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Semitonal Half-Stepping, Ever Sharper

Date: 05/05/2012 at 19:28:49
From: Ralph
Subject: Music: Circle of 5ths - question about key signatures

My question relates to the patterns on the keyboard. 

I have always wondered why, when you traverse the circle of fifths in
music -- for example, going up by 5ths -- each successive scale needs to
have one more sharp. Likewise, traversing it by 4ths in the other
direction, each successive scale has one more flat.

Can you tell me how to demonstrate mathematically why this is the case?

I studied advanced math in college, and later on piano and music theory in
conservatory. But I am afraid that math was learned too many years ago --
I graduated with an engineering degree ... in 1951! Now I do not have the
mental capacity to be able to "prove" this to my satisfaction.

I am sure there is some simple way to demonstrate why this is so, having
to do with the layout of the piano keys, and the distances between each
note of a scale, but I can't figure out how to demonstrate it
mathematically or in some logical -- and, I hope, simple -- manner.

By the way, my age is 83.5 -- not one of the choices on your pull-down
menu of ages! Think of me as 25 ...



Date: 05/18/2012 at 09:39:36
From: Doctor George
Subject: Re: Music: Circle of 5ths - question about key signatures

Hi Ralph,

Thanks for writing to Doctor Math.

I too have been interested in this question for a long time, so your
asking gives me an opportunity to think about it.

First, write the notes of the C major scale in the order that the flatted
notes occur in key signatures.

     B  E  A  D  G  C  F

Notice that the notes are separated by perfect fifths as we read from
right to left. Now move each note up a perfect fifth to the key of G major.

     F# B  E  A  D  G  C

The initial B moved to F# and the initial E has moved to B. So when we
move up another perfect fifth to the key of D major, we will get another F#
from the new B. We now get two sharps because the first F# has moved to
C#, like this:

     C# F# B  E  A  D  G

If we continue moving up by perfect fifths, we get this table of keys with
sharps:

    B  E  A  D  G  C  F
    F# B  E  A  D  G  C
    C# F# B  E  A  D  G
    G# C# F# B  E  A  D
    D# G# C# F# B  E  A
    A# D# G# C# F# B  E
    E# A# D# G# C# F# B
    B# E# A# D# G# C# F#

If you read down each column, you will see that every column follows the
same pattern, just shifted. Once the notes in the column receive a sharp
sign, the sign stays to the end of the column.

If you read across the rows, you will see that each sharp shifts across
the columns as we move down from one row to the next.

Now go back to C major and move down a perfect fifth. If you do this seven
times, you will see the table of keys with flats emerge:

    B  E  A  D  G  C  F
    E  A  D  G  C  F  Bb
    A  D  G  C  F  Bb Eb
    D  G  C  F  Bb Eb Ab
    G  C  F  Bb Eb Ab Db
    C  F  Bb Eb Ab Db Gb
    F  Bb Eb Ab Db Gb Cb
    Bb Eb Ab Db Gb Cb Fb

We can also reverse the rows of the flat keys table, and combine it with
the sharp keys table like this:

    Bb Eb Ab Db Gb Cb Fb
    F  Bb Eb Ab Db Gb Cb
    C  F  Bb Eb Ab Db Gb
    G  C  F  Bb Eb Ab Db
    D  G  C  F  Bb Eb Ab
    A  D  G  C  F  Bb Eb
    E  A  D  G  C  F  Bb
    B  E  A  D  G  C  F
    F# B  E  A  D  G  C
    C# F# B  E  A  D  G
    G# C# F# B  E  A  D
    D# G# C# F# B  E  A
    A# D# G# C# F# B  E
    E# A# D# G# C# F# B
    B# E# A# D# G# C# F#

In every case, the key signature is in the sixth column.

Now, how do we put some math to this? I expect that there is more elegant
math that can be applied, but here is my insight from what I have learned.

Let's number the notes of the scale, starting with 0 for C. I'll use two
rows for the black key names on the piano, and two rows for the white key
names, to show that some notes have two names.

      1   3       6   8   10
      C#  D#      F#  G#  A#
      Db  Eb      Gb  Ab  Bb
    C   D   E   F   G   A   B
    B#      Fb  E#          Cb
    0   2   4   5   7   9   11

In what comes next, I will make switches like A# to Bb or B to Cb, as
needed.

There are 12 notes in each octave, so we will be interested in modulo 12
arithmetic. A perfect fifth is an interval of 7 notes or half-steps. So
when we move up a perfect fifth, we get an equation like this:

    new_note = (old_note + 7)(mod 12)

We can apply this math to any note in the scale. For example, let's start
with B to produce the first column from the table of sharp keys above.

If old_note = 11 for B,  then new_note = 11 + 7 =  6(mod 12), or F#.
If old_note =  6 for F#, then new_note =  6 + 7 =  1(mod 12), or C#.
If old_note =  1 for C#, then new_note =  1 + 7 =  8(mod 12), or G#.
If old_note =  8 for G#, then new_note =  8 + 7 =  3(mod 12), or D#.
If old_note =  3 for D#, then new_note =  3 + 7 = 10(mod 12), or A#.
If old_note = 10 for A#, then new_note = 10 + 7 =  5(mod 12), or E#.
If old_note =  5 for E#, then new_note =  5 + 7 =  0(mod 12), or B#.

To move down a perfect fifth, we get an equation like this:

    new_note = (old_note - 7)(mod 12)

For example, let's start with F to produce the last column from the table
of flat keys above.

If old_note =  5 for F,  then new_note =  5 - 7 = 10(mod 12), or Bb.
If old_note = 10 for Bb, then new_note = 10 - 7 =  3(mod 12), or Eb.
If old_note =  3 for Eb, then new_note =  3 - 7 =  8(mod 12), or Ab.
If old_note =  8 for Ab, then new_note =  8 - 7 =  1(mod 12), or Db.
If old_note =  1 for Db, then new_note =  1 - 7 =  6(mod 12), or Gb.
If old_note =  6 for Gb, then new_note =  6 - 7 = 11(mod 12), or Cb.
If old_note = 11 for Cb, then new_note = 11 - 7 =  4(mod 12), or Fb.

From these patterns, you can see how the math produces the sequence of
sharps and flats as we move up and down by perfect fifths. Each note in
the scale moves through the same sequence. As a sharp note moves on to the
next sharp, it gets replaced by another note that moves to take its place
in the sequence. Likewise, as a flat key moves on to the next flat, it
gets replaced by another note that moves to take its place in the
sequence.

The 12 tone scale is really a remarkable achievement.

Here is an article from the archive that you may also find interesting:
  http://mathforum.org/library/drmath/view/52470.html 

I hope this helps.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 05/21/2012 at 10:22:13
From: Ralph
Subject: Thank you (Music: Circle of 5ths - question about key signatures)

Many thanks!

I am glad that someone else out there has wondered about this.

You gave me a lot to think about, and I need some time to review your
reply. I'll advise after I go through it!

Ralph Levy, Baltimore



Date: 05/21/2012 at 11:29:40
From: Doctor George
Subject: Re: Thank you (Music: Circle of 5ths - question about key signatures)

Hi Ralph,

I found another way to express this.

In my first reply we considered the notes of the C major scale in the
order B  E  A  D  G  C  F. Instead of viewing these notes as occurring in
the same octave, let's pick the low F on the bass clef, the C above that,
the G above that, and so on up to B.

Now let's move all of the notes up a perfect fifth, as before. But we will
write the new notes shifted over, as if at a piano, instead of straight
down the column as before.

    F  C  G  D  A  E  B
       C  G  D  A  E  B  F#

B moved to F#, E moved to B, A moved to E, and so on. When we shift
another perfect fifth, F# will move to C#, and we will get another F# from
the new B. So the sharp pattern begins to emerge.

If we keep shifting by perfect fifths up the keyboard, we get the same
tables of sharp keys as before, but organized a little differently.

    F  C  G  D  A  E  B
       C  G  D  A  E  B  F#
          G  D  A  E  B  F# C#
             D  A  E  B  F# C# G#
                A  E  B  F# C# G# D#
                   E  B  F# C# G# D# A#
                      B  F# C# G# D# A# E#
                         F# C# G# D# A# E# B#

The layout of the table in my first reply makes it easy to find the key
signatures, but this new layout provides a clearer demonstration of how
shifting up each fifth adds a new sharp, while keeping the previous ones.

We can do the same thing for the table of flats, too, and we can create
both tables using either perfect fourths or perfect fifths.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Number Theory

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