Well I'm running into many problems with the harmonics chart in Excel. The
programming does work to a point, however it is not portable to the web in any
quick format.
In addition, the entire chart itself is about 4 pages each with tables that
are 2000-3000 pixels wide to 4000 pixels tall and so forth. I covered all the
bases :)
Therefore I'm learning SQL databases with some PHP. I'm going to try to
produce an online chart customizeable to any horn or combinations of horns (F/Bb
double, F/Bb/High Eb triple, etc.). Also I'm going to try to get into Maple soon
and work up some nice formulaic proofs to accompany this.
If this works out for PHP I should be able to produce an optional color coded
one as well.
I'll post more about this later, however it will involve some neat tricks.
I'm associating all note names with a number 1 through 12.
0 ---- Fundamental note name (say A for example since I'm using A=440 by
example)
1 ---- half step above 0 (Bb)
2 ---- half step above 1 (B)
and so forth. With this I can assign octaves by listing notes by half step
from 0 (the fundamental, or rather A=440 divided by 16 which gives us the second
E below the Bass clef. This theoretical fundamental is 1 2 and 3 on the Eb
horn. This is why this division is chosen because one can simply move up by half
steps on the Eb horn to reach the realm of the F horn, and one can keep going
up the horns to Bb, Eb Alto, F Alto, and Bb Alto.
Now with this one can assign all the notes by half step from 1 to however
high you wish to go. The frequency values are attained by the following formula:
2^(1/12) * Fundamental.
Going up note by note we will have:
2^(1/12) * (2^(1/12) * Fundamental) for the 2nd note
2^(1/12) * [2^(1/12) * (2^(1/12) * Fundamental)] for the 3rd note, and so
forth.
Therefore one finds that the equation for any note's frequency is the
following:
{[2^(1/12)] ^ x } * F0
Where, x is the xth note above the Fundamental. Thus we can find that the
octave is the following:
{[2^(1/12)] ^ x } * F0
--- x = 12 --- this means that we are looking for the 12th note above F0
(fundamental).
{[2^(1/12)] ^ 12 } * F0
{[2^(12/12)]} * F0
(2^1) * F0
2 * F0
Which is the octave right above the fundamental. We can also determine the
note name and the octave of any note in our arbitrary half note scale (F0 to any
note you choose). All we have to do is divide the integer value of the note
in our list by 12 and correlate that with it's remainder. The 12th note from
the fundamental would be an even octave from our first note because if one
divides it by 12, one has 1 with no remainder. Thus:
int(Note_Number / 12) yields our octave number
Note_Number - [int(Note_Number/12) * Octave_Number] yields the remainder
(one could also use a floor function or even a remainder function to
determine both)
Now for the harmonics which are easy to determine. All one has to do is
multiply the Note_Number frequencies by the harmonic. F0 on the 4th Harmonic would
be F0*4 for example. The 14th Harmonic of the 45th note would be that
frequency times 14.
Now after this one must determine where these harmonics lie with regards to
note values. One simply CANNOT use the closest frequency value to the note
since the scale is logarhythmic. The quickest way is to find the closest note
frequency through programming is to find the closest value to the frequency (a
simple mathematical function can do this) and then going one step forward, one
step backward in note frequencies and determining which note has the smallest
intonation defficiency in cents and printing out the correllating note value.
Intonation in cents is determined by the following formula:
{ 1200*ln(T/V) } / ln(2)
Where
T = Testing value
V = Value you are comparing T to (T is sharp or flat to V)
With this you can determine the ideal pitch tendencies of every single note.
I hope this was informative. Soon I will be complete with these tables and
all this programming...
I wont spoil the surprises but there are some very fascinating things to this
math, and a lot of things that make it a lot simpler than you would think.
For example, if the instrument is in tune then it will not matter what A you
tune to, even if its 200Hz or 2000Hz, the note names of every harmonic and their
pitch tendencies will always be the same. Do the math, you'll be surprised :)
-William
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