On the music scale, why do half steps exist? Why is there no B sharp
or C flat? It seems as though sounds are created through frequencies
... why would there be an empty or missing frequency in just a couple
places; why aren't the sound frequencies regularly spaced? Is it a
physical thing or a notation thing?

In western music, the most commonly used scale is the "major diatonic
scale" - a C scale of this type corresponds to the white keys of a
piano.  The major diatonic scale is a subset of the "evenly tempered
chromatic scale".  This scale consists of the following notes:

A
A# (or A sharp, or Bb, or B flat)
B
C
C# (or C sharp, or Db, or D flat)
D
D# (or D sharp, or Eb, or E flat)
E
F
F# (or F sharp, or Gb, or G flat)
G
G# (or G sharp, or Ab, or A flat)

[A# and B is sometimes referred to as B and H respectively, but this
is not relevant when answering this question.]

These twelwe notes span over an octave.  An octave is the interval of
two tones where one has a frequency that is the double of the other. 
In western culture, the octave is dived into twelwe semitones - this
is partly a cultural artifact and partly due to human physiology.

Originally, these twelwe tones were tuned after what sounds well to
the human ear, "justly tempered".  This sounds good when music is
played in the particluar key that an instrument is tuned to, but the
sound will not sound very good if you attempt to play it in a
different key (without retunin the instrument).  This is because the
frequency differences are not eavenly spread out across the scale.

In the "evenly tempered" chromatic scale (which is commonly used
today) the twelwe notes are equally spread out across the scale. 
Mathematically, this implies that the frequency of two adjacent notes
are separated by a factor that is equal to the 12th root of 2;
2^(1/12) = 1.0594631.  You can read more on the differences between
just and equal tempered scales at
http://www.phy.mtu.edu/~suits/scales.html

If you look at a piano keyboard, you will notice that an octave
consists of 7 large white and 5 smaller black keys.  The frequency of
two adjacent keys (regardless of colour) will always differ by a
factor of the 12th root of 2 (there is an illustration of this at
http://www.chordwizard.com/hmw109.asp).  There is no B sharp because B
and C are only a semitone apart, just like C and C sharp only differ
by a semitone.

Any major diatonic scale has steps of 2,2,1,2,2,2 and 1 seminotes -
corresponding to C, D, E, F, G, A and B in a diatonic C major scale. 
There are no "missing frequencies" - it just happens that these seteps
have been selected as the steps of a major diatonic scale.  This is a
matter of cultural preference - other cultures have divised musical
scales that follow different patterns.

I hope this answers your question.  If not, please request an answer
clarification berfore you proceed to rate it.

Suggestions for further reading:
http://www.foundation.bw/MusicTheorySummary.htm gives a handy summary
of the theory behind tones, notes and scales.  If you'd like to get
into deeper detail about different scales used in western music,
please see http://www.phy.mtu.edu/~suits/west_scales.html.

Search term used on Google:
frequency OR frequencies "chromatic scale"

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Clarification of Answer by blazius-ga on 31 Mar 2004 03:58 PST

Unfortunately, some typing errors were included in the final answer:

- In the paragraph starting with "These twelwe notes", please read
'dived' as 'divided'

- In the next paragraph, 'the sound will not sound very good' should
have been 'the chords will not sound very good'

- Towards the end of the answer, 'steps' has become 'seteps'

I am sorry for these typos!

---

Clarification of Answer by blazius-ga on 31 Mar 2004 04:04 PST

By the way - you can find a table of the frequencies of the notes of
the evenly tempered chromatic scale at
http://www.phy.mtu.edu/~suits/notefreqs.html

---

Request for Answer Clarification by mirrormere-ga on 31 Mar 2004 12:56 PST

I don't feel this question has been answered yet.

You say it 'just happens' that these steps have been selected as a
major scale. How did it just happen, and why? It seems it would be
simpler to designate all notes an equal distance apart, with sharps
and flats for the frequencies in between. Yet B and C are "only a
semitone apart." To what purpose? It can't have been just a roll of
the dice.

To say that it is a cultural preference doesn't tell much ... the
notation division had to exist for some reason before it became a
cultural preference.

Also -- perhaps more importantly -- it seems as though sharps and
flats sound different, they have a different quality to the human ear.
So how could there be no no C-flat sound? or if there is a C-flat, why
does it not sound the same as the other family of flat note sounds,
and thus deserve a similar designation?

---

Clarification of Answer by blazius-ga on 01 Apr 2004 14:17 PST

I will address your last concern first:  Flats and sharps on a major
diatonic scale does *not* have a different sound from the other notes.
 The trained (and talented) ear may be able to tell that a tone is a
sharp or flat by being able to determine its exact pitch.  (Also, on a
piano, the black keys may give a slightly different sound than the
white ones because they have been played less!  This phenomenon has
nothing to do with the scale itself - it is a consequence of the
mechanics of the instrument.)

You are inquiring about the "C flat".  If it existed, the C flat would
be pitched a semitone lower than the C.  This frequency is already
occupied by the B.  You could argue that the B is actually a C flat,
but that would be generating a scale that is not consistent with what
most other people use.

Cultural preferences certainly play a major part in determining the
number and distribution of notes in a scale.  A scale is a method of
organizing sound into music and can be based on any number of notes. 
Among the many styles of music, a scale using seven tones (notes) is
the basis of Western music.  The Chinese and the Scots traditionally
use five-note scales and some Vedic music uses 22 notes.  Arab music
divides the octave into sixteen unequal intervals, making it entirely
different from Western scales.  (See
http://www.timestar.org/diatonic.htm and
http://www.ericweisstein.com/encyclopedias/music/Scale.html for
further details.)

And even if one settles for dividing the octave into 12 seminotes and
organizing these into seven "main" notes and five sharps/flats, it is
still possible to devise several different ways of tempering the
scale.  It is even possible to play a diatonic scale in different
modes - we are most familiar with the major and minor modes, but there
are several others - see
http://www.encyclopedia4u.com/m/musical-mode.html for more
information.

The selection of a particular scale, temperament and mode is a matter
of taste and indoctrination.  If you grew up in China in the 12th
century, our modern diatonic scale would probably sound as foreign to
you as a classical Chinese opera sound to you today.

However, there are models that try to explain why the major diatonic
scale has seven notes arranged in the particular 2-2-1-2-2-2-1
semitone pattern.  http://www.ericweisstein.com/encyclopedias/music/DiatonicScale.html
shows how the frequency of certain notes have a 4:5:6 relationship. 
This is a theory that tries to explain the scale - this reasoning was
certainly not behind the minds that evolved the diatonic scale to be
what it is today.

Explaining the layout of our diatonic scale is a bit like trying to
explain the distribution of letters in the alphabet.  Why is G the 6th
letter?  Why are the wowels not eavenly spread out between the
consonants?  A B C D E F... may sound like the only natural thing to
do for us, but why is it any more logical than Z Q H S I P...?

Thanks for allowing me to work with your intriguing question.  If you
have any further concerns, please post them here.  I'll go away on an
Easter break tomorrow, but I'll respond to any further requests as
soon as I am back.

---

Clarification of Answer by blazius-ga on 01 Apr 2004 14:22 PST

I forgot to address your idea of designating all notes an equal
distance apart, with sharps and flats for the frequencies in between. 
This is certainly possible.  However, you would have to spend some
time persuading your audience into thinking music made with such a
scale would be worth listening to.

---

Request for Answer Clarification by mirrormere-ga on 05 Apr 2004 01:49 PDT

Having read through the various links here, I haven't found an answer
to my question. All of the articles I've read simply present
half-steps in the diatonic scale as a given. There seems to be no
mathmatical or physical need for half-steps to exist, or to exist
precisely where they do.

So far, if someone asked me, "Why do half-steps exist on the music
scale?" I would, based on the information here, answer, "There is no
known reason why half steps exist or how they came about." Is this
correct?

---

Clarification of Answer by blazius-ga on 15 Apr 2004 07:51 PDT

There is certainly no mathmatical or physical need for half-steps to
exist.  If sound and harmony is regarded in a purely scientific
matter, there is no reason to count a certain combination of tones as
more "significant" or "neccessary" as any other combination.  Tones
with frequencies that are related by certain fractions might be
considered interesting because they might interfere with each other
(playing one string on a guitar will cause vibration in other
strings). The frequency of C2 is exactly the double of a C - a
doubling of frequency is the definition of an octave.

The particular note distribution of the major diatonic scale has
probably evolved out of musicians' desire to produce as many pleasing
(and playable!) harmonies as possible when using alimited number of
notes.  "Pleasing" is a pivotal point here, as the definition of what
is gentle to the ear varies between different cultures.

http://www.andymilne.dial.pipex.com/Diatonic.shtml goes into deeper
detail about the diatonic scales.  The text suggests that the diatonic
scales became popular because they have "a high number of consonant
intervals [and] the greatest possible number of major and minor
triads. [..] The diatonic scale is the only seven note scale that has
just one tritone (augmented fourth/diminished fifth).  All other
scales have two, or more, tritones.  [..] The diatonic scale is
therefore an ideal resource for both melodic and harmonic music - it
has lots of consonant triads, it has few dissonant intervals, and it
is melodically smooth with just two consecutive-step sizes."

I have to admit that this subject has started to become a bit too
tough to comprehend for a musical amateur like me.  However, I do
believe that your original question (which asked about the physical
properties of the notes) have been answered by now.  Scales, notes and
half-steps are an artistic tool created by humans.  Physical tones (as
opposed to notes) are only described by their frequencies, and they
are not limited by the restrictions impoesd by musical theory.

mirrormere --

Let me add one more suggestion for further reading on this subject. 
"Temperament: How Music Became a Battleground for the Great Minds of
Western Civilization," by Stuart Isacoff is a fascinating, accessible
account of how historical, physical, mathematical, philosophic,
aesthetic and religious factors were involved in the gradual adoption
of "equal temperament" in Western music during the 17th and 18th
centuries.  It is available in paperback from Amazon.com (and many
other bookstores):

http://www.amazon.com/exec/obidos/tg/detail/-/0375703306/qid=1080738788/sr=8-1/ref=pd_ka_1/104-7661571-3016744?v=glance&s=books&n=507846

markj-ga

The answers so far have covered most aspects, but I'd like to add a
few points which elaborate on the harmony section of the high-level
article at http://www.mfiles.co.uk/other-what-is-music.htm

A previously noted, the sequence of all white and black notes on a
keyboard represent a scale in half-tones or half-steps. Another way of
phrasing your original question is "Why are the white and black keys
not evenly distributed - for example alternative keys?" or "Why no
black notes between B/C and E/F?". Although other scales do exist and
it is hard to ignore the effect of familiarity or culture, the major
scale seems to the majority of people to be the most harmonious or
pleasant sounding. In the key of C the major scale consists of all
white notes so, out of the 12 divisions of each octave, 7 are white
and 5 black.

The next question is "Why should the major scale be the most
harmonious sounding scale?" and for the answer we should look to the
Harmonic Series which is the naturally occuring vibration modes which
produce notes from a string or a column of air. Mathematically notes
from the harmonic series arise when you take a whole object (e.g.
guitar string) and consider its vibration modes consisting of the
whole string (the fundamental) or subdivide it into equal fractions,
halves, thirds, quarters, fifths, sixths, etc. A natural brass
instrument (e.g. a bugle) will readily play the notes of a harmonic
series when its column of air vibrates in these fractions.

So how do the "white notes" arise from the harmonic series? Well, in
the key of C, the first non-C notes which arise from the series are G
and E in that order (although there are very slight differences due to
"equal temperament"). C, G and E form the major chord or triad. The
"5th note" or G is the next most natural or important note in the
scale of C (often called the dominant). A major triad or chord
starting on G also contains D and B. The relationship between C and G
is the same as the relationship between F and C (i.e. C is the
dominant or harmonically the next most important in the scale of F).
The major triad starting on F also contains C and A. Between these
three mathematically related triads (CEG, GBD and FAC) we have covered
all the white notes and no black notes. So dealing with small
fractions (the simplest modes of vibration) and small fractions of the
notes arising from those fractions, we cover all the white notes. It
is this "natural-ness" or "simplicity" which seems to make it also
"harmonious" or "pleasing".

Hi mirrormere-ga,

What you're really asking is why are there only 12 tones in an octave
as we are accustomed to.  19 tones actually works out pretty good too.
 There are mathematical reasons for it.  Since a half step is really
just a distance between notes or frequencies, it can be explained with
mathematics.

One question at a time:
Each half step is the next tone in the octave and a whole step is two
tones.  So you can take any note and sharp it or flat it.  If you can
picture a piano keyboard, there is no black key between B & C and E &
F.  So if you flat the F you have F-Flat which is the exact same note
as E.  You can also flat a note twice so you can have a double flat. 
So G-Double-Flat, for example, is the same note as F.  There actually
is a C-Flat, B-Sharp, F-Flat & E-Sharp.  The reason we don't usually
call them that is because of the key we choose to write and perform
music in.  As a formally educated instrumental musician, I have only
seen one piece of music in my lifetime of performance that uses the
C-Flat in a key signature.  It was difficult to play as everybody
struggled with the key signature.  Why use the key of C-Flat (7 flats
or everything flat in a scale) when the key of B has only 5 sharps? 
It's interesting to note that instrumentalists usually play music with
a key signature containing flats while much choral music contains
sharps!  A vocalist can adjust to a key signature fairly easily since
all they have to do is raise or lower their voice to match the pitch. 
For instrumentalists, adjusting to a rare key signature means a
different fingering or position and especially a different thinking
(double flats are difficult to think about on the fly).  Many
composers avoid the rare key signatures for this reason.

As for why there are 12 tones, in a nutshell it's because the
frequencies of 12 tones fits a ratio rather nicely.  For example, D
above middle C has a frequency of 293.6.  A above middle C has a
frequency of 440.  Using math, you can see that this is almost a
perfect 3:2 ratio.  The closer they are to a series of perfect ratios,
the more pleasant they sound to our ear.  4:3, 5:3, 5:4 and 6:5 ratios
work out pretty good for the 12 tone scale also.  There are overtones
that sound above the pure frequency that also contribute to how we
perceive sound but I won't get into that.  An octave is when a note
differs by a factor of 2 so all spacings or intervals between notes
need to fit between an octave.

Here are some interesting links that do a far better job of explaining
than I could:

http://users.frii.com/davejen/music/music.htm
http://www.math.niu.edu/~rusin/uses-math/music/