Math-sicians! Volume 2: Morse, Pythagorus and De-coding the Divine Frequency?

Pre-ramble:  Apparently, we at Very Us Mumblings have stumbled upon a greater problem in our home country. While talking about the various relationships between math and music, a real math problem has arisen in our schools. It was brought to our attention after we published our first Math-sicians! Volume 1. that apparently the 'New Math' the 'discovery'-based system adopted in many schools across Canada since about 2006 has left kids without the math skills necessary to move on past a generic high school education. Evidently, the problem becomes worse the further on up the grade-scale that you go, to the point where Universities are having to re-educate students in the basics of Arithmetic. Apparently those constant drills and practice that used to drive many students to utter boredom is just what this new generation is lacking. Suffice it to say, some of us didn't particularly like the 'old math'. Constantly doing the exact same problem over and over again seemed like doing the same crossword over and over again. And just because you tried to teach math a different way and it failed doesn't necessarily mean it wasn't worth the try.
    But practice is what's needed sometimes. Much like music, you can't just play something once and have it down pat, (unless you've plugged it into some software program like proTools) you have to get good at it. You learn and practice and practice until you truly understand it and can play an entire song through all the way to the end without problems or missteps. Only when you truly have learned to play something can you put your own feeling and emotions into it. Otherwise you're just going through the motions and not truly getting better. At the same time, we all have to understand that kids are not all the same and I would hate to think some children might not get a proper education just because he or she was so sick and tired of drilling and practicing the same thing over and over.
    With that in mind, let's set out about educating some of the adult students in this country, both young and not so much. 

If you missed Volume 1: Click Here

Part Eight: Music Code?

In the popular Urban Dictionary, which provides for several definitions of a term, one definition of the term "Math Rock" (ranked 5th best by urban dictionary) explains the genre as: 

A band or bands that use a formula to write a song. Usually referring to bands influenced by Rush, Helmet, King Crimson, etc, which use complicated repeating rhythms, or highly intellectualized beats based on some kind of math formula.

And the vernacular example given is:
Rush wrote a song called YYZ, in which the beat is Morse code for YYZ. That IS math-rock. 

Morse Code isn't math, specifically. Although the idea that a letter represents a number or a series of tones or notes, or dots and dashes, and then can be translated from one to the other or back again is not far off from the general idea of math or of the writing of music as a form of expression. So let's just go with this idea for a second...

The first part of this Math-Music Blog was espousing the idea that a mathematical formula could serve as the basis or formula for the creation of art. The second part debunked the Fibonacci formula or the phi formula as some kind of universal guiding principal (guiding principle, sure, but not universal). Perhaps it was wishful thinking on the part of several mathematicians that some universal truth could be derived from a simple set of numbers, and since Fibonacci is one of the great mathematicians, then it should be his formula and his 'golden mean' which should guide such art and music. For, of course, if there are two things that are universal it's math and music...right?

  (For arguments sake, we'll say Yes.)
  And then there's the song YYZ by Rush...to which the only formula is Morse code.

  YYZ is the airport code for Toronto's Pearson International Airport and the rhythm at the beginning of the song is the letters Y-Y-Z belted out in two notes of heavy-metal Morse-code. It sounds like this:

-.-- -.-- --..

The song is about the hustle and bustle of airports, and about returning home. However, YYZ is an instrumental. It has no lyrics to explain it's meaning, and yet somehow, with the help of a few time changes, slower portions and an exotic-sounding solo, everyone seems to 'get it'. In almost universal terms, the song is understood on some basic level. Despite complex timing, a musical proficiency of expert ability, as well requiring the listener to have some unconscious understanding of the staccato Morse code-driven rhythm, YYZ is probably one of the most famous rock instrumentals ever written. It is still a mainstay of Rush's live performances, a favourite to learn for those becoming proficient with an instrument, and it is a song that is synonymous with the sense of coming home,  international airports, and the band Rush itself.
    At the time it was released; YYZ had never topped the charts, nor reached any significant position. It was never expected to. It was nominated for a Grammy for best instrumental, but lost to a song by the Police. For years, YYZ was that other song that appeared on Rush's Moving Pictures album, beloved by fans, but otherwise sitting in the shadow of the bigger hit songs 'Tom Sawyer' and 'Limelight'. Somehow, decades later, It has become a song with international, if not universal appeal. More than twenty years after it was written, in 2002 Rush played YYZ for fans in Rio Di Janero in one of the biggest venues in all of Brazil and the reaction from the audience was more boisterous than if Rush had played some #1 chart-topping hit song. Not only did Brazilians recognize YYZ from the opening taps of the cymbal and greet it with a cheer, the audience also began singing along with the song, emulating the guitar parts, knowing all the changes and even the solos, and all while dancing in a rain-soaked stadium field. Clearly YYZ speaks to people around the world on some level that reaches beyond language and lyrics.
   So, is Morse-Code the key to connecting with people around the world?
     Not really. Unless you're thinking about sending a telegram, and you happen to travel back in time to some hundred or so years ago when international telephone lines hadn't quite been established yet and you have to get a message to someone faster than your average postal service could carry it. For conversation and casual communication, Morse code is used mostly by independent licensed radio operators; hobbyists and radio-enthusiasts. Ships and planes use morse code and radio technology mostly as back-up systems in case of emergency, not for regular conversation. Simply put, nowadays Morse Code is what people use only when more sophisticated systems of communication break down.

  So, perhaps we can celebrate Morse instead of Fibonacci? Well, not really. Samuel Morse was certainly educated in math among other things, but essentially, he was an artist and portrait-painter who later turned inventor. The Telegraph and his accompanying code system were his major contribution to the scientific and technological world. Unfortunately, Morse also supported slavery, was anti-immigration and even started a political campaign and ran for office on the promise of establishing laws that were anti-catholic. And, for the second time in this blog, I've suddenly crushed the reputation of a historical person of great talent and insight. It's shameful isn't it? What's more, Morse Code as we know it now, was not entirely his own design. The original code system designed by Morse himself contained several elements that could be confusing, and later had to be modified for speed and coherence of use. The original Morse Code contained a half-space in a series of dots in the code for a Y, and three dots with a half-space and two more dots in the letter Z, so YYZ would be: .. .. .. .. ... .. which would be pretty boring to play musically. So Samuel Morse can't actually be credited for Rush's international hit code-song, except very indirectly.  The actual YYZ code played by Rush is a more modern development called International Morse Code modified by the International Telegraph Union. So, I guess that YYZ is the first hit song that was inspired by an international skilled trades organization.

Part 13: Sonic Geometry, Pythagorean Tuning & the Universal Frequency of 432Hz

Feel free to watch the video to the right. It's about twenty minutes long, with about ten minutes of tonal music at the end.
I will sum up here, when you are done watching the movie part and you can leave the last five minutes of tonal music playing while you read the movie credits (Don't be silly! Watch the first 20min then turn it off and pay attention to this blog, it might save you hours of your life.) At one time in History, most musical instruments were tuned to a frequency of 432Hz and according to Pythagorean tuning. Whether Pythagorus actually suggested 432Hz or just happened to stumble upon a frequency already being used by numerous instruments is unclear, but Pythagorus invented and established a widespread musical instrument tuning system that was the dominant system of tuning for hundreds of years. At the same time, a traditional number system based on 12 and 60 led someone somewhere to base musical instruments on a calibration at 432Hz. Based on this video, the numbers 12, 60 and multiples of these numbers, including 432, are the basis of geometrical shapes, our date and timing systems, distances, vibrations, the geometrical 'flower of life' designs and perhaps everything in the universe. Also, according to this video: The Chords of F-sharp and C-sharp are wholly important and resonate with a power that matches that of the universe and perhaps all of life itself. The only problem is that somewhere between the time when Pythagorus was alive and now, Pythagorus-based tuning was abandoned and the 'standard' calibration of musical instruments was changed from 432Hz to about 440Hz and now all those numbers and musical instruments and all the frequencies don't resonate along with the rest of the cosmos the way they should. Essentially we're playing frequencies that are constantly out of tune with the natural tones of the world. We're losing our natural connections with the Universe!

   As a result of this idea, many people have 'conveniently' re-calibrated your favourite music to be 'in tune with the natural vibration-frequencies of the cosmos' and, with a quick search on YouTube, you can find hundreds of famous and/or classic songs that have been re-calibrated by about 8Hz for you to listen to according with the vibrations of the stars. You too can enjoy the one frequency that makes us all vibrate in unison with everything. Instruments digitally recalibrated to 432Hz, to be in tune with the universe. I've posted one such video here, so you can listen to John Lennon's masterpiece 'Imagine' in 432Hz. A perfect song perhaps made only more perfect by dropping the pitch 8hz without slowing the tempo. Feel free to find out for yourself. Compare to the original, or just sit back and relax and see if it makes you feel any more or less different or... you may feel much more at peace and at one with the universe...or....perhaps you've been paying attention to the overall methodology of this blog and you know what's coming next.

  The truth is Pythagorean tuning is not just a matter of re-calibrating instruments or songs to a frequency of 432Hz. It's an ancient tuning system based on a mathematical formula of each note being seperated by a ratio of 3:2 or what is called the 'untempered perfect fifth' (as if being untempered is perfect). The problem is that what is mathematically perfect doesn't equate to a perfect octave of frequencies. The A-flat and G-sharp are the same in modern tuning, but in Pythagorean tuning, the A-flat is simply left out because it doesn't match up with G-sharp. Also because the interval in between all the notes is the same, taking the example of the perfect chord in Part One, One (in Volume 1) playing the first, third, fifth and eightth notes together should make a perfect and harmonious chord (a la Fibonacci)...well, in Pythagorean tuning...it doesn't quite do that. Instead, Pythagorean chords are more of an approximation. The only two notes that are truly harmonious is the first(harmonious to itself) and the octave above, which is basically the first again. Also, the numbers don't match up with the angles suggested in the 'Sonic Geometry' video: take the Pythagorean tuning and the 3:2 rationale and work out the frequency for an F# and you get 364Hz, not 360. A C# becomes 546Hz not 540 (In modern tuning, it's actually closer at 544Hz). Follow out to the next octave and Pythagorean tuning only becomes further out of sync with this supposed 'geometric harmony'. Pythagorean tuning was once widespread, but the practice, and possibly the instruments too, were eventually phased out starting in the 1500s for a simple reason: Harmonies didn't work very well. While a good singer could harmonize with an instrument, another instrument would have difficulty. Dividing an octave into 12 sonically equivalent fifths (based on 3:2) leaves an uneven gap between the highest note and the next octave (called the Pythagorean Comma). Learning to play an instrument tuned to Pythagorean tuning in these modern times must be difficult, although there are still some people who can play it on such medieval instruments like a harp or specially-designed guitars with adjustable frets. Complex chords and music requiring a lot of strumming and chord changes would be difficult. Performances would probably revolve around smaller wind instruments with limited range like one-octave flutes or stringed instruments that are plucked rather than strummed, like a lyre or a harp. Harmonies of several different octaves and songs that changed keys would probably be very difficult.
     Pythagorean tuning was replaced with something called 12-TET Equal Temperament tuning. This invention pushed aside the mathematical formula in favour of the simplicity of dividing up an octave into twelve equal tones. This is why the frets on a modern guitar are all single straight frets across the board with the twelfth fret landing directly in the centre of the length of string. This invention was not only a good idea, it was a massive breakthrough that allowed the expansion of instruments to many multiples of octaves of tones and created most of the complexity associated with multiple harmonies in music today. The tuning and tones played in the 'Sonic Geometry' video, the notes used in the movie to make up the F-sharp or C-sharp chords are not Pythagorean tuning but closer to approximating what is commonly called 'Verdi tuning' which is the current modern Equal-Temperament-tuning with the 4th Octave 'A' set at 432Hz. So what the video claims as an ancient musical tuning method is actually based on a tuning structure invented in the Renaissance-period and more specifically associated with composer Giuseppi Verdi who passed away in 1901. Modern instruments are all still based on Equal-Temperament Tuning and as such, Converting John Lennon's 'Imagine' to Pythagorus-tuning at 432Hz would require a re-write and a new performance of the entire song. The associated videos on YouTube, including the version of 'Imagine' have simply dropped the pitch of songs by 8Hz to 432Hz, which is actually only approximating Verdi Tuning, not calling to the ancient wise-ones like Pythagorus. Verdi is a fairly well-known composer and musician, but maybe you might want to 'Imagine' listening to John Lennon the way that John Lennon intended.


Holy Fork!: How we Standardized Tuning
    Our crack team at Very Us Mumblings doesn't know exactly why standard tuning is set at 440Hz and despite our extensive research, we don't know who set the standard (Mostly the French, but not entirely). But we certainly think we have an idea why they felt they needed to set a standard of some kind. Before modern electronic tuners, musicians bought things called 'tuning forks'. These tuning forks were highly specialized, carefully calibrated, sensitive instruments that gave off a perfect tone when you smacked them against your favourite barfly or chosen enemy. The tone was usually an 'A' and you tuned your instrument accordingly. However, if you bought that fork from Wolfgang in the Dresden Opera House in 1815, it resonated at 432.2Hz , while your friend Giuseppi in Italy was tuning his spaghetti in the trattoria with a fork that resonated 451Hz. Well both Wolfgang and Giuseppi may have been kicked out of the orchestra for forking-up their shnitzel, so they got together at 'la boheme' and standardized their forks at 440Hz. (examples of different forks taken from this article in wikipedia. The names have been changed to protect the musicians.)
   However, do not be fooled! Now that tuning has been standardized doesn't mean that everyone has simply conformed and complied. Generally speaking; Classical musicians in the French, Italian and German styles have conspired to tune higher while Spanish, Jazz and Blues musicians have tuned lower. And no one is acting as musical law-enforcement to prevent anyone from changing their minds and going in the opposite direction. In order to dispel what may be a common misconception: Frequency-based tuning is nothing new, nor is it set in stone. 440Hz may be the current standard tuning for orchestras worldwide, but the truth is that musicians all over the world have been playing around with tuning frequencies ever since they first learned how to tune a fork. Higher tension strings and higher frequencies produce brighter, sharper notes, to the point where sometimes vibrating steel strings can almost produce bell-like tones. Lower frequencies produce more reverberations and "bouncey"-sounds. Everyone has known this for centuries and tuned their instruments accordingly to the sort of sound they want to achieve. You also have to realize that it may be impossible to re-calibrate a flute (how do you move a hole?), and re-calibrating a baby grand piano from 440Hz to 430Hz might take hours, but re-calibrating a guitar or bass takes only about fifteen minutes, even without modern tuning equipment. A modern electronic keyboard might be even easier, just a couple of buttons, depending on the functionality built into the unit. In fact, in modern times, It seems almost expected that musicians choose non-standard tunings, sometimes to suit the lead vocalist in their band or to match the mood of the songs that they wish to play. Stevie Ray Vaughan and Jimi Hendrix always tuned their guitars a half-step down from standard, guitarists refer to it as 'E-flat tuning', which would technically put the 'A', and hence most of their songs, calibrated at about 415Hz. (Which is ironic, because this version of Jimi Hendrix 'Little Wing' claims it is 432Hz  but it sounds like someone has re-calibrated the original song down to a lower pitch, rather than a higher one.) Soundgarden and many other bands of the Grunge-era went one step further, dropping the bottom string of their guitars to a D-tuning (which means one string might be vibrating at a lower tension from the rest.) So even If 432Hz is the frequency of the ancient musical instruments, I'll bet that even our musician-ancestors probably experimented with changing that frequency, too... with or without Pythagorus' permission.
    And finally, the important role of F-sharp or C-sharp played in the above video seems altogether unrealistic to us at Very Us Mumblings. Most of the greatest guitarists of the last century until now, including the African American Blues players like Muddy Waters and B.B. King have all shown a preference for writing songs in the key of E or A. Most popular-music piano players write in the key of C or D. Nobody is holding these musicians to these keys, they just gravitate to them based on how they feel and what they want to write. To think that F-sharp and C-sharp should play such an important role in sound of the universe seems somewhat odd. Why not pick two keys that are used more often in some of the greatest music? Beethoven wrote one of the most beautiful things ever heard in the key of D-minor, his Moonlight Sonata. Why should F-sharp be more important than D? Frequencies and Geometric angles are two different things. Match up frequencies according to a mathematical scale and they should coincide, but the number that you start with, however, doesn't have to be 12 or 60 or 180, for the simple reason that you don't have to listen to F-sharp and C-sharp. Why listen to harmonies of F-sharp or C-sharp for ten minutes on end in order to supposedly put you in some tune with the universe? I'm sure the universe contains many different tones, otherwise we wouldn't have ears that can hear different tones and brains that can enjoy them. So why fill your head with nothing but F-sharp or C-sharp for twenty-minutes or more when you can hear harmonies of G or A or even the uncommonly-used B? Variety is a good thing, we should make use of it.

    The whole point of this last portion of this math-sicians blog is to, at least partly, dispel the notion that it is somehow good for you to listen to a single tonality of music. The human brain and You, your brain, your ears are capable of hearing and enjoying an immense range of tones and frequencies as well the complexities that are involved in creating and arranging that music and the way that it makes you feel. Don't shut yourself in a room and listen to generic tones for twenty minutes just because somebody told you it was good for you. Listen to your heart, listen to good music, and find out what speaks to you and what's good for yourself and your soul and maybe the universe, too. Here at Very Us Mumblings, we thoroughly enjoy mental exercise and discovery and enlightenment, but we don't claim to have the answers to the whole universe. We don't have the key to Time, Space, Geometry and all your problems and the problems of mankind. We reject the idea that the universe is just a simple series of mathematical tones whether representing a spiral or a flower and we refuse to sell you bunk CDs and terrible musical cures for ailments you don't have. You don't have time to listen to bland, boring tones anymore than you have time to listen to 40 minutes or more of childish bubble-gum pop or annoying & repetitive house music. We think the universe is vast and powerful and wonderful and it doesn't sound like buzzing, annoying tones at geometrically-inferred frequencies. Neither does it sound like your favourite songs re-calibrated according to some ancient tuning method. And who's to say what the Universe sounds like? Maybe, if you listen closely, maybe the universe and all the eternal heavens might sound like Stevie Ray Vaughan at the El Mocambo one night in 1983!!...but that's just our opinion.
    And in any case, if you don't like Sonic Geometry and 432Hz, you can always check out 528Hz, We've heard it's the sound of the universe!!