source file: mills2.txt Date: Fri, 1 Mar 1996 07:45:06 -0800 Subject: Pythagorean scales From: COUL@ezh.nl (Manuel Op de Coul) With the current interest in Pythagorean scales, it's a good time to explain how the command in Scala works that creates a Pythagorean scale. Shameless plug alert: for those who missed it, Scala is a microtuning program written by me which can be downloaded from: ftp://ella.mills.edu/ccm/tuning/software/pc/scala/scala.zip The best $0 you will ever spend! Of course there's nothing against calculating a scale by hand, it's a good exercise. A program is usually quicker though. The command is called PYTHAGOREAN. It can calculate any Pythagorean scale. First it wants to know how many notes the scale must get. Type for instance: 26. Then, what the formal octave must be. I use the word "formal" to distinguish it from the octave of 2/1. It may be any value namely. But type for instance: 2/1. This will be the value of the last note in the scale and the interval that is subtracted every time the cycle of fifths passes over the last note. Instead of "formal octave", "interval of equivalence" is sometimes used. Then the scale degree for the formal fifth must be given (again to distinguish it from the perfect fifth). In 12-tET the degree of the fifth is 7. Here one can also enter 0, which means "I don't know" and then the program takes the nearest degree depending on the formal fifth to be given. So let's enter 0. Then enter the formal fifth, for instance: 7/4. Last, it is asked how many of the notes must result from fifths stacked in the downward direction. In this example they will be: 8/7, 64/49, etc. Going upwards you get 7/4, 49/32, etc. Let's enter: 10. That's all. The scale that results is this one: 1: 16807/16384 44.12955 2: 282475249/268435456 88.25910 3: 132.389 cents 132.3886 4: 131072/117649 187.0446 5: 8/7 231.1741 septimal whole tone 6: 2401/2048 275.3037 7: 40353607/33554432 319.4333 8: 363.563 cents 363.5628 9: 1048576/823543 418.2188 10: 64/49 462.3483 11: 343/256 506.4779 12: 5764801/4194304 550.6074 13: 594.737 cents 594.7370 14: 8388608/5764801 649.3930 15: 512/343 693.5225 16: 49/32 737.6521 17: 823543/524288 781.7817 18: 825.911 cents 825.9112 19: 67108864/40353607 880.5672 20: 4096/2401 924.6967 21: 7/4 968.8264 harmonic seventh 22: 117649/65536 1012.955 23: 1977326743/1073741824 1057.085 24: 536870912/282475249 1111.741 25: 32768/16807 1155.871 26: 2/1 1200.000 octave With the command SHOW CYCLE we can find out where this leaves the "wolf" 7/4 and see that it is placed between degrees 3 and 24. We can also see that the scale is fairly close to 26-tET. The largest difference is 6 cents. Masaaki Tsuji asks: > By the way, in Japan, a book called "MAGICAL MAX TOUR" came out this week. Is > someone trying retuned music on MAX? Is that kind of thing possible? I haven't used MAX but have read somewhere you have to use a pitch-bend object called "xbendout" and it can have a value of 0 .. 16383. Manuel Op de Coul coul@ezh.nl Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 1 Mar 1996 17:03 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA04657; Fri, 1 Mar 1996 08:03:19 -0800 Date: Fri, 1 Mar 1996 08:03:19 -0800 Message-Id: <9603011601.AA13972@delta1.deltanet.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu