In Opus 1, we announced our (royal we) intentions to look at the forest rather than the trees, so we could get a grasp of a whole work of music rather than let the pieces flit by, just a pretty wash of disconnected images and sounds.

Today, though, despite what I said, we ARE going to look at trees.  Okay, wait, not trees.  Pine needles!  In fact, we're going to get out a microscope and look at the beads of resin on pine needles!  We had to leave some questions unresolved after our first lesson, analyzing Mozart's Magic Flute Overture.  So let's get small.  And let's let Donald Duck lead the way!

I don't know how many times I saw that as a kid.  At Jane Adams Elementary School in Lakewood, California, when it rained, it was a traumatic event for everybody.  The outdoors lunch-baggers had to eat indoors with the tray-lunchers, but space being at a premium because of the Baby Boom, not everybody could be accommodated at once.  So school lunch, on such days, became a two hour or longer affair wherein we were all packed into the school auditorium and called out to eat lunch in shifts.  It was humid and noisy and disorganized.  When we got too rowdy, the authorities threatened to make us sit boy-girl-boy-girl, a terrible, terrible thing.  So they showed Donald Duck in Mathemagic Land while we waited.  And they showed it and they showed it!

In the film, Pythagoras demonstrates to Donald that there are whole number ratio relationships between notes of music.  Pinching the lyre string in certain ways gives us different notes, but pinching it in simple ratios, like 2/3 or 3/4 produces more interesting, musical sounding notes than just random pinches.

The Physics Hypertextbook puts it very elegantly:

The distinction between music and noise is mathematical form.

All sound is the result of vibrations in the air.  A single note of music is a single frequency of vibrations.  But musical relationships between those notes are mathematical.  They aren't random.  It's possible to make music out of random noises, like the Beatles' Revolution Number 9, but that is an aesthetics game.  What makes music magical and different is the way the hardware in our brains and auditory nerves manages to process those sounds and ferret out distinct frequencies and organize them into relationships without our having to understand any of this.

But let's try to understand anyway, so we can increase our understanding for fun things to come in future opuses.  (The real plural for opus is opera, but let's not confuse issues, eh?)

Let's talk about Do Re Mi.  (Doe, a deer, a female deer... la la la)  If you ever had chance to sit down to a piano, you, a complete novice, just as likely to break it and get yelled at as not, you probably figured out pretty quickly that the white keys are the nice ones and the black ones are evil.  If you wanted to plunk out something simple, like Chopsticks or Mary Had a Little Lamb, you did it using the white keys, and on a piano, that means C Major.

[The Julliard grads reading this might be bored about now, but cut the rest of us some slack.  We have to start with the basics before we can explain ninth chords and Phrygian modes and whole-tone scales.]

I had hoped to make a youtube to make this easier, but I haven't mastered the youtubing arts yet, so we'll have to do it this way.  Of course, everybody knows Do-Re-Mi, so we don't need to play that.  Let's look at the C Major chord, then.  To play a C Major chord (or just C chord), you press down on the C, the E, and the G keys.  Do + Mi + Sol.

It sounds very nice!  Why?  Mathematical ratios.  The sound frequency for middle C on a piano is 262 Hertz (vibrations per second).  The frequency for the next higher C, the one seven notes to the right on the keyboard above, is twice that, 524 hertz.  But G (Sol) has one of those nice whole-number ratios.  It is three halves of 262, 392 hertz.  3/2 is about as simple a ratio as you can get for two different musical notes.  What else has a nice easy ratio between 1 and 2?  4/3.  And that ratio gives us the frequency of F (Fa), 349 hertz.  How about one more easy ratio?  5/4 is the ratio that gives us the frequency of E (Mi), 330 hertz.

There is a nice table here showing the ratios between all twelve notes of what we call the just chromatic scale, the one that Pythagoras was demonstrating for Donald.

So the C chord has C and E and G... why not C, F and G?  F has an even sexier looking ratio than E, doesn't it?  F was 4/3, while E was 5/4, right?  Well, try playing it on a piano, and you'll notice, it doesn't sound as pretty.  It doesn't have that same oomph.  And the problem here is that F and G are too close together.  The ratio between F and G is 9/8.  But the ratio between E and G is 6/5.  So these three notes, C, E and G, (Do Mi Sol) give us the most elegant set of mathematical ratios between three different notes.  No brilliant mathematician invented this system and foisted it upon us.  It's based on the nature of audible sound itself and the way that our ears hear it, the way our brain processes it.

In 1977, NASA launched the Voyager probe, which, after surveying the planets of our system and sending back many beautiful pictures, left our solar system forever and is now on its way into the great woowoo beyond.  Inside the Voyager, there is an LP record made of gold which carries greetings from Earth in a host of different languages, as well as samples of Earth music.  You can listen to it here.

On the unlikely (it seem to me) chance that it should ever be discovered by another race of intelligent alien creatures, what would they think of it?  Assuming they did find it, and they determined that the grooves represented some kind of vibrating noises of importance to us, the people who created it, what are the odds that they would process sound information the same way that humans do?  The beautiful mathematics of the C Major chord would probably be beyond the capability of perception of such creatures.  They could analyze the patterns, the forms, the ratios, and they might find something interesting in beautiful in that, the same way some people find beauty in the formula for pi.  But that's not the same thing.  This ability to perceive musical beauty that we have is something basic to the way we humans are built.

How to Fake Your Way to Superstardom with Three Chords

When I gave my daughter her first guitar, I wasn't going to be around to instruct her, so I told her this:  All you really need to know how to do is play THREE CHORDS!  Three chords, and you can fake your way through thousands of songs, including most folk music.  All you have to do is strum the guitar with one of the three chords while you sing the song.

The piano is a little harder.  But since we started the discussion using piano chords in C, let's continue that way.  On a piano, the three chords that will let you fake your way to stardom are the C chord, the F chord, and the G chord.

C chord = C + E + G
F chord = F + A + C
G chord = G + B + D

Since a guitar is easiest to play in G instead of C, the three magic chords are different.  But just for kicks, you can look at this list of guitar chords and lyrics for miscellaneous children's songs.  They all use the same three chords, and some of them just two.  And the funny thing is, songs that require more than three often sound just as good with three if you smile big while playing them.

Songs are generally played according to a specific scale based on a home note.  That home note is called the key.  For instance, Do-re-mi, is in a way, a kind of song of its own, in the key of C as illustrated above, because C is the first note, the home note, the note that everything else resonates with and leads back to.  You can play other chords, like F and G, while you are in C, but the song is still C.

When we are in C, the chords C, F, and G acquire special names, and these are important, gonna be on the test double-damn-guaranteed.

The C Major chord (while playing a song in C major), is called the tonic.  The tonic chord means the same chord as the key of the song.  The tonic is obviously the chord that matters the most.  It's what you almost always must start with and end with, although there are rare, fancy-schmancy exceptions.

The G chord, (while playing a song in C) is called the dominant.  It is the second most important chord in music.  If there is a song that doesn't require it (I can't think of one off the top of my head), it's hardly a song at all.  Even Do-re-mi-fa-sol-la-ti-do, if you play the harmony for it, requires a dominant chord to sound right.

The F chord, (again, while playing a song in C) is called the subdominant.  And I've always thought that a curious name, too.  Is it a sub?  Is it a dominant?  Why can't it make up its mind?  The subdominant is a little less important than the dominant.  There are plenty of songs with no subdominant chord.

There are plenty of other important chords, but those are the three biggies to know if you want to stand there with a guitar on American Idol and con people into thinking you know what you're doing: The tonic, the dominant, and the subdominant.  There are lots of other chords with their own fancy names you may want to learn, especially if you take up an instrument.  But I'm not trying to make you a musician.  I'm laying the groundwork for understanding music, not just music in general, but specific pieces of music that we are going to hear in the future.  There were questions left unanswered in the OPUS 1 lesson about Mozart's Magic Flute that we can address better next time.

So why these three chords?  What makes them so special that they pop up everywhere with such frequency, that they need special names?

Here we go back to ratios.  The two notes with the simplest ratios to C were G (3/2) and F (4/3).  There is something rather magnetic about those notes that draw our attention to them.  They want to steal the spotlight away from C, the tonic, the master of the scale, the alpha and omega of the scale!  In a song, the music oftentimes temporarily just abandons C for another key, something more exciting like F and G.  In this case, it doesn't just change chords, but changes scale, with whole Do-re-mi schema being shifted several notes to the left or right on the keyboard.

Let's look at the G major KEY, the dominant of C, for a moment.  If you were to play in G, this would now be your Do-Re-Mi.

Do is now G, Re is now A...  What happened to Ti?  Well, it had to become one of those evil black keys.  Likewise, the scale for the F Major key has its own black key, with the B key reduced to B flat.  Any other major key other than C that we try to play in is going to require that we leave the lazy comfort of the white keys in some way.

But still, it's pretty close to the same, isn't it?  You only have to learn one black key.  It could have been worse.  It could have been B Major, in which case Do-Re-Mi would require we play five black keys and three white keys.  Dreadful.  And I'm running short of time so I can't make a picture of B Major for you.  Trust me on this.  B Major is the darkest, most evil circle of Hell.

But let's see.  G and F are both harmonically close to C, as shown by their ratios, and they have a lot of other notes in common.  It makes sense, does it not, that they would have a seductive quality to them that leads us astray from safe, sound, C Major?  Indeed they do,  First they lure us with their beautiful three-note chords, , then one thing leads to another, and we wake up in bed next to them with a hangover!  How did I get here? you ask yourself, as you try to slide out of bed without waking up this strange key.  Nope, no coffee, gotta get to work, I'm late as it is, besides, didn't you tell me you were just a subdominant?  You're nice, you deserve better than me, and I've got a beautiful C major waiting for me at home, I'd leave you my card, but I'm fresh out, so, oops, look at the time!

And that, my friends, [rewinding us all the way back to OPUS 1] is why the Magic Flute Overture starts in E-flat Major (the tonic), migrates to B-Flat Major (the dominant of E-flat) during the exposition, then gets lost in a host of strange keys before finally returning to E-flat major again in the recapitulation, properly chastened and bearing flowers and a shit-eating grin.

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#### Comment Preferences

• ##### Bravo!(6+ / 0-)

Tipped and rec'd...

• ##### Really glad I saw this. I went back to check (6+ / 0-)

and found your first two.  These are super.

Thank you so much.

"Never, desist till we ... extinguish this bloody traffic, of which our posterity, will scarce believe that it suffered a disgrace and dishonor to this country.

• ##### It's been a long time(5+ / 0-)

since I did this stuff myself. It all tumbles back into place so quickly though. A pleasure to read, and I look forward to the full explication of the Magic Flute...Masonic overtones and all!

Thank you.

• ##### Love it!(9+ / 0-)

I teach music and I've got a piano student who's interested in theory. I've been teaching him as we go along, but our latest hurdle has been equal temperament v. just temperament and why it makes a difference. I'll direct him to this diary for reference.

Thanks!

• ##### Oh, I'm not sure I deserve it.(6+ / 0-)

I linked to a couple of wonderful reference sites in the diary that do a better job.  I'm just more folksy.

Actually, here is a wonderful article by Leonard Bernstein from his Young People's Concert series that explains sonata form and tonic/dominant better than I did.

http://www.leonardbernstein.com/...

• ##### Oh, that is a great article!(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, slksfca

No, I really liked your perspective on it, and I think that's what he might need. I've been explaining it to him a couple of different ways, and he's getting it, but I think different perspectives often help get a difficult concept across. He's fascinated by the math part of music and he largely gets it, but when I start talking ratios he freaks out a bit. :)

I always find the switch from when the 4th, 5th, and octave were considered the "perfect" intervals to tertian harmony a fascinating time in the history of Western music. I've always thought it was no accident that it happened during the Renaissance.

• ##### The best resource for understanding...(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, euterpe

...that relationship is W.A. Mathieu's book, "Harmonic Experience."

Best music theory text in the world.

Freedom isn't "on the march." Freedom dances.

[ Parent ]

• ##### Oooh!(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, WarrenS

Thanks! I'm a huge, huge theory geek and am always looking for stuff I haven't read on it.

Currently finishing up "This Is Your Brain On Music." It's required reading for all my adult students now.

• ##### "Harmonic Experience" is (4+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, pico, euterpe

a huge book. Costs around \$40 or so.  Worth every penny.  It is the only music theory book I have ever read which talks about equal temperament as a psychoacoustic approximation to Just Intonation, and demonstrates how differently voiced chords on the piano can create the illusion of microtonally shifting pitches.

I heard Mathieu do that in a workshop over thirty years ago.

He noodled a minute, setting up a C tonality.  Then he began doinking on a high A, while shifting chords in the left hand between an F (IV) and a D minor (ii).

There were about 200 people in the room.  Every single one of us agreed that we had heard that A move downward in pitch when he shifted to D minor.

I have subsequently repeated that exercise in my classes, and the students all agree they hear it, too.

Freedom isn't "on the march." Freedom dances.

[ Parent ]

• ##### Awesome!(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, WarrenS

Going to order it tonight! Thanks for the recommendation!

Alas, the day job is over and must go teach now. Good night all!

• ##### absolutely(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, euterpe

bye-bye \$40

Love is the source, substance and future of all being. --St. Francis

[ Parent ]

• ##### Thank you!(4+ / 0-)

Very interesting!

Recommended by:
Dumbo, chingchongchinaman, cfk, pico, euterpe

The "Circle Of Fifths" and really blow their minds.  :-) p

"To do is to be." - Plato "To be is to do." - Aristotle "Dooby Dooby Do." - Sinatra

• ##### Remember...(5+ / 0-)

...that it's not really a circle, it's a spiral.

Because?

Because there is no such thing as a number that is simultaneously a power of 2 and a power of 3.

Freedom isn't "on the march." Freedom dances.

[ Parent ]

• ##### That is correct(4+ / 0-)

I haven't studied theory in decades but I spent my youth taking theory and instruction (woodwinds predominantly) in the 70's and 80's.  Then I picked up one of my brother's guitars and it was all over from there.

"To do is to be." - Plato "To be is to do." - Aristotle "Dooby Dooby Do." - Sinatra

[ Parent ]

• ##### love it love it love it(5+ / 0-)

PLEASE donate to a global children's PEACE project: Chalk 4 Peace

• ##### On a sidenote, music is now a drug:(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, euterpe

---
Toyota: Proof US Union Labor Still Does it Better

• ##### I forward these onto my adult kids...(4+ / 0-)

the replies come back, "I felt like I was back in music theory class in college" and from the other, "ya, me too"!

What if the hokey pokey is what it's all about?

• ##### Well...(4+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, pico, euterpe

...it looks like at some point I'm going to have to post a companion diary on the differences between 12-tone equal temperament and Just Intonation.

Nice work.

Freedom isn't "on the march." Freedom dances.

• ##### "Physics" is a term I may go along with(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, euterpe

I've always thought of music theory as an integration of math and language to some degree.

Where's zen bassoon dammit?

"To do is to be." - Plato "To be is to do." - Aristotle "Dooby Dooby Do." - Sinatra

• ##### I'm going to make a diary about just that(4+ / 0-)

somewhere much further down the line.  I'm thinking in particular of the formal language aspects that Hofstadter brings up in his wonderful book, Godel-Escher-Bach.

• ##### Agreed, although I would (5+ / 0-)

embellish that into music, for me, is a glorious marriage of math and language. I love that something that is so emotional can be quantified as well. A lot of my non-mathematically inclined musical friends say it takes some of the magic out of it for them, but to me it only makes it more magical.

• ##### Yeah...(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, euterpe

...just because I know the numbers doesn't mean I don't go "ooooh!" when the note is beautiful.

Freedom isn't "on the march." Freedom dances.

[ Parent ]

• ##### It's fascinating, when you think about it,(4+ / 0-)

that bisecting the octave perfectly gives you a dissonance instead of a consonance.  The devil-in-music, to use Medieval speak.

Hard to imagine modern music without it, though.  For me, the tritone is like a fine wine: it's an acquired taste, but once you get used to it, you become addicted.  Scriabin built an entire harmonic system off it.

Saint, n. A dead sinner revised and edited. - Ambrose Bierce

• ##### Tritones pull you away from the tonic key. (2+ / 0-)
Recommended by:
chingchongchinaman, pico

I'm going to have to do a second part on this next week.  I wanted to get into different modes, including Dick Dale playing Miserlou in Phrygian.

• ##### Well, assuming you have a tonic key.(2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

But in the more commonly used systems in the West, yes.

Saint, n. A dead sinner revised and edited. - Ambrose Bierce

[ Parent ]

• ##### richard lloyd(3+ / 0-)
Recommended by:
Dumbo, chingchongchinaman, pico

(television)

used to have free lessons on his site, and his first was on the

math/maxic

and the devil's tone

alas, he's sellin it now, but it's humorous and illuminatin

Love is the source, substance and future of all being. --St. Francis

[ Parent ]

• ##### eh...(2+ / 0-)
Recommended by:
chingchongchinaman, pico

I sort of changed tracks half way through that post.  I thought Phrygian had an augmented fourth in it, but it doesn't.

• ##### If you're going to do modal music(2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

consider including the first movement of Bloch's Concerto Grosso No. 1 for String Orchestra.  The first movement is in Dorian mode, and it's a rare example of a large ensemble movement staying fairly close to a traditional mode.

• ##### I don't know why but when I (2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

first looked at this diary, only the video loaded and nothing else. So, I thought WTF?

Then, I looked and saw about 30 people had commented and again, I thought WTF?

So, I came back and looked . . . and the rest of the page loaded in my browser and well, it makes much more sense now! ;-)

Very nicely done!

This ain't no party. This ain't no disco. This ain't no foolin' around!

• ##### heh...(2+ / 0-)
Recommended by:
chingchongchinaman, Snud

I had to "debug" the html of the diary a couple of times before it would post.  It took a few minutes.

• ##### To me, the most fascinating thing(2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

about music is that it converts physics to feeling, and that (musical) feeling is sensitive to the mathematical structure of the universe.

"I was a big supporter of waterboarding" - Dick Cheney 2/14/10

• ##### And if you really want to nitpick(2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

We start with the modern even tempered system, where every half step is 100c apart.  We then go back to the true ratios where the M3 is 386c, while the even tempered ratio is 400c.  Add the fifth to that, and we have a chord when played on a piano that sounds MUCH different than it should sound.  We hear the difference when the orchestra plays a chord--the musicians' ears attune themselves to the true chord, while the piano at the same time plays the even tempered chord.  It's subtle, but it's there.

This is why key relationships as discussed in these diaries are so important.  Back then, a keyboard would be tuned in the key of C, for example.  The tonic would be set, and all the other keys/strings would be tuned in that set of ratios.

THEREFORE, when the key modulated to the IV or V chords in other movements, the difference was more profound.

This is the foundation for modern music theory.  We all had to do our analyses of Bach Chorales as we learned "the rules" about chord progressions.  This is the reason.  Because the ratios link up.

"Nothing in all the world is more dangerous than sincere ignorance and conscientious stupidity." --M. L. King "You can't fix stupid" --Ron White

• ##### Ah, you'll have to keep telling me about it.(0+ / 0-)

I never took music theory. Everything I know about music I learned from Donald Duck.  And liner notes.

• ##### one nice thing about reading this is that....(2+ / 0-)
Recommended by:
Dumbo, jlms qkw

....it's much less financially stressful than taking a course :) .  Interesting to see what the next installment is.  How long do you plan to make the series?

BTW, just so you know, The First Night of The Proms tomorrow: Mahler 8.

"It's only in books that the officers of the detective force are superior to the weakness of making a mistake." (Wilkie Collins, The Moonstone)

• ##### I really hadn't planned to go into (2+ / 0-)
Recommended by:
chingchongchinaman, jlms qkw

music theory, but, well, look where we are, eh?  My goal is to first lay the groundwork so we can analyze larger works of music without dumbing it down to the point where it becomes "Beethoven's Seventh sounds like horses galloping in a fox hunt to me!"  It's hard to discuss a work like that without going at least a little into key relationships first.

• ##### yup, whether one wants to or not....(0+ / 0-)

....understanding at least the basics of music theory is needed for greater appreciation of music.  LvB, after all, was very careful in his tonal plans in his works, as was Mahler.  In fact, in The Mahler Companion, David Matthews talks about Mahler 6 and the controversy of the order of the middle movements, and comes down on the side of II=Scherzo, III=Andante, citing the tonal plan of the whole of Mahler 6, and comparing it to LvB 7.

"It's only in books that the officers of the detective force are superior to the weakness of making a mistake." (Wilkie Collins, The Moonstone)

[ Parent ]

• ##### Sweet! Now I can go play One!(0+ / 0-)

Especially the bit starting at 5:45 - http://www.youtube.com/...

:P

And I would've tipped this, but for some reason I couldn't :sadpanda:

I'm gonna go eat a steak. And fuck my wife. And pray to GOD - hatemailapalooza, 052210