source file: mills2.txt Date: Wed, 30 Oct 1996 22:19:38 -0800 Subject: Re: Intervals list (beating frequency) From: kollos@cavehill.dnet.co.uk (Jonathan Walker) Lorenzo Frizzera (Tue, 29 Oct) said: > Ciao. I'm working with a friend on a software for guitar and I need a > list of the intervals in 12-ET from the more consonant to more > dissonant. With "consonant" I mean beatless, empty. ... > Anyone can help me? Maybe with a mathematic > law on beating and interval spacing (spacing: 10M is more consonant > than 3M). Ciao. Beating is dependent upon frequency as well as interval size, so you'll have to set this according to your own purposes. The formula will vary according to the just (beatless) interval which a tempered interval most nearly approximates. Let's represent the just interval as "x/y", and indicate the size of the tempered interval close to this as "t"; the frequency of the lower pitch in the interval t we shall call "f". In this case the upper pitch of t will be ft. Take harmonic number y of the upper pitch, which will have the frequency fty, and subtract from this the harmonic number x of the lower pitch, which will have the frequency fx, i.e. f(ty - x). A positive result means that the tempered interval is sharper than the neighbouring just interval, and a negative result will signify flatness. To give an example: say you wish to find the rate at which the 12-TET fifth above A = 220Hz beats. In this case x = 3, y = 2, t = 2^(7/12) and f = 220. You will then have to calculate 220(2[2^(7/12)] - 3). The result is nearly -0.75, so you should listen for 3 beats every 4 seconds. I hope this enables you to complete your task, Lorenzo. Ciao! -- 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 08:59 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA03452; Thu, 31 Oct 1996 09:00:19 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA05136 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id AAA07518; Thu, 31 Oct 1996 00:00:17 -0800 Date: Thu, 31 Oct 1996 00:00:17 -0800 Message-Id: <9610310800.AA03250@arthaud.saclay.cea.fr> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu