Music Math (pt 1, major chord harmonics)

By now it’s out in the open, I am a music theory nerd and I spend far too much time over analyzing things.  With my engineering degree comes a large amount of math and mechanics, I’m not really that good at it, but I learned a lot about it. While my psychoacoustics post focused on the brains understanding of sound, this post will do the opposite, and I will talk about the physical properties of sound and how harmonics and chords can all be explained with math and physics.

A musical chord is two or more notes happening at the same time. Easy enough to understand, but some fiddling around with a piano will make you quickly realize that not any two notes can be played together and sound “good” to our ear. For example, playing two notes that are chromatically adjacent (one half step apart, the smallest unit of distance in harmonics) will produce a sound that, without context, sounds quite terrible. It doesn’t mean it isn’t a chord, it just means it doesn’t make sense mechanically.

http://www.phy.mtu.edu/~suits/notefreqs.html << I used this table for all of my information.

For example, lets say you play middle C, (C4, 4 referring to the octave it falls in across the entire range of musical notes) and at the same time play D flat or C sharp (which are sonically identical). These notes are chromatically adjacent, so it will not sound like a very good chord. C4 has a frequency of 261.63 Hz, D flat4 has a frequency of 277.18. Imagine the sound waves traveling through the air. As they oscillate, they are going to bump into each other a lot and interfere with each other path; this is what our brain perceives as dissonance. Our ear drum is attempting to resonate at two frequencies that don’t have many nodes or peaks in common; this is why it sounds “bad.”

Now let’s say you play C4 and G4 at the same time. C4 has a frequency of 261.63Hz, G4 has a frequency of 392Hz. At first this seems to be the same case as before, the waves will interfere with each other right? Actually, this is what we call a perfect 5th, the most magical and favorable chord to the human ear. Now it’s time for some math to explain why.

An octave is a doubling of frequency. C0 is 16.35 Hz and C1 is 32.7 Hz etc. When you play these two notes in unison, they sound different, but the same.  This is because these two notes have nodes and peaks that fall at exactly the same time. Two wavelengths will fit perfectly into one wavelength of the lower octave, so our ear is actually resonating exactly the same way, but faster. It works no matter how many octaves apart the notes are. So back to the perfect 5th, these frequencies don’t appear to have anything in common like the octave does. I digress, I have been deceptive. It becomes more obvious when you look at the lowest octave, and therefore the lowest frequencies each of the 12 distinct tones can occur at. C0 is 16.35, G0 is 24.5. These numbers do have something in common; they are both a multiple of 8.175. The next multiple of this number is 32.7, which is C1, the next octave up. To get to the next perfect fifth above C1, you have to double the distance between the perfect fifth of the previous octave, not only do C1 and G1 have a multiple of 8 in common, they also have the multiple of 16 in common. This doubling continues up as far as the human ear can perceive sound, with the amount of multiples in common increasing each time also.

This is why a perfect 5th sounds the way it does, as the waves are traveling through time and space, instead of the peaks and nodes interfering with each other, they fall within each other at certain points (depending on the interval). Since two notes that are an octave apart from each other fall perfectly in line with their peaks and nodes, the higher octave simply having twice as many, a perfect fifth has 1.5 wavelengths within one wavelength of the root or bottom note. So after three wavelengths of the root, the 5th will hit a node/peak at exactly the same time. This is why it sounds “good” to our ears. This is the most frequently re-occurring node/peak match that can happen in harmonics, hence perfect fifth.

So what about three notes at the same time? Well I’m sure you have already thought that far ahead. Let’s take a typical major chord, on the root of the musical key of C for consistency. This is C0 and G0 played at the same time, with the addition of E0 being the 3rd of the chord. E0 has a frequency of 20.6, which does not share the multiple of 8.125 with C0 and G0. This is actually just about halfway in-between the difference of the two frequencies, which again is always true no matter what octave you are in. We have 16.35Hz, 20.6Hz, and 24.5Hz.

This is where it gets a little weird. You may have gotten your calculator at this point to check my math, and you will notice that the 3rd is NOT perfectly in-between the root and the 5th. It’s really close (.3Hz off), but not exact. If you have studied math and mechanics like I have, this should bother you. The reason for this, is that there are 12 distinct pitches in music per octave, but there are only 7 distinct pitches that fall into each key (root, 2,3,4,5,6,7 then you are back to the same pitch as the root but an octave higher.)

7 is an odd number.

7 is also a prime number.

You may be wondering, who the hell decided that there should be 7 notes in a key when harmonics is based on notes sharing the same multiples of frequencies?

More on that next time.

Since there is an odd number of notes between a range of frequencies that doubles in value, the notes are not evenly spaced within an octave, except for the perfect 5th, hence perfect 5th.

Back to our typical major chord of C0, E0, and G0, they share a common multiple of ABOUT 4, which is ABOUT half of the multiple that C0 and G0. I won’t bore you with the math this time; I think you can see how those sound waves will line up pretty nicely. With every 3rd root wavelength sharing a node with the wavelength of the 5th, and every 6th wavelength of the root will share a node with all three notes. As long as every note in the chord shares at least one multiple, you can add as many notes as you want and it will still have the same satisfying resonating sound in your ear drum. However, the less multiples these notes have in common the more dissonant the chord will sound.

This brings us back to the two chromatically adjacent notes from the first example. Once you get to frequencies that high, every note is bound to share at least one multiple with the one next to it. Up until now, every interval has shared the multiple of 2 (4, 8, 16 etc.). 2 is a prime number, which means the only multiples it has are itself and 1 (2×1 = 2, there is no other way to combine two numbers and get 2. Sorry if you already know what a prime number is). What if the only shared multiple is 3, 5, 7 or any other prime number?

Find out next time.

Metal Monday #10

This one was a no brainer.

Of 50+ bands at Indianapolis Metalfest, most of which I did not know beforehand, this is the one which stuck the most over the weekend. They didn’t even play on Saturday, they played at a bar the Friday night before, which was technically part of the event, but had no paid entry.

Ever hear a band that is so good you are enraged by the fact you hadn’t heard them until that point? The writing is so eclectic, the transitions are so chaotix, and every single member is talented enough to carry their weight. When I saw them their singer was MIA, but even down a member they still sold me. This is exactly the type of music I listen to in my free time, thank you for existing FOTA.

Honorable mention:

So nobody every judges anybody by looking at them right? well I saw these guys climb up onto the tiny make-shift second stage that was set up in the hallway leading to the actual stage, and I almost laughed out loud. THIS BAND IS SO SCENE.

I didn’t exactly change my mind about that, but they at least caught my attention with their crisp playing and excellent stage presence. It follows the same songwriting formula that I hate and is used by everyone ever, their lyrics are sort of uninspired, but I just couldn’t help but enjoy myself. I won’t say they are good, I will probably forget about them by next week, but it was a good enough show to warrant some attention.

The Browning confirmed to me they are nothing more than breakdowns, which is OK when you want to mosh at a live show.
Affiance are totally awesome, although the vocalist carries them pretty hard.
Chimaira was Chimaira
I still don’t like 90’s death metal.

Overall, the event organizers did a good job running it, however I think it would have been a better economical decision to front load the show a little more as opposed to having 50+ bands and starting BEFORE NOON. It’s tour season and its hard to get good bands to come to the mid-west, especially Indy, but had they promised some bigger bands more stage time in exchange for less bands playing maybe they would have showed. Of course this is nothing against any of the groups that played, they all did a great job given the situation, but if you want to get people to go to a show you win them over with quality not quantity.

Indianapolis Metalfest!

http://www.indianapolismetalfest.com/#!lineup/c6n6

It’s time for another big metal event in the Midwest. Indianapolis doesn’t necessarily have a huge or good metal scene, nor do I have any idea what sort of crowd to expect, but never the less I am stoked that a show like this is even happening.

Once again I will be tweeting during the event tonight and most of the day tomorrow. I am most excited for Chimaira, Angel Vivaldi and Intronaut, but I hope to be pleasantly surprised by some of the regional bands that I am not familiar with. Follow me @metalmalarkey to get a glimpse of the action as it happens.

Metal Monday #9

Oh yes, another cliche deathcore album. I can’t wait to listen to it once and never listen to it again.

This album made me question exactly what it is that sucks about deathcore. This album has breakdowns galore and for the most part has one type of song on it.

WHY DO I LIKE THIS

Maybe it’s just because I recently saw this group live and have some bias going on in my subconscious, but there are legitimate feels in this album. THE FEELS. Deathcore lyrics are typically some of the most uninspired angry words in all metal, here each song has a distinct theme that I was able to identify by merely listening to it. These themes are in general about the current state of society, our endless wars (Hellbound), our obsession with processed food (Children of the Corn Syrup), the evil in human nature (Do You See Him), finally all culminating together in The Travelers, which explains that with all these subjects combined, we are all “Hellbound” because we make the society we live in into its own hell.

To go with these lyrics, the band incorporates the instrumentals to build and release with the lines of lyrics. As the vocalist reaches a main point in this song, it will almost always be followed immediately by a breakdown. The breakdowns once they hit avoid falling into repetitive rhythms on single notes, and instead usually transition very cleanly into the next part of the song.

Hellbound just came out last week. It isn’t the redefining album of deathcore, instead FFAA seems to have found their distinct character in a far to over-saturated genre. Its good to hear a band contribute to an old genre with new ideas. Hopefully there is a lot more to come.

The Solution (a small change to educational standards to improve the culture of our society)

Rite of Spring is one of the pieces of music that redefined old dying sonata form, and attempted to open peoples minds with its completely new style of composing.

The first time this piece was performed was in Paris’ Theatre des Champs-Elysees. People had been attending ballet’s for years, they know what to expect when they get there. They want to hear an up-beat, repetitive, sonata form program all the way through. Its all they knew, its all they could have known.

Perhaps as you begin to get further into the piece you may start to realize…this doesn’t sound like a ballet. This made people uncomfortable, the notes are dissonant, the rhythm isn’t metered in 4 or 3.
People had no idea what to do with these feelings that the music was making them feel. Plus, the upper class French attending that night felt entitled to the show they expected.

The crowed begain to protest, to yell, get out of their seats, throw things at the orchestra, they were confused and angry.

Today this piece is looked at as a masterpiece, one of the greatest compositions of the 20th century. How can people have such different opinions of it when nothing changed but time?

Because they didn’t understand.

When you were listening to it, I’m sure you never had the thoughts the audience in France did that day. You may have felt some discomfort as the dissonant chords refuse to resolve, and you could never quite tell which direction the song was going. The thing is, Stravinsky follows the same rules that Mozart did, the Beethoven did, that every romantic and classical era composer has followed since the method of writing music onto paper came about. He just applied music theory in a new and innovative way because it was the only way for him to express the message he wanted in his music.

Last week we found out that the largest consumer of music, parents buying music for their kids, are the ones who determine what the top 40 hits are and cause artists who make very simple and easy to make music to be all too successful.

http://www.vevo.com/watch/miley-cyrus/we-cant-stop/USRV81300234

(sorry, would not embed)

(NSFW, imminent Miley)

Sorry to jump on the make fun of Miley bandwagon but unfortunately for her she is the perfect example of what I am trying to point out.

Sing the chorus of Wrecking Ball during the chorus of We Can’t Stop, or vice versa.
Now I would like to remind you that these two songs are off of the SAME EXACT ALBUM BY THE SAME EXACT ARTIST. Same chord progression.

Since I have apparently already filled a page with words, I will get to the point. Our culture doesn’t care or notice these sorts of things because they have never learned anything about music.

The solution to our mainstream music problem? Education.

According to 2009 U.S. Census data, about 8 percent of Americans play musical instruments regularly and another 5 percent sing in a chorus or chorale. More than 9 percent attend symphonic concerts, and about 8 percent enjoy jazz concerts.”
http://www.wannaplaymusic.com/news/us-census-says-about-eight-percent-americans-play-musical-instruments

With this data I would like to make an assumption, having been in the music education system for 10 years. Of these ~13% of people who play instruments/sing, lessof them actually have taken music classes in school. Of that percentage of people who have taken music classes, far less of them have taken a music theory class. I apologize about not having any raw data to argue my point, so think of how many friends you have that have taken music theory related class in school. They are like vegans, they aren’t hard to find, anyone who has taken music theory will let you know within minutes of meeting them that they have done so regardless of the conversation topic.

Allow me to put this into something more measurable. At the university I attended, calculus 1 had over 2000 freshmen enrolled last fall semester.

Music Theory 1 had 64.

As an engineer I understand the mind-set that math and science are more important, that music theory is a waste of time that you will never use.

WHEN IS THE LAST TIME YOU DID A DOUBLE INTEGRAL?
WHEN IS THE LAST TIME YOU HAD TO LISTEN TO MUSIC?

I agree that music theory is not as important is the core classes, but our education system makes requirements on math and science, certain classes you must take to graduate with any degree. There is a minimum level of understanding you have to have to be a part of society. The minimum level for math and science is relatively high to other subjects, but universities often also require foreign language, humanities, history, and even english writing or speaking classes. These are things that are still important and necessary for our society to function.

Many universities also have a fine arts requirement, including the one I attended. What sort of classes do these entail though? Band, orchestra, singing, dancing, painting, drawing, the number of options is so vast that you really only get to/have to take a few of them if that. It is very un-structured and open ended as to what you can take, which causes many to take the easiest option available. Well yeah, I have to take CALC 161, CHEM 115, ENGR 195, thermodynamics etc. The attitude people have is that the fine arts credit is a waste of time, yet choose the easy classes which are in fact a waste of time.

The point is, obviously people aren’t going to take music theory or a related course when they can put minimal effort into something like an entry level band or dance class and get an A. Would 2000 people take calculus 1 by choice? From my experience, no, yet they are willing to select a major which requires difficult courses because it is worthwhile long term.

It’s difficult to make an argument for having a national standard for music education when it is not that important of a class to take to be able to be a functional member of society. It is a waste of time to teach our kids about music when we are still falling behind in test scores to other countries around the world in the subjects that kids ARE required to take.

My plea is not to get people to the same level of understanding that they have with the core classes, its simply to teach kids when they are young the fundamentals of music. I am convinced that even if our youth had a minimal understanding of notes and rhythms, the quality of the music we hear will improve.

What I am suggesting is the equivalent level of understanding that one 4 credit college class gives you by the age of 12.

You would be surprised just how little information this actually is, relative to what is available. All that I think everyone needs to know are things like scales, meter, the circle of fifths/fourths, and chord structure.

Thats it.

I’m not asking you to study in depth the methods of Bach, or the societal impacts of the romantic era, just to understand the language.

If the reason the popular music in our country is due to our youth not knowing anything better is out there, educating them about the simple concepts of music will help our society advance. Thats what education is about isn’t it? Improving out society constantly by teaching the future generations the knowledge we already know, so that one day they can add more knowledge to the endless cycle of education.

Education is supposed to improve our society, and neglecting to educate people about music is causing the music of our culture, the music which represents our society, to converge to the most simple and easy to write songs because it is the only thing that everyone is guaranteed to understand. The industry cannot evolve without our expectations to evolve first. If the point of education is to advance society, does that not spread to the realm of the arts? Are we becoming a people of only logic and reason? Do people today not care about the arts? Of course they do! EVERYONE LIKES MUSIC. I have never met a person who when prompted, said “no, I don’t like music.” Ask anyone, anyone on the whole civilized planet, its in our nature to like the fine arts, to seek them out and to appreciate them. If we can all agree on that, can’t we all agree it should always be getting better? We strive for our educational standards to improve, yet neglect to realize that cultural improvements are just as vital to our society.

If you cannot agree with my rantings, I offer an alternative argument. Lots of kids want to be a rock star because of how glamorized the industry is, yet most people (less than 85%, see census statistics above) never actually pursue music as a profession. How can they know they want to be a musician if they never know anything about it? How can you expect a person to consider studying music when they have never studied it before? There needs to be exposure, there needs to be the choice.

Ultimately the problem is that we are all just like the audience of the first playing of the Rite of Spring, we fight any change or new idea that comes to the music industry because we don’t understand. We don’t understand anything that takes us out of our 4-chord 4, 140bpm, verse-chorus-verse comfort zone. There are an infinite number of ways to write a song yet all these artists settle for only one. Doesn’t it bother you that they aren’t actually doing any creative work? Doesn’t it bother you that the industry can get away with taking advantage of our ignorance? I’m not asking for much, just a small push in the right direction.

Getting To The Bottom Of This (most people seem to dislike top 40’s music, yet it is the top 40)

Spending a lot of time alone in your single apartment gives you a lot of time to think and be mad at stuff. Maybe from time to time, I over analyze something and come to some profound conclusions – often based on nothing really. This time I think I’m am really on to something. This time even I can’t disprove myself.

People listen to ‘top 40’s’ music. In fact, so many people listen to it that it has adopted its own genre. Top 40’s is really an actual list by billboard who gauges how many people are buying what music and how often it is getting played. This creates somewhat of an endless loop of the same type of music going in and out, because once something is in the top 40 it’s getting played everywhere that plays top 40’s. The point is that people DO listen to it, they must to get this stuff onto these charts.

 

Many people I have talked to about this sort of thing almost always agree with me, the music that is getting into the top 40’s charts these days is derivative, uninspired, and engineered to make money instead of share the artists thoughts and feelings the way they intend to. If you are reading this, and have made it this far, you probably also agree. This is where I got confused.

Everyone listens to top 40’s

Everyone does not like top 40’s

These two statements are very broad generalizations, but in essence this is the problem. If everyone claims that they do not actually like the music they are being exposed to, why do they continue to expose themselves to it? Why is it that what I hear when I go to bars and night clubs, with people in my demographic, that this music is what is chosen to be played?

Well Mr. Malarkey, I know the answer to that dummy

NOBODY CARES WHAT MUSIC IS PLAYING

This I have found to be true. In most situations, aside from live concerts, people who are in these situations where this music is being played are not there to listen to the music, they want to hang out and not be distracted by the music, yet have a rhythm to dictate the mood. In previous posts I have talked about the recurring theme of going out at night and having a good time in today’s top 40’s hits. It makes sense, but the fact remains that these people are being exposed to this music that they did not get to choose themselves. Someone else, a DJ, has selected what they get to listen to.

Now imagine yourself a DJ, selecting songs for these foolish mortals to have to listen to, you don’t want any complaints because your boss will get upset, so you have to try and play things most people will like. Well geez I don’t even know these people! I guess I’ll play top 40’s to be safe.

The decision the DJ has made is perfectly logical, play the most popular music to have the highest chance of pleasing the highest number of people. Those of you who don’t like it will have to deal with it because more people probably like this.

I’m getting ahead of myself. Here is a diagram of the music industry to help clarify this process.

Record label finds artist with good looks/good enough voice à Artist begins to record album

Album is mixed and mastered by record label à Record label distributes record

Record is bought or streamed by consumers  à Music distribution platforms count total sales

DJ sees who is selling the most à plays song in the venue you are in

Where is the American public involved in this process to make a change to what ends up in the top 40’s? only one part.

Record is bought or streamed by consumers  à Music distribution platforms count total sales

What the American consumers are buying is what is being played. Why did I go through all this to prove something that you already know? Because here is what you may not know, who is buying the music.

Think back to before you were in college, before high school, you are 10. You are beginning to learn how to use computers, but not enough so to seek information on the internet. How are you going to get exposed to music? Either what your parents play for you, older siblings, or the radio. From person to person, family members are different, so the only one that is the same from case to case is the radio. You hear a song on the radio, you like it because you are 10 and don’t have that many higher level thoughts and the derivative, uninspired, very simplified music on the radio is the best thing ever to you, because you haven’t heard anything else. Parents buy it for you, adding to the statistics kept by record companies.

Now back to today, you hear a song you like, how do you get it so you can listen to it whenever you want?

Download it or stream it.

I don’t know about you but I haven’t bought a cd for over 5 years. It’s ludicrous to go to the store and buy an album when you can get it on the same device you are reading this with in under a few minutes. If done legally these all factor into industry stats and still affect the top 40.

If done legally.

I’m not accusing anyone of anything, but nearly half of Americans ADMIT to having pirated something in the past (http://piracy.americanassembly.org/the-copy-culture-survey-infringement-and-enforcement-in-the-us/). The actual number is probably even higher because of the people who think the internet police will come find them lying. I would like to think half is a considerable percentage. The percentage goes up to 70% when looking at only the demographic that includes me, 18-29 year olds. Aren’t these the same people I was talking about earlier that go to bars and nightclubs where top 40’s is played? 70% of them don’t contribute (at least not completely) to music industry statistics which determine these songs. Maybe Mr. DJ did not make such a well informed decision after all.

So who is buying music then, who is causing the same type of songs to continually make the top charts?

I now offer a very bad source as my argument, only because it is the shortest to read and most likely to actually get viewed by you. http://www.ask.com/question/what-demographic-buys-the-most-music
‘25% of all music sales are done by consumers who are over 45 years old, which is more than two times the share of any other age group’

This relates back to my anecdote about the 10 year old.

Obviously the 45+ demographic isn’t listening to anything released in the last 10 years, they grew up in a different generation with different music distribution methods. They are buying oldies for themselves, not the current top 40’s. Their 10 year old kid however, whom they must make all purchases for, really wants the new Justin Beiber song and won’t stop screaming until they buy it for them. Once you look at it that way, it’s no wonder all the 45+ year olds are buying all the music and controlling industry statistics.

So if you have been keeping up, the music we (the 19-29 demographic) are being exposed to is the music of people 10-20 years younger than us, the music of a people who are mostly only exposed to top 40’s in the first place.

TL;DR the top 40’s music of today is determined by children who don’t know any better.

This is the first half of an even longer post. Now that I have gotten to the bottom of the problem, in the next post I will figure out what our society needs to do to overcome this endless cycle and improve the quality of the music we are being exposed to against our will in public places.

Metal Monday (on a tuesday!) #8

What? It was a holiday….

I have scrolled  very deep into the endless abyss of the new Spotify Discover feature. You can keep scrolling forever and it will keep recommending you things, even based off of the things you just listened to as you were scrolling.

This leads to some very late nights.

somewhere down the line I ran into this. After listening to derivative breakdown bands to bands that don’t understand how to do dissonant chords, anything remotely good would have caught my ear. This was it, after listening to a few of their songs I finally felt like I had accomplished something so I closed my laptop and went to sleep.

I have figured out a way to describe what I like about a lot of the music I enjoy, it is all CLEAN. If you have ever been taught an instrument you may understand what I mean. When the whole band plays in unison, and nails their articulations, it sounds clean. If they play a more sludgy droning style where the notes are purposely placed slightly before or after the beat, it sounds sloppy. I am not saying that sloppy players are bad players necessarily, especially in metal. It is a style and a choice of the artist to play that way and appeal to that audience. That being said, I like it clean.
Other words that describe this song along with clean:
Technical
Brutal
Flowing
So I’ll end today with that vocabulary lesson. Now go listen to some more metal.