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Message: 4950 - Contents - Hide Contents Date: Sat, 01 Jun 2002 09:40:22 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: Carl Lumma>> >0-tone scale, e=24 c=4, in 72-tET >> (0 5 14 19 28 33 42 49 58 63) >> >> Connectivity seems so good, I'm not sure why we're not using it >> more often. >>It's also h10-epimorphic. This is clearly an important scale, and needs >a name.You've discovered it twice now, so the ball is clearly in your court. I have it as sa118. Does this address why we haven't just been using connectivity? Do you have code which searches for high c, or just code that measures c? -Carl

Message: 4951 - Contents - Hide Contents Date: Sun, 02 Jun 2002 07:15:26 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Does this address why we haven't just been using connectivity? Do > you have code which searches for high c, or just code that measures c?Maple will measure c, but it also will find intervals and triads even more quickly using the characteristic polynomial. The latter seemed to me more relevant, so I quit showing c--do you think I should reintroduce it?

Message: 4952 - Contents - Hide Contents Date: Sun, 02 Jun 2002 07:27:12 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> You've discovered it twice now, so the ball is clearly in your court. > I have it as sa118.How about gamelion?

Message: 4953 - Contents - Hide Contents Date: Sun, 02 Jun 2002 00:45:55 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: Carl Lumma>> >oes this address why we haven't just been using connectivity? Do >> you have code which searches for high c, or just code that measures c? >>Maple will measure c, but it also will find intervals and triads even more >quickly using the characteristic polynomial. The latter seemed to me more >relevant, so I quit showing c--do you think I should reintroduce it?I don't know... why don't we keep our eye on it for a while. (that's a yes). -Carl

Message: 4954 - Contents - Hide Contents Date: Sun, 02 Jun 2002 00:46:59 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: Carl Lumma>> >ou've discovered it twice now, so the ball is clearly in your court. >> I have it as sa118. >

Message: 4955 - Contents - Hide Contents Date: Sun, 2 Jun 2002 04:18:20 Subject: Scales for Starling (the 126/125 planar temperament) From: Gene W Smith Both star and nova are epimorphic for h8+v7, but I didn't find anything to match them among other scales of that sort I checked. However, I did find two interesting twelve note scales. I've given these in the 108-et; this is because while there are advantages to 46, etc, the true intonational exactness isn't reached there, and I at least find that the difference is noticable. I haven't heard about 108, which is sort of the crazy uncle of the 12-multiple family kept locked in the attic. It shares its {2,3,7} subgroup with 72, both of them having the 36-etvalues, but its different third makes it cover 126/125 instead of 225/224 (as well, of course, as 1029/1024, a comma of {2,3,7} it shares with 72.) Dave and George can ponder it, and see if they think it is garbage. :) [0, 7, 21, 28, 35, 45, 56, 63, 73, 80, 91, 101] 5: 23, 12 7: 36, 36 9: 42, 58 [0, 11, 17, 28, 35, 45, 52, 63, 73, 80, 91, 97] 5: 21, 10 7: 31, 28 9: 44, 65 star108 := [0, 7, 28, 35, 56, 63, 80, 91] 5: 13, 6 7: 21, 20 9: 23, 28 nova108 := [0, 11, 28, 35, 56, 63, 80, 91] 5: 13, 6 7: 20, 18 9: 24, 32

Message: 4956 - Contents - Hide Contents Date: Sun, 02 Jun 2002 00:06:12 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: >>> It also seems (from the to and fro between Gene and Paul in that >> thread) that the only justifications for using _log_-flat (and not >> something stronger) are that >> (a) it's easy to deal with mathematically, and >> (b) Gene likes it. >> (a) It has a rational basis; what else does?Plenty else. Merely having a rational basis is no guarantee of anything. It might be the _wrong_ rational basis. I think what we really want is a human perceptual/cognitive/motor basis; empirical, not purely mathematical. What's the point otherwise? To be done properly that would have to be done statistically by surveying lots of musicians who have tried different temperaments. But we don't have that luxury and must rely on the experience of the few of us on this list, and what we've learnt by listening to others. To me, that seems quite rational too.> (b) I tried g^3, leading to the grooviest 7-limit thread. I thoughtit showed a decided bias in favor of low complexities. This is a valid consideration. It's good that you expressed it _as_ an opinion and I suppose your opinion should carry as much weight as anyone else's.> (c) It works.That also is a matter of opinion. I must apologise for not remembering, and not checking the archives, regarding the use of complexity^3 for 5-limit and complexity^2 for 7-limit. OK. Continue using complexity^2 for 7-limit, but _please_ can we have a list with a higher badness cutoff and _weighted_ complexity. I expect you dislike using weighted complexity because the justification for it is not purely mathematical? Badness should be high enough to include: twin meantone (injera) [[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]] One per line, no blank lines between please. Surely this list won't be too big if you use a weighted complexity cutoff of 17 generators and an error cutoff of 25 cents. This is in no way intended to be the final list, this is just to make sure we've seen everything worth seeing before deciding on the final cutoffs. Regards, -- Dave Keenan

Message: 4958 - Contents - Hide Contents Date: Tue, 04 Jun 2002 06:57:26 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> In my opinion Porcupine is of far more interest at the 7-limit than >> double meantone. >> far more interest? how do you figure?Porcupine has about the same weighted complexity but a significantly smaller error. Porcupine [3,5,-6], 5.84 gens, 6.81 cents Twin meantone [1,4,4] with half-octave period, 5.34 gens, 11.22 cents>> Diminished and the other >> diashismic don't make it. >> none of the ways of extending diminished to 7-limit suggested by > herman miller here: > > Diminished temperament * [with cont.] (Wayb.) > > make it? can you detail why that is?I didn't say that. I was only referring to the one in my message, in my list of temperaments that weren't in Gene's list. The [1,1,1] mapping (which was in Gene's list) will make it. The [1,1,4] mapping won't because of its combination of 8.26 gens and 9.15 cents. Herman's page referred to above, only gives the [1,1,1] mapping. Are there really any others worth considering?

Message: 4959 - Contents - Hide Contents Date: Tue, 04 Jun 2002 01:38:49 Subject: Re: A 7-limit best list From: David C Keenan Pending Gene's next list with badness up to 350, based on weighted complexity, here's a list of 27 we might agree on. It's given in order of increasing complexity. It consists of all those I _know_of_ with weighted-complexity < 13 gens, error < 23 cents, Gene's-weighted-badness < 343, but there may well be some we don't know about. So are there any others that anyone thinks should be included? Periods Gens per Gen Wtd Compl RMS err Name per oct 1:3 1:5 1:7 (cents) (gens) (cents) ---------------------------------------------------------------------- dominant 7th 1 1 4 -2 498 3.13 20.16 (meantone with inaccurate 7s) diminished 4 1 1 1 85 3.14 19.14 quintuple thirds 5 0 1 0 91 3.29 15.82 (Blackwood's decatonic) augmented 3 1 0 2 111 3.68 16.60 with inaccurate 7s diaschismic (pajara) 2 1 -2 -2 108 3.94 10.90 tertiathirds 1 4 -3 2 126 4.31 12.19 (Negri's system) kleismic 1 6 5 3 317 4.37 12.27 with inaccurate 7s augmented 3 1 0 -2 89 4.63 8.10 meantone 1 1 4 10 504 5.32 3.67 twin meantone 2 1 4 4 94 5.34 11.22 (injera) twin wide 2 4 3 3 274 5.75 10.13 subminor thirds porcupine 1 3 5 -6 162 5.84 6.81 superpythagorean 1 1 9 -2 490 6.21 6.41 neutral thirds 1 2 -9 -4 356 6.52 6.25 with complex 5s magic 1 5 1 12 381 6.79 4.14 narrow minor thirds 1 10 9 7 310 7.40 3.32 semisixths 1 7 9 13 444 7.42 5.05 (tiny diesic) subminor thirds 1 7 -3 8 271 7.42 2.59 (orwell) miracle 1 6 -7 -2 117 7.61 1.64 quartaminorthirds 1 9 5 -3 78 7.66 3.07 supermajor seconds 1 3 12 -1 232 7.74 3.58 schismic 1 1 -8 -14 498 8.61 2.86 diaschismic 2 1 -2 -8 104 9.47 3.82 (15-limit variant) octafifths 1 8 18 11 88 10.50 2.06 (semi minimal-diesic) shrutar 2 2 -4 7 53 11.19 2.25 (double diaschismic) kleismic 1 6 5 22 317 11.41 1.61 with complex 7s (catakleismic) half wuerschmidt- 1 16 2 5 194 11.75 0.88 thirds ------------------------------------------------------------------- I tried hard to get a list that included ennealimmal, and that could be made to agree between our two methods. This is not because I think ennealimmal is particularly interesting, but because I know Gene does. So far, no matter how I tinker with my two rolloff parameters and Gene's complexity and error cutoffs, I can't make two lists that agree while including ennealimmal, even if I'm willing to include stuff that I would normally consider junk. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4960 - Contents - Hide Contents Date: Tue, 04 Jun 2002 02:10:23 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> Badness should be high enough to include: >> >> twin meantone (injera) [[2 . . .] [0 1 4 4]] > > yup. >>> 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]] >> i'm not sure about the rest of these. maybe diminished, but the rest > have never been advocated as 7-limit temperaments.Apparently you mean they have only been advocated at _higher_limits than 7. But that doesn't disqualify them from consideration at the 7-limit. So I've been forced to find their minimum rms errors myself, so I can calculate Gene's badness (and my own) for them. In my opinion Porcupine is of far more interest at the 7-limit than double meantone. Shrutar and the [1, -2, -8] diaschismic are of about equal interest, just slightly less interest than double-meantone, but are at (or just off) the bottom of the list. Diminished and the other diashismic don't make it.> gotcha. i'm just worried that there will be way too much to consider > here. why not just insist that injera makes it in and go from there?Well hey we could at least take a look at how many are on such a list instead of refusing to look 'cause we're worried. I don't understand yours or Gene's attitude in this regard. I guess all I can say is, I won't be at all confident that we haven't missed something unless we look at everything with complexity (weighted or otherwise) less than 17 gens and rms error less than 25 cents and Gene's badness less than 500 if unweighted rms complexity is used or less than 350 if weighted rms complexity is used. Of course weighted should be used. This is only slightly more than is required to ensure that double-meantone appears. Double meantone has a Gene's-badness of 320. The [1, -2, -8] diaschismic is 343. These are using weighted complexity.

Message: 4962 - Contents - Hide Contents Date: Wed, 05 Jun 2002 12:41:28 Subject: "The hypothesis" from a wedge product point of view From: genewardsmith In the prime limit with k primes, take k commas in order from largest to smallest,[c1, c2, ... , ck], defining a square matrix with determinant +-1 (i.e., a "notation".) Leave off c1 and wedge the rest, and you have a val "h" defining an et mapping, corresponding to a Fokker block with h(2) notes to the octave. Suppose h linearly orders the block; then leave off *both* c1 and c2 and wedge the rest, and you have a linear temperament, which when applied to the block gives (when the period is an octave) n contiguous (in the sense of the generator of the temperament) notes of a MOS, and similarly with periods which are a fraction of an octave.

Message: 4963 - Contents - Hide Contents Date: Wed, 05 Jun 2002 23:12:38 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >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).I suppose if the 23-comma interpretation is confined to such large ETs it might be ok, but I'd need to be convinced that there was no other way to do it and the ET is actually 1,3,9,23-consistent and preferably 1,3,9,...,23-consistent where "..." are the other primes used for its notation. 135, 147, 159, and 224-ET are all 17-limit notatable, although you'd probably go to 19 limit for 224-ET. 198-ET is not 19-limit notatable, but why do you feel any need to notate it? And is it 23-limit notatable anyway?>> 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.I'll go with your solution, for the single-shaft symbols at least. The lateral confusability is addressed by the large difference in widths, and the user will just have to learn that 11 = 13' and 11' = 13. I propose (||( for 4deg74 purely because it is the mirror image of )|) (plus a shaft). More about this later.>>> 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: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /|| >||\ /||~ /||) /||\Yes, for the single-shaft symbols at least.>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.You know, I won't really be happy until I have a spreadsheet that generates the notation for every ET, based on a bunch of rules, because that's the only way I'll be sure we're being consistent. The rules may of course end up being very complicated, but I wouldn't want to see any rule that only applied to a single ET. By the way, perhaps we should use >| and |< instead of )| and |( to make the concave flags more distinct from the convex ones in ASCII. What do you think?>>> 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 (|).Yes, I did. Sorry.>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.I guess if there is already a popular notation in use for some ET, and there is a fairly direct correspondence to that notation available in sagittal, then we should use it. So on that basis at least, I can tentatively agree with your proposal for 31. I reserve the right to change my mind on this on further investigation. :-) 31: /|\ /||\>>> 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;I understand wanting comma up and down symbols in 22-ET. That's what I want too. But what do you mean by /|| and \!! as comma up and down symbols? We're proposing _not_ to have those those double-shafters with a _left_ flag. I assume you meant the double-shafters to have right flags. In which case 22 is agreed. 22: /| ||\ /||\>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.In fact, I'm very happy that /| and ||\, |\ and /|| should always be complements and to _always_ have a mirrored sequence rather than _sometimes_ have a matched one.>Also, with the apotome divided into fewer than 5 parts, I would want >to use /|\ only when it is exactly half of /||\. Fair enough. >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.)I am instead inclined to totally ignore rational complements with regard to ETs, especially for the lower numbered ones. One reason is that I feel that the choice of double-shaft symbols cannot in any way be allowed to influence the choice of single-shaft. One must first choose the best set of single shaft symbols (ignoring complements) since some users will have no interest in the double-shaft symbols and should not be penalised for it. In fact, (and I've been making gentle noises about this possibility for a some time now), I'm willing to throw away everything we agonised over with regard to rational complements and instead adopt a simple system that applies automatically to all ETs and rational tunings. I propose that the complement of a|b is always b||a, except that the complement of |//| (natural) is /||\ and the complement of /|\ is /|\ if it represents the same number of steps and (|) if it represents a different number of steps. You will notice that this would require no change to the 72-ET version 3 notation, nor any change to most of the smaller ETs we've agreed on such as 12, 17, 19, 22, 24, 26, 31, 46, 53. By the way, are you collecting all those we've agreed on into one place? I haven't been. Why do I want to do this despite some obvious disadvantages? Because I realised while trying to consistently notate the whole n*12-ET family, that it required us to repeat the whole somewhat arbitrary process we went thru for rational tunings, to find complements with minimum offsets. And what's more, that this process would have to be repeated for every such family or small range of fifth-sizes across the whole range of ETs. For example the n*29-ET family is the next largest, followed by n*17. And every such family, or small range of fifth sizes, would have a completely different complement mapping. The cognitive load for anyone who uses more than two such systems would be enormous. Now for those obvious disadvantages: 1. The second shaft does not have a fixed comma value. This doesn't seem very important to me? 2. We lose the association of flag size with rational comma size in the second half-apotome. This is the biggie. It can be remedied to some degree by redesigning the double-shaft (and X-shaft) symbols so their concave flags are wider than their wavy flags which are wider than their straight and convex flags. However it will be difficult to make single flag symbols bigger than double-flag ones. What other disadvantages have I omitted? Advantages: Simple to remember. Covers all tunings. Flags are more strongly associated with particular primes because the flags don't change when the comma is complemented. No new flag types ever need to be introduced merely to handle complements. Doesn't require /| and ||\ as a special case.>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).I agree with the above for 43-ET, but 36-ET can be notated as every second note of 72-ET, which means I want: 36: |) (|| /||\ 43: |) (|\ /||\ Do you have some argument as to why 36 and 43 should be the same? I don't see it.>>> 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.I now realise I will need to consider the entire n*29 family 29 58 87 116 145 174 203 232 261, before agreeing on either 29 or 58.>>> 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.I prefer /|) because it introduces only one new flag where (|\ introduces two, for single-shaft-only users.>>> 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).I now agree that 48 should be every second note from 96 and will address all n*12-ETs elsewhere.>>> 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.I can only agree to your version 3. 72: /| |) /|\ (|| ||\ /||\ I prefer the above to version 4, with (|\ as 4 steps, because I think that in any given ET, (| and |) flags should either both be 13-based or both be 7-based, so that (|) is always 11' (whether it is used or not). Otherwise we have 3 different possible values for (|), the largest and smallest of which differ by 0.84 cents in rational tuning. It's bad enough that folks have to know whether the convex flags refer to 7 or 13 in ETs where the tridecimal schisma doesn't vanish. I wouldn't want them to have to worry about the two convex flags _independently_. Also, 72-ET is not terribly good at the 13-limit, the error hikes from 3.9 cents at the 11-limit to 7.2 cents at the 13-limit, and in any case folks can learn that the 7-comma symbol doubles as the 13-comma symbol in 72-ET, just as they must learn that the 11-comma symbol doubles as the 7-comma symbol in 31-ET.>>> 58: s| w|x s|s w||v ||s s||s (version 2 -- more>complicated, 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 don't see any need for 29 to be every second of 58, but I do want to look at the whole huge family first.>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 -- more>complicated, 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.I changed my mind on 96, as you will have seen in other posts, but might end up changing it back.>>> 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?)Yes. That's what I meant. Sorry. I now want one of 94: ~| /| (| /|\ (|) ||) ||\ ||~ /||\ 94: ~| /| ~|\ /|\ (|) /||~ ||\ ||~ /||\ and need to look at 94 188 282 to decide.>>> 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!Only now I want the fully-mirrored half-apotomes: 111 (37): ~| /| |\ ~|\ /|\ (|) /||~ /|| ||\ ||~ /||\>140: )| |~ /| )|\ /|~ /|) (|~ (|\ )|| ||~ ||\ ) >||\ /||~ /||\ > >This is the simplest set I could come up with that uses both /| and >||\.I'll leave the second half-apotome out of it for now. It seems we have 4 options: 140: )| |~ /| )|\ (| /|) (|~ 6 flags 140: )| |~ /| )|) (| /|) (|~ 5 flags 140: )| |~ /| )|\ /|~ /|) (|~ 6 flags monotonic flags per symb 140: )| |~ /| )|) /|~ /|) (|~ 5 flags monotonic flags per symb I prefer the last one, and with mirror complements it would be 140: )| |~ /| )|) /|~ /|) (|~ (|\ ~||\ (||( ||\ ~|| ||( /||\ Note that with mirror complements, (|\ is the same as (||\.>>> 152: |v |w s| |s s|w s|x s|s x|x x|s ||w s|| ||s>s||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!With mirror complements we have: 152: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /|| ||\ ~|| ||( /||\>>> 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 /||).I'll buy 1), but no longer care about 2). So I agree with the above, as far as the single-shaft symbols.>>> 181: |v w| w|v s| |s w|x w|s s|x s|s x|x w|| w||v>s|| ||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.It is still unclear to me what's best for 181, but you will realise that rational complements may no longer be of any relevance to me.>>> 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.Except for the mirror complement thingy that we need to thrash out now. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4967 - Contents - Hide Contents Date: Wed, 05 Jun 2002 23:29:29 Subject: Re: "The hypothesis" from a wedge product point of view From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: >>> Okay, I've been waiting for this one... now, could somebody, maybe >> somebody other than Gene, say the same thing and give some examples >> *slowly*! >> >> >> thanks, >> >> --Dan Stearns >> i wish i could. unfortunately i haven't had the time to assimilate > the whole wedge product business. i'm pretty sure graham breed and > dave keenan have a handle on it. i'd say dave keenan might be the > best candidate to study gene's work here and translate it into a form > lots of us (especially people like you and me, who are familiar with > dave's language at least) could understand and visualize (i'm > thinking of dave's skill with graphics and diagrams) . . .Whew! That's flattering, but I don't have the time right now (end of semester marking), but anyone who would hope to understand even my explanation should first learn what the wedge product (a.k.a exterior product) is all about, from another Australian, John Browne. Gene found this excellent introduction some time ago. See the thread starting at Yahoo groups: /tuning-math/message/3754 * [with cont.] or go straight to Grassmann Algebra Book * [with cont.] (Wayb.) You might only need to read the first 6 pages of the introduction, to get what it's all about.

Message: 4969 - Contents - Hide Contents Date: Thu, 06 Jun 2002 17:02:25 Subject: Fwd: scala questions From: Carl Lumma All; I've created a tuning-math list on freelists.org... Welcome to FreeLists - Free, No-hassle Mailing... * [with cont.] (Wayb.) ...It's the closest thing I've seen to my wishlist for a list service, but it isn't perfect. For example, there's no database for all the lists you belong to like yahoo's "my groups" feature -- you have to remember them. And you can't post directly from the site. And it doesn't seem to assign unique message numbers to posts. The messages are archived by thread and date, though, and are fully searchable. Completely ad-free. Check it out, if you like, see what you think! -Carl>Date: Thu, 06 Jun 2002 15:48:29 -0700 >To: tuning-math@xxxxxxxxx.xxx >From: Carl Lumma <carl@xxxxx.xxx> >Subject: scala questions > >Manuel, > >() What's the best way to get Scala to represent scales as degrees >of an et? > >() As far as inputting scales as et subsets, I do "equal n" and then >"select". Is that the Official Way? > >() Think we could get View -> rank-order matrix? (Yes, I'm using >2.05 now). > >-Carl

Message: 4970 - Contents - Hide Contents Date: Thu, 06 Jun 2002 17:03:25 Subject: Re: Fwd: scala questions From: Carl Lumma>All; > >I've created a tuning-math list on freelists.org... > >Welcome to FreeLists - Free, No-hassle Mailing... * [with cont.] (Wayb.)Oh yeah, and I haven't tested digest mode, so if anybody wants to volunteer for that... -C.

Message: 4971 - Contents - Hide Contents Date: Thu, 06 Jun 2002 01:28:35 Subject: Re: "The hypothesis" from a wedge product point of view From: genewardsmith --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Okay, I've been waiting for this one... now, could somebody, maybe > somebody other than Gene, say the same thing and give some examples > *slowly*!I cancelled that about five minutes after posting, and I don't think it's a very good place to start. I'm thinking about how to remove the errors and get the basic idea to work, but I'm not there.

Message: 4972 - Contents - Hide Contents Date: Thu, 06 Jun 2002 17:31:31 Subject: Re: Fwd: scala questions From: Carl Lumma>> >'ve created a tuning-math list on freelists.org... >> >> Welcome to FreeLists - Free, No-hassle Mailing... * [with cont.] (Wayb.) >>All I get is an error message.My mistake -- no slash allowed: Welcome to FreeLists - Free, No-hassle Mailing... * [with cont.] (Wayb.) -Carl

Message: 4973 - Contents - Hide Contents Date: Fri, 7 Jun 2002 11:08:24 Subject: Re: scala questions From: manuel.op.de.coul@xxxxxxxxxxx.xxx Carl, () What's the best way to get Scala to represent scales as degrees of an et? set attribute log_fract [octave] Maybe the format is not ideal. I could add it to show data if the scale is an ET subset. () As far as inputting scales as et subsets, I do "equal n" and then "select". Is that the Official Way? That's how I do it too. But if it's expressed in steps, I do mode/equal () Think we could get View -> rank-order matrix? (Yes, I'm using 2.05 now). Could you explain what exactly you have in mind? Thanks, Manuel

Message: 4974 - Contents - Hide Contents Date: Fri, 07 Jun 2002 15:49:29 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4405]:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> 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). >> I suppose if the 23-comma interpretation is confined to such large ETs it > might be ok, but I'd need to be convinced that there was no otherway to do> it and the ET is actually 1,3,9,23-consistent and preferably > 1,3,9,...,23-consistent where "..." are the other primes used for its > notation. > > 135, 147, 159, and 224-ET are all 17-limit notatable, although you'd > probably go to 19 limit for 224-ET.For that you would need to use a two-flag symbol ~|( for 2deg of 135, 147, and 159 and for 3deg224. I thought that you might prefer a single-flag symbol |~ for a small interval such as that, as with 217. However, if you think that we should use something such as the following, 135 (45 ss.): ~| ~|( /| (| (|( /|\ (|) ~|| ~||( ||\ (|| (||( /||\ 147: ~| ~|( /| |\ (|( /|) /|\ (|\ ~||( /|| ||\ ) ||\ /||) /||\ or 147: ~| ~|( /| |\ ~|\ /|) /|\ (|\ ~||( /|| ||\ ~||\ /||) /||\ 159: ~| ~|( /| |\ ~|\ (|~ /|\ (|) ~|| ~||( /|| ||\ ~||\ (||~ /||\ then I would have no problem with it.> 198-ET is not 19-limit notatable, but why do you feel any need to notate > it?I thought that we were notating everything that we possibly could. Who knows what the tuning scavengers might want to use?> And is it 23-limit notatable anyway?Only the 17 factor is so far removed from everything else that it would need to be avoided in the notation. (As it turns out I had to use ~|) for 7deg.) The 19 factor is actually more accurately represented that most everything else, and it's inconsistent only with respect to 15. When I did the notation, this is what I got: 198: )| |~ )|~ /| |\ )|\ ~|) /|) /|\ (|) (|\ )|| ||~ ) ||~ /|| ||\ )||\ ~||) /||) /||\ Aside from the mirroring issue (which I will adress below), do you have any problem with this?>>> 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 neveroccurs 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. >> I'll go with your solution, for the single-shaft symbols at least. The > lateral confusability is addressed by the large difference in widths, and > the user will just have to learn that 11 = 13' and 11' = 13. > > I propose (||( for 4deg74 purely because it is the mirror image of ) |) > (plus a shaft). More about this later. >>>>> 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: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /||||\ /||~ /||) /||\> > Yes, for the single-shaft symbols at least. >>> 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. >> You know, I won't really be happy until I have a spreadsheet that generates > the notation for every ET, based on a bunch of rules, because that's the > only way I'll be sure we're being consistent. The rules may of course end > up being very complicated, but I wouldn't want to see any rule that only > applied to a single ET.That sounds like a worthy goal. I just wonder how long it would take us to agree on all of the rules and the hierarchy.> By the way, perhaps we should use >| and |< instead of )| and |( to make > the concave flags more distinct from the convex ones in ASCII. What do you > think?I think that we would have a problem with that in e-mail when the lines wrap and an extra > character gets inserted right in the middle of a symbol. (Yes, Yahoo does break lines that way, and I try to clean them up before sending my replies -- there were several instances where I did it in this message -- and using > to represent a flag would make that more difficult.)>>>> 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 (|). >> Yes, I did. Sorry. >>> 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. >> I guess if there is already a popular notation in use for some ET, and > there is a fairly direct correspondence to that notation available in > sagittal, then we should use it. > > So on that basis at least, I can tentatively agree with your proposal for > 31. I reserve the right to change my mind on this on further investigation. > :-) > > 31: /|\ /||\ >>>>> 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; >> I understand wanting comma up and down symbols in 22-ET. That'swhat I want> too. But what do you mean by /|| and \!! as comma up and down symbols? > We're proposing _not_ to have those those double-shafters with a _left_ > flag. I assume you meant the double-shafters to have right flags. In which > case 22 is agreed. > > 22: /| ||\ /||\Sorry. I got laterally confused. I meant ||\ and !!/.>> 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. >> In fact, I'm very happy that /| and ||\, |\ and /|| should always be > complements and to _always_ have a mirrored sequence rather than > _sometimes_ have a matched one.As you said above, more about this below.>> Also, with the apotome divided into fewer than 5 parts, I would want >> to use /|\ only when it is exactly half of /||\. > > Fair enough.Conversely, should we not also specify that if /|\ *can* be used for exactly half of /||\, then it *should* be used. (I would consider this the primary justification for its use as 1deg31.) This would be one of the rules for deriving ET notations in a spreadsheet that I would like to see.>> 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.) >> I am instead inclined to totally ignore rational complements with regard to > ETs, especially for the lower numbered ones. One reason is that I feel that > the choice of double-shaft symbols cannot in any way be allowed to > influence the choice of single-shaft. One must first choose thebest set of> single shaft symbols (ignoring complements) since some users will have no > interest in the double-shaft symbols and should not be penalised for it. > > In fact, (and I've been making gentle noises about this possibility for a > some time now), I'm willing to throw away everything we agonised over with > regard to rational complements and instead adopt a simple system that > applies automatically to all ETs and rational tunings. > > I propose that the complement of a|b is always b||a, except that the > complement of |//| (natural) is /||\ and the complement of /|\is /|\ if it> represents the same number of steps and (|) if it represents a different > number of steps. > > You will notice that this would require no change to the 72-ET version 3 > notation, nor any change to most of the smaller ETs we've agreed on such as > 12, 17, 19, 22, 24, 26, 31, 46, 53. > > By the way, are you collecting all those we've agreed on into one place? I > haven't been.Yes, in two places.> Why do I want to do this despite some obvious disadvantages? Because I > realised while trying to consistently notate the whole n*12-ET family, that > it required us to repeat the whole somewhat arbitrary process we went thru > for rational tunings, to find complements with minimum offsets. And what's > more, that this process would have to be repeated for every such family or > small range of fifth-sizes across the whole range of ETs. For example the > n*29-ET family is the next largest, followed by n*17. And every such > family, or small range of fifth sizes, would have a completely different > complement mapping. The cognitive load for anyone who uses more than two > such systems would be enormous. > > Now for those obvious disadvantages: > > 1. The second shaft does not have a fixed comma value. > > This doesn't seem very important to me? > > 2. We lose the association of flag size with rational comma size in the > second half-apotome. > > This is the biggie. It can be remedied to some degree by redesigning the > double-shaft (and X-shaft) symbols so their concave flags are wider than > their wavy flags which are wider than their straight and convex flags. > However it will be difficult to make single flag symbols bigger than > double-flag ones. > > What other disadvantages have I omitted?One very big one that I will state below, when you give a couple of examples.> Advantages: > > Simple to remember. > Covers all tunings. > Flags are more strongly associated with particular primes because the flags > don't change when the comma is complemented. > No new flag types ever need to be introduced merely to handle complements. > Doesn't require /| and ||\ as a special case. >>> 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). >> I agree with the above for 43-ET, but 36-ET can be notated as every second > note of 72-ET, which means I want: > 36: |) (|| /||\ > 43: |) (|\ /||\ > > Do you have some argument as to why 36 and 43 should be the same? I don't > see it.I have no particular reason, other than notating these two the same way might make them easier to remember. Then we can agree on what you have above for 36 and 43, inasmuch as I will also be agreeing with you on 72, 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. >> I now realise I will need to consider the entire n*29 family > 29 58 87 116 145 174 203 232 261, before agreeing on either 29 or 58.Why bother with anything above 145?>>>> 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. >> I prefer /|) because it introduces only one new flag where (|\ introduces > two, for single-shaft-only users.As if this were already too many flags? Okay, let's use /|); it's not that big a deal, anyway.>>>> 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). >> I now agree that 48 should be every second note from 96 and will address > all n*12-ETs elsewhere. >>> ... 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, noconfusability, but (|| < ||\ )>> 72: /| |) /|\ (|\ ||\ /||\ (version 4 -- simple, noconfusability, consistent, harmonic-oriented)>>> ... I can only agree to your version 3. > > 72: /| |) /|\ (|| ||\ /||\ > > I prefer the above to version 4, with (|\ as 4 steps, because I think that > in any given ET, (| and |) flags should either both be 13-based or both be > 7-based, so that (|) is always 11' (whether it is used or not). Otherwise > we have 3 different possible values for (|), the largest and smallest of > which differ by 0.84 cents in rational tuning. > > It's bad enough that folks have to know whether the convex flagsrefer to 7> or 13 in ETs where the tridecimal schisma doesn't vanish. I wouldn't want > them to have to worry about the two convex flags _independently_.That makes sense. However, I would look at it a little differently. Adhering more closely to the one-comma-per-prime ideal, I would consider 715:729 the preferred ratio for (|, which gives an exact 26:27 for (|\, which I consider the principal 13 diesis (even if it exceeds a half-apotome by a few cents) that modifies a natural note (downward) to give 13/8. We already have |\ as 54:55, which gives an exact 11 diesis of 32:33 that modifies a natural note (upward) to give 11/8. So we would have one comma each for 11 and 13, not two commas for 11. As long as the tridecimal schisma vanishes, then (|) will be /||\ minus /|\, and we can leave well enough alone. In cases where it doesn't vanish, then we can redefine *one* of the two commas: either (| as the 11'-7 comma or |) as the 13-5 comma. (To avoid confusion, I have used your nomenclature, which calls 1024:1053 the 13 diesis, although I would prefer to label it the 13' diesis, with 26:27 as the 13 diesis, in which case the redefined |) would then be labeled the 13'-5 comma.) Anyway, however one chooses to look at it, we are in agreement on how to apply the symbols in the notation.> Also, 72-ET is not terribly good at the 13-limit, the error hikes from 3.9 > cents at the 11-limit to 7.2 cents at the 13-limit, and in any case folks > can learn that the 7-comma symbol doubles as the 13-comma symbol in 72-ET, > just as they must learn that the 11-comma symbol doubles as the 7- comma > symbol in 31-ET.I will agree to use version 3, then, with 36-ET being notated as a subset. One thing that I like about this version is that it closely resembles the sagittal notation as I originally presented it, with all straight flags, except that convex flags replace straight flags in the tones neighboring the half-apotome to eliminate lateral confusability. For the Miracle family of ET's, you will recall that I was advocating the possibility of reading uncomplicated music in 72-ET directly into 31 and 41-ET by a mental "rounding-off" process. The 72-ET symbols with straight flags can usually be read directly, and the convex flags would readily identify the symbols that would need to be "rounded off" to (straight-flag) 31 and 41-ET symbols. (Of course, this would work only if we use /|\ as 1deg31, which would be another reason for doing 31 that way.)>>>> 58: s| w|x s|s w||v ||s s||s (version 2 -- more>> complicated, 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 don't see any need for 29 to be every second of 58, but I do want to look > at the whole huge family first.My subsequent posting #4354 also addressed this in more detail. I will have to take a closer look at 116 and 145. I made a comment on those above 145 below.>> 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 17and 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. >> I changed my mind on 96, as you will have seen in other posts, but might > end up changing it back. >>>>> 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?) >> Yes. That's what I meant. Sorry. > > I now want one of > > 94: ~| /| (| /|\ (|) ||) ||\ ||~ /||\ > 94: ~| /| ~|\ /|\ (|) /||~ ||\ ||~ /||\ > > and need to look at 94 188 282 to decide.Very well! But I don't think that 188 has much going for it, and 282 is going to have gaps. Otherwise, if we think it advisable to keep the same sequence of symbols for both 87 and 94 (for ease of learning), then our choice may be influenced by what we do for multiples of 29.>>>> 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! >> Only now I want the fully-mirrored half-apotomes: > > 111 (37): ~| /| |\ ~|\ /|\ (|) /||~ /|| ||\ ||~ /||\Hmm. That makes /||~ is smaller than either /|| or ||~. Let me give this a little thought.>> 140: )| |~ /| )|\ /|~ /|) (|~ (|\ )|| ||~ ||\ )||\ /||~ /||\>> >> This is the simplest set I could come up with that uses both /| and >> ||\. >> I'll leave the second half-apotome out of it for now. It seems we have 4 > options: > 140: )| |~ /| )|\ (| /|) (|~ 6 flags > 140: )| |~ /| )|) (| /|) (|~ 5 flags > 140: )| |~ /| )|\ /|~ /|) (|~ 6 flags monotonic flags per symb > 140: )| |~ /| )|) /|~ /|) (|~ 5 flags monotonic flags per symb > > I prefer the last one, and with mirror complements it would be > > 140: )| |~ /| )|) /|~ /|) (|~ (|\ ~||\ (||( ||\ ~|| || ( /||\ > > Note that with mirror complements, (|\ is the same as (||\.In 70-ET )|\ is 2deg, whereas )|) is 1deg, so I prefer the former. However, with the mirrored symbols ~||\ is a smaller interval than either ~|| or ||\ and (||( is smaller than ||(. This goes counter to what I would expect.>>>> 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! >> With mirror complements we have: > 152: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ /|| ||\ ~|| || ( /||\I think you intended ~||\ for 10deg, which gives: 152: )| |~ /| |\ /|~ /|) /|\ (|) (|\ ~||\ /|| ||\ ~|| || ( /||\ So ~||\ is smaller than either ||\ or ~||. I need to think about this a little more.>>>> 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 /||). >> I'll buy 1), but no longer care about 2). So I agree with theabove, as far> as the single-shaft symbols. >>>>> 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. >> It is still unclear to me what's best for 181, but you will realise that > rational complements may no longer be of any relevance to me. >>>>> 217: |v w| |w s| |x |s w|x w|s s|x s|s x|x x|sw|| ||w s|| ||x ||s w||x w||s s||x s||s>>> >>> Agreed. >> Except for the mirror complement thingy that we need to thrash out now.In effect, mirroring gives the flags negative values, with the zero point being the apotome, which itself is notated as an exception, /||\ ,when its proper mirror should be ||. For the simpler ET's that use no concave or wavy flags, I don't see much of a problem, since the symbol arithmetic usually works in spite of the mirroring. But as soon as you introduce concave or wavy flags, particularly in two-flag symbols, the symbol arithmetic goes crazy. After all that we went through figuring out the rational complements, I can't see replacing that with something in which the order of symbols in the second half-apotome makes very little sense? All to "fix" a problem involving not-quite-matched symbols /| and ||\ in a few ET's? I say: "forget it." If we can get mirroring in the lower-numbered ET's by means of the complementation that we already worked out, then that's a commendable goal. But please, let's not dump the concept of consistent symbol arithmetic in the process. If you feel that the best choice of single-shaft symbols is in some instances compromised by the need to have double-shaft complements, then I'll work with you to address that problem. --George

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