http://www.microsoft.com/windows/ie/download/ie55.htm ======== > > > [me, Monz] > > > http://www.egroups.com/message/tuning/11867 > > > > > > this comparision of orbital periods *only of adjacent planets* > > > is similar to Paul Erlich's comparison of *dyadic* harmonic > > > entropy among members of a larger set (such as a chord or > > > scale). > > [David J. Finnamore] > > http://www.egroups.com/message/tuning/11894 > > > > Analogous, yes, but are they really similar in any musically > > relevant way? > [Paul Erlich] > http://www.egroups.com/message/tuning/11907 > > Well, the cycles of adjacent planets and moons do tend to lock > into simple-integer ratios, Which was pretty much my point. > [Paul] > > though they I don't know if they always manage to do so. > Non-adjacent planets or moons might have very complex > ratios as a result. Below I give tables showing all the orbital ratios as decimal values, to 4 significant digits. I also give the orbital periods in Earth-days, which means that the values can be calculated with much greater accuracy by anyone who wants to do it. (I've done it, but it won't fit here; listen to the piece to hear the accurate values.) In the 'addendum' at the bottom of this post, there are two charts showing how closely these values fit various low-integer ratios, with the percentage error. In my previous post on this, http://www.egroups.com/message/tuning/11867 I already pointed out the planetary relationships that had the closest fit to low-integer ratios (but I used 23-limit there); here they are quantified, and many other less-good rational approximations are shown. > [Paul] > > On the other hand, I've been considering dyadic harmonic > entropy for all the notes in a set, adjacent or otherwise; > if I only considered adjacent ones, it would be more analogous > to the solar system. I was going to add that statement to my original post, but I guess I thought it was obvious enough for anyone who's been following the harmonic entropy threads... that's why I used the emphases that I did. > [Paul Erlich] > http://www.egroups.com/message/tuning/11906 > > But what does this have to do with tuning? To answer David directly, and you too, Paul: this is musically relevant because I've suggested a collaboration on a 'solar system' piece here, and in fact as I was writing this post, it got so long and involved, and inspired me so strongly to work on the piece I've been suggesting, that I just decided to disconnect from the internet and work on the piece myself. I stayed up all night to write this post, and here's the piece: http://www.egroups.com/files/tuning/monz/solarsystem/solar.gif The basic orbital data ---------------------- Orbital periods of planets (in Earth-units): days years Mercury 87.97 0.24 Venus 224.70 0.62 Earth 365.26 1.00 Mars 686.98 1.88 Jupiter 4332.71 11.86 Saturn 10759.50 29.46 Uranus 30685.00 84.00 Neptune 60190.00 164.80 Pluto 90800 247.70 Ratios of orbital periods of the solar system: Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Mercury 1.000 2.554 4.152 7.809 49.24 122.3 348.8 684.3 1028 Venus 0.391 1.000 1.626 3.057 19.28 47.89 136.5 267.9 402.6 Earth 0.241 0.615 1.000 1.881 11.86 29.46 84.00 164.8 247.7 Mars 0.128 0.327 0.532 1.000 6.306 15.66 44.66 87.62 131.7 Jupiter 0.020 0.052 0.084 0.159 1.000 2.484 7.083 13.90 20.89 Saturn 0.008 0.021 0.034 0.064 0.403 1.000 2.851 5.594 8.408 Uranus 0.003 0.007 0.012 0.022 0.141 0.351 1.000 1.962 2.949 Neptune 0.001 0.004 0.006 0.011 0.072 0.179 0.510 1.000 1.503 Pluto 0.001 0.002 0.004 0.008 0.048 0.119 0.339 0.665 1.000 Notice anything familiar? Why, it's nothing other than an Interval Matrix! We're used to seeing tables like this filled with fractional ratios (one need go no farther than Partch's _Genesis of a Music_ for an example); this is the same thing but in decimal format. Given the fact that the planetary distances increase logarithmically, just like our perception of musical ratios, I have given all numbers to the same number of significant digits - in this case, 4 digits, so that it would fit here. 6 significant digits would provide pretty reasonable accuracy, and can be calculated from the table above this one. Each row in the chart shows orbital periods from the point of view of an observer on the planet listed on the left, in terms of that planet's 'years' or fractions thereof. Each column shows how many times the planets revolve around the Sun in terms of that planet's 'year'. So the numbers less than 1 are the reciprocals of the corresponding relationship given by the number greater than 1. For example, Jupiter revolves around the Sun in 11.86 Earth-years, and the Earth revolves around the Sun in 1/11.86 = ~0.08 Jupiter-years. Observe how 1.00 (the 1/1) flows diagonally down the center of the chart. In mapping the orbital frequencies as sounds, we can show all 36 planetary relationships as either an otonality or a utonality. The otonality representation utilizes all the numbers larger than 1, and the utonality all the numbers less than 1. My 'Solar System' Piece ----------------------- This piece is going to be a work in progress. I'm starting it off with a simple demonstration of the orbital periods of the 9 planets as a sustained 9-note chord. I use a reed-organ (harmonium) timbre, so as to be free from vibrato; the beating you hear is caused by slight differences in pitch. The obvious starting point was to take the frequency values from the 'Pluto' column, since they give a nice easy 1/1 at the bottom, and it made sense to me to have the slowest planet give the lowest note. So I used Pluto's orbital period as the 1/1 and all the closer planets as the higher pitches of the chord. (I'm using the greatest pitch-bend resolution, 4096 units per semitone.) '8ve' + Semitones = MIDI-note + pitch-bend Mercury 10 0.08 C 10 + 309 Venus 8 7.84 G# 8 - 653 Earth 7 11.43 B 7 +1759 Mars 7 0.49 C 7 +2020 Jupiter 4 4.61 F 4 -1585 Saturn 3 0.86 Db 3 - 569 Uranus 1 6.72 G 1 -1140 Neptune 0 7.05 G 0 + 223 Pluto 0 0.00 C 0 0 Thru my speakers, Pluto is inaudible, Neptune can just barely be heard, Uranus is a low growl, and Mercury can just barely be heard (it may be beyond the MIDI-note range). The greater-than-10-'8ve' ratio between Mercury and Pluto forced me to map each of them at the respective limits of the human audible frequency range. What's most surprising is that the overall sound rings out quite clearly as something resembling a major chord! Mercury, Earth, Mars, and Saturn all fall fairly close to '8ves' of Pluto's 1/1 (if you isolate Earth and Saturn they pretty much give a 'major 7th'); Venus, Uranus, and Neptune all give something close to the '8ves' of the 3rd harmonic, and Jupiter sounds like an '8ve' of the 5th harmonic. (see the '8ve'-reduced pitch graph) http://www.egroups.com/files/tuning/monz/solarsystem/solar.gif Using Pluto as 1/1 is rather arbitrary. It would make much more sense to use the Sun's period of rotation (25.38 Earth-days) as a 'guide tone' (in Fokker's sense), and calculate all the planet's periods as subharmonics of that: Sun 1.0 Mercury 0.288507446 Venus 0.112950601 Earth 0.069484751 Mars 0.036944307 Jupiter 0.005857766 Saturn 0.002358846 Uranus 0.000827114 Neptune 0.000421665 Pluto 0.00028052 Since the relative orbital periods are still the same, if the Sun is omitted the chord would have the same sound as the one I used, because the ratio between the Sun and Pluto is impossible to represent within the human audible range - either one of them must be given a note beyond the audible frequency limit to make all the other planets audible. Ideas for extending the piece ----------------------------- I'd also like to include the planetary distance data in the piece somehow, but since the orbital distances *are* periodic frequencies, it makes sense to map *them* as musical pitch frequencies. After a similar experiment, but one which mapped the planetary *distances* as pitch, David Finnamore agreed that orbital period made more sense: http://www.egroups.com/message/tuning/11894 Perhaps the planet's rate of rotation can be mapped as meter, with the frequency sounding at the rate of one note per day (of that particular planet's timescale, based on its rotation), and the precise distance from the sun can be mapped as some sort of timbral change. ...? So it would be a La Monte Young-type piece consisting of a droning chord whose notes have periodic timbral shifts. Also, one of the parameters should be mapped to amplitude. With a little more imagination, there are lots of other data that could be mapped, such as orbital inclination (above or below the Sun's equator) and obliquity (the planet's rotational tilt from its orbital plane). To make things really complicated, one could throw in stuff like precession and nutation. The list goes on and on... everything in the solar system (the universe?) that occurs over long periods of time, repeats at periodic frequencies. It would be really interesting to include the data about the satellites, asteroids, and comets too... and how about planetary impacts? Wow, what a fantastic sound! Talk about 'music of the spheres'! This may turn out to be a really useful endeavor, too: I've mentioned before how mapping data aurally can allow us to perceive patterns that we might not notice visually. Perhaps listening to an accurate aural mapping of the solar system's frequencies would allow us to hear patterns that would enable us to predict such important things as major asteroid or comet impacts on Earth. Addendum -------- Here is the interval matrix again, converted into semitones, for easy comparison with common low-integer musical ratios. Note the 841 cents (almost exactly a 13:8) of Venus/Earth, and the 705 cents (extremely close to 3:2) of Neptune/Pluto. Other relationships between adjacent planets which are close to musical ratios are Mercury/Venus (fairly close to 18:7, the '8ve' of the 9:7 'septimal 3rd'), Earth/Mars (just ~6 cents wider than the 15:8 'major 7th'), and Jupiter/Saturn (~11 narrower than 5:2, the '8ve' of the 5:4 'major 3rd'). SEMITONES Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Mercury 0.00 16.23 24.65 35.58 67.46 83.21 101.35 113.02 120.08 Venus - 16.23 0.00 8.41 19.35 51.23 66.98 85.12 96.79 103.84 Earth - 24.65 - 8.41 0.00 10.94 42.82 58.57 76.71 88.37 95.43 Mars - 35.58 - 19.35 -10.94 0.00 31.88 47.63 65.77 77.44 84.49 Jupiter - 67.46 - 51.23 -42.82 -31.88 0.00 15.75 33.89 45.56 52.61 Saturn - 83.21 - 66.98 -58.57 -47.63 -15.75 0.00 18.14 29.81 36.86 Uranus -101.35 - 85.12 -76.71 -65.77 -33.89 -18.14 0.00 11.67 18.72 Neptune -113.02 - 96.79 -88.37 -77.44 -45.56 -29.81 -11.67 0.00 7.05 Pluto -120.08 -103.84 -95.43 -84.49 -52.61 -36.86 -18.72 -7.05 0.00 For those who want to know some other planetary orbital data: (time is given in Earth-units; incl.=inclination, ecc.=eccentricity; obliquity and inclination measured in degrees) rotation obliquity incl. ecc. hours days Mercury 1403.49 58.65 0 7 .206 Venus 5814.99 243.0 178 3.39 .007 Earth 23.93 1.000 23.4 0.00 .017 Mars 24.55 1.026 25 1.85 .093 Jupiter 9.8 0.410 3.08 1.31 .048 Saturn 10.2 0.426 26.7 2.49 .056 Uranus 17.9 0.748 97.9 0.77 .047 Neptune 19.1 0.798 29.6 1.77 .009 Pluto 152.91 6.39 122.5 17.15 .248 Notes: Because Venus, Uranus, and Pluto are tilted more than 90 degrees, they seem to us to be rotating backwards, with apparent obliquities of 2, 82.1, and 57.5 degrees respectively. In either case, Uranus appears tilted onto its side. Charts showing percentage error from ratios ------------------------------------------- The percentage of error I obtained is greatly dependent on the integer-limit I chose for my rational denominators, which was 14. Including larger denominators, particularly primes, would result in some planetary relationships coming much closer to a low-integer ratio. I leave this as a project for the reader... Here is a table showing the smallest errors of rational approximations from the actual orbital periods, based on my 14-integer-denominator-limit, and ranked by amount of error from least to greatest. I chose 2.18% error as the arbitrary cut-off point. Note both the very low integers of the ratio as well as the tiny error for the Venus/Earth synchronicity, and the small errors from the approximations of 2:3 for Neptune/Pluto and 2:5 for Jupiter/Saturn. Also, even tho it's not a particularly small number, the Earth/Uranus ratio is so close to 1:84 that multiplying it by every integer thru 14 did not change the error from 1:84. Planets ratio % error uranus/earth 84:1 0.00 uranus/jupiter 85:12 0.01 saturn/mars 47:3 0.02 jupiter/mars 82:13 0.03 venus/earth 8:13 0.03 earth/mercury 54:13 0.04 venus/mercury 23:9 0.05 neptune/saturn 28:5 0.11 mars/mercury 86:11 0.11 uranus/jupiter 78:11 0.12 neptune/pluto 2:3 0.20 uranus/saturn 20:7 0.20 saturn/venus 48:1 0.23 pluto/saturn 59:7 0.24 uranus/jupiter 71:10 0.25 jupiter/venus 58:3 0.28 mars/mercury 47:6 0.31 earth/mercury 25:6 0.35 neptune/saturn 73:13 0.38 uranus/jupiter 64:9 0.40 mars/earth 17:9 0.43 neptune/mars 88:1 0.43 pluto/saturn 76:9 0.43 jupiter/mars 19:3 0.44 mars/venus 43:14 0.46 mars/mercury 102:13 0.47 pluto/jupiter 21:1 0.55 saturn/mars 63:4 0.55 neptune/saturn 45:8 0.55 uranus/jupiter 57:8 0.60 mars/mercury 55:7 0.61 mars/venus 40:13 0.64 saturn/jupiter 5:2 0.64 earth/venus 18:11 0.67 venus/mercury 18:7 0.67 earth/mercury 46:11 0.72 neptune/jupiter 14:1 0.75 neptune/saturn 62:11 0.76 uranus/mars 45:1 0.76 uranus/saturn 23:8 0.83 uranus/jupiter 50:7 0.85 mars/venus 37:12 0.85 mars/earth 19:10 1.02 earth/venus 23:14 1.06 pluto/saturn 17:2 1.09 mars/venus 34:11 1.10 venus/mercury 31:12 1.14 jupiter/venus 39:2 1.15 earth/mercury 21:5 1.15 jupiter/uranus 1:7 1.18 jupiter/earth 12:1 1.18 earth/mars 7:13 1.27 neptune/saturn 17:3 1.30 uranus/saturn 26:9 1.32 mars/venus 31:10 1.40 jupiter/mars 32:5 1.49 mars/earth 21:11 1.50 venus/earth 5:8 1.60 uranus/saturn 29:10 1.71 saturn/neptune 2:11 1.71 pluto/uranus 3:1 1.74 mars/venus 28:9 1.76 venus/mercury 13:5 1.79 saturn/uranus 5:14 1.83 mars/earth 23:12 1.91 venus/mars 1:3 1.91 neptune/uranus 2:1 1.94 saturn/mars 16:1 2.15 mercury/venus 2:5 2.17 ------ And here is a more comprehensive table showing all the significant low-integer approximations to the orbital periods, within my 14-integer-denominator-limit. Relationships are ranked by planetary order from the Sun. Comparisons between widely-separated planets (i.e., between any of the 'gas giants' and any of the smaller inner planets) are going to be very close to integer ratios, because one of the numbers is already extremely large; in most of these cases I have given only two ratios: the most accurate (within my 14-limit) and the smallest-integer. Planets ratio % error mercury/venus 5:13 -1.76 mercury/venus 2:5 2.17 mercury/venus 3:8 -4.21 mercury/earth 1:4 3.80 mercury/earth 3:13 -4.18 mercury/earth 2:9 -7.73 mercury/mars 1:8 -2.38 venus/mercury 23:9 0.05 venus/mercury 28:11 -0.35 venus/mercury 33:13 -0.62 venus/mercury 18:7 0.67 venus/mercury 31:12 1.14 venus/mercury 13:5 1.79 venus/mercury 26:10 1.79 venus/mercury 5:2 -2.13 venus/mercury 8:3 4.40 venus/earth 8:13 0.03 venus/earth 5:8 1.60 venus/earth 3:5 -2.47 venus/earth 7:11 3.44 venus/earth 9:14 4.50 venus/mars 1:3 1.91 venus/jupiter 1:14 37.71 earth/mercury 54:13 0.04 earth/mercury 29:7 -0.22 earth/mercury 25:6 0.35 earth/mercury 33:8 -0.65 earth/mercury 46:11 0.72 earth/mercury 37:9 -0.99 earth/mercury 21:5 1.15 earth/mercury 17:4 2.36 earth/mercury 4:1 -3.66 earth/venus 13:8 -0.03 earth/venus 21:13 -0.63 earth/venus 18:11 0.67 earth/venus 23:14 1.06 earth/venus 8:5 -1.57 earth/venus 5:3 2.53 earth/venus 11:7 -3.33 earth/venus 7:4 7.66 earth/venus 3:2 -7.72 earth/mars 7:13 1.27 earth/mars 6:11 2.59 earth/mars 5:9 4.49 earth/mars 1:2 -5.96 earth/mars 4:7 7.47 earth/jupiter 1:12 -1.17 earth/jupiter 1:11 7.82 earth/jupiter 1:13 -8.77 mars/mercury 86:11 0.11 mars/mercury 39:5 -0.12 mars/mercury 109:14 -0.30 mars/mercury 47:6 0.31 mars/mercury 70:9 -0.40 mars/mercury 102:13 0.47 mars/mercury 55:7 0.61 mars/mercury 31:4 -0.76 mars/mercury 23:3 -1.83 mars/mercury 8:1 2.44 mars/venus 43:14 0.46 mars/venus 40:13 0.64 mars/venus 37:12 0.85 mars/venus 34:11 1.10 mars/venus 31:10 1.40 mars/venus 28:9 1.76 mars/venus 3:1 -1.87 mars/earth 15:8 -0.31 mars/earth 17:9 0.43 mars/earth 19:10 1.02 mars/earth 13:7 -1.26 mars/earth 21:11 1.50 mars/earth 24:13 -1.84 mars/earth 23:12 1.91 mars/earth 11:6 -2.52 mars/earth 9:5 -4.30 mars/jupiter 2:13 -2.99 mars/jupiter 1:6 5.10 mars/jupiter 1:7 -9.92 mars/saturn 1:14 11.88 jupiter/mercury 197:4 0.01 jupiter/mercury 49:1 -0.50 jupiter/venus 212:11 -0.03 jupiter/venus 135:7 0.04 jupiter/venus 19:1 -1.45 jupiter/earth 83:7 -0.02 jupiter/earth 154:13 -0.12 jupiter/earth 95:8 0.13 jupiter/earth 71:6 -0.22 jupiter/earth 107:9 0.24 jupiter/earth 119:10 0.34 jupiter/earth 130:11 -0.35 jupiter/earth 59:5 -0.51 jupiter/earth 47:4 -0.93 jupiter/earth 12:1 1.18 jupiter/mars 82:13 0.03 jupiter/mars 63:10 -0.09 jupiter/mars 44:7 -0.32 jupiter/mars 19:3 0.44 jupiter/mars 69:11 -0.53 jupiter/mars 25:4 -0.89 jupiter/mars 32:5 1.49 jupiter/mars 13:2 3.08 jupiter/mars 6:1 -4.85 jupiter/saturn 2:5 -0.64 jupiter/saturn 5:12 3.50 jupiter/saturn 5:13 -4.46 jupiter/uranus 1:7 1.18 jupiter/uranus 2:13 8.96 jupiter/neptune 1:14 -0.75 jupiter/neptune 1:13 6.89 jupiter/pluto 1:14 49.18 saturn/mercury 367:3 0.01 saturn/mercury 489:4 -0.06 saturn/mercury 245:2 0.15 saturn/mercury 122:1 -0.26 saturn/venus 431:9 0.00 saturn/venus 48:1 0.23 saturn/earth 383:13 0.01 saturn/earth 324:11 -0.02 saturn/earth 265:9 -0.05 saturn/earth 206:7 -0.11 saturn/earth 59:2 0.14 saturn/earth 147:5 -0.20 saturn/earth 88:3 -0.43 saturn/earth 29:1 -1.56 saturn/mars 47:3 0.02 saturn/mars 219:14 -0.13 saturn/mars 172:11 -0.17 saturn/mars 204:13 0.18 saturn/mars 157:10 0.23 saturn/mars 125:8 -0.25 saturn/mars 110:7 0.32 saturn/mars 78:5 -0.41 saturn/mars 63:4 0.55 saturn/mars 31:2 -1.04 saturn/mars 16:1 2.15 saturn/jupiter 5:2 0.64 saturn/jupiter 32:13 -0.90 saturn/jupiter 27:11 -1.18 saturn/jupiter 22:9 -1.59 saturn/jupiter 17:7 -2.23 saturn/jupiter 12:5 -3.38 saturn/jupiter 7:3 -6.06 saturn/uranus 5:14 1.83 saturn/uranus 4:11 3.68 saturn/uranus 1:3 -4.96 saturn/uranus 3:8 6.92 saturn/neptune 2:11 1.71 saturn/neptune 1:6 -6.77 saturn/neptune 1:5 11.88 saturn/pluto 1:9 -6.58 saturn/pluto 1:8 5.10 uranus/mercury 1395:4 -0.01 uranus/mercury 1046:3 -0.03 uranus/mercury 349:1 0.06 uranus/venus 1502:11 0.00 uranus/venus 273:2 -0.03 uranus/venus 137:1 0.33 uranus/earth 84:1 0.00 uranus/mars 134:3 0.01 uranus/mars 357:8 -0.08 uranus/mars 223:5 -0.14 uranus/mars 89:2 -0.36 uranus/mars 45:1 0.76 uranus/jupiter 85:12 0.01 uranus/jupiter 92:13 -0.08 uranus/jupiter 78:11 0.12 uranus/jupiter 99:14 -0.16 uranus/jupiter 71:10 0.25 uranus/jupiter 64:9 0.40 uranus/jupiter 57:8 0.60 uranus/jupiter 50:7 0.85 uranus/jupiter 7:1 -1.17 uranus/saturn 37:13 -0.18 uranus/saturn 20:7 0.20 uranus/saturn 17:6 -0.63 uranus/saturn 23:8 0.83 uranus/saturn 31:11 -1.16 uranus/saturn 26:9 1.32 uranus/saturn 29:10 1.71 uranus/saturn 14:5 -1.80 uranus/saturn 11:4 -3.55 uranus/saturn 3:1 5.21 uranus/neptune 1:2 -1.90 uranus/neptune 7:13 5.64 uranus/pluto 1:3 -1.71 uranus/pluto 5:14 5.31 uranus/pluto 4:11 7.23 neptune/merc. 2737:4 0.00 neptune/mercury 684:1 -0.04 neptune/venus 2411:9 0.00 neptune/venus 2679:10 0.00 neptune/venus 268:1 0.04 neptune/earth 824:5 0.00 neptune/earth 165:1 0.12 neptune/mars 701:8 0.00 neptune/mars 263:3 0.05 neptune/mars 175:2 -0.14 neptune/mars 88:1 0.43 neptune/jupiter 139:10 0.03 neptune/jupiter 125:9 -0.05 neptune/jupiter 14:1 0.75 neptune/saturn 28:5 0.11 neptune/saturn 67:12 -0.19 neptune/saturn 73:13 0.38 neptune/saturn 39:7 -0.40 neptune/saturn 45:8 0.55 neptune/saturn 50:9 -0.69 neptune/saturn 62:11 0.76 neptune/saturn 17:3 1.30 neptune/saturn 11:2 -1.68 neptune/saturn 6:1 7.26 neptune/uranus 27:14 -1.70 neptune/uranus 2:1 1.94 neptune/pluto 2:3 0.20 neptune/pluto 9:14 -3.38 neptune/pluto 9:13 4.06 neptune/pluto 7:11 -4.35 pluto/mercury 2057:2 0.00 pluto/mercury 1028:1 -0.05 pluto/venus 805:2 -0.04 pluto/venus 2013:5 -0.01 pluto/venus 1208:3 0.00 pluto/venus 403:1 0.09 pluto/earth 2477:10 0.00 pluto/earth 248:1 0.12 pluto/mars 1317:10 0.00 pluto/mars 395:3 -0.02 pluto/mars 132:1 0.23 pluto/jupiter 188:9 0.02 pluto/jupiter 167:8 -0.05 pluto/jupiter 125:6 -0.25 pluto/jupiter 104:5 -0.41 pluto/jupiter 21:1 0.55 pluto/saturn 42:5 -0.10 pluto/saturn 101:12 0.10 pluto/saturn 59:7 0.24 pluto/saturn 67:8 -0.39 pluto/saturn 76:9 0.43 pluto/saturn 92:11 -0.53 pluto/saturn 17:2 1.09 pluto/saturn 25:3 -0.89 pluto/saturn 8:1 -4.85 pluto/uranus 41:14 -0.69 pluto/uranus 38:13 -0.87 pluto/uranus 35:12 -1.09 pluto/uranus 32:11 -1.35 pluto/uranus 29:10 -1.66 pluto/uranus 3:1 1.74 pluto/neptune 3:2 -0.20 -monz http://www.ixpres.com/interval/monzo/homepage.html