Today is as good a day as any to catch up on music-related topics -- specifically my first experience last week with the five-string banjo. Once again, what I'm about to write will sound silly to experienced banjoists, but I like writing from the perspective of someone trying to discover music as opposed to just looking things up.
As I found out on Wednesday, the five strings of a banjo are tuned GDGBD. You might ask -- if I haven't looked up anything up, how do I know that the five strings are GDGBD? Well, when I was younger, I developed perfect pitch. Lately, my perfect pitch has been corrupted by my broken out-of-tune guitar -- because my D string played flatly, I got confused as to what a D actually sounds like. Still, I think my pitch is good enough to identify the open strings as a G major chord.
There's one thing about the G here -- we'd expect that first G in GDGBD to be a low G, but in fact it sounds like a high G4, just like the unexpected G4 on the ukelele. In fact, the high G is shorter than all the other strings.
Let's try a barre chord (and I assume that with five strings, banjoists really do play barre chords, as opposed to ukelelists with only four strings). If we want to reach the first real fret of the high G -- a G# (or Ab), then this lines up with the sixth fret of the other four strings, as the first five frets are nonexistent on the short G. The resulting chord sounds as a Db major chord (note not as Ab major). I wonder whether banjoists would denote the fingering as 16666 or 66666.
Also, notice that while the five open strings sound as a G chord, the lowest bass note played isn't on the G, but on the D string. Thus guitarists would label this as a G/D chord. (Likewise the bass note for the Db barre chord shown above is Ab, so guitarists would label it as Db/Ab). I wonder whether banjoists are as insistent as guitarists in writing the bass note when it's not the root of the chord.
OK, so let's look at chords that can be played in actual songs. On Wednesday, one of the first songs I played on the banjo was Square One TV's "That's Math." Let's think back to when I played this song on my guitar during my long-term assignment in October. My D string was still broken, and so my guitar was tuned to EACGAE instead. This was, in fact, the very first song I played in EACGAE, and since I was still trying to figure out chords in this tuning, I played this entire song with one chord, a G chord -- fingered as a G/B chord (x22023). Instead of my fretting hand, I focused at my strumming hand -- when the words "That's Math!" were sung, I played a quick G chord, and then used my thumb to play three bass notes, namely B-D-D, on the two lowest strings without moving my fretting hand.
Since I used a G chord on the guitar, and since the open G chord is the easiest chord to play on the banjo, this song would seem like a natural fit. But now the fact that the open chord is G/D rather than G/B hurts us -- there is no low B to play with the thumb during the running bass line shown above. We could play the higher B and D strings, but then it would no longer be a running bass line.
My solution that day was to play the song in the key of C major rather than G. I was able to find a simple playable C chord -- 02012. This is technically a C/E chord, and thus it has the notes E and G that we need for the running bass. The trickiest part for me is counting the strings -- there are five strings for me to strum, but only four strings to fret, since the high G is short. Thus I often fretted or strummed the wrong string.
The next song I played were the "Row Row Row Your Boat" parodies. (I actually performed all three that I had -- measures of center, same sign, and pi). Since I already had a C chord, all I needed was a G7 chord, which was easily derived from the G chord as 00003. I also played a pi song that was a parody of "Twinkle Twinkle Little Star" -- in C major, this is often played with C/F/G7 chords, but I just played it with C and G7 only.
Of course, I could have played both of these in G major, but then I'd have needed a D7 chord. The tricky thing about this chord is, what should we do about that short high G? We wouldn't mind fretting it at the second fret to produce A, but that's really the seventh fret, and so our hand can't reach it if the other strings are fretted to anything resembling an open D7 chord.
Then again, think about how we play a D7 chord on the guitar (in standard tuning) -- that chord uses only four strings, xx0212. So we can try a four-string D7 on the banjo and just ignore the high G. Note that the middle strings DGB of the banjo match the corresponding strings on the guitar, and so it suffices only to adjust the finger on the high D string, x0214. (Notice that we can't just keep the D string open as x0210 as there would be no third -- it would be ambiguous between D7/Dm7). This is a bit of a stretch, but it's still playable (certainly more so than 20214, which is really 70214).
So now we see how to convert open guitar chords to banjo -- keep the middle DGB strings the same as on the guitar, advance the guitar high E finger two frets on the banjo high D string, and omit the high G unless the chord contains a G. Thus in the key of D major, the tonic chord becomes x0234 (which is a bit easier to play than D7) and A7 becomes 02022. But A major becomes x2222, which looks a lot more like a barre chord than an open chord, even though E7 is rather simple as x2100. At least E major can take advantage of the open B string -- it is playable as x2102, with B7 as x1204.
Meanwhile, the guitar F chord xx3211 becomes xx3213, which ironically might be slightly easier on the banjo than the guitar. On the guitar, we often play C7 as x32310, which is missing its fifth -- but on the banjo, it converts to 02312, with the high G supplying the fifth.
As for our minor chords, Dm becomes x0213, Am becomes x2212, and Em becomes 02002. Since the open strings are set to G major, we might be tempted to find a G minor chord, but of course it's not easy to play a note of Gm on the open B string. We might try something like 00330, even though this leaves the fretted note on the B string in unison with the high D string.
We've been working on EDL-fretted versions of the guitar and banjo, so we might consider what an EDL version of the banjo might look like. Since the open strings form a G major chord, it seems straightforward to tune the B string yellow (yo) and all the other strings white (wa). But the huge problem here is the short high G. Since it's five frets shorter than the other strings, it lines up with the Degree 13 notes on the other strings. Thus if we play it at its first fret, the resulting note is 13/12 above the open G, producing a 13-over (tho) A that clashes with the wa A's frettable on the other strings. Or conversely, we can tune the open G to 13-under (thu) G so that its first fret is wa A, but then the open strings no longer produce a G major chord as its thu G clashes with the open wa G string.
There are several ways out of this. We could fret the instrument to 20EDL instead of 18EDL -- then the fifth fret is 20/15 = 4/3 above the open string, so that open wa G's line up. The second alternative is to use 16EDL and line up the nut of the high G with the fourth fret, 16/12 = 4/3 above the open string. The last alternative is to keep 18EDL, but place the nut of the high G halfway between the fourth and fifth frets of the other strings. On this string, both wa G and thu G are playable, as these notes are now 13.5/13 or 27/26 apart.
OK, that's enough about banjo music for now.
Lesson 13-3 of the U of Chicago text is called "Ruling Out Possibilities." In the modern Third Edition of the text, ruling out possibilities appears in Lesson 11-1.
This is what I wrote about Lesson 13-3 two years ago:
Here are a few things that I want to point out. First of all, some texts refer to the Law of Ruling Out Possibilities in Section 13-3 by another Latin name, modus tollens. Here is a link to the Metamath reference to modus tollens.
http://us.metamath.org/mpegif/mt4.html
As we can observe in the proof at the above link, modus tollens is essentially modus ponens (The Law of Detachment) applied to the contrapositive (Law of the Contrapositive, or contraposition.)
Section 13-3 is another section that lends itself to an activity, since many of its questions are actually logic problems, like the ones that often appear in puzzle books. But unfortunately, today isn't an activity day.
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