Consonance and Dissonance

I tracked down the abstract for that University of Melbourne study on music that has been reported in various places and that I have written a couple of posts on. I often find that journalistic reporting on research is so confused it is hard to tell what is really going on. So let's have a look at that abstract. Here it is:

Consonance and Pitch.

By McLachlan, Neil; Marco, David; Light, Maria; Wilson, Sarah

Journal of Experimental Psychology: General, Jan 7 , 2013, No Pagination Specified.

Abstract

To date, no consensus exists in the literature as to theories of consonance and dissonance. Experimental data collected over the last century have raised questions about the dominant theories that are based on frequency relationships between the harmonics of music chords. This study provides experimental evidence that strongly challenges these theories and suggests a new theory of dissonance based on relationships between pitch perception and recognition. Experiment 1 shows that dissonance does not increase with increasing numbers of harmonics in chords as predicted by Helmholtz's (1863/1954) roughness theory, nor does it increase with fewer pitch-matching errors as predicted by Stumpf's (1898) tonal fusion theory. Dissonance was strongly correlated with pitch-matching error for chords, which in turn was reduced by chord familiarity and greater music training. This led to the proposition that long-term memory templates for common chords assist the perception of pitches in chords by providing an estimate of the chord intervals from spectral information. When recognition mechanisms based on these templates fail, the spectral pitch estimate is inconsistent with the period of the waveform, leading to cognitive incongruence and the negative affect of dissonance. The cognitive incongruence theory of dissonance was rigorously tested in Experiment 2, in which nonmusicians were trained to match the pitches of a random selection of 2-pitch chords. After 10 training sessions, they rated the chords they had learned to pitch match as less dissonant than the unlearned chords, irrespective of their tuning, providing strong support for a cognitive mechanism of dissonance.

 The first thing I notice is that I was correct in assuming Helmholtz was one of the people they were disagreeing with. I suppose this is all reasonable from a scientist's point of view? Perhaps my commentor Joel could weigh in here. But I find a lot of things perplexing. First of all, the opening sentence: "To date, no consensus exists in the literature as to theories of consonance and dissonance." What literature are they referring to? Certainly not the theoretical literature in music that abounds in discussion of consonance and dissonance. By "theories" of consonance and dissonance they must be referring to theories of the listener's reception or perception of consonance and dissonance. OK. Then they refer to the "frequency relationships between the harmonics of music chords". What this means is unclear. A chord in music is the simultaneous sounding of two or more notes. Typically chords are triads or tetrads. Yes, every note, in a chord or alone, has harmonics, that is, the overtone series, but what the "frequency relationships between the harmonics of music chords" is, is a mystery to me. Most of the rest I find unclear as well. But perhaps that is due to my ignorance of the scientific context.

Here is how a musician and composer looks at consonance and dissonance. All intervals are divided up into consonant and dissonant, but the line between them has changed over time. If we go back far enough, every interval except the perfect ones (fourth, fifth and octave) was considered more or less dissonant. In actual use this meant that while you could use other intervals, you couldn't end with anything other than a perfect one. Dissonances like thirds and sixths, had to be passing. Later on, thirds and sixths were accepted as consonant, though the minor third was often not considered suitable in a final chord, so a major third was substituted resulting in a tierce de Picardy. As chords with an added seventh developed, found very useful for the added tension they provided cadences, the seventh and tritone were used more and more--though again, they were used in a chord normally passing to a consonant tonic harmony: GBDF (the tritone lies between the B and the F) going to CEGC. The most dissonant intervals have always been the tritone and the minor second, but we find them in constant use during the whole common practice period. However, for most of this period they were used in specific ways and resolved in specific ways. In the late 19th and early 20th century, their use became more and more frequent and the resolution more and more ambiguous until finally the whole notion of consonant and dissonant was tossed out with the development of 12-tone music.

What I find problematic with the researcher's approach is that they seem to take no account of the context. But I can't imagine understanding anything much about consonance and dissonance without context. A C major triad can sound harsh and dissonant in just the right (or wrong!) context. Here is the beginning of the Piano Sonata op 106 by Beethoven, nicknamed the "Hammerklavier" which begins with a simple B flat chord, though sounding quite bold and 'crunchy':

It's all in the context...

Miscellanea

• A Stanford geneticist has research that seems to show human intelligence declining.

“I would wager that if an average citizen from Athens of 1000 BC were to appear suddenly among us, he or she would be among the brightest and most intellectually alive of our colleagues and companions, with a good memory, a broad range of ideas, and a clear-sighted view of important issues. Furthermore, I would guess that he or she would be among the most emotionally stable of our friends and colleagues. I would also make this wager for the ancient inhabitants of Africa, Asia, India or the Americas, of perhaps 2000–6000 years ago. The basis for my wager comes from new developments in genetics, anthropology, and neurobiology that make a clear prediction that our intellectual and emotional abilities are genetically surprisingly fragile.”

You know, that might explain some of the trends in recent music history... Naaah, just kiddin'.

• Here is an essay on the delights of good design--with the misleading headline Why We Love Beautiful Things. I say misleading, because as a comment on beauty, it is only perhaps a quarter true. The essay attributes 'beauty' to proportion (the Golden Section), certain colors and landscapes and fractal geometry. And, of course, all this is sparked by various samples of psychological research. Isn't it about time that talk about beauty--which is really aesthetics--was informed, just a bit, by actual understanding of aesthetics? I find these clumsy scientistic fumblings ever more tiresome... Here, let me mess things up by posing a nice philosophical question: do we love beautiful things because they are beautiful? Or are they beautiful because we love them?
• Here's a slightly more detailed account of that research on beauty in music from the University of Melborne. Here's the lead paragraph:

Our love of music and appreciation of musical harmony is learnt and not based on any innate natural ability according to a new study by researchers at the University of Melbourne.

Well, that certainly raises more questions. For one thing, it poses a false dichotomy. Our appreciation of harmony can both be learned AND based on some innate natural ability. Mozart learned harmony, but his abilities were enormously greater than the average person's because of some innate natural ability. Does anyone seriously question this? It continues:

Associate Professor Neil McLachlan from the Melbourne School of Psychological Sciences said that previous theories about how we appreciate music were based on the physical properties of sound, the ear itself and an innate ability to hear harmony.

Previous theories, being, probably, ones pointing out the relationship between the overtone series and basic harmonic progressions. The first overtone is the octave, the second overtone is the fifth. The fifth is an extremely important interval in harmony as witnessed by the basic teaching device, the "circle of fifths". Things like this indicate that the structure of harmony was, at some points in history, related to basic acoustic facts. Here is another claim:

The researchers used 66 volunteers with a wide range of musical training and tested their ability to hear combinations of notes to determine if they found the combinations familiar or pleasing. “What we found was that people needed to be familiar with sounds created by combinations of notes before they could hear the individual notes. If they couldn’t find the notes they found the sound dissonant or unpleasant,” he said. “This finding overturns centuries of theories that physical properties of the ear determine what we find appealing.”

Again, some very confused half-truths. It is certainly true that repeated exposure to certain combinations of notes enables most listeners to better hear what is going on and therefore to appreciate the music more. This simple discovery, known, by the way, to every music teacher for the last 1,000 years, overturns nothing! What is most puzzling here is the claim about how this is a discovery that "overturns centuries of theories". But they don't bother to mention what those theories were or who held them.

Here's some home truths for the folks at the University of Melborne: people's ability to appreciate and understand complex music is partially learned and partially based on natural ability. A lot of harmony is closely related to acoustic phenomena like the overtone series. It is hard to know what exactly they mean by the "physical properties of the ear", but it would certainly be impossible to appreciate music if your hearing were severely compromised.

I am at something of a loss here to see just what, if anything, the researchers managed to discover here.

• The recorder player Walter Bergmann offers us some charming rules of chamber-music playing. Hat tip to Norman Lebrecht.

Twenty Six Golden Rules of Ensemble Playing

by Walter Bergman (1903-1988)

1.  Never worry whether you play the same piece as the others;  they will soon find out.

2.  Stop at every repeat sign and enter into a palaver about whether you should, should not, would, would not, could, could not, want, or want not, to repeat.

3.  Which is the most important part in an ensemble?  The other one.

4.  Always aim for the highest number of n.p.s. (notes per second).

5.  If you play a wrong note, give your partner a dirty look.

6.  Always keep your fingering chart handy.  If, in the middle of the piece, you don’t know the fingering of a note, look it up, try the note, and then catch up with the others.

7.  If a passage is difficult, slow down; if it’s easy, speed up.  In the long run it all evens out.

8.  A right note, at the wrong time, is a wrong note (and vice versa).

9.  Do take your time turning a page – it gives everyone a nice rest.

10. Rests are difficult, especially on the recorder.  If you are not sure of their lengths, ignore them.

11. If you alone are right and everyone else is wrong, follow the wrong.

12. If you have irretrievably lost your place in the music, stop everyone and say, ‘I think you need to retune.’

13. Blessed are the poor in intonation, for theirs is the kingdom of music.

14. Memorize the following line, which you can have ready for a variety of situations: ‘I alway play in tune, because I play a Moeck (Coolsma, Aulos, Dolmetsch, Koch, Kung, von Huene, etc.) recorder.’

15. Tune carefully before playing, and then you can safely play out of tune for the entire evening.

16. Your conductor has been paid.  There is no need to look at him.

17. But be sure to _follow_ the conductor (don’t be together with him).

18. Spare the breath and spoil the tune!

19. Remember, vibrato _always_ starts on the upper frequency.

20. An ornament should be an embellishment and not an embarrassment.

21. Remember Shakespeare’s immortal lines:

“A rest is silence” (Hamlet)

“My kingdom for a semiquaver.” (Richard III)

“My foot my tutor?”  (The Tempest)

22. Pick out of old books (Quantz, etc.) what you like, and bypass what does not suit your preconceived ideas.

23. Authentic interpretation is not achieved until not a note of the original is left.

24. Do be careful to select the right edition. The best editor is he who writes _forte_ at the beginning of a fast movement, and _piano_ at the beginning of a slow one.  He puts breath marks over rests and omits them where they could be helpful. He also write prefaces that make the performance of a piece completely unnecessary and sometimes even undesirable.

25. Remember, _forte_ and _piano_ marks, dots, and crescendos and decrescendos are not there to be observed.  They are decorations for the eye, invented by frustrated engravers, and they have no special musical meaning. As communications from the composer they are equally unimportant, because composers are mostly dead and don’t understand their own compositions, anyhow.

There are, however, three exceptions to this rule:

a)  A dot over a note prolongs its duration by one half-step.

b)  Crescendo and decrescendo hairpins are essential over rests.

c)  In examples like the following, adhere carefully to directions: [six measures of tied whole notes, marked 'Nicht schleppen (do not drag)']

26. Thou shalt not play the little bit left over at the end… [Here there should be the little musical example. The little bit left over is the 2nd entrance of a repeat.]

Are There Musical and Unmusical Nations?

I'm not sure if this is a forbidden topic or not, so I will tread carefully. It seems very obvious, despite the efforts of some scientists to deny it, that musical talent is not evenly distributed in the population. Yes, I'm afraid that it really is the case that merely purchasing violin or piano lessons for your child will not guarantee that they will learn how to play well. On quite a few occasions in my decades-long career as a music teacher, I have had to tell parents that their child was not progressing and it was probably not worth it to continue to enroll them in lessons. I regarded this as my duty. I also think that it is the responsibility of the parents to find out where their child's skills may lie. Yes, put them in dance classes, tennis classes, music lessons, art camp--give them as many opportunities as possible to discover their talents. But be aware that in most areas they may have no particular talent.

I think all that is simple empirical observation and a little common sense. But this leads to another question: are all cultures and nations equally gifted musically? According to the doctrine of multiculturalism, the answer should be yes, the cultures of all nations are of equal worth. But this seems incorrect to me. For example, some nations have enormous and brilliant musical traditions while others do not. Take Italy and Switzerland, for example. Though next door neighbors, one has an unequaled musical tradition but the other does not. Italy has given the world opera, Vivaldi, Verdi, Rossini, Francesco da Milano, Stradivarius violins, La Scala, Pavarotti and the list goes on and on.

OK, now name one important composer from Switzerland.

(crickets...)

See what I mean?

In Latin America there are strong musical traditions in some nations like Argentina and Brazil, and very scanty musical traditions in other nations like Chile and Bolivia. By "scanty" I don't mean that there is no music--there is scads of music--but that little of it is of lasting quality. Mexico, where I live, seems to be a somewhat tone-deaf nation. True, there are some fine Mexican musicians like the composer Manuel M. Ponce or the singer Luis Miguel, and the mariachi tradition is an interesting one. But by and large, Mexican music is harsh in timbre and rhythmically stiff. Brazilian music, on the other hand, is typically fluid and charming.

Why is this? I'm not sure. The reasons are probably complex and different for every culture. One interesting phenomenon seems to be that oppressed sub-cultures sometimes develop strong musical traditions. Some that come to mind are the gypsy culture in Europe where the gypsy folk music traditions are very strong in places like Hungary, Romania and Spain. Flamenco is largely the creation of the gypsy culture of Andalusia in southern Spain. Another one is the Irish musical tradition, a Celtic one that has lived under English domination through most of modern history. A third one is the Black music culture in the US which has given the world blues, gospel and jazz.

How about some examples? First, Guinness-fueled Irish fiddling and picking:

Next, one of the original Delta bluesmen, Charley Patton with "Spoonful Blues":

Here are the virtuosic Romanian gypsy band Taraf de Haidouks:

You don't hear that kind of thing every day...

The Möbius Strip and Music

I discovered topology at a fairly young age. I think it was one of the very first things I discovered that truly seemed magical. An object in the real world with only one side? And one edge? That was the Möbius strip. The easiest thing in the world to construct: just take a strip of paper, twist it once and tape the ends together. Voila, something you can perplex your friends with. Here is what it looks like:

It has some interesting properties. Yes, it has just one side which you can demonstrate by taking a pencil and drawing a line down the middle. You can draw the line continuously without lifting the pencil on both sides, because there is only one side. Now, what do you think will happen if you take a pair of scissors and cut the strip in two down the middle? Well, you can't cut it in two! Instead you will get a single strip half as wide and twice as long with two twists.

Now what does this have to do with music? There is a YouTube video that demonstrates the relationship between a Möbius strip and a Bach canon. The canon is the type known as a 'crab' canon, meaning that the melody combines with itself played backwards:

The musical mathematician Vi Hart does even more in this mind-bending video where she demonstrates the folding of space-time with a music box:

If you are interesting in mathematics, music and Bach, this article is a useful introduction.

Let's end with a piece by Bach that has provoked a great deal of commentary. The St. Matthew Passion that Mendelssohn called "the greatest Christian work" is a mammoth depiction of the crucifixion of Christ set for two choruses, two orchestras and six vocal soloists. This article gives some idea of the scope and complexity of the structure. Here is a performance of Part I by Nicholas Harnoncourt:

Making Classical Music More Elitist!

UPDATE: I shortened the title of this post.

A few weeks ago an article was published in the Independent covering a talk the new head of Universal Music's classical music unit gave to the Association of British Orchestras. The appropriately named Max Hole was previously a rock band manager and the advice he gave was predictable:

He believes that classical music needs to be promoted beyond the existing core audience, not just young people but “people like me who would engage in classical music if they didn’t feel it was elitist or forbidding”.
“Musicians need to think about the way they dress, and need to appear more excited engaged with the audience,” he said. “There’s more to it than just taking a couple of bows at the end of a concert.”
He said that the traditions and institutions that seek to promote and preserve classical music “are in danger of causing the genre great harm and hinder its growth”. Even the term “classical” is in danger of alienating its audience, he said.
Conductors need to actually talk to the audiences, and Mr Hole pointed to the “sheer exuberance” of the Simon Bolivar Youth Orchestra. He said there should be screens showing the audience the conductor’s work, and “more theatrical” use of lighting.

Yes, of course, classical music should be promoted just like popular music.

I couldn't agree less! This is the conventional wisdom these days and perhaps it might gain Universal Music and some of the more trivial performers more sales, but this kind of attitude is likely to make classical music just as superficial and trivial as popular music.  Do I even need to point out that classical music is not the same as popular music? It attracts a different audience who listen in a different way. They really do not need to adopt all the ridiculous distractions that pop music uses to hide its vacuousness like chatting up the audience and theatrical lighting.

Max, I just can't emphasize enough how I dislike your whole attitude. In fact, it is with renewed vigor that I will set out to make classical music just as elitist and forbidding as possible in hopes of driving you as far away as possible. And furthermore, shut up!

Why did it seem like a good idea to appoint someone like Max Hole, who obviously dislikes classical music, to an important role in the business? Does Universal Music have a death-wish? It is people like Max Hole that are the ones that have the potential to cause classical music great harm.

Now, let's listen to some of that elitist, forbidding music played by a particularly elitist and forbidding performer, the great Grigory Sokolov:

Learning to Unappreciate Music!

In my last post I stumbled across an interesting cliché about listening to music: that learning more about music and how to listen to it inevitably leads to appreciating more music. Here is how one writer put it:

Researchers at Australia's University of Melbourne say that the more dissonance (which they describe as "perceived roughness, harshness, unpleasantness, or difficulty in listening to the sound") that we hear in music, the less we enjoy said music. Seems obvious enough, right?
Also falling into the "no kidding" category is the not-at-all new notion that the more we're exposed to a certain kind of music — either through intentional engagement or simple osmosis in whatever culture we're immersed in — the more we like that music.

 So, any exposure to any kind of music means we will like it more? I'd like to test that theory. Here, listen to this:

If you didn't appreciate it the first time, then I'm sure that just listening to it ten more times will do it?

How about this?

Just give it more time and I'm sure you will start to appreciate the subtleties...(!?!)

The truth is that most music is not very good and some is rather remarkably bad. If I were still teaching at university I would love to offer a course in "How to Unappreciate Music". We would learn to identify bad music by listening. This is valuable because good music, not to mention great music, is at one end of an aesthetic spectrum at the other end of which is bad music. The clues as to what constitutes aesthetic worth are as evident, by their absence, in bad music as good music. If you want to appreciate the subtle power of good repetition in, say, Beethoven, then I really think that you should be exposed to the grating unpleasantness of bad repetition.

Why are the virtues of unappreciating music so rarely mentioned? Is it that those who are trying to flog music appreciation courses are just scared of driving away business? Is it the fear of seeming negative? Is it just too subtle a concept? The truth is that the more comprehensive and profound your knowledge of music is, the fewer pieces of music you are likely to enjoy and the more music will drive you up the wall! Ah, that's probably why people don't want to mention it.

Well, let's clear the palate with some good music, shall we? Here is a late Haydn symphony, nicknamed the "Drumroll" for obvious reasons:

Learning to Listen

I just ran across an article that I thought I was going to be able to recommend whole-heartedly--at least until I read it! The Atlantic has a report on some music research done at the University of Melbourne. From the headline, I thought it was going to be wonderful stuff:

Study: Hearing Music as Beautiful Is a Learned Trait

It starts like this:

Why does the music that to some people is lovely, even transcendent, sound to others like a lot of noise?
Researchers at the University of Melbourne attribute to the amount of pleasure we take in music to how much dissonance we hear -- the degree of "perceived roughness, harshness, unpleasantness, or difficulty in listening to the sound."

I've put up a lot of posts before critiquing 'scientific' research into music, here, here and here. Often the problem starts with the basic assumptions: if the scientists have no real knowledge of music, then they will often craft experiments to test things that are too obvious, or simply too crude. For example, the assumption that the amount of pleasure we take in music is inversely related to the amount of dissonance, is a crude one. I'm having to deduce the assumption here as the writer seems unable to state it clearly and grammatically!

Trained musicians, perhaps predictably, were more sensitive to dissonance than lay listeners. But they also found that when listeners hadn't previously encountered a certain chord, they found it nearly impossible to hear the individual notes that comprised it. Where this ability was lacking, the chords sounded dissonant, and thus, unpleasant.

Yes, an interesting and valuable observation. A great deal of music takes repeated exposure to appreciate because you can't hear what is going on at first.

The ability to identify tones and thus enjoy harmonies was positively correlated with musical training. Said study co-author Sarah Wilson, "This showed us that even the ability to hear a musical pitch (or note) is learned."

Well, sort of. Here one of the fundamental prejudices of some scientists comes out. As Steven Pinker argued in The Blank Slate,  one of dogmas about human nature is that the mind has no innate traits. However, as every music teacher knows, there are those who have an inborn propensity to hear and make music and those who don't. Or, as a Czech violist of my acquaintance used to observe, "there is talent, and there is also anti-talent!"

From a practical standpoint, the results seem to suggest that we can train ourselves to better appreciate music. ... And in fact, the researchers conducted a second experiment to test the validity of that theory. They took 19 non-musicians and trained them to identify the pitches of certain chords. Ten sessions later, the participants were better at hearing notes. They also reported that they found those chords to be less dissonant than other chords that they hadn't been taught, regardless of how technically harmonious they were.

I think that they are confusing two different things here: comprehension on the part of the listener and dissonance in the music. If you have a trained ear, you are able to comprehend and enjoy complex music which can often be dissonant. It is still dissonant, but you understand the point of the dissonance, where it comes from and where it is going. An untrained ear just hears the dissonance and not its function. Oh, and the writer is leaving out the obvious corollary: as we become more trained to listen to music, we will begin to notice how vacuous, offensive, crude and maudlin much music is. Why is that never pointed out?

The more ambitious implication of the findings, according to lead author Neil McLachlan, is that it "overturns centuries of theories that physical properties of the ear determine what we find appealing."

Now that is just weird. What centuries of theories? Who thought this? The only candidate I can think of was the 19th century German scientist, Hermann von Helmholz. But hey, anything to puff up the significance of your research, right? In the conclusion, the writer goes completely off the rails--at least I assume it is the writer as it would be strange indeed if the researchers were this ignorant:

As they explain in their discussion, the basic, 12-tone do re mi scale isn't "naturally" harmonious. Instead, it was first introduced by Pythagoras (yes, he of the theorem), who developed a system of "tuning based on successive 2/3 proportions of string length." It was a logical, mathematical method that in turn gave us "the simple mathematical relationships [that] can be found between the harmonics of common Western chords" that we've since learned to love.

First of all, the do, re, mi scale has seven notes, not twelve. The chromatic scale has twelve notes. And scales are not "harmonious" naturally or otherwise because they are notes spaced out in time. Harmony, on the other hand, is notes sounded together. Yes, Pythagoras did observe some interesting things about the proportions of overtones in vibrating strings. But what is missing here is not only Pythagoras' own ideas about the effects of music, but any hint at all that how we relate to music, whether by love or hate, has something to do with musical aesthetics.

Much of the confusion that this study seems burdened with could be alleviated by just a tiny amount of research into music aesthetics. The concept they are desperately in need of is the 'sublime' which I talk about in this post. Much of the aesthetic function of dissonance in music is to evoke the sublime, which is often more interesting than the merely 'beautiful'.

Let's have an example. Here is Le Poème de l'Extase by Alexander Scriabin that, by means of calibrated dissonance and orchestration creates a somewhat threatening beauty: