________________________________________________________________________ Date: Mon, 21 Aug 2000 [with errata corrected by CKL] From: "Paul H. Erlich" Subject: locally concordant tetrads -- verifying Margo and George In case you have no clue what harmonic entropy is, see http://www.ixpres.com/interval/td/entropy.htm. It is, at this point, a dyadid discordance measure, which certainly conforms to Pierre Lamothe's partial ordering for simple ratios, but is a realistic, continuous function that is well-defined for all intervals, simple ratios or otherwise. For this exercise, I used a Farey series of order N=100 and an auditory resolution of 1%. Within the first two octaves, the resulting harmonic entropy curve shows significant local minima at the ratios 5:6, 4:5, 7:9, 3:4, 7:5, 2:3, 5:8, 5:3, 4:7, 1:2, 4:9, 3:7, 2:5, 3:8, 4:11, 1:3, 3:10, 2:7, 3:11, as well as hints of minima at 7:8, 6:7, 5:12, 5:13, 4:15, and 1026¢ (the last one results from being less than 1% from two different simple ratios -- 5:9 below, and 11:6 above). I'll make a graph of this curve available soon. Anyway, I calculated the total pairwise harmonic entropy for about 8 million tetrads, allowing each of the 3 intervals between adjacent notes to vary between 150¢ and 550¢ in 2¢ increments. By "total pairwise harmonic entropy" I mean I added the six harmonic entropy values for the six possible pairs of notes found within the tetrad. As I've mentioned before, this ignores effects that arise from the synergy of three or more notes. I found 140 tetrads that were local minima (i.e., changing one, two, or three notes by 2¢ would necessarily increase the total pairwise harmonic entropy), and ranked them according to total pairwise harmonic entropy (call it discordance, since that's what it's a crude model of). Going from lowest (most concordant) up the list: First there were a bunch of chords which were really triads with the lower note doubled an octave higher. The next most concordant tetrad was: 0 498 886 1384¢ or 9:12:15:20 or 1/1:4/3:5/3:20/9. It is an open-voiced JI minor seventh chord in third inversion, or an open-voiced JI major added sixth chord in second inversion. So, in a sense, Margo agrees with this model (note, HOWEVER, caveat in second-to-last paragraph below) that this "added sixth" chord is the most concordant tetrad with distinct pitch classes, though perhaps not in the voicing she would have suggested (of course, had I increased the upper bound for adjacent intervals, some other inversion of this chord, or some other chord, might have come out as most concordant). The next most concordant tetrad was one that George Kahrimanis brought up recently, and Margo and I have brought up in the past: 0 492 980 1472¢ It is the "stack of fourths" chord where the fourths are tempered narrow so that the outer interval approaches 3:7, and the bass-alto and tenor- soprano intervals approach 4:7. To the 2¢ accuracy of the calculation, this is the same as the 22-tET tuning of the "stack of fourths". This clearly shows that the relationship between concordance and JI is quite different when dealing with dyads vs. dealing with larger chords, even when the model only takes dyadic discordance, with local minima at JI ratios, into account. Also, since the triadic analogue to this calculation showed a local minimum at 0 498 996¢, this model agrees with George K. that while a stack of two fourths sounds best pure, a stack of three fourths can be improved with some temperament. The next most concordant tetrad was the JI major seventh chord, 0 386 702 1088¢ or 8:10:12:15 or 1/1:5/4:3/2:15/8. The next most concordant tetrads were (tied) 0 502 1002 1390¢ and 0 388 888 1390¢ which have their fourths and major thirds tempered a bit wide, as in a very mild meantone. The first one is the "Carole King" chord, the second is an add6, add9 (6/9) chord with no 5th, in typical jazz fashion. Relative to what would seem ideal just tunings of 9:12:16:20 (1/1:4/3:16/9:20/9) and 36:45:60:80 (1/1:5/4:5/3:20/9), respectively, the intervals are expanded a bit, probably to bring the outer 9:20 closer to the much more concordant 4:9. The next most concordant tetrad was the JI minor seventh chord, 0 316 702 1018¢ or 10:12:15:18 or 1/1:6/5:3/2:9/5. The next most concordant tetrads were (tied) the second-inversion JI major triad with the fourth in the bass, 0 204 702 1088¢ or 8:9:12:15 or 1/1:9/8:3/2:15/8, and the first-inversion JI minor triad with the ninth on top, 0 386 884 1088¢ or 24:30:40:45 or 1/1:5/4:5/3:15/8. The next most concordant tetrad was a surprise -- a very modern "augmented octave" chord, 0 388 886 1274¢ or 12:15:20:25 or 1/1:5/4:5/3:25/12. It contains two 4:5s, two 3:5s, and one 3:4s, all concordant enough to counteract the great discordance of the 12:25. While the usual, 5-prime-limit JI minor seventh chord has appeared in two different voicings so far in our list, at this point the next most concordant tetrad was the other JI minor seventh chord discussed at http://www.cix.co.uk/~gbreed/erlichs.htm, 0 268 702 970¢ or 12:14:18:21 or 1/1:7/6:3/2:7/4. The next most concordant tetrads (tied) were 0 318 818 1320¢ and 0 502 1002 1320¢ obtained by stacking a minor third and two fourths, or two fourths and a minor third. In JI, these would be 15:18:24:32 (1/1:6/5:8/5:32/15) and 45:60:80:96 (1/1:4/3:16/9:32/15), respectively, but the intervals are expanded a touch, probably to help alleviate the sharp discordance of the outer interval. The next most concordant tetrads (tied) are Margo's vote the JI added sixth, 0 388 702 886¢ or 12:15:18:20 or 1/1:5/4:3/2:5/3, and its second inversion 0 184 498 886¢ or 10:12:15:18 or 1/1:10/9:4/3:5/3. The next most concordant tetrads (tied) were 0 434 820 1320¢ and 0 500 886 1320¢ which exploit the fact that a 7:9 stacked with a 4:5 very nearly produces a 5:8 (exploiting the 225:224 comma), and add a 3:4 on either the low end or the high end (the 3:4 is adjacent to the 4:5, producing a 3:5). The next most concordant tetrads (tied) were 0 302 502 1004¢ and 0 502 702 1004¢ which are two of the ways a stack of fourths can be transposed to within one octave. A slight meantone-like tempering of the fourths is evident, in order to bring the 27:32 closer to a more concordant 5:6. The next most concordant tetrads (tied) were the first-inversion JI major triad with the ninth on top, 0 318 816 1020¢ or 5:6:8:9 or 1/1:6/5:8/5:9/5, and the second-inversion JI minor triad with the fourth in the bass, 0 204 702 1020¢ or 40:45:60:72 or 1/1:9/8:3/2:9/5. The next most concordant tetrads (tied) were also of the augmented octave type, 0 498 888 1282¢ 0 394 784 1282¢ where the 3:4 is attached to either the low end or the high end of a stack of two 4:5s (each slightly stretched to alleviate the discordance of the 25:16), producing a 3:5. The next most concordant tetrads (tied) exploit the fact that a major third plus two fifths is very nearly a 5:7 (i.e., exploiting the 225:224 "comma"): 0 384 588 1086¢ 0 498 702 1086¢ The major third is compressed a bit to make this tempering magic happen. The next most concordant tetrad is an open-voiced third-inversion major seventh chord, 0 500 816 1316¢ which would be 15:20:24:32 or 1/1:4/3:8/5:32/15 in JI, but the fourths are expanded a touch, probably to help alleviate the sharp discordance of the outer interval. The next most concordant tetrads (tied) are the JI major 7 (#5) chord, 0 388 776 1090¢ or 16:20:25:30 or 1/1:5/4:25/16:15/8, and the JI minor (maj7) chord, 0 314 702 1090¢ or 40:48:60:75 or 1/1:6/5:3/2:15/8. The next most concordant tetrads (tied) are the second-inversion JI major triad with ninth on top, 0 498 702 886¢ or 6:8:9:10 or 1/1:4/3:3/2:5/3, and the first-inversion JI minor triad with the fourth in the bass, 0 184 388 886¢ or 36:40:45:60 or 1/1:10/9:5/4:5/3. The next most concordant tetrad is 0 442 884 1326¢ which stacks three 442-cent intervals, each of which approximates 7:9 from the high side, and each pair of which approximates 3:5 very closely. In other words, an equal division of the 3:5! The next most concordant tetrads (tied) are a major triad with 5-prime- limit minor seventh in the bass, 0 186 576 888¢ or 18:20:25:30 or 1/1:10/9:25/18:5/3, and a JI minor triad with 5-limit major sixth on top, 0 312 702 888¢ or 30:36:45:50 or 1/1:6/5:3/2:5/3. The 18:25s in these chords are actually expanded relative to JI, to help them approximate the concordant 5:7. These are thus pretty close to inversions of the 7-odd- limit tetrads, but the major sixth between the outer voices is clearly 3:5 and not 7:12, and in fact the 7-limit versions do not show up as local minima of their own. The next most concordant tetrads (tied) are the "root position" 7-odd- limit tetrads, the otonal 0 388 702 970¢ or 4:5:6:7 or 1/1:5/4:3/2:7/4, and the utonal 0 268 582 970¢ or 60:70:84:105 or 1/1:7/6:7/5:7/4. Although for the purposes of my 22-tET paper, and Pierre's recent post, it might have been nice if these appeared on top of the list, they didn't. (HOWEVER: the order here is pretty arbitrary. The Farey-series-based harmonic entropy curve greatly favors large intervals over small ones, and no open voicings of these chords had an opportunity to make it into the comparison, since I limited the adjacent intervals to be under 550¢. Voicings such as 0 702 970 1588¢ and 0 618 886 1588¢ would handily defeat all the chords listed above, while even 0 498 886 1468¢ and 0 582 970 1468¢ would have come out ahead of the major seventh chord above. Other formulations of the model, though they might completely change the order, would have very little effect on _what_ these local minima are, so that's what's important here). Experientially speaking, I'm not too opposed to the ranking as regards the utonal 7-limit tetrad, while the otonal 7-limit tetrad is clearly more concordant than this ranking implies, which reflects the fact that this model ignores synergies between three or more notes. A true triadic or tetradic harmonic entropy model (rather than a sum of dyadic harmonic entropy models, which is what this is) would take this into account. I stop here. If anyone wants the full list of 140 local minima (cents only), I'll e-mail it to them. One could spend a considerable period of time just listening to the chords above, and tweaking the notes one by one to see how the particular tuning (just or tempered) is a local optimum for that chord. Many of the chords above come from a standard application of 5-limit JI to tetrads found in the diatonic scale (including melodic minor and harmonic minor variants), but many are tempered in a subtle way, and some are downright xenharmonic. An excellent exercise for a composer looking for points of stability in the infinite realm of microtonal harmony would be to spend some time with the chords above. ------------------------------------------------------------------------ Date: Thu, 24 Aug 2000 [with errata corrected by CKL] From: "Paul H. Erlich" Subject: moving on to "scales" Dan Stearns wrote, >I'd also be interested in some things like, relative to some of the >20-tET work that I've done, what say the most concordant stack of near >240¢ fifths of an octave are for instance. If you'd like to think of it as a 5-tone scale that continues many octaves up and down, I'd use an octave-equivalent diadic discordance measure. For that purpose, let me just simplistically "invert" the interval into first half-octave of the harmonic entropy curve, and then into the second half-octave, and average the two values. Then, starting with an equipentatonic (5-tET) scale, the local minimum of total diadic harmonic entropy is A C D E G 0 309 504 699 1008 which is a standard pentatonic scale consisting of the following fifths: C---699---G---696---D---696---A---699---E Both minor thirds are 309¢, and the major third is 390¢. For an equiheptatonic (7-tET) scale, the local minimum comes out as: G A B C D E F 0 191 387 503 696 888 1004 which has these fifths: F---699---C---697---G---696---D---695---A---697---E---699---B The two minor thirds involving D are 308-309¢, the other two minor thirds are 312¢, the C-E major third is 385¢, and the other two major thirds are 387¢. So, if you can imagine a 7-dimensional diadic discordance surface, and you place a ball at the point representing the equiheptatonic scale, it will roll down into one of the seven basins representing the seven modes of the diatonic scale, all tuned in this pseudo-meantone fashion. By the way, here are the results for putting the ball on various ETs (I've called these result "relaxed" ETs in the past) -- again, all modes of these scales are equally valid: 2: 0 616 3: 0 400 800 4: 0 314 574 888 5: 0 309 504 699 1008 6: 0 197 388 620 811 1008 7: 0 191 387 503 696 888 1004 8: 0 116 312 428 620 813 929 1124 10: 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3 12: 0 100 200 300 400 500 600 700 800 900 1000 1100 For 9, 11, and 13 and up, at least one pair of notes made it over the hump and settled into a unison. Wow. This is a very strong justification of the usual pentatonic and diatonic scales (in particular, pseudo-meantone tunings), of 12-tET, and of a few scales I've never seen before (let's start playing with them!), purely on grounds of diadic concordance. ------------------------------------------------------------------------ Date: Thu, 24 Aug 2000 From: "Paul H. Erlich" Subject: new decatonic scales and dodecatonic scales? Let's look at what I got by "relaxing" 10-tET: 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3 Basically, that's C C# D E F F# G G# A B in a pseudo-meantone -- a chain of fifths. Now let's try "relaxing" the 22-tET pentachordal decatonic scale. I get: 0 79 193 387 502 581 696 890 1003 1083 -- basically a mode of the above. Now let's try "relaxing" the 22-tET symmetrical decatonic scale. I get: 0 109 199 385 501 599 698 814 999 1089 Rather than slipping into meantone, it retains more of its essential 22-tET pattern: the two chains of fifths: (999)--702--(501)--699---(0)---698--(698)--701--(199) (385)--704-(1089)--710--(599)--710--(109)--705--(814) But while one of the chains is hanging on to the large fifths of 22-tET, the other one abandons them for 12-tET-like fifths. I'll have to take a closer look at how this symmetry-breaking improves the diadic concordance. . . ------------------------------------------------------------------------ Date: Tue, 29 Aug 2000 From: "Paul H. Erlich" Subject: RE: locally concordant tetrads -- verifying Margo and George Here are the results again, same as before but this time imposing octave equivalence (by using, for each interval, the average of the entropies of the smallest inversion (0-600¢) and the second-smallest inversion (600-1200¢)): 498 702 1200 21.016 498 996 1200 21.522 204 702 1200 21.522 498 884 1200 21.572 316 702 1200 21.572 498 812 1200 21.654 388 702 1200 21.654 386 884 1200 21.981 316 814 1200 21.981 498 934 1200 22.029 266 702 1200 22.029 388 812 1200 22.649 440 760 1200 22.967 196 502 698 23.507 196 698 894 23.507 502 698 1004 23.507 306 502 1004 23.507 502 1004 1506 23.507 386 702 884 23.632 316 498 814 23.632 316 702 1018 23.632 182 498 884 23.632 498 884 1382 23.632 386 702 1088 23.788 498 814 1312 23.788 498 702 884 23.811 498 814 996 23.811 316 498 702 23.811 316 814 1018 23.811 204 386 702 23.811 204 702 1018 23.811 498 996 1382 23.811 182 386 884 23.811 182 498 996 23.811 386 884 1382 23.811 386 884 1088 23.885 498 996 1312 23.885 204 702 1088 23.885 316 814 1312 23.885 498 702 1086 24.03 384 588 1086 24.03 266 498 764 24.052 232 498 934 24.052 498 934 1432 24.052 436 702 934 24.052 266 702 968 24.052 498 702 932 24.058 498 766 996 24.058 498 996 1430 24.058 268 498 702 24.058 268 766 970 24.058 204 434 702 24.058 204 702 970 24.058 230 434 932 24.058 230 498 996 24.058 434 932 1430 24.058 314 702 886 24.119 388 572 886 24.119 184 498 812 24.119 388 702 1016 24.119 184 572 886 24.119 314 498 886 24.119 314 628 1016 24.119 314 628 812 24.119 312 810 1122 24.21 390 888 1278 24.21 264 498 884 24.238 264 580 966 24.238 386 620 884 24.238 386 702 966 24.238 234 620 936 24.238 316 580 814 24.238 316 702 936 24.238 234 498 814 24.238 498 810 1120 24.276 310 622 1120 24.276 386 774 1088 24.296 314 702 1088 24.296 498 886 1312 24.296 426 814 1312 24.296 188 576 1074 24.328 498 886 1074 24.328 498 814 1072 24.33 258 574 1072 24.33 498 884 1110 24.36 226 612 1110 24.36 314 812 1078 24.384 436 934 1322 24.384 388 886 1322 24.384 266 764 1078 24.384 270 702 884 24.385 432 702 1018 24.385 270 586 768 24.385 182 498 768 24.385 182 614 884 24.385 432 614 930 24.385 316 498 930 24.385 316 586 1018 24.385 498 766 1080 24.447 314 582 1080 24.447 384 812 1086 24.476 274 702 1086 24.476 498 926 1314 24.476 388 816 1314 24.476 312 626 886 24.523 260 574 886 24.523 314 574 888 24.523 314 626 940 24.523 264 436 700 24.617 264 764 1028 24.617 436 936 1372 24.617 172 436 936 24.617 500 764 936 24.617 246 568 814 24.776 246 632 878 24.776 386 632 954 24.776 322 568 954 24.776 230 388 618 24.785 232 620 812 24.79 388 580 968 24.79 388 620 1008 24.79 192 580 812 24.79 390 622 808 24.82 390 782 968 24.82 392 782 1014 24.82 232 622 1014 24.82 392 578 810 24.82 186 418 808 24.82 186 578 968 24.82 232 418 810 24.82 390 618 784 24.834 390 806 972 24.834 416 806 1034 24.834 166 394 784 24.834 228 394 810 24.834 228 618 1034 24.834 416 582 810 24.834 166 582 972 24.834 190 388 578 24.863 186 390 576 24.864 182 392 574 24.865 390 814 1006 24.872 424 616 810 24.872 424 814 1008 24.872 192 616 1006 24.872 390 584 776 24.872 192 386 776 24.872 194 386 810 24.872 194 584 1008 24.872 316 764 1080 24.877 436 884 1320 24.877 180 620 940 24.932 320 580 760 24.932 320 760 940 24.932 440 620 880 24.932 440 760 1020 24.932 180 440 760 24.932 260 440 880 24.932 260 580 1020 24.932 192 582 774 24.934 390 582 1008 24.934 192 618 810 24.934 426 618 1008 24.934 176 616 1056 24.964 440 880 1056 24.964 ------------------------------------------------------------------------ Date: Tue, 29 Aug 2000 From: "Paul H. Erlich" Subject: A plethora of pentatonics! Carl wrote, >Regarding your entropy minimizer algorithm... have you reached any >conclusions as to how much the initial scale choice influences the >relaxed result? Wouldn't it be possible to find, for a given number >of tones, the one scale with the least total dyadic harmonic entropy? >I understand that this might not always be desirable -- when looking >for a local minimum around a certain chord, for example. But for >finding new scales -- how much did the initial choice of n-tET >influence the results? I promised another List member to seed the program with random scales, and I suspect that would address your concerns. Well, for 3 and 4 notes, we already know most of the terrain, so I'll start with 5. Throwing out the results with unisons, shifting to most compact mode, and sorting by discordance (which comes out to 21.092 for 5 of the same pitch), scale discordance #observations 0 195 390 699 891 39.506 3 0 192 501 696 891 39.506 2 0 112 316 498 814 39.829 2 0 316 498 702 814 39.829 2 0 118 314 619 815 39.99 2 0 196 501 697 815 39.99 0 197 383 502 698 40.01 3 0 196 315 501 698 40.01 2 0 196 315 501 699 40.01 0 115 317 702 816 40.048 0 114 499 701 816 40.048 3 0 314 497 628 812 40.193 2 0 184 315 498 812 40.193 2 0 311 389 702 887 40.228 0 185 498 576 887 40.228 2 0 184 388 572 886 40.243 0 197 501 586 699 40.296 0 113 198 502 699 40.296 2 0 91 205 589 703 40.393 0 229 317 498 814 40.398 0 316 497 585 814 40.398 0 314 390 702 812 40.411 3 0 386 499 701 814 40.412 0 113 315 428 814 40.412 2 0 268 498 703 885 40.473 0 112 235 498 814 40.476 2 0 203 387 586 702 40.478 2 0 116 315 499 702 40.478 0 113 428 611 815 40.484 3 0 388 702 780 886 40.49 0 125 312 623 703 40.505 0 313 497 624 703 40.533 2 0 201 316 584 701 40.548 2 0 116 431 701 818 40.551 0 81 390 498 703 40.572 0 205 313 622 703 40.572 0 86 316 583 703 40.605 0 202 431 700 817 40.629 3 0 313 497 628 886 40.681 2 0 386 498 615 703 40.703 2 0 268 384 499 767 40.712 0 117 223 498 616 40.719 0 112 424 499 611 40.796 2 0 187 498 581 812 40.816 2 0 313 426 626 811 40.822 0 313 426 625 811 40.823 0 127 312 441 625 40.86 0 184 313 498 625 40.86 0 234 497 623 812 40.879 0 237 386 622 883 40.895 0 125 314 440 702 40.944 0 129 311 392 626 40.972 0 271 387 586 770 40.998 0 311 438 621 702 41.05 0 127 234 314 625 41.051 0 183 267 497 581 41.06 2 0 117 233 431 616 41.082 0 102 183 319 497 41.098 0 178 314 395 497 41.098 0 232 387 498 620 41.119 0 75 185 388 499 41.156 0 264 388 497 581 41.198 0 258 320 496 577 41.225 0 80 235 392 498 41.239 0 88 193 272 585 41.346 0 229 311 394 623 41.457 That's probably missing a bunch, but at this point, if the "diatonic" pentatonic in pseudomeantone isn't the global minimum, I'll eat my shoe. 6&up . . . later ------------------------------------------------------------------------ Date: Tue, 12 Sep 2000 From: "Paul H. Erlich" Subject: a hodgepodge of hexatonics I repeated the process in "a plethora of pentatonics". I used 3000 random starting points. 359 did not collapse into scales of 5 or fewer notes: Scale (sample mode) total diadic h.e. #observations 0 196 310 506 697 1009 59.738 20 0 196 309 508 697 1010 59.746 1 0 192 314 501 697 891 60.164 16 0 196 316 501 698 815 60.176 22 0 112 316 498 702 814 60.181 10 0 193 501 580 697 892 60.319 18 0 197 501 585 698 1083 60.46 12 0 302 501 618 1003 1116 60.463 1 0 78 390 575 703 888 60.469 5 0 195 503 586 696 1004 60.491 8 0 103 408 602 908 1104 60.54 1 0 113 498 611 702 815 60.6 2 0 204 314 390 702 1016 60.601 10 0 311 388 702 886 1090 60.614 5 0 196 315 580 699 895 60.65 5 0 197 384 502 698 815 60.688 11 0 314 389 498 702 813 60.702 7 0 183 387 498 619 885 60.722 2 0 128 206 315 703 1017 70.728 2 0 77 186 574 888 1071 70.729 1 0 79 206 389 703 1091 70.3 1 0 115 385 500 701 816 60.76 12 0 96 400 496 800 896 60.8 1 0 105 210 601 705 809 60.804 3 0 103 600 704 805 1098 60.815 4 0 386 498 617 703 884 60.828 3 0 86 316 584 703 1018 60.841 8 0 203 315 389 702 813 60.848 3 0 312 424 810 923 1125 60.864 1 0 197 313 386 502 699 60.866 3 0 195 272 582 698 894 60.872 1 0 315 433 701 814 932 60.877 2 0 115 384 499 614 815 60.887 2 0 121 311 499 620 703 60.895 9 0 188 310 392 889 1011 60.896 1 0 117 502 698 1002 1086 60.912 8 0 111 314 701 812 890 60.917 4 0 197 313 501 623 699 60.939 4 0 233 312 624 809 1121 60.947 1 0 82 277 395 583 780 60.953 1 0 312 497 627 810 888 60.961 2 0 314 498 626 812 939 60.974 2 0 196 316 500 584 699 60.979 6 0 198 384 501 586 699 60.985 6 0 113 198 315 501 699 60.986 1 0 115 262 499 700 817 60.995 1 0 115 203 431 701 817 61.003 2 0 106 225 410 608 1109 61.014 1 0 79 312 393 576 890 61.024 3 0 315 497 588 702 1086 61.034 1 0 274 500 700 773 1086 61.048 4 0 80 390 578 703 966 61.068 3 0 196 274 384 503 698 61.092 1 0 106 226 420 923 1118 61.093 1 0 203 389 587 702 1015 61.1 11 0 85 392 496 583 704 61.112 4 0 87 315 497 703 889 61.114 3 0 201 272 387 701 973 61.129 1 0 127 206 388 703 1015 61.136 2 0 196 273 501 587 698 61.144 4 0 126 235 314 624 938 61.162 1 0 234 498 622 812 936 61.177 1 0 184 313 417 498 810 61.182 6 0 84 266 497 581 890 61.205 1 0 81 314 579 891 1016 61.21 1 0 124 205 437 702 1015 61.214 2 0 184 265 576 763 887 61.223 1 0 81 314 393 578 703 61.238 1 0 77 391 779 888 1014 61.269 1 0 114 231 428 613 815 61.283 4 0 105 183 414 497 994 61.299 2 0 249 386 497 631 883 61.306 1 0 78 186 264 576 888 61.312 1 0 74 387 498 649 885 61.319 1 0 202 315 390 583 702 61.34 2 0 311 389 498 622 703 61.342 4 0 72 251 324 569 954 61.349 1 0 111 196 315 427 813 61.415 1 0 263 579 701 817 966 61.426 3 0 123 309 389 500 622 61.437 3 0 119 188 310 500 619 61.443 1 0 266 381 646 766 1080 61.446 1 0 181 271 385 499 769 61.527 3 0 497 619 703 811 886 61.54 1 0 583 701 815 967 1082 61.559 2 0 76 190 777 890 1009 61.619 1 0 78 314 497 632 887 61.652 1 0 81 192 392 579 968 61.699 4 0 186 385 498 619 810 61.707 2 0 236 424 553 813 1049 61.723 1 0 156 266 389 580 968 61.73 3 0 106 187 316 498 1075 61.843 3 0 130 238 320 555 941 61.848 1 0 231 312 392 498 624 61.968 1 0 83 209 316 397 1019 61.984 1 0 172 319 442 560 942 61.986 1 0 80 182-3 263 395 577 62.029 2 0 76 251 318 391 956 62.082 1 3000 tries was not enough to find the result of relaxing the 6-tET scale that we found before, 0 197 388 620 811 1008 62.207 or the result of relaxing harmonics 7-12: 0 202 387 584 702 971 61.228 ------------------------------------------------------------------------