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Message: 11475 - Contents - Hide Contents Date: Sun, 18 Jul 2004 07:09:52 Subject: 15 microtemperaments supported by 270 From: Gene Ward Smith The point posting this isn't simply to annoy people; it seems to me that 270 is an obvious route to notating the 13-limit. I am sure thet it has a Saggital method worked out for it already, but I see that a generator of 44/39, and a period of 1/2 octave, for instance, might be a good way to work with the 13-limit. We also find the plausible ways to extend ennealimmal and hemiennealimmal to the 13 limit in this way, and it is nice to know they are out there. There's a lot of ten-nominal stuff lurking in the underbrush as well. On the bottom line is Graham complexity, TOP error, and the resultant logflat badness. [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61] [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] 102 .078096 173.897744 [50, 60, 110, 70, 0, -21, 34, -62, -185, 87, -45, -222, -184, -407, -259] [[10, 14, 21, 24, 32, 37], [0, 5, 6, 11, 7, 0]] 110 .093739 236.724371 [72, 108, 72, 36, -54, 4, -88, -192, -352, -136, -290, -525, -148, -418, -320] [[18, 28, 41, 50, 62, 67], [0, 4, 6, 4, 2, -3]] 198 .049207 330.980012 [12, 63, 102, 6, 81, 72, 128, -32, 84, 60, -204, -45, -336, -150, 258] [[3, 4, 3, 2, 10, 6], [0, 4, 21, 34, 2, 27]] 102 .149522 332.943641 Hemiennealimmal [36, 54, 36, 18, 108, 2, -44, -96, 38, -68, -145, 51, -74, 170, 307] [[18, 28, 41, 50, 62, 65], [0, 2, 3, 2, 1, 6]] 108 .144410 353.701213 [16, 84, 46, 98, 108, 96, 28, 100, 112, -129, -63, -60, 116, 133, 11] [[2, 4, 9, 8, 12, 13], [0, -8, -42, -23, -49, -54]] 108 .149522 366.221101 Ennealimmal [18, 27, 18, -126, -81, 1, -22, -262, -195, -34, -386, -288, -416, -294, 186] [[9, 15, 22, 26, 26, 30], [0, -2, -3, -2, 14, 9]] 162 .076135 366.527988 [2, -57, -28, 46, 81, -95, -50, 66, 121, 95, 304, 399, 226, 331, 110] [[1, 1, 19, 11, -10, -20], [0, 2, -57, -28, 46, 81]] 138 .102172 376.526707 [34, 111, 64, -28, 27, 97, 6, -162, -83, -163, -449, -348, -300, -161, 197] [[1, 7, 20, 13, -1, 8], [0, -34, -111, -64, 28, -27]] 139 .104131 388.394328 [92, 78, 62, -44, -54, -90, -160, -388, -426, -75, -372, -414, -338, -381, -24] [[2, 26, 24, 21, -4, -6], [0, -46, -39, -31, 22, 27]] 238 .044086 402.962652 [24, -9, -66, 12, 27, -70, -172, -64, -46, -128, 59, 96, 262, 320, 49] [[3, 2, 8, 16, 9, 8], [0, 8, -3, -22, 4, 9]] 114 .152174 407.862914 [52, 3, 82, 116, 81, -116, -16, 4, -64, 182, 259, 177, 42, -76, -149] [[1, -4, 2, -6, -9, -5], [0, 52, 3, 82, 116, 81]] 116 .152722 421.372278 [20, -30, -10, -80, 0, -94, -72, -196, -74, 61, -82, 111, -190, 37, 296] [[10, 16, 23, 28, 34, 37], [0, -2, 3, 1, 8, 0]] 120 .151497 442.287562 [74, 51, 44, 82, 27, -91, -138, -126, -231, -41, 14, -126, 78, -87, -210] [[1, -25, -16, -13, -26, -6], [0, 74, 51, 44, 82, 27]] 148 .110293 456.719330 [70, 30, 100, -10, 0, -115, -38, -258, -259, 148, -127, -111, -374, -370, 37] [[10, 13, 22, 24, 35, 37], [0, 7, 3, 10, -1, 0]] 150 .117374 497.041042
Message: 11476 - Contents - Hide Contents Date: Sun, 18 Jul 2004 09:09:02 Subject: Re: names and definitions: schismic From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: >>> 12&19 meantone isn't excessively complex... >> "Excessively complex" is a value judgment which only > makes sense in the context of some projected use; hence > it is really meaningless for us, since the possible uses > are varied. So long as the *commas* are not complex it > certainly might find a use in electronic music (and > this *is* the 21st century; this aspect is likely to > assume ever-increasing importance) if nowhere else. And > from a comma point of view, this so-called "excessively > complex temperament" is not that complex! It has commas > of 4375/4374 (Hahn size 7) and 32805/32768 (Hahn size 9.) > You are simply failing to think in terms of the 7-limit > lattice when you call it "excessively complex".please give me enough data (or point to it if it's already posted somewhere) to *make* that 7-limit lattice for my webpage. this is *precisely* the kind of thing i want to enhance the Encyclopaedia with, to help as many people as possible understand the amazing rapid developments that have occured in tuning theory within the last few years.>> and it's not a good idea to change established names >> in any case. >> We've *been* changing established names; I don't think > 7-limit schismic was ever as established as some Paul > wants to deep-six. Moreover, sticking to a consistent > scheme means we have a better idea what the name means; > in this case, that the name in the higher limit has a > tuning in accord with the lower limit.i've been thinking a lot about the naming thread lately, and about an hour before i read this post, i was thinking that "orwell" really provides a clue here. orwell is already the *family* name for these temperaments, and it looks like Gene made "george" a member of the family. so now we can have "emily", "mike", etc. maybe there will be a "bundy" family, with members like "al", "peggy", "kelly", and "bud". how about a "simpson" family, with "homer", "marge", "bart", "lisa", and "maggie"? or the small "petri" family, consisting only of "rob", "laura", and "ritchie" ... but with friends and co-workers like "buddy", "sally", "mel", "jerry & millie" (a pair of married temperaments), and occasional appearances by "alan" and "stacey". ... for the younger people on the list who might not get that last reference, try this: Dick Van Dyke Show * [with cont.] (Wayb.) all kidding aside ... i can sympathize with those who are alarmed by the "jargon explosion", but really, naming tunings is a good thing, because it makes it much easier to talk about them and remember stuff about them. putting the names together in families like this is a great way to keep them organized. i've created a webpage for "family" in the Encyclopaedia Tonalsoft Encyclopaedia of Tuning - family, (c... * [with cont.] (Wayb.) it contains a sample table of some 5-limit families and an incomplete list of EDOs which belong to them. (i got the data from my own bingo-card webpage) others are welcome to help fill out that table with other family names, and to construct whole new tables for other prime-limits.> This is the ***21st century***. I would *not* assume people > in the future are going to ignore complex tunings.you go, Gene! -monz
Message: 11477 - Contents - Hide Contents Date: Sun, 18 Jul 2004 00:39:23 Subject: Re: names and definitions: schismic From: Graham Breed Herman Miller wrote:> 12&19 meantone isn't excessively complex, and it's not a good idea to > change established names in any case. If you have to go all the way to > 118&171 for schismic, it's way too complex to be of much use to anyone; > very few people will bother with it at all, so why give it the familiar > name when there are better options? Even the 13-limit Cassandra 1 is > only 41&94.53&118 is simpler than 118&171. Graham
Message: 11478 - Contents - Hide Contents Date: Sun, 18 Jul 2004 09:22:06 Subject: Generators as names From: Gene Ward Smith We've been talking a little about this. I thought I'd point out that not only can you get a generator from two numbers specifying an equal division range, you can invert the process, deriving from a fraction two integers; simply take the next and the previous numbers on the row of the Farey sequence for that fractional generator, and the denominators are your two numbers. So from 18/31 one gets, for instance, [19, 12]; where the ordering means that 19 is on the flat side for this generator, and 12 on the sharp side. We can also interpret things in terms of periods other than an octave; 3/34 gives us [23, 11], which we can interpret in terms of 34-equal, but which when multiplied by two gives us [46, 22]; and 2/23 < 3/34 < 1/11, using half-octave periods, become shrutar generators. Hence if we have generator and period in the form [1/2, 3/31], we can obtain from them [46, 22], and if we have [46, 22], we can obtain from that [1/2, 3/31]. If now we interpret these as defining vals in some p-limit, we can obtain a temperament. Of course, this might not work out the way you want; from [19, 12] which we can derive from 18/31, we get a fine 5 and 7 limit version of the meantone mapping of 31, but we don't get either accurate version in the 11-limit. Instead we map 11 to six fifths, which means (3/2)^6/11 = 729/704 is a comma. This boils down to the system where 45/44 is a comma, which might be worth putting on Joe's meantone page, I suppose; 88 does better for this as a tuning than 31 in case anyone feels inspired to try it. This suggests to me that to name an 11-limit version of meantone, we don't use 18/31; 29/50 will give us meanpop, and 25/43 suffices for huygens. Hence "11-limit 29/50" could be a possible name or way of denoting meanpop, and likewise "11-limit 25/43" could denote huygens. It's really just another form of the 19&31, 12&31 method.
Message: 11479 - Contents - Hide Contents Date: Sun, 18 Jul 2004 00:19:46 Subject: Re: names and definitions: meantone From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> Gene, when you wrote, for example, >> "7 limit poptimal: 86/205", you *did* mean >> "7-limit poptimal *generator*", right? >> Right; the generator is 86/205 of an octave, or 2^(86/205).to be precise, you really mean 86/205 of the period, which in this case happens to be an octave ... correct? -monz
Message: 11480 - Contents - Hide Contents Date: Sun, 18 Jul 2004 09:37:58 Subject: Re: names and definitions: schismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> please give me enough data (or point to it if it's already > posted somewhere) to *make* that 7-limit lattice for my > webpage.The TM basis is {4375/4374, 32805/32768}, and the Hahn reduced basis is {4375/4374, 65625/65536}; these are the two most interesting block commas to use; of course {32805/32768, 65625/65536} is another possibility, and there are other commas out there, such as 95703125/95551488. Incidentally, if you are putting up the web page I don't know what you are going to use as a name, but whatever it is may need a "?" after it. However, instead of calling it 118&171 we should call it 53&118, since 53 and 118 are really the extreme ranges, and 171 is a good choice for tuning, and those are the rules for deciding the question. I just did it wrong.> i've been thinking a lot about the naming thread > lately, and about an hour before i read this post, > i was thinking that "orwell" really provides a clue here. > > orwell is already the *family* name for these temperaments, > and it looks like Gene made "george" a member of the > family. so now we can have "emily", "mike", etc.Fortunately I don't think too many more family members are likely to show up.> maybe there will be a "bundy" family, with members > like "al", "peggy", "kelly", and "bud".Don't forget Ted!> all kidding aside ... i can sympathize with those > who are alarmed by the "jargon explosion", but really, > naming tunings is a good thing, because it makes it > much easier to talk about them and remember stuff > about them. putting the names together in families > like this is a great way to keep them organized.As I say, naming tunings is not such a big deal for me, but I strongly resist the idea that concepts like "val" and "monzo" do not deeserve names. They are fundamental concepts.>> This is the ***21st century***. I would *not* assume people >> in the future are going to ignore complex tunings.> you go, Gene!Not everyone agrees; but I will inflict another piece of music on the world fairly soon now, and theory, despite what many think, does actually translate to practice.
Message: 11481 - Contents - Hide Contents
Date: Sun, 18 Jul 2004 00:28:54
Subject: Magic
From: Gene Ward Smith
family name: magic, 19&22
period: octave
generator: flat major third
5-limit
name: magic, 19&22
comma: 3125/3072
mapping: [<1 0 2|, <0 5 1|]
poptimal generator: 19/60
TOP period: 1201.276744: generator: 380.795718
MOS: 7, 10, 13, 16, 19, 22, 41, 60
7-limit
name: magic, 19&22
wedgie: <<5 1 12 -10 5 25||
mapping: [<1 0 2 -1|, <0 5 1 12|]
7-limit poptimal generator: 13/41
9-limit poptimal generator: 71/224
TOP period: 1201.276744 generator: 380.795718
TM basis: {225/224, 245/243}
MOS: 7, 10, 13, 16, 19, 22, 41
name: muggles 16&19
wedgie: <<5 1 -7 -10 -25 -19||
mapping: [<1 0 2 5|, <0 5 1 -7|]
7-limit poptimal generator: 62/197
9-limit poptimal generator: 53/168
TOP period: 1203.148010 generator: 379.393104
TM basis: {126/125, 525/512}
MOS: 7, 10, 13, 16, 19, 35, 54
name: witch
wedgie: <<5 1 9 -10 0 18||
mapping: [<1 0 2 0|, <0 5 1 9|]
7-limit poptimal generator: 39/125
9-limit poptimal generator: 6/19
TOP period: 1193.274911 generator: 377.328028
TM basis: {36/35, 1875/1792}
MOS: 7, 10, 13, 16, 19
11 limit
name: magic, 19&22
wedgie: <<5 1 12 -8 -10 5 -30 25 -22 -64||
mapping: [<1 0 2 -1 6|, <0 5 1 12 -8|]
poptimal generator: 33/104
TOP period: 1200.749423 generator: 380.922404
TM basis: {100/99, 225/224, 245/243}
MOS: 7, 10, 13, 16, 19, 22, 41, 63, 104
Message: 11482 - Contents - Hide Contents Date: Sun, 18 Jul 2004 09:47:07 Subject: Re: names and definitions: meantone From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>> Right; the generator is 86/205 of an octave, or 2^(86/205). >> >>>> to be precise, you really mean 86/205 of the period, >> which in this case happens to be an octave ... correct? >> No, I always intend the fractions as fractions of an octave.OK, that answers my question with respect to your own individual usage. but my real question was: shouldn't the reader assume that the generator size is given as a fraction of the equivalence-interval? so in all of your definitions, it is an octave, but the definitions don't necessarily have to make it an octave. am i wrong about this? -monz
Message: 11483 - Contents - Hide Contents Date: Sun, 18 Jul 2004 00:30:40 Subject: Re: names and definitions: schismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Herman Miller wrote: >>> 12&19 meantone isn't excessively complex, and it's not a good idea to >> change established names in any case. If you have to go all the way to >> 118&171 for schismic, it's way too complex to be of much use to anyone; >> very few people will bother with it at all, so why give it the familiar >> name when there are better options? Even the 13-limit Cassandra 1 is >> only 41&94. >> 53&118 is simpler than 118&171. > > > Graham
Message: 11484 - Contents - Hide Contents Date: Sun, 18 Jul 2004 09:55:28 Subject: very-high-cardinality ETs (was: names and definitions: schismic) From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> At some point a temperament's errors are so low and > its complexity so high, that you might as well use > strict rational intervals (or the best approximation > of them you can get on your instrument/computer). > If that's the case, why would anyone care about > the temperament?it's been a basis of my theories for a long time that humans make use of two fundametally different pitch-perception mechanisms, both of which recognize some equivalence-interval: one which is based on the harmonicity of the conglomerate sounds (ratios), and the other which is based on approximately equal logarithmic divisions of that equivalence-interval (ETs). from this perspective, it's *very* valuable to have ways of comparing and relating similar tunings by means of calculations based on either of the two mechanisms. -monz
Message: 11485 - Contents - Hide Contents Date: Sun, 18 Jul 2004 00:32:02 Subject: Re: names and definitions: schismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> 53&118 is simpler than 118&171.Perhaps that should be the name; it certainly covers anything you'd want to use for it, and 171 is a good version of it.
Message: 11486 - Contents - Hide Contents Date: Sun, 18 Jul 2004 10:03:56 Subject: Re: Extreme precison (Olympian) Sagittal From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> It would be easy enough if I had a table, in ascii, > posted to this list, which gave a rational number > followed by a 9-vector on each line.if you would furnish me with a table of ASCII-sagittal symbols with their ratios, i'll put it into the "sagittal" definition webpage. you might want to look at this and use the same format; you'll find many of your ratios in it: Tonalsoft Encyclopaedia of Tuning - big list o... * [with cont.] (Wayb.) -monz
Message: 11487 - Contents - Hide Contents Date: Sun, 18 Jul 2004 00:33:24 Subject: Re: names and definitions: meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>> Right; the generator is 86/205 of an octave, or 2^(86/205). > >> to be precise, you really mean 86/205 of the period, > which in this case happens to be an octave ... correct?No, I always intend the fractions as fractions of an octave.
Message: 11488 - Contents - Hide Contents Date: Sun, 18 Jul 2004 10:45:49 Subject: Alternative names for Paul's paper -- 5-limit LTs From: Dave Keenan This is an update of a table I sent to Paul. I've only done the 5- limit ones so far. Did I miss any? Did I get any wrong? Erlich name Other names used in discussions Descriptive name in terms of generators as fractions of just intervals Descriptive name in terms of ETs Dicot Neutral thirds Minor major thirds 7&10-LT Meantone 'Diatonic' Wide fourths, Narrow fifths 12&19-LT Augmented Diesic Triple minor thirds 12&27-LT Mavila Pelogic Extra-wide fourths 7&16-LT Porcupine Semi minor thirds 15&22-LT Blackwood Quintuple (major) thirds 15&25-LT Dimipent Diminished, 'Octatonic' Quadruple (major) thirds 12&28-LT Srutal Diaschismic, 5-limit Pajara Twin narrow fourths, Twin wide fifths 22&46-LT Magic 5-limit Magic, Major thirds (Simple) (major) thirds 19&22-LT Superchrome Superchrome Tri(sect) minor thirds 12&35-LT Hanson Kleismic (Simple) minor thirds 19&34-LT Negripent Negri Tri(sect) (major) thirds 19&29-LT Tetracot Quartafifths, Minimal diesic Quarter fifths 34&41-LT Superpyth Superpythagorean Narrow fourths, Wide fifths 22&27-LT Helmholtz Schismic, Helmholz/Groven Complex fourths, Complex fifths 12&41-LT Sensipent Semisixths, Tiny diesic Semi (major) sixths 19&46-LT Subchrome Quarter (major) thirds 12&37-LT Würschmidt Wuerschmidt Complex (major) thirds 31&34-LT Compton Aristoxenean 12-fold 15 cents 12&72-LT Amity AMT, Acute minor thirds 5-part elevenths, 339 cents 46&53-LT Orson 5-limit Orwell Tri(sect) minor sixths 22&31-LT
Message: 11489 - Contents - Hide Contents Date: Sun, 18 Jul 2004 01:17:51 Subject: Re: Atomic temperament as a basis for a valhalla-level notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > Here are some 11-limit intervals corresponding to certain quantities > of schismas in atomic. If we had symbols for these, it would be a start.Yes. We can supply a symbol for every multiple of the 5-schisma. I'll put the ASCII longhand version of the symbol before the ratio below that is it's definition or primary role. Any with an asterisk after the ratio is still open for discussion about whether that should be the symbol's defining ratio or not. There was only one for which the symbol of the right size doesn't currently have one of the commas you listed, as its primary role, and that's the two schismas symbol .|(, for which we have 385/384 as its primary role.> 1: '| 32805/32768, 703125/702464, 5632/5625 > 2: 441/440, 1375/1372, 6250/6237, .|( 385/384* > 3: 3136/3125, |( 5120/5103 > 4: '|( 225/224 > 5: .)~| 176/175*, )|( 896/891 > 7: 4000/3969, ')~| 126/125* > 9: .)|~ 99/98*, ~~| 100/99* > 10: ./| 2048/2025 > 11: /| 81/80, 3125/3087, 245/242 > 14: |) 64/63 > 16: .(| 56/55 > 18: '(| 50/49* > 20: (|( 45/44 > 21: .//| 128/125 > 25: /|) 36/35 > 29: '(/| 405/392 > 32: '(|) 648/625 > 39: ||~ 256/245* > 43: '/||360/343*
Message: 11490 - Contents - Hide Contents Date: Sun, 18 Jul 2004 10:47:02 Subject: Re: Extreme precison (Olympian) Sagittal From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> It would be easy enough if I had a table, in ascii, posted to this >> list, which gave a rational number followed by a 9-vector on each > line. >> The more of our work you duplicate, the more chance of finding our > mistakes. But here it is, tab-delimited, with column headings. > > num den '| )| (| /| ~| |~ |\ > |) |( symbol comma name > 32768 32805 1 0 0 0 0 0 0 > 0 0 '| 5-schisma > > etc. -- snip>i've put a neat copy of it, with monzos added, into the Encyclopaedia "sagittal" page. -monz
Message: 11491 - Contents - Hide Contents Date: Sun, 18 Jul 2004 01:46:28 Subject: Re: Atomic temperament as a basis for a valhalla-level notation From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> There was only one for which the symbol of the right size doesn't > currently have one of the commas you listed, as its primary role, > and that's the two schismas symbol .|(, for which we have 385/384 as > its primary role.This suggests there may be a pretty good fit to atomic. 385/384 is mapped by atomic to one semitone, down 49 schismas, so it isn't what we'd want to use for 2 schismas; but a 441/440 symbol fits nicely.
Message: 11492 - Contents - Hide Contents Date: Sun, 18 Jul 2004 10:50:34 Subject: definition needed: logflat badness From: monz Gene, please, please, PLEASE ... write an explanation of "logflat badness" that the layperson can understand ... feel free to include all the heavy math stuff at the end, but keep the beginning easy. thanks. -monz
Message: 11493 - Contents - Hide Contents Date: Sun, 18 Jul 2004 02:25:43 Subject: Re: Extreme precison (Olympian) Sagittal From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I was just hoping for a statement of the problem which did not require > me to read things I have to squint at and still can't make out. What > are the barbs, arcs, scrolls, boathooks and accent marks supposed to > do, in numerical terms?I understand it may be difficult to figure out the flag- decomposition of some of the symbols in Figure 3 of the paper, but those in Table 1 should be obvious enough. I realise now that you should ignore Figure 3, and see if you come up with the same new unaccented symbols that we did. If you're having to squint to figure out the flag-composition of the symbols in Table 1, then maybe you need to set the zoom on your PDF reader to exactly 100%, or maybe 200%. But just in case, here they are in the same order as in Table 1, in ASCII longhand: '| )| |( ~| )|( ~|( |~ ./| )|~ /| |) |\ (| (| ( //| /|) /|\ (|) (|\ So you could set up 9 columns, one for each flag, with a row for each symbol. The accent column would have either -1, 0 or +1, and all the others would have 0, 1 or 2. For any symbol, the sum of the numbers in the non-accent columns must never be greater than 2. Remember it's only the first comma definition for each that has to be preserved. And you only need to consider single-shaft upward symbols smaller than a semiapotome (56.84 c). I think you wrote elsewhere that you thought with this "Olympian" Sagittal we were on a quest for the ultimate notation. That's not really the case. It's just that we came up with symbols that did everything we thought anyone actually needed to do, and then we said, "Well we have these 9 flag types, there must be plenty of spare capacity here. How far can we go with what we've got (for JI and ETs)?" The rules of the game are: 0. No new flag types. 1. Every flag must be assigned an approximate value in cents, and every symbols must represent a comma close to its sum-of-flags value (no more than about 0.5 c away). 2. Start with the symbols in Table 1 of http://users.bigpond.net.au/d.keenan/sagittal/Sagittal.pdf - Type Ok * [with cont.] (Wayb.) and ensure they keep their primary comma roles (on same line as symbol, not necc. in bold). 3. Allow new symbols to be formed by adding either an up or down accent to any of the unaccented symbols in Table 1 or in the Athenian or Trojan sets. See below. 4. No new unaccented symbols are to be created (from new pairings of existing flags) if an accented version of an existing (Table 1 or Athenian or Trojan) symbol will do. 5. If new symbols are created, they must have no more than two ordinary flags (in the same direction) and one accent (up or down). I'm pretty sure the only symbols in Athenian or Trojan but not in Table 1 are )/| and /|~ Have fun.
Message: 11494 - Contents - Hide Contents Date: Sun, 18 Jul 2004 10:57:29 Subject: Stepping through generators of a fifth From: Gene Ward Smith In the table below, the fractions are from a portion of the 99th row of the Farey sequence; then we have the corresponding range of ets, flat to sharp, and the 7-limit wedgie obtained from the standard vals. It is interesting to see what lurks in regions which have recieved less attention in the vicinity of a fifth as generator; of course occasionally near the endpoints the fifth does not turn out to even be the generator, due to the use of standard vals. One could take the same list and work it up for other periods, other prime limits, or even other ways of deciding on the vals. 53/92 [33, 59] [34, -22, -9, -114, -110, 41] 34/59 [33, 26] [1, -22, -9, -37, -17, 41] 49/85 [59, 26] [25, 22, 9, -23, -56, -41] 15/26 [7, 19] [1, 4, -9, 4, -17, -32] 56/97 [26, 71] [9, 10, -3, -5, -30, -35] 41/71 [26, 45] [1, 4, -9, 4, -17, -32] 26/45 [26, 19] [1, 4, -9, 4, -17, -32] 37/64 [45, 19] [1, 4, -9, 4, -17, -32] 48/83 [64, 19] [1, -15, -28, -26, -47, -23] 11/19 [7, 12] [1, 4, -2, 4, -6, -16] 51/88 [19, 69] [1, 4, 29, 4, 43, 56] 40/69 [19, 50] [1, 4, 10, 4, 13, 12] 29/50 [19, 31] [1, 4, 10, 4, 13, 12] 47/81 [50, 31] [1, 4, 10, 4, 13, 12] 18/31 [19, 12] [1, 4, 10, 4, 13, 12] 43/74 [31, 43] [1, 4, 10, 4, 13, 12] 25/43 [31, 12] [1, 4, 10, 4, 13, 12] 57/98 [43, 55] [1, 4, -33, 4, -55, -88] 32/55 [43, 12] [1, 4, 10, 4, 13, 12] 39/67 [55, 12] [1, 4, 22, 4, 32, 40] 46/79 [67, 12] [1, 4, 22, 4, 32, 40] 53/91 [79, 12] [1, 16, 22, 23, 32, 6] 7/12 [7, 5] [1, 4, -2, 4, -6, -16] 52/89 [12, 77] [1, -8, -26, -15, -44, -38] 45/77 [12, 65] [1, -8, -26, -15, -44, -38] 38/65 [12, 53] [1, -8, -14, -15, -25, -10] 31/53 [12, 41] [1, -8, -14, -15, -25, -10] 55/94 [53, 41] [1, -8, -14, -15, -25, -10] 24/41 [12, 29] [1, -8, -14, -15, -25, -10] 41/70 [41, 29] [1, -8, -14, -15, -25, -10] 58/99 [70, 29] [1, -37, -43, -61, -71, 4] 17/29 [12, 17] [1, -8, -2, -15, -6, 18] 44/75 [29, 46] [1, 21, 15, 31, 21, -24] 27/46 [29, 17] [1, -8, 15, -15, 21, 57] 37/63 [46, 17] [1, -25, 15, -42, 21, 105] 47/80 [63, 17] [1, -25, 15, -42, 21, 105] 57/97 [80, 17] [1, -42, 15, -69, 21, 153] 10/17 [12, 5] [1, 4, -2, 4, -6, -16] 53/90 [17, 73] [1, 43, -19, 66, -33, -165] 43/73 [17, 56] [1, 26, -19, 39, -33, -117] 33/56 [17, 39] [1, 26, -19, 39, -33, -117] 56/95 [56, 39] [1, 26, -19, 39, -33, -117] 23/39 [17, 22] [1, 9, -2, 12, -6, -30] 36/61 [39, 22] [1, -13, 20, -23, 29, 83] 49/83 [61, 22] [1, -13, 20, -23, 29, 83] 13/22 [17, 5] [1, 9, -2, 12, -6, -30] 55/93 [22, 71] [1, 9, -24, 12, -41, -81] 42/71 [22, 49] [1, 9, -2, 12, -6, -30] 29/49 [22, 27] [1, 9, -2, 12, -6, -30] 45/76 [49, 27] [1, 9, -2, 12, -6, -30] 16/27 [22, 5] [1, 9, -2, 12, -6, -30] 51/86 [27, 59] [1, -18, -2, -31, -6, 46]
Message: 11495 - Contents - Hide Contents Date: Sun, 18 Jul 2004 02:41:04 Subject: Re: Atomic temperament as a basis for a valhalla-level notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: >>> There was only one for which the symbol of the right size doesn't >> currently have one of the commas you listed, as its primary role, >> and that's the two schismas symbol .|(, for which we have 385/384 as >> its primary role. >> This suggests there may be a pretty good fit to atomic. 385/384 is > mapped by atomic to one semitone, down 49 schismas, so it isn't what > we'd want to use for 2 schismas; but a 441/440 symbol fits nicely.OK. Well you saw that 385/384 isn't set in stone for this symbol. 441/440 was second in line. That tentative decision was based on the count of ratios in the Scala archive that would be notated exactly by these commas. I just looked up the actual numbers and there were 52 ocurrences for 385/384 and 51 for 441/440, so that's neither here nor there. I think that your "atomic" observation, plus the fact that 441/440 (which we'd call the 49:55-schisma) is closer to the sum-of-flags, means that it should indeed be the primary role of .|( . What do you think Mr Secor?
Message: 11496 - Contents - Hide Contents Date: Sun, 18 Jul 2004 11:00:01 Subject: Re: Alternative names for Paul's paper -- 5-limit LTs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> This is an update of a table I sent to Paul. I've only done the 5- > limit ones so far. Did I miss any? Did I get any wrong?Things seem to be getting more and more complicated.
Message: 11497 - Contents - Hide Contents Date: Sun, 18 Jul 2004 02:51:20 Subject: Re: Extreme precison (Olympian) Sagittal From: Dave Keenan Rule 5 may have been ambiguous.> 5. If new symbols are created, they must have no more than two > ordinary flags (in the same direction) and one accent (up or down).By "in the same direction" I meant "either both up or both down", i.e. both on the same end of the shaft. You are allowed to have two flags on the same side of the shaft, as in the 25-small-diesis symbol //|
Message: 11498 - Contents - Hide Contents Date: Sun, 18 Jul 2004 11:01:15 Subject: Re: Generators as names From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Of course, this might not work out the way you want; > from [19, 12] which we can derive from 18/31, we get a > fine 5 and 7 limit version of the meantone mapping of 31, > but we don't get either accurate version in the 11-limit. > Instead we map 11 to six fifths, which means (3/2)^6/11 = > 729/704 is a comma. This boils down to the system where > 45/44 is a comma, which might be worth putting on Joe's > meantone page, I suppose;hmm ... i remember that 45/44 popping up all over the place when i was trying to analyze Schoenberg's mappings to 12edo, but basing my research on the typo in Carter's translation of _Harmonielehre_. in that mistake, the 11th harmonic of "F" is given as "B" (+6 generators) instead of the correct "Bb" (-1 generator) that's in the German version (which is notated "B" ... i know, confusing...).> 88 does better for this as a tuning than 31 in case anyone > feels inspired to try it.88edo is LucyTuning ... close enough, anyway. -monz
Message: 11499 - Contents - Hide Contents Date: Sun, 18 Jul 2004 09:47:02 Subject: Re: Alternative names for Paul's paper From: Carl Lumma>This is an update of a table I sent to Paul. I've only done the 5- >limit ones so far. Did I miss any? Did I get any wrong?Are you aware that the e-mail RFC calls for hard reterns every 76 chars? Some servers put them in. Anyway, your table came wrapped. Why not upload it to the web? -Carl
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