Linear vs. Exp = Pitch vs. Frequency

[WeAreAs1 at aol.com]

Hello Logheads and Linheads,

In an earlier part of this thread, fa_diy at fa.camelot.de wrote something which 
indicated to me that he might need a little more explanation about the 
logarithmic nature of the relationship between Pitch and Frequency.  Jorgen 
Bergfors responded, but maybe not in great enough detail, I'm afraid.  So, 
here is a more detailed explanation of the concept, for fa_diy at fa.camelot.de, 
and for any other readers who might benefit from this, as well.  My apologies 
to those for whom this is "old news".

fa_diy at fa.camelot.de wrote:

>Sorry, but I do not understand what you mean. A Detune is an _constant_  
>offset and there is no relevance for logarithmic or linear thinking becaus  
>you dont have differential aspects.

Just to clarify things, a Detune Offset may be defined in two different ways. 
 One way is to define it in terms of its frequency offset, or number of 
Hertz.  The other way is to define it in terms of its musical interval or 
pitch.  It's important to remember that Pitch and Frequency are not (always) 
the same thing.

For instance, to go from the note A=220 to the octave above it (A=440), you 
simply raise the pitch by 12 half-steps, or *1200 Cents*, and to raise by 
another octave (A=880) you simply raise (or offset) the pitch by another 1200 
Cents.  This 1200-Cents-per-octave progression will continue up and down the 
entire musical range - you simply add or subtract 100 Cents for each 
half-step in the chromatic musical scale.  

However, please note that *frequency* does not work the same "additive" way.  
Yes, it is true that if you start at A=220 and add 220Hz to it, you will end 
up at A=440, exactly one octave above.  But if you try adding another 220Hz 
to that note, you do would not raise the pitch by another octave, you would 
just raise it to 660Hz (I don't what "note" this is, but rest assured that 
it's not an exact, in-tune pitch in our equal-tempered scale series).  To 
raise frequency so that the apparent pitch raises in an equally proportional 
manner, you must *multiply*, rather than add.  For instance, you must 
multiply 220Hz by 2 to raise it by one octave, and then further multiply the 
result by two again, to futher raise it by another octave.  Conversely, 
multiplying by 0.5 would lower any pitch by one octave.

This relationship of Frequency to Pitch is (surprise!) a logarithmic 
relationship.  This is one of the reasons why most analog synth oscillators 
are designed to work with exponential control voltage response.  Exponential 
VCO response makes it so that the logarithmic relationship between oscillator 
frequency (in Hertz) and keyboard control voltage (1/12th volt per half step) 
is the same as the logarithmic relationship between frequency (in Hertz) and 
musical pitch (expressed in notes or Cents).

OK, I'm sure all of this makes pretty good sense when looking at it on a 
"macro" scale, that is, when referring to musical octaves and note pitches.  
But now let's look at it on a "micro" scale, as it would pertain to a 
discussion of detuning VCO's by a very small amount.  For instance, slightly 
detuning two exponential synthesizer VCO's so that they "beat" against each 
other, creating a pleasant "fatness" of sound.  

If we tune the first oscillator to 440Hz, and the second to 441Hz, then there 
is a 1Hz difference in their frequencies, so we will hear them beat once 
every second.  If we raise the *pitch* of the oscillators by one octave, the 
frequencies will double, so the 440Hz VCO becomes 880Hz, and the 441Hz VCO 
becomes 882Hz.  The frequency difference between the two is now TWO Hertz, 
which means that the beating will now occur *two times* per second.  
Conversely, lowering the two by one octave, from 440Hz & 441Hz to 220Hz & 
220.5Hz, would result in a beat cycle that would repeat every TWO seconds.

This, getting back to the original discussion, is why Jorgen said:

>>Linear detuning in most cases sound far better than exponential, because 
the  
>>beating rate will stay the same, no matter what note you play.

With "Linear Detuning", that is, a detuning control that does not respond to 
control voltage in an exponential manner, you could slightly detune your 
VCO's by a specified amount of Hertz, for instance, 1Hz, as in the above 
example, and they would always beat at a rate of one time per second, no 
matter what note or what octave you played.  In most cases, this is more 
musically desirable than having the beat rate change with the pitch.

I hope this helps.

Michael Bacich

(BTW, this is a great thread!)