source file: mills2.txt Date: Fri, 5 Jan 1996 21:20:24 -0800 Subject: Minimum Chord Sets From: Mmcky@aol.com LCM: 2^9*3^4*5^2 12 675 15/ 8 Major Seventh 11 640 16/ 9 Grave Minor Seventh 10 600 5/ 3 Major Sixth 9 576 8/ 5 Minor Sixth 8 540 3/ 2 Perfect Fifth 7 512 64/ 45 Diminished Fifth 6 480 4/ 3 Perfect Fourth 5 450 5/ 4 Major Third 4 432 6/ 5 Minor Third 3 405 9/ 8 Major Tone 2 384 16/ 15 Semitone 1 360 1/ 1 Unison Minimum Chords: 1 60 3: 4: 5 1- 6- 10 2 60 3: 4: 5 2- 7- 11 3 60 3: 4: 5 3- 8- 12 4 60 4: 5: 6 1- 5- 8 5 60 4: 5: 6 2- 6- 9 6 60 10: 12: 15 1- 4- 8 Chord List: 1 60 3: 4: 5 1- 6- 10 2 60 3: 4: 5 2- 7- 11 3 60 3: 4: 5 3- 8- 12 4 60 4: 5: 6 1- 5- 8 5 60 4: 5: 6 2- 6- 9 6 60 10: 12: 15 1- 4- 8 7 60 10: 12: 15 5- 8- 12 8 60 12: 15: 20 1- 5- 10 9 60 12: 15: 20 2- 6- 11 10 72 6: 8: 9 1- 6- 8 11 72 6: 8: 9 2- 7- 9 12 72 6: 8: 9 5- 10- 12 13 72 8: 9: 12 1- 3- 8 14 72 8: 9: 12 2- 4- 9 15 90 6: 9: 10 1- 8- 10 16 90 6: 9: 10 2- 9- 11 17 90 9: 10: 15 3- 5- 12 The above scale has 9 lcm60 chords. Using the LCM as a criterion, all are equal in consonance. We might note that if only two out of the three notes are sounded, the 3:4:5, and 12:15:20 versions would seem to have an advantage. Possibly, there are 9 factorial or, 362,880 ways these 9 chords can be arranged, but some arrangements would lead to the same minimum chord set, so 9 factorial is an upper bound on the number of minimum chord sets that can be derived by rearranging this set of lcm60 chords. It takes at least five triads to span the 12 tones, so I suppose 5 factorial, or 120, would be a lower bound on the number of possible minimum chord sets, though I suspect the number is much higher than that. When first I started finding minimum chord sets, I simply sorted the chords according the their first notes. Then I used the method above, sorting them by their ratios. Now I have tried another method, which is sorting the chords by the LCM's of their constituent ratios and the fundamental. For the chord CEG in the above scale, the LCM of the C is 1/1, or one. The LCM of the E is 5/4 or 20, and the LCM of the G is 3/2 or 6 giving an aggregate of 27. Applying this algorithm to the above scale gives the minimum chord set: 60 27 ceg 60 28 cfa 60 37 cd#g 60 146 egb 60 198 dgb 60 292 c#fg# 60 396 c#fa# 60 3264 c#f#a# This is much closer to the results that musicians have derived from experience, and seems a better solution. Here are the results of applying this analysis to a scale that contains the 9/7 interval: LCM: 2^5*3^3*5*7 12 135 27/ 14 11 126 9/ 5 Minor Seventh 10 120 12/ 7 9 112 8/ 5 Minor Sixth 8 108 54/ 35 7 105 3/ 2 Perfect Fifth 6 96 48/ 35 5 90 9/ 7 4 84 6/ 5 Minor Third 3 80 8/ 7 2 72 36/ 35 1 70 1/ 1 Unison Minimum Chords: 60 37 cd#f# 60 81 d#f#a# 60 1407 c#ea 60 1820 dfa 60 2331 egb 72 115 d#g#a# I have not yet taken the trouble to extend this analysis to scales with more than 12 tones, but it is on my list. Marion Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 6 Jan 1996 18:27 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA25770; Sat, 6 Jan 1996 09:26:57 -0800 Date: Sat, 6 Jan 1996 09:26:57 -0800 Message-Id: <960106172319_71670.2576_HHB39-1@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu