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Message: 5925 - Contents - Hide Contents Date: Fri, 10 Jan 2003 12:36:13 Subject: Re: A common notation for JI and ETs From: David C Keenan Working down the ratio popularity list, of those we don't yet have a symbol for: There are two 125 commas of interest 125-diesis 125:128 41.06 c .//| exact, no symbol without 5' 125'-diesis 243:250 49.17 c /|) or (|~ two 49 commas 49-diesis 48:49 35.69 c ~|) 49'-diesis 3963:4096 54.53 c (/| or |)) one 7:25 comma 7:25-comma 224:225 7.71 c '|( exact, no symbol without 5' two 5:49 commas 5:49-comma 321489:327680 33.02 c (| 5:49'-diesis 392:405 56.48 c '(/| or '|)) exact, no symbol w/o 5' Perhaps we should ditch the (/| symbol entirely and use |)) for the 31' comma since |)) is the more obvious symbol for the 49'-diesis. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5926 - Contents - Hide Contents Date: Fri, 10 Jan 2003 04:40:05 Subject: Re: Notating Kleismic From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> I've been pondering this, and I think there is a strong argument infavor of using 53. The top of the poptimal range for septimal kleismic and the bottom of the poptimal range for 5-limit kleismic coincide at the minimax generator of 3^(1/6), which is the same for both. This is the only generator which is poptimal in both limits, but of course 53, which has a much better fifth than 72, comes a lot closer. Moreover, kleismic is more important as a 5-limit system (where it is very strong) than as a 7-limit system, and I think we should try to use one et for all of the versions of a temperament when we can. I vote (if that is how this is done) for 53. Anyone else care to chime in? < The way this standardisation stuff has been done in the past, is not by voting as such, but by consensus. That is, we keep presenting arguments for and against various options, with as little ego investment and as much praise of the other people's ideas as possible, until everyone who has ever expressed an opinion on it, either agrees or says they no longer care. So here goes me. While I liked the pattern that kleismic makes in 72-ET, as pointed out by George, I'm swayed by Gene's arguments above. By most people's reckoning kleismic is one of the top four 5-limit temperaments. At the 7-limit there are two extensions of kleismic that might be considered. Both are down past number ten on anyone's list. However, if you _were_ using kleismic for 7-limit with the least complex 7's, you would probably be dissatisfied with a 53-ET based notation. Lets try approaching it in an ET-independent manner, considering only the 5-limit map 2:3 is 6 gens 4:5 is 5 gens 5:6 is 1 gen BTW the two 7-limit extensions have 4:7 is 3 gens or 4:7 is 22 gens I'll use / and \ as 5-comma (80:81) up and down symbols and FCGDAEB and #b have their usual Pythagorean relationships. Then the following is clearly a correct notation for a chain of 19 notes of kleismic. Alternative names are given underneath some note names. A#\\\ C#\\ E\ G Bb/ Db// E#\\\ G#\\ B\ Bbb/// Fb/// D F/ Ab// Cb/// D#\\ F#\ A C/ Eb// Gb/// B#\\\ Fx\\\ Which ET notation best preserves this? 53: /| /|\ (|) or 72: /| |) /|\ or should we use something else such as /| //| .//| Since we've got symbols for 11 commas in both of those ET notations, we really should check whether they are valid in any sensible 11-limit extensions of kleismic. Gene or Graham, have any 11-limit kleismics turned up in your searches. If so, what maps?

Message: 5927 - Contents - Hide Contents Date: Fri, 10 Jan 2003 05:49:44 Subject: Re: Notating Kleismic From: Dave Keenan I wrote:> Which ET notation best preserves this? > 53: /| /|\ (|) > or > 72: /| |) /|\ > or should we use something else like > /| //| .//| > > Since we've got symbols for 11 commas in both of those ET notations, > we really should check whether they are valid in any sensible 11-limit > extensions of kleismic. > > Gene or Graham, have any 11-limit kleismics turned up in your > searches. If so, what maps?In the case of the 53-ET based notation it could also be a 1,3,5,11 temperament (no 7s).

Message: 5928 - Contents - Hide Contents Date: Fri, 10 Jan 2003 06:49:01 Subject: Notating Catakleismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:>> Gene or Graham, have any 11-limit kleismics turned up in your >> searches. If so, what maps?The more complex one, "catakleismic", has an 11-limit extension which I've also called "catakleismic", which uses -21 generators for 11. The 300th row of the Farey sequence goes 19/72 < 71/269 < 52/197 < 33/125 Of these, 52/197 is poptimal for the 7-limit, and 71/269 for the 11-limit; I would suggest 72-et for notating catakleismic, therefore, although 125 would be possible also.

Message: 5929 - Contents - Hide Contents Date: Fri, 10 Jan 2003 07:28:02 Subject: Re: Notating Kleismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> However, if you _were_ using kleismic for 7-limit with the least > complex 7's, you would probably be dissatisfied with a 53-ET based > notation.Why? Neither one is using the best value of the "7" of the et in question, and in both cases the 7-limit intervals are much more out of tune than the 5-limit intervals. It is cheesy no matter how you notate it, and I don't see why it is any *more* dissapointing for 53 than for 72.

Message: 5930 - Contents - Hide Contents Date: Fri, 10 Jan 2003 10:23:44 Subject: Re: thanks manuel From: manuel.op.de.coul@xxxxxxxxxxx.xxx>the triad player? . . . what's weird is that the idea above deals >with dyads, while i had a similar idea actually dealing with triads --Yes, that doesn't matter. The triad player also plays dyads by the way. It can only play triads with one fixed tone, 1/1 or the octave.> plotting all the triads in the scale on top of the snowflake, and >seeing how the points (scale's triads) move around as you fiddle with >the scale (possibly with an eye towards approximating a number of >otonal triads) . . .So this is quite another story. Manuel

Message: 5931 - Contents - Hide Contents Date: Fri, 10 Jan 2003 19:31:54 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote:>> At 01:16 AM 9/01/2003 +0000, Dave Keenan <d.keenan@u...> wrote:>>> I tried small arrowheads to indicate the 5' down and up symbols. >In>>> the 3rd staff I attached them to the point of an existing sagittal >>> symbol; for the up-arrow I removed the pixel at the end of the >shaft>>> to clarify the symbol. The big advantage here is that we would >avoid>>> having detached symbol elements. >>>> Yes. But unfortunately they make it look like you're modifying a >note>> aligned with the place between the 5' arrowhead and the rest of the >flags. >>I just gave them the same vertical placement that you used for >your "accent" marks.No. I was referring here to where you attached them to the point of the existing symbol. But I think we've both rejected these by now.>> Based on making symbols proportional to their size in cents >relative to>> strict Pythagorean, the 5' symbol should only have about 6 pixels >because>> the 19 comma flag has 10 and corresponds to 3.4 cents. The small >arrowheads>> (or circumflex and caron) contain 8 pixels. >>At least it's fewer pixels.Yes. The 8 pixels wasn't a big deal.>> Full arrowheads already have a sagittal association with the prime >11>> whereas the slanted lines preserve the association with 5. >>Also true. Another problem that I see with these full arrowheads is >how to represent them in ascii -- ^ and v would need to be used, and >although this doesn't pose any conflict with sagittal ascii, it would >pose a problem for those who want to use these as shorthand for the >11 diesis.Good point. But I don't think we should allow the limitations of ASCII to exert much influence, if any.>> Well, I'd go along with kerning the acute nearer to (the left of) >the>> symbol being modified, when that symbol has a left flag (as in the >> pythagorean comma symbol), but I'd still prefer that the 5' symbols >were>> defined as separate symbols in the font, for what are, I hope, >obvious >> reasons, > >Yes. >>> and I'd still prefer that the unkerned distance was two pixels >> (such as in the diaschisma symbol). >> >> Pythag comma '/| >> Diaschisma `/| >>To evaluate all of these issues, I added a fifth staff to my figure: > >Yahoo groups: /tuning- * [with cont.] >math/files/secor/notation/Schisma.gif > >Note: If you don't see 5 staves in the figure, then click on the >refresh button on your browser to ensure that you're looking at the >latest version of the file. > >I put on the fifth staff 5 different versions of symbols for each of >five commas, along with the 19 and 5:7 comma symbols for comparison. >The five versions are (left to right): > >1) Your 5' "accent marks" with the largest separation from the rest >of the symbol that I believe would be acceptable. > >The separation for some of these is still more than I would like, so >the next one is: > >2) Same as 1), but with 1 pixel less separation. > >One problem I have with your accent marks is that part of the mark is >lost because it coincides with a staff line when the note is on a >line, since the accent is 4 pixels high. This doesn't occur with my >arrowheads, which are 3 pixels high. Therefore in the next one: > >3) The accent mark is redrawn 3 pixels high by 4 wideGood. I prefer this to mine.> and given an >amount of separation that I judged to be best, which is never greater >than in 2), and sometimes less. > >Observe that with equal separation with 2) the 5' symbols (except for >the pythagorean comma) appear to have an amount of separation >intermediate between 1) and 2). For the pythagorean comma symbol the >separation is one pixel less than 2), such that the rightmost pixel >of the accent mark is aligned with the leftmost pixel of the 5 >comma. This would require a separate symbol in a font, much as some >fonts have the letter combination "fi" as a single character.You won't need to do that if the score software (e.g. Sibelius) recognises the font's kerning table. When designing the font you list specific pairs and their required negative offset relative to standard spacing. It doesn't matter if this causes them to overlap. But we could also do them as ligatures (single glyphs combining two characters) in some obscure location in the font, just in case.>The next two use my small arrowhead marks: > >4) To the left, using an amount of separation that I judged to be >best, and > >5) As in 4), but to the right. > >After studying these, I reached the conclusion that I like 3) best. >If you agree in principle, we would need to finalize what should be >the amount of separation between the accent mark and the rest of the >symbol.I definitely go for something very much like 3). I might prefer an extra half pixel of separation when it comes to the outline font, but otherwise great! -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5932 - Contents - Hide Contents Date: Fri, 10 Jan 2003 09:43:27 Subject: Notating Augmented and Tripletone From: Gene Ward Smith "Augmented" is the name I've given the 7-limit linear temperament with wedgie [3, 0, 6, 14, -1, -7] and TM basis [36/35, 128/125]; and "Tripletone" is my name for the system with wedgie [3, 0, -6, -14, 18, -7] and TM basis [64/63, 126/125]. Unless we are in 12-et these are different systems and need different names; they cannot both be called "augmented". Tripletone has a 7-limit poptimal generator of 2/27 and a 9-limit one of 5/66. I would suggest 2/27. Augmented has 8/33 for a 7-limit poptimal and 11/45 in the 9-limit. The 33 et seems a reasonable choice to me, but 12 would be an alterative. Any opinions on this?

Message: 5933 - Contents - Hide Contents Date: Fri, 10 Jan 2003 10:16:29 Subject: Notating Pajara From: Gene Ward Smith This is the system with wedgie [2, -4, -4, 2, 12, -11] which we used to call Paultone. It has [1/2, 5/56] as poptimal in both the 7-limit and the 9-limit, and my recommendation is that the 56-et be used to notate it. The alternative is 22, but with all due respect for Paul's favorite division, 56 isin much better tune. As a way of tuning a 22-tone MOS and playing Decatonic, it's something Paul might try if he hasn't already.

Message: 5934 - Contents - Hide Contents Date: Fri, 10 Jan 2003 10:16:29 Subject: Notating Pajara From: Gene Ward Smith This is the system with wedgie [2, -4, -4, 2, 12, -11] which we used to cal= l Paultone. It has [1/2, 5/56] as poptimal in both the 7-limit and the 9-li= mit, and my recommendation is that the 56-et be used to notate it. The alte= rnative is 22, but with all due respect for Paul's favorite division, 56 is= in much better tune. As a way of tuning a 22-tone MOS and playing Decatoni= c, it's something Paul might try if he hasn't already.

Message: 5935 - Contents - Hide Contents Date: Sat, 11 Jan 2003 11:53:44 Subject: Re: A common notation for JI and ETs From: David C Keenan At 11:13 PM 10/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote:>> Working down the ratio popularity list, of those we don't yet have >a symbol >> for: >> >> There are two 125 commas of interest >> 125-diesis 125:128 41.06 c .//| exact, no symbol without >5'>> 125'-diesis 243:250 49.17 c /|) or (|~ >>Now that we've agreed on the 5' comma symbols, may I suggest that the >ascii symbols for -5' and +5' be ' and ` respectively,I assume you meant to write ` and ' respectively?>regardless of >the direction of alteration of the main symbol (particularly since >the actual accents don't appear aligned with the point of the arrow >in the actual symbols)? I think that the period and comma are too >difficult to remember, especially the way you've done the 125-diesis >above (which is different than before), and I think `//| and '\\! >should be clear enough for a 125-diesis up and down, respectively.Yes it's different than before. I find that `//| and '\\! don't look like inverses of each other. My thinking is that, with these tiny ASCII symbols, the vertical position is a much stronger cue than the slope, particularly since neither ' nor . have any slope. I find that `//| and ,\\! look like inverses, but unfortunately position and slope cues conflict with each other in these two symbols. That only leaves .//| and '\\! So I'm proposing that the ascii symbols for -5' and +5' be . and ' respectively, regardless of the direction of alteration of the main symbol. Consider distinguishing the Pythagorean comma from the diaschisma. Which pair makes it clearer which is which. '/| `/| or '/| ./| and in the other direction `\! '/! or .\! '\! I have to say both options are pretty unsatisfactory.>For the 125' diesis, many divisions (including 171, 217, 224, 270, >282, 342, 388, and 612) would allow either /|) or (|~, but 53, 99, >and 494 all require /|), while 311 allows neither. So I believe >that /|) is the clear choice.That's fine by me, for a completely different reason. Namely that it would be bizarre to introduce a wavy flag at the 5-prime-limit when these generally correspond to primes 17, 19, 23 and only appear in very large ETs.>> two 49 commas >> 49-diesis 48:49 35.69 c ~|) >> 49'-diesis 3963:4096 54.53 c (/| or |)) >>For the 49 comma ~|) is obviously the right size. > >The 49' diesis should be 3969:4096. More on this one below.Yes. My typo. Sorry.>> one 7:25 comma >> 7:25-comma 224:225 7.71 c '|( exact, no symbol without >5' >>>> two 5:49 commas >> 5:49-comma 321489:327680 33.02 c (| >> 5:49'-diesis 392:405 56.48 c '(/| or '|)) exact, no symbol w/o >5' >>>> Perhaps we should ditch the (/| symbol entirely and use |)) for the >31'>> comma since |)) is the more obvious symbol for the 49'-diesis. >>For the 31' comma only the divisions that have any semblance of >consistency up to the 31 limit would have any practical bearing on >this decision. For 270 and 311 |)) is required, while for 217, 388, >and 653 either one is valid; 494 requires (/|, but is not 1,7,31,49- >consistent. It looks like |)) takes it. But this would require >other symbols for 23 and 24deg494; any ideas?I don't think that notating 494-ET is a high enough priority to delay the adoption of |)) as both the 49' and 31' diesis symbol. I can only think that we might be forced to use some symbols involving 5' for 494-ET. We could wait and see if suitable symbols come up as we work our way down the ratio popularity list. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5936 - Contents - Hide Contents Date: Sat, 11 Jan 2003 11:55:32 Subject: Re: A common notation for JI and ETs From: David C Keenan At 11:14 PM 10/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote:>>>> Are there any ETs in which we should now prefer )|( over some >other >>> symbol>>>> given that it now has such a low prime-limit or low product >>> complexity? >>>> >>>> I'll just note that neither of us have answered the above yet, in >case the>> way I edited things might have made it look like the following was >> answering it, which of course it is not. >>There are none that I see for this as a 7':11' comma (or whatever >we're going to call it). It has a dual role with the 7+5+19 comma in >212, 224, 311, 342, 612, and 624, where )|( has already been agreed >on or is the obvious choice. And it is not valid as the 7':11' comma >in either 217 or 494.OK. Good. Thanks for checking that. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5937 - Contents - Hide Contents Date: Sun, 12 Jan 2003 09:22:47 Subject: Notating linear temperaments From: David C Keenan Here are some more sagittal notations that are independent of any ET (as far as they go). They use only 5-comma and 5^2-comma symbols /| and //| in addition to sharps and flats. Since they only use 5-prime symbols they are valid for any 7-limit or higher extension of these 5-limit temperaments. Diaschismic (including Pajara) 3s 5s 0 1 periods 1 -2 gens ... Eb Bb F C G D A E B F# C# ... ... A\ E\ B\ F#\ C#\ G#\ Eb/ Bb/ F/ C/ G/ ... Ab/ Compatible with 22-ET notation. Augmented (including Tripletone) 3s 5s 0 1 periods 1 0 gens G#\\ ... C#\\G#\\C#\\Ab/ Eb/ Bb/ F/ C/ G/ D/ A/ ... ... Eb Bb F C G D A E B F# C# ... ... G\ D\ A\ E\ B\ F#\ C#\ G#\ Eb//Bb//F// ... Ab// Not compatible with any existing sagittal notation for an ET, but 27-ET notation is closest. We don't have a notation for 33-ET that uses its native fifth. We notate it as a subset of 99-ET. As a 7-limit temperament, I think Tripletone (the 7-limit version of Augmented that exists in 33-ET) is garbage anyway. However, the above scheme will still notate it. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5938 - Contents - Hide Contents Date: Sun, 12 Jan 2003 11:04:17 Subject: Notating linear temperaments From: David C Keenan Here are some more sagittal notations that are independent of any ET (as far as they go). They use only 5-comma and 5^2-comma symbols /| and //| (abbreviated below) in addition to sharps and flats. Since they only use 5-prime symbols they are valid for any 7-limit or higher extension of these 5-limit temperaments. Schismic 3s 5s 0 0 periods 1 -8 gens ... Eb\\Bb\\F\\ C\\ G\\ D\\ A\\ E\\ B\\ F#\\C#\\G#\\ Eb\ Bb\ F\ C\ G\ D\ A\ E\ B\ F#\ C#\ G#\ Eb Bb F C G D A E B F# C# G# Eb/ Bb/ F/ C/ G/ D/ A/ E/ B/ F#/ C#/ G#/ Eb//Bb//F// C// G// D// A// E// B// F#//C#//G#// ... Major thirds = Magic (including Narrow major thirds (= Muggles or Wizard?)) 3s 5s 0 0 periods 5 1 gens ... C E\ G#\\Cb//Eb/ G B\ D#\\Gb//Bb/ D F#\ A#\\Db//F/ A C#\ E#\\Ab//C/ E ...

Message: 5939 - Contents - Hide Contents Date: Mon, 13 Jan 2003 05:24:12 Subject: Notating Hemififths From: Gene Ward Smith This is an important temperament, so it should go onto the notation "to do" list. As a 13-limit temperament, it has the following wedgie: [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] We have generators (half of a fifth, or a neutral third) in the following range: 12/41<65/222<53/181<41/140<70/239<29/99<46/157<63/215<17/58 The vals corresponding to 58, 99, 140 and 239 which cover it are as follows: l58 := [58, 92, 135, 163, 201, 215] l99 := [99, 157, 230, 278, 343, 367] l140 := [140, 222, 325, 393, 485, 519] l239 := [239, 379, 555, 671, 828, 886] The errors for the first five odd primes using these vals are: err58 := [1.493275, 6.789735, 3.587887, 7.302747, 7.748202] err99 := [1.075302, 1.565075, .871064, 6.257815, 7.957188] err140 := [.902142, -.599427, -.254477, 5.824914, 8.043769] err239 := [.973869, .297166, .211751, 6.004233, 8.007905] 70/239 is poptimal in the 7 and 9 limits, 29/99 in the 11-limit, and 17/58 in the 13-limit. How to sort all of this out? How much would the notations differ?

Message: 5940 - Contents - Hide Contents Date: Mon, 13 Jan 2003 19:16:03 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Why not optimize the generator size for the map, and let >>> it target the consonances? Presumably because in some >>> tunings the errors for say 3 and 5 will cancel on consonances >>> like 5:3. >>>> i'm not following you, or where you differ from what's> "standard" around here . . . > > As I say, I don't know how much differing from what's > standard. Calculations are seldom posted here at the > undergrad level.well then you need to ask for clarification in such cases. there's no reason anyone should be left behind.>> why don't you post a complete calculation for the meantone >> case, or if you wish, some other, more contrived case . . . >> Map for 5-limit meantone... > > 2 3 5 > gen1 1 1 -2 > gen2 0 1 4hmm . . . gen1 is an octave, gen2 is a fifth . . . right?> Complexity for each identity... > > 2= 1 > 3= 2 > 5= 6 defined how? > Let's weight by 1/base2log(i)... > > 2= 1.00 > 3= 1.26 > 5= 2.58 > Now gen1 and gen2 are variables, and minimize... > > error(2) + 1.26(error(3)) + 2.58(error(5)) > > I don't know how to do such a calculation, or even > if it's guarenteed to have a minimum. It would > give us minimum-badness generators, not minimum > error gens.i'm not sure what you're getting at. given the mapping, there's no way to change the complexity, so we'd be holding complexity constant. so isn't minimizing badness then the same thing as minimizing error?

Message: 5941 - Contents - Hide Contents Date: Mon, 13 Jan 2003 07:17:39 Subject: Notating Supersupermajor From: Gene Ward Smith The 13-limit wedgie is [3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, 28] A Farey sequence of generators is 8/41 < 41/210 < 33/169 < 25/128 < 42/215 < 17/87 We have 128 as poptimal in the 7, 9, and 11 limits and 215 as poptimal in the 13 limit. Vals: l87 := [87, 138, 202, 244, 301, 322] l128 := [128, 203, 297, 359, 443, 474] l215 := [215, 341, 499, 603, 744, 796] Errors: err87 := [1.4933, -.1068, -3.3087, .4061, .8515] err128 := [1.1700, -1.9387, -3.2009, 1.8070, 3.2222] err215 := [1.3008, -1.1974, -3.2445, 1.2401, 2.2629] The 87 et probably sufficies; if not, the 128-et will do nicely.

Message: 5942 - Contents - Hide Contents Date: Mon, 13 Jan 2003 19:23:56 Subject: Re: thanks manuel From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:>> plotting all the triads in the scale on top of the snowflake, and >> seeing how the points (scale's triads) move around as you fiddle with >> the scale (possibly with an eye towards approximating a number of >> otonal triads) . . . >> So this is quite another story.yes, i wouldn't expect this anytime soon, while a dyad analysis tool like i suggested (with my colorful blackjack diagram) would be one of the first things i would expect from a program called scala :)

Message: 5943 - Contents - Hide Contents Date: Mon, 13 Jan 2003 19:29:18 Subject: Re: Notating Pajara From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> This is the system with wedgie [2, -4, -4, 2, 12, -11] which we >used to call Paultone. It has [1/2, 5/56] as poptimal in both the 7- >limit and the 9-limit, and my recommendation is that the 56-et be >used to notate it. The alternative is 22, but with all due respect >for Paul's favorite division, 56 is in much better tune.in my paper, i found that the fifth should be in the range 708.8143 to 710.0927 cents, with a preference for the former (equal-weighted RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while 56-equal is 707.1428, which is not even in the range.

Message: 5944 - Contents - Hide Contents Date: Mon, 13 Jan 2003 20:12:10 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus could i ask for periods and rms-optimal generators for each? --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <genewardsmith@j...> wrote:> Not that any list is really ultimate, but with rms error < 40,geometric complexity < 500, and badness < 3500, it covers a lot of ground.> > 27/25 3.739252 35.60924 1861.731473 > > 135/128 4.132031 18.077734 1275.36536 > > 256/243 5.493061 12.759741 2114.877638 > > 25/24 3.025593 28.851897 799.108711 > > 648/625 6.437752 11.06006 2950.938432 > > 16875/16384 8.17255 5.942563 3243.743713 > > 250/243 5.948286 7.975801 1678.609846 > > 128/125 4.828314 9.677666 1089.323984 > > 3125/3072 7.741412 4.569472 2119.95499 > > 20000/19683 9.785568 2.504205 2346.540676 > > 531441/524288 13.183347 1.382394 3167.444999 > > 81/80 4.132031 4.217731 297.556531 > > 2048/2025 6.271199 2.612822 644.408867 > > 67108864/66430125 15.510107 .905187 3377.402314 > > 78732/78125 12.192182 1.157498 2097.802867 > > 393216/390625 12.543123 1.07195 2115.395301 > > 2109375/2097152 12.772341 .80041 1667.723301 > > 4294967296/4271484375 18.573955 .483108 3095.692488 > > 15625/15552 9.338935 1.029625 838.631548 > > 1600000/1594323 13.7942 .383104 1005.555381 > > (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263 > > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 > > (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306 > > (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873 > > (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562 > > (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615 > > (3)^8*(5)/(2)^15 9.459948 .161693 136.885775 > > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 > > (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642 > > (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207 > > (2)*(5)^18/(3)^27 33.653272 .025593 975.428947 > > (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149 > > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 > > (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233 > > (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284 > > (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255 > > (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661 > > (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786 > > (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237 > > (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721 > > (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642 > > (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514 > > (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003 > > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 > > (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668 > > (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625 > > (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527 > > (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421 > > (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113 > > (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823 > > (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

Message: 5945 - Contents - Hide Contents Date: Mon, 13 Jan 2003 20:46:56 Subject: Re: Notating Hemififths From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> This is an important temperament, so it should go onto thenotation "to do" list.> > As a 13-limit temperament, it has the following wedgie: > > [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] > > We have generators (half of a fifth, or a neutral third) in the following range: > > 12/41<65/222<53/181<41/140<70/239<29/99<46/157<63/215<17/58 > > The vals corresponding to 58, 99, 140 and 239 which cover it are as follows: > > l58 := [58, 92, 135, 163, 201, 215] > l99 := [99, 157, 230, 278, 343, 367] > l140 := [140, 222, 325, 393, 485, 519] > l239 := [239, 379, 555, 671, 828, 886] > > The errors for the first five odd primes using these vals are: > > err58 := [1.493275, 6.789735, 3.587887, 7.302747, 7.748202] > err99 := [1.075302, 1.565075, .871064, 6.257815, 7.957188] > err140 := [.902142, -.599427, -.254477, 5.824914, 8.043769] > err239 := [.973869, .297166, .211751, 6.004233, 8.007905] > > 70/239 is poptimal in the 7 and 9 limits, 29/99 in the 11-limit, and > 17/58 in the 13-limit. How to sort all of this out? How much wouldthe notations differ?^ For 41, and 58: 3 = +2G, 11 = +5G, 13 = -1G 99, 140, 157 are not 1,3,11,13-consistent; they take 11 as one degree lower than +5G and 13 as one degree higher than -1G. 181 is 1,3,11,13-consistent, but takes 11 as one degree lower than +5G and 13 as one degree lower than -1G. To put it another way, none of those that are larger than 58-ET take either 9:11 or 13:16 as half of 2:3. It looks like we need something below the 11 limit. To digress for a moment, if you consider only 41 and 58, you might as well include all of these: 9deg31 < 7deg24 < 12deg41 < 17deg58 < 5deg17 < 3deg10 which lets you notate using the 11-diesis: 1/1 27/22 3/2 18/11 9/8 11/8 27/16 C E\!/ G B\!/ D F/|\ A But you're considering a narrower range that includes higher divisions, so you want a 7-limit just ratio that's approximately half of 2:3. The simplest one is 40:49, which is good for all the ET's you gave except for 215 (which is not 1,7,49-consistent). Fortunately, Dave has just proposed some symbols involving 7^2. I was also looking (previously) for a decent symbol that would notate half an apotome as nearly as possible (for such divisions as 99 and 311-ET), so it looks as if Dave's proposal of the symbol '|)) for the 5:49' diesis, 392:405, is going to be the answer for both. The hemififth notation would then be (if we agree to use a period to symbolize the -5' comma in ascii): C E.!)) G B.!)) D F'|)) A C'|)) E etc. --George

Message: 5946 - Contents - Hide Contents Date: Mon, 13 Jan 2003 20:57:34 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >ap for 5-limit meantone... >> >> 2 3 5 >> gen1 1 1 -2 >> gen2 0 1 4 >>hmm . . . gen1 is an octave, gen2 is a fifth . . . right? Right.>> Complexity for each identity... >> >> 2= 1 >> 3= 2 >> 5= 6 > >defined how?Those should be 2/1, 3/2, and 5/4. It's the taxicab distance on the rectangular lattice of generators. Which I cooked up as a generalization of Graham complexity for temperaments that don't necessarily have octaves. How have you been calculating Graham complexity for temperaments with more than one period to an octave?>> Let's weight by 1/base2log(i)... >> >> 2= 1.00 >> 3= 1.26 >> 5= 2.58 > >> Now gen1 and gen2 are variables, and minimize... >> >> error(2) + 1.26(error(3)) + 2.58(error(5)) >> >> I don't know how to do such a calculation, or even >> if it's guarenteed to have a minimum. It would >> give us minimum-badness generators, not minimum >> error gens. >>i'm not sure what you're getting at. given the mapping, >there's no way to change the complexity, so we'd be >holding complexity constant. so isn't minimizing >badness then the same thing as minimizing error?Doesn't the presence of the weighting factors change the result? How would you calculate complexity, error, badness, and optimum generators for 5-limit meantone? -Carl

Message: 5947 - Contents - Hide Contents Date: Mon, 13 Jan 2003 21:28:32 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Map for 5-limit meantone... >>> >>> 2 3 5 >>> gen1 1 1 -2 >>> gen2 0 1 4 >>>> hmm . . . gen1 is an octave, gen2 is a fifth . . . right? > > Right. >>>> Complexity for each identity... >>> >>> 2= 1 >>> 3= 2 >>> 5= 6 >> >> defined how? >> Those should be 2/1, 3/2, and 5/4. It's the taxicab distance > on the rectangular lattice of generators.ok . . .> Which I cooked up > as a generalization of Graham complexity for temperaments that > don't necessarily have octaves.there seems to be a problem, in that by defining the generators as an octave and a fifth, you get different numbers than by defining them as an octave and a twelfth, say. plus, graham complexity doesn't operate on a per-identity basis.> How have you been calculating > Graham complexity for temperaments with more than one period > to an octave?multiply the generator span of the otonal (or utonal) n-ad by the number of periods per octave.>>> Let's weight by 1/base2log(i)... >>> >>> 2= 1.00 >>> 3= 1.26 >>> 5= 2.58 >> >>> Now gen1 and gen2 are variables, and minimize... >>> >>> error(2) + 1.26(error(3)) + 2.58(error(5)) >>> >>> I don't know how to do such a calculation, or even >>> if it's guarenteed to have a minimum. It would >>> give us minimum-badness generators, not minimum >>> error gens. >>>> i'm not sure what you're getting at. given the mapping, >> there's no way to change the complexity, so we'd be >> holding complexity constant. so isn't minimizing >> badness then the same thing as minimizing error? >> Doesn't the presence of the weighting factors change the > result?well, if you mean minimizing the expression above (assuming you meant to the errors to be absolute values or squares), basically you've just come up with a different weighting scheme for the error. not one which i like, by the way, since you don't penalize the error in 3 and the error in 5 for being in opposite directions; that is, you don't take into account the error in 5:3.> How would you calculate complexity, error, badness, and > optimum generators for 5-limit meantone?well, certainly woolhouse's derivation is known by everyone by now, isn't it? that gives you the error and optimum generator, in the equal-weighted RMS case. the complexity is a function of the mapping, and can be defined in various ways (the graham complexity is 4), but does not depend on the precise choice of generator. badness also has several definitions -- log-flat badness is pretty much gene's territory -- but is typically error times complexity to some power.

Message: 5948 - Contents - Hide Contents Date: Mon, 13 Jan 2003 21:52:21 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >hich I cooked up as a generalization of Graham complexity for >> temperaments that don't necessarily have octaves. >>there seems to be a problem, in that by defining the generators >as an octave and a fifth, you get different numbers than by >defining them as an octave and a twelfth, say.I thought of that, but I thought also that as long as one always uses the same set of targets across temperaments, one is ok. Whaddya think?>plus, graham complexity doesn't operate on a per-identity basis.Indeed. That's part of my inquiry into the order of operations.>> How have you been calculating Graham complexity for temperaments >> with more than one period to an octave? >>multiply the generator span of the otonal (or utonal) n-ad by the >number of periods per octave.That's what I thought. How does this compare to the taxicab approach? Say, for Pajara.>>>> error(2) + 1.26(error(3)) + 2.58(error(5)) //>>> i'm not sure what you're getting at. given the mapping, >>> there's no way to change the complexity, so we'd be >>> holding complexity constant. so isn't minimizing >>> badness then the same thing as minimizing error? >>>> Doesn't the presence of the weighting factors change the >> result? >>well, if you mean minimizing the expression above (assuming you >meant to the errors to be absolute values or squares), yep. >basically you've just come up with a different weighting scheme >for the error. Ok. >not one which i like, by the way, since you don't penalize the >error in 3 and the error in 5 for being in opposite directions; >that is, you don't take into account the error in 5:3.That's what I said at the beginning of the thread. So how do you do weighted error? Do you weight the error for an entire limit by the limit, for intervals individually?>> How would you calculate complexity, error, badness, and >> optimum generators for 5-limit meantone? >>well, certainly woolhouse's derivation is known by everyone >by now, isn't it? that gives you the error and optimum generator, >in the equal-weighted RMS case. the complexity is a function of >the mapping, and can be defined in various ways (the graham >complexity is 4), but does not depend on the precise choice of >generator. badness also has several definitions -- log-flat >badness is pretty much gene's territory -- but is typically error >times complexity to some power.Okay, that's what I thought. -Carl

Message: 5949 - Contents - Hide Contents Date: Mon, 13 Jan 2003 22:35:56 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Which I cooked up as a generalization of Graham complexity for >>> temperaments that don't necessarily have octaves. >>>> there seems to be a problem, in that by defining the generators >> as an octave and a fifth, you get different numbers than by >> defining them as an octave and a twelfth, say. >> I thought of that, but I thought also that as long as one always > uses the same set of targets across temperaments, one is ok. > Whaddya think?what about temperaments without octaves, or without fifths? and anyway, why would keeping the same set of targets help? complexity shouldn't be this arbitrary!>> plus, graham complexity doesn't operate on a per-identity basis. >> Indeed. That's part of my inquiry into the order of operations. ?>>> How have you been calculating Graham complexity for temperaments >>> with more than one period to an octave? >>>> multiply the generator span of the otonal (or utonal) n-ad by the >> number of periods per octave. >> That's what I thought. How does this compare to the taxicab > approach? Say, for Pajara.i'm unclear on what taxicab approach you mean. be patient with me, i know this would be easier in person. but i have to go now.>>>>> error(2) + 1.26(error(3)) + 2.58(error(5)) > //>>>> i'm not sure what you're getting at. given the mapping, >>>> there's no way to change the complexity, so we'd be >>>> holding complexity constant. so isn't minimizing >>>> badness then the same thing as minimizing error? >>>>>> Doesn't the presence of the weighting factors change the >>> result? >>>> well, if you mean minimizing the expression above (assuming you >> meant to the errors to be absolute values or squares), > > yep. >>> basically you've just come up with a different weighting scheme >> for the error. > > Ok. >>> not one which i like, by the way, since you don't penalize the >> error in 3 and the error in 5 for being in opposite directions; >> that is, you don't take into account the error in 5:3. >> That's what I said at the beginning of the thread. So how do > you do weighted error? Do you weight the error for an entire > limit by the limit, for intervals individually?i don't think anyone's been doing weighted error on this list. but if you did, you'd minimize f(w3*error(3),w5*error(5),w5*error(5:3)) where f is either RMS or MAD or MAX or whatever, and w3 is your weight on ratios of 3, and w5 is your weight on ratios of 5.

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