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Mario Music of Golden Proportions

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You could say that video game music is becoming much more epic as the years go by.

A growing number of MIDI possibilities are at the disposal of game composers, and the number of streaming, fully orchestrated soundtracks is continually increasing. There are those that say newer game soundtracks are less memorable, because soundtracks for sequels and franchises seemingly rehash old music, while soundtracks for new IPs seem to be very contemporary, ambiguous, and often only ambient. However, I would disagree with anyone that says new video game soundtracks can't be just as memorable or even better than the soundtracks of old, as I'm going to turn to a recent adventure of everyone's favorite plumber to help demonstrate my case.


The music from the first Super Mario Galaxy takes some of the best traits in Mario music, which are then redefined and augmented in a process that ups the presentation to a level unparalleled by any other Nintendo game currently available. There are no doubts in my mind that many of you consider the music of Super Mario Galaxy to be one of the most impressive and memorable video game soundtracks in recent years. Today, we're going to look at the music a little more objectively, revisiting a concept I used over three years ago to illustrate a musical phenomenon that seemed to pop up in music from the Zelda series. I hope it both boggles your mind and challenges you to listen to video game music in new ways. Read on, my friends!


For those that weren't around last time, here's a brief summary of what the Golden Ratio (a.k.a Phi), the natural and mathematical phenomenon behind this article, is and why it has significance. If you think of the length of a line segment as being equal to 1.0, there lies a point at approximately 0.618 where the ratio of the smaller segment to the larger segment is equal to the ratio of the larger segment to the whole, as illustrated above. For clarification, Phi is actually approximated at 1.618 (where the relationship of .618 to 1.0 is the same as 1 to 1.618), but I'll be using its reciprocal of 0.618 because I find it easier to multiply by 0.618 rather than to divide the whole by 1.618. As long as the proper number is used with the proper operation, either can be used, as Phi is merely representative of a relationship between two numbers.

Why the big fuss? Well, it's a mathematical phenomenon, as demonstrated by the Fibonacci Series of Numbers above, that seems to be found in many aspects of nature such as the structures of plants, animals, and even chemical compounds. In addition, it has been applied to visual artwork such as paintings and architecture, and many believe that the human mind subconsciously perceives anything that exhibits the Golden Ratio in any notable fashion as an object of beauty. It's considered natural symmetry. In music, the theory lingers that compositions designed (both intentionally and unintentionally) with forms that stress the Golden Ratio are aurally more pleasing to the ear and mind. The most significant example is Béla Bartók's "Music for Strings, Percussion and Celesta," which lacks hard proof that Bartók consciously used the Golden Ratio, but nonetheless is littered with chord, rhythms, and structures echoing the Golden Ratio and the Fibonacci Series of Numbers.

That should do it for a crash course on the Golden Ratio, though remember that this is all just theory. Mario composer Koji Kondo doesn't consciously use the phenomenon in his compositions, though it continues to pop up in his work. We've already seen some prime examples of the ratio in music from The Legend of Zelda and other games from tha franchise, so let's turn around and launch right into the music (YouTube recordings from Artictoc) that helped propel our favorite plumber out into the heavens.

"Good Egg Galaxy"

One of the first fully orchestrated songs you'll hear upon playing a game of Super Mario Galaxy is the theme from Good Egg Galaxy. It's peppy, majestic, and has that "Go get ‘em!" feel to it. For this one, we need to recognize right away that this piece is meant to be looped. When you hear it as it is in-game, you'll realize that it just starts repeating endlessly until Mario (controlled by you) triggers an event such as encountering a boss, losing a life, or something of the sort. That spot in the music where the loop begins will mark the end of our measuring sample, and is located 52 measures into the song, or at roughly 1:22 into the recording above.


Now that we have determined the 52 measure chunk we need to look at, we can break it down into three sections. The first section is a two-bar brass fanfare followed by lengthy string melody, which lasts a total of 18 measures (or 17.75 measures, if you're picky and notice the flute's single pick-up quarter note before measure 21). Next, we introduce the woodwinds, evident by the flute melody that starts in measure 21. While the flute doesn't hold the melody for the whole 16 measures, this section technically does last for 16 measures and can be broken down into smaller, four-measure phrases. The key one here is when that fourth smaller phrase begins – after the flute melody ends, the timpani enters, and the strings dominate the crescendo into the next section. Finally, we have a 16 measure section featuring a forte ("loud") string melody to close out the loop.

If we take those 52 measures and use our special ratio to find the Golden Section, we should hope to find something significant happening roughly 32.14 measures in. Since we're measuring in musical units, this is interpreted as 32 whole measures being completed and something happening within the first quarter note of the 33rd measure. Remember that crescendo and timpani entrance that I mentioned was sort of a big deal? It happens right at measure 33 (about 51 seconds in).


Now, to make things even more exciting, let's take a look at the Golden Section OF the Golden Section. Multiply 32.14 by 0.618 and we get 19.86. Or, take the original 52 measures, and find the Golden Section from the end. 32.14 measures from the end. Same point? Same point. This is called the Negative Golden Section, and the 19.86 means that if we're lucky, something important happens at the end of measure 20 (since the 19.86 means we don't yet have 20 full measures), just before measure 21. Remember that flute pick-up note we said comes in right before measure 21? There it is, 32 seconds in!

"Gusty Gardens"

In addition to Good Egg Galaxy, Gusty Gardens is another one of the game's locations that managed to get a truly grand accompanying track. In fact, it's so special that many parts of this piece show up in various other arrangements throughout the game. For example, the "A" theme from Gusty Gardens (the oboe melody near the beginning) can be found in a beautiful piano solo towards the end of the game, titled "Birth." The "B" theme (the string melody around 1:26) recurs in the "Purple Comet" theme as well as the game's title track, both of which we'll look at on the next page. It's a clear indication as to how iconic the themes in this piece were meant to be.


The music for Gusty Gardens is another looped song, so we won't be looking at the entire recording above. The main portion of this piece begins at about 10 seconds into the YouTube clip above, and last for 64 measures until about 1:51 seconds into the recording. If you do the math, we should end up with a Golden Section at 39.552, or the end of measure 39 heading into measure 40 at about 1:13 into the recording. At this point, the cornet (trumpet) is joined by the oboe in a playful musical sequence in which the same musical idea is repeated a major third down as part of a transition into the next major section. (The sequence then begins continuing down an additional third, before interrupting the sequence to wrap up the transition.) This in itself doesn't seem like a particularly important point in the song, but let's take a look at the Negative Golden Section and see if there's any connection.

The Negative Golden Section should occur at measure 24.448, meaning in the first half of measure 25. In the clip above, this can be heard at 48 seconds into the song, where we hear a very prominent string climax on a half note (essentially the entire first half of measure 25). But wait! If you continue listening from this point, what do you hear? That string climax is the start of a short musical idea, which is then repeated it a step down, and then begins continuing down again before wrapping up this segment. Didn't we have something like this before? That's right, it's another musical sequence! It looks like sequences are the big aspect for this particular piece, which becomes evident if you go back and listen to the entire thing.


So far, we've looked primarily at structural Golden Sections by taking the length of a piece or loop and broken its duration down according to the Golden Ratio. Let's continue on and see what else we can find in Super Mario Galaxy!

"Purple Comet"

Next up is the "Purple Comet" theme. It plays in the game when Mario encounters the Purple Comet (obviously), and its quick tempo gives the player a sense of urgency, as if you have a clear mission to head for that comet. At the same time, the song combines the usual strings and percussion sounds with electronic sounds, a techno beat, and more into a surprisingly fun and catchy tune.


"Purple Comet" is another looped song, the main portion of which lasts 52 measures long from about 6.5 seconds into the recording above until just after the 1:31 mark. Some quick calculations give us a Golden Section at measure 32.14, or literally at the start of measure 32. In the recording above, measure 32 starts at 59 seconds in, where we find that the techno beat has kicked into high gear and we get that special "Gusty Gardens" theme! In addition, I personally find this to be the most fun part of the song (probably because of that catchy techno beat), and I'd bet that many of you do as well.

Unfortunately, the Negative Golden Section doesn't quite line up with anything here. We should expect something interesting on the last beat of measure 20 (as the Negative Golden Section calculates to 19.86), but nothing happens until 1 beat later, when we begin a new 4-bar transition. At this point, we get some interestingly weird chords – the bottom voice alternates between a Bb and C (2 half steps), and the voice above that alternates between an Eb and F above it (2 half steps), always separated by an interval of a "perfect fourth" (5 half steps), with the C and the Eb additionally separated by 3 half steps. All of these intervals are the prime numbers that begin the Fibonacci Series, and it would have been nice had this all occurred one beat earlier. Alas, being "close" just isn't as impressive… but it's worth a mention.


"Super Mario Galaxy"

Unlike the previous examples where the song's tempo stayed the same, the title track from this game (used for the game's ending credits) has what are called ritards, cesuras, and a tempos in music, i.e., points where the music slows down, pauses, and then returns to the original tempo, respectively. Because of this, our calculations for this composition will need to be done in terms of time instead of measures to account for the fact that our ears will hear something that measure numbers don't tell us. After all, music is an aural experience, isn't it?


We'll cut to the chase on this one, since I counted nine distinct sections in this song and I don't really feel like explaining them all. This song is 3:57 long, because we're going to cut out the almost one whole second of silence at the beginning and the several seconds of silence at the end – they're not written in the music and are just a product of the YouTube clips being padded with enough buffer time on the ends. From there, we can simply multiply our 237 seconds by our magic number of 0.618 and find the Golden Section at just past 2:26 into the song (or slightly before 2:27 into this recording because of that silence at the beginning). While we're at it, we'll find the Negative Golden Section by either multiplying the 146.47 seconds by 0.168 or just subtracting 146.47 seconds from the total length, and we get just over 90.5 seconds – or 1:30.5 into the song.

That gives us a Golden Section right where a ritard and that one cesura I mentioned occurs, again just before 2:27 into the recording above. It doesn't get much more obvious than that: a clear break in the music where the slowing tempo creates the tension and making you want a release of that tension. Next, the Negative Golden Section of the song may seem a little less significant, but it nonetheless occurs at a distinct point in the music where we switch from a piercing brass sound from the trumpets to the flooding sound of violins and violas, 1:30.5 in. It's not a major structural point in music (as the music is largely divided into 8-bar phrases, and the Negative Golden Section occurs smack dab in the middle of one of them rather than at a start/end), but it's a huge change in instrument/tonal color, which is always significant in music.


If you want even more oohs and ahhs, find the Golden Section in between those two points. Go on, do it. I happens around 125 seconds in, or just before 2:06 in this recording. You'll find a point where the flute/oboe double solo ends, the strings enter, and the harp comes in with a huge glissando (i.e., that thing where they strum the harp really fast), and everyone crescendos into the next section. On one side of it, you have the soft, soothing flute/oboe take on the "B" theme from "Gusty Gardens," and on the other side is the full-sounding, majestic string rendition of the same melody. The gliss and crescendo are a huge deal here, where they serve as the single turning point between the two halves of this segment, building the tension and anticipation across the entire orchestra.


After four examples of the Golden Ratio in the music of Super Mario Galaxy, many of you are probably thinking, so what? Yes, you realize that these are significant points in the music for one reason or another (new instruments entering, changes in tempo, style and texture differences, etc.), and you understand the concept behind the Golden Ratio… but why are these aspects of the music happening at these points a big deal? It all goes back to what theorists believe we perceive as natural symmetry, natural beauty. Some (hopefully very few) of you are hearing these songs for the first time, thinking yeah, these are pretty good songs for a Mario game, but you don't get the hype. Hopefully those of you familiar with the songs better understand how epic these songs are, especially given the history and evolution of music in the franchise.

Just to reiterate, I'm hoping to give you all another perspective as to why your mind likes these songs more than it normally would. If your brain really does react to the Golden Ratio, it's heard these songs, and it wants to hear impacts at certain points. Even as you go along, your brain might begin to recognize the Golden Ratio in those smaller portions of the song, or maybe even in the frequencies of the harmonies, and you begin to subconsciously enjoy the song more and more as more of the song unfolds and reveals more of these mathematical relationships.


For those who think, "Wouldn't you have to hear the whole song first to even know where the Golden Sections are?", you may be partially right – in order to consciously know where these points are, we would need to know the song's length or have heard it all before… but even when you first hear a piece, you'll notice those impact points as you go. Ever hear a song and think afterwards, "That was a bit too short/long," and not know why? Maybe that's your brain hearing those impact points at certain times and using that information to subconsciously figure out ideally how long the song should be based on hearing those impact points and looking for the Golden Ratio. Again, it's all just theory, but it's always good to keep an open mind.

Closing Thoughts

It's always difficult to find prime examples of the Golden Ratio in existing music. Game music and other soundtracks in particular are even harder to work with, since oftentimes those songs need to adhere to the length of a scene. For example, the final example above (Super Mario Galaxy) has significant moments at measures 3, 5, 8, 13, 21, 55, and 89 – all numbers in the Fibonacci Sequence. Yet, these segments are blocked from having that special golden relationship with the piece as a whole, due to the necessity of keeping the song to 121 measures to fit the length of the staff credits roll. As such, any examples of the Golden Ratio found in music, especially those found in significant numbers, are always interesting to come across.


The Golden Ratio with respect to music has always been controversial as to whether it really does have any significant impact on you, especially if a piece wasn't composed specifically to this ratio. The fact remains that there are indeed examples of significant musical moments at points defined by the Golden Ratio, be they impact points, structural boundaries, or changes in timbre. These examples are found in abundance in the acclaimed music of Super Mario Galaxy, which has since been heralded as one of the greatest Mario adventures in several years. Could this merely be coincidence? Possibly. But there may very well be an aesthetic connection that causes our brains to interpret this music as beautiful. We can only hope that when Super Mario Galaxy 2 arrives in stores next Sunday, it will be accompanied by a soundtrack just as beautiful as – or even more beautiful than – the last.

Republished with permission from The Tanooki.

Christian Ponte is a self-made gaming journalist and has served as the Editor-In-Chief of The Tanooki, a video game news and culture blog, since its inception. Also a lifelong musician, Christian is a graduate of the music education program at the University of Illinois at Urbana-Champaign. He continues to be a strong advocate for game music as both a legitimate art form and an effective vehicle for teaching music to kids today.