As the subtitle of this blog suggests, it is supposed to be about math, music, and me. So far, I have been writing mostly about math (which is indeed my main passion). However, I also have a great appreciation for music—especially for contemporary classical music. So far, music has been the main topic only of one blog post. To revive this theme, I am dedicating this blog post to my favorite composer—Philip Glass—and two of his piano pieces: Opening and Mad Rush. I also included recordings of me performing them (scroll down to listen).
Philip Glass is an American composer who has spent most of his life in New York City and is still living there. He describes himself as a composer of “music with repetitive structures” (which, I think, is the most accurate description one could possibly come up with) and is credited as one of the founders of the minimalism movement in music.
I came to know his music during a summer I spent in Princeton, where I shared house with several young composers studying at Princeton University (Troy Herion and Mark Dancingers were among them). I remember one evening they had a very heated debate about the music of Philip Glass and whether it should be considered music at all. After all, his music is so repetitive, he is not ashamed of copying himself, and some of his pieces consist of simply counting 1,2,3,4,… (if you start listening to this particular piece, I recommend you listen to it till the end).
At the time I did not know who Philip Glass is, so I decided to learn more about his music. I came to really love it and now consider him my favorite composer. I appreciate his music both emotionally and intellectually. In fact, I plan to have the 1,2,3,4,… piece (called Knee 5) performed at my wedding.
As a mathematician, I also really enjoy his unusual rhythmic structures, which I will briefly discuss in this post.
Opening is a piece for piano and is named so because it is the opening piece of Glassworks—an album that introduces his music to a general audience (it serves this purpose very well and I highly recommend it to those who are not familiar with his music). This piece does not have a clearly distinguishable melody. Instead, the melody arises from simple repetitive patterns that overlap and intertwine with each other. You can listen my rendition of this piece here (including a wrong key I hit in the beginning):
The piece is based on a simple periodic pattern: the right hand does three equally spaced beats while the left hand does only two within the same period of time. To understand how this works, you can try to tap the following rhythm with your hands—repeat it several times and count 1-2-3-4-5-6 while you are tapping (you can also say the phrase “not dif-fi-cult” or “cold cup of tea” instead):
Once you have mastered this, here is the next exercise! Use your thumb and middle finger of both hands to tap this rhythm (again, count 1-2-3-4-5-6 out loud while you do it):
|Right middle finger:||·||·||X||·||·||·||X||·||·||·||X||·|
|Left middle finger:||X||·||·||·||·||·||X||·||·||·||·||·|
What you end up doing is simply oscillate the two fingers of both hands at different speeds! It might help if you note that some taps are in-between the counts, and that on 1 and 4 both hands synchronize (on 1 you tap the right thumb and the left middle finger at the same time, but on 4 you tap both middle fingers simultaneously).
It is quite tricky to master this, but once you get it, you are ready to play the Opening! Indeed, you just have to put your hands on piano and repeat the same pattern on different keys and with different fingers. This is how the first line of the score looks like:
This is the simplest example of what is known as polyrhythm—when melodies with different meter are played at the same time. This particular polyrhythm (two beats with one hand and three with the other) is a trademark of Philip Glass—it appears in many of his piano pieces (including Mad Rush discussed below).
Glass’s understanding or rhythm and his distinctive musical style was strongly influenced by Indian classical music during his collaborations with the famous sitar player Ravi Shankar and tabla player Alla Rakha. Glass was also deeply impressed by Steve Reich’s Piano Phase when he first heard its performance. Given all this, it is no surprise that polyrhythmic structures are so prevalent in Glass’s music.
Philip Glass wrote this piece on the occasion of the first public appearance of the 14th Dalai Lama in New York City in 1979. At the time it was not clear exactly how long the wait before his arrival would be, so Glass was asked to compose a piece of somewhat indefinite length (which is not a problem for him…). Here is my rendition of this piece:
As you are listening to it, you can clearly notice that it alternates between two themes (you can even see this in the wave diagram above!). One theme is peaceful and meditative, but the other is fast and a bit frantic. Glass himself explained that the two themes represent the play of the wrathful and peaceful deities in Tibetan Buddhism.
Glass went to India in the 60s and came in contact with Tibetan refugees. During this time he started to gravitate towards Buddhism. Practicing Buddhism and meditation has become an important part of his life. He is also known to be a strong supporter of Tibetan independence.
Let’s take a look at the sheet music again. You can immediately recognize Glass’s signature in the peaceful theme—it is also based on the 2:3 polyrhythm that we are already familiar with (in fact, if you have learned how to play Opening, then playing this part is relatively straightforward):
The wrathful theme, however, consists of very fast arpeggios that are played symmetrically with both hands:
This differs from the peaceful theme not only in tempo, but also in both hands being completely synchronized (which is not the case at all when you are playing a polyrhythm). In fact, they are so synchronized that you even use the same finger with both hands on all notes!
To understand what difference this makes, you can do a simple experiment: try to play the above arpeggios (or just tap your fingers) so that the melody goes up/down for both hands at the same time. Alternatively, you can just play the top line with both hands simultaneously. While both hands are still synchronized, in this case you will always be using opposite fingers (e.g., when the right hand uses the thumb, the left hand uses the little finger). You may notice that this is slightly harder than the original mirror symmetric arpeggios! Indeed, it turns out that human motor system has a preference for mirror symmetric movements (especially at high speeds).
In any case, this part of the piece reminds me of New York City and it represents the title Mad Rush very well. If you listen carefully, you might even notice that the last two arpeggios break the rhythm (they include two extra notes). This is how the score looks like (notice also the unusual 14/16 time signature):
When I listen to or play these two bars, it almost feels as if my heart skips (or does an extra!) beat, because things get totally out of sync. It’s like running while carrying a big box and almost tripping over. When this tipping point is reached, the piece stops and goes back to the peaceful part again…
I hope you enjoyed reading this post and listening to the music. If you are interested to find out more about the music of Philip Glass, you can start with Glassworks. In fact, you might already have heard some of his music, since he has written lots of music for films (e.g., The Hours). If you like piano music, you can also check out Metamorphosis. I highly recommend his Qatsi trilogy: Koyaanisqatsi, Powaqqatsi, and Naqoyqatsi (all of which should be watched as movies if possible). If you are more adventurous, you can even try to listen to all of Einstein on the Beach in a single sitting (which I did a few years ago when I saw this opera in Toronto).
I am grateful to Steve Tulloch for giving me the opportunity to perform these two pieces at The View From Here and to Steve Errey for recording my performance. You can check out Steve Errey’s Studio S page on SoundCloud for more recordings of other talented young artists. Finally, I also want to thank Mohammad and Jesslyn for the photos, and Amandine for her feedback and the Scholarpedia reference on motor coordination.