Help with "bad" tuning...need help

[pianolover 88]

another way to look at it is "cycles". Each cycle is comprised of 4 cents. 
So, 100 cents flat, or 1/2 step flat is 25 cycles flat, or A 415.

Terry Peterson



----Original Message Follows----
From: Robin Hufford <hufford1@airmail.net>
Reply-To: Pianotech <pianotech@ptg.org>
To: Pianotech <pianotech@ptg.org>
Subject: Re: Help with "bad" tuning...need help
Date: Fri, 17 Dec 2004 14:58:39 -0600

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