A little puzzle

Hello again. I'm sure by now the endless talk about mathematical necklaces has left you gagging on the words Polya's Enumeration Theorem, but bear with me.

Consider a 12-bead necklace constructed of black and white beads, any number of each, so long as it adds to 12 exactly. No, this necklace cannot be taken off and put on the other way; you have to stick your head in the same way every time, otherwise

How many such necklaces are there? As is intuitively obvious even to the most casual observer, 352. But given a specific combination of black and white beads, how many can be constructed? Well, twelve white beads is easy: there's only one option, and it's the same with twelve black beads, and similarly there is only one way to arrange a single bead of one color along with eleven of another color. For counting (though not aesthetic) purposes, black and white are interchangeable: for instance, you'll get the same count if you're using ten black and two white as you will with ten white and two black. Take the latter option: how many necklaces can you form with two black beads and ten white? Well, the two black beads could be right next to each other, or one bead removed, two beads removed, and so on until they have five white beads between them on either side. This yields six necklaces of two black beads and ten white, and six more of the opposite arrangement.

The other arrangements are harder to count by inspection, but with the help of a visualization tool I found once, I cheated straight to the answer of how the 352 necklaces break out into the 13 possible combinations of white and black beads (see below). The first two ones represent using all beads of one color, and using eleven beads of one color and one of the other. The two ones at the end represent using just one bead of the one color, and using all beads of the other color (Make sense? Of course not. It's 10:30 PM and I was up at 4:20 this morning) In the case of using six and six, there are eighty possibilities:

1 + 1 + 6 + 19 + 43 + 66 + 80 + 66 + 43 + 19 + 6 + 1 + 1 = 352

What the heck does any of this have to do with music theory? Here's a hint: https://www.google.com/search?tbo=p&tbm=bks&q=isbn:1619114860

My apologies for being pedantic and cocky.

Why you should learn to read guitar in two clefs (part I)

I'm turning this blog into a forum for my musical interests. Right now I'd like to write a short post about why you should learn to read guitar in two clefs.

Grab your guitar and play the open fourth string in standard tuning. Now climb up the notes of the Dorian Mode (that's the C scale starting and ending on D): D, E F (moving to the 3rd string now) G, A, (and so on to the treble strings) B, C, D, E, F, G...

Stop!

You have just played *all* the notes in the key of C that either lie on or touch the guitar staff. Everything else must be notated with ledger lines. I should say here that guitar is notated with a treble staff, but sounded an octave lower than it would be on, say, a piano. I find reading these ledger lines to be endlessly frustrating as they lead to bothersome counting: "O wait, that's three lines down, so it's an F, not an A..." 

Now let's try that same exercise with the grand staff:

Your tour of the bass staff starts with the second lowest note on the instrument (in standard tuning): F on the first fret of the 6th string. Play the C scale starting there (i.e. the Lydian mode) and going all the way to the open B string. BOOM! The bass staff, like the guitar treble staff, overlaps entirely with the fretboard, and provides much more guidance for your bass playing than a bunch of ledger lines below the staff.

The treble clef is that same D Dorian mode from the first exercise, but played an octave up. I don't care how you play it, but it starts with the D on the fourth string 12th fret and ends on the first string 15th fret.

So with few exceptions (the open 6th string, middle C, and the very high 1st string) all the notes playable on the instrument are on one of the two clefs. That's why you need to learn to read guitar music in two clefs. Yay!

There are probably many reasonable complaints about what I've written here, but two main ones come to mind:

1) The best way to notate guitar is with tablature supported by standard notation. This provides the legibility of tablature and the ability to figure out the rhythms if you don't know them. Who, then, gives a shit about any of this?

2) It's not impossible to read standard guitar notation, and it's compact, with a legible (if cumbersome) notation that removes ambiguity if you just put in the time to learn it. For the few of you who can actually read guitar music, I'm sorry, you've wasted your time reading this.

No, actually, you haven't. While I personally believe that reading in two clefs easier and more fun than reading standard notation, the real reason to learn two clefs is to expand your repertoire of playable songs to anything written in a grand staff. That is quite an expansion--more on that later.