My Music Thoughts
If you have something to add, or if you simply disagree with me, feel free to comment below!
Starting from Sample Space
What is the music sample space? The largest (whole) music sample space is the set of all possible combinitions of musical elements. Actually, the whole musical set is far too large. Great composers can extract the exact subsets which are good compositions.
When composing music, we try to reduce our sample space, or eliminate the uncertainty in our mind. This process is like finding a winding path towards a beautiful hidden corner. For example, theory of harmony tells us what kind of chords are recommended to be used in certain musical contexts, and this reduces the chord sample space to a few so-called consonant chords.
Now, we may ask: what is the property of those good subsets, and how to reach them? Well, a musician can write a thousand-paged essay to explain that, maybe from the point of view of melody, harmony, counterpoint and music structures… I am not going to expand those topics here.
Instead, if we are inspired by information theory, we can find the commonplace of good musical works: they bring something unexpected. Those unexpected can be good melodies, satisfying chord prograssions or impressing performance skills etc.
This may be a little hard to understand. I will explain this “unexpectation theory” in the next section.
Music Appreciation: from Information Theory’s Perspective
Appreciation = Eliminate the Uncertainty
Appreciating music is like travelling: we aim to encounter the unknown world. From my point of view (and also an information theoretrical point of view), appreciation is centered in acquiring knowledge, eliminating uncertainty, or adding up regularity. (And no matter what we appreciate, like music, sculptures, maths etc.)
Therefore, just like famous attractions have some landmarks which can “surprise” the tourists (e.g. the spectacular landscape of Yello Stone Park which tourists had never experienced on the scene), musical works should contain something out of listeners’ expectations. This is the intuition from information theory, which defines information as the existance to eliminate uncertainty.
When appreciating music, we try to cumulate more information by decoding the music signal. Again, in information theory’s point of view, we are eliminating the uncertainty of the world. Our validation music sample space is then enlarged.
Why Listening to a Music Segment Again and Again
You may be curious about the case that we may listen to a piece of music again and again, enjoying an impressing music moment for many times even if we are familiar with that moment. Why are we so keen on the moment which seemingly cannot surprise us? Well… I explain this with our limited capacity of music decoding.
Remember one of the music pieces which did not impress you when you first hear, but later when you happened to listen to it again, and then it captivated you. You may have listened to a music segment in that piece over and over again, and each time you listened to that piece, you expected to reach that segment. This was because, when you first listened to that music, you did not successfully decode the music signal into exciting information. Later when your decoder worked better, you captured the moment that important information was decipherd, and you wanted to emphasize that moment over and over again until you could easily decipher the similar information.
You appreciation skills are improved during this process!
Then comes the question: how to evaluate “appreciation skills”?
Our Appreciation Skills Need Training
Different people are sensitive on different things, and this is because of the diversity in their capacities of (music) decoders. For example, many listeners cannot be moved by atonal music because they focus on the jarring sound and cannot acquire artistic information which is impressive to musicians, who may be impressed by the organization of the piece or fantastic chord tensions. This is similar to: illiterate people cannot appreciate the beauty of Euler’s $e^{i\pi}+1=0$ because they do not know the prior knowledge associated with this formula.
Again, our appreciation skills are in fact the capacity of our decoders.
How to train those skills? First, we need training data. The key is trying to appreciate unfamiliar music and look for the surprising moments, which is like visiting an unknown place on your own. Another choice is to check music reviews (some of which are logs of the beautiful moments of a music piece), and this process is similar to visiting an attraction with a tour guide.
Of course, even if we are well-trained to appreciate music, we cannot despise the music we have already familar with. After all, we upgraded out decoder system thanks to them. And more importantly, there may remain something we did not discover in the music even if we think we are well familiar with.
How About Music Composing
We have established the point that listeners are to be surprised when appreciating music. Therefore, composers have to surprise the listeners in order to create impressing works.
What does surprising the listeners mean? This is not necessarily mean that they have to use “$\pmb{pppp} \rightarrow \pmb{ffff}$” to startle the listeners. This is to say that they have to convey important information which is out of listeners’ expectations, just as we have already discussed.
The Case of Pop Music
For listeners who did not receive a lot of appreciation training, a fairly normal chord progression could grasp their attentions and make them feel surprised and happy. Hence, most pop music composers achieve this by applying easy-listening chord progressions and music structures to their music, and they focus on writing “surprising” melodies, which are normally easy to appreciate.
In this case, the music sample space of the composer may be greatly larger than listeners. Composers should find the music samples at the edge of listeners’ music sample spaces, and this is enough to entertain listeners.
The intersection of the music sample spaces of most people form the set of universal pop music. What about the rest (which may be a lot larger)?
The Case of All Kinds of Music
For listeners who are well-trained, they may seek for an unexpected music element combination (remember, it is actually a sample point of the music sample space). This challenges composers to explore their music sample spaces, and this is the fairly interesting: because not everyone share the same music sample space, the hidden corner you visited is not likely to be visited by others.
Before surprising the listeners, composers had better first surprise themselves, and this needs composers to have be skillful at appreciation, which means that they have to dig out more and more information. In other words, composers themselves have to own powerful decoders.
But where do the surprising things come from? You know that even if you have a good decoder, you may not have musical codes to feed into the decoder. And this is where generative models are crutial!
Randomness, Generative Models and Sampling
To be frank, we do generation by implementing randomness. We randomly sample music data points from our music sample spaces. Main differences between composers lie in 1. the structure of music sample spaces, 2. their sampling strategies and 3. their appreciation skills. Appreciation skills are analyzed in the previous section. Therefore, I mainly talk about the other two.
First talk about music sample spaces. Remember the first section? I put it here again…
When composing music, we try to reduce our sample space, or eliminate the uncertainty in our mind. This process is like finding a winding path towards a beautiful hidden corner.
Yeah, good composers know how to reduce their sample spaces. They may use harmony theory to exclude the bad chords, and thus the amount of remaining music data points are smaller. They also have other constraints on their sample spaces, like: melody alignments, music structures, instruments etc. You know, it is fairly easy to sample a data point from a small sample space according to their wills!
In order to be skillful at reducing the sample space, composers need to know what compact sample spaces look like. Therefore, they had better learn harmony theory, music structures, counterpoint and different music styles. All aspects lead to certain compact sample spaces, and great composers are skilled in pinpointing the required compact sample space.
Now let’s go to another question: even if we have a compact sample space, how to sample a music data point from that space?
Randomness and Sampling Stratagies for Music
Notice that the famous generative models (e.g. HMM, VAE, GAN) all depend on randomness, and they achieve generation by sampling their sample space in various ways. With a good sampling stratagy (e.g. conferring to the thinking pattern of first determining the music structure), you can have a good start point to converge your chaotic mind into a relatively small sample space.
Music is contextulized. Therefore, we do sampling according to contexts. Most composers may sample their music measure by measure, and the result might be that their music do not have a good structure. This is similar to the strategy of RNNs.
Another strategy is to first determine some global constraints and then do sampling over time. This is the case of most composers because this safe guards a good music structure. This is similar to the strategy of transformers.
What about music with unexpected beautiful organizations? Chances are that composers see the music structures as non-deterministic, and they have a unique subset of sample space for music organizations. They first sample a music organization, and then follow the traditional way as described before.
Of course there are many many other ways I have not mentioned. For example, some avant-guard compsers may record natural sounds and reorganize them as a music work, which is totally unconventional. This is where composers have to explore various of ways of sampling, and this is where to expand the border of art!
Go Practice
A last topic, how to practice composing music?
We have to know that, when practicing to compose music or perform music, we try to revisit the winding paths towards beautiful hidden corners. This is because we are humans rather than GODs (who know the global maxima), and we have to struggle to do optimization, just like training the machine learning models.
On the one hand, if you find an unvisited “winding path” (e.g. a unique melody), you can revisit it over and over again until you can introduce it to others, and then you become a composer! You must train yourself to be familiar with the process of encoding the “winding path” into audios or music language. Remember not all people have visited those winding paths. Therefore, practicing to efficiently encode your music minds into music signals can let a composer quickly impress the audience.
On the other hand, your decoder should also be sensitive enough to discover the winding paths in your chaotic mind or the colorful world! In this case, you should try to listen to more music you did not tried, and push yourself into digging out surprising information from them, or even imitate them. Just as I had explained previously.
Another thing you can really do is to optimize your sampling stratagy, or the composition process. You have to try a lot in order to find a desirable process. For me, I am kind of conventional that I use draft scores to compose music instead of use DAW. What about you? Try try try!
Finally, your chaotic mind, the generation source. Well, your mind can be influenced by external conditions like changing moods or spirits. Therefore, try to find different envirments of composing.
Go practice, think more, and appreciate more!
Algorithm Composing
In fact, all of my discussions above can be instructions for designing algorithm composing models.
Algorithm composing models face the same challenges: to constrain the sample space, to have a sample stratagy, to have a good decoder and a good discriminator, to be optimized in order to be familiar with the “winding paths”…
Interestingly, you can find corresponding topics in machine learning in terms of each of the aspect I mentioned above. Let’s go and see!
A Problem of My Theory Above
[Updated on 30-April-2021]
I might have taken it for granted that we tend to appreciate music which brings us surprise. How to explain the phenomenon that the music which sounds common is popular? For example, experiments have shown that car-radio listeners tend to prefer music which sounds typical (the so-called “sticky music”), rather than classical music which have abundant artistic information. And how can we explain that we listen to a piece of music again and again, even if we know that we are quite familiar with that? For example, after finishing composing a piece of music, I tend to listen to it again and again even if I am the composer and I know that I know a lot about it; and sometimes I listen to a pop song again and again. How to expain them?
In my previous opinion, I defined happiness as knowing something unexpected. However, a friend of mine inspired me that happiness might also come to us when some of our expectation are met. (We were discussing why people get unhappy, and we found that a great reason is that their expectations, from either subconscious or conscious, are not reached.)
What is expectation? Can we explain it with a definition, like “surprise is elimination of uncertainty, which is measured by entropy”? Well… For me, it is a big problem to be solved. But I have an idea that habit is a kind of expectation. Most of our habits work in our subconscious; therefore, when we find ourself appreciating a familiar old song, we may say that we find our subconscious is feeding on that old song by meeting its expectation.
Here, the two theories seem to fight with each other: when we define happiness as eliminating uncertainty, we may say that things are happening out of our expectation (if not, we may not be surprised), but we cannot explain the case when we are appreciating a piece of familiar music and enjoying one particular moment again and again; when we define happiness as meeting our expectation, we cannot explain the euphoria of listening to a great symphony. There must exist a theory which is able to blend those two into one (and I think the great theory of happiness exists because of Hegel’s dialectics).
I think the main difference between those two theories lie on the quality of happiness: the “surprise theory” dipicts a “tired but happy” case, while the “expectation theory” dipicts a “safe and sound” case. Our habits tend to listen to music which is familiar to us, and they do not consume much of our computation resource in our mind; when we find ourself being able to appreciate a symphony, we have consumed a lot in our mind, and we find that we did not consume it in vain because we found something unexpected.
How to explain when doing experiments, we get an everything ruined (e.g. unexpect error in coding), and get mad? How to explain when listening to a piece of music, we get startled by a wrong note played by the music performer and get angry? In those cases, our expectations are not met at all, but we consumed a lot of mental energy. It seems that, when knowing something unexpected, we get either happy or unhappy, and we really have to tell the difference between those two cases.
To sum up, we have four cases:
- we did not consume much mental energy, but get something expected;
- we consumed a lot of mental energy, and get something expected;
- we did not consume much mental energy, but get something unexpected;
- we consumed a lot of mental energy, and get something unexpected.
We did not discuss case 2 and case 3; but we demonstrated that case 1 brings happiness and case 4 could bring either happiness or unhappiness. When thinking about our experiences, we could find that case 3 could bring either happiness or unhappiness, while case 2 is complicated…
It seems that happiness is a rather complicated thing to be explained. I believe that it could be measured and decomposed into more elementary concepts. However, my assumption that “elimination of uncertainty brings happiness” might be incorrect; or I should endow this statement a more suitable definition.
本文作者： lucainiaoge
本文链接： https://lucainiaoge.github.io.git/2020/09/03/my-music-thoughts/
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