Friday, January 10, 2014

Math-sicians! Volume 1: The Fibonacci Time Shuffle!

Part One, One: Fibonacci and the Bridge-Time Ratio!!
Fibonacci!! Simply uttering his name causes the mathematicians around you to listen 2, 3 or 5 times more closely than previous! And his divine progression of numbers is enough to make you weep at the feet of your god (Maybe... if you watch this video.) But as it relates to music, well... there's two things that you have to understand first. The Fibonacci numbers 1,1,2,3,5, 8, 13, 21, 34, 55, 89 etc... are derived from the simple method of each number being the sum of the two numbers before it. Equally, the numbers of the notes on the musical scale correspond well with these numbers. A perfect, non-dissonant chord, played on piano or guitar would include the 1st; being the root note, the third, the fifth and the eighth, which is, of course, the first over again, one octave higher. However, flatten out all those flats and sharps and include them in the equation and once again it looks to match up with Fibonacci. A perfect chord includes the first, the fifth, the eighth, and the thirteenth, which is of course, the root note repeated 12 semitones higher. So far it all seems perfect.
     But wait, there's more! You see there's also this thing called Phi. Phi, the golden number is an irrational number much like pi, except it has a h in it, so you can't eat it. Phi (pronounced 'Fi' like WI-Fi not fee like 'Fifi') is 1.618 (and so on) and derived from dividing any of the Fibonacci numbers by the one before it (except one or two or 3 or 5 or any of the numbers before 89). So the Fibonacci numbers don't actually produce Phi, but they approach Phi and get closer to it as the numbers go up. The number Phi is derived from a quadratic equation (don't make me 'splain!), and so it has a reciprocal called phi (lower case p) which is 0.618(and so on). This is what is sometimes referred to as the 'Golden Ratio'. And using the number phi (0.618) a guy named Gary Ewer found a way to approximate where in a song, the 'break' or 'bridge' begins. This clever musical equation is simply a matter of the full length of a song, all in seconds, then multiply it by phi and convert back to minutes and bing, you have the time in the song where the bridge begins. So, for example. you take the song 'F**king Perfect' by P!nk. The video length is over 4min, but there's a fade-to-black ending and a few seconds of blank at the end, so let's even it out at 4 minutes:
4*60= 240 (seconds)
240 *0.618 = 148.32
convert all back to min and go to 2:28 and bang! There it is! Holeee Mother F-ing OMG!! Fibonacci was a genius! phi landed right at the bridge!! The Golden Ratio is for f-ing real!!

Part Two: Fibonacci: Not so Fast, Smartypants...
2 is also a Fibonacci number, And I often grab my guitar, eager to play the second note in a chord (in honour of Fibonacci and his contribution to music) on either the chromatic or diatonic scale and for some reason, it sounds incredibly dissonant and 'not so perfect' at all! Oh, sorry, was I just supposed to forget about the 2? I thought it was a Fibonacci number and thus 'Golden'! Gee, no one seemed to forget the one or the three or the five or the eight or even the thirteen, which is usually considered bad luck in North America! But what about the 2? Doesn't a great math-metician-musician like Fibonacci want to be associated with the background music from the movie 'JAWS'? No? Oh, I guess great mathematicians only want to be associated with the really perfect and non-dissonant beautiful inspiring music! Well, I guess that's only natural. If you were a smarty-pants mathematician like Fibonacci you'd want to be compared to an artist that's also another smarty-pants like Mozart or DaVinci or someone like that.
   Unfortunately, I also tried the corresponding phi formula on something other than your usual verse/chorus basic musical composition and found results more like the dissonant and imperfect Fibonacci 2: on Bohemian Rhapsody (6min*60=360*0.618=222.48/60=3.708 about 3min,40sec) by Queen the phi-calculated 'ideal place for a bridge'  landed me smack in the middle of the operatic section in Bohemian Rhapsody, well past the opening 'I see a little silhouetto of a man' and twenty five seconds away from the really cool guitar part. Basically neither here nor there. I also tried the formula on Stairway to Heaven (8 min = 460sec 460*0.618=296.64/60= 4.94 or about 5min) by Led Zeppelin and the phi calculation placed me just after the 'Spring clean for the May Queen' lyrical part of Stairway where it's rumoured that if you play it backwards, it contains messages from Satan. Maybe phi switched sides from the Divine to the Devil, but basically it landed in no specific place. So the phi formula works perfectly, so long as the song you choose doesn't happen to be a masterpiece. Either that or some musicians really are worshiping the anti-Christ and therefore using a different set of disturbing and non-divine numbers to write their songs.
    And then there's a few songs that would seem to be entirely friendly to Fibonacci, like Max Webster's Toronto Tontos, which, granted, sounds very weird at first listen, but it has Fibonacci numbers ALL OVER that mother!! The guitar riffs alone are a series of ones and fives. And yet, phi don't play too well in this context. Go ahead and try for yourself or take my word for it and go to 2:16. In either case, I tricked you: There is no bridge in this song. The second 'part' starts about ten seconds before phi and continues to the end. The phi formula however, lands just before the singer proudly announces: "I got no fire on me." Which can be very inspiring words if you're quitting smoking, but not so much if you're a mathematician. (If your phi wants a little more abuse, try the phi formula on 'The Crunge' by Led Zeppelin, but make sure you keep listening until the end for the bridge.)
    The Phi formula of where to put the break or bridge in a song is something thought up/calculated/figured-out by a guy named Gary Ewer, who teaches courses on musical composition. I have never taken his course and I don't know how much of a role that the number phi or the Fibonacci sequence plays in his teaching. But there is no organic numbers-math-magic involved or universality of phi in music. The phi calculation works, not because the golden-ratio is so 'golden' but because most pop songs are structured so similarly. Simply put: In a typical verse-chorus rock or pop song, with three verses, it's not unreasonable to assume that the break, or bridge, if there is any, is about 2/3 of the way through. Whether 0.618(phi) or 0.66(2/3) doesn't really matter because It's usually a difference of less than 2 seconds and I would go ahead and give 2 or 3 seconds leeway on either side. Once you change that basic pop-tune structure, however, phi no longer applies. This isn't because you're playing poor quality music, and it isn't because your music is antithetical to the golden ratio or the mathematical harmony of the universe. Simply put, you've changed the structure of the song to something other than the most typical form. Perhaps you're playing a song that has a longer chord progression, say five-chords instead of four or three, or no specific chorus, or no bridge, or perhaps it's arranged in two or four parts instead of the usual three. Or perhaps, you've made a pact with the Devil...
    We at Very Us Mumblings don't mean any offense to the great Fibonacci, but if everything in music was simply a matter of arranging a collection of notes, arrangements and bridges in order to fulfill some sort of divine progression or formula or mathematical aesthetic... music would be boring and predictable, much like listening to pop music all day long. It would all have the same structure and style, and would essentially be like hearing the same song over and over. And for some people, that's good enough. Phi, phi, the golden ratio and the Fibonacci numbers, and the 'spiral' that they represent may be pleasing to the eye or the ear and universally appealing to mathematicians and artists alike, but they are not the 'ideal' way to arrange a song. Given time, effort and creativity, I'm sure that a musician could come up with numerous musical pieces based on Fibonacci numbers or phi, but sometimes great music is about breaking with doctrine and formula. So although you can arrange a song according to Fibonacci numbers doesn't mean that you should.
    Part two of this blog-entry might be a little dissonant to the ears of a mathematician, but the number 2 is STILL A FIBONACCI, even if papa no like him a no-more!! And if you really like, you can still do things the Fibonacci-way, but there are many other ways to arrange a song!

Part Three: The Half-Time Shuffle: Math applied to music or Fool in the Rain?

Many people, music-lovers, listeners and even musicians themselves have wondered "Why should I learn or listen to that complex song/rhythm/solo? It's too difficult to understand or learn quickly and once you do, it only sounds as difficult and complex as it is to play." It's a question all musicians must face. How much time do you want to spend on learning something... and is it really worth it for the sound that you achieve at the end?
   Well, If any kind of rhythm suggests that complexity doesn't necessarily mean that the music is difficult to listen to, disturbing or unnerving, it's the Half-Time Shuffle!!
A complex rhythm that sounds easygoing, simple and fun. It contains eighth notes, sixteenths, triplets, counteracting ghost notes and one big whole-note. It's kind of a shuffle, but not really. What is it? It's the half-time shuffle, a rhythm used in at least three hit songs by Steely Dan, Toto and Led Zeppelin.
 Bernard Purdie may not strike a person as your typical mathematician, but clearly, in this video, he has his numbers down pat, switching from one  timing to another while describing a very complex rhythm. Bernard Purdie is a consummate drummer with an impeccable sense of time and a professional studio/session-musician. His list of musical associations is both stellar and extensive, having played with James Brown, Aretha Franklin and many others over the course of his career. He made history when he played on Gil Scott-Heron's Poetry-infused song 'The Revolution Will Not Be Televised'. And, in the mid-1970s, Purdie was hired by Steely Dan, a band known for having a laid-back sound while maintaining a high-level of musical skill. It was with Steely Dan that Bernard Purdie was able  to play his own backbeat/invention: the 'Purdie shuffle' on the song 'Home At Last' on the album 'Aja'.
  The Half-Time Shuffle consists of triplets played the whole time: two on the hi-hat and one on the snare drum. Ghost-notes or rebounds on the snare drum are allowed to come into play with the beat, and there's one added whole note on the third beat. The kick drum usually hits on the one and the half-beat before the three.
  And with all this action going on, you wonder how does it not sound busy and aggressive? How can it sound groovy, lighthearted and easygoing? And all I can answer is that it just does... provided it's done right.
 In Led Zeppelin's Fool in the Rain, the half-time shuffle switches to a calypso rhythm and then back again. Drummer Jeff Porcaro changes the bass drum to a Bo Diddley beat In the song "Rosanna" by Toto. With allowance for variation of the use of the bass drum and emphasis of the hi-hat, the Half-Time Shuffle is one of the more challenging and yet popular rhythms that drummers want to learn. Variations of this rhythm have been used in hundreds of other songs, and any drummer would be proud to have the Half-Time Shuffle in his or her repertoire of tricks.

Part Five: "Really Odd" Time Signatures:
In an earlier entry in this blog, we have discussed the basic idea of odd time signatures. In that entry, I may have suggested at some of the complexity that is possible from playing with the time of a song, but that didn't delve into the real complexity and nuance that can be achieved. The possibilities are varied and numerous... and often complicated and difficult. The simplest way to play the rhythm of an odd-time is to just count out the odd-time in notes and sometimes heavy rock bands resort to chunky rhythm-guitar sounds that basically emulate either the drumming or the count of the beats in order to keep time. The problem with this is that despite odd-timing, it's predictable and a little boring. But there are many and much more elegant ways to play an odd time signature such as 9/4 and one of the ways is to play one bar of 4/4 and another bar of 5/4 and to alternate back and forth. This keep the sound more interesting and provides a smoother transition than a full-on count-to-nine kind of rhythm. Another way is to create a proper riff or melody on the odd-time signature. One of your better examples of complex or 'really odd' timing is the song Schism by Tool. This song's odd-time is driven by a bass-riff that simply doesn't end with your basic 4/4. The band has stated that the time signature is 6.5/8, which doesn't seem to make sense, and could easily be corrected to 13/16, but I'll try to be a little more explanatory by saying that the beat is six eight-notes with an extra sixteenth note in there. The brief pause in the two-part bass-riff could be the sixteenth note that changes everything. so it's count is in 8th notes:1-2-3 (pause16th) 1-2-3 = 6.5/8 . This is a song somewhat designed to throw off a person's natural timing, so although the bass riff returns, the timing doesn't remain steady throughout the song.
  A song like 'Schism' is meant to unnerve a person, both its' timing and the musical structure of the song as well as the lyrics and the melody of the vocalist. This song is clearly using odd-timings to throw the listener into a mood of strangeness and disturbance.
  The real question, then, for musicians is how 'odd' or complex a time signature is possible before it just sounds too weird or disturbing to listen to? And can these strange timings be used for other stuff? Can you write a love song in an odd time? Can you dance to it? It seems unlikely but not impossible. Certainly not every song in an odd timing is an ominous and morose tune. Money by Pink Floyd, though critical, is not unnerving. So how complex can you get without forcing some weird or strange vibrations upon the listener? Perhaps only Bernard Purdie can help us by coming up with yet another amazingly complex but still up-beat kind of groove.

For release Next Week: The Very Us Mumblings Team have their thinking-caps on and are popping the brain-glucose tablets to get ready for the release of Volume 2 of Math-sicians!! Next time, we leave the subject of Time and the Golden Ratio and enter the world of Coding and Frequencies! Samuel Morse and the great Pythagorus and perhaps we'll even unlock the frequency of the Universe!

Volume 2: Morse, Pythagorus and De-coding The Divine Frequency!