source file: mills2.txt Date: Fri, 18 Oct 1996 08:43:41 -0700 Subject: Equal beating well temperaments, reply to John From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) This post is a reply to one long ago of John Chalmers of Tue, 4 Apr 95. See also the ones before that of 31 March '95 and 3 April '95. I was calculating some equal beating temperaments and found that John had done so too: [snip..] > I derived them as approximations to > Equal Beating tunings, a type described by Rudolf Rach in Music > Perception 1(3): 308-322 (1984). Rasch examined tunings in which > intervals and their inversions had the same beat rate (e.g. 1/4 comma > in which the major third and minor 6th have 0 beats). In 1991 I > decided to examine other temperaments in which pairs of non- > inversionally related intervals had the same beat rate. I was able > to find solutions for 9 such tunings. I also found fractional comma > approximations for them as well. I think only 2 are correct. > E.B. Intervals Fifths Approximation > 1. 4th = min 6th 695.80957 2/7-comma > 2. 4th = Maj 6th 696.29584 5/19-comma > 3. min 6th= Maj 6th 696.0063 5/18, 13/47 > 4. 5th = Maj 3rd 695.63044 5/17, 2/7 > 5. 5th = min 3rd 695.80957 2/7 > 6. Maj 3rd = min 3rd 695.72794 11/38, 9/31 > 7. min 3rd = min 6th 698.8780 1/7, 65/454 > 8. 4th = Maj 3rd 697.47472 5/24, 1/5 > 9. 5th = min 6th 697.0389 8/35, 3/13 1. 4th = min 6th 695.80957 not ok, 695.163 2. 4th = Maj 6th 696.29584 ok 3. min 6th= Maj 6th 696.0063 not ok, not computed 4. 5th = Maj 3rd 695.63044 not ok, 697.817 5. 5th = min 3rd 695.80957 not ok, 696.895 appr. 1/4-comma 6. Maj 3rd = min 3rd 695.72794 almost ok, not computed 7. min 3rd = min 6th 698.8780 ok 8. 4th = Maj 3rd 697.47472 not ok, 697.710 appr. 2/11 pyth. comma 9. 5th = min 6th 697.0389 not ok, 697.599 appr. 3/16 pyth. comma These are the other ones I did. E.B. Intervals Fifths Approximation 5/4 = twice 3/2 698.590 1/7 pyth. comma 3/2 = twice 5/4 697.278 1/5 comma 6/5 = 3/2 696.895 1/4 comma 6/5 = twice 3/2 698.048 2 schisma 3/2 = twice 6/5 696.022 5/18 comma 3/2 = 7/4 697.282 1/5 pyth. comma 5/4 = 3/2 other one 694.273 5/14 comma 6/5 = 3/2 other one 689.734 too extreme Computing these fifths involves finding the root of a polynomial of the fourth order or higher. To see beat rates in Scala, use SHOW DISTANCE. Manuel Op de Coul coul@ezh.nl Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 18 Oct 1996 19:55 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02828; Fri, 18 Oct 1996 19:57:01 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA02850 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA15044; Fri, 18 Oct 1996 10:56:58 -0700 Date: Fri, 18 Oct 1996 10:56:58 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu