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Message: 4825 - Contents - Hide Contents Date: Wed, 15 May 2002 23:11:15 Subject: Re: graham's linear temperament page From: monz> From: "emotionaljourney22" <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, May 15, 2002 2:36 PM > Subject: [tuning-math] Re: graham's linear temperament page > > >> Definitions of tuning terms: linear temperamen... * [with cont.] (Wayb.) >> hi joe, sorry if i seemed mean before . . . hope you can forgive me!no sweat, paul -- i'm used to your style by now. :) consider it water under the bridge. the main thing is that i want the Dictionary to be as correct as possible.> it's just that i've done a lot of work trying to help you correct > your et definition and the pages it links to, and you, quite > understandably, have had little time to consider my many e-mails, > posts, and IM comments to you on these issues. hope you've at least > saved my e-mails, and made note of the relevant posts, for later > consideration -- that's all i can ask. maybe in 10 years we can get > back to this -- i'd be very happy with that.well, i hope it doesn't take 10 years! but right now i am kind of tied up with other projects, and have pretty successfully weaned myself off the tuning list(s) addiction(s). i'm just checking in these days and fighting the urge to reply to everything.> > anyway, on the current topic, it seems you missed an important fix > that everyone here agrees on -- please replace "interval of > equivalence" with "period" in your definitions.ok, look at the linear temperament definition now. NOW -- regarding "equivalence interval" and "periodicity interval", you'll all see that i provided links to definitions for those two terms, but still don't have any content in them. that's because i'm *still* not clear on how to define them. can someone(s) *please* post simply a definition for each of them which i can put into those Dictionary entries. if others disagree, then i'll try to keep updating the webpages the way i've done with "linear temperament" -- but please give me something to start with. -monz

Message: 4834 - Contents - Hide Contents Date: Wed, 15 May 2002 00:05:04 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: >>> 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 Iwould 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. OK. I agree with Gene and Graham. Although it doesn't make any _sense_ to call it linear when there is no IoE, we will anyway. If only because it _can_ be treated the same mathematically (rank two) whether there is an IoE or not. But I wouldn't want a definition to be based on that since having no IoE is so rare. 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.

Message: 4835 - Contents - Hide Contents Date: Wed, 15 May 2002 04:10:04 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4242]:> 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. That's terrific!I made an updated version of your complementation worksheet: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/DKComp2.xls I removed a few no-longer-relevant rows and also added some, mostly at the bottom. I also, added (in column K) the number of degrees corresponding to an apotome minus column B, which will help in selecting symbol sets for various ET's.> 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.)> 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.) Now it looks as if we will need a |vv symbol for the complement of w|s. (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. 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.> 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.> 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||s 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).>> 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.Yes, you have. It was just another possibility that I wanted to check out before wrapping this up.> But you should go over my proposal with a fine-toothed comb.And I found only the one problem with the 23' comma.> 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. --George

Message: 4836 - Contents - Hide Contents Date: Thu, 16 May 2002 02:20:22 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >>> 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. >> hmm . . . is there going to be transparancy for cases like 76-equal, > which gives you so many good linear-temperament systems within it -- > can the notation show that? how about 152-equal, in which most of the > linear-temperament systems use different approximations to the primes > from what 152-equal as a whole would suggest?No. The whole basis of the notation is the chain of approximate fifths. If two temperaments available within a single ET use different sized fifths then how could they possibly be covered by a single notation for the ET. You have already seen, in your adaptive JI example, how 31-ET _notation_ cannot continue to exist within 217-ET, despite the fact that 31-ET exists within it. The quarter-commas become explicit instead of implicit. In exactly the same way, the 1/3 commas must become explicit in the notation for 152-ET. The native best-fifth of 76-ET is not suitable to be used a notational fifth because, among other reasons, it is not 1,3,9-consistent (i.e. its best 4:9 is not obtained by stacking two of its best 2:3s) and I figure folks have a right to expect C:D to be a best 4:9 when commas for primes greater than 9 are used in the notation. So 76-ET will be notated as every second note of 152-ET. Here's my proposal for 152-ET. Steps Symbol ------------ 1 )| 2 |~ 3 /| 4 |\ 5 ~|) 6 (|~ 7 /|\ 11 B:C, E:F 15 # 26 A:B, C:D, D:E, F:G, G:A Although it seems a minor problem that the 1/3 comma symbol of 152-ET is smaller in rational terms than the 1/4 comma symbol of 217-ET. We'll see what George comes up with for 152-ET.

Message: 4837 - Contents - Hide Contents Date: Thu, 16 May 2002 21:29:12 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote [#4274]:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >> Sorry. I forgot about the 4095:4096 schisma (which needs a >> distinctive name of some sort). >> How about "the 13-schisma" or the "tridecimal schisma".That sounds good. We should probably propose that term on the main tuning list, to see if anyone knows whether it has already been used for a different schisma.>> 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"). >> "Quasi" is fine, but (11'-7) is also a quasi-29-comma, so you can't > call (13'-(11-5)) _the_ ~29 comma.Since the (| flag is undoubtedly going to be used so much more often in connection with ratios of 11 and 13 -- as (|) and (|\ -- than for ratios of 29, I would prefer to keep its standard definition as other than 256:261 (the 29 comma). I would also prefer the 13'-(11-5) ratio to (11'-7) because, 1) The numbers in the ratio are smaller (715:729 vs. 45056:45927); and 2) The 13'-(11-5) comma (33.571 cents) is much closer in size to the 29 comma (33.487 cents) than is the (11'-7) comma (33.148 cents).>> I suggest that 37-ET be notated as a subset of 111-ET, with the >> latter having a symbol sequence as follows: >> Yes. That's also what I suggested in a later message (4188). >>> 111: w|, s|, |s, w|s, s|s, x|s, w||, s||, ||s, w||s, s||s. >> And that's almost the notation I proposed in the same message (with > its implied complements), except that I would use x|x (|) as the > complement of s|s /|\. Surely that is what you would want too, since > it represents a lower prime and is the rational complement?I used x|s (|\ as 6deg111 because x|x (|) calculates to 5deg111 and, in addition, 26:27 is closer in size to 6deg111 than is 704:729. However, if we think that there should be no problem in redefining x|x as 6deg111 (as it would seem to make more sense), then so be it!>> 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. >> I think 74-ET is garbage.Be careful when you say something like that around here -- do you remember my "tuning scavengers" postings?> But if someone insisted ... Due to its lack of 1,3,9 consistency and > the same going for 2*74 = 148, it would need to be notated as every > third note of 3*74 = 222-ET which is itself garbage and we can't > notate it anyway. We have no 11 step symbol for it without two flags a > side. > > I don't think we need to apologise for failing to notate 74-ET. We can > do every ET up to 72 and many useful ones beyond.The problem is not the fault of the notation so much as the weirdness of the division -- I hesitate to call it a tonal system. Any systematic notation is going to have problems with 74-ET.>> I think that we'll get more of a feel for this once we start trying >> to determine symbol sequences for various ET's. > > Yes. >>> 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. > > I agree.Yes, especially since we've just had an object lesson with 111-ET.> We'll just have to do them all individually. I can't imagine > there being much disagreement on those using their native fifths until > we get up to 38-ET. See my message 4188. The complements are implied > by them having the same sequence of flags in the second half apotome, > and the complement of /|\ always being itself or (|).We'll each work them out then and compare notes.>> 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. >> If you're talking about (|\ as the complement of /|\ then I must > disagree. In most ETs that use /|\ its complement would be itself or > (|) so I think (|) should be exempt from the consideration of too many > flag types.Yes, I agree with that. You'll notice that this wasn't among the reasons I gave above. Would you also now prefer my selection of the /|) symbol for 6deg111 to your choice of (|~ on the grounds that it is a more commonly used symbol, particularly in view of the probability that you might want to use (|\ instead of )|| or ||( for 9deg as its complement?>> 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. >> Certainly. It's only the question of how we tell "when appropriate" > that remains to be agreed. I've proposed two and only two reasons in > message 4188. You might say what you think of these.They sound reasonable enough. Until I thought of 7-ET, which seems to be a "natural" for the 7 naturals. Of course, a simple way around that is to put the modifying symbols from 56-ET into a key signature, a solution that would keep the manuscript clean and make everybody happy.>> 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.Here's what I did a couple of weeks ago for some of the ET's (in order of increasing complexity): 12, 19, 26: s||s 17, 24, 31: s|s s||s 22: s| ||s s||s 36, 43: |x ||x s||s 29: w|x w||v s||s 50: w|w x|s s||s 34, 41: s| s|s ||s s||s 27: s| x|s ||s s||s 48: |x s|s ||x s||s 46, 53: s| s|s x|x ||s s||s 58, 72: s| |s s|s s|| ||s s||s (version 1 -- simpler, but more confusability) 72: s| |x s|s ||x ||s s||s (version 2 -- more complicated, but less confusability) 58: s| w|x s|s w||v ||s s||s (version 2 -- more complicated, but less confusability) 96: s| |x |s s|s s|| ||x ||s s||s (version 1 -- simpler, but more confusability) 96: s| |x w|s s|s w|| ||x ||s s||s (version 2 -- more complicated, but less confusability) 94: w| s| w|s s|s x|x w|| ||s w||s s||s 111 (37 as subset): w| s| |s w|s s|s x|s w|| s|| ||s w||s s||s 140: |v |w s| |s s|w s|x s|s x|s ||w s|| ||s s||w s||x s||s 152: |v |w s| |s s|w s|x s|s x|x x|s ||w s|| ||s s||w s||x s||s 171: |v w|v s| |x |s w|s s|x s|s x|s w||v s|| ||x ||s w||s s||x s||s 183: |v w|v s| |x |s w|s s|x s|s x|x x|s w||v s|| ||x ||s w||s s||x s||s 181: |v w| w|v s| |s w|x w|s s|x s|s x|x w|| w||v s|| ||s w||x w||s s||x s||s 217: |v w| |w s| |x |s w|x w|s s|x s|s x|x x|s w|| ||w s|| ||x ||s w||x w||s s||x s||s I changed 217 to conform to our new standard set. As I said, I may want to change some of these in light of the new rational complements and to remedy (if possible) an inconsistency in symbol arithmetic that may be lurking somewhere. You will want to compare some of these with what you have in your message 4188. We did not even agree on something as simple as 31-ET. --George

Message: 4838 - Contents - Hide Contents Date: Thu, 16 May 2002 02:47:24 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau On second thoughts, here's my revised proposal for 152-ET. There were too many different flags in the previous one. Steps Symbol ------------ 1 )| 2 |~ 3 /| 4 |\ 5 /|~ 6 (|~ or //| 7 /|\ 11 B:C, E:F 15 # 26 A:B, C:D, D:E, F:G, G:A

Message: 4839 - Contents - Hide Contents Date: Thu, 16 May 2002 21:38:38 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote [#4283]:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote > [#4274]:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >>>>> 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. >> >> I agree. >> Yes, especially since we've just had an object lesson with 111-ET.I meant to say 152-ET.> ... > Would you also now prefer my selection of the /|) symbol for 6deg111 > to your choice of (|~ on the grounds that it is a more commonly used > symbol, particularly in view of the probability that you might want > to use (|\ instead of )|| or ||( for 9deg as its complement?Here, again, I meant to say 6deg152. Sorry if I caused any confusion. --George

Message: 4840 - Contents - Hide Contents Date: Thu, 16 May 2002 03:36:50 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: > Sorry. I forgot about the 4095:4096 schisma (which needs a > distinctive name of some sort).How about "the 13-schisma" or the "tridecimal schisma".> 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")."Quasi" is fine, but (11'-7) is also a quasi-29-comma, so you can't call (13'-(11-5)) _the_ ~29 comma.> I suggest that 37-ET be notated as a subset of 111-ET, with the > latter having a symbol sequence as follows:Yes. That's also what I suggested in a later message (4188).> 111: w|, s|, |s, w|s, s|s, x|s, w||, s||, ||s, w||s, s||s.And that's almost the notation I proposed in the same message (with its implied complements), except that I would use x|x (|) as the complement of s|s /|\. Surely that is what you would want too, since it represents a lower prime and is the rational complement?> 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.I think 74-ET is garbage. But if someone insisted ... Due to its lack of 1,3,9 consistency and the same going for 2*74 = 148, it would need to be notated as every third note of 3*74 = 222-ET which is itself garbage and we can't notate it anyway. We have no 11 step symbol for it without two flags a side. I don't think we need to apologise for failing to notate 74-ET. We can do every ET up to 72 and many useful ones beyond.> I think that we'll get more of a feel for this once we start trying > to determine symbol sequences for various ET's. Yes. > 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.I agree. We'll just have to do them all individually. I can't imagine there being much disagreement on those using their native fifths until we get up to 38-ET. See my message 4188. The complements are implied by them having the same sequence of flags in the second half apotome, and the complement of /|\ always being itself or (|).> 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.That's an excellent list of (often conflicting) criteria.> 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.Yes. A very good point> 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? OK. > 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.If you're talking about (|\ as the complement of /|\ then I must disagree. In most ETs that use /|\ its complement would be itself or (|) so I think (|) should be exempt from the consideration of too many flag types.> 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.Certainly. It's only the question of how we tell "when appropriate" that remains to be agreed. I've proposed two and only two reasons in message 4188. You might say what you think of these.

Message: 4842 - Contents - Hide Contents Date: Thu, 16 May 2002 12:04 +0 Subject: Re: graham's linear temperament page From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <000f01c1fca0$7d449ca0$af48620c@xxx.xxx.xxx> monz wrote:> ok, look at the linear temperament definition now. Fancy that! > NOW -- regarding "equivalence interval" and "periodicity interval", > you'll all see that i provided links to definitions for those two > terms, but still don't have any content in them. that's because > i'm *still* not clear on how to define them. can someone(s) > *please* post simply a definition for each of them which i can > put into those Dictionary entries. if others disagree, then > i'll try to keep updating the webpages the way i've done with > "linear temperament" -- but please give me something to start with.Dave gave some, and I didn't notice anybody disagreeing. The only quibble I have is that the period definition uses the terminology differently to the linear temperament one. And that it's circular, but that's probably unavoidable. """ "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. """ Graham

Message: 4843 - Contents - Hide Contents Date: Thu, 16 May 2002 12:04 +0 Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <abul01+9p4m@xxxxxxx.xxx> Dave:>> Note that those with an IoE _can_ be treated mathematically as rank >> one, provided all arithmetic is modulo the IoE. Paul:> i used to think so, but it seems gene was able to convince me > otherwise. i think that you can't handle torsion properly unless you > express the unison vectors in IoE-specific, rather than IoE- > equivalent/IoE-invariant terms.I've never been convinced of this. Every time it comes up people say they aren't interested enough to prove it either way. So you should state it as unproven and not use it to dismiss other ideas. The biggest outstanding problem is that we don't have an algorithm for calculating the optimal generator size for a given mapping and target consonances. But then nobody's seriously looked. There is a specific problem with contorsion, as different generator sizes give different, but equally valid results. But hey, we're not counting contorsion anyway. Adding a range of generator sizes to the definition solves this one if there's no other way. There's also the detail that IoE-equivalent algebra gives a period-equivalent result, but you can get round that if you know the (IoE equivalent) sizes of the intervals you're approximating. The real problem here is that the IoE-equivalence doesn't make it any simpler if you won't accept period-equivalence. I'm sure the torsion problem will go away when somebody looks at it properly. It doesn't matter anyway if we're only considering linear temperaments, because they don't have to be derived from unison vectors. Graham

Message: 4844 - Contents - Hide Contents Date: Thu, 16 May 2002 21:42:34 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >Here's what I did [for 152-ET] a couple of weeks back, and after looking at the >rational complements, I would still do it this way: > >Steps Symbol >------------ > 1 |( > 2 |~ > 3 /| > 4 |\ > 5 /|~ > 6 /|) > 7 /|\ > 8 (|) > 9 (|\ >10 ||~ >11 /|| >12 ||\ >13 /||~ >14 /||) >15 /||\ > >Something that you will notice immediately is that I have used the >(17'-17) comma as 1 degree. (In 152 it calculates to zero degrees >and would be unusable unless it were redefined as I have chosen to do >here.)But redefined it as what comma? I believe a fundamental tenet of this whole excercise, one that many people agree on, is that an accidental must never simply represent a number of steps of the ET, but must represent a rational comma in a manner consistent with the ETs best approximation of the primes involved.>Dave, your solution also redefines a flag, although it is not so >obvious: since |~ is 2deg and (|~ is 6deg, then (| must be 4deg. >This is so if it is calculated as the 29 comma, but it is 5deg if it >is calculated as the 715:729 comma, as I have done. (But this is not >the redefinition to which I refer.)That was unintentional. Thanks for spotting it. I now agree that using (|~ for 6 steps is completely wrong. It corresponds to 7 steps (but should not be used). The only possible symbols for 6 thru 9 steps are the ones you have given. And there's no question about 3 and 4 steps either. (And therefore also 11, 12, 14, 15). I've (re-)realised that there is no need to go beyond the 19-prime-limit for notating any ETs (that we _can_ notate). So, for notating ETs, all the symbols must be given their 19-limit definitions. The fact that some of them might also be 23, 29 or 31 commas, when used for rational scales, is utterly irrelevant (for notating ETs). Then the only remaining ambiguity is the one involving the 13-schisma. For notating ETs: v| is always 512:513 |v is always 288:289 w| is always 2176:2187 |w is always 722:729 s| is always 80:81 |s is always 54:55 |x is always 63:64 but x| can be either 45056:45927 or 715:729 In some ETs these will be the same number of steps and the choice doesn't need to be made. But when the choice _is_ made, the meaning of (|( (|~ (|\ and (|) all follow automatically from it. I now believe that the rational complements beyond the 19-limit (i.e. if either of the pair is outside the 19-limit in the rational conception) are very unimportant for notating ETs, and would only be used as a tie-breaker if all else fails. Here are the valid options for 1, 2 and 5 steps of 152-ET, from a 19-limit perspective. Steps Symbol Comma Comment ----------------------------------------------------------------------- 1 )| 19 1 )|( 19 + (17'-17) 2 ~| 17 2 ~|( 17' 2 |~ 19'-19 5 (| (11'-7) 5 or 4 (| 13'-(11-5) (5 steps for 1:13, 4 steps for 3:13) 5 )|\ 19+(11-5) 5 ~|) 17 + 7 5 /|~ 5+(19'-19) 5 (|( (11'-7)+(17'-17) 5 or 4 (|( (13'-(11-5))+(17'-17) (5 steps for 1:13, 4 steps for 3:13) So we see that 152-ET is not 1,3,13-consistent. I believe that, if for any prime p, the ET is not 1,3,p-consistent, then commas involving that prime should not be used for notational purposes unless there's no other option. I also think that, if possible, all notational commas should be mutually consistent and consistent with 1,3 and 9. And if the ET can be notated by using more than one such set (unlikely), then we should use the one with the lowest maximum error in its intervals. So here's the information about 152-ET that I find most relevant for deciding the notation. These are its maximal consistent sets of odds in the 19-limit, along with the maximum error of any interval in the set. By maximally consistent I mean that no other 19-limit odd number can be added to a set without making it inconsistent. {1, 13, 17} 3.7 c {1, 3, 5, 9, 11, 15, 17} 3.7 c {1, 3, 5, 7, 9, 11, 15, 19} 2.5 c I haven't listed any sets that do not include 1, because (a) I haven't computed them, and (b) they would only be relevant if all else fails, which seems very unlikely. The first set does not contain 3 or 9. The second set does not provide any way of notating a single step. The third step is just right, and Goldilocks ate it all up. The third set works beautifully and happens to have the lowest error. It says we shouldn't use any 13 or 17 commas, so our choice for 1, 2 and 5 steps is reduced to just these. Steps Symbol Comma ------------------------- 1 )| 19 2 |~ 19'-19 5 (| (11'-7) 5 )|\ 19+(11-5) 5 /|~ 5+(19'-19) None of the choices for 5 introduce any new flags, but I consider the introduction of a new flag prior to the half-apotome to be nearly as bad. So on that basis I reject (|. Also it seems like it is good to have more equal numbers of single and double-flag symbols. They are both 19-limit. None of them are the rational complement of |~. I choose /|~ because its size in a rational tuning is closer to 5/152 octave than is )|\. So here's the full set for 152-ET. 1 )| 2 |~ 3 /| 4 |\ 5 /|~ 6 /|) 7 /|\ 8 (|) 9 (|\ or )|| ? 10 ||~ 11 /|| 12 ||\ 13 /||~ 14 /||) 15 /||\ We now only disagree on the 1 step symbol.>You didn't give symbols for 8 and 9deg, but if I assume that 8deg >would be (|), so then |) would be 4deg. In 152 |) calculates to >3deg, which is where your redefinition occurs.As I say, this wasn't an intentional redefinition, it was just dumb.>So the difference between our two solutions is that the flag that I >redefined is associated with a higher prime.There is no need to redefine any, for 152-ET. And only ever a need to redefine x|.>May I assume that you would use matching symbols for the apotome >complements? Yes. > That being the case, we both chose a rational >complement for 1deg, but a 152-specific complement for 2deg.Now I choose no rational complement for either of them. It seems that the _only_ justification for using |v for 1 step is that it is the rational complement of s|x. To my mind this is not sufficient justification to violate the definition of |v as the 17 comma 2176:2187. In fact I don't think anything could be sufficient justification for that.>I came >to the conclusion that a simple (i.e., easy-to-remember) sequence of >symbols is more important than using rational complements.So did I. And I came to the conclusion that 19-limit-comma flag-definitions are more important than using rational complements. I don't see how changing 1 step from )| to |( improves ease of remembering. In fact with )| we have that property that you admired in 217-ET, that the flags alternate sides as you go up. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4845 - Contents - Hide Contents Date: Thu, 16 May 2002 08:29:23 Subject: updated Tuning Dictionary definitions From: monz> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, May 16, 2002 4:04 AM > Subject: [tuning-math] Re: graham's linear temperament page > > > In-Reply-To: <000f01c1fca0$7d449ca0$af48620c@xxx.xxx.xxx> > monz wrote: > >>> NOW -- regarding "equivalence interval" and "periodicity interval", >> you'll all see that i provided links to definitions for those two >> terms, but still don't have any content in them. that's because >> i'm *still* not clear on how to define them. can someone(s) >> *please* post simply a definition for each of them which i can >> put into those Dictionary entries. if others disagree, then >> i'll try to keep updating the webpages the way i've done with >> "linear temperament" -- but please give me something to start with. >> Dave gave some, and I didn't notice anybody disagreeing. The only quibble > I have is that the period definition uses the terminology differently to > the linear temperament one. And that it's circular, but that's probably > unavoidable. > > <Dave's definitions snipped -- look in the Dictionary!>thanks much, Graham! actually, i saw both of those the other day (in fact, i think Dave posted them in response to my own query), but as i've been keeping my distance from the tuning lists, i already lost track of them. the definitions have been updated: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) now, doesn't the "planar temperament" definition also need to be fixed to say "periodicity interval" instead of "equivalence interval"? Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) -monz

Message: 4846 - Contents - Hide Contents Date: Thu, 16 May 2002 17:12 +0 Subject: Re: updated Tuning Dictionary definitions From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <000b01c1fcee$75f9b680$af48620c@xxx.xxx.xxx> monz wrote:> now, doesn't the "planar temperament" definition also need to > be fixed to say "periodicity interval" instead of "equivalence > interval"? > > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)What? Yes, they're like linear temperaments but with one extra generator. And you could also say that the process can be extended to any number of generators (however you count them). Hopefully, anybody doing this will mention "planar temperament" somewhere. Graham

Message: 4847 - Contents - Hide Contents Date: Thu, 16 May 2002 20:00:46 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote [#4272]:> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >>>>> It seems to me that the notation for a linear temperamentshould 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. >>>> hmm . . . is there going to be transparancy for cases like 76- equal, >> which gives you so many good linear-temperament systems within it -- >> can the notation show that? how about 152-equal, in which most of the >> linear-temperament systems use different approximations to the primes >> from what 152-equal as a whole would suggest? >> No. The whole basis of the notation is the chain of approximate > fifths. ... > > Here's my proposal for 152-ET. >[This has been deleted and replaced with:]>--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote [#4273]:> On second thoughts, here's my revised proposal for 152-ET. There were > too many different flags in the previous one. > > Steps Symbol > ------------ > 1 )| > 2 |~ > 3 /| > 4 |\ > 5 /|~ > 6 (|~ or //| > 7 /|\ > 11 B:C, E:F > 15 # > 26 A:B, C:D, D:E, F:G, G:A > > Although it seems a minor problem that the 1/3 comma symbol of 152- ET > is smaller in rational terms than the 1/4 comma symbol of 217-ET. > We'll see what George comes up with for 152-ET.Here's what I did a couple of weeks back, and after looking at the rational complements, I would still do it this way: Steps Symbol ------------ 1 |( 2 |~ 3 /| 4 |\ 5 /|~ 6 /|) 7 /|\ 8 (|) 9 (|\ 10 ||~ 11 /|| 12 ||\ 13 /||~ 14 /||) 15 /||\ Something that you will notice immediately is that I have used the (17'-17) comma as 1 degree. (In 152 it calculates to zero degrees and would be unusable unless it were redefined as I have chosen to do here.) Dave, your solution also redefines a flag, although it is not so obvious: since |~ is 2deg and (|~ is 6deg, then (| must be 4deg. This is so if it is calculated as the 29 comma, but it is 5deg if it is calculated as the 715:729 comma, as I have done. (But this is not the redefinition to which I refer.) You didn't give symbols for 8 and 9deg, but if I assume that 8deg would be (|), so then |) would be 4deg. In 152 |) calculates to 3deg, which is where your redefinition occurs. So the difference between our two solutions is that the flag that I redefined is associated with a higher prime. May I assume that you would use matching symbols for the apotome complements? That being the case, we both chose a rational complement for 1deg, but a 152-specific complement for 2deg. I came to the conclusion that a simple (i.e., easy-to-remember) sequence of symbols is more important than using rational complements. --George

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