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Message: 4800 - Contents - Hide Contents Date: Mon, 13 May 2002 21:22:37 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> 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 just got this one when I was about to post my latest, so you can look at my proposal and I'll look at yours, and we'll see where that gets us. I would like to see the complementation used in 217-ET (and available for use in other ET's) compatible with the rational complementation scheme, i.e., if at all possible, all of the rational complements would be valid in 217-ET. It bothers me that (the way things are at present) wL doesn't have a decent 217-ET complement -- as you noted, the 4-cent offset of wL+sR is excessive, and I want to do better than that. But (as I believe you also indicated) I would prefer not to have a 3-flag complement in the 217-ET (or any other ET) notation. I wondered whether redefining one of the flags would help us to accomplish this. The following is a complementation scheme I determined by redefining the wR flag as 23-19: symbol complement comma offset comments ------------------------------------------------- v| xv||w 19 -0.22 cents Complement requires 3 flags |v s||x (17'-17) -1.50 w| x||w 17 -2.19 Better than before! v|v ss|| 0.88 |w w||s 23-19 -0.39 The new wR flag w|v x||v 17' -1.03 v|w w||x 0.73 vv|w x|| 19' -0.14 19' requires 3 flags s| ||s 5 0.00 w|w v||x 0.73 |x ||x 7 -1.26 s|v s||v -1.74 We probably won't need this v|x w||w 0.73 |s s|| (11-5) 0.00 x| vv||w ~29 -0.22 Complement requires 3 flags s|w ||vw -0.57 Not usable in 217 v|s vw||v 31 0.02 Not usable in 217 w|x v||w 0.73 x|v w||v -1.03 w|s w|| -0.39 ss| v||v 0.88 x|w w|| -2.19 s|x ||v 13 -1.50 xv|w v|| -0.22 I also did another speadsheet for this: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/GSComp2.xls With this change, the 23 comma (the former wR flag) is available as v|w, and previous 2-flag combinations using the wR flag could still be achievable using a 3-flag symbol. We would have to see whether either of the two 3-flag wR combinations that you presently have are achievable by other means. I also updated my file: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Symbols3.bmp in which I added some 3-flag symbols at the far right (and also fixed a few mistakes). I came to the conclusion that 4-flag symbols (2 flags per side) would probably not be a good idea (too difficult to read), but 3 flags are okay (provided, of course, that not all 3 flags are on the same side). Two complements that formerly could be achieved with 2-flag symbols now require 3-flag symbols, but these were not (and still would not be) standard 217-ET symbols. In fact, 217-ET could still be done with the standard symbols that we previously agreed on (since none of them contained a wR flag), although two pairs of these would be _faux complements_ that are consistent in 217-ET (but are not rational complements). Your previous complementation rules could then be called "217-ET faux complementation rules" for these standard symbols, since we would probably desire to keep things as simple as possible in 217. At present both concave flags in 217 are one degree, and with the wR flag as (23-19), both wavy flags in 217 would be two degrees, which would further simplify things in 217. Do you see any problems with this proposal? --George

Message: 4801 - Contents - Hide Contents Date: Mon, 13 May 2002 15:15:20 Subject: Re: graham's linear temperament page From: monz many thanks, Graham! so the Tuning Dictionary now has a couple of new definitions: linear temeperament Definitions of tuning terms: linear temperamen... * [with cont.] (Wayb.) planar temperament Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) as well as an important addition to an old definition: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) see Paul, that's the *right* way to do it! (and if anyone would care to explain that "matter of formalism" which differentiates a "generator" from an "equivalence interval", i'd sure like to read it! after much discussion of it a few months ago, i'm still not clear about it.) -monz ----- Original Message ----- From: <graham@xxxxxxxxxx.xx.xx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Monday, May 13, 2002 2:57 AM Subject: [tuning-math] Re: graham's linear temperament page> In-Reply-To: <abmetp+mul3@xxxxxxx.xxx> > jonszanto wrote: >>> So great and so important you couldn't take the time to include a link? >> Sure, we can find it from our collection of links, or do a search, etc. >> - I thought you were the consummate cross-referenced poster. >> Everybody does or doesn't include links according to their mood. This one > is <Catalogue of linear temperaments * [with cont.] (Wayb.)>. > > I personally would like somebody to include the definitions of the > temperaments I'm supposed to be adding. It's been quite difficult to keep > up over the past year. >>>> (by contrast, check out joe >>> monzo's definition of linear temperament if you want to turn red). >> After following the links, I thought the "temperament" definition was > lacking, and decided to clarify it. Then scrolled down, and saw I already > had. Although I now disagree with my own definition -- temperaments can > approximate "ideal" tunings other than just intonation. > > It looks like the problem with the "linear temperament" definition is that > there isn't one! You simply say it's "another term for unequal > temperaments" which isn't the case. Not all unequal temperaments are > linear. > > A linear temperament is fully described by a generator and equivalence > interval. The mapping from generators to consonances is fixed. For > example, meantones are defined by the generator approximating 3:2 and four > generators approximating 5:4. The generator can have different sizes, > giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same > temperament class. 53-equal, however, isn't a meantone because the best > 5:4 is 8 fourths rather than 4 fifths. > > Planar temperaments are like linear temperaments, but with two generators > and an equivalence interval. The difference between a generator and an > equivalence interval is only a matter of formalism. > > > Graham

Message: 4802 - Contents - Hide Contents Date: Mon, 13 May 2002 10:57 +0 Subject: Re: graham's linear temperament page From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <abmetp+mul3@xxxxxxx.xxx> jonszanto wrote:> So great and so important you couldn't take the time to include a link? > Sure, we can find it from our collection of links, or do a search, etc. > - I thought you were the consummate cross-referenced poster.Everybody does or doesn't include links according to their mood. This one is <Catalogue of linear temperaments * [with cont.] (Wayb.)>. I personally would like somebody to include the definitions of the temperaments I'm supposed to be adding. It's been quite difficult to keep up over the past year.>> (by contrast, check out joe >> monzo's definition of linear temperament if you want to turn red).After following the links, I thought the "temperament" definition was lacking, and decided to clarify it. Then scrolled down, and saw I already had. Although I now disagree with my own definition -- temperaments can approximate "ideal" tunings other than just intonation. It looks like the problem with the "linear temperament" definition is that there isn't one! You simply say it's "another term for unequal temperaments" which isn't the case. Not all unequal temperaments are linear. A linear temperament is fully described by a generator and equivalence interval. The mapping from generators to consonances is fixed. For example, meantones are defined by the generator approximating 3:2 and four generators approximating 5:4. The generator can have different sizes, giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same temperament class. 53-equal, however, isn't a meantone because the best 5:4 is 8 fourths rather than 4 fifths. Planar temperaments are like linear temperaments, but with two generators and an equivalence interval. The difference between a generator and an equivalence interval is only a matter of formalism. Graham

Message: 4803 - Contents - Hide Contents Date: Mon, 13 May 2002 22:56:20 Subject: Re: graham's linear temperament page From: dkeenanuqnetau --- In tuning-math@y..., "monz" <monz@a...> wrote:> (and if anyone would care to explain that "matter of formalism" > which differentiates a "generator" from an "equivalence interval", > i'd sure like to read it! after much discussion of it a few months > ago, i'm still not clear about it.)Unfortunately I have a slightly different take on this to Graham's recent statement. I'm wondering if Graham really meant to say the following. A linear temperament is generated by two "generating intervals" (or generators for short), one of which is distinguished as the "interval of periodicity" (or "period" for short) and the other is simply called "the generator". The "period" is distinguished from the other generator by the fact that the "interval of equivalence" is a whole number multiple of the period. The interval of equivalence is usually the octave. A particular scale within a linear temperament (e.g. a diatonic scale within meantone) will have a fixed number of (non period) generators but an unspecified number of intervals of equivalence.

Message: 4804 - Contents - Hide Contents Date: Mon, 13 May 2002 18:09:31 Subject: definitions of period, equivalence, etc. (was: Re: graham's linear temperament page) From: monz hi Dave,> From: "dkeenanuqnetau" <d.keenan@xx.xxx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, May 13, 2002 3:56 PM > Subject: [tuning-math] Re: graham's linear temperament page > > > --- In tuning-math@y..., "monz" <monz@a...> wrote:>> (and if anyone would care to explain that "matter of formalism" >> which differentiates a "generator" from an "equivalence interval", >> i'd sure like to read it! after much discussion of it a few months >> ago, i'm still not clear about it.) >> Unfortunately I have a slightly different take on this to Graham's > recent statement. I'm wondering if Graham really meant to say the > following. > > A linear temperament is generated by two "generating intervals" (or > generators for short), one of which is distinguished as the "interval > of periodicity" (or "period" for short) and the other is simply > called "the generator". The "period" is distinguished from the other > generator by the fact that the "interval of equivalence" is a whole > number multiple of the period. The interval of equivalence is usually > the octave. A particular scale within a linear temperament (e.g. a > diatonic scale within meantone) will have a fixed number of > (non period) generators but an unspecified number of intervals of > equivalence.hmmm... thanks for that. it seems to me that i should put this into a definition as an addendum, but the "linear temperament" definition is really only appropriate for the first part of it. i still don't have a definition for "equivalence interval" or "interval of equivalence", mainly because of my confusion over exactly what it is and why it's different from the "interval of periodicity". i think it's really best for me to stay out of it until you, Graham, Paul, and Gene define exactly what all these terms mean. then let me know, and i'll update the Dictionary. thanks. -monz

Message: 4805 - Contents - Hide Contents Date: Mon, 13 May 2002 18:39:53 Subject: Re: A common notation for JI and ETs From: David C Keenan At 22:41 13/05/02 -0000, you wrote:>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >I would like to see the complementation used in 217-ET (and available >for use in other ET's) compatible with the rational complementation >scheme, i.e., if at all possible, all of the rational complements >would be valid in 217-ET.You'll be pleased to know that my latest proposal has the above property. However, it is not consistent with the plan B notation for 217-ET that we agreed on earlier. Nor is it consistent with plan A. In particular, my proposal has no rational complement for w|s (w|s is no longer the 23' symbol, x|v is). 8 steps of 217-ET would need to be notated as ss| the 25 comma. I don't have a problem with that since it involves a lower prime and still has only 2 flags (it's just that they unfortunately have to be on the same side because they are the same flag). Also, the 217-ET 7 step symbol would need to become x|v to agree with the rational complement of w|v. Alternatively the 3 step symbol could be changed to |w and the 7 step symbol could remain as w|x. But the latter pair represent higher primes and introduce one more lateral confusable. But at least it doesn't introduce 2 more like the old plan A, and I still like the idea of not having a double-flag for 3 steps, when 4, 5 and 6 are single flags. What do you think?>It bothers me that (the way things are at >present) wL doesn't have a decent 217-ET complement -- as you noted, >the 4-cent offset of wL+sR is excessive, and I want to do better than >that.I believe the answer is to use ss|| as the complement of w| in 217-ET (and rationally).>But (as I believe you also indicated) I would prefer not to >have a 3-flag complement in the 217-ET (or any other ET) notation.I totally agree about no 3-flaggers in any ETs. What's more I don't want any 3-flaggers in the rational notation (including complements)!, now that I know it is possible to do so with only a tiny increase in the 23' schisma, and larger but still modest increases in the 31 and 37' schismas. Having the Reinhard property hold up to the 29 limit is good enough for me.>I wondered whether redefining one of the flags would help us to >accomplish this.Well it helped, but not enough. And I don't think it is necessary. ...>Do you see any problems with this proposal?Only that it needs symbols with 3 flags. I hope I have shown that this is not necessary. But you should go over my proposal with a fine-toothed comb. I've managed to fool myself into believing that various schemes would work so many times only to discover later that they wouldn't, that I no longer trust my own checking. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4807 - Contents - Hide Contents Date: Mon, 13 May 2002 21:20:48 Subject: Re: A common notation for JI and ETs From: David C Keenan I've updated Yahoo groups: /tuning-math/files/Dave/Compleme... * [with cont.] to show my latest proposal. Assuming you find it acceptable, the next major job before we go public is to agree on the notation of the important ETs. I made a first pass at this in Yahoo groups: /tuning-math/message/4188 * [with cont.] You might address that when you have time. I think we should use for ETs only those symbols that are necessary for rational tunings. i.e. we should not use v|v w|w s|v v|x v|s s|w w|s. We should also try to make the ET complements agree with the rational complements where possible. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4808 - Contents - Hide Contents Date: Tue, 14 May 2002 21:51:54 Subject: Re: A common notation for JI and ETs From: David C Keenan Gene, Working up sagittal notations for the most important regular temperaments is a great idea. I assume you are volunteering to investigate this and make some proposals. I expect you will find the table below useful, but I don't expect you will need to use any symbols beyond the 13-prime-limit (only straight or convex flags). It seems to me that the notation for a linear temperament should be the same as that for some large ET that represents it well. e.g meantone same as 31-ET, miracle same as 72 ET. Maybe it should be the largest compatible ET that we can notate without using any higher prime than is approximated by the temperament. I'd like to see them notated as a chain of generators centered on D natural. George, Regarding your suggestion of redefining the w| flag as 23 comma - 19 comma, while I see no benefit in doing that, it made me realise that w| could be defined as (19'-19) comma, (i.e. 722:729), in the same way that x| can be defined as (11'-7). i.e. using the lowest prime limit possible. I think you suggested that x| should be defined as 11'-7 instead of 29 comma, but you gave the ratio as 715:729 which is 13'-(11-5). 11'-7 is 45056:45927. Similarly x|v could be defined in lowest prime terms as (11'-7)+(17'-17) (1441792:1474767) instead of 23' (16384:16767), and s|x could be defined as 5+7 instead of 13. Ultimately everything could be defined in terms of 5 7 11 17 19, but what would be the point? Maybe we should not define _any_ symbol as being a _single_ comma, but adopt the attitude that what we've done is produce a bunch of symbols each of which may be used for a number of different commas/dieses very close together in cents. The user can say precisely which of these she means, but the consequences of not doing so are almost insignificant. Someone who wants to notate a strictly rational 29-limit scale, and is willing to use multiple symbols, will define x| as the 29 comma and x|s as the 13' comma, while someone notating an 11-limit temperament will define x| as 11'-7 and x|s as apotome-(5+7). Maybe we should produce a list that shows all the possible 19-limit interpretations of each of the 20 single-shaft symbols, plus the obvious 41-limit interpretations. But mostly all you want to know are (a) the simplest interpretation, i.e. the one that involves the fewest primes, and (b) any interpretations that have a lower prime limit than the simplest one. ------------- ASCII symbols ------------- I propose we start using the following more representational ASCII versions of our symbols in place of the "svwx|" versions (although I hope that later we can agree on a single-character-per-symbol version). I suggest we use slashes /\ for straight flags, parentheses () for convex and concave, and tildes ~ for wavy. And so that we can tell which way a symbol is pointing when it has no straight flag, I suggest we use exclamation marks ! for the shafts of down-pointing arrows. So here I list all 20 single-shaft up-symbols with their rational apotome complements (not yet approved by George) and their 31-limit comma names and ratios. Symbol Complement Comma names Ratios -------------------------------------------------------- |//| /||\ natural 1:1 )| (||~ 19 512:513 |( /||) (17'-17) 288:289 ~| //|| 17 2176:2187 ~|( (||( 17' 4096:4131 |~ ~||) 23 or (19'-19) 729:736 or 722:729 )|~ (|| 19' 19456:19683 /| ||\ 5 80:81 |) ||) 7 63:64 |\ /|| (11-5) 54:55 (| )||~ 29 or (11'-7) 256:261 or 45056:45927 or (13'-11+5) or 715:729 ~|) ||~ 31 or (7+17) 243:248 or 238:243 (|( ~||( 23' * 729:736 * //| ~|| 25 6400:6561 /|) ||( or (|\ 13 or (5+7) 1024:1053 or 35:36 (|~ )|| (29+23) * 648:667 * /|\ (|) 11 32:33 (/| |\) 31' * 31:32 * |\) (/| (7+11-5) 1701:1760 (|) /|\ 11' 704:729 (|\ /|) 13' 26:27 * Too many lower prime interpretations and of too little interest to list. You can see the bitmap version of the above symbols (and others which we may not use) in Yahoo groups: /tuning-math/files/secor/notatio... * [with cont.] Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4809 - Contents - Hide Contents Date: Tue, 14 May 2002 23:33:46 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4242]:>> However, it is not consistent with the plan B notation for 217-ET >that we>> agreed on earlier. Nor is it consistent with plan A. >>Since we don't have the individual flag-complement conversion rules >anymore, there's no point in being concerned about that; low-error >rational complements are more important. Anyone using these will >just have to memorize them. There are really only 8 pairs, since >memorizing a|b <--> c||d also gives you c|d <--> a||b. (Actually >there are 4 more if you count nat. <--> s||s, s|x <--> x|s, s|s <--> >x|x, and sx| <--> |sx, but these are fairy easy to remember.) Agreed.>> In particular, my >> proposal has no rational complement for w|s (w|s is no longer the >23'>> symbol, x|v is). >>This presents a problem, the only one I have found so far with your >proposal. (I'm sorry to have to bring this up, because aside from >this, I really like what you have.) > >The problem is that in 217 x|v is 7 degrees, whereas the 23' comma is >8, which is why we originally chose w|s for its symbol. (This is not >unique to 217 -- the same situation also occurs in both 311 and 494, >although those don't really matter for our purposes, since we aren't >notating them.)I think the solution is easy. 217-ET is not in fact 23-limit consistent, right? So if we use any possibly 23 comma symbol |w x|v w|s, i.e. |~ (|( ~|\, to notate it, then we just consider them to be (19'-19), (11'-7)+(17'-17), (11-5)+17, just as we will not consider w|x ~|) as 31, or |x |) as 29. So then it doesn't matter a whit to 217-ET, which of those symbols is the 23' comma in the rational system. We are then free to choose x|v as the 23' symbol because it fits into a rational complement scheme with no triple flags and low errors.>Now it looks as if we will need a |vv symbol for the complement of >w|s.I hope I've dispensed with that.>(That's consistent in 217, but not 311 or 494.) It depends on >how much we want to complicate the 217 notation to make it conform to >the rational notation.Not at all.>Allowing wL and wL+sR to be complements in >the 217 notation makes everything much simpler in that ET, and I >think this is one place where it just might be best to apply the >guideline that the versatility (i.e., complexity) of the rational >notation should not make the simpler 217-ET notation more complicated. Agreed.>> 8 steps of 217-ET would need to be notated as ss| the 25 comma. I >don't>> have a problem with that since it involves a lower prime and still >has only>> 2 flags (it's just that they unfortunately have to be on the same >side>> because they are the same flag). >>And, unfortunately, that's one more complication. I'd like to >restrict two flags on the same side to the rational notation. That >being the case, the only possibility for 8deg217 would be w|s. OK.>> Also, the 217-ET 7 step symbol would need to become x|v to agree >with the>> rational complement of w|v. Alternatively the 3 step symbol could be >> changed to |w and the 7 step symbol could remain as w|x. But the >latter>> pair represent higher primes and introduce one more lateral >confusable. But>> at least it doesn't introduce 2 more like the old plan A, and I >still like>> the idea of not having a double-flag for 3 steps, when 4, 5 and 6 >are>> single flags. What do you think? >>This is the sequence that I favor: > >217: |v w| |w s| |x |s w|x w|s s|x s|s x|x x|s w|| ||w >s|| ||x ||s w||x w||s s||x s||sFine. Let's go with that, AND keep the rational system as I've described it. There are bound to be _many_ ETs where the complements _cannot_ agree with the rational ones. In this case they can, but we choose not to make them so, for other reasons. I think that in a case like this, where the symbol we pass over actually relates to lower primes, we should tell the user that this option exists. It may make more sense to use a double 5-comma in a situation such as a major third where the root already has a 5-comma down. I've updated Yahoo groups: /tuning-math/files/Dave/Adaptive... * [with cont.]>Except for w| <--> w||s and w|s <--> w|| (to avoid two flags on the >same side for 8 & 19deg217), all of these are rational complements. >In fact, except for |w and ||w, this is the same as the plan B >notation (with that nice sequence of two-flag symbols), and now that >wR is the complement of wL+xR, your argument for its use is a very >persuasive one. Another thing that I like about it is that, in the >sequence of the first five symbols, the flags alternate from one side >to the other, which will work to good effect in your adaptive JI >example (which would need to be updated). Neat.>> I've managed to fool myself into believing that various schemes >would work>> so many times only to discover later that they wouldn't, that I no >longer>> trust my own checking. >>True words of wisdom, and a good reason why one person working alone >would have been hard pressed to come up with this notation. Agreed.-- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4810 - Contents - Hide Contents Date: Tue, 14 May 2002 02:13:33 Subject: definitions of period, equivalence, etc. (was: Re: graham's linear temperament page) From: dkeenanuqnetau --- In tuning-math@y..., "monz" <monz@a...> wrote:> i still don't have a definition for "equivalence interval" or > "interval of equivalence", mainly because of my confusion over > exactly what it is and why it's different from the "interval > of periodicity". > > i think it's really best for me to stay out of it until you, > Graham, Paul, and Gene define exactly what all these terms mean. > then let me know, and i'll update the Dictionary.Graham, Paul and Gene, please let Monz know if you agree with the following. "interval of equivalence" = "equivalence interval" = "formal octave" is that interval (much larger than a unison) which, when it occurs between two pitches, we consider them to be, in some sense, (formally if not perceptibly) the same note. For most scales this is the octave 1:2, and when it is not the octave it is usually some other highly consonant interval such as the "tritave" 1:3. But the essential feature of the interval of equivalence in relation to definitions of scales and types of scales is that when we describe a scale we describe only the pitches that fall within a single interval of equivalence, and we leave it up to the instrument builder to decide the range of the instrument and therefore how many times (including fractions) the interval of equivalence should be repeated. "interval of periodicity" = "periodic interval" = "period" is that generator of a regular temperament (whether linear, planar, or n-dimensional) which generates the interval of equivalence all by itself. This means that the period is either equal to the interval of equivalence or fits into the interval of equivalence a whole number of times.

Message: 4811 - Contents - Hide Contents Date: Tue, 14 May 2002 02:37:28 Subject: Re: graham's linear temperament page From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> A linear temperament is fully described by a generator and equivalence > interval.I disagree. I'd be a lot closer to agreeing if you had said "a generator and a periodicity interval" (which can be a fraction of the equivalence interval), but I think you want to know the equivalence interval as well as the period. But what if there isn't an equivalence interval but there are still two generators; does it still make sense to call it a "linear" temperament. That's a tough one.>The mapping from generators to consonances is fixed. For > example, meantones are defined by the generator approximating 3:2 and four > generators approximating 5:4. The generator can have different sizes, > giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same > temperament class. 53-equal, however, isn't a meantone because the best > 5:4 is 8 fourths rather than 4 fifths.So the mapping defines a "temperament class" while specific generators define a "temperament (instance)". I like that. But what do we call it when we have different numbers of notes per equivalence interval (typically MOS) for the same temperament. I guess we just call them things like "12 note meantone" or "miracle-21".> Planar temperaments are like linear temperaments, but with two generators > and an equivalence interval. The difference between a generator and an > equivalence interval is only a matter of formalism.Again, substitute "period" for "equivalence interval" and "only a matter of formalism" for the definitions in my preceeding post.

Message: 4812 - Contents - Hide Contents Date: Tue, 14 May 2002 03:27:34 Subject: Re: graham's linear temperament page From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> I personally would like somebody to include the definitions of the > temperaments I'm supposed to be adding. It's been quite difficult to keep > up over the past year.I think it would be nice to include links to music in the various temperaments discussed. I have examples for Orwell and Wonder, and am working on something in Magic. I also have 126/125 and 64/63 if you want to add planar temperaments. Porcupine, Miracle and of course Meantone are out there--what else?

Message: 4813 - Contents - Hide Contents Date: Tue, 14 May 2002 03:32:04 Subject: Re: graham's linear temperament page From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> A linear temperament is generated by two "generating intervals" (or > generators for short), one of which is distinguished as the "interval > of periodicity" (or "period" for short) and the other is simply > called "the generator".One of them may or may not be an interval of peridoicity--there's no requirement to have one, it's simply always possible to express them in that way. The key fact is that a linear temperament is rank two--ie, it has two generators.

Message: 4814 - Contents - Hide Contents Date: Tue, 14 May 2002 03:35:12 Subject: definitions of period, equivalence, etc. (was: Re: graham's linear temperament page) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Graham, Paul and Gene, please let Monz know if you agree with the > following.It's fine, but I think we should be clear that a linear temperament does not require an interval of equivalence, of an octave or anything else.

Message: 4815 - Contents - Hide Contents Date: Tue, 14 May 2002 03:38:36 Subject: Re: graham's linear temperament page From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> But what if there isn't an equivalence interval but there are still > two generators; does it still make sense to call it a "linear" > temperament. That's a tough one.That's more or less my definition of a linear temperament, except I would add that the generators are mapped to from some subgroup of the positive rationals. Making an interval of equivalence part of the definition gives us one meantone for 2, one for 3/2, one for 3, one for 5/2, another for 5/3 and so forth. I don't like it, and don't plan to use it.

Message: 4816 - Contents - Hide Contents Date: Tue, 14 May 2002 05:36:14 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> I've updated > Yahoo groups: /tuning-math/files/Dave/Compleme... * [with cont.] > to show my latest proposal. > > Assuming you find it acceptable, the next major job before we go public is > to agree on the notation of the important ETs.What about for the important temperaments?

Message: 4817 - Contents - Hide Contents Date: Tue, 14 May 2002 11:57 +0 Subject: Re: graham's linear temperament page From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <abq076+gjf0@xxxxxxx.xxx> genewardsmith wrote:> I think it would be nice to include links to music in the various > temperaments discussed. I have examples for Orwell and Wonder, and am > working on something in Magic. I also have 126/125 and 64/63 if you > want to add planar temperaments. Porcupine, Miracle and of course > Meantone are out there--what else?I'd prefer to link to a "home page" for each temperament. I already have these in place for meantone, miracle, schismic and diaschismic. If you can supply your own pages for these other temperaments, I can link to them and you can handle the links to examples. I should do something for magic and I plan to tune up the multiple-29 soon as well. The 29&19 temperament that we're apparently calling Negri does look promising on its own terms and as a way of slicing up multiple-29, maybe even giving a planar temperament. Graham

Message: 4818 - Contents - Hide Contents Date: Tue, 14 May 2002 11:57 +0 Subject: Re: graham's linear temperament page From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <abpt98+61vd@xxxxxxx.xxx> dkeenanuqnetau wrote:> I disagree. I'd be a lot closer to agreeing if you had said "a > generator and a periodicity interval" (which can be a fraction of the > equivalence interval), but I think you want to know the equivalence > interval as well as the period.I should have said period. It can be fully described by the period and generator, although it can also be convenient to think in terms of an equivalence interval. If you put aside contorsion, the number of periods to an equivalence is the GCD of the generator mapping, but that's not something the definition should concern itself with. Unless we want to pronounce on whether temperaments with contorsion really count as temperaments. But to cover them you need to state the full mapping, in terms of both period and generator, anyway. In which case we still have to specify that the mapping should only involve integers.> But what if there isn't an equivalence interval but there are still > two generators; does it still make sense to call it a "linear" > temperament. That's a tough one.You can always call one of the generators an equivalence interval. There's really little difference as far as the mathematics is concerned. But I expect musicians will find it easier to think if the period as a generalisation of an octave and the generator as the generalisation of a fifth. They'll find it obvious that a fifth and an octave function differently in meantone. So that's how the simple definition should be framed, and mathematical caveats added later. The word "linear" does strongly suggest a single generator, and music theory usually assumes octave equivalence, sometimes implicitly. If we define linear temperaments as having 2 generators, people will get confused by the off-by-one error.> But what do we call it when we have different numbers of notes per > equivalence interval (typically MOS) for the same temperament. I guess > we just call them things like "12 note meantone" or "miracle-21".I think those are called "scales". These ones in particular being MOS or Well Formed.> Again, substitute "period" for "equivalence interval" and "only a > matter of formalism" for the definitions in my preceeding post.Those definitions look fine to me. Oh, and Joe, when I said "consonances" I meant something much closer to your definition of "consonance" than "concordance". Musical context is important for this. Graham

Message: 4819 - Contents - Hide Contents Date: Tue, 14 May 2002 11:40:18 Subject: Re: graham's linear temperament page From: monz> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, May 14, 2002 3:57 AM > Subject: [tuning-math] Re: graham's linear temperament page > > > Oh, and Joe, when I said "consonances" I meant something much closer to > your definition of "consonance" than "concordance". Musical context is > important for this.oops ... OK, it's been fixed. Definitions of tuning terms: linear temperamen... * [with cont.] (Wayb.) and what's "contorsion"? that's a new one for me! -monz

Message: 4820 - Contents - Hide Contents Date: Tue, 14 May 2002 22:47 +0 Subject: Re: graham's linear temperament page From: graham@xxxxxxxxxx.xx.xx monz wrote:> and what's "contorsion"? that's a new one for me!If you define 5-limit meantone using neutral thirds instead of fifths, you find a fifth is two generators and a major third is two generators. Because all consonances involve an even number of steps it has a contorsion of 2. Whether it counts as a new temperament or an odd way of mapping meantone isn't currently addressed by the definition. Graham

Message: 4821 - Contents - Hide Contents Date: Tue, 14 May 2002 23:38:12 Subject: definitions of period, equivalence, etc. (was: Re: graham's linear temperament page) From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> Graham, Paul and Gene, please let Monz know if you agree with the >> following. >> It's fine, but I think we should be clear that a linear temperamentdoes not require an interval of equivalence, of an octave or anything else. Agreed. But I'd prefer to see the definition assume an IoE at first, so as not to lose the musicians entirely, and then add that caveat at the end. The situation of no IoE is extremely rare, and this fact is why it makes sense to call them "linear" when mathematically they are more simply treated as rank 2. Can someone name a popular or historical scale/tuning that doesn't have an interval of equivalence? i.e. one where every pitch must be listed, over the entire compass of the instrument. Note that those with an IoE _can_ be treated mathematically as rank one, provided all arithmetic is modulo the IoE.

Message: 4822 - Contents - Hide Contents Date: Tue, 14 May 2002 23:42:52 Subject: Re: graham's linear temperament page From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> A linear temperament is generated by two "generating intervals" (or >> generators for short), one of which is distinguished as the "interval >> of periodicity" (or "period" for short) and the other is simply >> called "the generator". >> One of them may or may not be an interval of peridoicity--there's norequirement to have one, it's simply always possible to express them in that way. The key fact is that a linear temperament is rank two--ie, it has two generators. I agree, but point out that this is a fairly recent way of looking at them, or at least a particularly mathematical way. To a musician it is rank 1 where the arithmetic is modulo the interval of equivalence. If it is "always possible to express them in that way" then they will always want you to.

Message: 4823 - Contents - Hide Contents Date: Wed, 15 May 2002 06:44:08 Subject: Re: graham's linear temperament page From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I'd prefer to ignore the question of whether the IoE or the other > consonances _must_ be rational ratios, and thereby avoid a whole can > of worms associated with temperaments for inharmonic timbres.We could leave off the mapping, though that would make it murky in some respects. Another region of murk are degenerate cases--what if we start with related generators, and find that that the actual rank is less than the number of generators?

Message: 4824 - Contents - Hide Contents Date: Wed, 15 May 2002 21:13:01 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> > George, > > Regarding your suggestion of redefining the w| flag as 23 comma - 19 comma, > while I see no benefit in doing that, it made me realise that w| could be > defined as (19'-19) comma, (i.e. 722:729), in the same way that x| can be > defined as (11'-7). i.e. using the lowest prime limit possible. > > I think you suggested that x| should be defined as 11'-7 instead of 29 > comma, but you gave the ratio as 715:729 which is 13'-(11-5). 11'-7 is > 45056:45927.Sorry. I forgot about the 4095:4096 schisma (which needs a distinctive name of some sort). I meant to say that we should redefine the xL flag as the 13'-(11-5) comma (which is how I've been calculating everything involving that flag up to this point), and if that's a bit unwieldy, then we could call it the ~29 comma (if we can figure out how to pronounce "~" (how about "quasi").> Similarly x|v could be defined in lowest prime terms as (11'-7)+ (17'-17) > (1441792:1474767) instead of 23' (16384:16767), and s|x could be defined as > 5+7 instead of 13. Ultimately everything could be defined in termsof 5 7> 11 17 19, but what would be the point? > > Maybe we should not define _any_ symbol as being a _single_ comma, but > adopt the attitude that what we've done is produce a bunch of symbols each > of which may be used for a number of different commas/dieses very close > together in cents. The user can say precisely which of these she means, but > the consequences of not doing so are almost insignificant.That is very possibly what may need to be done in order to notate some troublesome ET's. Allow me to quote from an earlier message of yours (#4009): << When we look at the ETs where 5-comma + 7-comma =/= 13-comma (among those we intend to notate) we find in most cases that we only need to use two of the 3 commas in notating the ET. e.g. In 27-ET the 7-comma vanishes and we use the 5 and 13 comma symbols. In 50-ET the 5-comma vanishes and we use the 7 and 13 comma symbols. 37-ET is a case I'm not too sure about. Here we have the 5-comma being 2 steps, the 13 comma being 3 steps and the 7-comma vanishing. There is no prime comma within the 41-limit that is consistently equal to 1 step (11-comma is 2 steps, same as 5-comma). We could notate 1 step as 13-comma up and 5-comma down, but if we insist on single symbols, is it ok to use the 7-comma symbol to mean 13-comma - 5 comma? Or should we use the 19-comma symbol for one step, even though it's 1,3,p-inconsistent? >> I suggest that 37-ET be notated as a subset of 111-ET, with the latter having a symbol sequence as follows: 111: w|, s|, |s, w|s, s|s, x|s, w||, s||, ||s, w||s, s||s. However, a more difficult problem is posed by 74-ET, and the idea of having redefinable symbols may be the only way to handle situations such as this. Should we do that, then there should probably be standard (i.e., default) ratios for the flags, and the specific conditions under which redefined ratios are to be used should be identified.> Someone who wants to notate a strictly rational 29-limit scale, and is > willing to use multiple symbols, will define x| as the 29 comma and x|s as > the 13' comma, while someone notating an 11-limit temperament will define > x| as 11'-7 and x|s as apotome-(5+7). > > Maybe we should produce a list that shows all the possible 19-limit > interpretations of each of the 20 single-shaft symbols, plus the obvious > 41-limit interpretations. But mostly all you want to know are > (a) the simplest interpretation, i.e. the one that involves the fewest > primes, and > (b) any interpretations that have a lower prime limit than the simplest one.I think that we'll get more of a feel for this once we start trying to determine symbol sequences for various ET's. There is something else I would like to quote from one of your earlier messages (#4188) that demonstrates a point that I would like to make: << [GS:] By the way, it's been bugging me that we've yet to agree on the spelling of confusable vs. confusible. I finally looked up the - able (etc.) [DK:] Unfortunately I find "confusability" and not "confusibility" in my Shorter Oxford. >> I guess we should consider the English to be best authority on how to spell English words and settle on "confusability". (Besides, even if this is merely a difference between English vs. American usage, since you were the one who first used the word in the present discussion, your preference should then take precedence.) << [GS:] Then I think that we should decide on standard (or preferred) sets of symbols for as many ET's as we can before doing this [taking the notation to the main tuning list]. [DK:] What would be even better is, after doing a few very different ones the hard way, and therefore thinking about what the issues are, if we could simply give an algorithm for choosing the notation for any ET. >> I tried selecting sets of symbols (including complements) for a number of ET's and came to the conclusion that it is not all that obvious what is best. Among the possible objectives I identified are: 1) Consistent symbol arithmetic (a top priority); 2) A matching symbol sequence in the half-apotomes; 3) Choose flags that represent the lower prime numbers; 4) Try not to use too many different types of flags; 5) Use rational complements where possible. In the same way that a difference of opinion occurs among experts or authorities in the matter of English spelling (as with the word "confusability"), a problem could result when different composers, using the same rules and guidelines, arrive at different sets of symbols for the same ET. Some composers won't want to use sagittal notation if in involves puzzling with how to notate an ET and uncertainty about the suitability of the outcome, say if, after composing a piece in a certain ET, it turns out that others were already using a different set of symbols. I suspect that, in order for us to figure out how the rules should be applied, we'll have to do all of the ET's anyway. So why not just do as many as possible and include the symbol sequences along with the specifications of the notation? Notice that in doing 111 (above), I found that giving objectives 2 and 4 a higher priority than objective 5 gave me the simplest notation. One thing that I thought should be taken into consideration is that, where appropriate, ET's that are subsets of others should make use of a subset of symbols of the larger ET. This would especially be advisable for ET's under 100 that are multiples of 12 -- if you learn 48-ET, you have already learned half of 96-ET. I previously did symbol sets for about 20 different ET's, but that was before the latest rational complements were determined, so I'll have to review all of those to see what I would now do differently.> ------------- > ASCII symbols > ------------- > I propose we start using the following more representational ASCII versions > of our symbols in place of the "svwx|" versions (although I hope that later > we can agree on a single-character-per-symbol version).Okay, we'll try it and see how it works out. --George

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