From: Dave Rusin Date: Fri, 26 Mar 1999 15:09:25 -0600 (CST) To: ajjmae@globalfrontiers.com Subject: Re: Why 12 tones per octave? > Do you have a website with some of your articles?? http://www.math.niu.edu/~rusin/uses-math/music/index.html >about the shape of the harp? If the frequency of a string were dependent only on the length (it's not) then we would need the lengths to double with each octave downward. If the harpist were at the origin and the harp on the positive x-axis, (the harpist sits at the high-note end) then the string at position x would have to have a length varying like y = A * 2^x (with distances along the x-axis measured in octaves). To make room for the harpist's body, the bottom of the string is not on the floor; rather, it's attached to the soundboard -- a straight but tilted plane. Thus the bottom of the string which is x octaves down from the highest note is elevated to a height of the form y = a * (1 - x/k) where a is the height of the bottom of the shortest string (a is about 5 feet) and k is the number of octaves until you reach the floor (about 6). OK, we know now where the bottom of each string is, and we know how long it is; where's the top? Must be at y = a * (1 - x/k) + A * 2^x. We can even determine A here if we assume that the top of the bass strings is about equal to the height of the top string: we need y(k) = y(0), which is roughly A*2^k = a. So the shape of the top of the harp is something like A*( 2^k*(1-x/k) + 2^x ). (This is a little bogus: to get deeper notes they don't just take longer strings but also make them thicker and, at the very bottom, change material (from gut to coiled metal). These affect the necessary lengths -- the lengths needn't grow so fast but rather grow like A * r^x for some r < 2, or so it appears by sight.) dave From rusin@math.niu.edu Tue Jan 16 01:51:44 2001 Received: from vesuvius.math.niu.edu (vesuvius.math.niu.edu [131.156.3.93]) by mail.math.niu.edu (8.9.1a/8.9.1) with ESMTP id BAA05965; Tue, 16 Jan 2001 01:51:43 -0600 (CST) From: Dave Rusin Received: (from rusin@localhost) by vesuvius.math.niu.edu (8.9.3/8.8.5) id BAA17108; Tue, 16 Jan 2001 01:51:43 -0600 (CST) Date: Tue, 16 Jan 2001 01:51:43 -0600 (CST) Message-Id: <200101160751.BAA17108@vesuvius.math.niu.edu> To: pirashennah@home.com Subject: Re: Help, please... Cc: rusin@math.niu.edu Status: R No, I never really thought about it, but a quick search of Google.com is informative. Just type in harp mathematics and it finds this, and more! Harp mathematics ... Harp mathematics. To: harp-l@garply.com; Subject: Harp mathematics; From: Snaruhn@aol.com; Date: Thu, 17 Feb 2000 03:28:50 EST; Sender: owner-harp-l. ... www.garply.com/harp-l/archives/2000/0002/msg00349.html - 3k - Magic Soloist ... Prev by Date: Harp mathematics; Next by Date: William Clarke LP's & CD's; Prev by thread: Harp mathematics; Next by thread: Re: Magic Soloist; Index(es): ... www.garply.com/harp-l/archives/2000/0002/msg00350.html - 4k - Harp Mailing List 1996 Digest Archive: Harp Digest #130 ... I have a centered place from which to venture into my cross-cult investigations of music, harp mathematics, and Austrian cuisine. I encourage everyone on the ... tns-www.lcs.mit.edu/harp/archives/digests/1996/0316.html - 31k - Harp Mailing List 1996 Digest Archive: Harp Digest #83 ... of Mittelshmerz, Austria is happy to announce is 20th Annual Fall Mathematics Retreat for Harp Practitioners and Musicians. Music is simply an acoustic ... tns-www.lcs.mit.edu/harp/archives/digests/1996/0263.html - 48k - MAVIS Mathematics Accessible to Visually Impaired Students ... ... index three" is read verbally. A harp twang at c5 indicates branch two ... of tabular structures, the rest of mathematics notation is similarly hierarchical, and ... www.dinf.org/csun_99/session0255.html - 8k - Andrew Lawrence-King ... and College Orchestra, my academic subject was Mathematics, which I followed up to MA level ... to a party given by the harp-maker, Tim Hobrough. It must have been ... ourworld.compuserve.com/homepages/Harp/bio.htm - 8k - Good luck dave