Tuning-Math Archive Section 5: 4776

Message: 4776 - Contents - Hide Contents Date: Wed, 08 May 2002 20:12:06 Subject: Re: A common notation for JI and ETs From: David C Keenan I've added one more rational complement, for vw|w, which may be of use as an alternate 7/5 comma. Symbol Comp Comma name Comments ------------------------------------ v| x||w 19 |v s||x (17'-17) w| ww||x 17 v|v vw||s w|v x||v 17' |w s||w 23 v|w x|| 19' s| ||s 5 ww|v v||x pythag comma (comp probably not required) |x ||x 7 vw|w w||w (7/5)' v|x ww||v (probably not required) |s s|| (11-5) x| v||w 29 or (11'-7) v|s vw||v 31 (comp probably not required) w|x ww|| (probably not required) s|w ||w (prob not required, 5 comma + 23 comma) vw|x none 11/5 (hope comp is not required) x|v w||v alt 23' (comp is good reason to make this standard 23') w|s vw|| 23' ss| ||vv 25 v|wx vv|| 37' (comp probably not required) s|x ||v 13 s|s x|x 11 sx| |sx 31' The above complements correspond to flags being the following numbers of steps of 665-ET. v| 2 3 |v w| 5 9 |w s| 12 15 |x x| 19 18 |s I note that, apart from a few exceptions below the resolution of 665-ET, we have the following complementary pairs of flags on the same side. Left side v ww w vw s (blank) x (none) Right side v x w w s (blank) -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4777 - Contents - Hide Contents Date: Wed, 08 May 2002 03:38:05 Subject: Re: A common notation for JI and ETs From: David C Keenan I wrote: "My current thinking is that the rational complements should be based on 665-ET, an ET with an extremely good 1:3 so there is no danger of any size cross-overs with any pairs of symbols, existing or future. We only need to introduce a |vv symbol (instead of my earlier proposed vw|) as the complement to ss|, the 25 comma symbol." The last paragraph was wrong. It seems that at least one other 3 flagger must be introduced as the complement of the 17 flag, and possibly some others, as shown below. Here's my latest proposal for rational apotome complements. Symbol Comp Comma name Comments ------------------------------------ v| x||w 19 |v s||x (17'-17) w| ww||x 17 (or vw||s, less preferred) w|v w||s 17' |w s||w 23 v|w x|| 19' s| ||s 5 ww|v v||x pythag comma (comp probably not required) |x ||x 7 |s s|| (11-5) x| v||w 29 or (11'-7) v|s vw||v 31 (comp probably not required) s|w ||w (prob not required, 5 comma + 23 comma) vw|x none 11/5 (hope comp is not required) w|s w||v 23' ss| ||vv 25 v|wx vv|| 37' (comp probably not required) s|x ||v 13 s|s x|x 11 sx| |sx 31' Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4778 - Contents - Hide Contents Date: Wed, 08 May 2002 21:08:50 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> Dave, > > I've put out a file containing my latest proposal for symbols for > alterations above the half-apotome. > > Yahoo groups: /tuning- * [with cont.] > math/files/secor/notation/Symbols3.bmp

The | and || shaft symbols look great, but I'm afraid the whole concept of ||| and X shaft symbols will have to be a minority report. I'd rather just stack a s||s beside the | and || symbols. What did you think of my suggestion to use V tails or single and double wavy tails?

> I've paid particular attention to scaling the width of the 2 & 3- > shaft and X symbols. For these I didn't think it was appropriate to > make the concave flags as small as you did in your examples > (comparing the size of the symbols at the left extreme with those at > the right extreme of the line above), and I find that these and the > wavy flags are quite readable this way.

Agreed. I may want to fiddle with a pixel here and there if I get time. But otherwise I think they are great.

> I realize that a few of > these symbols won't be used the way we presently have things figured > out, but I did all of these just to get a sense of continuity in the > progression of size moving vertically. > > I also tried my hand at ss||, ss|||, and ssX symbols at the far > right, just to see how those might look.

ss|| looks OK, but maybe you should try omitting the part of one shaft that appears between the two flags. Wanna try some of the other two-flags-on-one-side symbols I've proposed as rational complements? e.g. ww|x (17), vw||s (other possibility for 17), vw|| (23'), ||vv (25).

> (I hope that the meanings > of x and X don't get too confusing.)

I read them just fine. I'm keen to finalise the rational complement relationships and the single 11/5 comma symbol if any. Should we look at possible single symbols for 13/5, 13/7, 13/11 commas too, or is this getting too silly? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4780 - Contents - Hide Contents Date: Thu, 09 May 2002 18:36:12 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> Dave, >>

>> I've put out a file containing my latest proposal for symbols for >> alterations above the half-apotome. >> >> Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Symbols3.bmp >

> The | and || shaft symbols look great, but I'm afraid the whole > concept of ||| and X shaft symbols will have to be a minority report. > I'd rather just stack a s||s beside the | and || symbols.

I think that looks almost as bad as putting a couple of conventional sharp symbols next to each other -- there's a reason why the conventional double-sharp symbol was invented. In fact, what you're suggesting is worse than two conventional sharps next to each other: you've got one symbol with various flags and the other with two straight flags. In the latter symbol the flags are irrelevant -- the only information that the player really needs is given by the two shafts, so the straight flags are just extraneous information which can only generate annoyance and confusion. On the single-shaft symbols concave and wavy flags are rather tiny, whereas on a three-shaft symbol they are larger and, therefore, easier to read. Double symbols create a lot of clutter, particularly in keyboard music, where the process of reading the notation must be made as efficient as possible. Have I given enough reasons for single symbols?

> What did you think of my suggestion to use V tails or single and > double wavy tails?

I don't see any advantage in the V tails. And I see a real problem with wavy tails -- a performer would already be required to identify three different types of curved flags (in addition to straight flags), and wavy tails would probably generate confusion by hindering that process. I think that the straight shafts (and X) are sufficient to communicate what they are called upon to do without drawing undue attention to themselves.

>> I've paid particular attention to scaling the width of the 2 & 3- >> shaft and X symbols. For these I didn't think it was appropriate to >> make the concave flags as small as you did in your examples >> (comparing the size of the symbols at the left extreme with those at >> the right extreme of the line above), and I find that these and the >> wavy flags are quite readable this way. >

> Agreed. I may want to fiddle with a pixel here and there if I get > time. But otherwise I think they are great. Wonderful!

>> I realize that a few of >> these symbols won't be used the way we presently have things figured >> out, but I did all of these just to get a sense of continuity in the >> progression of size moving vertically. >> >> I also tried my hand at ss||, ss|||, and ssX symbols at the far >> right, just to see how those might look. >

> ss|| looks OK, but maybe you should try omitting the part of one shaft > that appears between the two flags.

My rationale for leaving it has to do with the ssX symbol: I tried making that one with the top of the X occurring at the lower flag. This left the two flags disjointed, so I connected them with parallel vertical lines. It didn't look good, and since I didn't think there was any problem in reading the symbol with the X terminating at the top flag, that was what I used. Extending all of the lines in the other symbols up to the top flag makes them consistent with the X symbol.

> Wanna try some of the other two-flags-on-one-side symbols I've > proposed as rational complements? e.g. ww|x (17), vw||s (other > possibility for 17), vw|| (23'), ||vv (25).

First I have to see what all of these are about, which will involve spending more time with the rational complements.

>> (I hope that the meanings >> of x and X don't get too confusing.) >

> I read them just fine. > > I'm keen to finalise the rational complement relationships and the > single 11/5 comma symbol if any.

I'll start to answer some of this in my next posting.

> Should we look at possible single symbols for 13/5, 13/7, 13/11 commas too, > or is this getting too silly?

I'm seriously wondering whether anyone is going to use any of those 2- flags/side symbols. Our whole reason for notating 217 was to get 19- limit-unique capability (which would handle these eventualities), but now that we have it we're still off on a quest into the great beyond - - 311, 1600, 453, 653, 665 (with which I find a serious problem that I will address in my next posting) -- what next? Stay tuned! --George

Message: 4781 - Contents - Hide Contents Date: Thu, 09 May 2002 20:27:56 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4213]:

> There were mistakes in my latest proposal for rational apotome complements. > The 17' and 23' comma complements were wrong. I'll give the whole thing > again with corrections and additions.

(You updated this in your next posting, which I will respond to below.)

> The above complements correspond to flags being the following numbers of > steps of 665-ET. > > v| 2 3 |v > w| 5 9 |w > s| 12 15 |x > x| 19 18 |s

I have a serious problem with using 665-ET as a basis for anything. It is only 9-limit consistent -- the 11 factor falls almost midway between degrees. Among other things, this causes xL+sR to be 37 degrees, whereas it should be 36. Below I will also give another problem with using 665 as a basis for rational complementation.

> By the way, 217-ET isn't the largest ET we can notate using symbols having > no more than one flag per side. We can do 306-ET as follows. I

think it is

> the largest.

Even if it isn't a very enticing one, with the 5 factor coming almost midway between degrees and the 11 factor almost as bad. --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4214]:

> I've added one more rational complement, for vw|w, which may be of use as > an alternate 7/5 comma. > > Symbol Comp Comma name Comments > ------------------------------------ > v| x||w 19 > |v s||x (17'-17) > w| ww||x 17 > v|v vw||s > w|v x||v 17' > |w s||w 23 > v|w x|| 19' > s| ||s 5 > ww|v v||x pythag comma (comp probably not required) > |x ||x 7 > vw|w w||w (7/5)' > v|x ww||v (probably not required) > |s s|| (11-5) > x| v||w 29 or (11'-7) > v|s vw||v 31 (comp probably not required) > w|x ww|| (probably not required) > s|w ||w (prob not required, 5 comma + 23 comma) > vw|x none 11/5 (hope comp is not required) > x|v w||v alt 23' (comp is good reason to make this standard 23') > w|s vw|| 23' > ss| ||vv 25 > v|wx vv|| 37' (comp probably not required) > s|x ||v 13 > s|s x|x 11 > sx| |sx 31'

Working only with symbols that have no more than one flag per side, I came up with the following: symbol complement comma offset comments ------------------------------------------------- v| x||w 19 -0.14 cents |v s||x (17'-17) -1.50 653-inconsistent w| w||s 17 4.05 653 & 665-inconsis. v|v none w|v s||w 17' 0.49 665-inconsistent |w w||x 23 0.73 665-inconsistent v|w x|| 19' -0.14 s| ||s 5 0.00 w|w s||v 0.49 665-inconsistent |x ||x 7 -1.26 653-inconsistent s|v w||w 0.49 665-inconsistent v|x none |s s|| (11-5) 0.00 x| v||w 29 -0.14 v|s none 31 w|x ||w 0.73 665-inconsistent s|w w||v 0.49 665-inconsistent x|v none w|s w|| 23' 4.05 653 & 665-inconsis. s|x ||v 13 -1.50 653-inconsistent x|w v|| -0.14 These were the possibilities that I didn't use: symbol complement comma offset comments ------------------------------------------------- v|v w||s 3.40 653 & 665-inconsistent w|v x||v 17' -0.95 |w s||w 23 -1.32 653-inconsistent w|w v||x -2.64 217, 653, 665-inconsis. v|x w||w -2.64 217, 653, 665-inconsis. s|w ||w -1.32 653-inconsistent x|v w||v -0.95 w|s v||v 3.40 653 & 665-inconsistent The term "offset" requires some explanation. I did a spreadsheet evaluating these complements based on your Complements.bmp illustration: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/GSCompls.xls in which I bolded my selections. I made a copy of it: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/DKCompls.xls in which I bolded your selections (not including any with 2 flags/side, which you may add if you wish). The column labeled "ud-compl cents" is the number of cents in the unidecimal diesis minus the single-shaft complementary symbol, as in your bitmap figure. The offset is the difference in cents between the original symbol and the ud-minus-complement. You can change the number in cell A8 to any ET to view the complementation inconsistencies of that ET in column I. You passed up some nice small-offset complements that are 665- inconsistent. The only ones that I was forced to pass up in 217-ET have an offset of over 2.6 cents. And you can see that 653-ET also has a number of inconsistent complements. This is due in large part to the fact that 653 and 665 are much finer divisions, so this is not surprising. However, this is a good reason not to base rational complementation on a particular division of the octave, but rather on the basis of a small offset. --George

Message: 4782 - Contents - Hide Contents Date: Thu, 09 May 2002 21:58:15 Subject: Re: 3D geometry From: dkeenanuqnetau --- In tuning-math@y..., "robstrange66" <robstrange@n...> wrote:

> Hi- > I am very interested in 3D & geometry. Anyone here study > Platonic solids, polyhedra, etc? > > Regards, > Rob

Yes, and unless it relates to music, we should probably talk about it in the ZomeWorld yahoogroup.

Message: 4784 - Contents - Hide Contents Date: Thu, 09 May 2002 04:15:21 Subject: I found it! From: genewardsmith I couldn't find this group, and wondered what had happened to it. It turns out that for some reason I'm now a moderator, and it moved positions in my list of groups.

Message: 4786 - Contents - Hide Contents Date: Thu, 09 May 2002 04:18:29 Subject: Re: Microtonal Fiction (Was: A common notation for JI and ETs) From: genewardsmith --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> So, Dave, where and when do we get started? (Off-list, I would say. > Tuning-math just doesn't seem to be the right place.)

What about spiritual-tuning?

Message: 4788 - Contents - Hide Contents Date: Fri, 10 May 2002 18:31:53 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> There were mistakes in my latest proposal for rational apotome complements. > ... I'll give the whole thing > again with corrections and additions. > > Symbol Comp Comma name Comments > ------------------------------------ > ... > x| v||w 29 or (11'-7) > ... > > The above complements correspond to flags being the following numbers of > steps of 665-ET. > > v| 2 3 |v > w| 5 9 |w > s| 12 15 |x > x| 19 18 |s

We need to define the xL flag strictly as 11'-7 (715:729), or we are going to run into problems with ET's in which the number of degrees is different than with the 29 (256:261) definition, which is quite a few of them (e.g., 27, 46, 53, 99, 140, and 152-ET, which all require the xL flag in their notation). In 665-ET xL should be 18 degrees. This is the real reason why xL+sR comes out as 37 degrees in 665 (whereas it should be 36), which is different from the one I gave in posting #4220, in which I atttributed the problem to the poor representation of the 11 factor. However, I still think using 665-ET as a basis for rational complementation is inadvisable because of its inconsistency. (This just makes one less reason.) --George

Message: 4789 - Contents - Hide Contents Date: Fri, 10 May 2002 19:07:53 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote [#4220]:

> We need to define the xL flag strictly as 11'-7 (715:729), or we are > going to run into problems with ET's in which the number of degrees > is different than with the 29 (256:261) definition ..., > > In 665-ET xL should be 18 degrees. ...

In addition, this affects the complementation table that I gave in message #4220, which results in additional inconsistencies both for 653 and 665. The table now should read: symbol complement comma offset comments ------------------------------------------------- v| x||w 19 -0.14 cents 653 & 665-inconsis. |v s||x (17'-17) -1.50 653-inconsistent w| w||s 17 4.05 653 & 665-inconsis. v|v none w|v s||w 17' 0.49 665-inconsistent |w w||x 23 0.73 665-inconsistent v|w x|| 19' -0.14 653 & 665-inconsis. s| ||s 5 0.00 w|w s||v 0.49 665-inconsistent |x ||x 7 -1.26 653-inconsistent s|v w||w 0.49 665-inconsistent v|x none |s s|| (11-5) 0.00 x| v||w 29 -0.14 653 & 665-inconsis. v|s none 31 w|x ||w 0.73 665-inconsistent s|w w||v 0.49 665-inconsistent x|v none w|s w|| 23' 4.05 653 & 665-inconsis. s|x ||v 13 -1.50 653-inconsistent x|w v|| -0.14 653 & 665-inconsis. These were the possibilities that I didn't use: symbol complement comma offset comments ------------------------------------------------- v|v w||s 3.40 653 & 665-inconsistent w|v x||v 17' -0.95 653 & 665-inconsistent |w s||w 23 -1.32 653-inconsistent w|w v||x -2.64 217, 653, 665-inconsis. v|x w||w -2.64 217, 653, 665-inconsis. s|w ||w -1.32 653-inconsistent x|v w||v -0.95 653 & 665-inconsistent w|s v||v 3.40 653 & 665-inconsistent I have also modified the files: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/GSCompls.xls and Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/DKCompls.xls to reflect this change. --George

Message: 4792 - Contents - Hide Contents Date: Fri, 10 May 2002 06:20:53 Subject: Re: 3D geometry From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: Him Paul--I was afraid you'd run off somewhere. :)

Message: 4793 - Contents - Hide Contents Date: Fri, 10 May 2002 00:56:39 Subject: Another Blackjack detempering From: Gene W Smith Here is the symmetrical JI scale I get from the Fokker block using <36/35,225/224,1029/1024> as commas. Does this correspond to any of your (Paul's) detemperings? Scale: 1, 49/48, 16/15, 35/32, 8/7, 7/6, 128/105, 5/4, 21/16, 4/3, 7/5, 10/7, 3/2, 32/21, 8/5, 105/64, 12/7, 7/4, 64/35, 15/8, 96/49 Steps: 49/48, 256/245, 525/512, 256/245, 49/48, 256/245, 525/512, 21/20, 64/63, 21/20, 50/49, 21/20, 64/63, 21/20, 525/512, 256/245, 49/48, 256/245, 525/512, 256/245, 49/48

Message: 4794 - Contents - Hide Contents Date: Sun, 12 May 2002 11:24:53 Subject: Re: graham's linear temperament page From: monz hi Paul,

> From: "emotionaljourney22" <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, May 10, 2002 3:25 PM > Subject: [tuning-math] graham's linear temperament page > > > this page is great and very important (by contrast, check out joe > monzo's definition of linear temperament if you want to turn red).

how about a more constructive criticism? post something that i can use to replace or supplement my definition to make it better. (and also please specify whether i should be replacing or supplementing!) thanks. (BTW, i'm only checking in on the lists sporadically these days. too much other "life" stuff happening...) -monz

Message: 4796 - Contents - Hide Contents Date: Sun, 12 May 2002 15:16:15 Subject: Re: A common notation for JI and ETs From: David C Keenan At 22:17 9/05/02 -0000, you wrote:

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >I have a serious problem with using 665-ET as a basis for anything. >It is only 9-limit consistent -- the 11 factor falls almost midway >between degrees. Among other things, this causes xL+sR to be 37 >degrees, whereas it should be 36.

True, but an error of a whole step of 665-ET is still 33% smaller than an error of a half step of 217-ET. Consistency relates to accuracy relative to step size, but surely absolute accuracy is more relevant here? I wasn't intending to notate 665-ET, it was just a way of ensuring that the flags (and the second shaft) could consistently be assigned fixed values (different kind of consistency) while minimising offsets. But you'll be pleased to know that I've abandoned 665-ET (and all ETs) as a basis for rational complements. I agree 306-ET is not very enticing. Thanks for those spreadsheets. I like the idea of ignoring ETs and just trying to minimise the offsets. We can take a set of symbol complements and treat them as a system of linear equations which can then be solved to obtain values in cents for the individual flags. It is possible to make a set that has no solution. This is a different (and more serious) kind of inconsistency than the kind we talk about when we say a certain ET is n-limit inconsistent. I think it is important that they be consistent in this sense. Of course the glaring problem with your recent proposal is the 4 cent offset for w| <-> w||s. All the others are less than half that. I'd be much happier if we could keep the max offset to 1.5 cents or less. But I also agree that the use of symbols with two-flags-a-side should be a last resort. The 25 comma symbol really does need a complement, e.g. C:G#\\. I don't think there's any problem with _its_ complement having two flags a side, in fact I think it would be expected. My favourite complement for ss| is ||vv. This works in your system as well as mine. For our system of linear equations, we can write this as ss| + ||vv = 113.685 cents If we insist on consistency (as in having a solution for the flag sizes) then the above implies s|v + s||v = 113.685 It seems very desirable to have |x + ||x = 113.685 We agree on that. Taken with the above, that implies s|v = |x and means that s||v is not available as a complement for anything else. I assume we want our rational complements to be uniqu. I would simply outlaw s|v and s||v. You have s||v as the complement of w|w. I don't think we actually need a complement for w|w. Do we? The above implies that s|v + ||x = 113.685 which implies |v + s||x = 113.685 another complement that we agree on. Of course there's no question that s| + ||s = 113.685 Another equivalent pair that we agree on is v| + x||w = 113.685 v|w + x|| = 113.685 with its 0.14 cent offset. I set up a speadsheet that allows one to enter these equations and then solves them for the size of each flag in cents. I've made the value of the second shaft (the difference between s|s and s||s) a free variable too, which is the equivalent of allowing the top and bottom parts of my complement.bmp diagrams to slide against each other to minimise offsets. Based on the solution for all the flags, I calculate the errors in all the commas (including some alternate symbols for 23', 31 and 37') and find the maximum error over all of them. I find that with all the complement equations I want, I still have two degrees of freedom left over. I use these to specify the values in cents of the 5 and 7 comm flags. I adjust these to minimise the maximum-absolute error. I found a set of complement relationships, which is a mixture of yours and my earlier ones, that lets me get the maximum error in any comma (i.e. sum of flag values minus comma value) down to 1.12 cents. This includes every comma up to 41 and the ones for 11/7 and 7/5. The best I can do with your complements is a max error of 2.0 cents. It makes sense that the max error would be half the max offset. Unfortunately mine requires that the complement of w| has 3 flags, x||vv. This is a consequence of the complement of w|v being x||v. I'm hoping you can find a set of complements that either have a lower max error, or a similar max error without needing a 3 flag symbol as the complement of a one-flag symbol, but it doesn't look too hopeful. Here's the system I'm talking about. I've put an asterisk against those that differ from yours. symbol complement comma offset --------------------------------------- v| x||w 19 -0.14 cents |v s||x (17'-17) -1.50 * w| x||vv 17 v|v none * w|v x||v 17' |w w||x 23 0.73 v|w x|| 19' -0.14 s| ||s 5 0.00 * w|w none |x ||x 7 -1.26 * s|v not used equiv to |x and so not used v|x none |s s|| (11-5) 0.00 x| v||w 29 -0.14 v|s none 31 w|x ||w alt 31 0.73 * s|w none * x|v w||v 23' was alt 23' * w|s none was 23' *ss| ||vv 25 s|x ||v 13 -1.50 alt 37' (replacing 3-flag symbol) x|w v|| -0.14

>You passed up some nice small-offset complements that are 665- >inconsistent. The only ones that I was forced to pass up in 217-ET >have an offset of over 2.6 cents. And you can see that 653-ET also >has a number of inconsistent complements. This is due in large part >to the fact that 653 and 665 are much finer divisions, so this is not >surprising.

Yes. Consistency is irrelevant here. It's the offsets (or errors) in cents that matter.

>However, this is a good reason not to base rational complementation >on a particular division of the octave, but rather on the basis of a >small offset. Agreed.

The spreadsheets I used for solving the two sets of equations are at Yahoo groups: /tuning-math/files/Dave/DKCompSo... * [with cont.] Yahoo groups: /tuning-math/files/Dave/GSCompSo... * [with cont.] Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4797 - Contents - Hide Contents Date: Sun, 12 May 2002 17:21:30 Subject: Re: A common notation for JI and ETs From: David C Keenan I wrote: "Consistency is irrelevant here. It's the offsets (or errors) in cents that matter." Of course I meant n-limit consistency of ETs is irrelevant here. I think consistency of the system of linear equations representing the complement rules is very important. Another problem I have with your proposal is a crossover of symbol sizes between single and double shafts when one includes the obvious complement for the 25 comma symbol. It is related to the 4 cent offset. Assuming we take the complement of ss| to be ||vv, and you have w| complement is w||s, then in order of increasing size (of commas represented, not of solutions to equations) we have w| |vv ... w|s ss| but in order of increasing size, the complements go ||vv w|| ... ss|| w||s Maybe what this is trying to tell us is that we should consider making ss|| the complement of w| (17 comma), and w|| the complement of ss| (25 comma). When I substitute that for the w| rule in your system I can get the max error down to 1.21 cents, provided I use w|x for 31 and s|x for 37'. I've put up the spreadsheet as Yahoo groups: /tuning-math/files/Dave/GS2CompS... * [with cont.] Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4798 - Contents - Hide Contents Date: Mon, 13 May 2002 09:19:55 Subject: Re: A common notation for JI and ETs From: David C Keenan I realised that the system of my previous message is no good because it didn't give a complement for the 23' comma with either the standard or alternate symbol. But now I think I've cracked it. I've found a system where every comma that needs a complement has one, and no new symbols are required, and the maximum error is 1.23 cents according to my spreadsheet. The maximum offset is 1.53 cents according to your spreadsheet. The system happens to be consistent with 494-ET. You can see it in Yahoo groups: /tuning-math/files/Dave/DK2Compl... * [with cont.] and Yahoo groups: /tuning-math/files/Dave/DK2CompS... * [with cont.] The 23' comma symbol is now x|v, not w|s, because w|s has no one-flag-per-side complement in this system. This involves a 0.52 cent schisma. It also frees w|s to be used as purely a 125-diesis symbol if we want (0.56 cent schisma). The 31 comma symbol is now w|x, not v|s, and the 37' symbol is s|x, not v|wx. These involve schismas of 0.73 cents and 0.88 cents respectively, but I consider these a price worth paying for the reduced number of symbols and the complete rational complements without any 3-flag symbols. As a bonus, all the symbols that do not have complements are not needed at all. We could take w|w and s|v as complementary but I don't think we need either of them. I believe we only need 20 single-shaft symbols and 16 double-shaft symbols. None of these symbols have more than 2 flags and only 4 have two flags on the same side (the 25 and 31' symbols ss| and sx| and the complements of the 17 and 31' symbols ss|| and |xs). Here it is. The differences from your most recent proposal are shown with asterisks. symbol complement comma offset (cents) --------------------------------------- natural s||s apotome 0.00 v| x||w 19 -0.14 |v s||x (17'-17) -1.50 * w| ss|| 17 1.53 * w|v x||v 17' -1.03 |w w||x 23 0.73 v|w x|| 19' -0.14 s| ||s 5 0.00 |x ||x 7 -1.26 |s s|| (11-5) 0.00 x| v||w 29 -0.14 w|x ||w 31 0.73 * x|v w||v 23' -1.03 *ss| w|| 25 1.53 s|x ||v 13 -1.50 s|x x|s 13 -1.50 alternative single-shaft complement x|w v|| -0.14 s|s x|x 11 0.00 sx| |sx 31' 0.00 In 494-ET the flags correspond to the following numbers of steps. v| 1 2 |v w| 4 7 |w s| 9 11 |x x| 14 13 |s Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)