Piano keyboard pattern
patrol
keyboard?
The typical 88 key piano keyboard is based on *repeating patterns* of
12 keys -- 7 white keys and 5 black keys.
As a simplified approximation, one could say that the white and black
keys alternate (w,b,w,b,...) but this is of course not completely
accurate (it couldn't be given the unequal numbers of black and white
keys in a pattern), but it's a good starting approximation, I think.
In terms of the number of occurrences of adjacent white keys, there
seems to be a dependency on the repeating pattern's *chosen starting
point*. For example, if you say the pattern starts at a "C" key, you
could say that the pattern is alternating w,b,w,b,... except between
the 5th and 6th keys, which are adjacent whites.
w,b,w,b,w,w,b,w,b,w,b,w
However, if you say the pattern starts on the "D" key, you would have
to say that the pattern is alternating w,b,w,b,... except between the
3rd and 4th keys *and* between the 10th and 11th keys, which are
adjacent whites.
w,b,w,w,b,w,b,w,b,w,w,b
It's even more confusing because a set of 12 keys, regardless of
starting point, contains exactly 7 white keys and 5 black keys, so how
can a repeating set starting at D have more pairs of adjacent whites
than a repeating set starting at C? It gives the appearance that there
are *more* white keys in a D set than a C set.
It seems to be a mathematical problem to describe the pattern of a
piano keyboard in a unified matter, one that doesn't depend on a
chosen starting position. How would it be done?
Henry
I assume you are not talking musical scales, but just the patterns on
the keyboard.
Even so, the word "octave" might give a clue as to what you are
missing: starting from C, you should be looking to the next C (8 white
keys with inclusive counting, 7 with half inclusive counting; 13 black
and white keys with inclusive counting, 12 with half inclusive
counting).
In terms of adjacent white keys, there need to be 12 adjacencies going
from C to C or from D to D. Two of these are white to white; five are
white to black; and five are black to white. What you seem to have
missed is that to complete your pattern starting from C, the 12th and
13th keys are adjacent whites. Starting from D they are black and
white.
alainv...@gmail.com
>
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Bonsoir,
We may distinguish three kinds of sequences :
c#d#e# f# g#a#b
......|wbwbw |wb|wbwbw|....
t1 d t2 ; t for tierce, d second.
a repeated motive t1dt2 worth an octave,
Alain
Ken Pledger
<9bca75c9-729a-4863...@g26g2000vbz.googlegroups.com>,
patrol <patro...@hotmail.com> wrote:
> ....
> In terms of the number of occurrences of adjacent white keys, there
> seems to be a dependency on the repeating pattern's *chosen starting
> point*. For example, if you say the pattern starts at a "C" key, you
> could say that the pattern is alternating w,b,w,b,... except between
> the 5th and 6th keys, which are adjacent whites.
>
> w,b,w,b,w,w,b,w,b,w,b,w
>
> However, if you say the pattern starts on the "D" key, you would have
> to say that the pattern is alternating w,b,w,b,... except between the
> 3rd and 4th keys *and* between the 10th and 11th keys, which are
> adjacent whites.
>
> w,b,w,w,b,w,b,w,b,w,w,b
> ....
Avoid having a starting point. Since the one-octave pattern is
repeated, represent it around a circle.
Ken Pledger.
Gerry Myerson
<9bca75c9-729a-4863...@g26g2000vbz.googlegroups.com>,
patrol <patro...@hotmail.com> wrote:
Consider the multiples of 12 / 7:
0, 12/7, 24/7, 36/7, 48/7, 60/7, 72/7, 84/7, 96/7, ...
Write each as a mixed fraction:
0, 1(5/7), 3(3/7), 5(1/7), 6(6/7), 8(4/7), 10(2/7), 12, 13(5/7), ....
(where, for example, 10(2/7) means "ten and two-sevenths').
Now keep the whole number and forget the fraction:
0, 1, 3, 5, 6, 8, 10, 12, 13, ....
Now replace each whole number in 0, 1, 2, 3, 4, ... with a w if it's in
that last sequence, a b if it isn't:
w, w, b, w, b, w, w, b, w, b, w, b, w, w, ....
and you have the piano keyboard pattern, starting at B.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Dan
Approach to Music Fundamentals" as a book on this topic . The first
thing we know about piano is its parted in groups of 12 keys , such
that the frequency of the 13-th key is always twice the frequency of
the first .
Pitches who's frequency ratios are rational number with a small
denominator (ex 3/2 or 4/3 ) sound good together (that would require
a separate explanation) . Initially instruments had a system known as
Pythagorean tuning http://en.wikipedia.org/wiki/Pythagorean_tuning
where you had a few "main" groups of pitches (witch today would be on
the white keys) sound perfect , at the dissonance of every other .
Today we use a system called equal temperament . The problem to be
solved was this :
You have pitches so that the 13th is twice the first , and you want :
-the ratio between any consecutive pitches to be equal . (p1 / p2 =
p2 / p3 = ... )
-the ratios be nice rational numbers .
As it turns out , satisfying both is impossible , equal temperament
goes for number 1 . The resulting system is one in witch all twelve
scales sound "equally a little bad" , as opposed to Pythagorean tuning
where the main groups of white keys sound perfect and the rest badder
(badness is unequally distributed ) . In equal temperament , the
pitch-ratio between any two consecutive keys is 2^(1/12) . (notice
that 2^(12/12)= 2 , 12 keys apart .) .
Now , we have equal temperament ,but the structure of the piano still
makes the C scale (all white keys) easier to play that the others .
So , the reason that piano is the way it is is partially historical .
The keys are made to show the structure of a scale (though its just
the C scale , not the others :) ) .
To understand the structure of a scale , imagine a round table with
12 black chairs , and 7 men dressed in white . They want to sit at the
table "as evenly as possible" . The only ways to do this are all a
"circulification" of the piano key pattern (.... W B W B W W B W B W
B W ... ) .
So that's the structure of a piano .
For the reasons above , I don't consider the piano layout to best
possible . More info on alternatives here .
http://improvise.free.fr/
http://improvise.free.fr/beanbut/bean.htm
Rob Johnson
I always think of C being a base point for an octave. This being
the case, let [x] = floor(x), then
12n + 5
[ ------- ]
7
maps
C: 0 -> 0
D: 1 -> 2
E: 2 -> 4
F: 3 -> 5
G: 4 -> 7
A: 5 -> 9
B: 6 -> 11
Of course, we could just as easily use A = 440 Hz as a base. This
yields
12n + 2
[ ------- ]
7
maps
A: 0 -> 0
B: 1 -> 2
C: 2 -> 3
D: 3 -> 5
E: 4 -> 7
F: 5 -> 8
G: 6 -> 10
Thus, the frequency of a note on the equal-tempered scale would be
12n + 2
[ ------- ]/12
440 x 2 7
Rob Johnson <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
Dan
Let's measure dot distances clockwise . There are two kinds of dot-
distances :
-the distance between two dots measured in dots .
-the distance between two dots measured in lines .
For example : The dot-distance between B and D is 1 dots (there is 1
dot between B and D) , but its 2 lines . Also , the distance between
B and C is 0 dots , or 0 lines . And the distance between C and D is 0
dots , or 1 lines .
An arrangement is called maximaly-even if :
for every possible value of distance-measured-in dots , there are at
most 2 possible values for line-distance , and if there are two , they
are consecutive .
In our example , for distance-in-dots 0 (B - C , C-D , G-A , etc ) ,
we have distance in lines 0 or 1 .
For distance-in dots 1 (B-D , C-E , A-C etc . ), we have distance in
lines 2 or 3 .
For distance-in dots 2 (F-B , B-E , etc.) , we have distance in
lines 4 or 5 ..
The piano key-pattern is the maximally-even arrangement of 7 points
over a 12-circle .
Dan
Bill Taylor
on the absurdity of an "octave" having 8
notes in it!
After all, (to look at a virtually identical situation),
a week has only seven days in it.
Unless you're French, of course....
-- British Bill
** Which would you hate the most,
** being color-blind or tone-deaf?
Gerry Myerson
<22778d03-d139-422d...@21g2000prv.googlegroups.com>,
Bill Taylor <wfc.t...@gmail.com> wrote:
> I'm surprised that no-one has yet commented
> on the absurdity of an "octave" having 8
> notes in it!
I'm sure that if it were up to the mathematicians
those intervals the musicians call third, fourth, fifth
would be called second, third, fourth, respectively.