The Logic Off Topic list

Hi Debbie,

I moved this to the OT-list, as per Joeri's request, and send you a 
CC privately as well.  I suppose you still read the OT-list, so 
subsequent posts will be OT only, ok?


Thoughts from the mind of DbbBrook@..., 21-11-2001:

>Thank you HJ.  I will download the CellSynth today.

If you only want to experiment with the harmonics stuff, you can 
avoid reading the entire manual.
Just open CellSynth with an empty matrix.  From the top line of 
objects, grab an oscillator (a blue-ish button-like object with a 
sine drawn in it).  Put it somewhere in the empty matrix.
Click the oscillator object.  A small window opens where you can 
adjust the osc.'s settings.  The bottom dial in that window is for 
selecting the waveform.  Turning it all the way, clockwise, gives you 
'harmonic'.  Now in the row of small icons, you see one that is a 
horizontal line, with a few vertical lines sticking out (the last one 
before the lock icon).  Click it, and you'll get a window in which 
you can simply draw or type the volumes of 32 harmonics.

In the main matrix window, click the speaker (below the VU meter), or 
press spacebar to listen to the sound.

Recording the sound is equally simple: open the Recording Transport 
from the Windows menu, and press the record button.

>I need to find the mathematical ratio in frequency between the first two
>notes of the melody 'The Last Post' which is keys CGCGCCGGGCECCCC?
>
>Do you have alittle bit of advice on how i can find the mathematical on this.
>We have note CG.  I'm getting abit lost on this.

This seemingly innocent question is a bit more complicated than it 
seems at first sight.  The answer depends on whether you're talking 
about "pure" tunings, where notes are as close to natural harmonics 
as possible, or whether you're talking about current tuning of e.g. 
piano's, where every note is a bit out of tune.

When talking pure tuning, a musical octave like C2-C3 has a frequency 
ratio of 1:2.  A fifth, like C-G, has a frequency ratio of 2:3 
(1:1.5).  So if C3 is, say, 200 Hz (which it isn't, but this is just 
an example), then G3 is 300 Hz.
 From these 2 ratios, all other ratios can be calculated.  Take the 
next fifth up from G3, i.e. D4.  Multiply 300 Hz by 1.5 once more: 
D4=450 Hz.  So D3 is half of that, or 225 Hz.  So the ratio C3:D3 (a 
major second) is 200:225, or 8:9.  Going on in a similar fashion, you 
find that C:E (major third) = 4:5 for example (some rounding is 
involved here, never mind).
This sort of tuning is perceptually (= for the human ear) the most 
pure, and is how you would tune an instrument "instinctively".  All 
basic intervals, like octaves, fifths, fourths and thirds are rather 
simple ratios, using small numbers (2:3 instead of 21:32).  The more 
complex the ratio, the less "harmonic" the interval sounds.

There is however a massive problem with this way of tuning e.g. a 
piano.  Suppose you tune a piano like this:
First tune some C.  Then tune all other C's by ear (easy: just an 
octave up -- anyone can hear this).  Now start with e.g. a low C0 and 
tune the next fifth, G0 in a 2:3 ratio (also easy to do by ear). 
This allows you to then tune all other G's as well -- again just 
octave-jumps.  Then again a fifth up: D1.  And again: A1.  Continuing 
this way, you get the sequence:
C0 G0 D1 A1 E2 B2 F#3 C#4 G#4 D#5 A#5 F6 C7 (circle of fifths).
I.e. each note has been tuned once, and all others of the same kind 
can be tuned using simple octave-steps.

BUT...  While going from C0 to C7, you've ascended 7 octaves -- which 
should be 7 steps of "times 2".  I.e. you ought to have multiplied by 
a factor of 2*2*2*2*2*2*2 = 2^7 (^ = to the power) = 128.
However, you did this using 12 fifth-steps, which is all 1:1.5 
ratios.  So in fact you multiplied by 1.5^12 instead of 2^7 -- and 
these are not the same!!  1.5^12 = 129.75, which is too much!

So using the circle of fifths, your C7 will be 129.75*C0 instead of 
the correct 128*C0, and will thus be tuned too high.

There's a whole body of literature on this problem and on the various 
ways people over the ages have tried to solve it.  Somewhere in 
Bach's time, our current-day tuning was invented, which is generally 
regarded to be the best solution to the problem (although some 
disagree).  The idea is rather simple: if an octave is "times 2" 
(which is musically unavoidable), then all other notes should be 
spaced equally apart.  I.e. the "times 2" distance should be divided 
in 12 equal parts.
This means mathematically that a semitone distance now becomes a 
multiplication by the 12th-root of 2, or 2^(1/12), which is approx. 
1.05946.  You can check that 12 of these steps, i.e. a factor of 
1.05946^12, is indeed 2.

So, to get back to your fifth interval, a fifth is 7 semitones, and 
would thus have a ratio of 1.05946^7 (or 2^(7/12)) = 1.498 -- just a 
little short of the ideal 1:1.5 ratio.
This indeed means that on a modern piano, _all_ intervals except the 
octave are a little bit out of tune.  Somehow we've learned to live 
with that...


Finally answering your question: a fifth is thus either a 1:1.5 
ratio, or a 1:1.498 ratio, depending on which system of tuning you 
consider.
When talking harmonics and such, you always are bound to whole 
multiples of the base frequency, and thus to "neat" ratios.  So if 
the base frequency is f, the next harmonics are 2*f (octave) and 3*f 
(fifth).  So indeed a fifth here is a 2:3 ratio.

OK, end of lecture for today...

>I'm reading and reading but i think my books are abit old.

Well, frequency ratios in a diatonic scale are probably a bit older 
than your books :-).


tata,
HJ
-- 
Hendrik Jan Veenstra  <h@...>
Omega Art: http://www.ision.nl/users/h/index.html