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Musical Scale Tone Frequencies
and the
People Who Love Them
Even Tempered Scale Ratio Derivation
and Frequencies
by Roland Wilhelm
Problem:
What is the value of the ratio between each half-step in an even tempered
scale?
Givens:
- the octave above any given
note is twice the frequency of that note – a ratio of 2:1.
- every pair of consecutive
notes in an octave must have the same ratio.
- there are 12 semitones in an
octave.
A generic picture of the problem would look like
where:
n = 2 = integer multiplier
x = base frequency at the start
of the octave.
y = 12 = the number of
subdivisions (or semitones, or half-steps) in an octave.
To relate the above variables to an octave we will also define:
ro = 2:1 = the
ratio for the entire octave.
rs = the ratio
between each subdivision in the octave.
So now we can see the following relationships
1) 
2) 
3) 
Somehow we have to reduce rs down to the values we
know: n and y.
If we stare at the right side of the third relationship long enough, we
might notice that if all the half-step ratios were multiplied together, lots of
terms cancel out. We might also go to sleep.
4a) 
The equivalent left side of equation 3 would look like
4b) 
5) 
Substituting in equation 2 yields
6) 
Solving for rs gives the result
 or
7) 
Substituting in the known values
8) 
So now all the frequencies can be calculated based on the A 440 Hz standard.
Notice that the frequencies are not evenly spaced, but logs of the frequency
are.
And if you’ve made it this far, you’re probably fairly spaced yourself!
|
note
|
frequency
|
ln(f)
|
freq
diff
|
ln
diff
|
|
|
C
|
261.63
|
5.5669
|
|
|
|
1.
|
C#
|
277.18
|
5.6247
|
15.5571
|
0.057762
|
|
2.
|
D
|
293.66
|
5.6824
|
16.4821
|
0.057762
|
|
3.
|
Eb
|
311.13
|
5.7402
|
17.4622
|
0.057762
|
|
4.
|
E
|
329.63
|
5.7980
|
18.5006
|
0.057762
|
|
5.
|
F
|
349.23
|
5.8557
|
19.6007
|
0.057762
|
|
6.
|
F#
|
369.99
|
5.9135
|
20.7662
|
0.057762
|
|
7.
|
G
|
392.00
|
5.9713
|
22.0010
|
0.057762
|
|
8.
|
G#
|
415.30
|
6.0290
|
23.3093
|
0.057762
|
|
9.
|
A
|
440.00
|
6.0868
|
24.6953
|
0.057762
|
|
10.
|
Bb
|
466.16
|
6.1445
|
26.1638
|
0.057762
|
|
11.
|
B
|
493.88
|
6.2023
|
27.7195
|
0.057762
|
|
12.
|
C
|
523.25
|
6.2601
|
29.3678
|
0.057762
|
 
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The above information was e-mailed to me Mr. Roland Wilhelm, who
had this to say:
Dear Richard,
Noticed you were missing a graphic for the Musical Scale Tone
Frequencies. Thought you might be interested in a couple of documents I
put together while trying to get a handle on this (yes, I'm one of
those people). If you don't have Word or Excel, you can download free
readers from
Microsoft.
And you're right, I don't care about the movie rights. I can't believe
I really read all that either! We both need a life?
Take care,
Roland Wilhelm
Note #1: the original files were in Word and Excel format;
they've been converted to html for your enjoyment.
Note #2: It is entirely likely that we both DO need a life.
Thank you, Roland, for sending these files!
The Wobbling of Pitch
Today, concert pitch A is 440Hz. Why is it 440Hz? That's an easy one...
back in 1939, a bunch of guys with a bunch of education got together, talked about it for
awhile, probably there was some shouting involved at one point, no doubt someone left mad
and basically they decided to plop their "A" down at 440Hz. Oh, those wacky
scholar types! It should be noted that there was a big, nasty war that ripped
through Europe right after this; some people just take stuff way too seriously.
So, you ask, what was it before that? I'm glad you asked that! In the
past, the note "A" has been defined everywhere from 466Hz (A#/Bb of today) all
the way down to what we call "G" today -- 392Hz. (I'll leave off the joke; it's
far too obvious.) So, for the couple of hundred years before 1939, note pitch was jumping
around all over the place. This caused problems for musicians when they went on tour and
also resulted in more than one bar fight, probably over the suggestion that pitch jumped
around so much so that the harp players wouldn't have to constantly retune.
But, what is the real reason for pitch snaking around up and down the
scale? To oversimplify some, throughout history notes have been defined according to the
tuning fork of whoever happened to be the most prestigious musician at the time.
So, "A" is 440Hz now and it's going to stay there and we can all
move on to better things, no? Well... No. Some orchestras are tuning their piano to A446;
The Chicago Symphony tunes to A442. Can a person really hear the difference between A440
and A446? Pretty much. Is it enough to get bent out of shape over? Only if you make your
living in the halls of academia (and they make an ointment for that).
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