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Message: 5150 - Contents - Hide Contents

Date: Mon, 19 Aug 2002 15:26:24

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote [#4586]:

> ... The foregoing was written before you pointed out that (|( is the true > 11'-5 and 13'-7 comma in your latest message. In light of this, I > would still assign ~|\ as the 23' comma, while making (|( a standard > symbol with rational complement ~||(, thereby eliminating /|~ from > the picture. (I was also using /|~ for 17/11 as Ab\!~ or A\!!!~, but > I'll see how well (|( works later.)
I checked how well (|( would work for 17/11. The commas we now have for (|( are: 11'-5 comma (44:45, ~38.906c) 13'-7 comma (1664:1701, ~38.073c) The comma for 17/11 is intermediate in size, so it works just fine: 11-17' comma (1377:1408, ~38.543c) So with 1/1 as C, 17/11 will be Ab(!( or A(!!!(. --George
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Message: 5152 - Contents - Hide Contents

Date: Fri, 23 Aug 2002 19:31:24

Subject: Re: no specific pattern of intervals

From: wallyesterpaulrus

--- In tuning-math@y..., <Josh@o...> wrote:
> Hi. > > I'm new here. > > I'm looking for an octave-equivalent > scale tuning system consisting of step sizes > no larger than 400 cents in which there is > no discernible mathematical pattern. > > How is this best acheived?
random number generator?
> Should I just make a list of available cent > approximations and then mark them off as I > find patterns they might reflect until I > get down to 5 tones?
not sure what you have in mind, but go for it!
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Message: 5153 - Contents - Hide Contents

Date: Fri, 23 Aug 2002 19:31:36

Subject: question for gene

From: wallyesterpaulrus

Yahoo groups: /tuning/message/38834 * [with cont.] 


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Message: 5155 - Contents - Hide Contents

Date: Sat, 24 Aug 2002 12:04:03

Subject: Re: Riemann Zeta Function and Tuning Systems...

From: genewardsmith

--- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...> wrote:
> Hello, > > Would like to stimulate further discussion on this posting from the > sci.math newsgroup. I am especially interested in the part that talks > about "streching or shrinking the octave to the nearest Gram point". > What does this mean exactly? I have also included, at the bottom, a > short explanation that I received from Gene a couple days ago.
Hi, Paul; I'm some minor computer problems so I'm afraid I'm a little tardy in my reply. Probably the best thing to do would be to take a look at my tuning-math postings on this first, and then go from there. The Gram point business arises because the Gram points are easily computed and close to the critical values of Z(t) in question. Here is something to start the discussion off with: Yahoo groups: /tuning-math/message/879 * [with cont.] Yahoo groups: /tuning-math/message/894 * [with cont.] Yahoo groups: /tuning-math/message/946 * [with cont.] As you can see, there is much, much more going on on this list than the Riemann Zeta function discussion, which I commend to your attention if you are interested in the musical aspect of all this.
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Message: 5156 - Contents - Hide Contents

Date: Mon, 26 Aug 2002 09:41:00

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Sorry for the long delay in replying.

At 11:56 AM 8/17/2002 +0000, George Secor wrote:
>In working out a spreadsheet to automatically assign the symbols for >ETs, one of the criteria I am using is to select ones that eliminate >(or at least minimize) the inconsistencies. This can get not only >complicated, but tricky.
Indeed. Good on you for doing this!
>So it looks like this will be the 217 standard set: > >217: |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~||( >/|| ||) ||\ (||( //|| /||) /||\ (new RCs)
Looks OK to me.
>> Reason enough to reject 282-ET as what? Reject it as a good way of >> having a fully notatable closed system that approximates 29-limit JI? >> I seriously disagree. It just means that we should use (| and |) with >> their non-13 meanings in 282-ET. >
>I guess I didn't get my point across. I want to be able to use a >large-numbered ET (217 or 282 or whatever) to notate *JI* when there >are no suitable rational symbols that will do the job. If (| or |) >don't have 13 meanings in 282, then there cannot be a good transition >between the rational notation and the large-ET notation -- symbols >would have to be converted from one to the other should a JI >composition suddenly require 282-ET symbols. This problem is minimized >with 217, because even the non-standard symbols such as )| and (| can >be kept, because they are all the correct number of degrees.
I see what you mean.
>>> 3) The following rational complements for the 15-limit symbols are >not
>>> consistent in 282: >>> >>> )|~ <--> (|| 19' comma >>> |( <--> /||) as 7-5 comma or 11-13 comma (but 17-17 is okay) >>> ~| <--> //|| 17 comma >>> |) <--> ||) 7 comma >>> //| <--> ~|| 25 comma >>> (| <--> )||~ 11'-7 comma >>> >>> And besides this, there are others that are inconsistent, such as: >>> >>> |~ <--> ~||) as both the 19-19 and 23 comma >>
>> All this means is that maybe we should consider making our rational >> complements consistent with 282-ET rather than 217-ET. >>
>>> What makes 217 so useful is that *everything* is consistent to the >19
>>> limit, and, except for 23, to the 29 limit. >>
>> I don't know what you mean by *everything* here. Isn't 282-ET >> consistent to the 29-limit with no exceptions? >
>It isn't consistent with the schismas that are essential to the >rational notation: > >1) The 5 comma /| (5deg) plus the 7 comma |) (6deg) doesn't equal the >13 comma /|) (12deg); this is the 4095:4096 schisma, ~0.423c. So you >can't notate ratios of 7 that are consistent with ratios of 13 in 282. > >2) The 17'-17 comma (2deg) doesn't equal the 7-5 (1deg), or put >another way, |) <> /|(; this is the 163840:163863 schisma, ~0.243c. So >you can't notate ratios of 17 that are consistent with ratios of 7 and >13 in 282. > >Or should we discard these and start over -- I think I would then be >entitled to say that you have either a 288-bias or an anti-217 bias.
OK. I understand now. Yes we definitely have a 217-ET bias (or rather a bias toward systems whose fifth is close to that of 217-ET, like 494) in the sense that we are only using schismas that vanish (I think we've been overloading or overusing the term "consistent") in 217-ET. And it may well be possible to start completely from scratch and build a different system where we only use sub-cent (or sub-half-cent) schimas that vanish in 282-ET. Then we'd have a 282-ET bias (not anti 217-ET). But then the 282-ET fifth _is_ closer to the precise 2:3 that the system is supposedly based on. This is a daunting prospect, having come this far with the current system. But wouldn't it be terrible if there was a _better_ system waiting to be discovered, based on 282-ET schismas, and we passed it over? Perhaps you can come up with a simple argument as to why this is not possible, short of a complete investigation?
>> I mean: What's the smallest one we've agreed on that uses |(, where >> the 7-5 comma interpretation of it would be a different number of >> steps from what we've used it for. >
>Our latest agreement has been on mostly ETs below 100, and I don't >think any of those even used |(. The larger-numbered ones were still >subject to review at the time you took your break, so they are still >open to review.
We agreed on |( for 1deg67 which is wrong (or at least not 1,3,5,7-consistently right) if |( is the 7-5 comma. I also proposed it for 93-ET (3*31) but we didn't agree on a notation for that.
>> OK. But this is not so, if we adopt (|( as the 7/5-comma symbol. >
>True (except that you meant the 11/5 comma, but I would prefer calling >it the 11'-5 comma for now).
Yes I did mean the 11/5 comma, and yes I will continue to call it the 11'-5 comma. At 08:17 AM 8/20/2002 +0000, George Secor wrote:
>(This is a continuation of my message #4580, which is in reply to Dave >Keenan's message #4543.) > >New Rational Complements Part 3 >--------------------------------- > >You previously mentioned that all of the rational complements are >consistent with 494-ET (as they are also with 217-ET). I would like to >define another pair of supplementary rational complements; we didn't >need these before, but they just might be useful when we're doing some >of the more obscure ETs. They're consistent in both 217 and 494, and >the offset is 0.49 cents. They are: > >~|~ <--> /||( and >/|( <--> ~||~
I have no objection to these at this stage.
>There are at least a couple of ratios that these can be used to notate: > >19/10 = Cb~|~ or C\!( >19/15 = Fb~|~ or F\!(
You could more generally just say that it can notate 19/5. We know that adding any number of factors of 2 or 3 doesn't change the saggital accidental required.
>Also, we might want to allow both /|( and ~|~ as their own alternate >complements in certain instances: > >/|( <--> /||( >~|~ <--> ~||~ > >This is just in case we need them. I would really not want to use >these unless it were a last resort. (After all, I want to keep the >number of symbols to a minimum.)
Definitely last resort.
>New Rational Complements Part 4 >--------------------------------- > >Now for what may be the most controversial issue -- actually, at the >last minute I came up with a very non-controversial solution to the >whole thing (almost a no-brainer), but I'll leave what I had here; just >don't reply to any of it until you get to the end -- I would like to >propose a definition of yet another supplementary pair of rational >complements: > >)|( <--> ~||\ and >~|\ <--> )||( > >Both of these are symbols that formerly lacked rational complements. >This is being done so that ~|\, which I am now proposing to be the 23' >comma instead of (|(, may have a rational complement. > >The reason that we did not previously use ~|\ as the 23' comma is that >it lacked a rational complement. Using ~|\ for this purpose has the >advantage of making the 23' comma consistent in the majority of the >best large-numbered ETs, including 152, 171, 217, 224, 270, 311, 494 >(yes, 494 too!!!), and 612, *none* of which use (|( consistently as the >23' comma. (This is one more thing that would make a transition >between rational notation and 217 notation for JI as easy and >consistent -- seamless might be a good word -- as possible.) > >Another advantage relates to the Reinhard property: The accuracy for >(|(, 1441792:1474767, ~39.149c, as the 23' comma, 16384:16767, >~40.004c, is contingent on the definition of (| as the 13'-(11-5) comma >(715:729) or as the 29 comma (256:261). But if (| is defined as the >11'-7 comma (45056:45927), then the schisma is 2023:2024, ~0.856 cents, >which is larger than what we have with ~|\, 4352:4455, ~40.496c, for a >schisma of 3519:3520, ~0.492c. Using ~|\ makes the schisma independent >of the size of (|. > >There are a couple of possible objections to this: > >1) The rational complementation offset is ~3.40 cents, which is >relatively large. (This would apply only to the single-symbol >notation.) I don't think this is much of a problem, because the >complement symbols are *defined* as rational intervals, not as the sum >of their component stems and flags. We wanted to keep the offsets low >in order to minimize the inconsistencies, but consider the alternative: >when we had (|( as the 23' comma we had an inconsistency for the symbol >itself in both 217 and 494; this new proposal eliminates that.
I really don't think I could have accepted a 3.4c offset.
>2) The rational complement being proposed is consistent in 217, but not >in 494. I checked consistency for a number of the better ETs in this >general neighborhood; most of those under 300 are consistent, and all >of those above 300 are inconsistent, so it's definitely related to the >offset. (Again, this would apply only to the single-symbol notation, >and the inconsistency occurs mostly in systems that we are not even >going to notate.) > >Is it all that important to have all of the rational complements >consistent with 494?
No. But to minimise offsets I think it needs to be consistent with _some_ similarly high numbered ET. 653-ET was a favourite of mine for this purpose at one time.
> If it is, then I just got an idea for what may be >an even better solution, one that you suggested, but with a twist: > ><< We could resurrect ~)||, with two left flags, as the complement of >the 23' comma. It isn't like a lot of people really care about ratios >of 23 anyway. We already made a good looking bitmap for ~)| with the >wavy and the concave making a loop. >> > >You were intending ~)|| to be the complement of (|(, which has the >following consequences: > >1) The complement has an offset of 1.59c with xL as the 13'-(11-5) >comma, which increases to 2.02 cents if you make xL the 11'-7 comma. > >2) The complement is inconsistent in 494, but consistent in 217. > >3) And as I said above, the 23' comma itself is inconsistent in both >217 and 494. > >But if we were to make ~)|| the rational complement of ~|\, then: > >1) The offset would be 0.67c, independent of the xL flag. > >2) The complement would be consistent in 494, but inconsistent in 217. > >3) And as I said above, the 23' comma itself would be consistent in >both 217 and 494. > >As for the inconsistency of the complement in 217, the ~)|| symbol >could either be replaced with the standard ~|| symbol or else with )||( >to specially designate the 23' complement. Thus only one obscure >complementary symbol would have to be changed in going from the strict >rational to the 217 quasi-rational version. > >The foregoing was written before you pointed out that (|( is the true >11'-5 and 13'-7 comma in your latest message. In light of this, I >would still assign ~|\ as the 23' comma, while making (|( a standard >symbol with rational complement ~||(, thereby eliminating /|~ from the >picture. (I was also using /|~ for 17/11 as Ab\!~ or A\!!!~, but I'll >see how well (|( works later.) One thing I am very happy about is that >the lateral confusability between /|~ and ~|\ is eliminated if one of >those two symbols is eliminated. > >So what do you think?
I think I'm confused, and I think I would have preferred you to spare me the foregoing and just given me the "almost no-brainer". So I think what you want to know is, do I think it is OK to have ~|\ as the 23' comma with a rational complement of ~)||, and (|( as the 11'-5 and 13'-7 commas with rational complement ~||(. And I've already agreed to ~|~ as the 5+19 comma with complement /||(. Well ~||( already was the complement of (|( because we needed (||( as the complement of ~|( which is the 17' comma. So that's no problem. And I also have no problem with ~)|| as the complement of ~|\ since the offset is so low and it interleaves nicely between the existing complements. Given this option I must totally reject )||( as a possible rational complement for ~|\ . Now the remaining question is whether I can accept ~|\ as the 23' comma. The answer is yes. But the whole 282-ET schisma question still haunts me. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5157 - Contents - Hide Contents

Date: Mon, 26 Aug 2002 08:40:51

Subject: Ringing the changes

From: genewardsmith

I found this just now on the Web:

404 Document not found * [with cont.]  (Wayb.)

The things one learns one never learned!


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Message: 5158 - Contents - Hide Contents

Date: Tue, 27 Aug 2002 17:10:36

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- David C Keenan <d.keenan@xx.xxx.xx> wrote:
> Sorry for the long delay in replying.
No problem -- take your time.
>> ... So it looks like this will be the 217 standard set: >> >> 217: |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~||
~||( /|| ||) ||\ (||( //|| /||) /||\ (new RCs)
> > Looks OK to me.
Good! This is point of agreement #1.
>>>> ... The following rational complements for the 15-limit
symbols are not
>>>> consistent in 282: >>>> >>>> )|~ <--> (|| 19' comma >>>> |( <--> /||) as 7-5 comma or 11-13 comma (but 17'-17 is okay) >>>> ~| <--> //|| 17 comma >>>> |) <--> ||) 7 comma >>>> //| <--> ~|| 25 comma >>>> (| <--> )||~ 11'-7 comma >>>> >>>> And besides this, there are others that are inconsistent, such as: >>>> >>>> |~ <--> ~||) as both the 19'-19 and 23 comma >>>
>>> All this means is that maybe we should consider making our rational >>> complements consistent with 282-ET rather than 217-ET. >>>
>>>> What makes 217 so useful is that *everything* is consistent to the 19 >>>> limit, and, except for 23, to the 29 limit. >>>
>>> I don't know what you mean by *everything* here. Isn't 282-ET >>> consistent to the 29-limit with no exceptions? >>
>> It isn't consistent with the schismas that are essential to the >> rational notation: >> >> 1) The 5 comma /| (5deg) plus the 7 comma |) (6deg) doesn't equal the >> 13 comma /|) (12deg); this is the 4095:4096 schisma, ~0.423c. So you >> can't notate ratios of 7 that are consistent with ratios of 13 in 282. >> >> 2) The 17'-17 comma (2deg) doesn't equal the 7-5 (1deg), or put >> another way, |) <> /|(; this is the 163840:163863 schisma, ~0.243c. So >> you can't notate ratios of 17 that are consistent with ratios of 7 and >> 13 in 282. >> >> Or should we discard these and start over -- I think I would then be >> entitled to say that you have either a 288-bias or an anti-217 bias. >
> OK. I understand now. Yes we definitely have a 217-ET bias (or rather a > bias toward systems whose fifth is close to that of 217-ET, like 494) in > the sense that we are only using schismas that vanish (I think we've been > overloading or overusing the term "consistent") in 217-ET. And it may well > be possible to start completely from scratch and build a different system > where we only use sub-cent (or sub-half-cent) schimas that vanish in > 282-ET. Then we'd have a 282-ET bias (not anti 217-ET). But then the 282-ET > fifth _is_ closer to the precise 2:3 that the system is supposedly based on.
But then the fifth of 494 is closer to an exact 2:3 than that of 282: 217: ~702.304c -- 0.349c or 0.063deg wide 288: ~702.128c -- 0.173c or 0.041deg wide 494: ~702.024c -- 0.069c or 0.029deg wide 2:3 ~701.955c Yet 494 uses the virtually the same schismas as 217.
> This is a daunting prospect, having come this far with the current system. > But wouldn't it be terrible if there was a _better_ system waiting to be > discovered, based on 282-ET schismas, and we passed it over? Perhaps you > can come up with a simple argument as to why this is not possible, short of > a complete investigation?
To answer this, let me begin by quoting from prior correspondence: [gs] << >I never considered 282 before, but I do see some problems with it:
> >1) 11 is almost 1.9 cents in error, and 13 is over 2 cents; these >errors approach the maximum possible error for the system. (This is >the same sort of problem that we have with 13 in 72-ET.) [dk]
You're only looking at the primes themselves. What about the ratios between them. 217-ET has a 2.8 cent error in its 7:11 whereas 282-ET never gets worse than that 2.0 cents in the 1:13. >> This is only because the error in either one can never exceed half a system degree, which for 282 is ~2.127 cents, but in 217 is ~2.765c. So whatever advantage 282 has is only because it divides the octave into more parts, which would be an advantage in itself. But there is more than this to take into consideration -- something that will demonstrate that it is better to have the error of the primes distributed in both directions rather than in a single direction, given that prime-limit consistency is maintained in each case. For situations involving schisma consistency, sometimes the error of two primes will accumulate rather than cancel, so that large unidirectional errors added together exceed 1/2 degree, resulting in an inconsistency. Both 72 and 282 are consistent to at least the 17 limit (as are 217 and 494). Since the tridecimal schisma (4095:4096, ~0.423c) vanishes in our notation but in neither ET, we cannot notate *both* ratios of 7 and 13 consistently in either one. I found this schisma at least a week before I considered 217 as a basis for mapping out the symbols, so we can't say that its selection was 217-biased; indeed it vanishes in a majority of the best ETs above 100. The fact that it doesn't vanish in either 72 or 282 is a consequence of the relatively large error for 13 (approaching the maximum) that I referred to above. Since the functional 13 diesis (1024:1053) is computed as the number of degrees (rounded) in the best fifth times 4, less the number of degrees in 3 octaves, plus the number of degrees (rounded) for 8:13, we can calculate the number of degrees for each of four divisions as follows: Interval deg72 deg282 deg217 deg494 -------- ------ ------- ------- ------- fifth (2:3) 42.117 164.959 126.937 288.971 rounded 42 165 127 289 times 4 168 660 508 1156 less 3 octaves -216 -846 -651 -1482 equals -48 -186 -143 -326 plus 8:13 rounded 50 198 152 346 equals 13 diesis 2 12 9 20 We then calculate the number of degrees in the 5+7 comma for each: Interval deg72 deg282 deg217 deg494 -------- ------ ------- ------- ------- 5 comma 80:81 1 5 4 9 7 comma 63:64 2 6 5 11 5+7 diesis 35:36 3 11 9 20 and compare these with the actual (as opposed to functional) number of degrees for 1024:1053, the ratio of the 13 diesis: Interval deg72 deg282 deg217 deg494 -------- ------ ------- ------- ------- actual 13 diesis 2.901 11.362 8.743 19.903 rounded 3 11 9 20 for which we find complete agreement in all four divisions, as opposed to the functional values calculated above: Interval deg72 deg282 deg217 deg494 -------- ------ ------- ------- ------- funct'l 13 diesis 2 12 9 20 We see that there is indeed an inconsistency in both the 72 and 282 divisions in that the number of degrees in the functional 13 diesis does not agree with the number of degrees for the actual interval; this inconsistency exists apart from the tridecimal schisma, but it happens to cause this schisma not to vanish. This is due principally to the excessive relative error in the representation of 13: Interval deg72 deg282 deg217 deg494 -------- ------ ------- ------- ------- actual 8:13 50.432 197.524 151.995 346.017 8:13 rounded 50 198 152 346 error in degrees -0.432 0.476 0.005 -0.017 I don't think that we would want to devise a system of notation in order to work around an inconsistency such as this, because I expect that we would then have some problems notating those ETs in which the tridecimal schisma *does* vanish. Our goal should be to make the smallest schismas vanish. As for what schismas do vanish in 282, maybe Gene would best be able to answer that. I thought that it was most productive to start with rational intervals, find the most useful schismas that can vanish, and then look for ETs that are consistent with those schismas. Working backwards by starting with a large-number ET and then finding the schismas that vanish in that ET is something that I don't have much experience with, and I have a feeling that we're not going to find anything better in 282 that will be useful in devising a notation that offers a better economy of symbols.
>> Our latest agreement has been on mostly ETs below 100, and I don't >> think any of those even used |(. The larger-numbered ones were still >> subject to review at the time you took your break, so they are still >> open to review. >
> We agreed on |( for 1deg67 which is wrong (or at least not > 1,3,5,7-consistently right) if |( is the 7-5 comma. I also proposed it for > 93-ET (3*31) but we didn't agree on a notation for that.
It is valid as 1deg67 for the 11-13 comma, but I would prefer not to use |( here (or elsewhere) unless it were valid for *both* the 7-5 and 11-13 commas. In addition, if *both* the 17 ~| and 17' ~|( symbols were to occur in an ET notation, then it would also have to be valid as the 17'-17 comma in order to maintain consistent symbol arithmetic. (At least that's the ideal I'm shooting for.) Anyway, after looking at 67 again, I don't see any clear choice for 1deg among several possibilities. I would prefer to do the easier ETs first (again) and in the process establish a hierarchy of rules for choosing the symbols. As we attempt to do increasingly difficult ones, we should get a better perspective on how to handle problems such as this one. I'll be discussing these issues in more detail in my next message, when I will again address the hows and whys of notating some of the less difficult ETs.
>> ... You previously mentioned that all of the rational complements are >> consistent with 494-ET (as they are also with 217-ET). I would like to >> define another pair of supplementary rational complements; we didn't >> need these before, but they just might be useful when we're doing some >> of the more obscure ETs. They're consistent in both 217 and 494, and >> the offset is 0.49 cents. They are: >> >> ~|~ <--> /||( and >> /|( <--> ~||~ >
> I have no objection to these at this stage.
Good! This is point of agreement #2.
>> New Rational Complements Part 4 >> --------------------------------- >> >> Now for what may be the most controversial issue -- actually, at the >> last minute I came up with a very non-controversial solution to the >> whole thing (almost a no-brainer), but I'll leave what I had here; just >> don't reply to any of it until you get to the end -- I would like to >> propose a definition of yet another supplementary pair of rational >> complements: >> >> )|( <--> ~||\ and >> ~|\ <--> )||( >> ... >
> I think I'm confused, and I think I would have preferred you to spare me > the foregoing and just given me the "almost no-brainer".
I was just trying to compare it with the alternatives, because until I did that, I didn't realize how much of a no-brainer it was.
> So I think what you want to know is, do I think it is OK to have
~|\ as the
> 23' comma with a rational complement of ~)||, and (|( as the 11'-5 and > 13'-7 commas with rational complement ~||(. And I've already agreed to ~|~ > as the 5+19 comma with complement /||(. > > Well ~||( already was the complement of (|( because we needed (||( as the > complement of ~|( which is the 17' comma. So that's no problem. > > And I also have no problem with ~)|| as the complement of ~|\ since the > offset is so low and it interleaves nicely between the existing > complements. Given this option I must totally reject )||( as a possible > rational complement for ~|\ .
I meant )||( to be only a 217-specific alternative complement; ~)|| would be the true rational complement.
> Now the remaining question is whether I can > accept ~|\ as the 23' comma. The answer is yes.
Good! Then this is point of agreement #3.
> But the whole 282-ET schisma question still haunts me.
Did I deal with it above adequately? --George
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Message: 5159 - Contents - Hide Contents

Date: Thu, 29 Aug 2002 18:33:12

Subject: Re: A common notation for JI and ETs

From: gdsecor

(This is a continuation of my message #4586, which is in reply to 
Dave Keenan's message #4543.)

Summary of Additional Rational Complements
------------------------------------------

In addition to the seven 217 standard symbol RC pairs and the 
supplementary pair of RCs I listed previously, there are then four 
additional pairs of supplementary symbols in my proposal.  These 
rational complements are used for some of the ratios of 17, 19, and 
23:

19 comma and (11'-7)+(19'-19) comma:  )| <--> (||~ and (|~ <--> )||
23 comma and 7+17 comma or 5+17' comma:  |~ <--> ~||) and ~|) <--> ||~
17+19 comma and 11-5+17 comma or 23' comma:  ~)| <--> ~||\ and ~|\ <--
> ~)||
17+23 comma and 5+(17-17') comma: ~|~ <--> /||( and /|( <--> ~||~ Here is how the ratios from 17 through the 21 limit are notated: 17/9 = B~! 18/17 = Db~| or D//!! or Cb~|( or C(!!( C#~!( or C(||( 17/10 = Bbb~|) or Bx~ 20/17 = D#~!) or D||~ 17/11 = Ab(!( or A(!!!( 22/17 = E(|( 17/12 = F#~! or F//|| 24/17 = Gb~| or G//!! or Gb~|( or G(!!( F#~!( or F(||( 17/13 = 17/13 = F(! 26/17 = G(| 17/14 = Eb//| or E~!! 28/17 = A\\! 17/15 = Ebb~|) or Ex~ 30/17 = A#~!) or A||~ 17/16 = C#~! or C//|| 32/17 = Cb~| or C//!! or Db~|( or D(!!( B~!( 19/10 = Cb~|~ or C\!( 20/19 = C#~!~ or C/|( or B or Db or D\||/ 19/11 = Bb(!~ or B(!!!~ 22/19 = D(|~ 19/12 = Ab)| or A(!!~ 24/19 = E)| or G#)!~ or G(!! or Dx(! or D)X~ 19/13 = G\\! 26/19 = F//| 19/14 = F)|) 28/19 = G)!) (no rational complement defined; use /|| as alternate complement) 19/15 = Fb~|~ or F\!( 30/19 = G#~!~ or G/|( or E or Ab or A\||/ 19/16 = Eb)| or E(!!~ 32/19 = A)| or D#)!~ or D(!! or Gx(! or G)X~ 19/17 = D~)! 34/19 = Bb~)| or B~!!/ 19/18 = Db)| or D(!!~ 36/19 = B)| or C#)!~ or C(!! or Ax(! or A)X~ 21/11 = Cb(| or C)!!~ 22/21 = C#(! or C)||~ 21/13 = Ab(|( or A~!!( 26/21 = E(!( 21/16 = F!) 32/21 = G|) 21/17 = E\\! 34/21 = Ab//| or A~!! 21/19 = )!) 38/21 = Bb)|) or B\!! 21/20 = Db!( or D!!!( 40/21 = B|( We will have to prepare a comprehensive listing of these in some form. It would be nice if we could have a spreadsheet in which you could input a letter-plus-symbol(s) for a tone and a ratio up or down for a second tone, and letter-plus-symbol options for the second tone would be displayed in both single and double-symbol versions. (Something like this would be useful for ETs as well.) Notation of ETs --------------- Since we would want to see how well the proposals I have made for modifying the RCs would work for various ETs, following are some that I have tried. First, for reference I am listing symbol sequences for some of the ETs on which we have most recently agreed. These are the ones that will not change as a result of the latest proposals. 12, 19, 26: /||\ (RC) 17, 24, 31, 38: /|\ /||\ (RC) 45: /|) /||\ (RC) 22, 29: /| ||\ /||\ (RC) 36: |) ||) /||\ (RC & MS) 43, 50, 57, 64: /|) (|\ /||\ (RC) 27: /| /|) ||\ /||\ (RC) 34, 41: /| /|\ ||\ /||\ (RC) 62: /|) /|\ (|\ /||\ (RC) 39, 46, 53: /| /|\ (|) ||\ /||\ (RC) 51: |) /| /|) ||\ ||) /||\ (RC) 65, 72, 79: /| |) /|\ ||) ||\ /||\ (RC; ISA ||) 65,72,79) 58: /| |\ /|\ /|| ||\ /||\ (RC & MS) 84: /| |) /|) (|\ ||) ||\ /||\ (RC) RC = rational complementation AC = alternate complementation MS = matching symbol sequence MM = most memorable sequence ISA = inconsistent symbol arithmetic Some of the conditions that I needed to get the above symbols in my spreadsheet-under-construction are: 1) The 5 comma and 7 comma must each be less than 2/5 apotome. 2) The 11 diesis must be greater than 1/3 apotome. 3) The 11 diesis must be less than the 11' diesis if both symbols are used. This avoids results such as: 17: /| /||\ 36: (|) /|\ /||\ 27: /|\ /|) (|) /||\ 60: /| |) ||) ||\ /||\ In the meantime I have discovered that a couple of those that we did agree on have properties that now persuade me either to question or reject outright the symbol sequences: 52: /|) /||\ (RC) 32: )| /|\ (|) (||~ /||\ (RC) for the following reasons. The 13 comma /|) is not valid as 1deg52. Instead I propose the half- apotome symbol of last resort that can usually be made to work when nothing else will: 52a: (|~ /||\ [(11-7)+23 comma] (RC) However, after doing 69, 76, 86, 93, and 100 (see below), where )|\ is quite useful for the half-apotome, I thought that this might also be a possibility: 52b: )|\ /||\ (RC) With 32 the best we could do for 1deg was the 19 comma, which is quite a bit smaller than 1deg52, 37.5 cents. We have subsequently defined (|( as the 11'-5 comma (~38.9 cents), which would give us this: 32: (|( /|\ (|) ~||( /||\ [11'-5 comma] (RC) ~||( would be 3deg32 with |( as the 7-5 comma, but since ~| is not being used as the 17 comma (or as anything else, for that matter), there is no inconsistency in symbol arithmetic, and the symbol can be justified as 4deg simply on the basis of its being the rational complement of (|(, which is the same thing we did before for (||~. I have a question. In doing the symbol selection spreadsheet, my logic gives this for both 36 and 43: 43: |) ||) /||\ but it gives 50, 57, and 64 with the 13 commas (as we agreed on above), because the 7 comma |) is not 1deg for those systems. You said that you wanted 2deg43 (~55.8c) to be a single-shaft symbol (|\, but I don't know what sort of test to introduce to give this result for 43 without giving the 13 diesis precedence over the 7 comma for other ETs in which both are valid (such as 36). Why is it so important to have (|\ as 2deg43? Now for some of the larger ETs that we need to reconsider. With most of those over 100 tones, I have found that matching symbol sequences or, where that's not possible, most memorable symbol selections will be the most important factor. Most memorable means minimizing the number of flags while keeping the symbol arithmetic consistent. I have found that the easiest ETs between 100 and 217 can be done with both matching symbol sequences and rational complementation: 118a (59 ss.): ~| /| |\ //| /|\ (|) ~|| /|| ||\ //|| /||\ (RC & MS) 130 (65 ss.): |( /| |) |\ /|) /|\ (|\ /|| ||) ||\ /||) /||\ (RC & MS - 7-5 comma) 142 (71 ss): |( /| |) |\ /|) /|\ (|) (|\ /|| ||) ||\ /||) /||\ (RC & MS - 7-5 comma) 176a (88 ss.): |( |~ /| |) |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||) ||\ ~||) /||) /||\ (RC & MS) 183: |( ~|( /| |) |\ (|( /|) /|\ (|) (|\ ~||( /|| ||) ||\ /||~ /||) /||\ (RC & MS) 217: |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~|| ( /|| ||) ||\ (||( //|| /||) /||\ (RC & MS) Even if you're not interested in the single-symbol notation, the rational complementation ideal still has validity in these larger divisions, because in each half-apotome pairs of symbols are also related as unidecimal-diesis complements such as ~| = /|\ - //| and | ( = /|\ - /|). In all of the above, in every case where a particular symbol is used, it is valid as all of the comma aliases for which it is called upon: |( is valid as both the 7-5 (or 7/5) and 11-13 (or 13/11) comma in all of these divisions (and also as the 17'-17 comma in 183 & 217) and is consistent as the rational complement of /||). ~| is valid as the 17 comma and //| as the both the 5+5 comma (1,5,25- consistent) and 13'-5 (or 13/5) comma, and the two symbols are consistent with their rational complements. ~|( is valid as the 17' comma and (|( as both the 11'-5 (or 11/5) and 13'-7 (or 13/7) commas and also the 11-17' (or 17/11) comma, and the two symbols are consistent with their rational complements. |~ is valid as the 23 comma and ~|) as both the 7+17 and 5+17' (or 17/5) commas, and the two are consistent with their rational complements. (If the |~ flag is used alone, I want to use it only if it is valid as the 23 comma; I would use it as the 19'-19 comma only if it is being used in combination so that it would not be misinterpreted as the 23 comma.) So everything works perfectly for these select half-dozen ETs above 100. For 176 ~| was not used for 2deg for two reasons: 1) //|| is not consistent as its rational complement, which would have also necessitated //| for 6deg to match flags in the half-apotomes; 2) also, //| is compromised because 176 is not 1,5,25 consistent. I chose not to use ~|( for 2deg because |( was used as the 7-5 comma for 1deg, but this is not valid as the 17'-17 comma, which usage would be required for ~|( as 2deg. So it is by process of elimination that I arrived at |~ for 2deg176, which is not bad, because 23 is represented much better than 17 in this ET; the only problem is that ~|) is not valid as the 5+17' (or 17/5) comma. However, if you don't like this many flags for 176, I have another solution below. This next one was not quite perfect: 125: ~|( /| |\ (|( /|\ (|) ~||( /|| ||\ (||( /||\ (RC & MS) ~|( is valid as the 17' comma but (|( is valid as the 11'-5 (or 11/5) and 11-17' (or 17/11) commas, and the two symbols are consistent as rational complements. However, (|( is not valid as the 13'-7 (or 13/7) comma, so 13 usage must be excluded from the notation, which is not inappropriate, since 13 is not well represented in this ET. I don't think we can complain about the number of flags in this one. The next easiest ETs can be done with matching sequences and mostly rational complementation with a little bit of alternate complementation: 171: |( ~|( /| |) |\ ~|\ /|) /|\ (|\ ~||( /|| ||) ||\ ~||\ /||) /||\ (MS; 10,14deg AC) I'll spare you the details, other than to say that ~|\, which is valid here as the 23' comma, serves nicely to keep the number of flags down. Here's the other solution for 176, which keeps the flags to a minimum: 176b (88 ss.): |( ~| /| |) |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||) ||\ ~||) /||) /||\ (MS; 11,15deg AC) I really don't know if I prefer this to version a; 23 is much better represented than 17 in 176, which would justify using |~ for 2deg. For 152 I have three solutions: 152a (76 ss.): )| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ (||( /||) /||\ (MS; 14deg AC) 152b (76 ss.): )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~ /|| ||\ ~||) /||) /||\ (MS; 14deg AC) 152c (76 ss.): )| ~| /| |\ ~|) /|) /|\ (|) (|\ ~|| /|| ||\ ~||) /||) /||\ (MS; 10,13,14deg AC) In version a, (|( as 6deg152 is valid as the 11'-5 (or 11/5) and 11- 17' (or 17/11) commas, but not the 13'-7 (or 13/7) comma. The replacements in version b result in higher primes and more flags; here ~|) is valid as both the 7+17 and 5+17 (or 17/5) commas. Version c uses the simplest matching symbols, and I am inclined to go with that. (I have reached the conclusion that if a set of symbols isn't close to flawless with rational complements, then we should just go for the most memorable set, with matching symbols in the half- apotomes where possible.) For some of the more difficult ETs above 100 I have matching sequences and consistent symbol complementation using as few flags as possible. In order to avoid using invalid 13-limit symbol indications (such as (|( not being valid for both the 11'-5 and 13'-7 commas), I found that ~|) and ~|\ come in very handy, particularly because they introduce no new flags in instances where either the 17 or 17' comma is also being used. 111 (37 ss): ~| /| |\ ~|\ /|\ (|) ~|| /|| ||\ ~||\ /||\ (MS) 144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ (MS) 193: )| ~| ~|( /| |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ~|| ( /|| ||\ ~||) ~||\ /||) /||\ (MS) 207: |( ~|( /| /|( (| |\ ~|\ /|) /|\ (|) (|\ ~||( /|| /| ( (|| ||\ ~||\ /||) /||\ (MS) 224: )| |~ )|~ /| |) |\ /|~ //| /|) /|\ (|) (|\ ||~ ) ||~ /|| ||) ||\ /||~ //|| /||) /||\ (MS - 19 commas) The symbol set for 111 is one that we previously agreed upon; I didn't use //| for 111 because it is not 1,5,25 consistent, whereas 118 and 125 are. The |) flag in 144 is the 13-5 comma, so )|) is 3deg. I'm considering this without regard to a comprehensive multiples-of-12 plan for the time being. For the most difficult ETs (RC, AC, and MS not possible), it's a matter of doing them any way you can to find the most memorable selection of symbols: 128b (64 ss.): )| ~|( /| (|( ~|\ /|\ (|) )|| ~||( ||\ (|| ( ~||\ /||\ (MM) 135a (45 ss.): ~| |~ /| (| /|~ /|\ (|) ~|| ||~ ||\ (|| /||~ /||\ (MM) 135b (45 ss.): ~| ~|( /| (| /|~ /|\ (|) ~|| ~||( ||\ (|| /||~ /||\ (MM) 140 (70 ss.): )| ~|( /| )|\ ~|\ /|) (|~ (|\ )|| ~||( ||\ ) ||\ ~||\ /||\ (MM) 181a: |( ~| |~ /| /|( ~|) /|~ /|) (|~ (|\ ||( ~|| ||~ ||\ /||( ~||) /||~ /||\ (MM) 181b: |( ~| ~|( /| /|( (| /|~ /|) (|~ (|\ ||( ~|| ~||( ||\ /||( (|| /||~ /||\ (MM) I have another version for 128 below, which uses rational complementation without matching symbols in the half-apotomes. I have two options for 135, with no choice other than /|~ for 5deg. Take your pick. 70 is given above as a subset of 140, but I found that it works better on its own: 70: /| |\ /|\ (|) /|| ||\ /||\ (RC & MS) 77: /| |) /|\ (|) ||) ||\ /||\ (RC; ISA-5deg) While 77 can be done like 70, my spreadsheet selects |) in preference to |\ to eliminate lateral confusability. I have come to the conclusion that there is not much point in trying to notate ETs 7 tones apart alike if one of them can be done a better way (hence 111, 118, and 125 were all done differently above). For ETs below 100 (for which matching sequences are often not possible) and for those above 100 for which /| and |\ are the same number of degrees (and therefore for which matching sequences are not possible), I think that rational complementation is the most important principle, including those that we already agreed upon (which I summarized above). Even for the double-symbol notation (where rational complementation is of little concern), some of the flags in the second apotome will also occur in the first apotome, where symbols are paired as unidecimal-diesis complements. Here are some more of the ET notations that I am proposing: 68: |\ /| /|\ /|) (|) ||\ /|| /||\ (RC & MS) if we permit /|\ < /|) 80: )| /| (|~ /|\ (|) )|| ||\ (||~ /||\ [13'-(11-5)+23 = 11-19 diesis] (RC) 87a: |~ /| ~|) /|\ (|) ||~ ||\ ~||) /||\ (RC) 94a: ~|( /| (|( /|\ (|) ~||( ||\ (||( /||\ (RC) 99a: |~ /| ~|) /|) (|~ (|\ ||~ ||\ ~||) /||\ (RC) 108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ (RC; ~|| as RC) 108b: /| (|( |) /|) (|\ ||) ~||( ||\ /||\ (RC) 104a: )| |) /| (| /|\ (|) )||~ ||\ ||) (||~ /||\ [~| as 23 comma] (RC) 104b: )| |) /| (|~ /|\ (|) )|| ||\ ||) (||~ /||\ [~| as 23 comma] (RC) 128a (64 ss.): )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (|| ( (||~ /||\ [~| as 23 comma] (RC) The familiar flags can be used for 68 if we allow some the symbols to be used in an unusual order. (Also see 51 above, which we previously agreed on.) I think that if the apotome is 10 or more degrees and if the symbols in the half-apotomes *can* be made to match, then they *should* be made to match, even if they are not all strict rational complements -- it is easier to remember them that way. I want to make special mention of 94, since it is such an important division. For 3deg94 (|( is valid as the 11'-5, 13'-7, and 11-17' commas, and I chose ~|( for 1deg as its unidecimal-diesis complement; this pairing works perfectly, much better than ~| and //|, which is plagued with inconsistencies. I would like to use these rational complement pairs whenever they work this well, but perhaps you would prefer another option (to follow). One problem I find with 108 version a is that //| is not valid as the 13'-5 comma, which is significant, since we are using the 13 commas in the notation. In version b I have written off 11 for excessive error and used (|( as the 13'-7 comma (even though it is invalid as the 11'-7 comma) and have used )||~ as its rational complement. For some divisions under 100 (with matching sequences not possible), we would have to decide whether we prefer strict rational complement sets to a "most memorable" selection of symbols, such as these (including ones for 67 and 81): 87b, 94b: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ (MM) 87c, 94c: |~ /| /|~ /|\ (|) ||~ ||\ /||~ /||\ (MM) 87d, 94d: |~ /| /|~ /|\ (|) ~|| ||\ ~||\ /||\ (MM) 99b: ~| /| ~|\ /|) (|~ (|\ ~|| ||\ ~||\ /||\ (MM) 99c: |~ /| /|~ /|) (|~ (|\ ||~ ||\ /||~ /||\ (MM) An important principle that I have employed in arriving at these symbols sets is that the more difficult or obscure ETs should not dictate what the notation for the easier and more-likely-to-be-used ETs should be. (This is a corollary to the principle that the more difficult things should not make the simpler things more difficult.) So I have considered each division on its own without attempting to use the same symbols in ETs differing by 7. I have shown sets for both 108 and 144, which I will discuss further when I reply to your proposal for the symbol sets for multiples of 12, where the two principles mentioned in the previous paragraph will be relevant. --George (To be continued.)
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Message: 5160 - Contents - Hide Contents

Date: Fri, 30 Aug 2002 19:44:51

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> Just a quick reply to one question. > > At 11:35 AM 29/08/2002 -0700, George Secor wrote:
>> I have a question. In doing the symbol selection spreadsheet, my logic >> gives this for both 36 and 43: >> >> 43: |) ||) /||\ >> >> but it gives 50, 57, and 64 with the 13 commas (as we agreed on above), >> because the 7 comma |) is not 1deg for those systems. You said that >> you wanted 2deg43 (~55.8c) to be a single-shaft symbol (|\, but I don't >> know what sort of test to introduce to give this result for 43 without >> giving the 13 diesis precedence over the 7 comma for other ETs in which >> both are valid (such as 36). Why is it so important to have (|\ as >> 2deg43? >
> What seems important to me, is to be able to notate any ET using only > single-shaft symbols in combination with # and b. > > In that case, the largest number of steps to need a single-shaft symbol in > an ET is given by > =TRUNC(MAX(steps_in_tone, steps_in_diatonic semitone)/2) > in some cases the largest number of steps will be catered for by
the # or b
> itself. > -- Dave Keenan
I don't understand this at all. For 43, steps_in_tone=7 and diatonic_semitone=4, for which your formula gives 3. Did you mean TRUNC(MAX(steps_in_tone/2, steps_in_diatonic_semitone)/2), for which your formula gives 2? (However, I found that doesn't work either, because it gives 1 for 27, 34, and 41-ET, but we want 2.) TRUNC (steps_in_apotome/2), which gives 1, is what I think it should be; we can still notate 43 with single-shaft symbols using only |): 0 1 2 3 4 5 6 7 C C|) C#!) C# C#|) Cx!) Cx Dbb Dbb|) Db!) Db D!) D This is how it would be with the 13-comma symbols: C C/|) C(|\ C# C#/|) C#(|\ Cx Dbb Db(!\ Db/!) Db(!\ D/!) D I don't recall that we previously objected to having a 7 comma alter in the opposite direction in combination with a sharp or flat. So I am at a loss as to what to do. --George
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Message: 5161 - Contents - Hide Contents

Date: Fri, 30 Aug 2002 09:22:21

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Just a quick reply to one question.

At 11:35 AM 29/08/2002 -0700, George Secor wrote:
>I have a question. In doing the symbol selection spreadsheet, my logic >gives this for both 36 and 43: > >43: |) ||) /||\ > >but it gives 50, 57, and 64 with the 13 commas (as we agreed on above), >because the 7 comma |) is not 1deg for those systems. You said that >you wanted 2deg43 (~55.8c) to be a single-shaft symbol (|\, but I don't >know what sort of test to introduce to give this result for 43 without >giving the 13 diesis precedence over the 7 comma for other ETs in which >both are valid (such as 36). Why is it so important to have (|\ as >2deg43?
What seems important to me, is to be able to notate any ET using only single-shaft symbols in combination with # and b. In that case, the largest number of steps to need a single-shaft symbol in an ET is given by =TRUNC(MAX(steps_in_tone, steps_in_diatonic semitone)/2) in some cases the largest number of steps will be catered for by the # or b itself. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5162 - Contents - Hide Contents

Date: Mon, 02 Sep 2002 11:13:17

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 12:47 PM 30/08/2002 -0700, George Secor wrote:
>> In that case, the largest number of steps to need a single-shaft >symbol in
>> an ET is given by >> =TRUNC(MAX(steps_in_tone, steps_in_diatonic semitone)/2) >> in some cases the largest number of steps will be catered for by the
># or b >> itself.
>> -- Dave Keenan >
>I don't understand this at all. For 43, steps_in_tone=7 and >diatonic_semitone=4, for which your formula gives 3. Did you mean >TRUNC(MAX(steps_in_tone/2, steps_in_diatonic_semitone)/2), for which >your formula gives 2? (However, I found that doesn't work either, >because it gives 1 for 27, 34, and 41-ET, but we want 2.) >TRUNC(steps_in_apotome/2), which gives 1, is what I think it should be; >we can still notate 43 with single-shaft symbols using only |): > >0 1 2 3 4 5 6 7 > >C C|) C#!) C# C#|) Cx!) Cx > Dbb Dbb|) Db!) Db D!) D > >This is how it would be with the 13-comma symbols: > >C C/|) C(|\ C# C#/|) C#(|\ Cx > Dbb Db(!\ Db/!) Db(!\ D/!) D > >I don't recall that we previously objected to having a 7 comma alter in >the opposite direction in combination with a sharp or flat. > >So I am at a loss as to what to do. Sorry George,
I screwed up. You nearly got it. What I meant to say was =TRUNC(MAX(steps_in_apotome, steps_in_Pythagorean_limma)/2) apotome = 2187:2048 Pythagorean limma = 243:256 (i.e. the Pythagorean versions of the chromatic and diatonic semitones) and sure, it doesn't matter if you put the divide-by-twos before the MAX. And there's certainly no objection to having a 7 comma alter in the opposite direction in combination with a sharp or flat. By the way, you left out the Db|) in your first example and the Db in your second. The way of thinking that will favour using saggitals in combination with # and b, is one that thinks of C# as a single symbol, and would rather not have to accept Db as being a different pitch. In this person's mind there are not 7 but 12 basic symbols which are to be modified by the saggitals. For example, when the key is nominally C or Am then the 12 symbols are Eb Bb F C G D A E B F# C# G# So it could be: 0 1 2 3 4 5 6 7 C C|) C#!) C# C#|) C#(|\ D(!/ D!) D So you see it's the 4 step _limma_ (between C# and D) that causes the problem here. Similarly: 0 1 2 3 4 B B|) B(|\ C(!/ C!) C -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5163 - Contents - Hide Contents

Date: Mon, 02 Sep 2002 14:32:10

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I wrote:

"The way of thinking that will favour using saggitals in combination with # 
and b, is one that thinks of C# as a single symbol, and would rather not 
have to accept Db as being a different pitch. In this person's mind there 
are not 7 but 12 basic symbols which are to be modified by the saggitals. 
For example, when the key is nominally C or Am then the 12 symbols are Eb 
Bb F C G D A E B F# C# G#"

I should have said "_One_ way of thinking that will favour using sagittals 
in combination with # and b ...", since some folks will prefer it even 
though they don't prescribe to this way of thinking. However I think that 
many trained musicians, who have never before had to deal with tunings 
other than 12-ET, will think this way, in particular keyboard players and 
players of other fixed pitch instruments where all 12 equally-spaced 
pitches are almost equally playable. I became convinced of this through 
discussions with Paul Erlich and Joseph Pehrson.

It's clear that you and I have trouble seeing things from this perspective, 
immersed as we have been, in tuning theory, for many years.

I realised after sending the previous message that I have not followed it 
consistently either. A person who does not want to see C# and Db as 
different pitches (and therefore should use only one of them at a time to 
avoid inconsistencies) will need a single shaft symbol for 
TRUNC(steps_in_Pythagorean_limma/2) even if this is the same as
steps_in_apotome and could therefore be symbolised by # or b, e.g in 19-ET, 
26-ET, 38-ET and 45-ET.

I certainly wouldn't expect you to _replace_ /||\ and \!!/ with single 
shaft symbols in these (the extreme meantones), but I do feel that we must 
provide single-shaft _alternatives_ for them, when used with a 
chain-of-twelve-fifths basis (as opposed to a chain-of-seven-fifths). The 
same goes for 2deg43, with an alternative to ||).

(|\ is a sensible alternative for 1deg19 and 1deg26, but 2deg38 presents a 
problem. I can find no consistent candidate below the 23 limit, but it 
seems like we should use (|\ on the basis that 2deg38 is the same as 1deg19.

|) is 2deg45 but it doesn't seem wise to use this symbol for something that 
large and again I fall back on (|\. Neither 38 nor 45 are 
1,3,13-consistent, but a 2 step shift does at least give the best 3:13 in 
both cases.

A single shaft alternative for ||) as 2deg43 is no problem. It's fine to 
use both |) as 1deg43 and (|\ as 2deg43, since the 13-schisma vanishes.

2deg50 is already the single-shaft (|\ as standard.

(|\ also works for 3deg62, 3deg67, 3deg69, 3deg74, 4deg86, 4deg91.

But I can't see any possibility of meaningful single-shaft alternatives for:
3deg52, 3deg57, 3deg64, 4deg76, 4deg81, 4deg88, 4deg93 etc., so I'm 
prepared to give up on them. These ETs are all 1,3,9-inconsistent and will 
be better notated as subsets anyway.

Here's a proposed rule:
if TRUNC(steps_in_Pythagorean_limma/2) > TRUNC(steps_in_apotome/2) then
the alternative single-shaft symbol for
degree[TRUNC(steps_in_apotome/2) + 1] is (|\.

Here's a slightly more restrictive version of it.

if TRUNC(steps_in_Pythagorean_limma/2) - TRUNC(steps_in_apotome/2) = 1 then
the alternative single-shaft symbol for
degree[TRUNC(steps_in_Pythagorean_limma/2)] is (|\.

Let me know what anomalies these produce, if any. I think 93-ET (3*31) 
might be a problem.

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)


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Message: 5164 - Contents - Hide Contents

Date: Mon, 02 Sep 2002 18:11:24

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 11:35 AM 29/08/2002 -0700, George Secor wrote:
>From: George Secor, 8/28/2002 (#4596) >Subject: A common notation for JI and ETs > >(This is a continuation of my message #4586, which is in reply to Dave >Keenan's message #4543.) > Th >Summary of Additional Rational Complements >------------------------------------------ > >In addition to the seven 217 standard symbol RC pairs and the >supplementary pair of RCs I listed previously, there are then four >additional pairs of supplementary symbols in my proposal. These >rational complements are used for some of the ratios of 17, 19, and 23: > >19 comma and (11'-7)+(19'-19) comma: )| <--> (||~ and (|~ <--> )|| >23 comma and 7+17 comma or 5+17' comma: |~ <--> ~||) and ~|) <--> ||~ >17+19 comma and 11-5+17 comma or 23' comma: ~)| <--> ~||\ and ~|\ <--> >~)|| >17+23 comma and 5+(17-17') comma: ~|~ <--> /||( and /|( <--> ~||~
Just to confirm: I have no problem with these complementary symbol pairs. They have all either been agreed before or fit nicely between what has been agreed before. They also all agree with a suitably large numbered ET, 494-ET, in at least one of their comma interpretations. Do they actually agree with 494 in all the comma interpretations you have given them (not that it matters very much)?
>Here is how the ratios from 17 through the 21 limit are notated: ... >19/14 = F)|) 28/19 = G)!) > (no rational complement defined; use /|| as alternate complement)
Yes, there's no other choice for the complement. We have to stop generating new symbols somewhere. I think getting to the 35th harmonic and 17-limit diamond is pretty impressive.
>We will have to prepare a comprehensive listing of these in some form.
Ultimately I think maybe we should have a series of staves one under the other. The first should show all the (octave-reduced) odd harmonics of G (as 1/1) that we can notate. Then next shows all the odd harmonics of the 3rd subharmonic of G (i.e. C), then the odd harmonics of the 5th subharmonic of G (i.e. Eb/), and so on to the 35th subharmonic of G. There will be lots of holes. I expect most of the lower right triangle to be missing, but I hope we have a full upper right triangle. Which ratios can we actually notate uniquely without going to multiple saggitals? i.e. for each symbol what is the ratio with the lowest product complexity when all factors of 2 and 3 are removed. Product complexity of a ratio a:b being simply a*b.
>It would be nice if we could have a spreadsheet in which you could >input a letter-plus-symbol(s) for a tone and a ratio up or down for a >second tone, and letter-plus-symbol options for the second tone would >be displayed in both single and double-symbol versions. (Something >like this would be useful for ETs as well.) Yes. >Notation of ETs >--------------- > >Since we would want to see how well the proposals I have made for >modifying the RCs would work for various ETs, following are some that I >have tried. > >First, for reference I am listing symbol sequences for some of the ETs >on which we have most recently agreed. These are the ones that will >not change as a result of the latest proposals. > >12, 19, 26: /||\ (RC) >17, 24, 31, 38: /|\ /||\ (RC) >45: /|) /||\ (RC) >22, 29: /| ||\ /||\ (RC) >36: |) ||) /||\ (RC & MS) >43, 50, 57, 64: /|) (|\ /||\ (RC) >27: /| /|) ||\ /||\ (RC) >34, 41: /| /|\ ||\ /||\ (RC) >62: /|) /|\ (|\ /||\ (RC) >39, 46, 53: /| /|\ (|) ||\ /||\ (RC) >51: |) /| /|) ||\ ||) /||\ (RC) >65, 72, 79: /| |) /|\ ||) ||\ /||\ (RC; ISA ||) 65,72,79) >58: /| |\ /|\ /|| ||\ /||\ (RC & MS) >84: /| |) /|) (|\ ||) ||\ /||\ (RC) > >RC = rational complementation >AC = alternate complementation >MS = matching symbol sequence >MM = most memorable sequence >ISA = inconsistent symbol arithmetic
I'm glad these remain unchanged. Between them they probably cover 99% of what anyone will ever want to do with ETs other than 12.
>Some of the conditions that I needed to get the above symbols in my >spreadsheet-under-construction are: > >1) The 5 comma and 7 comma must each be less than 2/5 apotome. >2) The 11 diesis must be greater than 1/3 apotome. >3) The 11 diesis must be less than the 11' diesis if both symbols are >used.
Perfectly reasonable constraints.
>In the meantime I have discovered that a couple of those that we did >agree on have properties that now persuade me either to question or >reject outright the symbol sequences: > >52: /|) /||\ (RC) >32: )| /|\ (|) (||~ /||\ (RC)
I have no strong attachments to these. As ETs go, they are probably of marginal interest, and they should be primarily notated as subsets (of 96 and 104).
>for the following reasons. > >The 13 comma /|) is not valid as 1deg52. Instead I propose the >half-apotome symbol of last resort that can usually be made to work >when nothing else will: > >52a: (|~ /||\ [(11-7)+23 comma] (RC)
I don't see how (|~ is any more valid than /|). What comma (or combination of commas) did you have in mind? I suggest (|( as the 11'-5 comma for 1deg52. And we also have (|\ as the single shaft (alternative) symbol for 2deg52, although 1:7's are so good in 52-ET is almost seems a shame not to use |) for 2deg52. There's no sensible single-shafter for 3deg52 (to reach the half-limma without an unwanted # or b when using a 12 note base), although |( is valid as the 7-5 comma.
>However, after doing 69, 76, 86, 93, and 100 (see below), where )|\ is >quite useful for the half-apotome, I thought that this might also be a >possibility: > >52b: )|\ /||\ (RC)
Tell me why you'd prefer this 19+(11-5) comma )|\ to the 11'-5 comma (|(.
>With 32 the best we could do for 1deg was the 19 comma, which is quite >a bit smaller than 1deg52, 37.5 cents. We have subsequently defined >(|( as the 11'-5 comma (~38.9 cents), which would give us this: > >32: (|( /|\ (|) ~||( /||\ [11'-5 comma] (RC)
Yes. I like that.
>Now for some of the larger ETs that we need to reconsider. ...
I'll have to respond to these another time. In general I'm happy to leave the big ETs to you. But I guess you'd like me to check your results. I'm more interested in what you think re notation relative to 12-ET or at least notation of n*12-ETs. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5165 - Contents - Hide Contents

Date: Tue, 03 Sep 2002 10:49:38

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Hi Klaus,

At 12:14 PM 2/09/2002 +0200, you wrote:
>dear dave, >i sent the attached mail to george secor because i thought he forwarded it >to the list. seeing this ain't so, here's copy for you, the truly intended >recipient. > >klaus > > >Message-ID: <3D7338EC.3010703@x.xxx.xx> >Date: Mon, 02 Sep 2002 12:09:48 +0200 >From: klaus schmirler <KSchmir@x.xxx.xx> >User-Agent: Mozilla/5.0 (Windows; U; Win98; en-US; rv:1.1) Gecko/20020826 >X-Accept-Language: en-us, en >MIME-Version: 1.0 >To: gdsecor@xxxxx.xxx >Subject: re: 1 of dave keenans mails >Content-Type: text/plain; charset=us-ascii; format=flowed >Content-Transfer-Encoding: 7bit > >David C Keenan wrote: >> I wrote: >>
>> "The way of thinking that will favour using saggitals in combination with # >> and b, is one that thinks of C# as a single symbol, and would rather not >> have to accept Db as being a different pitch. In this person's mind there >> are not 7 but 12 basic symbols which are to be modified by the saggitals. >> For example, when the key is nominally C or Am then the 12 symbols are Eb >> Bb F C G D A E B F# C# G#" >> >> I should have said "_One_ way of thinking that will favour using sagittals >> in combination with # and b ...", since some folks will prefer it even >> though they don't prescribe to this way of thinking. However I think that >> many trained musicians, who have never before had to deal with tunings >> other than 12-ET, will think this way, in particular keyboard players and >> players of other fixed pitch instruments where all 12 equally-spaced >> pitches are almost equally playable. I became convinced of this through >> discussions with Paul Erlich and Joseph Pehrson. > >
>please stick to the strict pythagorean for the olden accidentals.
After reading the rest of your email, I suspect you mean stick to notating the chain of fifths (approximate 2:3s) in a tuning as ... Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# ... even when those fifths are not precise 2:3s. Otherwise, I would normally take "strict Pythagorean" to refer only to a chain of precise 2:3s (say 702.0 +- 0.5 c).
>i was not very advanced when i've been told not to confuse f# and gb, even >though i used the same fingerings (on the clarinet). i filed this under >orthography when i learned what a chord is. and when i discovered >alternate fingerings on my -super cheap- clarinet, it puzzled me that >pitches were _not_ the same. (i doubt that this was intentional, however, >nowadays i wish for a a clarinet that is actually built to produce 17- or >19-et).
Many folks can easily accept that F# might be different from Gb in non-12-ET tunings. But they find it odd that in 19-ET for example, F# is flatter than Gb, while in 17-ET it's the other way 'round. I've seen music texts that insist on one of these possibilities but ignore the other (meantone vs. strict Pythagorean).
>the concept is hard to understand only if you insist that c# and db be the >same pitch, and if people claim not to understand anything about this, you >only consolidate their confusion by sticking to a 12-et frame.
We are using the chain of native fifths as or frame, not 12-ET (although this may be a future option for those that feel they must have it). Your main concern seems to be that one shouldn't use say C# and Db to refer to the same pitch in tunings where, on the basis of the chain of best fifths, they are quite different. Have no fear. I would never propose such a thing. What I am proposing is merely that when C# and Db _are_ different pitches, the notation shouldn't _force_ us to use both names. For example, there will be an alternate way of referring to the Db pitch, that does not involve sharps or flats.
>plus you might just as well use johnny reinhardt's system of notating >12-et offsets, where i as a trombone player am not able to follow -- i can >learn to play 7/6 or 15/14 offsets from a 3/2 or 5/4 harmonic, or to >divide 9/8 into 20/19/18 (or divide small and easily intervals like 9/8 or >10/8 into divisions that i think to be equal, trusting that the >differences don't matter), but my hair rises at the thought of a large >interval divided into a huge number of equal parts. for me. this does not >work as a reference.
As a trombone player you fall way outside the category I mentioned. i.e. players of fixed pitch instruments, so I am not surprised that you would prefer small whole number ratios as your reference points. Rest assured that the proposed notation allows for tunings based on ratios to be notated without reference to _any_ equal temperament, but with reference to strict Pythagorean.
>so, pleasepleasepleeeeease, don't make the saggitals inbetweenies, but >true offsets of true reference pitches (as i think you wanted to do from >the outset).
The thing is that this is a common notation for JI and ETs. The unifying principle is that the same notation should always (or at least as far as possible) correspond to the best available approximation of the same ratio, regardless of whether the tuning is JI or whatever ET. To achieve this, the reference pitches (by which I assume you mean the ones that are notatable with only # or b and no saggitals) are always a chain of the best available 2:3 approximations (unless these are really poor approximations). That means that these reference pitches must be differ between JI and the different ETs. This means that to read the notation one must know what size the fifth is, or some equivalent piece of information such as what size the whole tone is, or whether it is rational (JI/RI) or what ET or linear temperament it is based on. But I don't think you are objecting to this.
>klaus schmirler > >who in general is unable to follow you in detail. i hope you end up >producing a couple of simple lookup tables comparing a couple of gamuts in >different notations (considering different reference intervals: i
Yeah we'll get around to it eventually. I feel we've really settled the notation as far as 90% of possible uses of it go. We are almost into counting angels on the head of a pin, but not quite. We've certainly been counting the angles on the head of a ping.
>can imagine i'd prefer a pythagorean notation for 19-et, but would like >different notations for 5/4 and 81/64 in 31et).
Not quite sure what you mean here. 81/64 (407.8 cents) is only very poorly approximated in 31-ET. Relative to C, its best approximation would be notated as E/|\ (really an upward arrow, which can be read as "half-sharp") and would be 425.8 cents. While 5/4 would be quite accurate as simply E. If however, by 81/64 you mean the note which is (an octave reduced chain of) four fifths away from 1/1, then this would also be notated as simply E, since it is exactly the same note as best approxiates 5/4. I hope I have set your mind at ease. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5166 - Contents - Hide Contents

Date: Tue, 03 Sep 2002 18:50:45

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> At 11:35 AM 29/08/2002 -0700, George Secor wrote:
>> From: George Secor, 8/28/2002 (#4596) >> Subject: A common notation for JI and ETs >> >> (This is a continuation of my message #4586, which is in reply to Dave >> Keenan's message #4543 [should be #4532].) >> Summary of Additional Rational Complements >> ------------------------------------------ >> >> In addition to the seven 217 standard symbol RC pairs and the >> supplementary pair of RCs I listed previously, there are then four >> additional pairs of supplementary symbols in my proposal. These >> rational complements are used for some of the ratios of 17, 19, and 23: >> >> 19 comma and (11'-7)+(19'-19) comma: )| <--> (||~ and (|~ <--> )|| >> 23 comma and 7+17 comma or 5+17' comma: |~ <--> ~||) and ~|) <--> ||~ >> 17+19 comma and 11-5+17 comma or 23' comma: ~)| <--> ~||\ and ~|\ <--> >> ~)|| >> 17+23 comma and 5+(17-17') comma: ~|~ <--> /||( and /|( <--> ~||~ >
> Just to confirm: I have no problem with these complementary symbol pairs. > They have all either been agreed before or fit nicely between what has been > agreed before. They also all agree with a suitably large numbered ET, > 494-ET, in at least one of their comma interpretations. Do they actually > agree with 494 in all the comma interpretations you have given them (not > that it matters very much)?
Within the 17 limit all of the comma aliases agree in 494 without exception, including (| as the 11/7, 17/13, and 29 commas and (|( as the 11/5, 13/7, 17/11, and 19/11 commas. The only difficulty I find in 494 is a relatively minor one: the 23 comma is 7deg, whereas the 19'-19 comma is 8deg, but this is not one of the latest definitions.
>
>> Notation of ETs >> --------------- >> ... >> In the meantime I have discovered that a couple of those that we did >> agree on have properties that now persuade me either to question or >> reject outright the symbol sequences: >> >> 52: /|) /||\ (RC) >> 32: )| /|\ (|) (||~ /||\ (RC) >
> I have no strong attachments to these. As ETs go, they are probably of > marginal interest, and they should be primarily notated as subsets (of 96 > and 104). >
>> for the following reasons. >> >> The 13 comma /|) is not valid as 1deg52. Instead I propose the >> half-apotome symbol of last resort that can usually be made to work >> when nothing else will: >> >> 52a: (|~ /||\ [(11-7)+23 comma] (RC) > >
> I don't see how (|~ is any more valid than /|). What comma (or combination > of commas) did you have in mind?
At first this was what I had for a reply: In 52 /|) is valid only as a 5+7 comma, since the 13 comma (1024:1053) vanishes. But I don't want to use /|) as a 5+7 comma unless it's also valid as the 13 comma, because the symbol will most usually be interpreted as the 13 comma, and its appearance here would be misleading. But I changed my mind (see below).
> I suggest (|( as the 11'-5 comma for > 1deg52.
Again, this is what I first had as a reply: I would want to use (|( only if it were valid as both the 11'-5 and 13'-7 commas, unless there were no other option. My intention is to avoid symbols that could be misleading. My reason for proposing (|~ is twofold: 1) It is approximately a half-apotome and should therefore be a leading choice for that function if neither the 11 or 13 comma can be used; 2) it isn't used to notate any consonances within the 15 limit (or 17 limit for that matter), so its strangeness could be considered an asset in cases such as this. (The first use I find for it is as the 11-19 or 19/11 comma.)
> And we also have (|\ as the single shaft (alternative) symbol for > 2deg52, although 1:7's are so good in 52-ET is almost seems a shame not to > use |) for 2deg52. There's no sensible single-shafter for 3deg52 (to reach > the half-limma without an unwanted # or b when using a 12 note base), > although |( is valid as the 7-5 comma. >
>> However, after doing 69, 76, 86, 93, and 100 (see below), where ) |\ is >> quite useful for the half-apotome, I thought that this might also be a >> possibility: >> >> 52b: )|\ /||\ (RC) >
> Tell me why you'd prefer this 19+(11-5) comma )|\ to the 11'-5 comma (|(.
Again, because it won't be misinterpreted for a 15-limit consonance. However, I just read what I did for 32 (immediately following) and why I did it, and I now agree that (|( could be justified for 52 on the same basis. So we can go with this: 52: (|( /||\ (11-7 comma)
>> With 32 the best we could do for 1deg was the 19 comma, which is quite >> a bit smaller than 1deg52, 37.5 cents. We have subsequently defined >> (|( as the 11'-5 comma (~38.9 cents), which would give us this: >> >> 32: (|( /|\ (|) ~||( /||\ [11'-5 comma] (RC) >
> Yes. I like that.
I consider 32 to be an 11-limit system at best, so I don't think that misinterpretation of (|( as the 13'-7 comma would be a problem here. This is what made me change my mind about 52, above, since 52 could also be considered at best an 11-limit system.
>> Now for some of the larger ETs that we need to reconsider. > ... >
> I'll have to respond to these another time. In general I'm happy to leave > the big ETs to you. But I guess you'd like me to check your results. I'm > more interested in what you think re notation relative to 12-ET or at least > notation of n*12-ETs.
That's the subject of my very next message. --George
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Message: 5167 - Contents - Hide Contents

Date: Tue, 03 Sep 2002 18:58:45

Subject: Re: A common notation for JI and ETs

From: gdsecor

(This is a continuation of my message #4596, which is in reply to 
Dave Keenan's message #4532 [erroneously designated as #4543 in a 
couple of previous postings].)

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> ... > [dk:] > Now to start on the others with 6 or less steps per apotome. > > I won't necessarily include the double-shaft symbols from here on.
You should assume they correspond to the rational complements.
> > We are really having problems with 1deg48 aren't we? > > You wrote:
>> I think that 48 and 55 have sufficiently different properties that >> there would be no reason to insist on doing them alike. Since I >> would do 96 this way: >> >> 96: /| |) /|) /|\ (|\ ||) ||\ /||\ >> >> I wouldn't see any problem with doing 48 as a subset of 96, >> particularly since 7 and 11 are among the best factors represented in >> 48: >> >> 48: |) /|\ ||) /||\ >
> We agree 48 should be every second step of 96, but we haven't
agreed on 96 yet.
> > I agree 48 doesn't _need_ to be the same as either 41 or 55, but it
would be good to minimise the number of different notations for all the scales with 4 steps to the apotome.
> > Both ~|) and ~|\ are consistently 1 degree of 48, 55 and 62-ET, but
of these only ~|) is also 2 degrees of 96-ET.
> > That's one reason why I favour ~|). > > But lets forget 55 and 62 for now. You propose to use |) which is
certainly correct as the 7-comma for both 48 and 96-ET. Why would I want to add the ~| 17-flag to it when this is zero steps?
> > One problem is that we're already using |) as one degree of 36-ET
and 2 degrees of 72-ET. People will naturally attach the meaning of 1/3 semitone to it in this application, and may find it confusing if 48 and 96-ET use it for 1/4 semitone. They are already going to have to get used to the idea that /| can represent anything from 1/5 semitone (in 60-ET) to 1/10 semitone (in 120-ET), not to mention a half semitone in 22-ET, so it's not as if this is something completely unexpected; the idea of the comma sizes changing in different systems is a basic characteristic of the notation.
> This opens a whole other can of worms regarding notation relative
to 12-ET. Lots of people would like to notate their tunings (even those which are not n*12-ETs) as deviations from 12-ET, rather than as deviations from a chain of the tuning's own native fifths (or it may have none).
> > Since people are going to try to do it anyway, shouldn't we look at
standardising a consistent way of doing it? There's no question that this would be well worth doing if all of the flags were fixed sizes, but they aren't. So those who will arrive at their pitches in this way will still have to remember the number of cents a symbol represents in a particular division.
> Some time ago I investigated this in depth and I now offer a first
pass at a spreadsheet that does it semi-automatically. And, you guessed it, it requires 1deg48 and 2deg96 to be ~|).
> > Yahoo groups: /tuning- * [with cont.] math/files/Dave/Notating12ETDeviations.xls.zip > > If you examine the formulae in this spreadsheet you will see that
the principle is that each symbol is given, in a lookup table, a range of cents deviations that it covers. In general the ranges overlap, but there is a strict order of precedence to resolve the cases where more than one symbol could notate the same degree. Determining the ranges was quite tedious, but the main requirement is to ensure that the symbols actually agree with their comma values, given 12-ET fifths. e.g. the changover between one symbol and the next, at the same precedence level, occurs at the point equidistant from their two comma values relative to a chain of 12-ET fifths.
> > But how did I choose which symbols to use in the first place? It's
so long ago I've almost forgotten, but the basic idea was for example, to look at all the n*12-ETs that contained a 25c step and find which symbol corresponded to 25 cents in all of them, and so on.
> > Here's what it gives for all the n*12-ETs whose best fifth is the
12-ET fifth. The dots indicate degrees that cannot be notated.
> > 12: > 24: /|\ > 36: |) > 48: ~|) /|\ > 60: /| |\ > 72: /| |) /|\ > 84: /| |) /|) > 96: /| ~|) |\ /|\ > 108: /| /|( |) /|) > 120: /| (| |) |\ /|\ > 132: ~|( /| |) |\ /|) > 144: ~|( /| ~|) |) /|) /|\ > 156: ~|( /| ~|) |) |\ /|) > 168: ~|( /| /|( |) |\ /|) /|\ > 180: ~|( /| (| ~|) |) |\ /|) > 192: ~|( /| (| ~|) |) |\ /|) /|\ > 204: ~|( /| (| ~|) |) |\ (|) /|\ > 216: ~|( /| (| /|( ~|) |) |\ /|) /|\ > 228: ~|( |( /| /|( ~|) |) |\ /|) (|) /|\ > 240: ~|( |( /| (| ~|) |) ~|\ |\ /|) /|\ > 252: ~|( |( /| (| ~|) |) ~|\ |\ /|) (|) /|\ > 264: ~|( |( /| (| /|( ~|) |) |\ . /|) /|\ > 276: ~|( |( /| (| /|( ~|) |) ~|\ |\ /|) (|) /|\ > 288: ~|( |( /| . (| ~|) |) ~|\ |\ /|) . (|) > 300: ~|( |( /| . (| ~|) |) ~|\ |\ /|) . (|) /|\ > > Notice that this scheme only uses 6 types of flag since it doesn't
go beyond 17-limit. Of course one has to get used to the fact that ~| is negative (-5.0 cents). So that's why ~|) is used for a smaller number of degrees than |). I have a lot of trouble with that and have serious doubts that something of that sort will be acceptable to others. (I could easily imagine someone on the tuning list jumping all over us about that.) The rational complement of ~|) is ||~. Is that what you propose to use in the second half-apotome? (That would require both left and right wavy flags in the 48 and 96 notation, when neither of these is really necessary.) I think that, of the multiples of 12, the ones under 100 tones will be by far the most frequently used. Is it all that necessary to have compatibility between, say 48-ET and 144-ET? This is an example of one of the more complicated things (144) making one of the simpler things (48) more complicated -- I wanted to keep the simpler things simple. I find it much simpler to use only the most familiar symbols for everything below 100: 12 /||\ 24 /|\ /||\ 36: |) ||) /||\ 48: |) /|\ ||) /||\ 60: /| /|) (|\ ||\ /||\ 72: /| |) /|\ ||) ||\ /||\ 84: /| |) /|) (|\ ||) ||\ /||\ 96: /| |) /|) /|\ (|\ ||) ||\ /||\ This is nice and orderly, except for the inconsistent symbol arithmetic in 72-ET, which I don't think will bother anyone (should they even notice). Am I correct in assuming that this pretty well covers all of the multiples of 12 used by any 20th-century composers worthy of mention? As soon as you get to 108, things immediately start getting complicated. How did you arrive at /|( for 2deg108? (I would guess that you treated it as a subset of 216, but I don't find that very appropriate -- 108 is a much better division, relatively speaking.) At first I found that the only thing that works is //|, which gave me the following (using all rational complements): 108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ (RC; ~|| as RC) Even by itself this is rather weird in that //| is a larger interval than |), yet is used for fewer degrees. But at least it's compatible with 36-ET. Both ||) and ~|| are justified as rational complements. And then I came up with this (from the previous message): 108b: /| (|( |) /|) (|\ ||) ~||( ||\ /||\ (RC) Why not just accept the fact that some of the multiples of 12 above 100 are going to be strange instead of passing the strangeness (along with less-used symbols) down to 48 and 96, where you have ~|) as a smaller number of degrees than |\? (I don't even know why we need to consider those above 144 -- they're not 1,3,9 consistent.) Even below 144 you have 120, which is not 5-limit consistent, and 132, which is not 7-limit consistent. In spite of that, I found that 120 could be notated with rational complements: 120: /| (| |) /|) /|\ (|\ ||) )||~ ||\ /||\ (RC) However, 132 is something else. Neither 11 or 13 commas can be used for 5 and 6deg -- I wanted to keep /|\ smaller than (|) -- so I had to use (|~ as 5deg with (| as the 11-7 comma and |~ as the 23 comma and just do the rest with a matched sequence: 132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\ (||~ /||\ (MS) However, if we permit /|\ to be larger than half an apotome, then it could still be done without (|) like this: 132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ (MS) One more thing: I tried notating 144 without referring to any other multiple of 12 and came up with this: 144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ (MS) whereas this is what you have above: 144: ~|( /| ~|) |) /|) /|\ (DK) One problem I have with this is your 4deg symbol -- it doesn't agree with the 5deg symbol, which we both employ as the 13 comma, making |) the 13-5 comma of 3deg. (This is in addition to the problem of the negative value for the wavy left flag.)
> Notice that 276-ET is the largest that can be fully notated, and
that 12,24,36,72 are as previously agreed. We haven't agreed on 60-ET yet, but the proposal above is different from what either of us suggested recently. Your 60-ET proposal has an excessive amount of lateral confusability (although this is not a highly important division): 60: /| |\ /|| ||\ /||\ (DK) One thing that I notice that your grand proposal doesn't do is to make all of the subsets compatible, e.g. 60 relative to 120, so its primary purpose seems to be to ensure consistency in assigning the symbols for the various divisions, which is only one goal among many. In my spreadsheet I placed a higher priority to assigning the 13 comma than the 11-5 comma, so I got the following: 60: /| /|) (|\ ||\ /||\ [13 commas] (RC)
> Notice that 144-ET has bad flag arithmetic, since /| and |) [7
flag] are 2 and 4 steps respectively and thereby agree with 72-ET, but /|) is 5 steps and must be interpreted as the 13 flag. If we are not willing to do this, then we must accept that 144-ET cannot be fully notated in a manner consistent with 72-ET, simply because we don't have a separate symbol for the 13-comma, and the 13-schisma doesn't vanish. I'm not overly excited about 144. Those who want to use it complain about how bad 13 is in 72-ET, yet are willing to overlook the fact that 9 is proportionally even worse in 144, while 13 is inconsistent. Anyway, we still have to be able to notate it. I think it would be best to use |\ instead of |) for 144, which is easy enough to understand (anything with straight flags would belong to 72). For 72 and a few other divisions we allowed ||) to be a rational complement of |), even if the symbol arithmetic was inconsistent by 1 degree. I would disallow it if the inconsistency is greater than 1 degree; in 144 (|)+|) is 10deg, whereas /||\-|) is 8deg. When I did 144, I treated it as a stand-alone division in the simplest way possible, which is repeated here: 144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ (MS) So those are my proposals for the multiples of 12 up to 144. I would say skip those above 144. --George (To be continued.)
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Message: 5168 - Contents - Hide Contents

Date: Wed, 04 Sep 2002 21:10:07

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4605]:
> At 12:03 PM 3/09/2002 -0700, George Secor wrote: >> Dave Keenan:
>> ... Lots of people would like to notate their tunings (even those >> which are not n*12-ETs) as deviations from 12-ET, rather than as >> deviations from a chain of the tuning's own native fifths (or it may >> have none). >>>
>>> Since people are going to try to do it anyway, shouldn't we look at
>> standardising a consistent way of doing it? >> >> There's no question that this would be well worth doing if all of the >> flags were fixed sizes, but they aren't. >
> But they are! Or at least can be. If the notational fifths are always > exactly 700 cents, then every comma can be assigned a fixed size in cents > (different from its size when the fifths are 701.955 cents). And if every > flag is assigned a fixed comma interpretation, then every flag will have a > fixed size in cents. However I'm not insisting totally on that, but merely > that every sagittal _symbol_ has a fixed comma interpretation and hence a > fixed size in cents (except for the messiness with the 13-comma in 144-ET). > > For example: > /| as 5 comma is 13.7 cents. > |) as 7-comma is 31.2 cents. > |\ as 11-5 comma is 37.6 cents. > (| as 11'-7 comma is 17.5 cents. > ~| as 17 comma is -5.0 cents. > |( as 17'-17 comma is 9.9 cents. > |( as 7-5 comma is 17.5 cents. > > In other words, by using this alternative system, even rational tunings > could be notated, relative to 12-ET, instead of relative to Pythagorean.
Okay, now I get your point. That would be a very useful capability for a notation, particularly if conventional instruments are used. But there's a problem, which I will address below.
> > ... I think you've nearly talked me out of using ~|) in 48. In fact > I'd like to avoid ~| altogether if I can, but I notice you're using it in > ~|( in 132 and 144-ET.
I don't know how to avoid it in 144, short of using |( as the 11-13 (or 13/11) comma and disregarding the possibility of its being interpreted as the 7-5 (or 7/5) comma. (I'm starting to appreciate your fractional comma notation now and will be using it more.) It's not a very good division, so maybe we could get away with it. In 132 the situation with the two commas is reversed, which would make it easier to justify (especially considering how badly the primes above 3 are represented, relative to a single degree).
>> I >> find it much simpler to use only the most familiar symbols for >> everything below 100: >> >> 12 /||\ >> 24 /|\ /||\ >> 36: |) ||) /||\ >> 48: |) /|\ ||) /||\ >> 60: /| /|) (|\ ||\ /||\ >> 72: /| |) /|\ ||) ||\ /||\ >> 84: /| |) /|) (|\ ||) ||\ /||\ >> 96: /| |) /|) /|\ (|\ ||) ||\ /||\ >> >> This is nice and orderly, except for the inconsistent symbol arithmetic >> in 72-ET, which I don't think will bother anyone (should they even >> notice). >
> Yes. This looks pretty good (without having considered it in detail).
The only ones that differ from your proposal are 48, 60, and 96. I think it's important not to have any complicated symbols for these, so that would not be an obstacle that would preclude the notation from being considered by European microtonalists.
>> Am I correct in assuming that this pretty well covers all of >> the multiples of 12 used by any 20th-century composers worthy of >> mention? >
> I expect so. But you probably know more about that than me.
Julian Carrillo went up to 96, and I haven't heard of anyone else going past that, except for suggestions on the tuning list to use 144 for the 13 limit to remedy a deficiency of 72.
>> As soon as you get to 108, things immediately start getting >> complicated. How did you arrive at /|( for 2deg108? (I would guess >> that you treated it as a subset of 216, >
> Apparently I arrived at it by making a mistake. There was no intention to > make it a subset of 216. > > By the way, I just noticed that if |( is the 7-5 comma then /|( is the 7 > comma, same as |).
Yes, I remember that if |( is the 17'-17 comma the two intervals differ by only ~0.2 cents and that at first we had |) and /||( as rational complements.
>> but I don't find that very >> appropriate -- 108 is a much better division, relatively speaking.) At >> first I found that the only thing that works is //|, which gave me the >> following (using all rational complements): >> >> 108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ (RC; ~|| as RC) >
> Yes. This looks good to me, (ignoring complements for now). At the time you > had convinced me to avoid using //|. >
>> Even by itself this is rather weird in that //| is a larger interval >> than |), yet is used for fewer degrees. But at least it's compatible >> with 36-ET. Both ||) and ~|| are justified as rational complements. >> >> And then I came up with this (from the previous message): >> >> 108b: /| (|( |) /|) (|\ ||) ~||( ||\ /||\ (RC) >
> I don't understand. What comma would make (|( valid as 2deg108?
The 13'-7 comma; to do this we would have to ignore that this symbol also represents the 11'-5 comma by writing off 11 on account of excessive error. It also involves skipping over a prime (11) in favor of another prime (13) that has almost as great an error, which is not very good. Okay, I agree with you that version 108a with //| is better; after all, 108 is 1,5,25 consistent!
>> Why not just accept the fact that some of the multiples of 12 above 100 >> are going to be strange instead of passing the strangeness (along with >> less-used symbols) down to 48 and 96, where you have ~|) as a smaller >> number of degrees than |\? (I don't even know why we need to consider >> those above 144 -- they're not 1,3,9 consistent.) >
> Yes. I agree we should accept strangeness and avoid passing it
down. And we
> don't really need to notate n*12-ETs above 144. But you're looking at it > purely from the point of view of notating these ETs under the standard > system, while I'm looking for a way to notate almost anything relative to > 12-ET, but which still agrees as much as possible with the standard system. > Do you think such a goal worthwhile? Of course! > Maybe it can be done in a way that agrees with all that you propose here > for the n*12-ETs. Care to put your mind to it? My spreadsheet might be made > to generate all the notations you suggest, by tweaking ranges and precedences.
Okay, I'll have to give this some thought. (But I'm a bit skeptical about anything above 144.)
>> Even below 144 you have 120, which is not 5-limit consistent, and 132, >> which is not 7-limit consistent. In spite of that, I found that 120 >> could be notated with rational complements: >> >> 120: /| (| |) /|) /|\ (|\ ||) )||~ ||\ /||\ (RC) >
> Yes, that works well. >
>> However, 132 is something else. Neither 11 or 13 commas can be used >> for 5 and 6deg -- I wanted to keep /|\ smaller than (|) -- so I had to >> use (|~ as 5deg with (| as the 11-7 comma and |~ as the 23 comma and >> just do the rest with a matched sequence: >> >> 132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\ (||~ /||\ (MS) >> >> However, if we permit /|\ to be larger than half an apotome, then it >> could still be done without (|) like this: >> >> 132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ (MS) >
> What comma interpretation of (|~ could possibly make it valid as 5deg132? > For 5deg132 I see only (|( as the (13'-(11-5))+(17'-17) comma which is > incompatible with |) as 7 comma, and /|) as 5+7 comma which we agreed not > to use, and (|) as 11' comma which you don't want to use if it's smaller > than a half-apotome. > > Wait a minute, I guess you're proposing (11'-7)+23 for (|~. I really hate > to go to 23 limit to notate ETs, but I guess this is one case where it > could be justified. Does it validly replace (|) everywhere I've proposed > it, for n*12-ETs?
I consider (|~ the half-apotome symbol of last resort. When you're doing the difficult ETs you can usually get the required number of degrees with one of the following: a) (11'-7)+23 diesis; b) (11'-7)+(19'-19) diesis; c) (13'-(11-5))+23 diesis; d) (13'-(11-5))+(19'-19) diesis; e) 11-19 (or 19/11) diesis provided, of course, that the flag usage does not conflict with any other symbols being used. But to answer your question, I started to reply: In 204 (|~ validly replaces (|) as the (11'-7)+23 diesis. Then I got no farther, because I noticed that your notation for 204 has a degree missing: 204: ~|( /| (| ~|) |) ?|? |\ (|~ /|\ for which I suggest: 204: ~|( /| (| ~|) |) ~|\ |\ (|~ /|\ Then I found that you have an inconsistency in 228; the flags for /|( don't add up: 228: ~|( |( /| /|( ~|) |) |\ /|) (|) /|\ Anything above 144 is 1,3,9 inconsistent, and the higher you go the worse it gets. So I don't see much point in trying to notate any of these divisions.
>> ... One thing that I notice that your grand proposal doesn't do
is to make
>> all of the subsets compatible, e.g. 60 relative to 120, so its primary >> purpose seems to be to ensure consistency in assigning the symbols for >> the various divisions, which is only one goal among many. >
> Its primary purpose is to be able to notate anything (not merely the > multiples of 12-ET) relative to 12-ET. One simply substitutes the 12-ET > fifth for the precise 2:3, as the backbone of the notation, and then > carries on as before, while keeping the same comma interpretations for the > symbols. > > One consequence of this, that I've been ignoring until now, is that to > properly notate rational tunings in this system, you would need a symbol > for a 3-comma of 1.955 cents. We could redefine )| to serve this purpose, > and limit this alternative 12-ET-based system to the 17-limit.
It's not that simple. Once you establish your base pitch -- G, for example -- then C will be raised by a 3-comma, F by two 3-commas, B- flat by three, E-flat by four, etc. You would therefore need a way to notate multiple 3-commas. And for every n-ET that's not a multiple of 12 you would also need a pseudo-3-comma in the notation corresponding to the difference between the fifth of n-ET and 12-ET -- or am I missing something? (You mentioned rounding off the symbols to a fixed size above, but that was for multiples of 12.) --George
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Message: 5169 - Contents - Hide Contents

Date: Wed, 04 Sep 2002 12:54:27

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 12:03 PM 3/09/2002 -0700, George Secor wrote:
>Dave Keenan:
>> One problem is that we're already using |) as one degree of 36-ET and
>2 degrees of 72-ET. People will naturally attach the meaning of 1/3 >semitone to it in this application, and may find it confusing if 48 and >96-ET use it for 1/4 semitone. > >They are already going to have to get used to the idea that /| can >represent anything from 1/5 semitone (in 60-ET) to 1/10 semitone (in >120-ET),
OK, yes. The latter is a variation of 10 cents where the former is a variation of only 8.3 cents, so my argument falls down there, except that 1/3 and 1/4 semitone are likely to be more commonly used than 1/5 and certainly more than 1/10.
>not to mention a half semitone in 22-ET, > so it's not as if >this is something completely unexpected; the idea of the comma sizes >changing in different systems is a basic characteristic of the >notation.
22-ET is irrelevant here. Certainly when the fifth changes size we expect the commas to be different sizes, but we're talking here about the same 12-ET fifth all the way through.
>> This opens a whole other can of worms regarding notation relative to
>12-ET. Lots of people would like to notate their tunings (even those >which are not n*12-ETs) as deviations from 12-ET, rather than as >deviations from a chain of the tuning's own native fifths (or it may >have none). >>
>> Since people are going to try to do it anyway, shouldn't we look at
>standardising a consistent way of doing it? > >There's no question that this would be well worth doing if all of the >flags were fixed sizes, but they aren't.
But they are! Or at least can be. If the notational fifths are always exactly 700 cents, then every comma can be assigned a fixed size in cents (different from its size when the fifths are 701.955 cents). And if every flag is assigned a fixed comma interpretation, then every flag will have a fixed size in cents. However I'm not insisting totally on that, but merely that every sagittal _symbol_ has a fixed comma interpretation and hence a fixed size in cents (except for the messiness with the 13-comma in 144-ET). For example: /| as 5 comma is 13.7 cents. |) as 7-comma is 31.2 cents. |\ as 11-5 comma is 37.6 cents. (| as 11'-7 comma is 17.5 cents. ~| as 17 comma is -5.0 cents. |( as 17'-17 comma is 9.9 cents. |( as 7-5 comma is 17.5 cents. In other words, by using this alternative system, even rational tunings could be notated, relative to 12-ET, instead of relative to Pythagorean.
> So those who will arrive at >their pitches in this way will still have to remember the number of >cents a symbol represents in a particular division.
Yes but it will simply be the rounding off of the symbol's fixed size, to the nearest whole division. As I wrote before:
>> e.g. the changover between one symbol and the next, at the same
>precedence level, occurs at the point equidistant from their two comma >values relative to a chain of 12-ET fifths. >>
Actually, this is true in many cases, even when they are not at the same precedence level.
>> Here's what it gives for all the n*12-ETs whose best fifth is the
>12-ET fifth. The dots indicate degrees that cannot be notated. >> >> 12: >> 24: /|\ >> 36: |)
>> 48: ~|) /|\ >> 60: /| |\ >> 72: /| |) /|\ >> 84: /| |) /|) >> 96: /| ~|) |\ /|\ >> 108: /| /|( |) /|) >> 120: /| (| |) |\ /|\ >> 132: ~|( /| |) |\ /|) >> 144: ~|( /| ~|) |) /|) /|\ >> 156: ~|( /| ~|) |) |\ /|) >> 168: ~|( /| /|( |) |\ /|) /|\ >> 180: ~|( /| (| ~|) |) |\ /|) >> 192: ~|( /| (| ~|) |) |\ /|) /|\ >> 204: ~|( /| (| ~|) |) |\ (|) /|\ >> 216: ~|( /| (| /|( ~|) |) |\ /|) /|\ >> 228: ~|( |( /| /|( ~|) |) |\ /|) (|) /|\ >> 240: ~|( |( /| (| ~|) |) ~|\ |\ /|) /|\ >> 252: ~|( |( /| (| ~|) |) ~|\ |\ /|) (|) /|\ >> 264: ~|( |( /| (| /|( ~|) |) |\ . /|) /|\ >> 276: ~|( |( /| (| /|( ~|) |) ~|\ |\ /|) (|) /|\ >> 288: ~|( |( /| . (| ~|) |) ~|\ |\ /|) . (|) >> 300: ~|( |( /| . (| ~|) |) ~|\ |\ /|) . (|) /|\ >> >> Notice that this scheme only uses 6 types of flag since it doesn't go
>beyond 17-limit. Of course one has to get used to the fact that ~| is >negative (-5.0 cents).
This scheme may have to change anyway, given the redefinition of |( as the 7-5 comma.
>So that's why ~|) is used for a smaller number of degrees than |). I >have a lot of trouble with that and have serious doubts that something >of that sort will be acceptable to others. (I could easily imagine >someone on the tuning list jumping all over us about that.)
So can I. But they don't appear together until you get to 144-ET. Maybe there's a better solution that still preserves my goal.
>The rational complement of ~|) is ||~. Is that what you propose to use >in the second half-apotome? (That would require both left and right >wavy flags in the 48 and 96 notation, when neither of these is really >necessary.)
I agree this is probably a bad idea. I actually worked out a set of complementary pairs based on minimising the offsets of rational complements, but using their cent values relative to chains of 12-ET fifths, rather than 2:3s. It introduces no new flags nor any new flag combinations. ~|( (|) |( /|) /| |\ (| ~|\ /|( |) ~|) ~|) These are meant to indicate complementary pairs when a shaft is added to one side or the other. also /|) (|\ without adding a shaft to any side. These agree with 276-ET the largest notatable multiple of 12.
>I think that, of the multiples of 12, the ones under 100 tones will be >by far the most frequently used. Sure. >Is it all that necessary to have >compatibility between, say 48-ET and 144-ET? No. > This is an example of one >of the more complicated things (144) making one of the simpler things >(48) more complicated -- I wanted to keep the simpler things simple.
Good point. I think you've nearly talked me out of using ~|) in 48. In fact I'd like to avoid ~| altogether if I can, but I notice you're using it in ~|( in 132 and 144-ET.
> I >find it much simpler to use only the most familiar symbols for >everything below 100: > >12 /||\ >24 /|\ /||\ >36: |) ||) /||\ >48: |) /|\ ||) /||\ >60: /| /|) (|\ ||\ /||\ >72: /| |) /|\ ||) ||\ /||\ >84: /| |) /|) (|\ ||) ||\ /||\ >96: /| |) /|) /|\ (|\ ||) ||\ /||\ > >This is nice and orderly, except for the inconsistent symbol arithmetic >in 72-ET, which I don't think will bother anyone (should they even >notice).
Yes. This looks pretty good (without having considered it in detail).
> Am I correct in assuming that this pretty well covers all of >the multiples of 12 used by any 20th-century composers worthy of >mention?
I expect so. But you probably know more about that than me.
>As soon as you get to 108, things immediately start getting >complicated. How did you arrive at /|( for 2deg108? (I would guess >that you treated it as a subset of 216,
Apparently I arrived at it by making a mistake. There was no intention to make it a subset of 216. By the way, I just noticed that if |( is the 7-5 comma then /|( is the 7 comma, same as |).
>but I don't find that very >appropriate -- 108 is a much better division, relatively speaking.) At >first I found that the only thing that works is //|, which gave me the >following (using all rational complements): > >108a: /| //| |) /|) (|\ ||) ~|| ||\ /||\ (RC; ~|| as RC)
Yes. This looks good to me, (ignoring complements for now). At the time you had convinced me to avoid using //|.
>Even by itself this is rather weird in that //| is a larger interval >than |), yet is used for fewer degrees. But at least it's compatible >with 36-ET. Both ||) and ~|| are justified as rational complements. > >And then I came up with this (from the previous message): > >108b: /| (|( |) /|) (|\ ||) ~||( ||\ /||\ (RC)
I don't understand. What comma would make (|( valid as 2deg108?
>Why not just accept the fact that some of the multiples of 12 above 100 >are going to be strange instead of passing the strangeness (along with >less-used symbols) down to 48 and 96, where you have ~|) as a smaller >number of degrees than |\? (I don't even know why we need to consider >those above 144 -- they're not 1,3,9 consistent.)
Yes. I agree we should accept strangeness and avoid passing it down. And we don't really need to notate n*12-ETs above 144. But you're looking at it purely from the point of view of notating these ETs under the standard system, while I'm looking for a way to notate almost anything relative to 12-ET, but which still agrees as much as possible with the standard system. Do you think such a goal worthwhile? Maybe it can be done in a way that agrees with all that you propose here for the n*12-ETs. Care to put your mind to it? My spreadsheet might be made to generate all the notations you suggest, by tweaking ranges and precedences.
>Even below 144 you have 120, which is not 5-limit consistent, and 132, >which is not 7-limit consistent. In spite of that, I found that 120 >could be notated with rational complements: > >120: /| (| |) /|) /|\ (|\ ||) )||~ ||\ /||\ (RC)
Yes, that works well.
>However, 132 is something else. Neither 11 or 13 commas can be used >for 5 and 6deg -- I wanted to keep /|\ smaller than (|) -- so I had to >use (|~ as 5deg with (| as the 11-7 comma and |~ as the 23 comma and >just do the rest with a matched sequence: > >132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\ (||~ /||\ (MS) > >However, if we permit /|\ to be larger than half an apotome, then it >could still be done without (|) like this: > >132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ (MS)
What comma interpretation of (|~ could possibly make it valid as 5deg132? For 5deg132 I see only (|( as the (13'-(11-5))+(17'-17) comma which is incompatible with |) as 7 comma, and /|) as 5+7 comma which we agreed not to use, and (|) as 11' comma which you don't want to use if it's smaller than a half-apotome. Wait a minute, I guess you're proposing (11'-7)+23 for (|~. I really hate to go to 23 limit to notate ETs, but I guess this is one case where it could be justified. Does it validly replace (|) everywhere I've proposed it, for n*12-ETs?
>One more thing: I tried notating 144 without referring to any other >multiple of 12 and came up with this: > >144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ >(MS)
I agree this is valid.
>whereas this is what you have above: > >144: ~|( /| ~|) |) /|) /|\ (DK) > >One problem I have with this is your 4deg symbol -- it doesn't agree >with the 5deg symbol, which we both employ as the 13 comma, making |) >the 13-5 comma of 3deg.
I pointed that out myself and explained why I'd done it. You respond to it below. I now agree it was a bad idea.
> (This is in addition to the problem of the >negative value for the wavy left flag.) >
>> Notice that 276-ET is the largest that can be fully notated, and that
>12,24,36,72 are as previously agreed. We haven't agreed on 60-ET yet, >but the proposal above is different from what either of us suggested >recently. > >Your 60-ET proposal has an excessive amount of lateral confusability >(although this is not a highly important division): > >60: /| |\ /|| ||\ /||\ (DK) > >One thing that I notice that your grand proposal doesn't do is to make >all of the subsets compatible, e.g. 60 relative to 120, so its primary >purpose seems to be to ensure consistency in assigning the symbols for >the various divisions, which is only one goal among many.
Its primary purpose is to be able to notate anything (not merely the multiples of 12-ET) relative to 12-ET. One simply substitutes the 12-ET fifth for the precise 2:3, as the backbone of the notation, and then carries on as before, while keeping the same comma interpretations for the symbols. One consequence of this, that I've been ignoring until now, is that to properly notate rational tunings in this system, you would need a symbol for a 3-comma of 1.955 cents. We could redefine )| to serve this purpose, and limit this alternative 12-ET-based system to the 17-limit.
> In my >spreadsheet I placed a higher priority to assigning the 13 comma than >the 11-5 comma, so I got the following: > >60: /| /|) (|\ ||\ /||\ [13 commas] (RC) >
>> Notice that 144-ET has bad flag arithmetic, since /| and |) [7 flag]
>are 2 and 4 steps respectively and thereby agree with 72-ET, but /|) is >5 steps and must be interpreted as the 13 flag. If we are not willing >to do this, then we must accept that 144-ET cannot be fully notated in >a manner consistent with 72-ET, simply because we don't have a separate >symbol for the 13-comma, and the 13-schisma doesn't vanish. > >I'm not overly excited about 144. Those who want to use it complain >about how bad 13 is in 72-ET, yet are willing to overlook the fact that >9 is proportionally even worse in 144, while 13 is inconsistent. >Anyway, we still have to be able to notate it. > >I think it would be best to use |\ instead of |) for 144, which is easy >enough to understand (anything with straight flags would belong to 72). OK. > For 72 and a few other divisions we allowed ||) to be a rational >complement of |), even if the symbol arithmetic was inconsistent by 1 >degree. I would disallow it if the inconsistency is greater than 1 >degree; in 144 (|)+|) is 10deg, whereas /||\-|) is 8deg. > >When I did 144, I treated it as a stand-alone division in the simplest >way possible, which is repeated here: > >144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ >(MS)
I agree this is valid. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5170 - Contents - Hide Contents

Date: Wed, 04 Sep 2002 13:18:34

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 12:02 PM 3/09/2002 -0700, George Secor wrote:
>>> The 13 comma /|) is not valid as 1deg52. Instead I propose the >>> half-apotome symbol of last resort that can usually be made to work >>> when nothing else will: >>> >>> 52a: (|~ /||\ [(11-7)+23 comma] (RC) >> >>
>> I don't see how (|~ is any more valid than /|). What comma (or >combination
>> of commas) did you have in mind? >
>At first this was what I had for a reply: In 52 /|) is valid only as a >5+7 comma, since the 13 comma (1024:1053) vanishes. But I don't want >to use /|) as a 5+7 comma unless it's also valid as the 13 comma, >because the symbol will most usually be interpreted as the 13 comma, >and its appearance here would be misleading. But I changed my mind >(see below).
I actually meant, "What comma did you have in mind for (|~"? I've figured it out myself now. It's (11'-7)+23 right?
>> I suggest (|( as the 11'-5 comma for >> 1deg52. >
>Again, this is what I first had as a reply: I would want to use (|( >only if it were valid as both the 11'-5 and 13'-7 commas, unless there >were no other option. My intention is to avoid symbols that could be >misleading.
Sure but we're allowed to use /|) when it is only the 13 comma and not the 5+7 comma, so why not similarly prefer 11'-5 for (|( because it has the lowest product complexity?
>My reason for proposing (|~ is twofold: 1) It is >approximately a half-apotome and should therefore be a leading choice >for that function if neither the 11 or 13 comma can be used; 2) it >isn't used to notate any consonances within the 15 limit (or 17 limit >for that matter), so its strangeness could be considered an asset in >cases such as this. (The first use I find for it is as the 11-19 or >19/11 comma.)
That's a new one for me. But wait a minute, the 11-19 comma is 3deg52, not 1deg52. So you must be using (11'-7)+23.
>> And we also have (|\ as the single shaft (alternative) symbol for >> 2deg52, although 1:7's are so good in 52-ET is almost seems a shame >not to
>> use |) for 2deg52. There's no sensible single-shafter for 3deg52 (to >reach
>> the half-limma without an unwanted # or b when using a 12 note base), > >> although |( is valid as the 7-5 comma. >>
>>> However, after doing 69, 76, 86, 93, and 100 (see below), where )|\ >is
>>> quite useful for the half-apotome, I thought that this might also be >a >>> possibility: >>> >>> 52b: )|\ /||\ (RC) >>
>> Tell me why you'd prefer this 19+(11-5) comma )|\ to the 11'-5 comma >(|(. >
>Again, because it won't be misinterpreted for a 15-limit consonance. >However, I just read what I did for 32 (immediately following) and why >I did it, and I now agree that (|( could be justified for 52 on the >same basis. So we can go with this: > >52: (|( /||\ (11-7 comma)
OK, but don't you mean 11'-5 comma?
>>> With 32 the best we could do for 1deg was the 19 comma, which is >quite
>>> a bit smaller than 1deg52, 37.5 cents. We have subsequently defined >>> (|( as the 11'-5 comma (~38.9 cents), which would give us this: >>> >>> 32: (|( /|\ (|) ~||( /||\ [11'-5 comma] (RC) >>
>> Yes. I like that. >
>I consider 32 to be an 11-limit system at best, so I don't think that >misinterpretation of (|( as the 13'-7 comma would be a problem here. >This is what made me change my mind about 52, above, since 52 could >also be considered at best an 11-limit system. OK.
-- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5171 - Contents - Hide Contents

Date: Wed, 04 Sep 2002 14:56:33

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 10:14 AM 27/08/2002 -0700, George Secor wrote:
>But then the fifth of 494 is closer to an exact 2:3 than that of 282: > >217: ~702.304c -- 0.349c or 0.063deg wide >288: ~702.128c -- 0.173c or 0.041deg wide >494: ~702.024c -- 0.069c or 0.029deg wide >2:3 ~701.955c > >Yet 494 uses the virtually the same schismas as 217.
You keep writing 288 when it's 282, but you've got the right fifth size for 282, so this is very curious. I'll need to wait until I have more time, to understand it.
>[dk] >You're only looking at the primes themselves. What about the ratios >between them. 217-ET has a 2.8 cent error in its 7:11 whereas 282-ET >never gets worse than that 2.0 cents in the 1:13. >> > >This is only because the error in either one can never exceed half a >system degree, which for 282 is ~2.127 cents, but in 217 is ~2.765c. >So whatever advantage 282 has is only because it divides the octave >into more parts, which would be an advantage in itself. Yes! > But there is >more than this to take into consideration -- something that will >demonstrate that it is better to have the error of the primes >distributed in both directions rather than in a single direction, given >that prime-limit consistency is maintained in each case. For >situations involving schisma consistency, sometimes the error of two >primes will accumulate rather than cancel, so that large unidirectional >errors added together exceed 1/2 degree, resulting in an inconsistency.
You lost me. Maybe if you try to explain it without using the word consistency? I don't understand what schisma coinsistency is.
>Both 72 and 282 are consistent to at least the 17 limit (as are 217 and >494). Since the tridecimal schisma (4095:4096, ~0.423c) vanishes in >our notation but in neither ET, we cannot notate *both* ratios of 7 and >13 consistently in either one. I found this schisma at least a week >before I considered 217 as a basis for mapping out the symbols, so we >can't say that its selection was 217-biased; indeed it vanishes in a >majority of the best ETs above 100.
Might not your decision as to which ETs above 100 are best, be biased towards those in which this schisma vanishes?
>The fact that it doesn't vanish in either 72 or 282 is a consequence of >the relatively large error for 13 (approaching the maximum) that I >referred to above.
Are there none which have good 13s (relative to their step size) without this schisma vanishing?
> Since the functional 13 diesis (1024:1053) is >computed as the number of degrees (rounded) in the best fifth times 4, >less the number of degrees in 3 octaves, plus the number of degrees >(rounded) for 8:13, we can calculate the number of degrees for each of >four divisions as follows: > >Interval deg72 deg282 deg217 deg494 >-------- ------ ------- ------- ------- >fifth (2:3) 42.117 164.959 126.937 288.971 >rounded 42 165 127 289 >times 4 168 660 508 1156 >less 3 octaves -216 -846 -651 -1482 >equals -48 -186 -143 -326 >plus 8:13 rounded 50 198 152 346 >equals 13 diesis 2 12 9 20 > >We then calculate the number of degrees in the 5+7 comma for each: > >Interval deg72 deg282 deg217 deg494 >-------- ------ ------- ------- ------- >5 comma 80:81 1 5 4 9 >7 comma 63:64 2 6 5 11 >5+7 diesis 35:36 3 11 9 20 > >and compare these with the actual (as opposed to functional) number of >degrees for 1024:1053, the ratio of the 13 diesis: > >Interval deg72 deg282 deg217 deg494 >-------- ------ ------- ------- ------- >actual 13 diesis 2.901 11.362 8.743 19.903 >rounded 3 11 9 20 > >for which we find complete agreement in all four divisions, as opposed >to the functional values calculated above: > >Interval deg72 deg282 deg217 deg494 >-------- ------ ------- ------- ------- >funct'l 13 diesis 2 12 9 20 > >We see that there is indeed an inconsistency in both the 72 and 282 >divisions in that the number of degrees in the functional 13 diesis >does not agree with the number of degrees for the actual interval; this >inconsistency exists apart from the tridecimal schisma, but it happens >to cause this schisma not to vanish. This is due principally to the >excessive relative error in the representation of 13: > >Interval deg72 deg282 deg217 deg494 >-------- ------ ------- ------- ------- >actual 8:13 50.432 197.524 151.995 346.017 >8:13 rounded 50 198 152 346 >error in degrees -0.432 0.476 0.005 -0.017 > >I don't think that we would want to devise a system of notation in >order to work around an inconsistency such as this, because I expect >that we would then have some problems notating those ETs in which the >tridecimal schisma *does* vanish. Our goal should be to make the >smallest schismas vanish. > >As for what schismas do vanish in 282, maybe Gene would best be able to >answer that. I thought that it was most productive to start with >rational intervals, find the most useful schismas that can vanish, and >then look for ETs that are consistent with those schismas. Working >backwards by starting with a large-number ET and then finding the >schismas that vanish in that ET is something that I don't have much >experience with, and I have a feeling that we're not going to find >anything better in 282 that will be useful in devising a notation that >offers a better economy of symbols.
I dusted off a spreadsheet I made way back near the start of this project. It comes at it from the direction you suggested. I figure a schisma is unlikely to be useful for notation if any prime has too high a power or if it involves too many primes (with non-zero powers). So I first found all the 31 limit schismas smaller than 1 cent that have no exponent with an absolute value greater than 1 for the primes 7 thru 31, and none greater than 2 for the prime 5. I then whittled that down to those where the sum of the absolute exponents of the primes 5 to 31 is no greater than 4. I then look at a selection of ETs to see in which of them each schisma vanishes. Let me know if you want a copy of it. The lowest prime-limit schisma I found that vanishes in 282-ET but not in 217-ET is 452608:452709 = 2^-11 * 3^9 * 13^-1 * 17^-1 * 23^-1 0.39 cents This says that the 13 comma is approximately equal to the 17 comma plus the 23' comma. For this to be useful, the notation would have to have both the 17 comma and the 23' comma as single flags. The 23' comma is 40.0 cents. It doesn't seem like a single flag of 40 cents would lead to a very economical notation. So now that I've investigated it, I think the vanishing of the 13-schisma 4095:4096 has a big impact on making the notation economical. So your discovery of this fact is very significant. It's also bloody annoying at times, not being able to have a 7 comma at the same time as a 13 comma in ETs where this schisma doesn't vanish.
>>> Our latest agreement has been on mostly ETs below 100, and I don't >>> think any of those even used |(. The larger-numbered ones were >still
>>> subject to review at the time you took your break, so they are still >>> open to review. >>
>> We agreed on |( for 1deg67 which is wrong (or at least not >> 1,3,5,7-consistently right) if |( is the 7-5 comma. I also proposed >it for
>> 93-ET (3*31) but we didn't agree on a notation for that. >
>It is valid as 1deg67 for the 11-13 comma, but I would prefer not to >use |( here (or elsewhere) unless it were valid for *both* the 7-5 and >11-13 commas.
I'd be happy to use it if it were only valid as the 7-5 comma (lowest product complexity) just like we are allowed to use /|) when it is only the 13 comma and not the 5+7 comma, provided we do not also use |) or (| as 7 commas. I'm not sure what the corresponding proviso is for |( as only the 7-5 comma.
> In addition, if *both* the 17 ~| and 17' ~|( symbols >were to occur in an ET notation, then it would also have to be valid as >the 17'-17 comma in order to maintain consistent symbol arithmetic. >(At least that's the ideal I'm shooting for.)
Oh yeah, that's the sort of thing.
>Anyway, after looking at 67 again, I don't see any clear choice for >1deg among several possibilities. I would prefer to do the easier ETs >first (again) and in the process establish a hierarchy of rules for >choosing the symbols. As we attempt to do increasingly difficult ones, >we should get a better perspective on how to handle problems such as >this one. > >I'll be discussing these issues in more detail in my next message, when >I will again address the hows and whys of notating some of the less >difficult ETs. OK.
>> But the whole 282-ET schisma question still haunts me. >
>Did I deal with it above adequately?
I didn't really follow it, but thanks for trying. You obviously put a lot of effort into it. That, and my own investigation with the abovementioned spreadsheet have convinced me that 282-ET schismas (that do not also vanish in 217-ET) are extremely unlikely to produce a more economical notation. The 13-schisma is very significant in this regard, because it kicks in at such a low prime limit. It is the one with the lowest prime-limit of all those I found in my search, as described above. So we can forget about 282-ET schismas. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5173 - Contents - Hide Contents

Date: Thu, 05 Sep 2002 09:01:40

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 02:13 PM 4/09/2002 -0700, George Secor wrote:
>> For example: >> /| as 5 comma is 13.7 cents. >> |) as 7-comma is 31.2 cents. >> |\ as 11-5 comma is 37.6 cents. >> (| as 11'-7 comma is 17.5 cents. >> ~| as 17 comma is -5.0 cents. >> |( as 17'-17 comma is 9.9 cents. >> |( as 7-5 comma is 17.5 cents. >> >> In other words, by using this alternative system, even rational >tunings
>> could be notated, relative to 12-ET, instead of relative to >Pythagorean. >
>Okay, now I get your point. That would be a very useful capability for >a notation, particularly if conventional instruments are used. But >there's a problem, which I will address below. OK.
>> ... I think you've nearly talked me out of using ~|) in 48. In fact >> I'd like to avoid ~| altogether if I can, but I notice you're using >it in
>> ~|( in 132 and 144-ET. >
>I don't know how to avoid it in 144, short of using |( as the 11-13 (or >13/11) comma and disregarding the possibility of its being interpreted >as the 7-5 (or 7/5) comma. (I'm starting to appreciate your fractional >comma notation now and will be using it more.) It's not a very good >division, so maybe we could get away with it.
I don't think there's really any alternative to using ~| in combination with other flags.
>The only ones that differ from your proposal are 48, 60, and 96. I >think it's important not to have any complicated symbols for these, so >that would not be an obstacle that would preclude the notation from >being considered by European microtonalists.
Fair enough. But lets wait until we look at my less ambitious proposal below,before finalising the n*12-ETs.
>>> Am I correct in assuming that this pretty well covers all of >>> the multiples of 12 used by any 20th-century composers worthy of >>> mention? >>
>> I expect so. But you probably know more about that than me. >
>Julian Carrillo went up to 96, and I haven't heard of anyone else going >past that, except for suggestions on the tuning list to use 144 for the >13 limit to remedy a deficiency of 72.
That's my understanding as well. But you might check Joe Monzos Equal Temperament web page. Sorry I don't have the URL handy.
>> I don't understand. What comma would make (|( valid as 2deg108? >
>The 13'-7 comma; to do this we would have to ignore that this symbol >also represents the 11'-5 comma by writing off 11 on account of >excessive error. It also involves skipping over a prime (11) in favor >of another prime (13) that has almost as great an error, which is not >very good. Okay, I agree with you that version 108a with //| is >better; after all, 108 is 1,5,25 consistent!
Good. In general I would prefer to use (|( to represents 11'-5.
>> ... I'm looking for a way to notate almost anything >relative to
>> 12-ET, but which still agrees as much as possible with the standard >system.
>> Do you think such a goal worthwhile? > >Of course! >
>> Maybe it can be done in a way that agrees with all that you propose >here
>> for the n*12-ETs. Care to put your mind to it? My spreadsheet might >be made
>> to generate all the notations you suggest, by tweaking ranges and >precedences. >
>Okay, I'll have to give this some thought. (But I'm a bit skeptical >about anything above 144.) ... >I consider (|~ the half-apotome symbol of last resort. When you're >doing the difficult ETs you can usually get the required number of >degrees with one of the following: > >a) (11'-7)+23 diesis; >b) (11'-7)+(19'-19) diesis; >c) (13'-(11-5))+23 diesis; >d) (13'-(11-5))+(19'-19) diesis; >e) 11-19 (or 19/11) diesis
Yikes! With so many possible interpretations it becomes so ambiguous as to be meaningless.
>provided, of course, that the flag usage does not conflict with any >other symbols being used. > >But to answer your question, I started to reply: In 204 (|~ validly >replaces (|) as the (11'-7)+23 diesis. Then I got no farther, because >I noticed that your notation for 204 has a degree missing: > >204: ~|( /| (| ~|) |) ?|? |\ (|~ /|\ > >for which I suggest: > >204: ~|( /| (| ~|) |) ~|\ |\ (|~ /|\ > >Then I found that you have an inconsistency in 228; the flags for /|( >don't add up: > >228: ~|( |( /| /|( ~|) |) |\ /|) (|) /|\ > >Anything above 144 is 1,3,9 inconsistent, and the higher you go the >worse it gets. So I don't see much point in trying to notate any of >these divisions.
OK. Forget 'em.
>> One consequence of this, that I've been ignoring until now, is that >to
>> properly notate rational tunings in this system, you would need a >symbol
>> for a 3-comma of 1.955 cents. We could redefine )| to serve this >purpose,
>> and limit this alternative 12-ET-based system to the 17-limit. >
>It's not that simple. Once you establish your base pitch -- G, for >example -- then C will be raised by a 3-comma, F by two 3-commas, >B-flat by three, E-flat by four, etc. You would therefore need a way >to notate multiple 3-commas. And for every n-ET that's not a multiple >of 12 you would also need a pseudo-3-comma in the notation >corresponding to the difference between the fifth of n-ET and 12-ET -- >or am I missing something? (You mentioned rounding off the symbols to >a fixed size above, but that was for multiples of 12.)
Yeah. You're right. You could possibly get away with symbols for only 3, 9 and 27 commas with many rational tunings, but we'd also need 5/3, 7/3, 11/3, 13/3, 17/3 comma symbols and 5/9, 7/9, 11/9, 13/9, 17/9 etc. OK. I see now that that's way too ambitious. I'm happy to forget being able to notate rational tunings precisely in this system and reduce the goal to one of being able to notate any tuning to within about 2.5 cents. So, in other words, we should have a way of interpreting a certain set of single-shaft symbols (about 13 of them) as specific offsets from 12-ET between about 2.5 and 60 cents (an alternative to writing plus or minus cents next to the notes) while preserving their (preferably lowest product complexity) comma meanings. Do you want to propose a set of symbols to do that? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5174 - Contents - Hide Contents

Date: Thu, 5 Sep 2002 01:25:54

Subject: Re: A common notation for JI and ETs

From: monz

> From: "David C Keenan" <d.keenan@xx.xxx.xx> > To: "George Secor" <gdsecor@xxxxx.xxx> > Cc: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, September 04, 2002 4:01 PM > Subject: [tuning-math] Re: A common notation for JI and ETs > At 02:13 PM 4/09/2002 -0700, George Secor wrote: > > ... >
>> Julian Carrillo went up to 96, and I haven't heard >> of anyone else going past that, except for suggestions >> on the tuning list to use 144 for the 13 limit to >> remedy a deficiency of 72. >
> That's my understanding as well. But you might check > Joe Monzos Equal Temperament web page. Sorry I don't > have the URL handy. Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.)
also note that Dan Stearns was a prominent advocate of 144 for a period back around 1999, and i joined with him. both of us liked Dan's 144 notation not specifically for its remediation of the 13-limit deficiency of 72, but rather because we both felt that 144 was a useful representation of the entire virtual pitch continuum. for example, i used it as an aid in notating the very complex JI tuning in my piece _A Noiseless Patient Spider_. Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) see my 144-EDO page: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) -monz "all roads lead to n^0"
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