source file: mills2.txt Date: Mon, 18 Dec 1995 07:31:02 -0800 From: "John H. Chalmers" From: mclaren Subject: a new iterated function for generating non-just non-equal-tempered tunings --- While boring holes for resonators in a pentatonic percussion instrument to be installed at the Exploratorium, a new iteration function occurred to me. Since these functions are a fertile breeding ground for non-just non-equal-tempered scales, this one might prove of interest to forum subscribers. Operating an industrial drill press is extremely peaceful work--excellent for mathematical contemplation. The function is an alternating series: start with a number, take the tan(x), and whenever it drops below 1.0, take e^x. The first 10 terms of the function are: i[1] = abs(tan(sqrt(2))) = 6.334119167... i[2] = abs(tan(6.334119167)) = 0.05097795... i[3] = abs(e^(0.05097795)) = 1.05229964... i[4] = abs(tan(1.05229964)) = 1.752641506... i[5] = abs(tan(1.752641506)) = 5.438434336... i[6] = abs(tan(5.438434336)) = 1.126353452... i[7] = abs(tan(1.126353452)) = 2.099871982... i[8] = abs(tan(2.099871982)) = 1.710348942... i[9] = abs(tan(1.710348942)) = 7.1119178021... i[10] = abs(tan(7.1119178021)) = 1.106677438... I believe but cannot prove that all of these numbers are transcendental. Numbers > 2.0 when octave-reduced, and < 1.0 when added to 1.0, produce a musical scale: p[1] = 795.7728117 cents p[2] = 86.07888146 cents p[3] = 88.25468083 cents p[4] = 971.4371163 cents p[5] = 531.8296507 cents p[6] = 205.991463 cents p[7] = 164.8027358 cents p[8] = 929.1488291 cents p[9] = 996.2863799 cents p[10] = 175.4817393 cents As usual, there does not appear to be any obvious pattern in these numbers. Another number arises from this series: 1 + the number of successive iterations required for switchover between e^x and abs(tan(x)), or vice versa. That number is: 1.21311151312121913141313111313... This number also appears to be a transcendental. Can you prove it? As a musical interval this equates to a neutral third of 334.4597716 cents. This is a third which has not to my knowledge appeared previously in the musical literature. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 18 Dec 1995 16:33 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id HAA04502; Mon, 18 Dec 1995 07:33:07 -0800 Date: Mon, 18 Dec 1995 07:33:07 -0800 Message-Id: <9512180731.aa02233@cyber.cyber.net> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu