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Message: 4500 - Contents - Hide Contents Date: Wed, 03 Apr 2002 20:43:42 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:>>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>>>>>> Which is one reason we have both 26:27 ; |, and 1024:1053 { } > as>>>> 13-commas. Are you saying we shouldn't have 1024:1053 at all? >>>>>> so the rule is that every comma and its 2187:2048 complement has a >>> unique symbol? >>>> No, not every comma. So far George and I have only agreed on the >> desirability of apotome complements of those commas which are close > to>> the half-apotome, say those between 1/3 and 2/3 apotome. So far > that's>> only the 11 and 13 commas (dieses). >> but these 'sizes' won't come out anything like that in many, if not > most, equal temperaments. right?Not really. When notating those strange and difficult ET's that I must assume you have in mind (inasmuch as the twelve ET's under 100 that I have tried so far behave quite nicely, thank you*), you can pick and choose the most appropriate symbols to use for the various degrees of alteration. There are at present (in the 217-ET-based notation) 13 different symbols less than half an apotome in size from which to select. Only the two largest of these have dedicated apotome-complement symbols. These four symbols taken as a group are the four varieties of "semisharps" or "semiflats" representing ratios of 11 and 13, and their distinctive appearance would make it relatively easy to translate them at sight from JI to ET notation -- a primary reason for providing the dedicated symbols. By the way, the approximate 1/3 to 2/3 apotome range given above has turned out to be narrower: more like 4/10 to 6/10 apotome. --George *The twelve ET's are 22, 27, 34, 41, 43, 46, 50, 53, 58, 72, 94, and 96. I didn't bother to count the more trivial ones, such as 12, 17, 19, and 31.

Message: 4504 - Contents - Hide Contents Date: Thu, 04 Apr 2002 07:43:08 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "annelizkeenan" <annelizkeenan@y...> wrote: > >>>> -- Dave Keenan >> who's anneliz and does she like to annelize things as much as you > do? :) Tee hee.She's Anne Elizabeth and she's my sister and no she certainly does not analyze. If you saw her fingers flying on her violin fingerboard during some of the Irish jigs she plays, you'd know that if she thought for a millisecond about what she was actually doing, the whole thing would just explode into a shower of wood, catgut and horsehair. :-)

Message: 4505 - Contents - Hide Contents Date: Thu, 04 Apr 2002 21:44:29 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > With the 29 factor we have passed the point (in both 217 and 311) > where all of the ratios within the harmonic limit are a unique number > of degrees, so some bridging is inevitable. However, what's most > important is that the number of degrees for the 29 comma is > consistent with this flag in both 217 (6 deg) and 311 (9 deg).I had mistakenly thought that 311-ET was 31-limit-unique until Graham corrected me. What's the smallest ET that has a 31-limit-unique mapping, even if it's not consistent? Could it be 624-ET. How about a 35-limit unique mapping? Never mind.> I prefer to consider the xL flag as 715:729 with the additional > meaning of 45056:45927. ... > Or am I just splitting hairs, inasmuch as conflating 4095:4096 would > allow us to look at it either way (or both ways at once)?Yes. Splitting hairs. All I'm saying is, if someone only needs an 11-limit notation then they shouldn't have to know anything about xL also being 715:729 (13-limit).> As I mentioned above, in the process of counting sharp-vs.-flat > occurrences for 23 in my JI heptatonic scales, I also counted the > number of sharp-vs.-flat occurrences for 17 (3#, 3b), 19 (4#, 5b), 25 > (5#, 3b), and 29 (no#, 4b). I conclude that there has to be a > provision for spelling any interval in at least two different ways, > which is a compelling reason for providing a complete set of symbols > for whatever division we settle on for the JI notation.Oh dear, I think I've agreed with certain things you proposed which have turned out to be the beginning of a slippery slope that I don't want to go down. One was to accept certain schismas vanishing in order to minimise the number of different flags making up the symbols. The other was to accept the need for more than one comma for certain primes. Now you say we need two commas for _every_ prime. Why stop there? Shouldn't we provide for anyone who wants to notate the prime intervals from C as varieties of any of the following: 5 E,Fb 7 A#,Bb 11 E#,F,F#,Gb 13 G#,Ab,A 17 C#,Db 19 D#,Eb 23 F#,Gb 29 A#,Bb,Cb 31 B,Cb,C These all involve commas smaller than 3/5 apotome. There are _only_ 23 of them. I am of course being facetious. I'm more inclined to go back to a strict single comma per prime. In all the lower ETs where either an 11-comma or a 13-comma is a semisharp or semiflat, its apotome complement is the same number of steps, so there isn't any need for both. In rational tunings, I don't see why we have to cater for all the base notes (powers of 3 to which the prime commas are applied) remaining within a single diatonic scale. Actually it isn't even possible, unless you allow commas much larger than 3/5 apotome. I see no problem with requiring a chromatic scale for the base notes, or worse, a few enharmonics such as Ab and G# together. Remember I'm happy to use up to 3 symbols against a note (for rational tunings and very large ETs) rather than have too many new symbols for people to learn.> In addition, being able to notate all of the degrees would ensure > that no matter how much modulation is done in JI, at least one would > never run out of symbols. If this were not done, then we would lose > one of the principal advantages of mapping the JI notation to a > specific division. ... > So the question now becomes: Are we left with any good reason for > basing the JI notation on 311 instead of 217?From your point of view, I would say that you are better off with 217-ET. However I do not wish to base a JI (rational) notation on _any_ temperament that has errors larger than 0.5 c. For me, 217-ET and 311-ET were merely a way of looking for schismas that might be notationally usable (less than 0.5 c), and of checking that things were working sensibly, and it was nice to actually be able to notate those ETs themselves. But I'm taking Johnny Reinhard at his word when he says (or implies) that nothing less than 1200-ET is good enough as an ET-based JI notation. What is the first ET above 1200 that is 35-limit consistent? 35-limit unique? But I am more interested in a 31-limit (or 35-limit) temperament (not equal or linear or even planar, but maybe 6D or 7D or 8D) with as many sub-half-cent schismas as possible, vanishing. Hence the challenge which no-one's taken up yet, except in part. I don't share your obsession with packing all the necessary information into a single symbol. Some folks may well be willing to notate their JI piece by mapping to the nearest degree of 217-ET, and they may well be glad of the possibility to do it with only one accidental per note, but that won't be everyone's cup of tea. Given that it may involve errors of up to 3 cents, it must be a conscious decision, not something forced upon us by the notation. However, it seems that most would be willing to wear it if a 0.5 c error were forced upon them by the notation. The only objections to this will be "philosophical" ones, not ones that anyone can hear (except in something like Lamonte Young's Dream House). -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4506 - Contents - Hide Contents Date: Thu, 4 Apr 2002 12:50 +01 Subject: Re: A common notation for JI and ETs From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a8fkev+5808@xxxxxxx.xxx> gdsecor wrote:> Something else in favor of 217: I noted earlier that it is 7 times > 31. If you make instruments for 31-ET (or train string players to > play 31-ET), then your JI can be reckoned in alterations of +/- 1 to > 3 increments of 217-ET (i.e., multiples of ~5.5 cents), to a maximum > of 16.6 cents, which is not an unreasonable amount of intonation > adjustment for instruments of flexible pitch. This would be a decent > practical alternative to alterations reckoned relative to 12-ET, > whether for (too-coarse) 72-ET or Johnny Reinhard's (too-small) one- > cent increments. Plus you get 31-ET in the bargain. And I should > mention that a series of 41 fifths in 217 (kept within an octave) > brings you only one degree away from your starting point, giving you > a very close approximation of 41-ET.31-ET for the 31-limit, great! In fact, a temperament with a period of 1/31 octaves is already my top inconsistent 21-limit temperament: 1/3, 16.0 cent generator basis: (0.032258064516129031, 0.013300385023233607) mapping by period and generator: [(31, 0), (49, 0), (72, 0), (87, 0), (107, 0), (115, -1), (127, -1), (132, -1)] mapping by steps: [(62, 31), (98, 49), (144, 72), (174, 87), (214, 107), (229, 115), (253, 127), (263, 132)] highest interval width: 1 complexity measure: 31 (62 for smallest MOS) highest error: 0.009287 (11.145 cents) Without a history of usage of such temperaments, it's really impossible to say if that 11 cent error is good or bad. It's lower in absolute terms than 12-equal in the 5-limit, which many put up with. But if the 21-limit is taken to mean the difference between 22:21 (80.5 cents) and 21:20 (84.5) cents is significant, 11 cents is huge. Of course, any inconsistent temperament will remove such distinctions. Here are the 21-limit equivalences for this particular temperament: 15:14 =~ 16:15 10:9 =~ 19:17 =~ 9:8 12:11 =~ 11:10 22:19 =~ 15:13 13:11 =~ 19:16 16:13 =~ 21:17 9:7 =~ 14:11 21:16 =~ 17:13 17:16 =~ 19:18 15:11 =~ 11:8 =~ 26:19 22:21 =~ 21:20 20:19 =~ 18:17 The online scripts and results won't handle the 31-limit, but I can run it locally (it's a trivial change). The temperament George describes is roughly>>> temper.Temperament(31, 217, temper.primes)1/8, 5.3 cent generator basis: (0.032258064516129031, 0.0043924816210988982) mapping by period and generator: [(31, 0), (49, 1), (72, 0), (87, 0), (107, 2), (115, -2), (127, -2), (132, -2), (140, 2), (151, -3), (154, -3)] mapping by steps: [(217, 31), (344, 49), (504, 72), (609, 87), (751, 107), (803, 115), (887, 127), (922, 132), (982, 140), (1054, 151), (1075, 154)] highest interval width: 6 complexity measure: 186 (217 for smallest MOS) highest error: 0.002255 (2.706 cents) 217-equal is only accurate to 3.2 cents, so there is some kind of improvement here. The complexity may be too high for the tradeoff to be worthwhile. An interesting alternative is>>> temper.Temperament(62,217,temper.primes)4/9, 16.7 cent generator basis: (0.032258064516129031, 0.013944526171363311) mapping by period and generator: [(31, 0), (50, -2), (72, 0), (87, 0), (109, -4), (116, -3), (128, -3), (133, -3) , (142, -4), (151, -1), (154, -1)] mapping by steps: [(217, 62), (344, 98), (504, 144), (609, 174), (751, 214), (803, 229), (887, 253 ), (922, 263), (982, 280), (1054, 301), (1075, 307)] highest interval width: 6 complexity measure: 186 (217 for smallest MOS) highest error: 0.002263 (2.715 cents) If you're prepared to forego ratios involving 27, the complexity is only 4*26, compared to 5*26 for the other version. One disadvantage is that the 3:1 has a more complex approximation. But that may not be important, as you can modulate by as many 31-equal fifths as you like, and they're good enough in another context. Again, there isn't enough (any!) history of tempered 31-limit music to pronounce on the importance of these differences. Graham

Message: 4509 - Contents - Hide Contents Date: Fri, 5 Apr 2002 10:12 +01 Subject: Re: A common notation for JI and ETs From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <3.0.6.32.20020404214429.00b1bb90@xx.xxx.xx> David C Keenan wrote:> What is the first ET above 1200 that is 35-limit consistent? 35-limit > unique?Above 1200? 1600. The only ones below that are 311 and 388. If I can remember how I did uniqueness for the 11-limit, I'll try a search on that as well. Anyway, the simplest linear temperament from consistent ETs is 236/699, 405.2 cent generator basis: (1.0, 0.3376257079318315) mapping by period and generator: [(1, 0), (8, -19), (-23, 75), (18, -45), (46, -126), (1, 8), (-29, 98), (11, -20 ), (42, -111), (16, -33), (61, -166)] mapping by steps: [(388, 311), (615, 493), (901, 722), (1089, 873), (1342, 1076), (1436, 1151), (1 586, 1271), (1648, 1321), (1755, 1407), (1885, 1511), (1922, 1541)] highest interval width: 316 complexity measure: 316 (388 for smallest MOS) highest error: 0.001078 (1.293 cents) The next two are unique 41/497, 24.7 cent generator basis: (0.25, 0.020624732047905801) mapping by period and generator: [(4, 0), (7, -8), (13, -45), (1, 124), (6, 95), (20, -63), (18, -20), (8, 109), (11, 86), (21, -19), (10, 119)] mapping by steps: [(1600, 388), (2536, 615), (3715, 901), (4492, 1089), (5535, 1342), (5921, 1436) , (6540, 1586), (6797, 1648), (7238, 1755), (7773, 1885), (7927, 1922)] highest interval width: 214 complexity measure: 856 (1212 for smallest MOS) highest error: 0.000284 (0.341 cents) unique 467/1911, 293.3 cent generator basis: (1.0, 0.24437509095621543) mapping by period and generator: [(1, 0), (27, -104), (-38, 165), (-49, 212), (56, -215), (45, -169), (-79, 340), (-61, 267), (49, -182), (53, -197), (79, -303)] mapping by steps: [(1600, 311), (2536, 493), (3715, 722), (4492, 873), (5535, 1076), (5921, 1151), (6540, 1271), (6797, 1321), (7238, 1407), (7773, 1511), (7927, 1541)] highest interval width: 696 complexity measure: 696 (978 for smallest MOS) highest error: 0.000298 (0.358 cents) unique which isn't enough to prove that 1600-equal is unique. Graham

Message: 4510 - Contents - Hide Contents Date: Fri, 05 Apr 2002 15:06:51 Subject: Re: A common notation for JI and ETs From: gdsecor I double-checked the following and found it to be in error: --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> > To further complicate things in 311, I also noticed in my notes that, > while I favored using 19 as an E-flat (where 1/1 is C), there were > almost as many instances where a heptatonic scale called for a D- > sharp (using the 19-comma 19456:19683, 3^9:2^14*19, ~20.082 cents). > This usage turns out to be *inconsistent* in 311 (but consistent in > 217), a problem that I didn't expect to find. While I compared the > inconsistency of a few ratios of 23 in 217 with driving a car > slightly onto the shoulder, the 311 problem (affecting all ratios of > 19) is more like attempting to drive in the less-traveled direction > on route 19 and finding yourself in the wrong lane moving against > traffic. So I am beginning to have serious doubts about going to 311.The 19-comma 19456:19683 is 2^10*19:3^9 (~20.082 cents), and its usage is consistent in *both* 217 and 311. I'm sorry about the misinformation, but I'm very delighted that I was wrong about this. --George

Message: 4511 - Contents - Hide Contents Date: Fri, 5 Apr 2002 17:34:50 Subject: Re: A common notation for JI and ETs From: manuel.op.de.coul@xxxxxxxxxxx.xxx Dave wrote:>I had mistakenly thought that 311-ET was 31-limit-unique until Graham >corrected me. What's the smallest ET that has a 31-limit-unique mapping, >even if it's not consistent? Could it be 624-ET.Yes, 624 to 633-tET are all 35-limit unique. 1600-tET is 37-limit consistent and 55-limit unique. Manuel

Message: 4513 - Contents - Hide Contents Date: Fri, 05 Apr 2002 21:50:41 Subject: Re: A common notation for JI and ETs From: gdsecor Note: I posted this more than a couple of hours ago, but it still hasn't shown up, so, after having made a correction, I am trying again. Please disregard the duplicate, as I suspect that it will eventually show up. --gs --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> With the 29 factor we have passed the point (in both 217 and 311) >> where all of the ratios within the harmonic limit are a unique number >> of degrees, so some bridging is inevitable. However, what's most >> important is that the number of degrees for the 29 comma is >> consistent with this flag in both 217 (6 deg) and 311 (9 deg). >> I had mistakenly thought that 311-ET was 31-limit-unique until Graham > corrected me. ...>> ... I conclude that there has to be a >> provision for spelling any interval in at least two different ways, >> which is a compelling reason for providing a complete set of symbols >> for whatever division we settle on for the JI notation. >> Oh dear, I think I've agreed with certain things you proposed which have > turned out to be the beginning of a slippery slope that I don'twant to go> down. One was to accept certain schismas vanishing in order to minimise the > number of different flags making up the symbols. The other was to accept > the need for more than one comma for certain primes. > > Now you say we need two commas for _every_ prime. Why stop there? Shouldn't > we provide for anyone who wants to notate the prime intervals from C as > varieties of any of the following: > 5 E,Fb > 7 A#,Bb > 11 E#,F,F#,Gb > 13 G#,Ab,A > 17 C#,Db > 19 D#,Eb > 23 F#,Gb > 29 A#,Bb,Cb > 31 B,Cb,C > > These all involve commas smaller than 3/5 apotome. There are _only_ 23 of > them. I am of course being facetious. I'm more inclined to go back to a > strict single comma per prime. In all the lower ETs where either an > 11-comma or a 13-comma is a semisharp or semiflat, its apotome complement > is the same number of steps, so there isn't any need for both.That is correct if you are indicating "semisharp" and "semiflat" to be exactly half an apotome in those ET's (or is this what you really meant to say?). In 39, 46, and 53-ET this is not the case, and you either need to have both 11-commas for the two different kinds of semisharps/semiflats, or else have the 11 comma for one and the 13 comma for the other (although these would then be faux complements). I think that I have probably misunderstood what you were trying to say and that my going on about this is sort of pointless, considering that we are getting the notation for the secondary 11 and 13 commas as a freebie with the 4095:4096 schisma. I think that your point is that each pair of 11 and 13 commas consists of apotome-complements, so that the larger of each could be notated as an apotome minus the smaller; hence we really have only one comma for each prime in the semantics of the notation; but for convenience in handling the semisharps & semiflats we have symbols that make it appear as if there were two. Anyway, in pointing out the need for being able to respell ratios at will, my primary objective was to indicate why we needed to notate all of the degrees in the ET, not to open up a can of worms whereby we need to have multiple commas per prime. One apiece should do.>> ... So the question now becomes: Are we left with any good reason for >> basing the JI notation on 311 instead of 217? >> From your point of view, I would say that you are better off with 217-ET.This amounts, then, to a 19-limit-unique-&-consistent, polyphonic- readable sagittal notation with non-unique capability up to the 35- odd limit. That sounds like something that fulfills (and in some ways exceeds) our original objective (as I understood it).> However I do not wish to base a JI (rational) notation on _any_ temperament > that has errors larger than 0.5 c. For me, 217-ET and 311-ET were merely a > way of looking for schismas that might be notationally usable (less than > 0.5 c), and of checking that things were working sensibly, and it was nice > to actually be able to notate those ETs themselves. But I'm taking Johnny > Reinhard at his word when he says (or implies) that nothing less than > 1200-ET is good enough as an ET-based JI notation. > > What is the first ET above 1200 that is 35-limit consistent? 35- limit unique?Graham suggests 1600-ET in his message #3947. It looks like a good choice, inasmuch as: 1) It is 37-limit consistent; 2) The largest error for any 37-limit consonance is ~0.36441 cents (for 19:25); 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and 20735:20736 (but not the 31-schisma that I tried, 59024:59049, which was also unusable in 311); 4) And I strongly suspect that it will be found to be at least 37- limit unique, inasmuch as the largest superparticular ratios that are not unique are 54:55 and 55:56 (both 42deg1600).> But I am more interested in a 31-limit (or 35-limit) temperament (not equal > or linear or even planar, but maybe 6D or 7D or 8D) with as many > sub-half-cent schismas as possible, vanishing. Hence the challenge which > no-one's taken up yet, except in part.Well, good luck on that one!> I don't share your obsession with packing all the necessary information > into a single symbol. Some folks may well be willing to notate their JI > piece by mapping to the nearest degree of 217-ET, and they may well be glad > of the possibility to do it with only one accidental per note, but that > won't be everyone's cup of tea. Given that it may involve errors ofup to 3> cents, it must be a conscious decision, not something forced uponus by the> notation.You're talking about two separate issues here: 1) One altering symbol per note: Since we seem to have concluded that the sagittal JI notation is going with 217, we can now determine what each ET notation is going to look like. This means that I can now go ahead and present my Sims vs. sagittal comparison on the main tuning list without having to issue a caveat that the 72-ET sagittal notation might not be the final version. In that comparison I will show an instance in which double symbols (Sims or otherwise) could present some confusion. I will also make the point that, once they are learned, single symbols can be read more quickly than double ones (particularly in chords), since there is less to read. 2) Mapping to 217-ET: There is a question that needs to be asked: are we notating JI or are we notating 217-ET? I understood that we were notating JI (mapped onto 217 for convenience in understanding some of the size relationships among the various ratios), which makes discussion about 3-cent errors a bit irrelevant. Now one may also want to make use of the 217 mapping as a convenience in conceptualizing a way of arriving at the approximate pitches represented by those ratios (which are in turn represented by symbols that correlate with a 217 mapping), and without any fine-tuning (by ear) you would be entitled to contemplate 3-cent errors. (Come to think of it, anyone coming within 3 cents -- less than the Miracle tuning minimax deviation -- is doing pretty well by almost anybody's standard.) But this is more of a matter of how the composer is going to treat the notation: 1) Either sticking with a specific set of ratios (in which case the ratios and symbols could, for reference, be listed in a table alongside each other, with cents values, if that helps), in which case the 217 mapping (and error thereof) would have little or no relevance; 2) Or else freely employing whatever intervals are permitted by the notation, with little regard to keeping track of ratios, in which case it could very well turn into (at best) a 217-ET performance. Putting this another way: If I write a piece for 13-limit JI using just 12 tones per octave and map the tones (consistently) into 12, specifying the ratios that I want for each position, would you be entitled to claim that I would be getting errors close to 50 cents for some of the tones if I used the 12-ET notation? I believe the problem is more of a matter of how the composer and performer understand the notation than with the notation itself. --George

Message: 4514 - Contents - Hide Contents Date: Fri, 05 Apr 2002 14:17:58 Subject: Fwd: Re: deriving classical major/minor scales From: Carl Lumma Gene, don't know if you saw this on the main list, or for that matter, if it showed up. Am re-posting here.>Date: Sun, 31 Mar 2002 17:24:45 -0800 >To: tuning@xxxxxxxxxxx.xxx >From: Carl Lumma <carl@xxxxx.xxx> >Subject: Re: deriving classical major/minor scales >>> The periodicity block business, in suitable generality, ends >> up saying that scales tend to be convex, which seems reasonable. > >Gene, >

Message: 4515 - Contents - Hide Contents Date: Sat, 06 Apr 2002 21:13:08 Subject: Re: Blocks and convexity From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> i'd like to better understand the structure of the balls of various > diameters, given the tenney norm, and extensions to octave-equivalent > lattices . . . can you help?You keep bringing up the Tenney norm in this context; it isn't priviledged. Fokker blocks have paralleopiped balls and are of L-infinity type; the Tenney norm is of L1 type and would have cross polytope (generalized octahedron) type balls. Other norms are possible, and given a convex set you can cook up the corresponding norm, and vice-versa. Let's look at an example. Using the sxymmetrical lattice norm on octave classes in the 7-limit, so that ||3^a 5^b 7^c|| = sqrt(a^2+b^2+c^2+ab+ac+bc) we find that the hexany 1-15/14-5/4-10/7-3/2-12/7 is contained in a spherical (L2) ball centered at [1/2, 1/3, -1/3]. Is it a block? We set up the five equations in four unknowns [a,b,c,d] which define the val, if there is one, such that the nth degree is given value n; solving this system of equations gives the unique solution {a=6,b=10,c=14,d=17}, so that the val in question exists and is h6; hence the hexany is a block. The L2 norm is not the only norm which would define the hexany as a block; one can always use the convec hull to generate a ball--in this case we would get an L1 norm in that way.

Message: 4516 - Contents - Hide Contents Date: Sat, 06 Apr 2002 22:04:18 Subject: Re: Blocks and convexity From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> we find that the hexany 1-15/14-5/4-10/7-3/2-12/7 is contained in a spherical (L2) ball centered at [1/2, 1/3, -1/3].Sorry, this should be [1/2,1/2,-1/2].

Message: 4517 - Contents - Hide Contents Date: Sat, 06 Apr 2002 23:15:51 Subject: Re: New Member-Question about ratios From: dkeenanuqnetau --- In tuning-math@y..., Alexandros Papadopoulos <alexmoog@o...> wrote:> Hello , I am Alex Papadopoulos from Greece. > > My mathematics knowledge is pretty low , so I wonder if this list is > accepting amateur-ish mathematics questions related to tunings. > If not please tell me , so I will not bother you. Hi Alex,We accept any mathematics questions related to tunings.> If yes , my question is: > > In Helmholtz's book "the sensations of tone" in page 15 he states ,that > since we know the ratios of the C ,E and G notes , we can find > the rest of the C scale notes , by associating this C triad with the F > and G triad. > So I thought that since G to B is a Major 3rd , I can add 5/4(Major 3rd) > to G which has a ratio of 3/2. > The B ratio is 15/8 , which is not what I find. > > I must have something very wrong in my mind![If you are reading this on Yahoo's web interface you will need to choose Message Index and then Expand Messages to see the following correctly formatted. Or simply choose Reply.] What is it that you find? 15/8 is the correct ratio for B in a _justly_intoned_ C major scale whose lattice follows. A---E---B / \ / \ / \ F---C---G---D 5/3-5/4-15/8 / \ / \ / \ 4/3-1/1-3/2-9/8 Of course this is not the same as its tuning in equal temperament (the most common tuning today). There all the notes are mapped to the nearest power of the twelfth root of two. In the case of B this is 2^(11/12) ~= 1.888, whereas 15/8 is 1.875. -- Dave Keenan

Message: 4518 - Contents - Hide Contents Date: Sat, 06 Apr 2002 02:23:31 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>> In all the lower ETs where either an >> 11-comma or a 13-comma is a semisharp or semiflat, its apotome > complement>> is the same number of steps, so there isn't any need for both. >> That is correct if you are indicating "semisharp" and "semiflat" to > be exactly half an apotome in those ET's (or is this what you really > meant to say?).I was being sloppy. I didn't check the facts before I wrote. Sorry.> In 39, 46, and 53-ET this is not the case, and you > either need to have both 11-commas for the two different kinds of > semisharps/semiflats, or else have the 11 comma for one and the 13 > comma for the other (although these would then be faux complements).Thanks. I agree that 46 and 53 are the first where this is a problem. 39-ET is 1,3,13-inconsistent and one can choose a number of steps for 1024:1053 that is the apotome complement of 32:33, but this is not possible for 46 and 53. So I guess you're assuming 26:27 as _the_ 13 comma. Or are you assumng 704:729 as _the_ 11 comma?> I think that I have probably misunderstood what you were trying to > say and that my going on about this is sort of pointless, considering > that we are getting the notation for the secondary 11 and 13 commas > as a freebie with the 4095:4096 schisma. I think that your point is > that each pair of 11 and 13 commas consists of apotome-complements, > so that the larger of each could be notated as an apotome minus the > smaller; hence we really have only one comma for each prime in the > semantics of the notation; but for convenience in handling the > semisharps & semiflats we have symbols that make it appear as if > there were two.That wasn't my point. As I say, I was just careless. But it's a good point. Thanks for making it.> Anyway, in pointing out the need for being able to respell ratios at > will, my primary objective was to indicate why we needed to notate > all of the degrees in the ET, not to open up a can of worms whereby > we need to have multiple commas per prime. One apiece should do.Ok. Good. But hopefully we can have less than one apiece, with the other sub-half-cent schismas you found.>>> ... So the question now becomes: Are we left with any good > reason for>>> basing the JI notation on 311 instead of 217? >>>> From your point of view, I would say that you are better off with > 217-ET. >> This amounts, then, to a 19-limit-unique-&-consistent, polyphonic- > readable sagittal notation with non-unique capability up to the 35- > odd limit. That sounds like something that fulfills (and in some > ways exceeds) our original objective (as I understood it).Sure. But I don't understand what 217-ET or 311-ET have to do with it. 217-ET just happens to be the highest ET that you can notate with it. The definitions of the symbols must be based on the commas, not the degrees of 217-ET. What do you mean by "polyphonic-readable"? As opposed to what?>> However I do not wish to base a JI (rational) notation on _any_ > temperament>> that has errors larger than 0.5 c. For me, 217-ET and 311-ET were > merely a>> way of looking for schismas that might be notationally usable (less > than>> 0.5 c), and of checking that things were working sensibly, and it > was nice>> to actually be able to notate those ETs themselves. But I'm taking > Johnny>> Reinhard at his word when he says (or implies) that nothing less > than>> 1200-ET is good enough as an ET-based JI notation. >> >> What is the first ET above 1200 that is 35-limit consistent? 35- > limit unique? >> Graham suggests 1600-ET in his message #3947. It looks like a good > choice, inasmuch as: > > 1) It is 37-limit consistent; > > 2) The largest error for any 37-limit consonance is ~0.36441 cents > (for 19:25); > > 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and > 20735:20736 (but not the 31-schisma that I tried, 59024:59049, which > was also unusable in 311); > > 4) And I strongly suspect that it will be found to be at least 37- > limit unique, inasmuch as the largest superparticular ratios that are > not unique are 54:55 and 55:56 (both 42deg1600). >>> But I am more interested in a 31-limit (or 35-limit) temperament > (not equal>> or linear or even planar, but maybe 6D or 7D or 8D) with as many >> sub-half-cent schismas as possible, vanishing. Hence the challenge > which>> no-one's taken up yet, except in part. >> Well, good luck on that one!You have in effect found one of these, with your 3 schismas. I'd just like to be sure that there isn't a whole 'nother larger set of 31-limit schismas that give an even better compression of the number of flags, without exceeding the 0.5 cent. But I guess I'll just assume 1600-ET. In other words, any schisma that vanishes in 1600-ET is acceptable to be built into the notation, but certainly not all the schismas that vanish in 217-ET (or 311-ET or 388-ET). So in that sense only, I could say that the notation is "based on" 1600-ET, but there is certainly no desire to notate every degree of 1600-ET using a single accidental per note, or even a single accidental in addition to a sharp or flat. In fact there is no desire to notate 1600-ET at all, and it is fine that 217-ET is the highest ET that can be so notated.>> I don't share your obsession with packing all the necessary > information>> into a single symbol. Some folks may well be willing to notate > their JI>> piece by mapping to the nearest degree of 217-ET, and they may well > be glad>> of the possibility to do it with only one accidental per note, but > that>> won't be everyone's cup of tea. Given that it may involve errors of> up to 3>> cents, it must be a conscious decision, not something forced upon> us by the >> notation. >> You're talking about two separate issues here: > > 1) One altering symbol per note: > > Since we seem to have concluded that the sagittal JI notation is > going with 217, we can now determine what each ET notation is going > to look like. This means that I can now go ahead and present my Sims > vs. sagittal comparison on the main tuning list without having to > issue a caveat that the 72-ET sagittal notation might not be the > final version.Agreed. I'm pretty sure it's only the concave flags whose meaning could still change.> In that comparison I will show an instance in which > double symbols (Sims or otherwise) could present some confusion. I > will also make the point that, once they are learned, single symbols > can be read more quickly than double ones (particularly in chords), > since there is less to read. > > 2) Mapping to 217-ET: > > There is a question that needs to be asked: are we notating JI or are > we notating 217-ET? I understood that we were notating JI (mapped > onto 217 for convenience in understanding some of the size > relationships among the various ratios), which makes discussion about > 3-cent errors a bit irrelevant.OK. Good. So I wish you'd stop talking about it being "based on" or "going with" 217-ET, or any other ET with larger than 0.5 cent errors.> Now one may also want to make use of the 217 mapping as a convenience > in conceptualizing a way of arriving at the approximate pitches > represented by those ratios (which are in turn represented by symbols > that correlate with a 217 mapping), and without any fine-tuning (by > ear) you would be entitled to contemplate 3-cent errors. (Come to > think of it, anyone coming within 3 cents -- less than the Miracle > tuning minimax deviation -- is doing pretty well by almost anybody's > standard.) But this is more of a matter of how the composer is going > to treat the notation: > > 1) Either sticking with a specific set of ratios (in which case the > ratios and symbols could, for reference, be listed in a table > alongside each other, with cents values, if that helps), in which > case the 217 mapping (and error thereof) would have little or no > relevance;Yes. This is more fundamental to the notation.> 2) Or else freely employing whatever intervals are permitted by the > notation, with little regard to keeping track of ratios, in which > case it could very well turn into (at best) a 217-ET performance.Yes, so 217-ET is just one ET that could be used in this way. The notation is not based on it. It just happens to be the highest one that is fully notatable with single symbols.> Putting this another way: If I write a piece for 13-limit JI using > just 12 tones per octave and map the tones (consistently) into 12, > specifying the ratios that I want for each position, would you be > entitled to claim that I would be getting errors close to 50 cents > for some of the tones if I used the 12-ET notation? No. > I believe the problem is more of a matter of how the composer and > performer understand the notation than with the notation itself.Yes. I'm glad we understand each other. Showing how the notation maps to 217-ET is no different from showing how it maps to 22-ET. The _definition_ of the symbols is in terms of the commas and schismas.

Message: 4519 - Contents - Hide Contents Date: Sat, 06 Apr 2002 07:28:18 Subject: Blocks and convexity From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> Will this suitable generality be covered in your paper? Or >> would you care to explain it on these lists?The metric on a vector space defined from a norm d(x,y) = ||x-y|| has the property (not shared by metrics in general) of convexity--the open and closed balls defined by the metric are convex. On the other hand, given a convex closed set S containing the origin in its interior, we can define a norm such that S is a unit ball. If we define a block as a set of octave-equivalence lattice points which are epimorphic and convex, from the above we can see this ends up equivalent to a definition in terms of normed vector spaces, where a block is an epimorphic set of lattice points of minimal diameter. The Fokker block is simply then the special case of a linearlly transformed L-infinity norm. I don't know if any of this needs to be expounded.

Message: 4521 - Contents - Hide Contents Date: Sun, 07 Apr 2002 01:49:56 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and > 20735:20736 (but not the 31-schisma that I tried, 59024:59049, which > was also unusable in 311);59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test anyway, being 0.73 c, however it might tempt me if it could be combined with other suitable schismas, as per my challenge.

Message: 4522 - Contents - Hide Contents Date: Sun, 07 Apr 2002 21:12:52 Subject: Re: A common notation for JI and ETs From: David C Keenan In case anyone cares; there's a 37-schisma that vanishes in 1600-ET (and 388-ET) that lets us make a symbol for a 37-comma, with no new flags. The 37 comma is 999:1024 (3^3*37 : 2^10) 42.8 cents, so a 1:37 from C is a lowered Eb. The schisma is 570236193:570425344 (3^12*29*37 : 2^25*17) 0.57 cents, which means the 37 arrowhead combines the 17-flag and the 29-flag. Only trouble is, these are presently both left flags, concave and convex. I believe we have a rule that says that at each prime limit, the symbols should be as simple as possible and that no higher prime should be allowed to "reach down" and cause us to change the way we do the flags for a lower limit. However, the only reason so far, to make the 17 flag a left flag is to make a symbol for the large 23 comma by combining the 17 flag with the 11-5 flag. It isn't essential to have a symbol for the large 23 comma, so the 17 flag and (small) 23 flag (both concave) could swap sides. 8 steps of 217-ET could still be notated as either the 25 symbol (sL+sL) or 7 flag + 23 flag (vL+xR). 1:37 from C is very close to halfway between D and Eb, a pythagorean limma apart, so there is a good argument for needing the large 37 comma of (36:37) 47.4 cents as well. There is a 1600-ET schisma that gives us the large 37 comma without new flags, provided we're willing to combine 3 of them, 19 flag + 23 flag + 7 flag (cL+vL+xR), which actually do combine ok. The schisma is 6992:6993 (2^4*19*23 : 3^3*7*37) 0.25 cents. This is probably all pretty silly, catering for 37, and we should probably just forget it and keep the large 23 comma symbol, but here's a pass at a full set of 37-limit symbols anyway. [If you're reading this on the yahoogroups website you will need to choose Reply or Message Index, Expand Messages, to see the following symbols rendered correctly.] 5-comma 80:81 /| / | | \ / | | 7-comma 63:64 _ | \ | | | L P | | 11-comma 32:33 /|\ / | \ | v ^ | | 13-comma 1024:1053 _ /| \ / | | | { } flags based on vanishing of schisma 4095:4096 | | 17-comma 2176:2187 | |\_ | j f | | 19-comma 512:513 _ (_) | | o * | | 23-comma 729:736 | _/| | w m | | 29-comma 256:261 _ / | | | | q d flag based on vanishing of schisma 20735:20736 | | 31-comma 243:248 _ (_)\ | \ | y h flags based on vanishing of schisma 253935:253952 | | 37-comma 999:1024 _ / | | |\_ | flags based on vanishing of schisma 570236193:570425344 | | We also have optional symbols for larger 11, 13 and 37 commas. 11'-comma 704:729 _ _ / | \ | | | | [ ] flags based on vanishing of schisma 5103:5104 | | 13'-comma 26:27 _ / |\ | | \ | ; | flags based on vanishing of schisma 20735:20736 | | 37'-comma 36:37 _ _ (_) \ _/| | | flags based on vanishing of schisma 6992:6993 | | If we really still wanted a symbol for the large 23 comma I guess we could still combine the 17 and 11-5 flags like this: 23'-comma 16384:16767 | |\_ | \ W M flags based on vanishing of schisma 3519:3520 | | -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4523 - Contents - Hide Contents Date: Sun, 07 Apr 2002 03:20:13 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and >> 20735:20736 (but not the 31-schisma that I tried, 59024:59049, which >> was also unusable in 311); >> 59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test anyway, > being 0.73 c, however it might tempt me if it could be combined with > other suitable schismas, as per my challenge.We can forget about that 31-schisma. What's wrong with 253935:253952 (3^5*5*11*19 : 2^13*31) 0.12 cents. Consistent with 311-ET 388-ET 1600-ET, but not 217-ET. 31 comma = (11 comma - 5 comma) + 19 comma Since (11 comma - 5 comma) is a single flag and 19 comma is a single flag (or blob) then this 31 comma can be represented by a pair of flags. The fact that it doesn't work in 217-ET doesn't matter because the notation is not "based on" 217-ET and the 31 comma is not needed in order to notate 217-ET.

Message: 4524 - Contents - Hide Contents Date: Sun, 7 Apr 2002 01:28:36 Subject: Re: Blocks and convexity From: Pierre Lamothe Perhaps what follows is sufficiently correlated to your interest in this thread. I failed until date to generalize something that seemed very simple in 3D for I believed that the parrallelopiped was the elementary shape of periodicity blocks in which thefundamental domain around the unison (corresponding to a set of steps) was decomposable. I waswrong. I found the generalization way using vectorial drawing. I don't want to useneither words nor math formalism to explain that now. I leave you to discover that by intuition with a simple 4D example. I used animated images. The image fd4.swf shows the decomposition with 4 periodicity blocks of the 4D fundamental domain associated with 4 steps. For comparison only, theimage fd3.swf shows the decomposition with 3 periodicity blocks of the 3D fundamental domain having only 3 steps. (I already showed similar hexagonal shapes decomposed with 3 parallelograms.) I guess that Gene will see immediately the link with norms and unit balls. I could show later how matrices are nicely correlated with that. Particularly, I have now a new tool, the TS-matrix, which is a variant of the S-matrix. (S means srutis - TS means steps and srutis) permitting to look anew, geometrically, at the correlation between steps and unison vectors. Don't forget to press "enter" or use "play" to start the animation. http://www.aei.ca/~plamothe/tuning/fd4.swf - Type Ok * [with cont.] (Wayb.) http://www.aei.ca/~plamothe/tuning/fd3.swf - Type Ok * [with cont.] (Wayb.) Pierre [This message contained attachments]

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