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Message: 4900 - Contents - Hide Contents Date: Mon, 27 May 2002 12:59:08 Subject: Re: A 7-limit best list From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> [[1, 0, 4, 1], [0, 1, -1, 1]] [1200, 2025] 1.354 154.263This one slipped by me--ignore it.

Message: 4901 - Contents - Hide Contents Date: Tue, 28 May 2002 10:07:31 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> This is a first pass at a 7-limit best list. The first entry is themapping matrix, the second period and generator, the third (unweighted) rms generator steps to consonances, the fourth (unweighted) rms error. Badness is not listed to save space, but is less than 300; generator steps are less than 40, and rms error less than 50 cents. The ordering is by badness, lowest to highest.>For future reference, the blank lines between temperaments were a nuisance re importing to Excel. And I think we should have at least one decimal place for the rms-optimum generators, since we're sometimes stacking 10 or more of them. Everything else about the format was just fine, thanks. Gene, I'm puzzled. How come we didn't see the 5-limit versions of these four in any of your earlier 5-limit lists.> [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016 > [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453 > [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410 > [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132In lowest terms the generators are 310, 356, 490 and 274 cents respectively. These are the only three of possible interest that I don't have names for, except I tentatively call the 490 cent one superpythagorean (since its generator can be considered to be a 710 cent fifth). For your next pass, you can limit the complexity (preferably odd-limit-weighted) to 17, and the rms error to 30 cents, but let your badness go up to about 500. Some of those that barely made it onto your list are reasonably high on mine.

Message: 4902 - Contents - Hide Contents Date: Tue, 28 May 2002 11:24:51 Subject: Re: A 7-limit best list From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Gene, I'm puzzled. How come we didn't see the 5-limit versions of > these four in any of your earlier 5-limit lists. >>> [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016 >> [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453 >> [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410 >> [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132That's because as 5-limit systems they are all pretty bad. On the other hand, here is something I should have included on my 7s list: [19, 19, 57, 79, -37, -14] [[19, 0, 14, -37], [0, 1, 1, 3]] badness 160.2710884 g 33.81074780 rms .1401992317 generators [63.15789474, 1901.874626]

Message: 4905 - Contents - Hide Contents Date: Tue, 28 May 2002 22:06 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> That seems to be saying the tone group is a circle group, R/1200R if we > use cents. This means that the group is not ordered and its image under > the log map does not embed into a field, both of which don't help you. > On the plus side, it is a topological group with an invariant metric, > which gives us a notion of closeness. Um, right. > I'd say from a mathematician's point of view, having made things harder > in this way, we would want a payoff of some kind.It doesn't matter if there's a payoff or not, so long as we can do it.>> But there you're giving an octave-specific definition of a >> temperament you say won't work in octave-equivalent space. >> I give octave-specific definitions of everything; you are the one > saying it might be better not to.You're being totally obtuse here. I don't even know how to answer that. I asked for an example that wouldn't work in octave-equivalent terms, and you give me an octave-specific one. How much less relevant could that be?> Obviously an octave equivalent>> system can't do octave-specific things, but it works fine on its own >> terms. >> It works when it works? It seems to me it works because you can lift it > to octave-specific in cases of practical interest. Why bother to do the > heavy lifting? What's the payoff?How many times to I have to repeat that neither I nor anybody else who has expressed an opinion cares about this? It doesn't matter. Nobody thinks it would be an improvement. You can talk in octave-equivalent terms and add the octaves to do the calculations. All I'm saying is that the octave equivalent algebra+whatever works fine in it's own terms and you do *not* have to add in the octaves. Graham

Message: 4906 - Contents - Hide Contents Date: Wed, 29 May 2002 18:48:41 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4297]:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>>> How about "the 13-schisma" or the "tridecimal schisma". >>>> That sounds good. We should probably propose that term on the main >> tuning list, to see if anyone knows whether it has already been used >> for a different schisma. >> Go ahead. I'm sure enough that it hasn't, that I can't be bothered.If you're sure about that, then I'll take your word for it: 4095:4096 gets the name "tridecimal schisma.">> Since the (| flag is undoubtedly going to be used so much more often >> in connection with ratios of 11 and 13 -- as (|) and (|\ -- than for >> ratios of 29, I would prefer to keep its standard definition as other >> than 256:261 (the 29 comma). I would also prefer the 13'-(11-5) >> ratio to (11'-7) because, >> >> 1) The numbers in the ratio are smaller (715:729 vs. 45056:45927); and >> >> 2) The 13'-(11-5) comma (33.571 cents) is much closer in size to the >> 29 comma (33.487 cents) than is the (11'-7) comma (33.148 cents). >> I say we can totally forget the 29 comma definition of (| for notating ETs. > But I think we need to decide, for every ET individually, whether x| is > defined as 13'-(11-5) or (11'-7) (or both, when they are the same number of > steps). Agreed!>>>> I suggest that 37-ET be notated as a subset of 111-ET, with the >>>> latter having a symbol sequence as follows: >>>>>> Yes. That's also what I suggested in a later message (4188). >>>>>>> 111: w|, s|, |s, w|s, s|s, x|s, w||, s||, ||s, w||s, s||s. >>>>>> And that's almost the notation I proposed in the same message (with >>> its implied complements), except that I would use x|x (|) as the >>> complement of s|s /|\. Surely that is what you would want too, since >>> it represents a lower prime and is the rational complement? >>>> I used x|s (|\ as 6deg111 because x|x (|) calculates to 5deg111 and, >> in addition, 26:27 is closer in size to 6deg111 than is 704:729. >> However, if we think that there should be no problem in redefining >> x|x as 6deg111 (as it would seem to make more sense), then so be it! >> (|\ is only 6deg111 if you define (| as 13'-(11-5), in which case you > should probably also use /|) for 5deg111 instead of /|\. In this case /|) > is defined as the 13 comma, not 5+7 comma. This is something else that we > need to define on an ET by ET basis, whether |) is the 7 comma or the 13-5 > comma. If we favout 7 over (13-5) in 111-ET then we probably shouldn't use > any commas involving 13, and should therefore define (| as (11'-7). In this > case we have /|\ for 5deg111 and (|) for 6deg111.Yes, your notation for 111 is best and is in agreement with my latest choice. But I arrived at (|) as 6deg111 by keeping |) as the 7 comma (of 2deg) and defining (| as the 11'-7 comma of 4deg. Same difference, I guess! It's taken me a little time to appreciate the value of your proposal for dual roles for (| and the 19'-19 role of |~. However, I believe that a dual role should be retained for |~ also; it is quite useful as the 23 comma for notating 135, 147, 159, 198, and 224-ET (particularly 198).>>>> However, a more difficult problem is posed by 74-ET, and the idea of >>>> having redefinable symbols may be the only way to handle situations >>>> such as this. Should we do that, then there should probably be >>>> standard (i.e., default) ratios for the flags, and the specific >>>> conditions under which redefined ratios are to be used should be >>>> identified. >>>>>> I think 74-ET is garbage. >>>> Be careful when you say something like that around here -- do you >> remember my "tuning scavengers" postings? >> Yes, I remember. That's why I said it. So I'd get corrected as quickly as > possible if it _wasn't_ garbage. :-) It isn't. See the topic "74-EDO > challenge" on the main tuning list. >>> The problem is not the fault of the notation so much as the weirdness >> of the division -- I hesitate to call it a tonal system. Any >> systematic notation is going to have problems with 74-ET. >> Here's my proposal for notating 74-ET using its native fifth (since it's a > meantone), despite the 1,3,9 inconsistency. > Steps Symbol Comma > ---------------------- > 1 )|) 19+7 > 2 )|\ 19+(11-5) > 3 /|\ 11 > 4 )||\ > 5 /||\ > > The )| flag actually has a value of -1 steps, but it never occurs alone, so > it doesn't really matter.While I was away, I worked on the notation for a number of ET's. I decided to tackle 74 on my own, since it seemed to be a challenge. The solution I came up with minimizes the use of flags with non- positive values: 74: )|) /|) (|\ /||) /||\ Your solution is simpler in that it uses fewer flags and has no lateral confusability, so it would probably be preferable on that basis. However, I mention below that I would rather not use /|\ for anything greater than half of /||\ unless absolutely necessary. On the other hand, the 11 factor is almost exact in 74, so it would be a shame not to represent it in the notation. So what do you think? (I'm not going to lose any sleep over this one.)>> Would you also now prefer my selection of the /|) symbol for [6deg152] >> to your choice of (|~ on the grounds that it is a more commonly used >> symbol, particularly in view of the probability that you might want >> to use (|\ instead of )|| or ||( for 9deg as its complement? >> Yes, but not on those grounds.Then we agree on the following (cf. below): 152: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /|| ||\ /||~ /||) /||\> The case of 74-ET has shown me that my requirement of not using the native > fifth if it is 1,3,9-inconsistent, unless we don't use any flags for any > prime greater than 9, may need to be relaxed in some cases.Perhaps "guideline" would be a better term than "requirement." Applying this notation to different systems is as much an art as a science in that you need to decide which guidelines take priority over the others to achieve the most user-friendly result.>>>> I previously did symbol sets for about 20 different ET's, but that >>>> was before the latest rational complements were determined, so I'll >>>> have to review all of those to see what I would now do differently.It turns out that I did quite a few things differently this past week.>> Here's what I did a couple of weeks ago for some of the ET's (in >> order of increasing complexity): >> >> 12, 19, 26: s||s >> Agreed. My 19 and 26 were wrong. >>> 17, 24, 31: s|s s||s >> 17 and 24 agreed. I guess you want (|) for 1deg31 because it is closer in > cents than |), but I feel folks are more interested in its approximations > of 7, than 11.I think you meant /|\ instead of (|). As with 17 and 24, I think it's more intuitive to use /|\ (semisharp) for half of /||\ (sharp) where it's exactly half the number of degrees. Anyone who has used the Tartini/Fokker notation already calls an alteration of 1deg31 a semisharp or semiflat and would expect to see this symbol used. Besides, if there is no problem with lateral confusability, I think that straight flags are the simplest way to go.>> 22: s| ||s s||s >> I agree, but how come you didn't want s|s for 1deg22?If you did that, then you wouldn't have the comma-up /| /|| and comma- down \! \!! symbols that are one of the principal features of this notation; this is something that I would want to have in every ET in which 80:81 does not vanish, even if that doesn't result in a completely matched sequence of symbols in the half-apotomes. I believe the matched sequence is more imporant once the number of tones gets above 100, by which point /| and |\ are usually a different number of degrees. Also, with the apotome divided into fewer than 5 parts, I would want to use /|\ only when it is exactly half of /||\. In essence, what I am proposing here is that, for the lower-numbered ET's, we should place a higher priority on the use of rational complements than on a matching sequence of symbols. (Note that virtually everything that we agree on below follows this principle.)> It's also arguable > that it could be s| s|| s||s, making the second half-apotome follow the > same pattern of flags as the first,/|| is 3deg22 (since |\ = 0deg).> but what you've got makes more sense to > me.So 22 is settled, then.>> 36, 43: |x ||x s||s >> Agreed for 36. But I wanted a single-shaft symbol for 2deg43 so it is > possible to notate it with monotonic letter names and without double-shaft > symbols when using a notation that combines standard sharp and flat symbols > with sagittals. One could use either /|\ or (|\. e.g I want to be able to > notate the steps between B and C as B|), B/|\ or B(|\, C/|\ or C (|\, C!).I would rather not use /|\ for anything greater than half of /||\ unless absolutely necessary. How about using 36, 43: |) (|\ /||\ for both? Since I re-evaluated 72-ET, I changed my mind about 36, which hinges on how 72 is done (see below).>> 29: w|x w||v s||s >> Why wouldn't you use the same notation as for 22-ET? There's no need to > bring in primes higher than 5.I was making it compatible with my non-confusable version of 58, which I no longer favor. When I discuss 58 (below), I will give another version, which would result in this: 29: /|) (|\ /||\ But if you prefer version 1 of 58 (with all straight flags), then we might as well do 29 like 22-ET.>> 50: w|w x|s s||s >> For 50-ET, {1, 3, 5, 7, 9, 13, 15, 17, 19} is the maximal consistent > (19-limit) set containing 1,3,9. So I like x|s for 2 steps (as13'), and if> it's OK here, why not also in 43-ET? But w|w as 17+(19'-19) is actually -1 > steps of 50-ET. > > The only options for 1deg50, that don't involve 11 are )|) as 19+7 or ~|) > as 7+17 and /|) as 13. /|) seems the obvious choice to me.This is my latest proposal: 50: /|) (|\ /||\ So we agree!>> 34, 41: s| s|s ||s s||s > > Agreed. >>> 27: s| x|s ||s s||s >> Why do you prefer (|\ to /|)?2deg27 is almost 90 cents, so (|\ is nearer in size than /|). Otherwise, it's a tossup.>> 48: |x s|s ||x s||s >> In 48-ET, {1, 3, 7, 9, 11} has only slightly lower errors than {1,3, 5, 9,> 11}, 10 cents versus 11 cents. Why prefer the above to the lower prime scheme > 48: /| /|s ||\ /||\ ?To make 48 compatible with 96-ET (see below).>> 46, 53: s| s|s x|x ||s s||s > > Agreed. >>> 58, 72: s| |s s|s s|| ||s s||s (version 1 -- simpler, but more confusability) >> 72: s| |x s|s ||x ||s s||s (version 2 -- more complicated,but less confusability)> > Of course, I prefer version 2 for 72-ET, since I started the whole > confusability thing. It isn't significantly more complicated.To further confuse the issue, I now have even more options for 72-ET: 72: /| |\ /|\ /|| ||\ /||\ (simplest, but most confusability) 72: /| |) /|\ ||) ||\ /||\ (version 2 -- more complicated, no confusability, inconsistent) 72: /| |) /|\ (|| ||\ /||\ (version 3 -- simpler, no confusability, but (|| < ||\ ) 72: /| |) /|\ (|\ ||\ /||\ (version 4 -- simple, no confusability, consistent, harmonic-oriented) The symbol arithmetic in version 2 is inconsistent: /|\ minus |) equals 1deg72, but /||\ minus ||) equals 2deg72 This is remedied in version 3, which also has a problem in that the symbol for 4deg72 is a larger rational interval than that for 5deg72, something I would rather not see in a division as important as 72, although the difference between (|| and ||\ is rather small. This leaves me with version 4 as my choice. Notice that the first 4 symbols are, in order, the 5 comma, the 7 comma, the 11 diesis, and the 13' diesis, all of which are the rational symbols used for a 13- limit otonal scale: C D E\! F/|\ G A(!/ Bb!) or B!!!) C. This option should also be considered in connection with our discussion of 36 and 43 above.>> 58: s| w|x s|s w||v ||s s||s (version 2 -- morecomplicated, but less confusability)> > I'm inclined to go with version 1 despite the increased lateral > confusability, rather than introduce 17-flags. Version 2 is a _lot_ more > complicated.These are my latest options for both 58 and 65-ET: 58, 65, 72: /| |\ /|\ /|| ||\ /||\ (simplest, but most confusability) 58: /| ~|) /|\ ~|| ||\ /||\ (version 2 -- more complicated, no confusability) 65: /| /|~ /|\ ||~ ||\ /||\ (version 2 -- more complicated, no confusability) 58: /| /|) /|\ (|\ ||\ /||\ (version 3 -- simpler, some confusability) Version 3 could offer 29/58 compatibility, but the straight flags of version 1 are the simplest. I also threw 65-ET in there. Below I have a proposal for 130-ET, which results in 65 having all straight flags (as in the first version above), so I believe I would prefer that.>> 96: s| |x |s s|s s|| ||x ||s s||s (version 1 -- simpler,but more confusability)>> 96: s| |x w|s s|s w|| ||x ||s s||s (version 2 -- morecomplicated, but less confusability)> > The only maximal 1,3,9-consistent 19-limit set for 96-ET is {1, 3, 5, 9, > 11, 13, 15, 17}. It is not 1,3,7-consistent so the |) flag should be > defined as the 13-5 comma (64:65) if it's used at all. The 17 and 19 commas > vanish, so we should avoid )| |( ~| and |~. So I end up with > 96: /| |) /|) /|\ /|| ||) /||) /||\ > Simple _and_ non confusable.My latest proposal for 96 is: 96: /| |) /|) /|\ (|\ ||) ||\ /||\ As I mentioned above, I would like to see both /| and ||\ used whenever possible. At least we agree on 48, if that is to be notated as a subset of 96.>> 94: w| s| w|s s|s x|x w|| ||s w||s s||s >> Why do you prefer that to >>> 94: ~| /| |) /|\ (|) ~|| ||\ ||) /||\ >> Surely we're more interested in the 7-comma than the 17+(11-5) comma. > > Also, it makes sense that /| + ||\ = /||\, but it makes the second half > apotome have a different sequence of flags to the first. Which should we > use, /|| or ||\ ?My proposal above for a matched sequence being subordinate to having ||\ and rational complements would apply here. While ~| and ~||\ are not rational complements, they are the 217-ET complements -- the nearest we can get to a rational complement for 1deg94. I calculate both |) and |\ as 2deg94, so I needed something else for 3deg. The best possibilities were (| and ~|\ -- neither one uses a new flag. My choice was: 87, 94: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ The symbol sequence is fairly simple, particularly in the second half- apotome. Or is the other option: 87, 94: ~| /| (| /|\ (|) ~|| ||\ (|| /||\ better? (Perhaps this is what you meant?)>> 111 (37 as subset): w| s| |s w|s s|s x|s w|| s|| ||s w||s s||s >> Dealt with above. I'd prefer (|) for 6deg111. Yes. Agreed!>> 140: |v |w s| |s s|w s|x s|s x|s ||w s|| ||s s||w s||x s||s >You made no comment about this one, but it's no good: /|\ should be 6deg140, not 7deg (wishful thinking on my part), even though /||\ is 14deg. I now propose: 140: )| |~ /| )|\ /|~ /|) (|~ (|\ )|| ||~ ||\ ) ||\ /||~ /||\ This is the simplest set I could come up with that uses both /| and ||\.>> 152: |v |w s| |s s|w s|x s|s x|x x|s ||w s|| ||ss||w s||x s||s> > Dealt with elsewhere. I see no reason to use |( which is really zero steps, > when )| is 1 step. Yes. Agreed!>> 171: |v w|v s| |x |s w|s s|x s|s x|s w||v s|| ||x||s w||s s||x s||s> > Why not ~| for 1 step? >>> 183: |v w|v s| |x |s w|s s|x s|s x|x x|s w||v s||||x ||s w||s s||x s||s> > Why not use w| for 1deg183, being a simpler comma than |v? 17 vs. 17'-17.After re-evaluating, I would keep what I had above for both 171 and 183. The choice between |( and ~| is almost a tossup, but I found two reasons to prefer |(: 1) It is closer in size to both 1deg171 and 1deg183; and 2) It is the rational complement of /||).>> 181: |v w| w|v s| |s w|x w|s s|x s|s x|x w|| w||vs|| ||s w||x w||s s||x s||s> > I don't see how |) can be 5deg181 or how /|\ can be 9deg181.More wishful thinking on my part that /|\ should be half of /||\ -- I guess I was getting tired.> The only > symbol that can give 9deg181 with 19-limit commas is (|~. Here's my proposal. > > 181: |( ~| |~ /| /|( (| (|( /|) (|~ (|\ ~||||~ /|| /||( (|| (||( /||) /||\ And here's my new proposal. 181: |( ~| |~ /| /|( ~|) /|~ /|) (|~ (|\ ||( ~|| ||~ ||\ /||( ~||) /||~ /||\ We don't agree on the symbol arithmetic in the second half-apotome. Both /| and |\ are 4deg181, so /||\ minus /| equals /||\ minus |\ equals 4deg. You have /|| as 5deg less than /||\. My choice for 6deg ~|) was on the basis of its being the rational complement of 12deg ||~; 7deg /|~ logically followed as 3deg plus 4deg.>> 217: |v w| |w s| |x |s w|x w|s s|x s|s x|x x|s w||||w s|| ||x ||s w||x w||s s||x s||s> > Agreed.And here is what I came up with over the past week, prior to reading your latest. It's easier to put the whole thing here than trying to sort through them to figure out what wasn't covered above. 12, 19, 26: /||\ 17, 24, 31, 38: /|\ /||\ 22: /| ||\ /||\ 36, 43: |) ||) /||\ (version 1) 36, 43: |) (|\ /||\ (version 2) 29: ~|) ~|| /||\ 50: /|) (|\ /||\ 34, 41: /| /|\ ||\ /||\ 27: /| (|\ ||\ /||\ 48: |) /|\ ||) /||\ 55: |( /|\ ||( /||\ 39, 46, 53: /| /|\ (|) ||\ /||\ 60: /| |) (|\ ||\ /||\ 67: |( /|) (|\ /||) /||\ 74: )|) /|) (|\ /||) /||\ 58, 65, 72: /| |\ /|\ /|| ||\ /||\ (simplest, but most confusability) 72: /| |) /|\ ||) ||\ /||\ (version 2 -- more complicated, no confusability, inconsistent) 72: /| |) /|\ (|| ||\ /||\ (version 3 -- simpler, no confusability, but (|| < ||\ ) 72: /| |) /|\ (|\ ||\ /||\ (version 4 -- simple, no confusability, consistent, harmonic-oriented) 58: /| ~|) /|\ ~|| ||\ /||\ (version 2 -- more complicated, no confusability) 58: /| /|) /|\ (|\ ||\ /||\ (version 3 -- simpler, some confusability) 65: /| /|~ /|\ ||~ ||\ /||\ (version 2 -- more complicated, no confusability) 51: |) /| /|) ||\ ||) /||\ 56, 63: )| /| /|\ (|) ||\ )||\ /||\ 70, 77: /| |\ /|\ (|) /|| ||\ /||\ (simplest, but most confusability) 77: /| |) /|\ (|) /|| ||\ /||\ 84: /| |) /|) (|\ ||) ||\ /||\ 68: |) /| /|) (|) ||( ||\ /||( /||\ 96: /| |) /|) /|\ (|\ ||) ||\ /||\ 80: |) /| /|) /|\ (|) (|\ ||\ /||) /||\ 87, 94: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ 108: /| //| |) /|) (|\ //|| ||) ||\ /||\ 99: |~ /| /|~ /|) (|~ (|\ ||~ ||\ /||~ /||\ 104: )| |) )|) (| /|\ (|) )|| ||) )||) (|| /||\ 111 (37 as subset), 118 (59 ss.): ~| /| |\ ~|\ /|\ (|) ~|| /|| ||\ ~||\ /||\ 125: |( /| |\ (|( /|\ (|) ||( /|| ||\ (||( /||\ 132: ~|( /| |) |\ /|) /|\ /|| ||) ||\ /||) /||\ 130 (65 ss.): ~| /| |) |\ /|) /|\ (|\ /|| ||) ||\ /||) /||\ 128 (64 ss.): )| ~| /| ~|\ ~|\ /|\ (|) )|| ~|| ||\ )||\ ~||\ /||\ 135 (45 ss.): |( |~ /| (| /|~ /|\ (|) ||( ||~ ||\ (|| /||~ /||\ 142: ~| /| |) |\ /|) /|\ (|) (|\ /|| ||) ||\ /||) /||\ 149: |( /| /|( |\ /|~ /|) (|~ (|\ /|| /||( ||\ /||~ /||\ 140 (70 ss.): )| |~ /| )|\ /|~ /|) (|~ (|\ )|| ||~ ||\ ) ||\ /||~ /||\ 147: |( |~ /| |\ /|~ /|) /|\ (|\ ||~ /|| ||\ ) ||\ /||~ /||\ 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /|| ||\ /||~ /||) /||\ 159: |( |~ /| |\ /|~ (|~ /|\ (|) ||( ||~ /|| ||\ /||~ (||~ /||\ 171: |( ~|( /| |) |\ ~|\ /|) /|\ (|\ ~||( /|| ||) ||\ ~||\ /||) /||\ 183: |( ~|( /| |) |\ ~|\ /|) /|\ (|) (|\ ~||( /|| ||) ||\ ~||\ /||) /||\ 181: |( ~| |~ /| /|( ~|) /|~ /|) (|~ (|\ ||( ~|| ||~ ||\ /||( ~||) /||~ /||\ 193: |( ~| ~|( /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ~|| ( /|| ||\ ~||) ~||\ /||) /||\ 207: |( ~|( /| /|( |) |\ ~|\ /|) /|\ (|) (|\ ~||( /|| /| ( ||) ||\ ~||\ /||) /||\ 198: )| |~ )|~ /| |\ )|\ ~|) /|) /|\ (|) (|\ )|| ||~ ) ||~ /|| ||\ )||\ ~||) /||) /||\ 217: |( ~| |~ /| |) |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ||~ /|| ||) ||\ ~||) ~||\ /||) /||\ 224: )| |( |~ /| |) |\ /|~ (|( /|) /|\ (|) (|\ ||( ||~ /|| ||) ||\ /||~ (||( /||) /||\ I thought that if these were listed in order of increasing complexity, perhaps we could spot a few patterns in the arrangement of symbols that might help to resolve differences of opinion in instances where the best choice of symbols is not clear. --George

Message: 4907 - Contents - Hide Contents Date: Wed, 29 May 2002 13:05:58 Subject: scale collection From: Carl Lumma Gene, you had also given Qm(2) previously (e=15, c=4). What's this: 10-tone scale, e=24 c=4, in 72-tET (0 5 14 19 28 33 42 49 58 63 72) ? -Carl

Message: 4908 - Contents - Hide Contents Date: Wed, 29 May 2002 13:20:46 Subject: Re: scale collection From: Carl Lumma Dave, this is one of yours: ! Single-chain MOS of 7:4's in 41-tet. 11 ! 204.878 !......7 234.146 !......8 439.024 !......15 468.293 !......16 673.171 !......23 702.439 !......24 907.317 !......31 936.585 !......32 1141.463 !.....39 1170.732 !.....40 1200.000 !.....41 ! I don't see any generators near 7:4 in the catalog. What is this? -Carl

Message: 4910 - Contents - Hide Contents Date: Wed, 29 May 2002 00:28:02 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> Gene, I'm puzzled. How come we didn't see the 5-limit versions of >> these four in any of your earlier 5-limit lists. >>>>> [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016 >>> [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453 >>> [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410 >>> [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132 >> That's because as 5-limit systems they are all pretty bad.Sure they wouldn't have made the final list, but they are way better than many others on the combined list I made of all those you generated back then.> On the > other hand, here is something I should have included on my 7s list:>> [19, 19, 57, 79, -37, -14] [[19, 0, 14, -37], [0, 1, 1, 3]] >> badness 160.2710884 g 33.81074780 rms .1401992317 > > generators [63.15789474, 1901.874626]You did include it. However your list does not include the following (from Graham's catalog): twin meantone (double diatonic) [[2 . . .] [0 1 4 4]] shrutar (double diaschismic) [[2 . . .] [0 2 -4 7]] porcupine [[1 . . .] [0 3 5 -6]] diminished [[4 . . .] [0 1 1 4]] diaschismic (15-limit variant) [[2 . . .] [0 1 -2 -8]] diaschismic (56-ET variant) [[2 . . .] [0 1 -2 9]] It seems that your current badness is too lenient on complexity and too hard on error. Can you explain why you used complexity^3 * error for 5-limit and are now using complexity^2 * error for 7-limit?

Message: 4911 - Contents - Hide Contents Date: Wed, 29 May 2002 01:53:22 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> Gene, I'm puzzled. How come we didn't see the 5-limit versions of >> these four in any of your earlier 5-limit lists. >>>>> [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016 >>> [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453 >>> [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410 >>> [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132 >> That's because as 5-limit systems they are all pretty bad.Having found their 5-limit rms optima, I can see that a cutoff on your badness (= complexity^3 * error) would explain the non-appearance of all of them except the superpythagorean one (the 3rd one above). It's non-appearance can only be explained by such a cutoff if your badness uses unweighted complexity. Using weighted complexity it has a badness of 1202 which is better than 1220703125/1207959552 semiminorsixths badness 1352 48828125/47775744 quintaminorthirds badness 1902 Maybe these two were only generated when you were using unweighted complexity and maybe you always had your badness limit set below 1202 once you started using weighted complexity. In my own badness ranking, superpythagorean (where 19683/20480 vanishes) comes 23rd (after septathirds and parakleismic). Gene, I afraid we might have missed other 5-limit temperaments that were too bad by your measure, but not mine. Could you please rerun your 5-limit temperament generator with an rms error cutoff of 35 cents, weighted rms complexity cutoff of 10 generators, and badness = weighted_complexity^3*error cutoff of 1900.

Message: 4912 - Contents - Hide Contents Date: Thu, 30 May 2002 00:01:55 Subject: Re: scale collection From: Carl Lumma>It's definitely not one of mine. What made you think it was?I have it in my records as being yours. It might be my mistake; I don't remember a source.>A mistake? It would take quite a stretch of my imagination to consider >the generator of this scale as a 4:7. It has a 32 cent error. If it >has a 7-limit mapping at all, it must be [2, -6, 1] in which case the >generator can be considered to be (the inversion of) half the pelogic >fifth. That temperament would be better represented in 37-ET, but 11 >notes would no longer be MOS there. Even with the optimum generator >(259.64 cents) it has a max abs error of 28.5 cents.I'll delete it. -Carl

Message: 4913 - Contents - Hide Contents Date: Thu, 30 May 2002 12:39:04 Subject: Re: A 7-limit best list From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> i'm not seeing the injera temperament here (generators should be 600, > 694); that concerns me . . .The feedback I'm getting from both you and Dave is that my badness cutoff was set too low. If we go all the way out to ennealimmal with a higher cutoff, we'll have a lot of temperaments. Perhaps we need to do either one of Dave's rolloffs or two different lists.

Message: 4915 - Contents - Hide Contents Date: Thu, 30 May 2002 02:02:04 Subject: Re: scale collection From: dkeenanuqnetau --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Dave, this is one of yours: > > ! > Single-chain MOS of 7:4's in 41-tet. > 11 > ! > 204.878 !......7 > 234.146 !......8 > 439.024 !......15 > 468.293 !......16 > 673.171 !......23 > 702.439 !......24 > 907.317 !......31 > 936.585 !......32 > 1141.463 !.....39 > 1170.732 !.....40 > 1200.000 !.....41 > !It's definitely not one of mine. What made you think it was?> I don't see any generators near 7:4 in the catalog.It might appear with a 7:8 generator in the catalog. But it doesn't and shouldn't.> What is this?A mistake? It would take quite a stretch of my imagination to consider the generator of this scale as a 4:7. It has a 32 cent error. If it has a 7-limit mapping at all, it must be [2, -6, 1] in which case the generator can be considered to be (the inversion of) half the pelogic fifth. That temperament would be better represented in 37-ET, but 11 notes would no longer be MOS there. Even with the optimum generator (259.64 cents) it has a max abs error of 28.5 cents.

Message: 4916 - Contents - Hide Contents Date: Thu, 30 May 2002 15:18 +0 Subject: neutral third scale From: graham@xxxxxxxxxx.xx.xx Mark Gould wrote:> Thoughts please on the following scale > (in cents) > 0 150 300 500 650 850 1000 (0 > > Interesting properties under transposition, it has. Can anyone name it, > or > provide any other infoThat's neutral Phrygian in Manuel's list. Another mode of it is Mohajira, an uncommon Arabic scale. I've got a page about these kind of things at <7+3 scales * [with cont.] (Wayb.)>. Graham

Message: 4917 - Contents - Hide Contents Date: Thu, 30 May 2002 17:21:02 Subject: Re: A common notation for JI and ETs From: gdsecor Dave, I'm taking a little time to elaborate on the issue of correlating symbols within each of two groups of systems. 36, 43, & 72 ------------ If we wish to completely correlate 36, 43, & 72, then the choice should be clear. There are three ways to do it; only one of these has no inconsistencies. 1) Rational complements, but ||) is inconsistent in 72: 36, 43: |) ||) /||\ 72: /| |) /|\ ||) ||\ /||\ If you think we can justify ||) for 4deg72 on the basis of rational complementation, I'm willing to consider it, but I think the symbol arithmetic is sloppy. Also, if you wanted a single-shaft symbol for 2deg43, then we can forget about 36-43 correlation (which may not be all that important). 2) Mirrored flags, but (|| is inconsistent in 36 & 43: 36, 43: |) (|| /||\ 72: /| |) /|\ (|| ||\ /||\ This one is not my choice. 3) Use of 13-limit symbols is consistent in all three: 36, 43: |) (|\ /||\ 72: /| |) /|\ (|\ ||\ /||\ This is my choice: complete correlation, with a single-shaft symbol for 2deg43. If you are going to use only single-shaft symbols in combination with conventional sharps and flats, I think you would still have the option to notate something as C#!} instead of C(|\, should that tone be used in a 7 relationship. 29, 58, 87, & 94 ---------------- For these it depends on how much symbol correlation is desired among the systems. 1) Complete symbol correlation, with correlated symbols ~|) and ~|| being rational complements: 29: ~|) ~|| /||\ 58: /| ~|) /|\ ~|| ||\ /||\ 87, 94: ~| /| ~|) /|\ (|) ~|| ||\ ~||) /||\ 2) Complete symbol correlation with less use of ~| flag: 29: (| ~|| /||\ 58: /| (| /|\ ~|| ||\ /||\ 87, 94: ~| /| (| /|\ (|) ~|| ||\ (|| /||\ 3) More memorable order of symbols for 87 & 94 gives almost complete correlation and maximizes rational complementation: 29: ~|) ~|| /||\ 58: /| ~|) /|\ ~|| ||\ /||\ 87, 94: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ 4) Using simpler flags in 29 & 58 maintains correlation only between those two, but still maximizes rational complementation: 29: /|) (|\ /||\ 58: /| /|) /|\ (|\ ||\ /||\ 87, 94: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ * 5) Using only straight flags for 29 & 58 eliminates all correlation, but still maximizes rational complementation: 29: /| ||\ /||\ 58: /| |\ /|\ /|| ||\ /||\ 87, 94: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ * *Since 87 & 94 are not correlated with 29 & 58, they may be considered separately. Option 3 is my first choice in that it is least disorienting to anyone who is going to use two or more of these systems. (Consider that a piece in 58 or 87 might have a section entirely in 29-ET.) Regarding the use of the 17 flag: In 87 it is the best of three not- very-good choices. Its use in 58 may be appropriate, considering how well 17 is represented in that division. It can be justified in 29 only as a subset of the other two. I don't see any reason not to use the same symbols for 87 and 94. Otherwise, options 4 or 5 would be okay with me. --George

Message: 4921 - Contents - Hide Contents Date: Thu, 30 May 2002 22:01:11 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: >>>>> i'm not seeing the injera temperament here (generators should be > 600,>>> 694); that concerns me . . . >>>> The feedback I'm getting from both you and Dave is that my badness >> cutoff was set too low. If we go all the way out to ennealimmal with >> a higher cutoff, we'll have a lot of temperaments. Perhaps we need >> to do either one of Dave's rolloffs or two different lists. >> if you're happy doing one of dave's rolloffs, then by all means (and > for 5-limit too!) . . . but i don't agree with dave that temperaments > like Shrutar necessarily have to show up in the 7-limit list . . .They don't have to show up in the final list, but the fact that they are not showing up in Gene's wide-open list leads to fears that we are missing other possibly interesting ones that we haven't seen before.

Message: 4922 - Contents - Hide Contents Date: Thu, 30 May 2002 22:06:48 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: >>> i'm not seeing the injera temperament here (generators should be 600, >> 694); that concerns me . . . >> The feedback I'm getting from both you and Dave is that my badnesscutoff was set too low. If we go all the way out to ennealimmal with a higher cutoff, we'll have a lot of temperaments. Perhaps we need to do either one of Dave's rolloffs or two different lists. You haven't addressed my question: Why complexity^2 * error now, when you used complexity^3 * error for 5-limit? I think if you change that, and put cutoffs at 17 gens for weighted complexity and 25 or 30 cents for error, the list shouldn't be too big. My earlier suggestion of 2000 or so badness cutoff is only if you insist on continuing to use complexity^2 * error.

Message: 4924 - Contents - Hide Contents Date: Thu, 30 May 2002 23:49 +0 Subject: Re: neutral third scale From: graham@xxxxxxxxxx.xx.xx emotionaljourney22 wrote:> i think mark would also be interested to know (if he doesn't already) > about the academic paper considering this in a class of "anti- > diatonic" scales. anyone have the reference handy?The only reference I know of for "anti-diatonic" scales is the page on my website that I've already given. I assume you're thinking of Carey&Clampitt's "Self-Similar Pitch Structures, Their Duals, Rhythmic Analogues" that appeared in Perspectives of New Music, probably the issue before Mark's own paper. The neutral third scales are only described in a footnote, as the mean of two different diatonic scales. In the body of the text a matrix is given which is a dual to the diatonic one, but not in a particularly important way (it comes down to them both being a 7 note MOS). So it's not really a good place to find out about neutral third scales. I got the paper online. I think it's still available somewhere, maybe the files area for one of these groups. Graham

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