"Cycles Per Sec vs. Cents/Ratio

[Kent Swafford]

Daleboy@aol.com wrote:

>Dear List:
>    A discussion among some musician friends has put me at the task of 
>finding some hard evidence pertaining to "European Concert Pitch" and just 
>where that is on the cycles per second scale. 
>    I have been attempting to search the PTG archives for some information 
>concerning the ratio of cycles per second and cents deviation as it pertains 
>to Concert Pitch, whether it be A440, A442, A435.....etc.  The question in 
>the group was  if the cents of deviation was equally spaced between each 
>change of cycles per second deviation, ie; A441= 3.6, 4.1 or 5.0 cents above 
>A440........A442= 8.0 9.0 or 10 cents above A440. And additionally which 
>cents deviation is correct.
>    I had a chart on it at one time...lost it in a house fire. The archives 
>is reacting to me like talking to the Monolith in 2001.
>    Please e-mail me privately to save on the redundancy of bandwidth. 
>THANKS!
>DALE WHITEHEAD 
>P.S. Actually....me thinks the real question was the ol' SATII vs. Guitar 
>tuner thang...


                        CENTS/HZ  CONVERSION CHART

               Frequency(Hz) to Cents Deviation (Offset)
                          A4 = 415Hz =   -101.3 cents
                          A4 = 420Hz =    -80.5 cents
                          A4 = 425Hz =    -60.0 cents
                          A4 = 428Hz =    -47.9 cents
                          A4 = 430Hz =    -39.8 cents
                          A4 = 435Hz =    -19.8 cents
                          A4 = 440Hz =      0.0 cents
                          A4 = 441Hz =      3.9 cents
                          A4 = 442Hz =      7.9 cents
                          A4 = 443Hz =     11.8 cents
                          A4 = 444Hz =     15.7 cents

                    Cents Deviation to Frequency(Hz)
                         A4 =  -50.0 cents = 427.5Hz
                         A4 =  -25.0 cents = 433.7Hz
                         A4 =  -15.0 cents = 436.2Hz
                         A4 =  -10.0 cents = 437.5Hz
                         A4 =   -5.0 cents = 438.7Hz
                         A4 =    0.0 cents = 440.0Hz
                         A4 =    5.0 cents = 441.3Hz
                         A4 =   10.0 cents = 442.5Hz
                         A4 =   15.0 cents = 443.8Hz

Following is a description of how the above chart was calculated:

There are 100 cents to a half-step and 1200 cents to an octave in the 
mathematical model of 12-tone to the octave equal temperament.

Steve Fairchild, practical teacher that he is, does not assume that every 
piano tech who might want to figure some cents deviations knows enough 
math to do so. Good assumption. Steve provided button-by-button 
directions about how to use a scientific calculator to do cents/Hertz 
conversions. These are cumbersome in ASCII, but I'll give it a try.

To figure the number of cents between any two frequencies (such as 440 
and 442):

Enter the larger number in the window.
Press "division" button.
Enter the smaller number in the window.
Press "equals" button.
Press "LOG" button.
Press "division" button.
Enter the number "2" in the window.
Press "LOG" button.
Press "multiplication" button.
Enter the number "1200" in the window.
Press the "equals" button.


To figure what frequency is a given number of cents higher than a 
beginning frequency:

Enter the number "2" in the window.
Press the "y to the x" button.
Press the ( left parentheses button.
Insert the variable, that is, enter the number of cents into the window.
Press "division" button.
Enter the number "1200" in the window.
Press the ) right parentheses button.
Press the "equals" button.
Press "multiplication" button.
Enter the original frequency into the window.
Press the "equals" button.


To figure what frequency is a given number of cents lower than a 
beginning frequency:

Enter the number "2" in the window.
Press the "y to the x" button.
Press the ( left parenthesis button.
Insert the variable, that is, enter the number of cents into the window.
Press "division" button.
Enter the number "1200" in the window.
Press the ) right parenthesis button.
Press the "equals" button.
Write down the number in the window or press a "Memory Store" button.
Enter the original frequency into the window.
Press "division" button.
Enter the number that you just wrote down or press the "Memory Recall" 
button.
Press the "equals" button.


Kent Swafford