source file: mills2.txt Date: Fri, 1 Mar 1996 06:00:15 -0800 Subject: Pythagorean scales From: Kami Rousseau >What I had in mind was the potentiality of using the circle of 7:4s instead of 3:2s. I invented a formula that will help you a lot (I someone else know this trick, email me so I stop claiming I invented it, but I am pretty sure that I am the first one to use it). Here goes: { (x,y) E N* | x>y} ---> x= 1, 2, 3... ; y= 2, 3, ... n E Z ---> n= ..., -2, -1, 0, 1, 2, ... 1 <= (x^n)/(y^p) < y where p= | n*log(x) / log(y) | | | means interger part of a number. So, for your question, x=7, y=2 Say you want to find the 5th interval of the "7/4 stack" serie, to build a Pythagorean scale. n=5, p= | 5*log(7) / log(2) | p= | 14.036...| p= 14 7^5 / 2^14 = 1.0258... n=45, p= | 45*log(7) / log(2) | p= | 126.33 | p= 126 7^45 / 2^126 = 1.2579... It works, the final number will always be smaller than y (2 in our exemple, so we needn't worry about normalizing intervals to a 2/1) If anyone would like to have the demonstration for this theorem, email me. Here are some arbitrary values: The human can ear from 20 30 000 Hz, The smallest interval one can perceive is 2 cents, The maximum number of beatings per second is 20, because 20 Hz is the lower limit of audition. We cannot hear beats that are further than 16 seconds apart (because the slowest tempo is 30 and the longest note is the square (2 whole notes)) I anyone disagrees with these values, email me. But if everyone agrees, then these values become the list's standard. Bye. Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 1 Mar 1996 16:11 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id HAA02817; Fri, 1 Mar 1996 07:11:13 -0800 Date: Fri, 1 Mar 1996 07:11:13 -0800 Message-Id: <960301100832_157370998@mail02.mail.aol.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu