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Message: 11325 - Contents - Hide Contents Date: Wed, 07 Jul 2004 05:40:29 Subject: Re: Ptolemy's genera (was: from linear to equal) From: monz hi Paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Hi Monz, > > Why is it that you're always creating new webpages and ignoring > corrections to your old ones? This seems to be a pattern with you.believe me, i've been making absurd numbers of corrections. also, i really try to put a webpage together on the spot if i have a few hours available when the inspiration strikes. this last one is a good example of that. i've done a lot of research into Ptolemy's tuning treatise, and have still presented only a half-baked version of my work on it in my book. i got the idea to collect this data on a webpage about Ptolemy, and as the years pass i'll be stuffing it full of ridiculous amounts of tables, graphs, lattices, and long rambling text connecting Ptolemy with the Sumerians and modern San Diego new-age UFOlogists. :) for now, those two simple little tables are but the seed from which a large fruitful tree will grow. i really am sorry about that. my computer was infected with a virus and now i have extra work to do just to keep webpages working properly and try to remember the stuff i lost that wasn't backed up. rest assured, soon enough i'll be asking you (and everyone else) to please itemize every error and broken link that can be found on the tonalsoft site. but i have a huge load of work to do for about the next month. -monz
Message: 11326 - Contents - Hide Contents Date: Wed, 07 Jul 2004 05:44:05 Subject: Re: Ptolemy and leapday From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> well, of course neither Ptolemy nor Partch advocated >> equal-temperament ... but if it's any help, these are >> three xenharmonic-bridges that i've posited for Ptolemy >> (in my book): >> >> monzo ratio ~cents >> 3 5 7 11 23 >> >> [-4, 0, 1,-1, 0 > 896 / 891 9.687960643 >> [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203 >> [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439 >> If these three are related, so they define an 11-limit > planar temperament. If you add 3388/3375 to this, you get > Graham's mystery, with a 1/29 period; if you add 385/384, > rodan; if 243/242 an 11-limit hemififths; and if 100/99, > an 11-limit version of garibaldi (schismic family) which > I have listed as "garybald". > > If we add 121/120 we get the 11-limit reduction of what > Herman dubbed "leapday" in the 13 limit. The TOP tunings > are not identical but they are close, and I suggest giving > them the same name, and perhaps "ptolemy" could be that name. > The 11 and 13 limit temperaments also have a common poptimal > generator of 19/46, which again supports giving them the > same name, and could be another naming idea along the lines > of 19/84 I suppose. The corresponding fifth is 27/46; > 2.39 cents sharp. I figured a Hellenistic Greek might like > a fifth as a generator, if introduced to temperament; but > more importantly, this (and "garybald"), is a "brigable" > temperament since it has a 1 as the first wedgie element. > The xenharmonic bridge to 5 is |31 -21 1>.awesome, Gene !!! thanks !!! this is exactly the kind of thing i was hoping for when i put out my bait. :) seriously, i'm having so much trouble following almost everything you post here ... by relating some of your work to some of my own (which is obviously nice and familiar to me), it helps me at least get an idea where this train is going. -monz
Message: 11327 - Contents - Hide Contents Date: Wed, 07 Jul 2004 05:47:37 Subject: Re: Ptolemy and leapday From: monz hi Paul and Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> The TOP tunings are not identical but they are close, >> and I suggest giving them the same name, and perhaps >> "ptolemy" could be that name. >> Note that Monz is only listing a small percentage of the > "ptolemy commas" that he found.*real* small percentage. completing these tables is the first thing i'm going to add to this page, so stay tuned. (he he)> You guys are playing real fast and loose! > (Don't take that as a complaint.)i'm having fun at a party i probably shouldn't have even crashed. -monz
Message: 11328 - Contents - Hide Contents Date: Wed, 07 Jul 2004 05:59:10 Subject: Re: Beethoven's Appassionata comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> How am I ever going to make it through 500+ pages though?I'll find out for you. I have in on interlibrary loan.
Message: 11329 - Contents - Hide Contents
Date: Wed, 07 Jul 2004 09:04:17
Subject: Kwai
From: Gene Ward Smith
"Kwai" because it is bridgable. The 7 and 11 limit have the same TOP
tuning, and 5 is close, so I would give them the same name.
11-limit Kwai {540/539, 1375/1372, 5120/5103}
poptimal generator 291/497
[1, 33, 27, -18, 50, 40, -32, -30, -156, -144]
[[1, 2, 16, 14, -4], [0, -1, -33, -27, 18]]
7-limit Kwai
Number 53 Kwai {5120/5103, 16875/16807}
7&9 copop fifth generator 493/842
[1, 33, 27, 50, 40, -30]
[[1, 2, 16, 14], [0, -1, -33, -27]]
TOP tuning [1199.680495, 1902.108988, 2785.571846, 3369.722869]
TOP generators [1199.680495, 497.2520023]
bad: 64.536886 comp: 14.212326 err: .319505
MOS 7, 12, 17, 29, 41, 70
The 41 note MOS seems logical. 152, in its universality, handles it
well.
Message: 11330 - Contents - Hide Contents Date: Wed, 07 Jul 2004 11:02:58 Subject: Re: Beethoven's Appassionata comma From: Graham Breed jjensen142000 wrote:> I *almost* got that book today from the music library... I had > the call number on a slip of paper in my pocket and everything, > but I was just too busy :( > > How am I ever going to make it through 500+ pages though?My copy covers Appassionata on p.349, and not in the detail Paul describes. Could he have a different edition? Amazon doesn't mention it. Oh, has everybody seen Eytan Agmon's Scarlatti analysis? Graham
Message: 11331 - Contents - Hide Contents Date: Wed, 07 Jul 2004 18:10:39 Subject: Re: Beethoven's Appassionata comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Oh, has everybody seen Eytan Agmon's Scarlatti analysis?No, where is that?
Message: 11332 - Contents - Hide Contents Date: Wed, 07 Jul 2004 19:47:01 Subject: Re: Beethoven's Appassionata comma From: Graham Breed Gene Ward Smith wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: > >>> Oh, has everybody seen Eytan Agmon's Scarlatti analysis? > >> No, where is that?In Theory Only, vol. 11, no. 5, pp.1-8. It's not as interesting as it looks from the title (Equal Division of the Octave in a Scarlatti Sonata) because the equal divisions are only 12 to the octave :( But it's an interesting example of an octatonic scale and enharmonic modulation from the 18th Century. Graham ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11333 - Contents - Hide Contents
Date: Thu, 08 Jul 2004 19:22:21
Subject: Re: 3-d ("planar") temperaments request
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>
>> I'd like to include a table in my paper which summarizes a bunch
of
> 7-
>> limit, codimension-1 temperaments.
>
> Fortunately for you, I now have a new computer which doesn't crash
> all the time. What's the time frame here?
ASAP.
>> As you'd guess, for each, I want a set of three generators, one
of
>> which generates 2:1 all by itself. And then the mappings from
these
>> generators to primes.
>
> The easiest approach is simply to take the Hermite reduction of a
set
> of generators for the vals. This would mean if 2 is a generator, it
> will give it, otherwise a fraction of an octave. If 2, 3, and 5
will
> work, it will always give that; and so forth if you wanted a
complete
> description of the decision proceedure. This would be easiest for
me,
> and it seems to me it has a good claim to be the best choice.
Sure . . . just make sure that you do give a complete description of
the decision procedure, because I do want one.
> Hermite reduction would result in a criterion you could explain.
Tell
> me if that would be acceptable.
Sure -- as long as the explanation ends up being something I can
understand, then I should be able to explain it in the paper.
Message: 11334 - Contents - Hide Contents
Date: Thu, 08 Jul 2004 19:25:03
Subject: Re: 3-d ("planar") temperaments request
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>> And then the mappings from these
>> generators to primes.
> By "give the generators", do you mean a TOP tuning plus a mapping?
Right, TOP tuning, and as I said above, the mappings from these
generators to (tempered) primes. I guess I could figure it out from
your RMS values, since TOP tuning for the (tempered) primes is easy
to calculate in these cases, and then I can just solve the system of
equations given by your mappings. But it would take me some time . . .
Message: 11335 - Contents - Hide Contents Date: Thu, 08 Jul 2004 23:43:48 Subject: Re: monz back to math school From: Carl Lumma>i've finally decided to enroll in school again >to study the math that i'm sorely lacking. > >i know that ultimately i want to take a course >in linear algebra, and that Grassmann algebra in >particular is something i want to be familiar with. > >the college catalog lists trigonometry and calculus >as prerequisites for linear algebra, so this is going >to be a long haul if i can stick with it. i nearly >bombed out of algebra II in high school, and never >studied math again after that ... except the bits >and pieces i picked up as a tuning theorist. > >i feel like i'm missing out on too much new discovery >here on tuning-math, and want to get up to speed. > >i haven't selected any particular classes yet. >i will certainly have to start with regular algebra >all over again, and will probably only be able to take >one course per semester. advice is appreciated.Congratulations, monz!! Best of luck. -Carl
Message: 11336 - Contents - Hide Contents
Date: Thu, 08 Jul 2004 00:42:55
Subject: Re: 3-d ("planar") temperaments request
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> I'd like to include a table in my paper which summarizes a bunch of
7-
> limit, codimension-1 temperaments.
Fortunately for you, I now have a new computer which doesn't crash
all the time. What's the time frame here?
> As you'd guess, for each, I want a set of three generators, one of
> which generates 2:1 all by itself. And then the mappings from these
> generators to primes.
The easiest approach is simply to take the Hermite reduction of a set
of generators for the vals. This would mean if 2 is a generator, it
will give it, otherwise a fraction of an octave. If 2, 3, and 5 will
work, it will always give that; and so forth if you wanted a complete
description of the decision proceedure. This would be easiest for me,
and it seems to me it has a good claim to be the best choice.
By "give the generators", do you mean a TOP tuning plus a mapping?
> The criteria for choosing the generators should be something I can
> explain, like, "the second generator is constrained to be narrower
> than the first generator, and given that constraint, is chosen so
as
> to map to the simplest ratio possible. The third generator is
> constrained to be smaller than the second generator, and given that
> constraint, is chosen so as to map to the simplest ratio possible."
Hermite reduction would result in a criterion you could explain. Tell
me if that would be acceptable.
Message: 11337 - Contents - Hide Contents
Date: Thu, 08 Jul 2004 00:59:06
Subject: Re: 3-d ("planar") temperaments request
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> Hi Gene,
>
> I'd like to include a table in my paper which summarizes a bunch of
7-
> limit, codimension-1 temperaments.
Here's what I can do with programs I've already written; I give the
comma, the Hermite reduced mapping, and the rms tuning of the
corresponding generators.
28/27
[[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]]
[1200., 1924.441037, 2795.308127]
36/35
[[1, 0, 0, 2], [0, 1, 0, 2], [0, 0, 1, -1]]
[1200., 1885.698206, 2794.442111]
49/48
[[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]]
[1200., 950.9775006, 2780.364245]
50/49
[[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]]
[600.0000000, 1901.955001, 2777.569810]
64/63
[[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]]
[1200., 1911.692177, 2792.156018]
81/80
[[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]]
[1200., 1896.164845, 3366.344411]
126/125
[[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]
[1200., 1899.984322, 2789.269735]
128/125
[[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]]
[400.0000000, 1908.798145, 3375.669050]
225/224
[[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]]
[1200., 1899.812912, 2784.171625]
245/243
[[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]]
[1200., 1904.876579, 440.9272508]
250/243
[[1, 2, 3, 0], [0, 3, 5, 0], [0, 0, 0, 1]]
[1200., -162.9960265, 3371.413597]
256/245
[[1, 0, 0, 4], [0, 1, 0, 0], [0, 0, 2, -1]]
[1200., 1918.248103, 1404.018924]
405/392
[[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]]
[1200., 1888.775891, 789.3913963]
525/512
[[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]]
[1200., 1892.089471, 2774.475077]
648/625
[[4, 0, 3, 0], [0, 1, 1, 0], [0, 0, 0, 1]]
[300.0000000, 1894.134355, 3368.825906]
686/675
[[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]]
[1200., 1907.336788, 130.2063806]
729/700
[[1, 0, 0, -2], [0, 1, 0, 6], [0, 0, 1, -2]]
[1200., 1889.305200, 2784.908179]
875/864
[[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]]
[1200., 1904.145204, 2781.933305]
1029/1000
[[1, 0, 0, 1], [0, 3, 0, -1], [0, 0, 1, 1]]
[1200., 632.3352811, 2791.262872]
1029/1024
[[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]]
[1200., 233.4444415, 2785.016372]
1323/1280
[[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]]
[1200., 1888.607610, 445.9928964]
1728/1715
[[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]]
[1200., 1900.647644, 929.2070233]
2048/2025
[[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]]
[600.0000000, 1905.446531, 3370.920823]
2240/2187
[[1, 0, 0, -6], [0, 1, 0, 7], [0, 0, 1, -1]]
[1200., 1908.500466, 2788.495535]
2401/2400
[[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]]
[1200., 350.9775007, 617.6844971]
2430/2401
[[1, 0, 3, 1], [0, 1, 3, 2], [0, 0, 4, 1]]
[1200., 1898.792052, -1627.854498]
3125/3024
[[1, 0, 0, -4], [0, 1, 0, -3], [0, 0, 1, 5]]
[1200., 1905.114871, 2776.834104]
3125/3072
[[1, 0, 2, 0], [0, 5, 1, 0], [0, 0, 0, 1]]
[1200., 379.9679494, 3366.005567]
3125/3087
[[1, 0, 0, 0], [0, 1, 1, 1], [0, 0, 3, 5]]
[1200., 1903.069782, 293.4856589]
3136/3125
[[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]]
[1200., 1902.435257, 1393.797198]
3645/3584
[[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]]
[1200., 1897.216978, 2783.549864]
4000/3969
[[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]]
[1200., 1904.436192, 793.1568564]
4375/4374
[[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]]
[1200., 1902.005884, 2786.302405]
5103/5000
[[1, 0, 0, 3], [0, 1, 0, -6], [0, 0, 1, 4]]
[1200., 1896.830653, 2786.883085]
5120/5103
[[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]]
[1200., 1902.888698, 2786.702752]
5625/5488
[[1, 0, 1, 0], [0, 1, 1, 2], [0, 0, 3, 4]]
[1200., 1896.338273, -105.9626599]
6144/6125
[[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, 2, -3]]
[1200., 1902.491206, -157.4631742]
8748/8575
[[1, 0, 1, 0], [0, 1, 2, 1], [0, 0, 3, -2]]
[1200., 1897.239564, -736.0551389]
10976/10935
[[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, 3, 1]]
[1200., 1902.880572, -1139.661549]
15625/15552
[[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]]
[1200., 317.0796754, 3368.695142]
16875/16807
[[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, 5, 4]]
[1200., 1901.307749, -583.6772476]
19683/19600
[[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]]
[1200., 950.5282864, 2786.121192]
32805/32768
[[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]]
[1200., 1901.727514, 3368.705472]
Message: 11338 - Contents - Hide Contents Date: Thu, 08 Jul 2004 07:50:30 Subject: Bearings and bridges From: Gene Ward Smith Here's an old post relevant to the question of xenharmonic bridges: Yahoo groups: /tuning-math/message/4147 * [with cont.]
Message: 11339 - Contents - Hide Contents
Date: Thu, 08 Jul 2004 18:47:26
Subject: 2500/2187
From: Gene Ward Smith
Pondering "bearings", and thinking about creating 5-limit Fokker
blocks for 4375/4374 to temper, I was led to consider the (more or
less) comma 8/7 * 4375/4374. This is a curiosity, with mapping
[<1 2 3|, <0 -4 -7|] giving a temperament with a secor as generator
(7/72 being the first poptimal generator we find.)
Maybe 6561/6250 could be useful from a bearing point of view also,
not to mention 15/14 * 4375/4374, 250/243 = 36/35 * 4375/4374, or
20000/19683 = 64/63 * 4375/4374.
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Message: 11340 - Contents - Hide Contents Date: Fri, 09 Jul 2004 19:47:30 Subject: Re: 43 7-limit planar temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Below I give the comma, mapping, and TOP generators for Paul's list > of 43 planar temperaments. The generator and mapping result from a > modified Hermite reduction, the modification being to change signs > when needed to ensure the generators are all positive.Thanks so much, Gene. If you could describe in common, non-technical language the criteria you used to choose the set of generators, I'll be able to explain it in my paper.> > 28/27 > [[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]] > [1193.415676, 1912.390908, 2786.313714] > > 36/35 > [[1, 0, 0, 2], [0, 1, 0, 2], [0, 0, 1, -1]] > [1195.264647, 1894.449645, 2797.308862] > > 49/48 > [[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]] > [1203.187309, 953.5033827, 2786.313714] > > 50/49 > [[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]] > [598.4467109, 1901.955001, 2779.100463] > > 64/63 > [[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]] > [1197.723683, 1905.562879, 2786.313714] > > 81/80 > [[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]] > [1201.698520, 1899.262910, 3368.825906] > > 126/125 > [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]] > [1199.010636, 1900.386896, 2788.610946] > > 128/125 > [[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]] > [399.0200131, 1901.955001, 3368.825906] > > 225/224 > [[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]] > [1200.493660, 1901.172569, 2785.167472] > > 245/243 > [[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]] > [1200., 1903.372995, 440.4316973] > > 250/243 > [[1, 2, 3, 0], [0, -3, -5, 0], [0, 0, 0, 1]] > [1196.905960, 162.3176609, 3368.825906] > > 256/245 > [[1, 0, 0, 4], [0, 1, 0, 0], [0, 0, 2, -1]] > [1195.228951, 1901.955001, 1398.695873] > > 405/392 > [[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]] > [1203.269293, 1896.773294, 787.7266785] > > 525/512 > [[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]] > [1202.406737, 1898.140412, 2780.725442] > > 648/625 > [[4, 0, 3, 0], [0, 1, 1, 0], [0, 0, 0, 1]] > [299.1603149, 1896.631523, 3368.825906] > > 686/675 > [[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]] > [1198.513067, 1904.311735, 130.9133777] > > 729/700 > [[1, 0, 0, -2], [0, 1, 0, 6], [0, 0, 1, -2]] > [1203.706383, 1896.080523, 2794.919668] > > 875/864 > [[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]] > [1201.121570, 1903.732647, 2783.709509] > > 1029/1000 > [[1, 0, 0, 1], [0, 3, 0, -1], [0, 0, 1, 1]] > [1202.477948, 632.6758490, 2792.067330] > > 1029/1024 > [[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]] > [1200.421488, 233.6218235, 2786.313714] > > 1323/1280 > [[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]] > [1202.764567, 1897.573266, 447.5797863] > > 1728/1715 > [[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]] > [1199.391895, 1900.991178, 929.2418964] > > 2048/2025 > [[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]] > [599.5552941, 1903.364685, 3368.825906] > > 2240/2187 > [[1, 0, 0, -6], [0, 1, 0, 7], [0, 0, 1, -1]] > [1198.134693, 1904.911442, 2781.982606] > > 2401/2400 > [[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]] > [1200.032113, 350.9868928, 617.6846359] > > 2430/2401 > [[1, 0, 3, 1], [0, 1, 3, 2], [0, 0, -4, -1]] > [1199.075238, 1900.489288, 1628.631774] > > 3125/3024 > [[1, 0, 0, -4], [0, 1, 0, -3], [0, 0, 1, 5]] > [1202.454598, 1905.845447, 2780.614314] > > 3125/3072 > [[1, 0, 2, 0], [0, 5, 1, 0], [0, 0, 0, 1]] > [1201.276744, 380.7957184, 3368.825906] > > 3125/3087 > [[1, 0, 0, 0], [0, 1, 1, 1], [0, 0, 3, 5]] > [1200., 1903.401919, 293.5973664] > > 3136/3125 > [[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]] > [1199.738066, 1901.955001, 1393.460953] > > 3645/3584 > [[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]] > [1201.235997, 1899.995991, 2783.443817] > > 4000/3969 > [[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]] > [1199.436909, 1902.847479, 792.7846742] > > 4375/4374 > [[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]] > [1200.016360, 1901.980932, 2786.275726] > > 5103/5000 > [[1, 0, 0, 3], [0, 1, 0, -6], [0, 0, 1, 4]] > [1201.434720, 1899.681024, 2789.645030] > > 5120/5103 > [[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]] > [1199.766314, 1902.325384, 2785.771112] > > 5625/5488 > [[1, 0, 1, 0], [0, 1, 1, 2], [0, 0, -3, -4]] > [1201.715742, 1899.235615, 106.2071570] > > 6144/6125 > [[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, -2, 3]] > [1199.786928, 1901.617290, 157.2978838] > > 8748/8575 > [[1, 0, 1, 0], [0, 1, 2, 1], [0, 0, -3, 2]] > [1198.678173, 1899.859955, 736.3383942] > > 10976/10935 > [[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, -3, -1]] > [1199.758595, 1902.337618, 1139.106063] > > 15625/15552 > [[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]] > [1200.291038, 317.0693810, 3368.825906] > > 16875/16807 > [[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, -5, -4]] > [1200., 1901.560426, 583.7891213] > > 19683/19600 > [[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]] > [1200.256485, 950.7742412, 2786.909253] > > 32805/32768 > [[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]] > [1200.065120, 1901.851787, 3368.825906]
Message: 11341 - Contents - Hide Contents Date: Fri, 09 Jul 2004 19:49:20 Subject: Re: monz back to math school From: Paul Erlich Do it, Monz. Now that you have the motivation, you'll do much better than you did in high school. In fact, I have a feeling you'll ace these classes . . . --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hey guys, > > > i've finally decided to enroll in school again > to study the math that i'm sorely lacking. > > i know that ultimately i want to take a course > in linear algebra, and that Grassmann algebra in > particular is something i want to be familiar with. > > the college catalog lists trigonometry and calculus > as prerequisites for linear algebra, so this is going > to be a long haul if i can stick with it. i nearly > bombed out of algebra II in high school, and never > studied math again after that ... except the bits > and pieces i picked up as a tuning theorist. > > i feel like i'm missing out on too much new discovery > here on tuning-math, and want to get up to speed. > > i haven't selected any particular classes yet. > i will certainly have to start with regular algebra > all over again, and will probably only be able to take > one course per semester. advice is appreciated. > > > > > -monz
Message: 11342 - Contents - Hide Contents Date: Fri, 09 Jul 2004 04:34:54 Subject: 43 7-limit planar temperaments From: Gene Ward Smith Below I give the comma, mapping, and TOP generators for Paul's list of 43 planar temperaments. The generator and mapping result from a modified Hermite reduction, the modification being to change signs when needed to ensure the generators are all positive. 28/27 [[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]] [1193.415676, 1912.390908, 2786.313714] 36/35 [[1, 0, 0, 2], [0, 1, 0, 2], [0, 0, 1, -1]] [1195.264647, 1894.449645, 2797.308862] 49/48 [[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]] [1203.187309, 953.5033827, 2786.313714] 50/49 [[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]] [598.4467109, 1901.955001, 2779.100463] 64/63 [[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]] [1197.723683, 1905.562879, 2786.313714] 81/80 [[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]] [1201.698520, 1899.262910, 3368.825906] 126/125 [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]] [1199.010636, 1900.386896, 2788.610946] 128/125 [[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]] [399.0200131, 1901.955001, 3368.825906] 225/224 [[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]] [1200.493660, 1901.172569, 2785.167472] 245/243 [[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]] [1200., 1903.372995, 440.4316973] 250/243 [[1, 2, 3, 0], [0, -3, -5, 0], [0, 0, 0, 1]] [1196.905960, 162.3176609, 3368.825906] 256/245 [[1, 0, 0, 4], [0, 1, 0, 0], [0, 0, 2, -1]] [1195.228951, 1901.955001, 1398.695873] 405/392 [[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]] [1203.269293, 1896.773294, 787.7266785] 525/512 [[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]] [1202.406737, 1898.140412, 2780.725442] 648/625 [[4, 0, 3, 0], [0, 1, 1, 0], [0, 0, 0, 1]] [299.1603149, 1896.631523, 3368.825906] 686/675 [[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]] [1198.513067, 1904.311735, 130.9133777] 729/700 [[1, 0, 0, -2], [0, 1, 0, 6], [0, 0, 1, -2]] [1203.706383, 1896.080523, 2794.919668] 875/864 [[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]] [1201.121570, 1903.732647, 2783.709509] 1029/1000 [[1, 0, 0, 1], [0, 3, 0, -1], [0, 0, 1, 1]] [1202.477948, 632.6758490, 2792.067330] 1029/1024 [[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]] [1200.421488, 233.6218235, 2786.313714] 1323/1280 [[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]] [1202.764567, 1897.573266, 447.5797863] 1728/1715 [[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]] [1199.391895, 1900.991178, 929.2418964] 2048/2025 [[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]] [599.5552941, 1903.364685, 3368.825906] 2240/2187 [[1, 0, 0, -6], [0, 1, 0, 7], [0, 0, 1, -1]] [1198.134693, 1904.911442, 2781.982606] 2401/2400 [[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]] [1200.032113, 350.9868928, 617.6846359] 2430/2401 [[1, 0, 3, 1], [0, 1, 3, 2], [0, 0, -4, -1]] [1199.075238, 1900.489288, 1628.631774] 3125/3024 [[1, 0, 0, -4], [0, 1, 0, -3], [0, 0, 1, 5]] [1202.454598, 1905.845447, 2780.614314] 3125/3072 [[1, 0, 2, 0], [0, 5, 1, 0], [0, 0, 0, 1]] [1201.276744, 380.7957184, 3368.825906] 3125/3087 [[1, 0, 0, 0], [0, 1, 1, 1], [0, 0, 3, 5]] [1200., 1903.401919, 293.5973664] 3136/3125 [[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]] [1199.738066, 1901.955001, 1393.460953] 3645/3584 [[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]] [1201.235997, 1899.995991, 2783.443817] 4000/3969 [[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]] [1199.436909, 1902.847479, 792.7846742] 4375/4374 [[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]] [1200.016360, 1901.980932, 2786.275726] 5103/5000 [[1, 0, 0, 3], [0, 1, 0, -6], [0, 0, 1, 4]] [1201.434720, 1899.681024, 2789.645030] 5120/5103 [[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]] [1199.766314, 1902.325384, 2785.771112] 5625/5488 [[1, 0, 1, 0], [0, 1, 1, 2], [0, 0, -3, -4]] [1201.715742, 1899.235615, 106.2071570] 6144/6125 [[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, -2, 3]] [1199.786928, 1901.617290, 157.2978838] 8748/8575 [[1, 0, 1, 0], [0, 1, 2, 1], [0, 0, -3, 2]] [1198.678173, 1899.859955, 736.3383942] 10976/10935 [[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, -3, -1]] [1199.758595, 1902.337618, 1139.106063] 15625/15552 [[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]] [1200.291038, 317.0693810, 3368.825906] 16875/16807 [[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, -5, -4]] [1200., 1901.560426, 583.7891213] 19683/19600 [[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]] [1200.256485, 950.7742412, 2786.909253] 32805/32768 [[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]] [1200.065120, 1901.851787, 3368.825906]
Message: 11343 - Contents - Hide Contents Date: Fri, 09 Jul 2004 19:53:08 Subject: Re: Joining Post From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "M Gould" <mark.gould@a...> wrote:> Hi all, > > some of you know me from tuning list and Make Micro Music. Just to say, > I'll be listening on this list for while. > > Mark G Hi Mark,Glad to see you back here! I'm about to publish a paper which includes a host of what you might call "generalized diatonic" scales. Some time ago I posted these horagrams, which show the scales as concentric rings: Yahoo groups: /tuning_files/files/miracle.gif * [with cont.] Yahoo groups: /tuning_files/files/pajara.gif * [with cont.] Yahoo groups: /tuning_files/files/Erlich/seven... * [with cont.] as well as most of the files in Sign In - * [with cont.] (Wayb.) with "horagram" as the description (these are 5-limit ones). A draft of the paper, about 75% complete, is here: Yahoo groups: /tuning/files/perlich/coyotepape... * [with cont.] If you have time, I'd appreciate your comments, because I'd like to make this paper as clear as possible . . . Thanks, Paul
Message: 11344 - Contents - Hide Contents Date: Fri, 09 Jul 2004 05:28:34 Subject: 43 planars logflat badness ordered From: Gene Ward Smith Here they are again in logflat badness order. The last line gives Tenney distance , TOP error, and logflat badness. 4375/4374 [[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]] [1200.016360, 1901.980932, 2786.275726] 24.189805 .016361 4.668292 2401/2400 [[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]] [1200.032113, 350.9868928, 617.6846359] 22.458238 .032113 6.807744 225/224 [[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]] [1200.493660, 1901.172569, 2785.167472] 15.621136 .493660 24.496112 126/125 [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]] [1199.010636, 1900.386896, 2788.610946] 13.943064 .989364 31.160753 81/80 [[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]] [1201.698520, 1899.262910, 3368.825906] 12.661778 1.698521 36.380484 64/63 [[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]] [1197.723683, 1905.562879, 2786.313714] 11.977280 2.276318 39.037716 50/49 [[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]] [598.4467109, 1901.955001, 2779.100463] 11.258566 3.106578 41.594258 49/48 [[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]] [1203.187309, 953.5033827, 2786.313714] 11.199672 3.187309 41.789211 32805/32768 [[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]] [1200.065120, 1901.851787, 3368.825906] 30.001628 .065120 43.965846 36/35 [[1, 0, 0, 2], [0, 1, 0, 2], [0, 0, 1, -1]] [1195.264647, 1894.449645, 2797.308862] 10.299208 4.735353 44.400354 28/27 [[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]] [1193.415676, 1912.390908, 2786.313714] 9.562242 6.584324 45.874248 245/243 [[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]] [1200., 1903.372995, 440.4316973] 15.861450 .894655 47.189574 1029/1024 [[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]] [1200.421488, 233.6218235, 2786.313714] 20.007027 .421488 56.277426 3136/3125 [[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]] [1199.738066, 1901.955001, 1393.460953] 23.224350 .261934 63.501663 6144/6125 [[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, -2, 3]] [1199.786928, 1901.617290, 157.2978838] 25.165457 .213072 71.213871 5120/5103 [[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]] [1199.766314, 1902.325384, 2785.771112] 24.639058 .233686 71.770898 128/125 [[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]] [399.0200131, 1901.955001, 3368.825906] 13.965784 2.939961 93.201243 10976/10935 [[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, -3, -1]] [1199.758595, 1902.337618, 1139.106063] 26.838730 .241405 104.378975 1728/1715 [[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]] [1199.391895, 1900.991178, 929.2418964] 21.498880 .608105 108.258175 16875/16807 [[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, -5, -4]] [1200., 1901.560426, 583.7891213] 28.079374 .248949 128.967040 875/864 [[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]] [1201.121570, 1903.732647, 2783.709509] 19.528027 1.121570 135.918456 19683/19600 [[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]] [1200.256485, 950.7742412, 2786.909253] 28.523229 .256485 141.473578 15625/15552 [[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]] [1200.291038, 317.0693810, 3368.825906] 27.856381 .291038 146.038672 4000/3969 [[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]] [1199.436909, 1902.847479, 792.7846742] 23.920344 .563091 153.626817 686/675 [[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]] [1198.513067, 1904.311735, 130.9133777] 18.820808 1.486934 155.476025 250/243 [[1, 2, 3, 0], [0, -3, -5, 0], [0, 0, 0, 1]] [1196.905960, 162.3176609, 3368.825906] 15.890597 3.094040 164.401387 2048/2025 [[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]] [599.5552941, 1903.364685, 3368.825906] 21.983706 .889412 173.111327 2430/2401 [[1, 0, 3, 1], [0, 1, 3, 2], [0, 0, -4, -1]] [1199.075238, 1900.489288, 1628.631774] 22.476160 .924762 196.669589 525/512 [[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]] [1202.406737, 1898.140412, 2780.725442] 18.036174 2.406738 212.238977 3125/3087 [[1, 0, 0, 0], [0, 1, 1, 1], [0, 0, 3, 5]] [1200., 1903.401919, 293.5973664] 23.201630 .912903 220.453810 405/392 [[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]] [1203.269293, 1896.773294, 787.7266785] 17.276488 3.269293 242.713818 256/245 [[1, 0, 0, 4], [0, 1, 0, 0], [0, 0, 2, -1]] [1195.228951, 1901.955001, 1398.695873] 15.936638 4.771049 256.459893 3125/3072 [[1, 0, 2, 0], [0, 5, 1, 0], [0, 0, 0, 1]] [1201.276744, 380.7957184, 3368.825906] 23.194603 1.276744 307.943011 3645/3584 [[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]] [1201.235997, 1899.995991, 2783.443817] 23.639058 1.235998 321.630472 1029/1000 [[1, 0, 0, 1], [0, 3, 0, -1], [0, 0, 1, 1]] [1202.477948, 632.6758490, 2792.067330] 19.972812 2.477948 328.600180 648/625 [[4, 0, 3, 0], [0, 1, 1, 0], [0, 0, 0, 1]] [299.1603149, 1896.631523, 3368.825906] 18.627562 3.358741 336.991833 2240/2187 [[1, 0, 0, -6], [0, 1, 0, 7], [0, 0, 1, -1]] [1198.134693, 1904.911442, 2781.982606] 22.224021 1.865307 379.192372 729/700 [[1, 0, 0, -2], [0, 1, 0, 6], [0, 0, 1, -2]] [1203.706383, 1896.080523, 2794.919668] 18.960986 3.706383 399.220421 1323/1280 [[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]] [1202.764567, 1897.573266, 447.5797863] 20.691525 2.764567 422.294972 5103/5000 [[1, 0, 0, 3], [0, 1, 0, -6], [0, 0, 1, 4]] [1201.434720, 1899.681024, 2789.645030] 24.604842 1.434720 438.195840 8748/8575 [[1, 0, 1, 0], [0, 1, 2, 1], [0, 0, -3, 2]] [1198.678173, 1899.859955, 736.3383942] 26.160658 1.321827 515.926826 5625/5488 [[1, 0, 1, 0], [0, 1, 1, 2], [0, 0, -3, -4]] [1201.715742, 1899.235615, 106.2071570] 24.879702 1.715742 547.837029 3125/3024 [[1, 0, 0, -4], [0, 1, 0, -3], [0, 0, 1, 5]] [1202.454598, 1905.845447, 2780.614314] 23.171883 2.454598 589.718096
Message: 11345 - Contents - Hide Contents Date: Fri, 09 Jul 2004 22:23:47 Subject: Re: Joining Post From: Graham Breed Paul Erlich wrote:> A draft of the paper, about 75% complete, is here: > > Yahoo groups: /tuning/files/perlich/coyotepape... * [with cont.]Hi! Can I have a copy of this without the lattice diagrams? They take an insane amount of time to draw on my machine. Graham
Message: 11346 - Contents - Hide Contents Date: Fri, 09 Jul 2004 06:22:08 Subject: monz back to math school From: monz hey guys, i've finally decided to enroll in school again to study the math that i'm sorely lacking. i know that ultimately i want to take a course in linear algebra, and that Grassmann algebra in particular is something i want to be familiar with. the college catalog lists trigonometry and calculus as prerequisites for linear algebra, so this is going to be a long haul if i can stick with it. i nearly bombed out of algebra II in high school, and never studied math again after that ... except the bits and pieces i picked up as a tuning theorist. i feel like i'm missing out on too much new discovery here on tuning-math, and want to get up to speed. i haven't selected any particular classes yet. i will certainly have to start with regular algebra all over again, and will probably only be able to take one course per semester. advice is appreciated. -monz
Message: 11347 - Contents - Hide Contents Date: Fri, 09 Jul 2004 21:40:48 Subject: Re: Joining Post From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> A draft of the paper, about 75% complete, is here: >> >> Yahoo groups: /tuning/files/perlich/coyotepape... * [with cont.] >> Hi! Can I have a copy of this without the lattice diagrams? They take > an insane amount of time to draw on my machine. > > > GrahamSure, I'll e-mail it to ya. But I don't have your e-mail address, so please e-mail me first -- ASAP! :)
Message: 11348 - Contents - Hide Contents Date: Fri, 09 Jul 2004 23:18:53 Subject: Re: 43 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Thanks so much, Gene. If you could describe in common, non- technical > language the criteria you used to choose the set of generators, I'll > be able to explain it in my paper.See if this will work for you: Rules 1. The mapping matrix is a M[i,j] 4x3 integral matrix, meaning four rows and three columns with integer values. For i from 1 to 3 (the first three rows) we have the following rules if the comma is 7-limit; in case it is a 5-limit comma the rules apply only to the first two rows. Then the last row is [0,0,1] and the last column is [0,0,0,1], giving a 7 as generator. Also only the 2x2 matrix obtained by deleting the last column and the last two rows needs to have a determinant of +-1. 2. If j>i then M[i,j] = 0. 3. If j=i then M[i,j] is not equal to 0. 4. If j<i then |M[i,j]| < |M[i,i]| 5. The 3x3 matrix obtained by leaving off the last row has determinant +-1. 6. The generators are three real numbers A, B, C greater than 1 (ie in terms of cents, such that their value in cents is positive.) 7. A^M[i,1] * B^M[i,2] * C^M[i,3] gives the TOP tuning of the ith prime, 1<=i<=4. A, B, and C are associated to 2, 3, and 5 respectively, and these rules result in A being an approximate octave if possible, B being an approximate 3 if that is consistent with what we have for A, and C being an approximate 5 if consistent with A and B.
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