This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 1211000 11050 11100 11150 11200 11250 11300 11350 11400 11450 11500
11450 - 11475 -
Message: 11450 - Contents - Hide Contents Date: Sat, 17 Jul 2004 14:31:52 Subject: Re: need definition: wedgie From: Herman Miller Herman Miller wrote:> By convention, if the first number is negative, the wedgie is normalized > by multiplying each element by -1. So the normalized wedgie for meantone > is <<1, 4, 10, 4, 13, 12||.To be more precise, "if the first nonzero element is negative".
Message: 11451 - Contents - Hide Contents Date: Sat, 17 Jul 2004 05:28:41 Subject: names and definitions: meantone From: monz let's start the whole project with perhaps the most familiar family of tunings. i already have a page about meantone at Frame Index for Tuning Dictionary * [with cont.] (Wayb.) i realize that some of the categories below require data for specific flavors of meantone, and not the whole family itself. but if there is a way to put in a range of data that does cover the whole family, it think that is good. please keep in mind that this is an extremely rough draft that's coming right off the top of my head. i'm real busy with other things but while i'm in the thick of working on the Encyclopaedia, i might as well use the opportunity to create a whole slew of webpages covering the names that everyone is complaining about. so that at least then familiarity can breed contempt ... ;-) fill in the blanks, and adjust, correct, argue etc. as much as possible ... family name: meantone period: 2:1 ratio generator: wedge product: wedgie: unison-vectors: monzos, multimonzos: vals, multivals: badness: MOS: DE: propriety: consistency: characteristic interval(s): x-chordal interval structure (tetrachord, pentachord, etc.): -monz
Message: 11452 - Contents - Hide Contents Date: Sat, 17 Jul 2004 19:38:10 Subject: Re: 136edo From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> on this page (which can also be reached via a link on > the "meantone" defintion) > > Definitions of tuning terms: meantone-from-JI ... * [with cont.] (Wayb.) > > i found that 136edo is some sort of optimal meantone for > the 11-limit, regarding the amount of error from JI. > (at least, using the prime-mappings i used)It's very near the rms optimum and hence the poptimal range for huygens. Since I'm on semiconvergents, here is what you get for rms (exponent 2), exponent 3, and exponent 4 optimums: rms 7, 12, 19, 31, 43, 74, 105, 136, 167, 198, 365, 532, ... p3 7, 12, 19, 31, 43, 74, 105, 136, 167, 198, 365, 563, ... p4 7, 12, 19, 31, 43, 74, 105, 136, 167, 198, 229, 427, ... 167 is a convergent for both rms and p3, however, which 136 is not. A while back we had a discussion of whether 74 was not, in fact, totally useless; here we see it showing up on the huygens lists.
Message: 11453 - Contents - Hide Contents Date: Sat, 17 Jul 2004 22:49:24 Subject: Re: names and definitions: schismic From: Herman Miller Gene Ward Smith wrote:> and it's not a good idea to >>> change established names in any case. > >> We've *been* changing established names; I don't think 7-limit > schismic was ever as established as some Paul wants to deep-six.I was referring to "meantone".
Message: 11454 - Contents - Hide Contents Date: Sat, 17 Jul 2004 05:29:56 Subject: names and definitions: aristoxenean From: monz i already have a page about aristoxenean at Frame Index for Tuning Dictionary * [with cont.] (Wayb.) fill in the blanks, and adjust, correct, argue etc. as much as possible. feel free to add new categories and descriptive text commentary as needed. family name: meantone period: 2:1 ratio generator: wedge product: wedgie: unison-vectors: monzos, multimonzos: vals, multivals: badness: MOS: DE: propriety: consistency: characteristic interval(s): x-chordal interval structure (tetrachord, pentachord, etc.): -monz
Message: 11455 - Contents - Hide Contents Date: Sat, 17 Jul 2004 19:46:32 Subject: Re: names and definitions: schismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:>> This should be "schismic". It's consistent with all those ETs except 17 >> (which isn't 7-limit consistent in itself). >> I agree; "schismic" is essentially a 5-limit name (based on tempering > out the [-15 8 1> "schisma"), so the 7-limit extension (if any) > shouldn't be more complex than it needs to be.This reasoning is backwards, and leads to bizarre conclusions. If schismic is essentially a 5-limit temperament, then you'd better keep close to the 5-limit tuning seems like the way to reason from your premise. Moreover, it leads to the conclusion that dominant should get the name "meantone" in the 7-limit, and that is a conclusion no one seems to buy. We'd also end up renaming pajara to diaschismic, I suppose.>>> name: schism, 12&17 >>> wedgie: <<1 -8 -2 -15 -6 18|| >>> mapping: [<1 2 -1 2|, <0 -1 8 2|] >>> 7 limit poptimal generator: 27/65 >>> 9 limit poptimal generator: 22/53 >>> TM basis: {64/63, 360/343} >>> MOS: 12, 17, 29, 41, 53> This looks like a strange hybrid of schismic and dominant; something you > might use if you don't have enough notes for schismic, but you're > willing to substitute a 16/9 for a 7/4. It would be nice to see some > error values for comparison purposes.It would indeed be what you'd get if you were in Schismic[12] and wanted something to serve as a 7; people would be happy with it if for no other reason than because they are used to noisy dominant seventh chords resolving; this would resolve something in the manner of meantone, where a brash V7 moves to a smooth I. With 17 notes of course, it would be like Meantone[12], with both choices available for certain chords.
Message: 11456 - Contents - Hide Contents Date: Sat, 17 Jul 2004 05:31:08 Subject: names and definitions: schismic From: monz NOTE: i propose that we drop "schismatic" as a synonymous term, or at least always mention that it is a synonym. (it is, right?) i already have a page about schismic at Frame Index for Tuning Dictionary * [with cont.] (Wayb.) fill in the blanks, and adjust, correct, argue etc. as much as possible. feel free to add new categories and descriptive text commentary as needed. family name: meantone period: 2:1 ratio generator: wedge product: wedgie: unison-vectors: monzos, multimonzos: vals, multivals: badness: MOS: DE: propriety: consistency: characteristic interval(s): x-chordal interval structure (tetrachord, pentachord, etc.): -monz
Message: 11457 - Contents - Hide Contents Date: Sat, 17 Jul 2004 19:47:44 Subject: Re: names and definitions: meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> Gene, when you wrote, for example, > "7 limit poptimal: 86/205", you *did* mean > "7-limit poptimal *generator*", right?Right; the generator is 86/205 of an octave, or 2^(86/205).
Message: 11458 - Contents - Hide Contents Date: Sat, 17 Jul 2004 23:29:26 Subject: Re: names and definitions: schismic From: Herman Miller Gene Ward Smith wrote:> We've *been* changing established names; I don't think 7-limit > schismic was ever as established as some Paul wants to deep-six. > Moreover, sticking to a consistent scheme means we have a better idea > what the name means; in this case, that the name in the higher limit > has a tuning in accord with the lower limit.Looking over this reply, I'm wondering if my meaning wasn't clear enough. I'm not referring to Paul's substitution of new names for the 7- and 5-limit versions of temperaments; those substitutions don't change the meanings of the existing names. If for consistency we wanted to give [<1, 2, 4, 2|, <0, -1, -4, 2|] the same name as [<1, 2, 4|, <0, -1, -4|], it would be confusing to use the name "meantone", since that name is already associated with [<1, 2, 4, 7|, <0, -1, -4, -10|]. So we'd have to come up with a new name, like "syntonic". But of course there could be occasional cases where there's a good reason to change a higher-limit name. I've argued for #56 [<1, 1, 2, 4|, <0, 2, 1, -4|] to get the name "dicot" in place of #23 [<1, 1, 2, 1|, <0, 2, 1, 6|], which would be "pseudo-dicot". It sounds like there are good arguments for changing "schismic" as well. I'm just not convinced that 118&171 is the best candidate for the name, and I'm dubious about applying the same criteria to other temperaments.
Message: 11459 - Contents - Hide Contents Date: Sat, 17 Jul 2004 05:32:37 Subject: names and definitions: orwell From: monz i already have a page about orwell at Frame Index for Tuning Dictionary * [with cont.] (Wayb.) but it's very much in need of amending. fill in the blanks, and adjust, correct, argue etc. as much as possible. feel free to add new categories and descriptive text commentary as needed. family name: meantone period: 2:1 ratio generator: wedge product: wedgie: unison-vectors: monzos, multimonzos: vals, multivals: badness: MOS: DE: propriety: consistency: characteristic interval(s): x-chordal interval structure (tetrachord, pentachord, etc.): -monz
Message: 11460 - Contents - Hide Contents Date: Sat, 17 Jul 2004 19:51:25 Subject: Re: need definition: copoptimal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> someone please write a definition of copoptimal for > inclusion into the Encyclopaedia.A generator is "copoptimal if" it is poptimal for more than one odd limit. For instance, for a 7-limit temperament we might wish for one poptimal in both the 7 and 9 odd limits; if one exists it would be 7&9 copoptimal, or simply copoptimal if the meaning is clear by context.
Message: 11461 - Contents - Hide Contents Date: Sat, 17 Jul 2004 00:32:25 Subject: Re: Naming temperaments From: Herman Miller Dave Keenan wrote:>> 11/27 [<1, 2, 6, 2, 1|, <0, -1, -9, 2, 6|] >> 20/49 [<1, 2, 6, 2, 10|, <0, -1, -9, 2, -16|] > >> It's not only different limits that cause this sort of problem, it's > also different optimisation criteria at the same limit.This isn't a "problem"; these really are different temperaments. It's only a problem when two different g/p ratios give the same temperament and you need to decide between them.>> 4/46 gives you [<2, 3, 5, 7|, <0, 1, -2, -8|], not pajara (see >> A third kind of linear temperament * [with cont.] (Wayb.)). > >> Oops. Sorry. That should have been 8/46. But even so, that's only > good for "5-limit pajara" or diaschismic.Ah; "pajara" and "diaschismic" are different 7-limit temperaments; I was under the impression that the 5-limit one has only been called "diaschismic" and not "pajara". But since pajara looks like the simplest (0, 1, -2) scale, it does make sense to use the same name for the 5-limit version.> My question was whether we all agree we should use the smallest > possible value of the generator ( the one that's less than half the > period) in these octave-fraction-type names? I note that Paul is not > doing this in his paper when he gives generators in cents. He is > using whatever falls out of a simple algorithm for deriving the > mapping from a set of vanishing commas.I think this is a reasonable convention.> As I've said elsewhere, in the two-ET/MOS/DE method (the two- > cardinalities method?) of naming, I'd like the two numbers to give > the denominators of two convergents (or semi-convergents) of the > generator as an octave fraction, such that one is near the minimum > useful generator size and the other is near the maximum.The issue here is quantifying "useful" in some easily defined / non-arbitrary way.> It would be ideal if you could also obtain the typical MOS/DE > cardinality by subtracting these two numbers, and obtain a good > approximation of an optimum generator by adding them. > > For example, calling meantone the "12&19-LT" works perfectly. 12-ET > and 19-ET are very near the extreme generator values re "harmonic > waste". The typical MOS cardinality is 19-12 = 7, and a near optimum > occurs at 12+19 = 31-ET. > > I think at least one of the two numbers should be a convergent, i.e. > it should give the cardinality of a Rothenberg-proper MOS/DE for > most optimum generator sizes.Is that the "step size ratio <= 2:1" criterion?> Don't you recognise your own zipcode and those of people you > regularly send mail to, and similarly phone numbers, although these > have many more digits than we're talking about here. It's more like > Australian postcodes, which are only four digits. But not even as > bad as that.I know my own ZIP code, but that's just 5 digits (if I live in one place long enough, I'll eventually get to the point where I can remember the full 9 digits). I always have to look up other ZIP codes and phone numbers. I keep my own phone number on a slip of paper so I'll have it if I need it; I don't need to fill it in on forms often enough to have it memorized yet. The point is that it's easier to recognize names, not that numbers are impossible to learn. The only ZIP codes I have memorized are the ones where I've lived.> They are generally pairs of only 2 digit numbers taken from a small > set of n-limit consistent ETs, whose numbers already have many > associations for us. So it's really just associating pairs of > already familiar things that we're already used to representing as > numbers. And they have the enormous advantage that they are not > totally opaque jargon to a newcomer, as are names like sensipent, > orson, amity, subchrome, wurschmidt, compton. > > At least when I see a postcode I've never seen before, I can > immediately tell what state it's in, and sometimes I can figure out > some towns I know that it must be near.You could probably do the same with ZIP codes if you have large numbers of them memorized. You can roughly tell how far west a place is, at any rate. Or at least that seems to be the pattern. But I know that Topeka is in Kansas; I don't have a clue what its ZIP code might be.>> assumption that most people who've heard of this temperament will >> recognize that name. > >> Only if they have been reading the tuning lists, and even then they > may only know that they seen the name but have no way of picking it > out of the jargon-diarrhoea that we're swimming in.It's possible that someone independently discovered the temperament we're calling "orwell", but would they be more likely to recognize it by the wedgie or the map? These conventions are also pretty much limited to the tuning lists and web sites of tuning list members. If anything, the generator size might give them the best clue. But we managed to figure out that Erv Wilson's "meta-mavila" was the same as what we'd been calling "pelogic"; differences in terminology aren't necessarily a problem.> Gene complains that some of these descriptive names are "a > mouthfull". So what? How many times a day do you find yourself > having to say or type them? Anyway, what's a few extra keystrokes > for one person in exchange for a whole lot of extra understanding on > the part of a whole lot of readers.I don't have any objection to supplementing the names with descriptions for those who are unfamiliar with the names. But if the descriptions themselves are used as a substitute for names, they could end up being confusingly similar.> I have been looking at "myna" in Paul's table and I never once > associated it with "starling". To me it was just another random > assembly of syllables. And even if I had, the name starling gives me > no clues as to the identity of _that_ temperament.The "Japanese monster" names weren't immediately apparent to me either, but once pointed out, they have mnemonic value.> The point is, we can do a lot better. We can actually have names > that give someone a clue, even when they have not been initaiated > into the Smith-Erlich-Miller mysteries. > > And in a situation where the names are a priori meaningless in > musical terms, and so any one is good as another, why the heck do > you guys have the need to keep changing them!!!!? It's tempting to > assume it's sheer arrogance or egotism, such as I was (more or less) > accused of when I wanted to use descriptive names in my > Microtempered Guitar article for Xenharmonikon. So I included the > cryptic/meaningless names as well, and now half of them are probably > obsolete.There are so many temperaments described in these huge lists of wedgies and maps that it's easy to forget (or be unaware) that one of them already has a name; I suspect this might be what happened with "bug" vs. "beep". Sometimes the name wasn't a very good one to begin with, and a better one comes along, as with "pelogic" vs. "mavila". Sometimes there never really was any agreement on a name to begin with, as in the case of 12&60. Then the whole issue of whether 5-limit and 7-limit temperaments should have the same name came up. But other times a perfectly good name like "tripletone" seems to have been set aside in favor of one that isn't really any better. I'd just as soon keep "tripletone".> OK. Well It seems we aren't that far apart in our thoughts on this, > but based on the colours thing, I wouldn't like to see non- > descriptive names for more than about the best 20. We've already got > more than 50, and we've only got to the 7-limit.I agree that most temperaments don't need specific names. It's probably not a good idea to name a temperament without even hearing it. I probably shouldn't have named "grackle" (12&77); I don't even know if it's any good. But I do think that anything that looks promising should at least get a provisional name of some kind to distinguish it from the crowd of identical-looking but very different temperaments. In my case, I'm also trying to build a music theory for my fictional culture, which has been developing scales and temperaments for thousands of years. I don't intend for any of these fictional names to replace existing ones, but in the case of "lemba", I don't know of an existing name for this temperament, so I just use the Yasaro name. Can you tell which of these are good temperaments? [<1, 2, 2, 3|, <0, -4, 3, -2|] [<1, 2, 3, 3|, <0, -2, -3, -1|] [<1, 2, 1, 5|, <0, -1, 3, -5|] [<3, 5, 7, 9|, <0, -1, 0, -2|] [<1, 1, 2, 4|, <0, 2, 1, -4|] [<2, 3, 5, 6|, <0, 1, -2, -2|] [<2, 4, 5, 6|, <0, -2, -1, -1|] I think that even short lists like this are confusing without some way to recognize individual temperaments. I can only keep track of them by consulting a list with names and searching for them. I've ignored many posts full of numbers because it's too much trouble to find out if there's anything interesting in there.
Message: 11462 - Contents - Hide Contents Date: Sat, 17 Jul 2004 21:01:13 Subject: Re: names and definitions: orwell From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: I'd revise it like this; I ditched 9/40 as a suggested generator; it is outside of the 22&31 range and not very good. family name: orwell, 22&31 period: octave generator: a sharp subminor third (7/6) or flat 75/64 5 limit name: orwell, semicomma comma: |-21 3 7> mapping: [<1 0 3|, <0 7 -3|] poptimal generator: 43/190 TOP period: 1200.0 generator: 271.599422 MOS: 9, 13, 22, 31, 53, 84 7 limit name: orwell, 22&31 wedgie: <<7 -3 8 -21 -7 27|| mapping: [<1 0 3 1|, <0 7 -3 8|] 7 limit poptimal: 26/115 9 limit poptimal: 43/190 other generators: 5/22, 7/31, 12/53, 17/75, 19/84 TOP period: 1199.532657 generator: 271.493647 MOS: 9, 13, 22, 31, 53, 84 11-limit name: george, 22&31 wedgie: <<7 -3 8 2 -21 -7 -21 27 15 -22|| mapping: [<1 0 3 1 3|, <0 7 -3 8 2|] poptimal generator: 19/84 TOP period: 1201.251092 generator: 271.425083 MOS: 9, 13, 22, 31, 53, 84 name: orwell, 31&84 wedgie: <<7 -3 8 33 -21 -7 28 27 87 65|| mapping: [<1 0 3 1 -4|, <0 7 -3 8 33|] TOP period: 1200.564417 generator: 271.443068 poptimal generator: 85/376 MOS: 9, 13, 22, 31, 53, 84, 115, 146 9-note MOS: -4 Eb+ (16/15) -3 F#- (5/4) -2 A-< (35/24) -1 B> (12/7) 0 D (1/1) +1 F< (7/6) +2 G+> (48/35) +3 Bb+ (8/5) +4 C#- (15/8)
Message: 11463 - Contents - Hide Contents Date: Sat, 17 Jul 2004 00:41:07 Subject: Re: Naming temperaments From: Herman Miller Gene Ward Smith wrote:> I'm not sure how you are using the 12+16 notation.In the context of talking about MOS scales, 12 steps of one size and 16 steps of another size. 12L+16s is more specific: 12 large steps and 16 small steps.
Message: 11464 - Contents - Hide Contents Date: Sat, 17 Jul 2004 21:04:58 Subject: Re: need definition: wedgie From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> By convention, if the first number is negative, the wedgie is normalized > by multiplying each element by -1.Another convention is that if there is a common divisor, it is divided out. Thanks, Herman.
Message: 11465 - Contents - Hide Contents Date: Sat, 17 Jul 2004 06:36:59 Subject: Re: names and definitions: meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> family name: meantone > period: 2:1 ratio > generator: a flat fifth or sharp fourth > wedge product: eh? 5-limit: 81/80mapping: [<1 2 4|, <0 -1 -4|] poptimal generator: 34/81 MOS: 5, 7, 12, 19, 31, 50 7-limit: name: meantone, standard septimal meantone, 12&19 wedgie: <1 4 10 4 13 12| mapping: [<1 2 4 7|, <0 -1 -4 -10|] (fourth generaor) 7&9 limit copoptimal generator: 5^(1/4) fifth (1/4 comma meantone) 7 limit poptimal: 86/205 9 limit poptimal: 47/112 TM basis: {81/80, 126/125} MOS: 5, 7, 12, 19, 31, 50 name: dominant, dominant sevenths, 5&12 wedgie: <1 4 -2 4 -6 -16| mapping: [<1 2 4 2|, <0 -1 -4 2|] 7&9 limit copoptimal generator: 29/70 TM basis: {36/35, 64/63} MOS: 5, 7, 12, 17, 29 name: flattone, 19&26 wedgie: <1 4 -9 4 -17 -32| mapping: [<1 2 4 -1|, <0 -1 -4 9|] 7 limit poptimal: 46/109 9 limit poptimal: 27/64 TM basis: {81/80, 525/512} MOS: 5, 7, 12, 19, 26, 45 11 limit name: meantone, meanpop, 31&19 wedgie: <1 4 10 -13 4 13 -24 12 -44 -71| mapping: [<1 2 4 7 -2|, <0 -1 -4 -10 13|] poptimal generator: 47/112 TM basis: {81/80, 126/125, 385/384} MOS: 5, 7, 12, 19, 31, 50, 81 name: huygens, fokker, 31&12 wedgie: <1 4 10 18 4 13 25 12 28 16| mapping: [<1 2 4 7 11|, <0 -1 -4 -10 -18|] poptimal generator: 13/31 TM basis: {81/80, 126/125, 99/98} MOS: 5, 7, 12, 19, 31, 43 name: meanertone wedgie: <1 4 3 -1 4 2 -5 -4 -16 -13| mapping: [<1 2 4 4 3|, <0 -1 -4 -3 1|] poptimal generator: 13/31 TM basis: {27/25, 21/20, 33/32} MOS: 5, 7, 12, 19, 31, 43
Message: 11466 - Contents - Hide Contents Date: Sat, 17 Jul 2004 21:09:36 Subject: Re: 136edo From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>> on this page (which can also be reached via a link on >> the "meantone" defintion) >> >> Definitions of tuning terms: meantone-from-JI ... * [with cont.] (Wayb.) >> >> i found that 136edo is some sort of optimal meantone for >> the 11-limit, regarding the amount of error from JI. >> (at least, using the prime-mappings i used) >> It's very near the rms optimum and hence the poptimal range for > huygens.Joe's web page says ``It can be seen that the lowest overall error in the 11-limit is given by meantones in the area of 3/13-comma to 136edo (in the version of 136edo where the "5th" is mapped to 79 degrees).'' Joe, this needs to be corrected to make clear that it is assuming the huygens mapping; another and equally valid choice is the meantone/meanpop mapping.
Message: 11467 - Contents - Hide Contents Date: Sat, 17 Jul 2004 07:13:20 Subject: Re: names and definitions: meantone From: monz thanks, Gene ... this is great! it looks to me like this tells the whole story ... but others out there will know better than i. can you make some like this for all of the other named temperaments? i'm in no hurry, but folks who are confused about the names might appreciate it. -monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> family name: meantone >> period: 2:1 ratio >> generator: a flat fifth or sharp fourth >> wedge product: eh? > > 5-limit: 81/80> mapping: [<1 2 4|, <0 -1 -4|] > poptimal generator: 34/81 > MOS: 5, 7, 12, 19, 31, 50 > > 7-limit: > > name: meantone, standard septimal meantone, 12&19 > wedgie: <1 4 10 4 13 12| > mapping: [<1 2 4 7|, <0 -1 -4 -10|] (fourth generaor) > 7&9 limit copoptimal generator: 5^(1/4) fifth (1/4 comma meantone) > 7 limit poptimal: 86/205 > 9 limit poptimal: 47/112 > TM basis: {81/80, 126/125} > MOS: 5, 7, 12, 19, 31, 50 > > name: dominant, dominant sevenths, 5&12 > wedgie: <1 4 -2 4 -6 -16| > mapping: [<1 2 4 2|, <0 -1 -4 2|] > 7&9 limit copoptimal generator: 29/70 > TM basis: {36/35, 64/63} > MOS: 5, 7, 12, 17, 29 > > name: flattone, 19&26 > wedgie: <1 4 -9 4 -17 -32| > mapping: [<1 2 4 -1|, <0 -1 -4 9|] > 7 limit poptimal: 46/109 > 9 limit poptimal: 27/64 > TM basis: {81/80, 525/512} > MOS: 5, 7, 12, 19, 26, 45 > > 11 limit > > name: meantone, meanpop, 31&19 > wedgie: <1 4 10 -13 4 13 -24 12 -44 -71| > mapping: [<1 2 4 7 -2|, <0 -1 -4 -10 13|] > poptimal generator: 47/112 > TM basis: {81/80, 126/125, 385/384} > MOS: 5, 7, 12, 19, 31, 50, 81 > > name: huygens, fokker, 31&12 > wedgie: <1 4 10 18 4 13 25 12 28 16| > mapping: [<1 2 4 7 11|, <0 -1 -4 -10 -18|] > poptimal generator: 13/31 > TM basis: {81/80, 126/125, 99/98} > MOS: 5, 7, 12, 19, 31, 43 > > name: meanertone > wedgie: <1 4 3 -1 4 2 -5 -4 -16 -13| > mapping: [<1 2 4 4 3|, <0 -1 -4 -3 1|] > poptimal generator: 13/31 > TM basis: {27/25, 21/20, 33/32} > MOS: 5, 7, 12, 19, 31, 43
Message: 11468 - Contents - Hide Contents Date: Sat, 17 Jul 2004 21:23:21 Subject: Re: 136edo From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Joe's web page says ``It can be seen that the lowest overall error in > the 11-limit is given by meantones in the area of 3/13-comma to 136edo > (in the version of 136edo where the "5th" is mapped to 79 degrees).'' > Joe, this needs to be corrected to make clear that it is assuming the > huygens mapping; another and equally valid choice is the > meantone/meanpop mapping.On the meantone page, I think the error for both meanpop and huygens should be graphed; comparing them on a graph of just the 11-limit versions would be instructive.
Message: 11469 - Contents - Hide Contents Date: Sat, 17 Jul 2004 07:15:22 Subject: Re: names and definitions: meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: This is fixed up. family name: meantone period: octave generator: a flat fifth or sharp fourth 5-limit name: meantone, 12&19 comma: 81/80 mapping: [<1 2 4|, <0 -1 -4|] poptimal generator: 34/81 MOS: 5, 7, 12, 19, 31, 50, 81 7-limit name: meantone, standard septimal meantone, 12&19 wedgie: <<1 4 10 4 13 12|| mapping: [<1 2 4 7|, <0 -1 -4 -10|] (fourth generaor) 7&9 limit copoptimal generator: 5^(1/4) fifth (1/4 comma meantone) 7 limit poptimal: 86/205 9 limit poptimal: 47/112 TM basis: {81/80, 126/125} MOS: 5, 7, 12, 19, 31, 50, 81 name: dominant, dominant sevenths, 5&12 wedgie: <<1 4 -2 4 -6 -16|| mapping: [<1 2 4 2|, <0 -1 -4 2|] 7&9 limit copoptimal generator: 29/70 TM basis: {36/35, 64/63} MOS: 5, 7, 12, 17, 29 name: flattone, 19&26 wedgie: <<1 4 -9 4 -17 -32|| mapping: [<1 2 4 -1|, <0 -1 -4 9|] 7 limit poptimal: 46/109 9 limit poptimal: 27/64 TM basis: {81/80, 525/512} MOS: 5, 7, 12, 19, 26, 45 11 limit name: meantone, meanpop, 19&31 wedgie: <<1 4 10 -13 4 13 -24 12 -44 -71|| mapping: [<1 2 4 7 -2|, <0 -1 -4 -10 13|] poptimal generator: 47/112 TM basis: {81/80, 126/125, 385/384} MOS: 5, 7, 12, 19, 31, 50, 81 name: huygens, fokker, 12&31 wedgie: <<1 4 10 18 4 13 25 12 28 16|| mapping: [<1 2 4 7 11|, <0 -1 -4 -10 -18|] poptimal generator: 13/31 TM basis: {81/80, 126/125, 99/98} MOS: 5, 7, 12, 19, 31, 43 name: meanertone wedgie: <<1 4 3 -1 4 2 -5 -4 -16 -13|| mapping: [<1 2 4 4 3|, <0 -1 -4 -3 1|] poptimal generator: 13/31 TM basis: {21/20, 28/27, 33/32} MOS: 5, 7, 12, 19, 31, 43
Message: 11470 - Contents - Hide Contents Date: Sat, 17 Jul 2004 22:03:20 Subject: Re: names and definitions: orwell From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: I forgot to add the TM basis> 5 limit > > name: orwell, semicomma > comma: |-21 3 7> > mapping: [<1 0 3|, <0 7 -3|] > poptimal generator: 43/190 > TOP period: 1200.0 generator: 271.599422 > MOS: 9, 13, 22, 31, 53, 84 > > 7 limit > > name: orwell, 22&31 > wedgie: <<7 -3 8 -21 -7 27|| > mapping: [<1 0 3 1|, <0 7 -3 8|] > 7 limit poptimal: 26/115 > 9 limit poptimal: 43/190 > other generators: 5/22, 7/31, 12/53, 17/75, 19/84 > TOP period: 1199.532657 generator: 271.493647TM basis: {225/224, 1728/1715}> MOS: 9, 13, 22, 31, 53, 84 > > 11-limit > > name: george, 22&31 > wedgie: <<7 -3 8 2 -21 -7 -21 27 15 -22|| > mapping: [<1 0 3 1 3|, <0 7 -3 8 2|] > poptimal generator: 19/84 > TOP period: 1201.251092 generator: 271.425083TM basis: {99/98, 121/120, 176/175}> MOS: 9, 13, 22, 31, 53, 84 > > name: orwell, 31&84 > wedgie: <<7 -3 8 33 -21 -7 28 27 87 65|| > mapping: [<1 0 3 1 -4|, <0 7 -3 8 33|] > poptimal generator: 85/376 > TOP period: 1200.564417 generator: 271.443068TM basis: {225/224, 441/440, 1728/1715}> MOS: 9, 13, 22, 31, 53, 84, 115, 146 > > 9-note MOS: > -4 Eb+ (16/15) > -3 F#- (5/4) > -2 A-< (35/24) > -1 B> (12/7) > 0 D (1/1) > +1 F< (7/6) > +2 G+> (48/35) > +3 Bb+ (8/5) > +4 C#- (15/8)
Message: 11471 - Contents - Hide Contents Date: Sat, 17 Jul 2004 07:16:10 Subject: Re: names and definitions: meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> thanks, Gene ... this is great!But please use the revised version. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11472 - Contents - Hide Contents Date: Sat, 17 Jul 2004 22:10:17 Subject: A template for temperaments From: Gene Ward Smith Here's a sort of form to fill in for these. One question is whether error/complexity/badness should be added, and if so, using what measures. family name: period: generator: 5-limit name: comma: mapping: poptimal generator: TOP period: mapping: TM basis: MOS: 7-limit name: wedgie: mapping: poptimal generator(s): TOP period: generator: TM basis: MOS: Higher limits are the same ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11473 - Contents - Hide Contents Date: Sun, 18 Jul 2004 06:58:32 Subject: Re: Atomic notation again From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> Here is what we find if we want to take it up to the 23-limit: >> >> 3: [19, 1] 1:32805/32768, 5632/5625 >> 5: [28, -7] 7: 126/125 >> 7: [34, -16] 16: 56/55 >> 11: [42, -25] 25: 36/35 >> 13: [43, 72] 72: 6561/6050 >> 17: [47, 105] 105: 260/231 >> 19: [50, 50] 50: 128/121 ~ (36/35)^2 >> 23: [52, 117] 117: 416/363> I haven't a clue what you're on about here.The [19, 1] for 3 says that you need 19 semitones, or 1900 cents exactly, plus a schisma to get to 3. The 5632/5625 says this will serve as a schisma. We have [19, 1] = [19, 0] + [0, 1]. The rest of the primes are the same. It indicates what complexites and what kind of 13-limit versions of n schismas it takes to get to the primes up to 23.
Message: 11474 - Contents - Hide Contents Date: Sun, 18 Jul 2004 06:59:53 Subject: Re: Extreme precison (Olympian) Sagittal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: