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Message: 4350 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 06:07:25

Subject: A group "best" listt

From: genewardsmith

This was done in a way exactly comparable to the most recent 5-limit
search.

567/512

map   [[0, -1, 4], [1, 2, 1]]

generators   535.897694119756   1200.

badness   670.137142647868   rms   27.2570556594034   g_w   2.90778025274862


243/224

map   [[0, -1, -5], [1, 2, 5]]

generators   528.248377247002   1200.

badness   534.715045110858   rms   21.7489179735939   g_w   2.90778025274862


343/324

map   [[0, 3, 4], [1, 1, 2]]

generators   240.309212849092   1200.

badness   493.563423167166   rms   19.3483699528574   g_w   2.94374053640771


256/243

map   [[0, 0, -1], [5, 8, 15]]

generators   222.151593963576   240.

badness   378.244754205085   rms   12.7597412545895   g_w   3.09487905565715


28/27

map   [[0, -1, -3], [1, 2, 4]]

generators   475.558962037126   1200.

badness   91.4596348840926   rms   16.8270093610171   g_w   1.75822388550304


16807/16384

map   [[0, -1, 0], [5, 10, 14]]

generators   502.457952369172   240.

badness   550.435696708892   rms   6.24085831443580   g_w   4.45130571562582


19683/19208

map   [[0, -4, -9], [1, 3, 6]]

generators   425.724527133587   1200.

badness   626.695380808806   rms   3.82888181257598   g_w   5.47009468947274


49/48

map   [[0, -2, -1], [1, 2, 3]]

generators   249.022499567306   1200.

badness   67.8093075322853   rms   14.5731625037549   g_w   1.66946954361805


64/63

map   [[0, -1, 2], [1, 2, 2]]

generators   488.307823491718   1200.

badness   39.6049568556885   rms   7.28663503411012   g_w   1.75822388550304


531441/524288

map   [[0, 0, -1], [12, 19, 36]]

generators   232.151593963576   100.

badness   566.495842771669   rms   1.38239436914426   g_w   7.42770973357716


537824/531441

map   [[0, 5, 12], [1, 0, -1]]

generators   380.751294592580   1200.

badness   524.893579245064   rms   1.39991667198979   g_w   7.21090363091141


65536/64827

map   [[0, 4, -3], [1, 0, 4]]

generators   476.124945200378   1200.

badness   206.568540713651   rms   2.18910247435947   g_w   4.55266851389582


177147/175616

map   [[0, -3, -11], [1, 3, 8]]

generators   566.505581347493   1200.

badness   281.673403211886   rms   1.07889655377572   g_w   6.39129227512076


1605632/1594323

map   [[0, 2, 13], [1, 1, -1]]

generators   351.476961726092   1200.

badness   314.893972274088   rms   .713664767134272   g_w   7.61301520051980


67108864/66706983

map   [[0, -7, 4], [1, 3, 2]]

generators   242.458056889570   1200.

badness   316.903243699017   rms   .762466103296028   g_w   7.46280420851729


13841287201/13759414272

map   [[0, 3, 2], [4, 3, 9]]

generators   334.046136423623   300.

badness   705.581672022668   rms   .686251415126464   g_w   10.0930249317794


1029/1024

map   [[0, 3, -1], [1, 1, 3]]

generators   233.444441296619   1200.

badness   42.1870934730102   rms   1.65379250859746   g_w   2.94374053640771


118098/117649

map   [[0, 3, 5], [2, 1, 2]]

generators   433.782535424977   600.

badness   155.357296461064   rms   .534890245526000   g_w   6.62250546977042


70632088586703/70368744177664

map   [[0, 13, -6], [1, -1, 4]]

generators   238.589951176615   1200.

badness   643.457389814532   rms   .271819428385752   g_w   13.3274306917460


94450499584/94143178827

map   [[0, -8, -23], [1, 2, 4]]

generators   62.2228613205755   1200.

badness   487.989402670565   rms   .197276010332117   g_w   13.5242141244344


4760622968832/4747561509943

map   [[0, 15, 19], [1, -3, -3]]

generators   366.784885146694   1200.

badness   579.210669002310   rms   .193857182595423   g_w   14.4029532849298


34451725707/34359738368

map   [[0, 4, -15], [1, 1, 5]]

generators   175.423395680988   1200.

badness   253.815642297510   rms   .188649236250683   g_w   11.0396281212545


33554432/33480783

map   [[0, -1, 14], [1, 2, -3]]

generators   497.783581217437   1200.

badness   135.621744048494   rms   .185179785930646   g_w   9.01388298147804


282429536481/281974669312

map   [[0, 5, 24], [1, 1, 0]]

generators   140.366054018771   1200.

badness   244.087415237322   rms   .899689577077160e-1   g_w   13.
9471102220917


68719476736/68641485507

map   [[0, 2, -1], [5, 6, 15]]

generators   231.033669889318   240.

badness   117.928158169825   rms   .105083431144134   g_w   10.3918871916168


40353607/40310784

map   [[0, -1, -1], [9, 18, 29]]

generators   497.942879666076   133.333333333333

badness   74.2853761882641   rms   .144418737385944   g_w   8.01235028812647


5559060566555523/5556003412779008

map   [[0, 14, 33], [1, -3, -8]]

generators   392.994636775254   1200.

badness   184.673444690140   rms   .234734105085574e-1   g_w   19.
8888383853365


1153470752371588581/1152921504606846976

map   [[0, 5, -29], [1, 0, 12]]

generators   380.385861981302   1200.

badness   145.368173525748   rms   .183370867638466e-1   g_w   19.
9394418821855


11399736556781568/11398895185373143

map   [[0, 19, 4], [1, -7, 1]]

generators   542.208280827679   1200.

badness   22.2288637273498   rms   .520794634506544e-2   g_w   16.
2212227246308
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Message: 4351 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 06:17:03

Subject: Re: A group "best" listt

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> This was done in a way exactly comparable to the most recent 5-limit
> search.



I'll have a much better feel for this if we do the full 7-limit first. 
But have we agreed on the 5-limit list yet. Do you agree the weighted 
complexity cutoff can be reduced to 13, to give us a list of 20?
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Message: 4352 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 10:44 +0

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a7eg9l+omel@xxxxxxx.xxx>
Paul:


>> a lower complexity seems desirable now but we'll regret it next year 
>> when we're working on the 19-limit.



You don't plan on getting to the 19-limit until next year?

Dave K:


> I have always assumed we would allow higher complexities for higher 
> primes. We might well be looking at stuff with up to 25 rms generators 
> for 19-limit. But 13 is plenty for 5-limit.



31&62 has an RMS complexity of 22.5 and it's the best one I've got below 
25.  It's got a worst error of 11 cents (or RMS of 5 cents) so I'm not 
sure it counts as 19-limit anything.  There's also 31&72 with RMS 
complexity of 24.8 and 8.8 cent minimax, and 62&72 with a 23.5 complexity 
and 7.5 cent minimax.


                            Graham
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Message: 4353 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 11:49:48

Subject: A group "best" listt

From: genewardsmith

Here we find, of course, the 3136/3125 system I mentioned before, but I'd also like to draw attention to the last temperament on the list--a
mathematically (if not practically) amazing temperament, associated to the rather amazing 789 et.


686/625

map   [[0, 3, 4], [1, 1, 1]]

generators   539.1060370   1200.

badness   489.0718160   rms   31.61840515   g_w   2.491595620


343/320

map   [[0, -3, -1], [1, 3, 3]]

generators   274.0898101   1200.

badness   232.0702794   rms   32.11518107   g_w   1.933316912


32768/30625

map   [[0, 1, -2], [2, 4, 7]]

generators   407.2232374   600.

badness   643.0205101   rms   15.64725500   g_w   3.450872300


4375/4096

map   [[0, 1, -4], [1, 2, 4]]

generators   361.8678362   1200.

badness   454.9331633   rms   17.60304342   g_w   2.956559423


2560/2401

map   [[0, -4, -1], [1, 3, 3]]

generators   201.2867627   1200.

badness   392.3271620   rms   21.77086970   g_w   2.621747521


16807/16000

map   [[0, 5, 3], [1, 1, 2]]

generators   317.7111025   1200.

badness   450.1531173   rms   13.81938198   g_w   3.193673737


8575/8192

map   [[0, 3, -2], [1, 2, 3]]

generators   123.9139044   1200.

badness   328.0735082   rms   12.83254896   g_w   2.945909568


256/245

map   [[0, 2, -1], [1, 2, 3]]

generators   204.0189242   1200.

badness   119.7408063   rms   20.32106851   g_w   1.806197400


823543/800000

map   [[0, -7, -5], [1, 4, 4]]

generators   287.3935752   1200.

badness   536.0657103   rms   5.685477364   g_w   4.551456166


401408/390625

map   [[0, 1, 4], [2, 4, 3]]

generators   392.6597660   600.

badness   470.1905708   rms   4.622664052   g_w   4.667960736


16807/16384

map   [[0, 1, 0], [5, 10, 14]]

generators   381.9007604   240.

badness   281.4513766   rms   6.240857928   g_w   3.559478388


128/125

map   [[0, 0, -1], [3, 7, 9]]

generators   224.3309516   400.

badness   69.03052049   rms   9.677665952   g_w   1.924967967


1071875/1048576

map   [[0, 3, -5], [1, 1, 5]]

generators   527.0889187   1200.

badness   370.6464204   rms   3.843254849   g_w   4.585853840


268435456/262609375

map   [[0, -5, 6], [1, 3, 2]]

generators   162.0275880   1200.

badness   708.1906883   rms   2.815872080   g_w   6.312175959


50/49

map   [[0, -1, -1], [2, 5, 6]]

generators   222.4301906   600.

badness   35.69105715   rms   12.36574700   g_w   1.423791355


68359375/67108864

map   [[0, 1, -10], [1, 2, 6]]

generators   383.2901826   1200.

badness   664.8390958   rms   2.145218464   g_w   6.767293431


4194304/4117715

map   [[0, 7, -1], [1, 1, 3]]

generators   226.9760748   1200.

badness   445.7435327   rms   2.988171840   g_w   5.303466469


78125/76832

map   [[0, -4, -7], [1, 3, 4]]

generators   204.3976654   1200.

badness   230.3456068   rms   3.358669391   g_w   4.093274712


40960000/40353607

map   [[0, -9, -4], [1, 5, 4]]

generators   357.0527370   1200.

badness   439.4738808   rms   2.337786935   g_w   5.728523696


244140625/240945152

map   [[0, 1, 2], [6, 12, 13]]

generators   384.4129525   200.

badness   514.3166613   rms   1.551964221   g_w   6.920162556


1977326743/1953125000

map   [[0, 11, 12], [1, -3, -3]]

generators   580.6686988   1200.

badness   705.1161663   rms   1.307238490   g_w   8.140203868


2000000000/1977326743

map   [[0, -11, -9], [1, 5, 5]]

generators   292.2142738   1200.

badness   544.3754097   rms   1.375242977   g_w   7.342427853


762939453125/755603996672

map   [[0, 8, 17], [1, 1, 0]]

generators   198.1639627   1200.

badness   748.4465765   rms   .8028678840   g_w   9.768748008


40353607/40000000

map   [[0, 9, 7], [1, 0, 1]]

generators   309.6535849   1200.

badness   275.9060105   rms   1.316290184   g_w   5.940228799


97656250000/96889010407

map   [[0, 13, 14], [1, -4, -4]]

generators   583.5195442   1200.

badness   623.8314928   rms   .7137704924   g_w   9.560990833


69206436005/68719476736

map   [[0, 12, -1], [1, 0, 3]]

generators   232.1473890   1200.

badness   478.2656671   rms   .6898731204   g_w   8.850476649


1638400000000000/1628413597910449

map   [[0, 18, 11], [1, 1, 2]]

generators   88.12377826   1200.

badness   726.3196762   rms   .4762208496   g_w   11.51082544


823543/819200

map   [[0, -7, -2], [1, 3, 3]]

generators   116.2911944   1200.

badness   97.72611226   rms   1.036476664   g_w   4.551456166


8589934592/8544921875

map   [[0, 1, -13], [1, 2, 7]]

generators   386.9847164   1200.

badness   311.5353743   rms   .4754474072   g_w   8.685636061


19073486328125/18990246039772

map   [[0, -15, -19], [1, 8, 10]]

generators   454.2650388   1200.

badness   540.6271280   rms   .3086102069   g_w   12.05486958


3136/3125

map   [[0, 2, 5], [1, 2, 2]]

generators   193.7971982   1200.

badness   23.18483462   rms   .9868326492   g_w   2.864091121


65712362363534280139543/65536000000000000000000

map   [[0, 3, 2], [9, 10, 18]]

generators   484.3391018   133.3333333

badness   727.8282850   rms   .1381631578   g_w   17.39985221


186759498717153574912/186264514923095703125

map   [[0, -19, -29], [1, 9, 13]]

generators   421.7657192   1200.

badness   669.4337469   rms   .1273318109   g_w   17.38829083


7450580596923828125/7430984482854797312

map   [[0, 4, 9], [3, 1, -5]]

generators   596.5348276   400.

badness   510.7162100   rms   .1375962522   g_w   15.48309146


80000000000000000/79792266297612001

map   [[0, -5, -4], [4, 19, 19]]

generators   582.7533331   300.

badness   406.7075158   rms   .1736411704   g_w   13.28035949


591363588909912109375/590295810358705651712

map   [[0, -13, 14], [1, 4, 1]]

generators   154.9079648   1200.

badness   354.5227097   rms   .9459396236e-1   g_w   15.53317425


2100875/2097152

map   [[0, 5, -3], [1, 2, 3]]

generators   77.19380915   1200.

badness   33.21811689   rms   .3101843093   g_w   4.748812924


134217728/133984375

map   [[0, -3, 8], [1, 3, 1]]

generators   271.1304136   1200.

badness   56.35222399   rms   .2162902226   g_w   6.386918744


37778931862957161709568/37714514598846435546875

map   [[0, 11, -19], [1, -3, 12]]

generators   580.5834970   1200.

badness   404.7327302   rms   .7947342068e-1   g_w   17.20476745


22539340290692258087863249/22517998136852480000000000

map   [[0, -3, -1], [10, 29, 30]]

generators   231.2326670   120.

badness   316.7381823   rms   .4383199824e-1   g_w   19.33316912


3909821048582988049/3906250000000000000

map   [[0, -22, -21], [1, 13, 13]]

generators   582.4387327   1200.

badness   189.2472322   rms   .5198637212e-1   g_w   15.38319960


23303567338232644370432/23283064365386962890625

map   [[0, 7, 16], [2, 0, -5]]

generators   398.0518660   600.

badness   239.3677518   rms   .3878099806e-1   g_w   18.34350066


59604644775390625/59553411580724992

map   [[0, -17, -24], [1, 6, 8]]

generators   259.6315753   1200.

badness   155.9226447   rms   .4924199004e-1   g_w   14.68445611


1342177280000000000000/1341068619663964900807

map   [[0, 25, 13], [1, -5, -1]]

generators   351.4524874   1200.

badness   187.4404574   rms   .4671147988e-1   g_w   15.89082956


281484423828125/281474976710656

map   [[0, 8, -11], [1, 0, 6]]

generators   348.2888153   1200.

badness   3.206369973   rms   .2486103265e-2   g_w   10.88507660
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Message: 4354 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 12:16:13

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., graham@m... wrote:


> In-Reply-To: <a7eg9l+omel@e...>
> Paul:


>>> a lower complexity seems desirable now but we'll regret it next 
year 
>>> when we're working on the 19-limit.
> 


> You don't plan on getting to the 19-limit until next year?
> 
> Dave K:


>> I have always assumed we would allow higher complexities for 
higher 
>> primes. We might well be looking at stuff with up to 25 rms 
generators 
>> for 19-limit. But 13 is plenty for 5-limit.
> 


> 31&62 has an RMS complexity of 22.5 and it's the best one I've got 
below 
> 25.  It's got a worst error of 11 cents (or RMS of 5 cents) so I'm 
not 
> sure it counts as 19-limit anything.  There's also 31&72 with RMS 
> complexity of 24.8 and 8.8 cent minimax, and 62&72 with a 23.5 
complexity 
> and 7.5 cent minimax.



Thanks. So we'll definitely be going _beyond_ weighted rms complexity 
of 25 for 19-limit. Or maybe it just says that 19-limit really isn't 
worth the trouble.
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Message: 4355 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 12:23:11

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


> neutral thirds
> meantone
> pelogic
> augmented
> semiminorthirds (porcupine)
> quintuple thirds (Blackwood's decatonic)
> diminished
> diaschismic
> magic
> tertiathirds (quadrafourths)
> kleismic
> quartafifths (minimal diesic)
> schismic
> wuerschmidt
> semisixths (tiny diesic)
> subminor thirds (orwell)
> quintelevenths (AMT)
> septathirds (4294967296/4271484375)
> twin tertiatenths (semisuper)
> parakleismic (1224440064/1220703125)



I find that I can also agree with Gene on a best 17, he just needs to 
change his complexity cutoff to 10 and we lose the last three off the 
list above, including septathirds which Paul objected to, and I agreed 
was boring. I would prefer this list of 17, to the list of 20. There 
is quite a big gap in complexity between quintelevenths and 
septathirds.
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Message: 4356 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 22:16:50

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>> 


>> I have been dealing with this issue in evaluating ways to notate 
>> ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded 
that 
>> the alterations for these should not be in the opposite direction 
>> from an associated sharp or flat.  In other words, relative to C, 
I 
>> would prefer to see these as varieties of E-semiflat rather than 
E-
>> flat with varieties of semisharps.  But (for other intervals) 
>> something no larger than 2 Didymus commas (~43 cents or ~3/8 
>> apotome) altering in the opposite direction would be okay with me.
> 


> I totally agree, when the 242:243 or 507:512 vanishes (as in many 
> ETs), but I don't see how it is possible when notating strict 
ratios. 
> If two of the above come out as E] and E}, then the other two must 
be 
> Eb{ and Eb[, where ] and } represent an increase, and { and [ a 
> decrease, by the 11 and 13 comma respectively. Also which ever have 
no 
> sharp or flat from F,C or G _will_ have a sharp or flat from A, E 
or B 
> and vice versa.



In the 17-limit approach that I outline below, you will see how I 
handle this.  I am also outlining a 23-limit approach; I went for the 
19 limit and got 23 as a bonus when I found that I could approximate 
it using a very small comma.  The two approaches could be combined, 
in which case you could have the 11-13 semiflat varieties along with 
the 19 or 23 limit, but the symbols may get a bit complicated -- more 
about that below.



>> I spent some time wrestling with 27-ET last night, and it proved 
to 
>> be a formidable opponent that severely limited my options.  There 
is 
>> one approach that allows me to do it justice (using 13 -- what 
else 
>> is there?) ...
> 


> The only other option I could see was to notate it as every fourth 
> note of 108-ET (= 9*12-ET), using a trinary notation where the 5-
comma 
> is one step, the 7-comma is 3 steps, and the apotome is 9 steps, 
but 
> that would be to deny that it has a (just barely) usable fifth of 
its 
> own.



I thought more about this and now realize that the problem with 27-ET 
is not as formidable as it seemed.  If we use the 80:81 comma for a 
single degree and the 1024:1053 comma for two degrees of alteration, 
we will do just fine, even if the *symbol* for 1024:1053 happens to 
be a combination of the 80:81 and 63:64 symbols (by conflating 
4095:4096).  For the 27-ET notation we can simply define the 
combination of the two symbols as the 13 comma alteration, and there 
would be no inconsistency in usage, since the 63:64 symbol is *never 
used by itself* in the 27-ET notation.  The same could be said about 
50-ET.  Are there any troublesome divisions above 100 that we should 
be concerned about in this regard?



>> With this it looks as if I am going to be stopping at the 17 
limit,
> 


> This might be made to work for ETs, but not JI. The 16:19:24 minor 
> triad has a following.



Yes, I can appreciate that, and there are other uses of ratios of 19 
that I value.  However, the 19 comma gets us down to less than 3.4 
cents deviation (32/27 from 19/16), which is about the same as the 
minimax deviation for the Miracle tuning, which is not bad.  However, 
I anticipate that you believe that the JI purists would still want to 
have this distinction, so we should go for 19.



>  


>> with intervals measurable in degrees of 183-ET.  
> 


> I don't understand how this can work.

See below.



>> Once I have made a 
>> final decision regarding the symbols, I hope to have something to 
>> show you in about a week or so.
> 


> I'm more interested in the sematics than the symbols at this stage. 
I 
> wouldn't spend too much time on the symbols yet. I expect serious 
> problems with the semantics.



I don't know what problems you are anticipating, but let me outline 
what I have, both in the way of semantics and symbols.  As I said, I 
have both a 17-limit and 23-limit approach, although I expect that 
you will be interested in only the latter.  Since there is not much 
difference between the two, it is not any trouble to give both.

I have found that the semantics and symbols are so closely connected 
that I could not address one without the other, given the limitation 
of no more than one symbol in conjunction with a sharp or flat 
(possible in my 23-limit approach) or of no more than one symbol of 
any sort (accomplished in my 17-limit approach).  For the sake of 
simplicity, I will outline only how the modifications to natural 
notes are accomplished by single symbols, leaving off the problem of 
how these are to be combined with sharps and flats for another time.

Both the 17-limit and 23-limit approaches use 6 sizes of 
alterations.  In the sagittal notation these are paired into left and 
right flags that are affixed to a vertical stem, to the top for 
upward alteration and to the bottom for downward alteration.  These 
pairs of flags consist of straight lines, convex curved lines, and 
concave curved lines.  With this arrangement there is a limitation 
that two left or two right flags cannot be used simultaneously.

*17-LIMIT APPROACH*

The 17-limit arrangement is:

Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (3 degrees 
of 183)
Straight right flag (sR): 54:55, ~31.8 cents (5deg183)
Convex left flag (xL): 715:729 (3^6:5*11*13), ~33.6 cents (5deg183)
Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (4deg183)
Concave left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat 
comma), ~14.7 cents (2deg183)
Concave right flag (vR): 2187:2176 (3^7:2^7*17, the 17-as-sharp 
comma), ~8.7 cents (1deg183)

I was a bit hesitant to use two different 17-commas, but I saw that 
16:17 could be used as either an augmented prime or a minor second 
(e.g., in a scale with a tonic triad of 14:17:21).  Rather than 
choose between the two, I found that it is handy to have both 
intervals, especially when going to the 19 limit.

With the above used in combination, the following useful intervals 
are available:

sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (8deg183)
sL+xR: 35:36, ~48.8 cents, which approximates
 ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents (7deg183)
sR+xL: 26:27 (the 13-as-semiflat comma), ~65.3 cents (10deg183)
xL+xR: 5005:5184 (5*7*11*13:2^6*3^4), ~60.8 cents, which approximates
 ~xL+xR: 704:729 (the 11-as-semiflat comma), ~60.4 cents (9deg183)
vL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates
 ~vL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents 
(6deg183)

That is how I make the distinction between all the different ratios 
of 11 and 13.  Note that ratios of 11 have like left and right flags, 
while ratios of 13 have dissimilar flags, making them easy to tell 
apart.  With the single-symbol sagittal notation it is necessary to 
have alterations exceeding half an apotome, since there is no way to 
notate a (two-flag) sagittal sharp/flat *less* a (two-flag) 
~semiflat/~semisharp alteration.

The following combinations are probably not as useful, but are 
available anyway:

xL+vR: 1555840:1594323 (2^7*5*11*13*17:3^3*13), ~42.3 cents, which 
also approximates
 ~xL+vR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents 
(6deg183)
vL+vR: 524277:531441 (2^19:3^12, the Pythagorean comma), ~23.5 cents 
(3deg183)

There are also two other combinations that I didn't see any use for:

vL+sR, ~46.5 cents (7deg183)
sL+vR, ~30.2 cents (4deg183)

Why 183?  Since I'm going only to the 17 limit here, it's one in 
which the 19 comma (512:513) vanishes, and it represents the building 
blocks of the notation rather nicely as approximate multiples of 7 
cents.  The apotome in 183 is 17 degrees.

*23-LIMIT APPROACH*

And here is the 23-limit arrangement, which correlates well with 217-
ET (apotome of 21 degrees):

Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4 degrees 
of 217)
Straight right flag (sR): 54:55, ~31.8 cents (6deg217)
Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat comma), 
~14.7 cents (3deg217)
Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217)
Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17-as-sharp 
comma), ~8.7 cents (2deg217)
Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4 cents 
(1deg217)

The difference between this and the 17-limit approach is that I have 
removed the 715:729 alteration and added the 512:513 alteration, 
while reassigning the 17-commas to different flags.  No combination 
of flags will now exceed half of an apotome.

With the above used in combination, the following useful intervals 
are available:

sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (10deg217)
sL+xR: 35:36, ~48.8 cents, which approximates
 ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents (9deg217)
vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which approximates
 ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents 
(8deg217)
xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates
 ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents 
(8deg217)

The vL+sR approximation of the 23 comma deviates by 3519:3520 (~0.492 
cents).

All of the above provide a continuous range of intervals in 217-ET, 
which I selected because it is consistent to the 21-limit and 
represents the building blocks of the notation as approximate 
multiples of 5.5 cents.

--George
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Message: 4357 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 22:56:54

Subject: List of 20

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:



> I'll have a much better feel for this if we do the full 7-limit first. 
> But have we agreed on the 5-limit list yet. Do you agree the weighted 
> complexity cutoff can be reduced to 13, to give us a list of 20?



It looks OK; hemithirds, ennealimmal and the 10485760000/10460353203
system are really best thought of as 7-limit temperaments.
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Message: 4358 - Contents - Hide Contents

Date: Sat, 23 Mar 2002 23:37:41

Subject: Re: Diatonics

From: genewardsmith

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:



> 5 0 7 2 9 4 11 reordered 0 2 4 5 7 9 11 (0,   has no points at which 3
> adjacent PCs lie together.



What do you mean by a "PC"?



> I know I use the tterm generators differently, but in the context of the
> group theoretic structures proposed initially by Balzano. 



"Generators" is a term taken from mathematics, and I hope when we say "generator" or "group" we
are all on the same page, and using the mathematical definitions. What
are the two uses of "generator" you see around here? The looseness
I've sometimes observed is "generator"
as opposed to octaves or a division thereof, but that should be clear
from context.
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Message: 4360 - Contents - Hide Contents

Date: Sat, 23 Mar 2002 23:41:19

Subject: Re: A common notation for JI and ETs

From: genewardsmith

If we have more than one comma per prime, we lose the very desireable
property of uniqueness, and automatic translation becomes harder.
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Message: 4361 - Contents - Hide Contents

Date: Sat, 23 Mar 2002 18:08:21

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:


> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>>> 


>>> I have been dealing with this issue in evaluating ways to notate 
>>> ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded 
> that 


>>> the alterations for these should not be in the opposite 
direction 
>>> from an associated sharp or flat.  In other words, relative to 
C, 
> I 


>>> would prefer to see these as varieties of E-semiflat rather than 
> E-


>>> flat with varieties of semisharps.  But (for other intervals) 
>>> something no larger than 2 Didymus commas (~43 cents or ~3/8 
>>> apotome) altering in the opposite direction would be okay with 
me.
>> 


>> I totally agree, when the 242:243 or 507:512 vanishes (as in many 
>> ETs), but I don't see how it is possible when notating strict 
> ratios. 


>> If two of the above come out as E] and E}, then the other two must 
> be 


>> Eb{ and Eb[, where ] and } represent an increase, and { and [ a 
>> decrease, by the 11 and 13 comma respectively. Also which ever 
have 
> no 


>> sharp or flat from F,C or G _will_ have a sharp or flat from A, E 
> or B 


>> and vice versa.
> 


> In the 17-limit approach that I outline below, you will see how I 
> handle this.



OK. So you have gone outside of one-comma-per-prime and 
one-(sub)symbol-per-prime. But you have given fair reasons for doing 
so in the case of 11 and 13.



>  I am also outlining a 23-limit approach; I went for 
the 
> 19 limit and got 23 as a bonus when I found that I could approximate 
> it using a very small comma.  The two approaches could be combined, 
> in which case you could have the 11-13 semiflat varieties along with 
> the 19 or 23 limit, but the symbols may get a bit complicated -- 
more 
> about that below.



You don't actually give more below about combining these approaches. 
But I had fun working it out for myself. I'll give my solution later.



> I thought more about this and now realize that the problem with 
27-ET 
> is not as formidable as it seemed.  If we use the 80:81 comma for a 
> single degree and the 1024:1053 comma for two degrees of alteration, 
> we will do just fine, even if the *symbol* for 1024:1053 happens to 
> be a combination of the 80:81 and 63:64 symbols (by conflating 
> 4095:4096).  For the 27-ET notation we can simply define the 
> combination of the two symbols as the 13 comma alteration, and there 
> would be no inconsistency in usage, since the 63:64 symbol is *never 
> used by itself* in the 27-ET notation.  The same could be said about 
> 50-ET.  



You're absolutely right.



> Are there any troublesome divisions above 100 that we should 
> be concerned about in this regard?



Not that I can find on a cursory examination.



> I anticipate that you believe that the JI purists would still want 
to 
> have this distinction, so we should go for 19.



Correct. I'll skip the 183-ET based one.



> I 


>> wouldn't spend too much time on the symbols yet. I expect serious 
>> problems with the semantics.
> 


> I don't know what problems you are anticipating, ...



Well none have materialised yet. :-)



> I have found that the semantics and symbols are so closely connected 
> that I could not address one without the other, 



Yes. I see that now.



> Both the 17-limit and 23-limit approaches use 6 sizes of 
> alterations.  In the sagittal notation these are paired into left 
and 
> right flags that are affixed to a vertical stem, to the top for 
> upward alteration and to the bottom for downward alteration.  These 
> pairs of flags consist of straight lines, convex curved lines, and 
> concave curved lines.  With this arrangement there is a limitation 
> that two left or two right flags cannot be used simultaneously.



Given these constraints I think your solution is brilliant.



> *23-LIMIT APPROACH*
> 
> And here is the 23-limit arrangement, which correlates well with 
217-
> ET (apotome of 21 degrees):



I don't think you can call this a 23-limit notation, since 217-ET is 
not 23-limit consistent. But it is certainly 19-prime-limit.



> Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4 degrees 
> of 217)
> Straight right flag (sR): 54:55, ~31.8 cents (6deg217)
> Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat 
comma), 
> ~14.7 cents (3deg217)
> Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217)
> Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17-as-sharp 
> comma), ~8.7 cents (2deg217)
> Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4 cents 
> (1deg217)
> 
> The difference between this and the 17-limit approach is that I have 
> removed the 715:729 alteration and added the 512:513 alteration, 
> while reassigning the 17-commas to different flags.  No combination 
> of flags will now exceed half of an apotome.
> 
> With the above used in combination, the following useful intervals 
> are available:
> 
> sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (10deg217)
> sL+xR: 35:36, ~48.8 cents, which approximates
>  ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents 
(9deg217)
> vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which approximates
>  ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents 
> (8deg217)
> xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates
>  ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents 
> (8deg217)
> 
> The vL+sR approximation of the 23 comma deviates by 3519:3520 
(~0.492 
> cents).
> 
> All of the above provide a continuous range of intervals in 217-ET, 
> which I selected because it is consistent to the 21-limit and 
> represents the building blocks of the notation as approximate 
> multiples of 5.5 cents.



Once I understood your constraints, I spent hours looking at the 
problem. I see that you can push it as far as 29-limit in 282-ET if 
you want both sets of 11 and 13 commas, and 31-limit in 311-ET if you 
can live with only the smaller 11 and 13 commas. But to make these 
work you have to violate what is probably an implicit constraint, that 
the 5 and 7 commas must correspond to single flags. Neither of them 
can map to a single flag in either 282-ET or 311-ET and so the mapping 
of commas to arrows is just way too obscure.

217-ET is definitely the highest ET you can use with the above 
additional constraint.

I notice that left-right confusability has gone out the window. But 
maybe that's ok, if we accept that this is not a notation for 
sight-reading by performers. However, it is possible to improve the 
situation by making the left-right confusable pairs of symbols either 
map to the same number of steps of 217-ET or only differ by one step, 
so a mistake will not be so disastrous. At the same time as we do this 
we can reinstate your larger 11 and 13 commas, so you have both sizes 
of these available. The 13 commas will have similar flags on left and 
right, while the 11 commas will have dissimilar flags. It seems better 
that the 11 commas should be confused with each other than the 13 
commas, since the 11 commas are closer together in size.

To do this you make the following changes to your 217-ET-based scheme.
1. Swap the flags for the 17 comma and 19 comma (vL and vR)
2. Swap the flags for the 7 comma and 11-as-semisharp/5 comma. (xR and 
sR)
3. Make the xL flag the 11-as-semiflat/7 comma instead of the 
17-as-flat comma. 

So we have
Steps  Flags  Commas
------------------------
1  vL     19-comma  512:513
2  vR     17-comma  2187:2176
3  vL+vR  17-as-flat-comma  4131:4096
4  sL     5-comma  80:81
5  sR     7-comma  63:64
6  xL or xR
7  vL+sR or xL+vR
8  vL+xR
9  sL+sR  13-as-semisharp comma  1024:1053
10 sL+xR  11-as-semisharp comma  32:33
11 xL+sR  11-as-semiflat comma  704:729
12 xL+xR  13-as-semiflat comma  26:27
21 apotome
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Message: 4362 - Contents - Hide Contents

Date: Sat, 23 Mar 2002 18:16:40

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


> So we have
> Steps  Flags  Commas
> ------------------------
> 1  vL     19-comma  512:513
> 2  vR     17-comma  2187:2176



Oops! That should have been:

Steps  Flags  Commas
------------------------
1  vR     19-comma  512:513
2  vL     17-comma  2187:2176
3  vL+vR  17-as-flat-comma  4131:4096
4  sL     5-comma  80:81
5  sR     7-comma  63:64
6  xL or xR
7  vL+sR or xL+vR
8  vL+xR
9  sL+sR  13-as-semisharp comma  1024:1053
10 sL+xR  11-as-semisharp comma  32:33
11 xL+sR  11-as-semiflat comma  704:729
12 xL+xR  13-as-semiflat comma  26:27


> 21 apotome
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Message: 4363 - Contents - Hide Contents

Date: Sat, 23 Mar 2002 21:04 +0

Subject: Re: Diatonics

From: graham@xxxxxxxxxx.xx.xx

Mark Gould wrote:



> 5 0 7 2 9 4 11 6 1 reordered 0 1 2 4 5 6 7 9 11 (0, has several places 
> where
> three adjacent PCs lie together, Viz, 0 1 2 or 11 0 1 or 4 5 6 or 5 6 7.



Ah, so this is relative to that base scale.  And the equivalent 9 note 
scale from 19-equal would still pass.  Doesn't this rule get taken care of 
by the later one rejecting "pentatonics"?  At least along with another one 
rejecting "degenerate" scales which are simply an equal scale.



> 2. The F F-sharp rule was discussed in section III, and from that 
> dicussion,
> it is simply the result of transposing the diatonic generated by the 
> sum of
> the two 'generators' (or the interval that behaves structurally like 
> the 3:2
> fifth, or 'generalised fifth', as Paul Zweifel dubs it) It must be noted
> that the new tone in he diatonic shall have the following property:
> 
> It should be the 'leading tone' of the new diatonic
> 
> It can also be equivalent to adding 1 to the PC that lies a 'generalized
> fifth' below the old tonic.



But doesn't this always work for an MOS?  Or is it this rule that 
specifies an MOS?



> If the raised tone lies in the 'supertonic' position then the scale is 
> an
> anhemitonic scale, in that it contains no tones that are separated by 
> the
> difference between the two generators of the scale.



And the 7 note diatonic in more than 12 notes would have this property.



> I know I use the tterm generators differently, but in the context of the
> group theoretic structures proposed initially by Balzano. I am merely
> continuing to use the term, and did not invent it myself.



It sounds consistent with Gene's ideas, so long as you can describe the 
octave in terms of the generators.



> Incidentally, can anyone point me to Paul Ehrlich's decatonics, a web
> address etc...?



I have it as <A few items from Paul Erlich * [with cont.]  (Wayb.)>.  Only 
one h in Erlich.  It's also in Xenharmonikon.


                   Graham
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Message: 4364 - Contents - Hide Contents

Date: Sun, 24 Mar 2002 11:41:30

Subject: Hermite normal form

From: genewardsmith

The normal form I mentioned, which "fixed" my first try, turns out on
reflection to be simply Hermite normal form. Hence, no special wedge
products are required--one could, for instance, start with columns
which are et maps. I propose therefore that we standardize the mapping
to be the (column) Hermite normal form, and the generators to be the
generators (in order) that this gives us.
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Message: 4365 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 00:27:12

Subject: Re: Hermite normal form

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> The normal form I mentioned, which "fixed" my first try, turns out 


on reflection to be simply Hermite normal form. Hence, no special 
wedge products are required--one could, for instance, start with 
columns which are et maps. I propose therefore that we standardize the 
mapping to be the (column) Hermite normal form, and the generators to 
be the generators (in order) that this gives us.


>



I'm not familiar with this, and Mathworld was no help to me (again).
Hermite Normal Form -- from MathWorld * [with cont.] 
It doesn't give me any clue as to what the HNF matrix looks like 
(except that it has a determinant of +-1) or how to compute it.

Would you be so kind as to give as examples, the HNF matrices and 
consequent generators for Meantone, Diaschismic and Augmented, as 
5-limit linear temperaments, Twin meantone and Half meantone fifth 
as degenerate 5-limit whatevers, and Starling as the 7-limit planar 
temperament where the 125:126 vanishes?
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Message: 4366 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 09:38:29

Subject: Re: Pitch Class and Generators

From: genewardsmith

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:



My disguised followup has shown up yet, let's see if this works.
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Message: 4367 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 04:22:59

Subject: Re: Hermite normal form

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:



> Would you be so kind as to give as examples, the HNF matrices and 
> consequent generators for Meantone, Diaschismic and Augmented, as 
> 5-limit linear temperaments, Twin meantone and Half meantone fifth 
> as degenerate 5-limit whatevers, and Starling as the 7-limit planar 
> temperament where the 125:126 vanishes?



Do you want this if half-fourth doesn't work? Hermite form seems to
allow twin meantone and schismic, and half-fifth meantome and schismic, but not the half-fourth versions.
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Message: 4368 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 09:34:45

Subject: Followups

From: genewardsmith

Has anyone noticed that followups to other postings have been appearing, but things using the "Post" command
have not? This is a followup disguised by removing what it follows up
to--let's see if it appears. 

I've been posting a little about Hermite normal form, but if it
doesn't appear, it slows the conversation down. How are other people
doing?
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Message: 4370 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 15:06:17

Subject: Re: Hermite normal form version of "25 best"

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> Here's the same list, this time using Hermite normal form. 



How about dropping the g_w cutoff to 13 for the best 20 like we 
agreed, or even to 10 for the best 17? Who wants the 20 and who wants 
the 17? Paul, Graham, Carl, Herman, anyone?



>The idea 


of this is to have a standard form which generalizes to higher 
dimension temperaments and could allow us to measure badness for them. 
It also is conceptually not wedded to octave-equivalence, but works 
well in that context. The disadvantage is that you might not like it!


> 



You're right. I don't like it. Generators bigger than an octave (in 
one case bigger than two octaves) and negative generators. This is not 
a form that allows a temperament's musical ramifications to be easily 
understood. However, I like the fact that the octave-related (or 
lowest-prime-related) generator comes first. If it facilitates badness 
calculations, well and good, but I wouldn't publish a list of 
temperaments in this form in a pink fit (unless it was not intended to 
be read by musicians).

So far I stand by my earlier proposal that minimises (positive) 
generator sizes while having zeros in the lower left triangle of the 
matrix (with columns corresponding to increasing primes from left to 
right). Each generator is less than half the size of the preceeding 
one. This gives maximum information about the melodic structure of the 
scale.



> I also changed the weighted "g" measure to one which is a weighted 


mean, since Dave complained bitterly that my adjustment wasn't one.


>



Thanks for doing that.
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Message: 4371 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 09:31:36

Subject: Re: Pitch Class and Generators

From: genewardsmith

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:



> Generators I have assumed is as follows:
> 
> assume three equivalence classes a b c lie linearly adjacent, such that the
> Next predicate follows this sequence
> 
> b = Next (a)
> c = Next (b)
> 
> thus c = Next(Next(a)) or Next^2(a)
> 
> If we (after Balzano), abbreviate Next to N,
> 
> we can write N^i, representinh powers of N, where i is an integer mod n.



This is a little strange, though mathematically correct. Normally
you
would write the cyclic group of order 12 additively unless it was
something coming from a multiplicative structure, such as the
multiplicative group of nonzero mod 13 classes. Writing it additively,
we can indentify "N" with 1, and then b=a+1, c=b+1. Then N^i becomes
simply i, and N^(jn+k) is jn+k. This raises the question of whether "j" is a generator for C12--which is the same as asking if j is
relatively prime to 12.



> A generator for Cn is an N^i, such that successive applications of
> N^i to the starting tone will generate the full Cn.



Or we could say, a generator for Cn is i, such that (i,n)=1, meaning
i is relatively prime to n. It is an "n-unit"; an invertible element
of Cn, taken as a ring.



> That's my definition too.



That's one usage; in general, a group generator for a finitely
generated abelian group can mean more kinds of things, some of them
not involving groups with torsion, and actually that's what's been
discussed the most around here recently.
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Message: 4372 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 20:49:49

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>> neutral thirds
>> meantone
>> pelogic
>> augmented
>> semiminorthirds (porcupine)


>> quintuple thirds (Blackwood's decatonic)
>> diminished
>> diaschismic
>> magic
>> tertiathirds (quadrafourths)
>> kleismic
>> quartafifths (minimal diesic)
>> schismic
>> wuerschmidt
>> semisixths (tiny diesic)
>> subminor thirds (orwell)
>> quintelevenths (AMT)
>> septathirds (4294967296/4271484375)
>> twin tertiatenths (semisuper)
>> parakleismic (1224440064/1220703125)
> 


> I find that I can also agree with Gene on a best 17, he just needs 
to 
> change his complexity cutoff to 10 and we lose the last three off 
the 
> list above, including septathirds which Paul objected to, and I 
agreed 
> was boring. I would prefer this list of 17, to the list of 20. 
There 
> is quite a big gap in complexity between quintelevenths and 
> septathirds.



let's use the name 'negri' for negri's system, shall we?
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Message: 4374 - Contents - Hide Contents

Date: Mon, 25 Mar 2002 11:34 +0

Subject: Re: Pitch Class and Generators

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <B8C4883F.38C4%mark.gould@xxxxxxx.xx.xx>
Mark Gould wrote:



> A generator for Cn is an N^i, such that successive applications of
> N^i to the starting tone will generate the full Cn.
> 
> For any n i=1 is a generator, as is i = -1
> 
> For C12, i = 5 and its mod 12 complement 7 are generators.



But I thought the major and minor thirds were also the twin generators for 
a diatonic scale.  They don't qualify in 12-equal by this definition.


                 Graham
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