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Message: 3950 - Contents - Hide Contents Date: Fri, 22 Feb 2002 19:55:31 Subject: Re: monz's et graph (from my lumma.gif) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., Carl Lumma <carl@l...> wrote: >>> The generator could be 98 cents... 6/73 gives MOS of 61, 49, 37, 25, >> 13, and 12 according to Scala. >> Right--the comma is 262144/253125, and the rms generator 98.317 cents.so monz should have a [18 -4 -5] label on the 12-73-61-49-37 line.
Message: 3951 - Contents - Hide Contents Date: Fri, 22 Feb 2002 00:58:02 Subject: Re: monz's et graph (from my lumma.gif) From: Carl Lumma>He also gives the 5-limit comma for this series as [-4 -5].And shows a couple of series that don't have lines on monz's chart: """ (3 4) : 28 47 19 48 29 (-2 7) : 26 29 32 """ -Carl
Message: 3952 - Contents - Hide Contents Date: Fri, 22 Feb 2002 19:56:59 Subject: Re: handy text breakdown of monz's chart From: paulerlich --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Lines > ------ > (no name) 250:243 [-5 3] 29, 22, 59, 37, 15that's porcupine, as in miller's mizarian porcupine overture.
Message: 3953 - Contents - Hide Contents
Date: Fri, 22 Feb 2002 01:39:17
Subject: handy text breakdown of monz's chart
From: Carl Lumma
Lines
------
(no name) 250:243 [-5 3] 29, 22, 59, 37, 15
diaschismic 2048:2025 [-4 -2] 22, 78, 56, 90, 34, 80, 46, 58, 70, 12
(no name) ? [-8 -7] 59, 71, 83, 95, 12
(no name) 262144:253125 [-4 -5] 25, 37, 49, 61, 73, 12
diesic 128:125 [0 -3] 15, 42, 27, 39, 12
schismic 32805:32768 [8 1] 29, 41, 94, 53, 65, 77, 89, 12
magic 3125:3072 [-1 5] 22, 63, 41, 60, 79, 19, 35
kleismic ? [-5 6] 15, 49, 83, 34, 87, 53, 72, 91, 19, 23
(no name) ? [-9 4] 48, 41, 75, 34, 95, 61, 27
(no name) ? [5 -9] 47, 31, 77, 46, 61, 15
(no name) ? [8 -5] 23, 35, 47, 12
wuerschmidt ? [1 -8] 28, 31, 96, 65, 99, 34, 71, 37
orwell ? [-8 -7] 31, 84, 53, 75, 97, 22
(no name) ? [4 -4] 28, 40, 52, 64, 12
(no name) ? [3 4] 29, 48, 19, 47, 28
Intersections (by eye)
----------------------
12 - 8
31 - 5
22, 37, 15, 34, 19 - 4
29, 41, 75, 61, 53, 72, 23 - 3
-Carl
Message: 3954 - Contents - Hide Contents Date: Fri, 22 Feb 2002 12:06:50 Subject: Re: handy text breakdown of monz's chart From: monz paul, i think your "favorite page on the internet" just got even better! see my latest post to the tuning list. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3955 - Contents - Hide Contents Date: Fri, 22 Feb 2002 10:05:46 Subject: Re: handy text breakdown of monz's chart From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Lines > ------ > (no name) 250:243 [-5 3] 29, 22, 59, 37, 15The comma is the maximal diesis, but Paul threated to die sick if we used all these diesis names at once for temperaments.
Message: 3956 - Contents - Hide Contents Date: Fri, 22 Feb 2002 21:02:34 Subject: spin cycle From: jpehrson2 --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/3366 * [with cont.] ***Well, this was an interesting read, but I'm dubious at best. In order to be convinced, I'd like to see more Mozart examples on "regular staff notation" like a "real" music theory article. It's quite possible that the examples were selected which just *happened* to prove the "formula." I'd need to see vastly more instances of this so-called Mozart "formula" before I'd believe it really exists and that Mozart used it to generate his works... Besides, due to the nature of the way themes are *normally* generated in "common practice" Western music, with small units of 2, 4, expanding to 8, 16, etc., almost *all* themes could be expressed by some kind of common "compound," yes? It's saying more about the generation of themes in common practice (i.e. dead white male) Western music than anything else, so it seems. Regarding the Beethoven, that seems even *more* specious. I don't know much about "Group Theory" but I can guess that it's expansive enough that you could included *just about anything* if you angle it the right way. This seems like a "spin cycle" worthy of the greatest of washing machines... JP
Message: 3957 - Contents - Hide Contents Date: Fri, 22 Feb 2002 12:24 +0 Subject: Re: comments sought From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <003501c1bb52$449655e0$af48620c@xxx.xxx.xxx> monz wrote:> in my MIDI rendition of the beginning of the piece in 55edo > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > i was careful to tune the sharps and flats differently to > reflect Mozart's notation, which had to be done by hand > because none of the programs i know of (Manuel's Scala, > Graham's Midiconv, John deLaubenfels adaptune) can retune > to more than 12 tones per octave.What do you mean? Midiconv doesn't have any 12-fetishism! I don't think Scale does either. Graham
Message: 3958 - Contents - Hide Contents Date: Fri, 22 Feb 2002 14:20:50 Subject: Re: comments sought From: manuel.op.de.coul@xxxxxxxxxxx.xxx>What do you mean? Midiconv doesn't have any 12-fetishism! >I don't think Scala does either.They don't, but Joe means the MIDI 12-fetishism. If there's a F# and a Gb in the score, we aren't able to guess which because they have the same note number in the MIDI-file. Still doing _all_ notes by hand what Joe does is a waste of time. Manuel
Message: 3959 - Contents - Hide Contents Date: Fri, 22 Feb 2002 01:41:32 Subject: Re: magma From: paulerlich --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>>> carl, why won't you answer us on the tuning list? we're asking about >> the cd you made for me of various a capella groups. could you discuss >> them please on tuning? >> Gee, I thought I did respond... here it is: 34636.i actually posted that message a long time ago -- look at the date in that message. the list is blowing its nose again.> BTW Gene, those ad blocking services work by filtering all your > http traffic through their server.are you serious?? someone tell me this isn't so.
Message: 3961 - Contents - Hide Contents Date: Tue, 26 Feb 2002 23:25:37 Subject: Re: Past Paul Post From: paulerlich --- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:> Mr Erlich... Sometime in the last week or so you asked me something about > the ³cradling² technique Iıd coined, in relation to something you do... Iım > sorry Iım lost here in catching up posts I might have even deleted it by > mistake but I havenıt been able to find it and I donıt rememberwhat list it> was even on. > > Might you reask, sir? > > mji have no clue.
Message: 3962 - Contents - Hide Contents Date: Fri, 01 Mar 2002 05:27:32 Subject: blast from the past re: microtempering partch From: paulerlich sent from me to dave keenan Thu 1/13/00 5:26 PM . . . You wrote,>In earlier messages I gave the following simultaneous distribution of the >224:225 and 384:385 commas or "unison vectors".Just for fun, here are the numbers of notes given by various pairs of 11-limit subdiaschismic unison vectors, keeping 224:225 and 384:385 as the other two unison vectors: 98:99, 99:100 -- 22 notes 98:99, 125:126 -- 31 notes 98:99, 242:243 -- 31 notes 98:99, 243:245 -- 22 notes 98:99, 440:441 -- 31 notes 98:99, 891:896 -- 22 notes 98:99, 1024:1029 -- 31 notes 98:99, 1323:1331 -- 31 notes 98:99, 2400:2401 -- 31 notes 98:99, 3024:3025 -- 31 notes 98:99, 3993:4000 -- 22 notes 98:99, 4374:4375 -- 53 notes 98:99, 9800:9801 -- 22 notes 99:100, 120:121 -- 22 notes 99:100, 125:126 -- 19 notes 99:100, 175:176 -- 22 notes 99:100, 242:243 -- 41 notes 99:100, 440:441 -- 41 notes 99:100, 1024:1029 -- 41 notes 99:100, 1323:1331 -- 63 notes 99:100, 2400:2401 -- 41 notes 99:100, 3024:3025 -- 41 notes 99:100, 3993:4000 -- 22 notes 99:100, 4374:4375 -- 19 notes 99:100, 5625:5632 -- 22 notes 99:100, 9800:9801 -- 22 notes 120:121, 125:126 -- 31 notes 120:121, 242:243 -- 31 notes 120:121, 243:245 -- 22 notes 120:121, 440:441 -- 31 notes 120:121, 891:896 -- 22 notes 120:121, 1024:1029 -- 31 notes 120:121, 1323:1331 -- 31 notes 120:121, 2400:2401 -- 31 notes 120:121, 3024:3025 -- 31 notes 120:121, 3993:4000 -- 22 notes 120:121, 4374:4375 -- 53 notes 120:121, 9800:9801 -- 22 notes 125:126, 175:176 -- 31 notes 125:126, 242:243 -- 31 notes 125:126, 243:245 -- 19 notes 125:126, 441:440 -- 31 notes 125:126, 891:896 -- 19 notes 125:126, 1024:1029 -- 31 notes 125:126, 1323:1331 -- 62 notes 125:126. 2400:2401 -- 31 notes 125:126, 3024:3025 -- 31 notes 125:126, 3993:4000 -- 50 notes 125:126, 4374:4375 -- 19 notes 125:126, 5625:5632 -- 31 notes 125:126, 9800:9801 -- 50 notes 175:176, 243:245 -- 22 notes 175:176, 891:896 -- 22 notes 175:176, 1024:1029 -- 31 notes 175:176, 1323:1331 -- 31 notes 175:176, 2400:2401 -- 31 notes 175:176, 3024:3025 -- 31 notes 175:176, 3993:4000 -- 22 notes 175:176, 9800:9801 -- 22 notes 242:243, 243:245 -- 41 notes 242:243, 891:896 -- 41 notes 242:243, 1323:1331 -- 31 notes 242:243, 3993:4000 -- 72 notes 242:243, 4374:4375 -- 72 notes 242:243, 5625:5632 -- 31 notes 242:243, 9800:9801 -- 72 notes 243:245, 440:441 -- 41 notes 243:245, 1024:1029 -- 41 notes 243:245, 1323:1331 -- 63 notes 243:245, 2400:2401 -- 41 notes 243:245, 3024:3025 -- 41 notes 243:245, 3993:4000 -- 22 notes 243:245, 4374:4375 -- 19 notes 243:245, 5625:5632 -- 22 notes 243:243, 9800:9801 -- 72 notes 440:441, 891:896 -- 41 notes 440:441, 1323:1331 -- 31 notes 440:441, 3993:4000 -- 72 notes 440:441, 4374:4375 -- 72 notes 440:441, 5625:5632 -- 31 notes 440:441, 9800:9801 -- 72 notes 891:896, 1024:1029 -- 41 notes 891:896, 1323:1331 -- 63 notes 891:896, 2400:2401 -- 41 notes 891:896, 3024:3025 -- 41 notes 891:896, 3993:4000 -- 22 notes 891:896, 4374:4375 -- 19 notes 891:896, 5625:5632 -- 22 notes 891:896, 9800:9801 -- 22 notes 1024:1029, 1323:1331 -- 31 notes 1024:1029, 3993:4000 -- 72 notes 1024:1029, 4374:4375 -- 72 notes 1024:1029, 5625:5632 -- 31 notes 1024:1029, 9800:9801 -- 72 notes 1323:1331, 2400:2401 -- 31 notes 1323:1331, 3024:3025 -- 31 notes 1323:1331, 3993:4000 -- 94 notes 1323:1331, 4374:4375 -- 125 notes 1323:1331, 5625:5632 -- 31 notes 1323:1331, 9800:9801 -- 94 notes 2400:2401, 3993:4000 -- 72 notes 2400:2401, 4374:4375 -- 72 notes 2400:2401, 5625:5632 -- 31 notes 2400:2401, 9801:9800 -- 72 notes 3024:3025, 3993:4000 -- 72 notes 3024:3025, 4374:4375 -- 72 notes 3024:3025, 5625:5632 -- 31 notes 3024:3025, 9800:9801 -- 72 notes 3993:4000, 4374:4375 -- 72 notes 3993:4000, 5625:5632 -- 22 notes 4374:4375, 5625:5632 -- 53 notes 4374:4375, 9800:9801 -- 72 notes 5625:5632, 9800:9801 -- 22 notes
Message: 3963 - Contents - Hide Contents Date: Fri, 01 Mar 2002 09:36:42 Subject: Re: blast from the past re: microtempering partch From: paulerlich we haven't heard much about these 63-tone systems -- any comments on them, gene?
Message: 3964 - Contents - Hide Contents Date: Fri, 01 Mar 2002 08:24:32 Subject: Re: blast from the past re: microtempering partch From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Just for fun, here are the numbers of notes given by various pairs of > 11-limit subdiaschismic unison vectors, keeping 224:225 and 384:385 > as the other two unison vectors:Where did this list of 11-limit commas come from?
Message: 3965 - Contents - Hide Contents Date: Fri, 01 Mar 2002 10:30:05 Subject: Re: blast from the past re: microtempering partch From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> we haven't heard much about these 63-tone systems -- any comments on > them, gene?If you wedge together 100/99^225/224^385/384 or 100/99^225/224^540/539 or 100/99^385/384^540/539 you get the 11-limit version of magic, but there doesn't seem to be any reason to prefer the 63 version over the 41. However, it extends to a different and more accurate 13-limit version of magic in the 63 form.
Message: 3966 - Contents - Hide Contents Date: Fri, 01 Mar 2002 08:54:44 Subject: Re: blast from the past re: microtempering partch From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Just for fun, here are the numbers of notes given by various pairs of >> 11-limit subdiaschismic unison vectors, keeping 224:225 and 384:385 >> as the other two unison vectors: >> Where did this list of 11-limit commas come from?i'll be darned if i remember. any glaring omissions?
Message: 3967 - Contents - Hide Contents Date: Fri, 01 Mar 2002 10:47:15 Subject: Re: blast from the past re: microtempering partch From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: I do get something other than magic for some combinations; for instance 100/99^225/224^1331/1323 gives a temperament with diesis-sized steps: 1/31, 2/63, 3/94 or 4/125.
Message: 3968 - Contents - Hide Contents Date: Fri, 01 Mar 2002 10:48:57 Subject: Re: blast from the past re: microtempering partch From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> we haven't heard much about these 63-tone systems -- any comments on >> them, gene? >> If you wedge together 100/99^225/224^385/384 or100/99^225/224^540/539 or 100/99^385/384^540/539 you get the> 11-limit version of magic, but there doesn't seem to be any reasonto prefer the 63 version over the 41. However, it extends to a different and more accurate 13-limit version of magic in the 63 form. cool! here's one of the 63-tone fokker periodicity blocks: cents numerator denominator 0 1 1 14.367 121 120 38.906 45 44 53.273 33 32 80.537 22 21 84.467 21 20 119.44 15 14 123.37 189 176 150.64 12 11 165 11 10 189.54 135 121 203.91 9 8 235.1 63 55 235.68 55 48 274.58 75 64 284.45 33 28 305.78 105 88 315.64 6 5 347.41 11 9 354.55 27 22 386.31 5 4 396.18 44 35 417.51 14 11 435.08 9 7 466.85 55 42 470.78 21 16 498.04 4 3 509.69 945 704 536.95 15 11 551.32 11 8 568.15 168 121 590.22 45 32 617.49 10 7 621.42 63 44 648.68 16 11 670.76 165 112 687.59 180 121 701.96 3 2 716.32 121 80 733.15 84 55 755.23 99 64 782.49 11 7 786.42 63 40 813.69 8 5 835.76 363 224 852.59 18 11 866.96 33 20 894.22 176 105 905.87 27 16 933.13 12 7 937.06 189 110 968.83 7 4 986.4 99 56 1007.7 315 176 1017.6 9 5 1049.4 11 6 1056.5 81 44 1088.3 15 8 1098.1 66 35 1119.5 21 11 1129.3 48 25 1168.2 108 55 1168.8 55 28 it looks pretty 'fishy' . . .
Message: 3969 - Contents - Hide Contents Date: Sun, 03 Mar 2002 14:00:34 Subject: maps, uvs From: Carl Lumma Can someone post a general method for transforming a list of unison vectors into a map and vice versa? -Carl
Message: 3970 - Contents - Hide Contents Date: Sun, 3 Mar 2002 22:36 +00 Subject: Re: maps, uvs From: graham@xxxxxxxxxx.xx.xx Carl Lumma wrote:> Can someone post a general method for transforming a > list of unison vectors into a map and vice versa? See <Automatically generated temperaments * [with cont.] (Wayb.)>and work out the code. I can see the value in a longer explanation of that, but I haven't done it and certainly won't at this time of night. The basic method is to take the wedge product of the vectors, and the octave-equivalent part is the mapping by generator. Wedge it with some chromatic unison vector and you get an example ET mapping. You can combine the two to get either kind of mapping I print out, but you'll have to check the code to see how. There's also <Unison vector to MOS script * [with cont.] (Wayb.)> that uses matrix operations instead of wedge products. There, you put the octave at the top, the chromatic UV second, and the commatic UVs below in a matrix. Take the adjoint, and the left hand column is your example ET and the next column is the mapping by generator. They're both mathematically equivalent to the same things you get from wedge products. Graham
Message: 3971 - Contents - Hide Contents Date: Sun, 03 Mar 2002 22:38:27 Subject: Re: maps, uvs From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Can someone post a general method for transforming a > list of unison vectors into a map and vice versa?(1) Wedge the unisons together to get a wedgie (2) If you have a linear temperament wedgie, wedge this with 2 to get the map to steps of the generator, and then solve the linear equations to get corresponding octaves for the generator map (3) If you aren't dealing with a linear temperament, you need to find a basis for the generators--wedging with the elements of this basis will give the map (4) I've started writing up a paper on the mathematics of temperament, so I should have this explained in detail. When it's ready I'd like some comments on it!
Message: 3972 - Contents - Hide Contents Date: Sun, 3 Mar 2002 16:34:23 Subject: Re: maps, uvs From: monz hi Gene and Carl,> From: Carl Lumma <carl@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, March 03, 2002 2:00 PM > Subject: [tuning-math] maps, uvs > > > Can someone post a general method for transforming a > list of unison vectors into a map and vice versa? > > -Carlsee the second section of: Tuning Dictionary, "periodicity block" Definitions of tuning terms: periodicity block... * [with cont.] (Wayb.) where i quote Gene's method of finding a notation which maps to JI pitches. (BTW, there were errors in this originally ... did i get them all out, Gene?) to find the mapping to EDOs, put the unison-vectors in vector form into a matrix, then calculate the determinant and the inverse. if the inverse is unimodular (= has a determinant = 1), then it gives the mapping to EDOs, the cardinality of which (i.e., mapping of prime-factor 2) is in the top row. see: Tuning Dictionary, "matrix" Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3973 - Contents - Hide Contents Date: Sun, 3 Mar 2002 17:05:49 Subject: Re: maps, uvs From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, March 03, 2002 4:34 PM > Subject: Re: [tuning-math] maps, uvs > > > to find the mapping to EDOs, put the unison-vectors > in vector form into a matrix, then calculate the > determinant and the inverse. if the inverse is > unimodular (= has a determinant = 1), then it gives > the mapping to EDOs, the cardinality of which (i.e., > mapping of prime-factor 2) is in the top row. see: > > Tuning Dictionary, "matrix" > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)oops ... the matrix is unimodular if the determinant is +1 or -1. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3974 - Contents - Hide Contents Date: Sun, 03 Mar 2002 17:15:09 Subject: listing linear temperaments From: Carl Lumma //This is long. May I humbly suggest we do it up old-school, like //in the Classic onelist years, and reply to everything until we //agree on everything? Let's get something export-quality! Dave //Keenan, activate your magic power ring, Voltron is needed once //again! Monz, break out the colored chalks! Paul, I totally //understand you wanting to take a break, and I've always been //behind a book from you, but why not finish the paper on linear //temperamenst first? A paper. I think it's a great idea. And, the 569 of us who don't have a computer set up to do calculations on linear temperaments need a list! Graham's catalog, "The grooviest 7-limit temperaments", Monzo's lines, and Herman Miller's "Carl's favorite page on the internet" Warped Canons page are huge, huge, huge. But wouldn't it be cool to really get the goat? --------------------------- Selection criteria (1) Badness --------------------------- ~~~~~~~~~~~~~~~~ (1a) Gene's list ~~~~~~~~~~~~~~~~ Paul wrote...>Gene, who's way, way ahead of any other theorist on this list (and >possibly anywhere) has (like Dave Keenan and Graham Breed before >him) completed a comprehensive search for linear temperaments for >7-limit music. He proposed a 'badness' measure defined as: > >step^3 cent > >where step is a measure of the typical number of notes in a scale >for this temperament (given any desired degree of harmonic depth), >and cent is a measure of the deviation from JI 'consonances' in >cents.Is this still state-of-the-art-badness? I seem to remember something about different exponents for each prime/odd (?) identity, taken from coefficients of Diophantine equations, or some such? Don't need details, just want to know if we need a new top 20. Paul wrote...>He then ranked his 505 temperaments by 'goodness'. The familiar >ones don't come in until later, so bear with me . . .Gene, you initially stopped after listing 20. Did you ever list the requested next few needed to uncover meantone? Are you still happy with your list? ~~~~~~~~~~~~~~~~~~~~~ (1b) The slippery six ~~~~~~~~~~~~~~~~~~~~~ I wasn't reading the tuning-math very closely back then, but Gene, your top 20 is generated by starting with some large number of ets and then seeing what temperaments they share, sort of like a more precise version of looking for lines on Herman Miller's / Paul's charts, right? But you found that some temperaments only hit a single et up to your cutoff -- those were the slippery six, right? Do we have a general solution to this problem -- making the cutoff really high, an entirely different method, etc.? Speaking of this method, nobody ever answered this: Carl wrote...>More to the point, every line on this plane is a linear temperament, >right? So what makes low-numbered (less than 100) equal >temperaments cluster on some of them?What makes some linear temperaments belong to more than one et, out of ets as high as some given number? They would have to share a common generator... Is sharing a common generator related to the un-even distribution of the rationals on the number line (such as makes harmonic entropy work)? Carl wrote...>Intersections (by eye) >---------------------- >12 - 8 >31 - 5 >22, 37, 15, 34, 19 - 4 >29, 41, 75, 61, 53, 72, 23 - 3Why are some of the 'best' ets (ones that have gotten so much attention on these lists for so many different reasons, for so long) here? Is it because we've often defined "best" as "consistent", and where two lines cross the same tuning is being reached two different ways (via two different maps), which requires consistency? --------------------------- Selection criteria (2) Maps and commas -------------------------- A map uniquely defines a linear temperament? Or do you also need period? Looking at Graham's catalog, I'm not sure how to use maps with non-octave periods. Carl wrote...>I say the most powerful maps are the ones with the smallest >numbers in them. Sum of abs value would work.Or, maybe the sum of the abs values of the max and min numbers in the map, for a given limit (or divided by the card of the map, if you want to compare across limits). Which is better? There's definitely some overlap with badness here, but by not considering the quality of the approximations, doesn't this tell us more about the abstract musical-theoretic properties of a temperament? Carl wrote...>Finally, re the jumping jacks / ideal comma question... what's the >question? How are we defining "most powerful" comma? ? Carl wrote... >What's the relationship between a comma vanishing and a map? ? -------------------------The contents of the list ------------------------- Paul wrote...>> Generators on the table Carl wrote...>Yes, I completely agree. Who can furnish rms optimums? Paul wrote... >i bet gene can do this in a jiffy. maybe graham too. >and oh, we need the period as well as the generator.I completely agree. Paul wrote...>actually, gene already did this back in december.I looked. My eyes! The searching did nothing! Paul wrote...>i'm making a graph that includes these as well as the ets. > >well, i tried to, but the points get too crowded near the >center for me to label them.F the graph. Let's have a list! Paul wrote...>but it's easy to see the optimum point on the graph on monz's >page already. simply look at the line representing the >temperament you're interested in, and the point on that line >that comes closest to the center ('origin') of the graph is >the optimal one. so optimal meantone is near 50-equal, and >optimal magic (in 5-limit at least) is near 60-equal, etc.I figured as much. But what if the nearest et below 100 is off the optimum some? Why not do it right? Paul wrote...>Topping off Gene's list are some very funky simple temperaments, /.../ >For these, I quote the simplest pair of unison vectors: > > (1) <21/20,27/25> > (2) <8/7,15/14> > (3) <9/8,15/14> > (4) <25/24,49/48> > (5) <15/14,25/24> > (6) <21/20,25/24> > (7) <15/14,35/32> > (8) <7/6,16/15> > (9) <16/15,21/20>Can we get a list with optimum generator, et series, commas, maps, periods for these (and the rest of the top 20)? Are any of the "Monzo's lines" temperaments in here? -Carl
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