Octave equivalence? Not so fast...

Sometimes I just feel like such an iconoclast in the world of music theory and pedagogy. Today I was inspired to make a video that attacks one of the most basic precepts of Western music theory: the idea of octave equivalence. To be sure, I'm far from the first to question the nature of octaves. I was inspired because today during a break I was practicing and came up with a very straightforward and accessible way to present one angle on the subject... read on and see the video below:

There are so many problems with the generic idea of octaves as taught in Western music that I can't get into them all here. The question of why do we treat octaves as equivalent has to be qualified by questions of when. In Bill Sethares' notable book Tuning Timbre Spectrum Scale (the link includes summary and samples), he describes and provides audio examples of how the idea of equivalence and blend is related to the particular vibrations in a sound's spectrum. Among the simplest examples, as piano tuners know, the inharmonicity of hammered (or plucked) strings can lead to stretched octaves. Bill goes on to discuss harmony for completely inharmonic sounds as well, which is a whole further (and very interesting) issue.

There have been many attempts to explain why the octave is so significant, some based on carefully controlled scientific experiments. What we know about octaves is that they are, at the most basic, a doubling (or halving) of vibration speed (or close). Because of that, they blend very well and fit into the same overall periodicity. Men and women have likely been singing octaves since prehistoric times. The vast majority of all the world's music treats the octave specially, usually with equivalency, giving the same note name to pitches at an octave.

While the significance of octaves must be acknowledged, there is more than enough evidence to disprove any claim that octaves are absolutely universally equivalent. Octaves are not fully equivalent. They sound different. Harmony does not work the same way at all octaves. A close position major chord sounds great in middle to high ranges, but move it down some octaves into bass ranges and it sounds muddy even if the tuning is not tempered. The vast majority of all guitar methods teach that octaves are equivalent and so any combination of C, E, and G makes a C chord; but the very same books near-universally teach students not to play the low E when holding a C chord. That E doesn't fit as well the harmonic series of the rest of the C chord, so it sounds rougher. If they explain at all, most books and teachers just say that while E is technically part of the chord, we just don't play it because it sounds bad. Some books say it is because the lowest note should be C for a C chord. And yet there is widespread acceptance of the same chord with a low G bass note (which fits the harmonic series in that octave better than the E). Explaining all this is simple once we drop the idea that octaves are totally equivalent.

Another great example is Diana Deutch's Mysterious Melody. She shows how if the octaves are mixed up in a melody it is unrecognizable. Significantly, however, once the melody is known by hearing it normally, then one can still hear it within the mixed-up-octaves version! So octaves have some equivalence: they can substitute for one another if our expectations are clear. But they aren't fully equivalent; it is based on expectations and context.

A common question says, "if octaves are equivalent because they fit into the same periodicity and are a simple 1:2 ratio and part of the harmonic series, wouldn't the 1:3 ratio be comparable and therefore also be equivalent?"

Maybe octaves are more significant because of cultural reinforcement. Maybe it's because men and women don't have such different ranges that they would sing at an even further 1:3 ratio. Maybe it's because 1:3 can be divided by 1:2, thus making 1:1.5 which is not as simple and thus 1:2 is more absolutely basic... but maybe we shouldn't even assume that 1:3 can't be equivalent. Maybe it can be.

I was playing around with this today and decided to make a video showing how effective it is to play at a set 1:3 ratio (called a twelfth in standard Western music theory terms counting letters; also called a tritave in harmonic terms because it's a multiple of 3)

Notice how the final note in the video sounds fully resolved (it does to me anyway). Both the low and high notes each feel not just like part of a tonic harmony but actually feel like the main tonal center, even though one is E and the other is B. But this isn't bitonality to me, though it might be arguably similar. I think this is more like twelfths/tritave equivalency and feels about the same as octave equivalency, just lacking the life-long cultural reinforcement. Maybe in a 12ths/tritave-based theory the E and B would actually get the same name, like we usually do with octaves (for example, the Bohlen-Pierce tuning is tritave-based). Sure, 12ths/tritaves don't sound really identical, but remember that octaves don't either...

EDIT update 11/15/10:
I should mention that the technique I used (playing some music and then checking whether an isolated tone seems to fit) has technical terms. The preparation listening is called "priming" and the isolated tone is called a "probe tone." These are some of the standard methods used in empirical studies of music cognition. My simple demonstration could easily be repeated in controlled testing administered to a number of listeners from different backgrounds. Different priming and different probe tones could be used. The results could better clarify my hypotheses about the potential for listeners to learn 12ths-equivalence (or other alternate equivalences), though additional varieties of tests would be needed to truly be conclusive.

Getting old stuff in order still

I've reworked the old content from my previous website to be at least functional finally. While most stuff is now included in this blog, the site has some autobiographical details on my music background, information about my compositions and published recordings, and a good selection of streaming music.

Check it out here: www.wolftune.com/music

First Day With Fretless Guitar (re-post from old site)

I have made a total of one YouTube video so far. Nearly a year and a half ago I got a fretless guitar, and I recorded my very first experiments in the first hour of trying it and edited a quick video.

I am now consolidating my web content into this updated blog, so here's this now-old video.

Further thoughts & a link on purpose, meaning, motivation

My personal struggles with ethics and purpose are described a couple posts ago:
http://wolftune.blogspot.com/2010/04/massive-blog-post-revalation-on.html

There I described my questions about Kant's categorical imperative and the stress of responsibility I may have had due to the implication that one should be a role model for the world. I decided to let that go.

However, I have not become comfortable with the idea of simply pursuing purely selfish desires. I recently was referred to an excellent video featuring Dan Pink. This particular one happens to be an entertaining and accessible, illustrated version of a talk he has given a few times that is also available in live video such as from TED. The illustrated version is here:

He describes the well-studied, essentially proven fact that external rewards actually reduce achievement for tasks that are anything more than mindlessly mechanical. When people are faced with complex challenges with no obvious solution, they are motivated primarily by three things: A. autonomy, B. desire for mastery, B. purpose (no particular order).

This isn't dramatically new to me, but it always helps to be reminded of things, especially with such clarity. Here's how these relate to my situation:

A. I am autonomous. I'm self-employed. Despite mixed issues with that, I'm very hesitant to give it up to go join some business or academic bureaucracy. I appreciate my autonomy and am definitely motivated by my ability to make of my business whatever I can. I do have desire for collaboration though, and I'm still working to figure out the best decisions in that direction.

B. Desire for mastery: the challenge that music presented is a large part of why I ended up in music. Mathematics and science were interesting but very easy for me as a young student. Being more understanding of mathematics than most of my peers or teachers, I felt unchallenged and had low motivation. Music was an area where I felt just talented enough that I believed it wasn't impossible for me to achieve something but still felt very challenged and humbled. On a side-note, I read that renowned biologist Richard Dawkins claims the reverse: that he had a knack for music and could pick up tunes on any instrument. He liked the music but found it un-motivating and never really pursued it with any seriousness. Science he found more challenging.
Anyway, today I have an issue with this mastery/challenge. I have progressed to the point where I am no longer completed awed by musical skills. I know that if I invest certain levels of time and energy I can achieve things I once found baffling and impossible. I am definitely still challenged. If I decide to pursue a musical skill, I do not find it boring and easy. But I have lost some of the passion I once had now that I know enough about what to do and what it is like to master something.

This brings me to the most important element:
C. Purpose? What purpose does music have? For what purpose am I working at what I do?
In previous posts here I have described music's functions and benefits. Great, that's something, but what purpose do I have in that? I am unconvinced that anyone needs me to make new music in order for those functions and benefits of music to be available to the public. I do realize that my particular students appreciate and gain from my teaching and that's motivating to a certain extent. If I have a student who is happy, well-adjusted, and not especially desperate for the benefits of music but is motivated and interested enough to still work hard... well, it's fun to teach them but it doesn't seem of extreme importance in the grand scheme of things. I don't think of music as clearly superior to most other fields as some musicians seem to believe.
Perhaps music therapy would be worth pursuing. There are other angles I've thought of as well. Perhaps I just need to develop my teaching in a direction that is even more purposeful and focused and promote it as such. Sometimes I think I really should have gone into science or engineering after all, and maybe I still should...

At this point, I don't know all the answers, but it is nice to have an answer as to a framework for thinking about it all. The need for purpose is what drove me to think about Kant's categorical imperative, and maybe if I interpret the imperative as just meaning that everyone should be purposeful and consider the greater public interest in their actions, then maybe I can still hold on to that value without feeling like I'm responsible for anyone besides myself. For now, autonomy and mastery are successfully keeping me going and I'm fulfilling my desire for purpose by putting in time and energy exploring the question.

One simple way to connect to students and be a more humble, patient teacher: practice upside-down...

Try doing something backwards or different than usual. For instance, as a typical right-handed guitarist, try flipping the guitar around and learning to play it left-handed and upside-down. One can get a sense of the experience of beginners.

This is distinct from just learning an entirely new skill. When a teacher learns a new skill, they have no way to evaluate it and are truly just a beginning student. Of course, it is important for teachers to learn new things and that helps their teaching; but it isn't the same as playing upside-down.

When learning a new skill, it is hard to judge what to expect and whether it is being taught well; it is hard to judge the value, and hard to know what it will feel like when mastered. When reversing a guitar (or whatever is comparable for any other skill), all of those things are clear. Thus, the teacher may evaluate and understand the challenges in a way that no other experience can elucidate.

massive blog post: revelation on morality, welcome to new version of blog

This is the first blog post I've made in over 18 months.

This morning I had a revelation that seems potentially life-changing.

tags/labels revolutionize style and genre in music

A "genre" is just a set of expectations you could reasonably apply to a large set of items. If all or most of your expectations when you hear a "blues" song are met, then obviously it was useful to call it "blues." If it had a few elements that remind you of that, then maybe "bluesy" would make sense. If there is almost nothing that anyone could expect based on hearing other music before then the piece in question HAS NO GENRE. Genre is not a quality absolutely inherent to everything. It only happens when a large enough quantity of something is able to be identified with a set of distinguishing characteristics. When a new genre starts, it isn't actually a genre until we've heard enough of it to know what to expect.

Categorizing things into styles and genres is a natural and useful part of human culture. Part of artistic, creative interest is in deviation from an existing norm. Those norms must be identified and embedded in the listener's unconscious in order for deviation to have any effect. On the other hand, some art is not intending to deviate, but is merely not related to any known genre, and people's openness to such new things is quite varying.

The future is promising. By using tags/labels in the digital computer world (in programs, like this blog site, that do not limit the number of tags versus the awful mp3 genre id tag that requires a single listing for any mp3), a piece with blues elements can be tagged as "blues" or as "bluesy", without sticking into any box. Tags are not exclusionary or restrictive. We may use as many or as few tags as are useful. Hopefully we will see the extinction of arguments that say, "this piece isn't rock, it's funk-blues." Instead the discussion can be "does this have enough rock elements to have a rock tag? I know it should be tagged with funk and blues, but maybe rock too?" In the past, a record had to be put in either the rock or the blues section at the record store, unless they had copies in both, which was very impractical. Now, an mp3 at an online store can be in both categories at once, and others as well. And we can choose to search for only songs that have both rock and blues tags if we like. Lots of people have tried to put things into one box or another, and those who rebelled tended to reject the entire concept of genres and labeling. I think tagging and digital technology liberates the discussion. I hope this will also encourage musicians to take a much more fluid view of their own stylistic identities.

What and why need to come before how

It is all too easy to spend immense time learning and preparing how to teach something. Teachers can fail to question the value or accuracy of what is being taught. Certainly, it is better to fail at an attempt to teach the most valuable lessons than it is to succeed at teaching incorrect ideas.

The challenge of the most dedicated teacher is to balance these questions. People who spend all their time on the facts and ideas will certainly be lucky to have even a moderate ability to express them accessibly. On the other hand teachers can sometimes be so focused on how that they actually become more and more dogmatic, close-minded, and even defensive regarding "what" type questions. Those questions could undermine all the work they've done preparing the unquestioned subject.

Of course, we need to be practical. I would simply encourage all teachers and parents, (well, everyone actually) to try to return to what and why every now and then. And if those can't able to be answered with certainty and you can't afford the time to explore further, then at least keep the question open - allow for it to be questioned by others - and just don't teach it as absolute.

Students should not only question what and why but they need to remember sometimes to think about how things are taught. If a student questions how things are taught, then sometimes ideas that are poorly presented can be figured out anyway -- by understanding what is wrong with the teaching style. Or they can figure out how to learn for themselves. They may even empathize with the challenges the teacher faces. Finally, if they are like me, they may be inspired to become teachers themselves, if partly to do a better job than some of the teaching they received.

There is never anything wrong with questions. Everything should be questioned. We just have to realize that questioning everything means a very small proportion of questions are able to be answered. So instead of randomly questioning, we need priorities. If we ask the big what and why questions first, we'll have the best shot at everything else making sense later.