Help with "bad" tuning...need help

[Robin Hufford]

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List,
    A half step is an adjacent key in the context of a piano keyboard.   
It may be white to white as in the case of e-f, or b-c.  It may be a 
white key to an adjacent black key, or a black key to a white one.  
There are two adjacent pairs of white keys on the piano, those mentioned 
above.  There are no black to black half steps as a white key will 
always be in between any two black keys chosen so as to be as close 
together as is possible.     Two half steps means an interval of a whole 
step, usually referred to as a whole tone.  This may be white to white, 
black to black, white to black or black to white.
     The foregoing emanates from the layout of the keys on the keyboard 
itself without reference to notation.  A second consideration obtains 
when notation is taken into account.  Half steps must have adjacent 
letters names, that is they must be proximate to one another in the 
sequence of the musical alphabet which is (a, b , c, d, e, f, g, a, g, 
......).   The same requirement occurs for whole tones, that is the 
interval must be named with proximate letters.  (A to b) is one such  
proximate pair, as is ( b to c), along with the others.   Well, which is 
it?  Half or whole?  This is determined by reference to the natural 
layout of the keyboard and the use or absence of a sharp or flat sign to 
indicate the half steps found on the keyboard referred to in the first 
paragraph above.
      Intervals are named for the number of letter names they contain:  
For example, counting upwards,  (a-a) is a unison, (a-b) is a second as 
it contains two letter names.  Similarly (a-c) is a third;(a-d), a 
fourth. etc.  You can count up and name any interval you wish, although 
some are more standard than others. 
       The number of half steps contained in the interval determines, in 
the case of the second, whether it is a major or a minor second, that is 
a half tone or whole tone, or, alternatively, a half step or a whole 
step.  ALL WHOLE TONES MUST CONTAIN ONLY TWO HALF STEPS. B to c is 
intrinsincally a half step on the keyboard, as mentioned above.  B- 
c(sharp) is now a hole tone as it contains two half steps.  So is 
b(flat) to c.  There are other somewhat arcane complexities, for example 
what is b(flat) to c(sharp)?  This is an augmented, major second.  For 
technicians who are not musicians, it is best to ignore such things.  
This method of naming may be applied similarly to any note on the 
keyboard subject to certain limitations which are in the nature of 
definitions.  The reference to whole tones contained above is just such 
one definition.  There are others some of which are ALL MAJOR THIRDS 
MUST CONTAIN FOUR HALF STEPS, MINORS THIRDS MUST BE ONLY THREE HALF 
STEPS. etc.
      There is no major third, as technicians are sometimes wont to do, 
which can correctly be referred to as a to D(flat).  This would, as it 
contains  four letter names, be a contracted, or diminished  fourth, 
even though acoustically, it would be, in fact, the sound of the major 
third, which should correctly be referred to as a to c(sharp).  This 
seems paradoxical but there is an underlying logic and utility in these 
rules of naming as they correspond, in an amazingly logical way 
considering that they have developed from musicians, to the harmonic 
motion inherent in the actual music which the notation attempts to 
express. 
     In the cents notation, which expresses the logarithmic aspect of 
equal temperament, one octave itself comprises 1200 cents, which 
encompasses an actual doubling of frequency.  Obviously each half step 
contains 100 cents, which means  a whole tone or step comprises 200, a 
whole tone and a half step, 300, etc.  These are equal ratios and not 
counts of frequencies per second.  One can not impose upon the frequency 
difference of any two adjacent notes, by definition a half step or a 
hundred cents, an equal division of the frequency difference and arrive 
at a value for a cent,  as 100 cents are, in actuality, not an equal 
division by a hundred, but, rather, a hundred equal ratios, as Bob Scott 
pointed out only a few weeks ago. 
     This means, for example, if you could find a half step comprised of 
a hundred hertz, or arbitrarily defined it such,  that a cent does not 
equal one hertz.  Rather, a cent is the number, which, when multiplied 
by the frequency of the lower note and, done, 99 more times, will 
produce the frequency of the upper note.  These are equal ratios, not 
equal divisions.
Regards, Robin Hufford   
   in iannaman@aol.com wrote:

> In a message dated 12/16/04 4:06:28 AM Pacific Standard Time, 
> pianoman@accessus.net writes:
>
>
>>
>> I think part of the problem is that we are calling these half steps. 
>>  The
>> distance between E and F is no larger than between F and F#.  It is 
>> still
>> 100 cents.  Why do we insist on calling those things whole steps 
>> anyway.?
>
>
>
> James,
>
> These are half-steps(not whole steps), aka half-tones, semi-tones or 
> minor seconds, and there are 100 equal divisions between them.  Each 
> one of these miniscule portions is called a cent.
>
> Dave Stahl


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