with 20, 40, or even 2.5. If it's a little bit off it produces a sort of wave effect.

The same with sound frequencies. An octave is double the frequency so it sounds like the same note. When its slightly off you get that wavelike effect. It sounds out of tune until you get to a third or a fifth, at which point its a chord.

When I used to tune pianos I used an equal temperament. In the middle octave you tune everything perfectly even. When you tune the lower keys, you get them to the "correct" pitch and then go a hair lower. The further down the keyboard you go the lower you adjust. When you go higher up the keys you tune them slightly higher than "correct". If you don't do this the piano sounds incredibly dull and lifeless even though its "correct". This is call stretching and is especially important on smaller upright pianos.
posted by a2a87 at 9:24 PM on December 1, 2014 [1 favorite]


So while it's useful to know what combinations of notes sound good and how to manipulate them, applying the sort of math you see in the hard sciences is pointless

It's worth pointing out that math is basically the 'science of patterns' and your going to be able to find all sorts of different (interesting!) patterns in any kind of structured and patterned activity like music.

You can find all sorts of interesting math patterns in western music, traditional Chinese music, gamelan music, West African drumming, or any other type of music you choose.

I think you're right, however, to be suspicious of how *central* these patterns might be to the underlying music. Music is, at its heart, a human art built on human perception. The math patterns are some combination of a byproduct of that (patterns built by humans to create sound are likely to have a number of interesting mathematical patterns as a byproduct of that process) and the fact that the underlying physics of sound as it necessarily interacts with our underlying perceptual processes and senses (ie, hearing) is bound to have some interesting mathematics associated with it.

Studying this type of 'mathematics of music' might be interesting and helpful in some ways but isn't likely to be necessary to understand the basics of how music works.

An additional factor is that sometimes (often?) musicians consciously choose math patterns as the basis of their musical systems. IE, the ancient Greeks were obsessed with integers, ratios, etc, and their ideas about how to translate those ratios into sound influenced both ancient Greek music and later Western music pretty dramatically. Another example--Twelve-tone composers rather deliberately used mathematical ideas from set theory, group theory, combinatorics, etc, as compositional devices.

If you really want to understand how these particular types of music work, you're likely to have to dive into the mathematics of it to a degree, simply because these systems are deliberately built on certain mathematical principles.

But as a rule, musicians and composers are working in the realm of sound and sound perception, not math. So the sound part of it, not that math part, is the part you MUST understand!
posted by flug at 9:38 PM on December 1, 2014 [1 favorite]


It's actually totally helpful to know the math if you want to tune your stringed instrument (most typically bass) by overtones, as it'll let you know which overtones to try it with. Basically you bow weakly and only touch the string and you can make it sing its overtone; you can get the same overtone the next string over by dividing the string in a different place. Then all you have to do is match the overtones, which you can HEAR wobbling in and out of phase with each other and is sometimes easier than pitch matching the true string note.
posted by Eyebrows McGee at 9:54 PM on December 1, 2014


The Well-Tuned Piano, LaMonte Young.
posted by Joseph Gurl at 10:16 PM on December 1, 2014 [1 favorite]


It's worth mentioning that only higher frequencies (>1500hz, if I recall) are heard via individual cilia on the cochlea (which, because it varies in density, experiences resonances in different locations according to incoming frequencies).

Lower frequencies (<3000hz) come into the brain via transmission of the overall vibration of the cochlea itself.

Yes, there's overlap, and indeed most of speech is between 1500hz and 3000hz.

Regarding audio perception, the brain's fundamental tactic is reducing complex information streams into the behavior of modeled objects. Instead of random colors floating about, you see a scene with chairs and windows and skies. Instead of random frequencies hustling and bustling, you hear a wind instrument playing a note. The hierarchy from costimulating frequency sensors collapsing into an instrument with a given timbre, does not stop there -- from there is the note, then measures, passages, songs.
posted by effugas at 12:07 AM on December 2, 2014


I guess it depends on if you mean harmonizing with the fourth and the fifth or moving to them in relation to the tonic in some key. Certainly the fourth was a popular harmonizing interval in very early music, but you rarely hear it as a harmonizing interval by the baroque period. Same with the fifth outside of its triadic use. Certainly you would never hear fifths moving together in counterpoint, and rarely would you even land on a fifth. Depending on who you talk to, the fifth as a sort of destination point has been the major theme throughout Western music, but it's role, especially in like a discrete sense, has evolved quite a bit.

You almost certainly know more music theory than I do but as I understand parallel fifths were avoided because the rules of counterpoint encouraged the independent movement of voices and fifths linked them too closely. I thought you were suggesting the interval evolved from dissonant to consonant and on the contrary my interpretation has been that their well-known consonance became seen as too simple, something of a cheap trick. As you say fifths - I'm talking about as a harmonizing interval, the V chord is a whole subject in itself- were popular before the baroque period and as far as I know even consecutive fifths never really fell out of favor in folk music - plus I would say that triads are evidence of the continued importance of the interval in all kinds of music if in a consciously more sophisticated way than a bare fifth. Certainly the role has changed, but my point is that there's an importance in Western music that points to a significance to the 3:2 ratio second only to to 2:1 - which is either inherent or something cultural that happened a *very* long time ago. If it's the latter I would be interested in counterexamples from other musical traditions.
posted by atoxyl at 1:12 AM on December 2, 2014


Maybe it's just been a while since I've studied music, but lutoslawski's comments are baffling to me. I haven't studied ears directly, but notes an octave apart should definitely have a large intersection of overtones.
For example, A 440 would resonate with cilia tuned to 440, 880, 1320, 1760, 2200, 2640, etc...
While an A 880 up an octave would resonate with cilia tuned to 880, 1760, 2640, 3520, 4400, 5280, etc...

There's an interesting experiment you can try if you have a non-synth piano. Softly press down on, say, center C, so that it doesn't make any noise. Now, press down the C an octave lower to make a clear sound, then release the lower C. The higher C key should now be emitting a "ghost" of a sound, because its string has begun vibrating in resonance with the overtones shared with the lower C (all of them). This also also happens if you silently hold down the G above middle C, then press lower C. However, the sound is much, much fainter if you silently hold down the G above middle C, then audibly press the middle C key. This is because the amount of overtones shared is much lower.

I remember one of my professors mentioning a colleague who wrote pieces for a 19-tone piano. I'll look them up when I can.
posted by halifix at 4:14 AM on December 2, 2014


Oh, and a lot of old baroque instruments were tuned specifically for overtones on the... A key, I believe. They'd make some funky sounds if you transposed to a different key.
posted by halifix at 4:15 AM on December 2, 2014


I had a prof who was fond of reminding us that Music Theory is not theory at all, but mostly nomenclature and heuristics.
posted by Jode at 4:59 AM on December 2, 2014 [2 favorites]


Music theory is tied to our perception of music, and our perception of music is tied to the mechanical physics of vibration and resonance. It's certainly possible that we could have our audio perceptions set up so that we heard stuff randomly mapped onto the actual spectral composition of the stuff that enters our ears, but it's hard to think why such a system would be useful or efficient - and I don't think there's any evidence for this anywhere in nature.

On the other hand, the natural parsimony of evolved systems would suggest that we'll have auditory systems which get the most information out of the air that they can with the least effort - which means making good use of natural effects such as vibration and resonance.

There'll be other stuff going on, of course. But resonance is one of the macro-scale physical effects where integers are a huge part of the equation, so it'd be very odd if they weren't a huge part of auditory perception.
posted by Devonian at 6:25 AM on December 2, 2014 [1 favorite]


I recall my own disappointment over the incomensurability of fifths and octaves. I now wonder what a synth with moving pitch would sound like. My thought was that each chord would recalculate based on its fundamental and each fundamental based on the note played before it so that at any given time all the ratios would be consonant but the pitch would drift around. (I'm sure it exists but I haven't found any recordings).
posted by Octaviuz at 7:54 AM on December 2, 2014 [1 favorite]


Well, yeah it's true that the octave is almost always a prominent overtone when you have some periodic wave in a tube or on a string attached to some resonant thing. But that doesn't have anything to do with why we perceive octaves as being particularly consonant sounding, or phenomenally the 'same pitch' as their fundamental. I mean, consider a pure sine wave. You're absolutely still going to perceive a 1000 Hz sine wave as having something inherently the same about it as a 2000 Hz sine wave that has nothing to do with an overtone series. The phenomenon isn't limited to instruments at all

It may not have anything to do with the overtone series present in these particular tones, but I would be surprised if it had nothing to do with the training your brain has received from all of the (even non-musical) sounds for which plentiful overtones are present.

The overtones don't have to necessarily be present in the sound itself to influence the perception of harmony; they just have to be present in the brain's representation of the sounds. If (and I realize this is speculative) the frequent co-occurrence of overtones has caused a commonality of neural representation, then even those frequencies presented in isolation to someone so trained would seem more similar.
posted by a snickering nuthatch at 7:57 AM on December 2, 2014


I guess I should further qualify the 'fourth rarely heard as a harmonizing interval by the baroque period' by saying that rarely might have been strong, and of course you hear it as a harmonizing interval at times in a fugue, and not just in passing, but I think that's a special case.

I'm not sure what you mean by "harmonizing interval". Do you mean that fourths and fifths aren't the basis for the harmony, which is built on stacks of thirds (and inversions of stacks of thirds)?

I can't really argue with any sort of citations, because it's been way to long since I've done any serious reading, and my books are all pack, but my general impression of common period practice theoretical thought was that the Root and Fifth determined the chord, and the third determined the flavor. Both intervals were important equally important to the construction of the harmony. I seem to remember the stack of thirds way of thinking is a by product of how we teach chords in isolation, rather than a function of how harmonies were actual thought of when they were being built. Which makes sense considering that the thinking of the Baroque grew out of a tradition (early church and court music) of thinking about harmony that absolutely was built on fifths and fourths rather than thirds.
posted by Gygesringtone at 8:06 AM on December 2, 2014


Pyry: My big question with music theory has always been whether there is some intrinsic perceptual reason to prefer e.g., octaves and perfect fifths, or whether our preference for these is instead the product of familiarity with these conventions. And it's irritating because it seems that none of the texts will go into this question, but rather assume that this is some obvious fact that is beyond discussion.
Octave harmonics are easy: they add their electric & magnetic field strengths at their harmonic maxima and minima, and likewise "zero out" the field strengths at the same time. Power is the square of field strength, so when you add 1 W at 100 Hz and 1 W at 200 Hz, the 100 Hz peaks will be (momentarily) 4 times as high: (1+1)^2. And the 100-Hz zeroes will truly zero out, every time, on a regular, detectable frequency.

When nonharmonic waves are added, the component field maxima and zeroes rarely coincide, so that the power as a function of time will remain lower (mostly under 2 W), and the zeroes will occur randomly, leaving a less-strong pattern of beats for the brain to recognize.

So, harmonic frequencies are easier to describe/detect as a regular-varying signal. As we add pure octave harmonics (300 Hz, 400 Hz, etc), the 100 Hz signal is not washed out by the extra signals at all. Essentially, the ear is still able to notice separate signals within the overall sound, and so one might say the signal-to-noise ratio (SNR) is very high.

That's EM wave theory.

There have been experiments showing that a variety of animals react differently to harmonics. 70% of bird calls use simple harmonics like 3/5. Dog sounds vary from highly harmonic puppy sounds to highly non-harmonic howls of adult dogs - clearly harmony plays some role in a dog's brain. However, wolves are less likely than domesticated dogs to use harmonic vocalizations, suggesting we have bred for that distinction in the past. There's no evidence to suggest this is a recent trait, since it's shared across most breeds, so our desire to select for these "pleasing" harmonic sounds probably predates modern civilization.

That's evolutionary evidence.
posted by IAmBroom at 8:45 AM on December 2, 2014 [1 favorite]


Chimes: you hear only the difference. That's more than my stupid electric tuner can do.

But I like the way a 3rd lets you pause on the way to the 4th, as in Tennessee Waltz, to let you reflect on the night they were playing....then there's fourth the fifth, the minor fall, the major lift. Truly: hallelujah.

I am so grateful to never need to tune a goddam piano.
posted by mule98J at 9:14 AM on December 2, 2014 [1 favorite]


I'm not sure what you mean by "harmonizing interval". Do you mean that fourths and fifths aren't the basis for the harmony, which is built on stacks of thirds (and inversions of stacks of thirds)?

I thought he meant that you wouldn't often hear a "chord" of just the fifth interval anymore? And something about parallel fifths. Unless this is about why we don't stack fifths instead of thirds.

I can't really argue with any sort of citations, because it's been way to long since I've done any serious reading, and my books are all pack, but my general impression of common period practice theoretical thought was that the Root and Fifth determined the chord, and the third determined the flavor.

That's pretty much how I was taught, though I don't live in the common practice period. When you start getting into 9, 11, 13 chords maybe that's a different story, that's out of my depth.

The main thing I find questionable is this statment:

the intervals we have sensed as being consonant have changed over time (namely the fifth)

Because what I know about the evolving use of the fifth suggests it's been recognized as a naturally pleasing interval for ages as in folk and church music but that counterpoint composers intentionally limited its use in pursuit of "higher" artistic goals.

I've also seen references from time to time to fourths being considered dissonant, but in context this seems to have to have to do with the interaction of fourths with triads - and I don't think the interpretation of something like a suspended fourth has changed that much except that composers are more likely to leave it intentionally unresolved.
posted by atoxyl at 11:54 AM on December 2, 2014


I thought he meant that you wouldn't often hear a "chord" of just the fifth interval anymore?

this is basically the chord that made The Police multi-millionaires - stacked fifths ("Message In A Bottle", "Every Breath You Take", etc).
posted by thelonius at 12:08 PM on December 2, 2014


I'm also intrigued by Lutoslawski's comments. It's not surprising that a ratio of 2 should be the best ratio for innate whatever. It's much more striking that there should be a bright line between that and a ratio of 3.

Lutoslawski: do you have reading recommendations on this? In particular, some kind of cross-cultural survey where the special status of the octave is discussed?

I'm wary of the putative bright line. For comparison, in linguistics (the human science with which I'm most familiar), there's a trope of declaring structural feature X of language to be a "universal", to be putatively true of every language; but these universals have a relatively bad track record for actually being universal, there's too often a counterexample. And there may well be good reasons to do with processibility or learnability or ease of use or possible historical origins or ... or all the above that feature X should be a likely one to have; but this shows there can't be anything categorical.
posted by finka at 12:24 PM on December 2, 2014 [1 favorite]


> some of the harmonic series is sharp to the fundamental

This. Maybe someone else mentioned it, but real-world physical instruments have slightly out-of-tune harmonics. Wendy Carlos theorized that's why Bach avoided parallel 5ths & 8ves in Tuning: At the Crossroads [JSTOR], then went on to come up with new instruments, octave divisions, tuning schemes, and voice leading rules and compose Beauty in the Beast.
posted by morganw at 1:34 PM on December 2, 2014 [1 favorite]


this is basically the chord that made The Police multi-millionaires - stacked fifths ("Message In A Bottle", "Every Breath You Take", etc).

Makes a sus2 I guess? I love those kind of half-dissonant chords but then I'm not from the 17th century.

I didn't mean that nobody does it, we're talking about trends in classical music. Actually what I mean by "just the fifth interval" is the root-fifth or root-fifth-octave chord which of course is everywhere and played up and down - i.e. parallel fifths - with no compunction. That's kind of my argument, that people (in Europe but I know it's in many musical traditions - can't speak for all) have always loved that sound but common practice classical composers intentionally disavowed or complicated it because they wanted to distinguish their music from "lower" forms.
posted by atoxyl at 1:49 PM on December 2, 2014


I now wonder what a synth with moving pitch would sound like.

Octavius, there's something called the Dewanatron Swarmatron that may be what you're looking for.
posted by infinitewindow at 4:47 PM on December 2, 2014 [2 favorites]


You could call it a sus2 I think, yeah. Maybe something like "A5 add9" would be better.
posted by thelonius at 7:51 PM on December 2, 2014


Best to avoid any ambiguity and just refer to it by its pitch class set designation of 3-9.
posted by invitapriore at 9:53 PM on December 2, 2014


I recall my own disappointment over the incomensurability of fifths and octaves. I now wonder what a synth with moving pitch would sound like. My thought was that each chord would recalculate based on its fundamental and each fundamental based on the note played before it so that at any given time all the ratios would be consonant but the pitch would drift around. (I'm sure it exists but I haven't found any recordings).

You can do this sort of thing with any off the shelf soft-synth if you feel like it. It's super easy to play notes at whatever frequency you want.
posted by empath at 7:19 AM on December 3, 2014


You could call it a sus2 I think, yeah. Maybe something like "A5 add9" would be better.

I would be absolutely amazed if a tour through the stacks of any school with a good theory department didn't turn up a couple of thesis on a new way of analyzing quartal harmony. Ives, Hindemith, Orff, and several others beat Sting to the punch by a half century or more.
posted by Gygesringtone at 1:48 PM on December 4, 2014


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