<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD> <META http-equiv=Content-Type content="text/html; = charset=iso-8859-1"> <META content="MSHTML 6.00.2800.1400" name=GENERATOR> <STYLE></STYLE> </HEAD> <BODY bgColor=#ffffff> <DIV><FONT face=Arial size=2>Hi Julia,</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>To get mathematical on ya', pitch (e.g. = the note#) and frequency (cps or Hz) are exponentially related, where</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>F = 440 * 2^(n/12),</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>where n is the number of half steps = above (positive value) or below (negative value) A440.  This can be = represented on a log scale:</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>log(F) = log(440) + n/12 * = log(2)</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>In this form of the equation, we see = there is a linear relationship between log(F) and n.  Similarly, there's a = linear relationship between log(F) and n/100 (which is a cent).  The = frequency value of a cent is greater in the upper end of a note's nominal range = and smaller in the lower end.  However, there really isn't *much* = difference from lower to upper.  For instance, consider A440:</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>For this note, F=440 = (obviously)</FONT></DIV> <DIV><FONT face=Arial size=2>A cent at this frequency would be .254 Hz</FONT></DIV> <DIV><FONT face=Arial size=2>One beat/sec would occur at a = pitch difference of 3.9 cents.</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>For a half of a semitone higher (or 50 = cents -- the division between A and A#):</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>F=452.9 Hz</FONT></DIV> <DIV><FONT face=Arial size=2>A cent at this frequency would be .262 Hz</FONT></DIV> <DIV><FONT face=Arial size=2>One beat/sec would occur at a pitch = difference of 3.8 cents.</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>At the lower end of the nominal range = for A, 50 cents lower:</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>F=</FONT> <FONT face=Arial = size=2>427.5 Hz</FONT></DIV> <DIV><FONT face=Arial size=2>A cent at this frequency would be .247 Hz</FONT></DIV> <DIV><FONT face=Arial size=2>One beat/sec would occur at a pitch = difference of 4 cents.</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>So you see, there's really very little difference.  Furthermore, the frequency difference across this = range is 25.4 Hz.  (Nothing implied here about conversion between imperial = and metric -- just coincidence!)  A hundredth of that value is .254 Hz, = which you should notice is the calculated value of a cent at 440 = Hz.  It is also 97% of a cent at a half of a semitone higher, and 103% of a cent at = a half of a semitone lower.</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>So TECHNICALLY a cent is NOT 1/100 of a = the frequency span of a semitone, except at the center frequency.  But = unless you are good enough to distinguish beat rate differences on the order of = 0.0075 cps (and I challenge *anyone* on this list, even Andre, to be that accurate!), then yes, Julia, a cent is indeed 1/100 of the frequency = difference between semitones (at least approximately enough for tuning).  = ;-)</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>Happy tuning, and...</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2>Peace,</FONT></DIV> <DIV><FONT face=Arial size=2>Sarah</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV>----- Original Message ----- </DIV> <BLOCKQUOTE style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; = BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: = black"><B>From:</B> <A title=Alpha88x@aol.com = href="mailto:Alpha88x@aol.com">Alpha88x@aol.com</A> </DIV> <DIV style="FONT: 10pt arial"><B>To:</B> <A = title=pianotech@ptg.org href="mailto:pianotech@ptg.org">pianotech@ptg.org</A> </DIV> <DIV style="FONT: 10pt arial"><B>Sent:</B> Thursday, December 02, = 2004 1:22 PM</DIV> <DIV style="FONT: 10pt arial"><B>Subject:</B> Re: Beats vs cycles vs = cents</DIV> <DIV><FONT face=Arial size=2></FONT><FONT face=Arial size=2></FONT><BR></DIV><FONT face=arial,helvetica><FONT lang=0 = face=Arial size=2 FAMILY="SANSSERIF" = PTSIZE="10">Greetings,<BR><BR>      &nbsp= ;     I know this is a delayed response on this thread,  but I am = confused. <BR><BR>           = Granted that 1/100th of a half step is a different numeric value for each note = of the piano, but 1/100th of a half step is always 100 divided into the = munber of cycles (or distance) from one half step to the next succesive half = step up (or down) isnt it? <BR><BR>Julia<BR>PA<BR><BR>In a message dated 3/15/2004 = 7:08:48 PM Eastern Standard Time, eromlignod@kc.rr.com writes:<BR><BR> <BLOCKQUOTE style="PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #0000ff 2px = solid; MARGIN-RIGHT: 0px" TYPE="CITE">There are 100 cents in a half-step (semitone), but a = cent is *not* 1/100th<BR>of a half-step.</BLOCKQUOTE><BR><BR>(100 cents is = always a half-step no matter how high or low the<BR>frequency is.<BR><BR></BLOCKQUOTE></FONT></FONT></BODY></HTML>