source file: mills2.txt Date: Thu, 31 Oct 1996 12:21:09 -0800 Subject: Re: Intervals list (beating frequency)2 From: kollos@cavehill.dnet.co.uk (Jonathan Walker) After my last message in reply to Lorenzo, I thought I'd finish the task. First I'll set out the method more clearly: f = frequency of lower pitch t = size of tempered interval p/q = ratio of (small number) just interval t approximates B = |f(q.t - p)| = frequency of beating |x| is the absolute value of x, i.e. any negative sign is disregarded. So for a 12-TET M3, the calculation runs as follows: B = |f(4.2^1/3 - 5)| = 0.03968f (to five places after the point) The list of intervals Lorenzo wants, from the slowest to the fastest beating, runs as follows: 12-TET interval p/q B/f P5 3/2 0.00339 P4 4/3 0.00452 M2 9/8 0.02030 m7 16/9 0.03618 M3 5/4 0.03968 M6 5/3 0.04538 m3 6/5 0.05396 m6 8/5 0.06299 M7 15/8 0.10199 m2 16/15 0.10805 a4 45/32 0.25483 I should stress the theoretical nature of the above results: the value of f, and the harmonic spectrum of the tones involved can affect the result in practice. Moreover, the beating given for the aug. fourth is something of a fiction, since the harmonics that govern the beating are very high. The M7 will also beat with 2/1 (0.11225) and the m2 with 1/1 (0.05946). I certainly wouldn't recommend that the beating frequency of tempered intervals, whether in theory or practice, should be employed as a criterion of consonance and dissonance. -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 31 Oct 1996 22:35 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05439; Thu, 31 Oct 1996 22:36:42 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA29548 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA24337; Thu, 31 Oct 1996 13:36:39 -0800 Date: Thu, 31 Oct 1996 13:36:39 -0800 Message-Id: <009AAB038BD0A780.6694@vbv40.ezh.nl> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu