source file: mills3.txt Date: Tue, 25 Nov 1997 11:04:09 +0100 Subject: Re: Thrice the Severity From: gbreed@cix.compulink.co.uk (Graham Breed) Carl Lumma wrote > 1. If we detune by hertz, a direct arithmatic variation in beating occurs, > right? > > 2. Detune in cents and you say we get a "linear" change in beating? > "Linear" and "geometric" are the same? "Linear" and "geometric" are approximately the same when they're small enough ... Pitch (p) and frequency (f) are related as follows: p = a*ln(f)/ln(2) For cents, a=1200, but I'll set it to ln(2) for simplicity. For a small change in frequency, df, we have: p = ln(f+df) dp = ln(f+df) - ln(f) dp = ln[f(1+df/f)] - ln(f) dp = ln(f) + ln(1+df/f) - ln(f) dp = ln(1+df/f) Now, it's a standard approximation that ln(1+x)=~x when x<<1. For example: (ln(441)-ln(440))*440=0.9989 (ln(442)-ln(440))*440=1.9955 It follows that dp = df/(f*ln(2)) oct 2 Hz at 440 is 2/440*1200/ln(2)=7.87 cents So, detuning by a few cents gives approximately linear beating. The approximation gets worse the larger the detuning. In fact, you can approximate the discrepancy in cents as: 600/ln(2) * (df/f)^2 =~ ln(2)/2400 * dp^2 At 20 cents, this comes out as: ln(2)/2400*20*20=0.12 cents So, if you want accuracy to the nearest tenth of a cent, this won't do. At 50 cents, it comes out as: ln(2)/2400*50*50=0.72 cents ln(2)/2400*50*100=1.44405662617 I think 1.4% is close enough, and you won't need much larger detunings in dissonance calculations. SMTPOriginator: tuning@eartha.mills.edu From: Daniel Wolf Subject: Material Press Catalog PostedDate: 25-11-97 11:05:02 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 25-11-97 11:03:30-25-11-97 11:03:30,25-11-97 11:03:31-25-11-97 11:03:32 DeliveredDate: 25-11-97 11:03:32 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C125655A.00373DDA; Tue, 25 Nov 1997 11:03:23 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00605; Tue, 25 Nov 1997 11:05:02 +0100 Date: Tue, 25 Nov 1997 11:05:02 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA00599 Received: (qmail 25047 invoked from network); 24 Nov 1997 13:00:25 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 24 Nov 1997 13:00:25 -0800 Message-Id: <199711241558_MC2-2974-3D20@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu