SudokuSolver Forum

Found this on my Facebook wall this morning.

I'm generally not a fan of 12x12 sudoku, and my music sight-reading isn't all it should be, but I'm going to give this one a shot. I may have to "cheat" and rewrite it in letters instead of notes on the staff before I solve it.

Enjoy!

Let's call this variant Schönbergoku

http://en.wikipedia.org/wiki/Arnold_Schoenberg

Finished this one tonight. Ironically, I did most of it during my kids' music lessons this morning. I did have to rewrite it with note names rather than staff notes, or I would have surely messed up more than once.

If, instead of Equal Temperament tuning, you use Mean Tone, then you'd have many more digits to choose from. For example, G-sharp would be different from A-flat, etc.

You'd probably want both sharp and flat versions of all the black keys, plus four of the whte keys -- C-natural and B-sharp, F-natural and E-sharp, B-natural and C-flat, and E-natural and F-flat.

Bill Smythe

Sorry--you'd have to talk to my kids the music theory experts on that one. LOL! Maybe one of them can convert a "normal" puzzle (using digits) into a music puzzle like that.

Like I said, I just found this puzzle and passed it on. Hope you enjoyed it!

Now that has completely lost me Bill.

I did maths at Uni and lots of people expect you to be good at music as well. I had a friend there who did maths and music - so it was true for him. The difference between us was that he excelled at algebra (Group theory etc.) which I just scraped through on. My best area was analysis (Differentiation etc.) and I've always felt that the linkage with music is through algebra.

That's an interesting idea, Maurice. Curiously enough, although all my children are musicians (as are hubby and myself), several have struggled with algebra (although they did fine with lower math). Hubby and I, on the other hand, had no trouble with algebra (my degree is in math). Made things really frustrating when I was teaching them algebra (we homeschool)--I had to actually change math curricula after I finally figured out their brains aren't wired for algebra like mine is.

For those desiring a little edification (or for those who just can't turn away, such as when observing a plane crash):

In pretty much any reasonable tuning system known to man, each octave represents a 2:1 ratio of frequencies. If A above middle C is 440 hertz, then A below middle C is 220 hertz, and the A below that is 110 hertz, etc.

In equal temperament, the octave is divided into twelve equal semitones. "Equal" means equal frequency ratios. Thus, the ratio between any two adjacent notes on the piano keyboard (counting both white and black keys) is the twelfth root of 2, or approximately 1.059.

The reason certain intervals sound good is that the ratio of their frequencies is simple. An octave is 2:1, a pure fifth is (approximately) 3:2, and a pure third is (approximately) 5:4.

I say "approximately" because the math doesn't quite work. For example, if you place 12 fifths end-to-end on the keyboard, it comes out 7 octaves: C - G - D - A - E - B - F# - C# - G# -D# - A# - F - C. This should mean that 3/2 to the twelfth power equals 2 to the seventh power. But it just misses. 2 to the seventh is 128, but 3/2 to the twelfth is just a tad more than that.

The situation with major thirds is even worse. 3 thirds placed end-to-end, for example C - E - G# - C, should come out equal to 1 octave. But when you do the math, 5/4 to the third power is 125/64, considerably short of 128/64 which would be a pure 2:1 octave.

It is convenient to use a logarithmic scale, since the human ear hears tones that way. Don't be scared away by the word "logarithm". Logarithms are simply a nice way of converting ratios into differences, so that what you hear is what you get. If, on this logarithmic scale, you divide the octave into 1200 equal parts (usually called "cents"), and use C as your starting point, then of course the result is:

C - 0 cents
C# - 100 cents
D - 200 cents
D# - 300 cents
E - 400 cents
F - 500 cents
F# - 600 cents
G - 700 cents
G# - 800 cents
A - 900 cents
A# - 1000 cents
B - 1100 cents
C - 1200 cents

The formula for converting ratios into cents is:

cents = log(ratio,2) x 1200

-- where log(ratio,2) means the base-2 logarithm of ratio. You can do this in Excel, if you want to. For example, if the ratio is 2:1 (i.e. a pure octave), then the base-2 logarithm of 2 is 1, and then times 1200 is 1200 cents.

Now, how do 3 major thirds stack up against 1 octave? If, in Excel, you type =log(5/4,2)*1200, you don't get 400. You get 386. (I'm rounding off a bit here.)

Likewise, it turns out that a pure fifth (ratio 3/2) is not 700 cents, but rather 702.

So, in equal temperament, the fifths are almost perfect (just 2 cents off), but the major thirds are horrible (14 cents off). They sound rough and beat quickly.

So, "mean tone" temperament has a different idea. Tune the C-E third as a pure 5:4 ratio, and E-G# likewise. This leaves the remaining third, G#-C, extremely wide, so that the key of Ab, for example, is unusable. But the keys that are useable sound really nice.

Note that, if instead of tuning G# as a pure third above E, you tune it as a pure third below C (in which case it should really be called Ab instead of G#), now the E-G# (oops, E-Ab) third is now horrible, so that now you can't satisfactorily play in the key of E-natural, among others.

Now, to keep things simple, suppose we tune D as the mean between C and E, so that the C-D and D-E intervals are the same size. Suppose we also tune the F-G-A-B foursome so that the three intervals F-G, G-A, A-B are all the same size as the C-D and D-E intervals. And let's also adjust the F-G-A-B foursome relative to the C-D-E threesome in such a way that the two diatonic semitones, E-F and B-C, come out the same size as each other. This gives us, in mean tone temperament, the following scale on the white keys:

C - 0 cents
D - 193 cents
E - 386 cents
F - 503.5 cents
G - 696.5 cents
A - 889.5 cents
B - 1182.5 cents
C - 1200 cents

What about the black keys? In mean tone, G# is not the same as Ab, so we have to pick and choose. In the days when mean tone was widespread, it was traditional to allow for three sharps, F#, C#, G#, and two flats, Bb and Eb. Tuning F#, for example, as a pure third (386 cents) above D, we have:

C - 0 cents
C# - 75.5 cents
D - 193 cents
Eb - 310.5 cents
E - 386 cents
F - 503.5 cents
F# - 579 cents
G - 696.5 cents
G# - 772 cents
A - 889.5 cents
Bb - 1007 cents
B - 1182.5 cents
C - 1200 cents

If you tune your piano in this way, you'll have an instrument that sounds really great (pure, beatless major thirds) in the keys of A, D, G, C, F, and Bb, but unacceptable in other keys, because of missing notes, wolf thirds, and a wolf fifth (G#-Eb).

Sudoku that!

Bill Smythe