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ItalianoBase 16
Base 16 has been used extensively in computer science and programming, and its digits are commonly denoted 0123456789ABCDEF. However, what usually matters is the structure of bits of a number and this structure is not apparent from its common symbol. For this reason, I decided to create new symbols for hexadecimal digits (but don’t take it too seriously, this really isn’t an important concept):
The structure is probably apparent – the top dot denotes the first bit, the middle dot the second bit, and the very bottom dot which is present only in digits above 8 denotes the fourth bit. After I created the font for these symbols, I found on Wikipedia that Bruce A. Martin created a notation very similar to this one. Nevertheless, I believe that my notation has certain advantages in comparison with Martin’s. In his notation, some digits may become hard to distinguish in certain circumstances – this is resolved in my version by making the digits lower than 8 the same size as an ordinary number, digits higher than 8 the same size as the letter j.
But lets stop talking and show some examples. Here are some random 4-digit numbers:
AFFA – 
56B3 – 
5BCE – 
FFFF – 
Of course, it’s also possible to express hexadecimal expansions of rational (or irational) numbers. For example, the first few digits of pi would be 
(standardly written as 3.243F6A8885A30), e (Euler number) would be 
(standardly 2.B7E151628AED2).
And to make this boring concept at least a little less boring – who will be the first one to find out what the following monstrous number denotes?
The inspiration was a window. http://yxulyoj.com
It has just occurred to me that if you play piano, and you are interested in a symmetric keyboard, you should have a look at the Axis-49 from C-Thru music, http://www.c-thru-music.com/cgi/?page=prod_axis-49. I can’t play piano but I can easily construct simple melodies and chords on it, and I use it for midi programming all the time.
That was off-topic, sorry.
I don’t play piano either, but I was also thinking about a piano-like symmetric keyboard. I actually came to this: http://up.kubaz.cz/piano.png even before I started using the major third tuning. The logic of it is the same as that of the major third tuning—half steps in the horizontal direction, up and left is the major third, up and right is the perfect fourth (at first the buttons were square as on the guitar, but I then thought that this “slanted” design would make it easier to play chords). What a pity I don’t have enough money to construct such a thing.
After I drew this I started exploring the market to find similar devices that already exist; the Axis one was one of them and I was even considering buying the Chromatone keyboard (http://www.weblogsinc.com/common/images/0895681712491002.JPG?0.12966241855699623), but such things cost a lot of money for a student of mathematics who is not even a pianist.
I also looked at the Chromatone and the Janko keyboard, as well as the Thummer. What I don’t understand is why nobody has made a keyboard where every second key is black (what I mean is for every two white keys there is a key in between them, positioned in the same way as a black key, as shown here, http://balanced-keyboard.com/ContextLinks.aspx). There is pretty much zero development required to put such an instrument on the market, and yet everybody sticks to the traditional design. We just have to realise that the vast majority of musicians deep down are very, very conservative despite their rebel attitudes and silly haircuts.
I think it’s not that much about conservativeness; it is more about their intentions. Most of the people don’t want to deeply understand musical principles and to improvise. Most people just play pieces written by other composers and those are written for the standard layout. Since the standard musical notation is basically just “tabs” for the piano, they are not really concerned; they have their tabs that allow them to sight-read music easily.
Another problem is that once you have learned the balanced layout, you have to bring your piano with you every time you want to play somewhere outside your home. If you play either a traditional piano or a quality digital piano, this might be a serious problem since they are pretty unmovable.
So, basically, the possible customers would be non-professional composers and players for whom this device would serve (presumably) as a midi input device. Those people usually don’t really appreciate possible advantages of a symmetric layout; they would be probably more scared by the fact that they wouldn’t be able to play any other piano.
And finally, the bottom line: What is the point of changing the layout without changing the notation? If you continue to think in the traditional musical notation, then a symmetric layout gives you almost no advantage. Unfortunately, to persuade people to use a different musical notation is like to persuade them to learn a revised English spelling. There’s only very few people who are willing to “forget” everything they know and learn something better.
I found it… but I cheated. I searched for the first 15 decimal digits and got a reference to a paper on group theory. I know Galois fields and some number theory but it is the first time I have heard about the Monster.
Well, that’s not really a cheating; I would probably do the same thing
. I am glad to see a fellow M3 guitarist commenting on my mathematical concepts – one perhaps has to be close to mathematics to even start thinking about something like this tuning.
Your site turned up on my radar because my stats show that some of my visitors come from your site. Very few people link to me so it is not difficult to keep track of new customers. The m3 club is small and exclusive — far too small, and far too exclusive if you ask me — and worldwide there is a tiny audience for this sort of thing. Alternative tunings appeal mainly to people with an analytical and curious mind, and I suppose mathematicians fit the bill better than any other group.
Anyway, you have created some great content, keep up the good work.
The number seems to be a factorial because of trailing zeros, but no factorial has this value actually.
The number indeed isn’t a factorial. Nevertheless, you probably know the thing to which this number is related.
Oh, I’ve heard of it, but I wouldn’t find out the number’s context myself. It’s really a monstrous number.
Ok, now a bonus task: Write the same number in base 60 (using the digits 0-9, a-z for 10-35 and A-X for 36-59)
. I would like to see how this number would look using my digits for base 60 (maybe it would uncover the whole structure of universe or so), but I am too lazy to convert it 
.
Ok, just for you, it’s 3DhQpy3Gql0koTvVA2JRgM000000000. Just few lines in Python interactvie shell.
Note the substrings “py3” for Python 3000 or “0ko”.
Well, another problem is, that I am too lazy to check that this is actually correct
, but I believe you. Unfortunately, no interesting patterns seem to arise in this base 
. Nevertheless, I should learn Python anyway, for it seems to be the most useful language for such little tasks. I usually write such little programs in PHP, but it is often inconvenient.
Here is a source code in Python, I use “–” for ident here.
def digits(n, base):
–while n > 0:
––n, digit = divmod(n, base)
––yield digit
DIGITS_REPR = string.digits + string.ascii_loweralpha + string.ascii_upperalpha
“”.join(DIGITS_REPR[digit] for digit in reversed(digits(n, 60)))
I don’t pretend I understand all the syntactical constructions used in your example, but overall it looks pretty “high-level”, and that’s exactly what I was looking for. I think I’ll give Python a try.