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Message: 5225 - Contents - Hide Contents

Date: Thu, 19 Sep 2002 10:20:55

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 11:45 AM 18/09/2002 -0700, George Secor wrote:

>After reading through them again a couple of times (to make sure I
>understand you correctly), I'm ready to throw up my hands.  I never
>imagined that anyone would have a problem with the notation of 19-ET,
>but now you're saying that sometimes a sharp won't do for 1 degree, so
>we will need a sagittal symbol for this in addition (and also for some
>other divisions).
>
>Okay, I can go along with that, but then my question is, why does it
>have to be a single-shaft symbol, for which we may have to take
>extraordinary measures to justify?  Why not use a double-shaft symbol
>instead?  To us it may seem strange to blend the single and
>double-symbol versions of the notation, but if we are going to have to
>deal with the problem that some people find it difficult to accept the
>fact that sharps and flats can differ in pitch, then why do we have to
>bend over backwards catering to their difficulties by using
>single-shaft symbols in highly unorthodox ways when we already have
>another alternative available.  Won't /||\ serve at least as well as
>(if not better than) (|\ for 1deg19?  And why not use /||\ for 3deg57,
>or /||) for 4deg81, or even (|||( for 3deg52?  To anyone new to this
>notation they're new symbols, just like the others.  (Or are you so set
>against the single-symbol notation that you'll go to great lengths to
>avoid it, even if it makes a lot of sense to use it for this?)
>
>I rest my case.


I'll settle out of court. You win. If they were going to be single-shaft 
symbols for low prime commas like 5, 7 or 11 then I would have preferred 
them to /||\, but now that they (nearly) all turn out to be /|), I see 
there's little or no point.

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

Message: 5226 - Contents - Hide Contents

Date: Thu, 19 Sep 2002 00:23:17

Subject: [tuning] Re: Proposal: a high-order septimal schisma

From: wallyesterpaulrus

--- In tuning-math@y..., paul.hjelmstad@u... wrote:

> 
> This is really interesting. What is the magic that connects
>(126/125 and
> 81/80) with (12et and 19et)?


i don't know about magic, but 12-equal and 19-equal are both tunings 
where 126;125 and 81;80 vanish, i.e., are tempered out. you can think 
of the septimal meantone temperament (which is defined by the wedgie 
in question) as defined either by the family of temperaments where 
both 126;125 and 81;80 are tempered out, or as the family of 
temperaments lying along the straight line connecting 12-equal and 19-
equal in 7-limit space.


> And also with the wedging [1,1,0,-3] with
> [0,1,4,10]?


the second of these tells you that the generator of septimal meantone 
temperament catches the third harmonic with 1 generator, the fifth 
harmonic with 4 generators, and the seventh harmonic with 10 
generators. the first tells you the number of octaves you need to 
correct for register.

Message: 5227 - Contents - Hide Contents

Date: Thu, 19 Sep 2002 11:12:01

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 06:19 PM 17/09/2002 -0700, George Secor wrote:

>From:  George Secor (9/17/02, #4626)
>Subject:  A common notation for JI and ETs
>
>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>> At 10:24 AM 13/09/2002 -0700, George Secor wrote:

>>> ET Notation Agreed Upon
>>> -----------------------
>>> ...

>> Thanks for collecting those. I haven't checked them.
>> 
>> I thought that in cases where we propose both a native fifth notation
>and a

>> subset notation for the same ET, we agreed that we would indicate
>which was

>> preferred. I also thought we agreed to always prefer the subset
>notation.

>> Do you have a reason to change this?
>

>At this point, no.  But I'm open to the possibility that my opinion may
>change once we get feedback from someone who actually tries to use the
>notation for one of these ETs and comes to a different conclusion.


I'm open to that too.


>> I also realise we need to say _which_ subset to use. I think we
>should

>> always specify the subset that contains D natural, for reasons I
>expect are

>> obvious to you.
>

>Well, D is the center of symmetry for the 7 naturals.  But pitch
>standards are usually set for A or C, so how did you intend to handle
>that if the ET doesn't have either A or C in the notation?  Do you have
>a particular pitch standard for D in mind that the rest of the world
>might be willing to accept?


Yes, the D of 12-equal when its A is 440 Hz.


>   (Come to think about it, D isn't a bad
>choice for a pitch standard once you consider the 5 notes corresponding
>to the open strings of the violin family.)


Yes except cello is CGDA (not a problem). Guitars are GDAEB (not a problem).


>> 

>>> ET Notation Proposals
>>> ---------------------
>

>Before I comment on any particular division, I want to discuss some of
>the principles which I used to select some of the symbols.
>
>I first assigned the 5 comma, 11 diesis, 7 comma, and 13 diesis, where
>possible, along with their respective rational complements (including
>the 11' and 13' dieses).  For the larger divisions (for which we would
>want matching symbols in the half-apotomes) I also assigned the 11-5
>comma, where possible.
>
>For the remaining degrees I evaluated other symbols in the following
>order for their suitability (along with their rational complements).
>Except for the first one, I have put these in pairs, which facilitates
>the process of achieving matching symbols in the half-apotomes.
>
>   |( <-->  /||)    5:7 comma and 11:13 comma (also 17'-17)
>
>  (|( <-->  ~||(    5:11 and 7:13 comma (also 11:17)
>  ~|( <-->  (||(    17' comma
>
>//|  <-->  ~||     5+5, 25, and 5:13 comma
>  ~|  <--> //||     17 comma
>
>  (|  <-->  )||~    7:11 comma (also 13:17)
>  )|~ <-->  (||     19' comma
>
>  )|  <-->  (||~    19' comma


You mean 19 comma.


>  (|~ <-->  )||     various complex dieses


The 11:19 comma, 171;176, is probably the simplest of them. But really this 
symbol is just the half-apotome of last resort. I agree it's pointless to 
give its comma value. We can always come up with one if challenged.


>   |~ <-->  ~||)    23 comma

Also 19'-19.

>  ~|) <-->   ||~    7+17 comma
>
>  ~|\ <--> ~)||     23' comma


Also the 11':19' comma, 297;304.


>~)|  <-->  ~||\    17+19 comma


Can also be described as 17:19 comma for what that's worth.


>  /|( <-->  ~||~    5+(17'-17) comma
>  ~|~ <-->  /||(    17+23 comma


Primarily the 5:19 comma.


>If all of the degrees aren't assigned by this point, then desperation
>begins setting in, and I start looking for just about anything else
>that will work.
>
>My first choice for assignment is |(, for two reasons:  1) it has the
>simplest comma ratio of any of these (5:7), and 2) its rational
>complement has the same flags as the 13 diesis (which takes advantage
>of an opportunity to match flags in the half-apotomes).  If |) is valid
>as both the 5:7 and 11:13 commas, then I will almost certainly assign
>it for the notation (and definitely if the 17'-17 comma is also the
>same number of degrees.  Otherwise I will defer assignment of this
>symbol until I have evaluated the other alternatives.  When I assign a
>comma, I will also assign its rational complement, in this case /||),
>if it is valid for the division.

Sounds good.

>My next choice is (|(, which is the next simplest comma (5:11), which I
>will also check to see if it is valid in its other major role as the
>7:13 comma.  I will also check to see if its unidecimal-diesis
>complement ~|( is valid as /|\ minus (|(.  If all of these are valid, I
>will assign both of these symbols.  Otherwise the decision is deferred.


Sounds good, except ...


>Next will be //|, which I will consider similarly.  As we discussed,
>the assignment of this symbol depends upon its being valid as the 5+5
>comma.  I will defer assignment if it is not also valid as both the 25
>comma (i.e., 1,5,25 consistency) and the 5:13 comma.


I would still assign it if it is not the 5:13 comma. 5*13 is much greater 
than 1*25. And stacked major thirds are common enough that people should 
get the //| symbol for them if no less-complex comma symbol can been used. 
I'd also assign this before (|( since 1*25 < 5*11.


>   I will also check
>to see if ~| is valid as /|\ minus //|.  If all of these are okay, then
>I will assign both //| and ~|, as well as their rational complements.
>
>So if there is a choice between (|( and //|, for example, it will come
>down to how many of their assigned roles they are able to play.


I disagree. I think that //| is so obvious a symbol for a double 5 comma, 
and double 5 commas will be in far greater demand than any ratio of 11, 
that I think it should have priority. I'm even prepared to use it when it 
isn't the 25-comma, i.e. when the ET isn't 1,5,25 consistent.


>The above order causes the 17 comma ~| to be considered after the 17'
>comma ~|(.  Even though the 17-comma symbol is simpler in appearance, I
>consider the two to be approximately equal in priority, differing only
>in whether a note such as 17/16 is going to be notated as an altered
>sharp or flat.


As I said befeore, I would prefer not allow the consideration of 
complements to affect the choice of single-shaft symbols unless the cost is 
minimal. In this case it seems it _is_ minimal.


>As I go down the list I find that the less desirable symbols have not
>only fewer but also less important roles to play, which makes their
>validation both easier and less critical.  It is important to observe
>whether alternate interpretations of certain flags such as |) or )|
>result in different numbers of degrees and to make the symbol
>assignment arithmetically consistent.

Agreed.

>If satisfactory rational and unidecimal-diesis complements are not
>valid for a division, then I look for alternate complements that
>minimize the number of flags.  In general, I would seek to retain
>symbols that are valid in all (or at least the most important) of their
>comma-roles and to replace the ones that don't fulfill those roles with
>alternate complement symbols.
>
>In setting up a spreadsheet to make these evaluations, I have not
>attempted to evaluate divisions differing by 7 simultaneously, so the
>process does not attempt to assign these divisions the same set of
>symbols.  One thing that *does* result from this is that a division is
>not forced to accept a less desirable set of symbols that would be
>shared with a second division if a better set is possible for the first
>one.  (This principle comes into effect in evaluating 87 vs. 94,
>discussed below.)
>
>Keeping these things in mind, I will now consider the following.
>

>>> 80:  )|  /|  (|~  /|\  (|)  )||  ||\  (||~  /||\    [13'-(11-5)+23 =
>11-19 diesis]
>> 

>> I'd prefer the single-shaft symbols to be
>> 80b:  |)  /|  (|(  /|\  (|)  ?  ||\  ?  /||\
>> since it stays within the 11-limit. It isn't nice to have |) smaller
>than

>> |\, but we've done it elsewhere.
>

>I really wasn't very happy with any of the choices for 3deg80.  I agree
>that (|( is definitely the most familiar symbol, but I place a higher
>value on ratios of 13 than you do, and I wanted to use it only if it is
>valid as both a 5:11 and 7:13 comma.  But the alternatives aren't
>really any better, so I guess I can go along with this.  The choice
>that I made had something to do with what I have to say next.
>
>The problem I had with |) for 1deg80 was only indirectly related to its
>size relative to /|:  this unusual placement results in using ||) for
>8deg as a rational complement -- a two-degree discrepancy in symbol
>arithmetic (whereas I wanted to allow no more than one degree off, as
>we allowed for 72).


I think that's generally a good rule, except I wouldn't let complements 
dictate the single-shaft symbols, so I suppose I just wouldn't use ||) as 
its complement.


>   I justified using the 19 comma because it's better
>represented in this division.  That caused me to use (||~ as its
>rational complement for 8deg, and I used the (|~ for 3deg because it
>matched.
>
>With your proposal I don't know what to do for apotome complements.
>This isn't a very good division, and I personally don't care very much
>what we use for it.  With so many problems involving the more familiar
>symbols, my solution was to use less familiar ones.  I guess you could
>say that I thought that the division and the symbols deserved each
>other!
>
>So unless you have any more ideas, a decision on this one would best be
>deferred.


80-ET is of interest for being the smallest 19-limit-consistent division, 
however its 7s are relatively bad, so I could accept
80c:  )|  /|  (|(  /|\  (|)  )||  ||\  (||(  /||\   (MM)
80d:  )|  /|  (|(  /|\  (|)  (||~  ||\  ~||(  /||\  (RC)


>>> 87a:  |~  /|  ~|)  /|\  (|)  ||~  ||\  ~||)  /||\    (RC)
>>> 94a:  ~|(  /|  (|(  /|\  (|)  ~||(  ||\  (||(  /||\    (RC)
>>> 87b, 94b:  ~|  /|  ~|\  /|\  (|)  ~||  ||\  ~||\  /||\    (MM)
>>> 87c, 94c:  |~  /|  /|~  /|\  (|)  ||~  ||\  /||~  /||\    (MM)
>>> 87d, 94d:  |~  /|  /|~  /|\  (|)  ~||  ||\  ~||\  /||\    (MM)
>> 

>> I'd prefer the single-shaft symbols to be
>> 87e, 94e: ~|  /|  (|  /|\  (|)  ~||( ||\  ?  /||\
>

>Since |\ is not used, there is no opportunity to have matching symbols
>in the half-apotomes, so I assumed that rational complementation should
>be the organizing principle, if possible.  For what you have, the
>following rational complements would be indicated:
>
>87e, 94e: ~|  /|  (|  /|\  (|)  )|~  ||\  //||  /||\
>
>Neither )|~ nor //|| is the correct number of degrees for the flags,
>whereas my 87a and 94a choices were determined on the basis of which
>pairs of symbols would work best as rational complements.  However, I
>can appreciate your desire that the single-shaft symbol choices not be
>compromised by the need to get rational complements, so I will plead my
>case on that basis.
>
>For 1deg87 I now see that |~ is a rather poor choice; the only
>advantage it had was that it had a valid rational complement.  But I
>won't pursue that any further.  For 1deg94 ~|( is not as simple a
>symbol as ~|, but the two different 17 commas are equally useful.  I
>chose ~|( because it has a good rational complement in 94, which,
>however, is of no use for 87.  I would have to agree on ~| for 87, but
>I think ~|( is better for 94.  However, I will keep ~| for 94 for now
>so I can continue to discuss the two divisions together.
>
>For 3deg I think that (|( has a distinct advantage over (| because it
>will be a more frequently used symbol (e.g., as one of those in the 217
>standard set), especially since it is valid as *both* the 5:11 and 7:13
>commas in *both* 87 and 94), whereas (| represents only the 7:11 comma.
>  Besides this, its rational complement ~||( avoids the |~ flag in the
>notation, introduces no other additional flags, and is the correct
>number of degrees in both 87 and 94.  So this would give us:
>
>87f, 94f: ~|  /|  (|(  /|\  (|)  ~||(  ||\  //||  /||\    (RC)
>
>The only problem I have with this is whether we can get away with
>forcing //|| as 8deg.  If not, then I would use ~||\ as an alternate
>complement (valid in both 87 and 94):
>
>87g, 94g: ~|  /|  (|(  /|\  (|)  ~||(  ||\  ~||\  /||\    (8degAC)
>
>But if I consider 94 apart from 87, I would prefer my first version,
>because all of the flag usages, comma roles, and rational complements
>are free of any problems:


I accept 87g.


>94a:  ~|(  /|  (|(  /|\  (|)  ~||(  ||\  (||(  /||\    (RC)
>
>Should we let the lesser division drag the better one down?


No. I accept 94a.


>>> 111:  ~|  /|  |\  ~|\  /|\  (|)  ~||  /||  ||\  ~||\  /||\
>> 
>> I prefer

>> 111b:  ~|  /|  |\  //|  /|\  (|)  ~||  /||  ||\  //||  /||\
>> which is the same as 118-ET below.
>

>Here //| is valid as the 5+5 and 7:13 commas


You mean 5:13, not 7:13.


>, but 111 is not 1,5,25
>consistent.  My choice of ~|\ was based on using no new flags in
>addition to the fact that it is valid as the 23' comma.  However, I am
>willing to go with 2 valid out of 3 comma roles for //|, plus the fact
>that it is also valid as the rational complement of ~||, and therefore
>//|| as RC of ~|.  So 111b it is, with rational complementation and
>matching symbols!

OK.


>>> 125:  ~|(  /|  |\  (|(  /|\  (|)  ~||(  /||  ||\  (||(  /||\
>> 
>> I prefer

>> 125b:  |(  /|  |)  //|  /|\  (|)  ~||  ||)  ||\  /||)  /||\
>

>I notice that I passed over |(, which isn't valid in the secondary role
>as the 11:13 comma, yet I used (|(, which isn't valid in the secondary
>role of 7:13 comma, so I see that wasn't the reason for my choice.  I
>now see that my objective was to have both matching symbols and
>rational complementation.
>
>In both of our versions rational complementation is maintained, but you
>forsook matching symbols by using a 7-comma symbol.  I made it a
>principle that, if there were over 10 symbols to the apotome, that
>matching symbols should be used wherever possible.
>
>So now what is your preference?


125-ET has very good 7s. Wouldn't it be a travesty not to provide the 7 
comma symbol, and instead to use a symbol that only means 11 comma minus 5 
comma? Likewise, if //| fulfills all its possible roles, how can we use (|( 
in its place?

Even when the mutishaft version of the notation is used, surely the 
single-shaft symbols will be used more often than any others and so should 
not be made less useful or memorable thereby. Isn't this just an 
application of your own principle that you mustn't make the simple things 
more complex in the process of making the complex things simpler?

Do what you like with the complements but I still prefer the single shaft 
symbols in 125b.


>>> 132a:  ~|(  /|  |)  |\  (|~  ~||(  /||  ||)  ||\  (||~  /||\    (MS)
>>> 132b:  ~|(  /|  |)  |\  (|~  /|\  /||  ||)  ||\  (||~  /||\    (MS)
>> 

>> I prefer 132b, but why not |( as 5:7-comma for 1deg132?
>

>I try to choose symbols that are as valid in as many roles as possible.
>  |( is valid only as the 5:7 comma and not as the 11:13 or 17'-17
>commas (1 out of 3), whereas ~|( needs to be valid only as the 17'
>comma (1 out of 1).  This is another one that I don't have strong
>feelings about, and in the course of working on the spreadsheet I might
>change my mind.  Even if we don't get any final agreement at this point
>about some of these less common divisions, at least our discussion of
>these will provide some examples from which I can arrive at general
>principles for choosing symbols.

OK


>>> 152a:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\

>(||(  /||)  /||\    (MS; 14deg AC)

>>> 152b:  )|  |~  /|  |\  ~|)  /|)  /|\  (|)  (|\  ||~  /||  ||\  ~||)

>/||)  /||\    (MS; 14deg AC)

>>> 152c:  )|  ~|  /|  |\  ~|)  /|)  /|\  (|)  (|\  ~||  /||  ||\  ~||)

>/||)  /||\    (MS; 10,13,14deg AC)
>> 

>> I prefer 152b.
>

>I'm rather surprised by your choice -- one that uses both wavy flags
>and that prefers the 23 comma over either of the 17 commas.  It looks
>very much like the set you chose in your message (#4272) of 15 May
>(which you quickly revised).  So you need to explain this one to me.


I think I screwed up.


>I discussed the above options in a previous message (#4596 of 28 Aug),
>which I will repeat here (with comma designations updated):
>
><<  In version a, (|( as 6deg152 is valid as the 5:11 and 11:17 commas,
>but not the 7:13 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
>5:17) commas.  Version c uses the simplest matching symbols, and I am
>inclined to go with that.


Now I'm liking this one.
152d:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  ||\  /||~
/||)  /||\    (MS)
or maybe this
152e:  )|  |~  /|  |\  (|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||~(
/||)  /||\    (MS)


>(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.)  >>


I think I agree with that principle. But choose the best single-shafters 
first and only let complements alter that choice if it does very little damage.


>>> 159a:  |(  ~|(  /|  |\  ~|\  /|)  /|\  (|)  (|\  ~||(  /||  ||\  ~||\
>  /||)  /||\
>> 
>> I prefer

>> 159b:  ~|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  (||(

>/||)  /||\ (RC & MS)
>
>The (| flag is not the same number of degrees in (|( and (|\, so (|( is
>not valid.
>
>I prefer |( because it is valid as the 5:7, 11:13, and 17'-17 commas,
>hence is more desirable for its lower-prime applications than a
>17-comma symbol.  In addition, it is consistent as the rational
>complement of /||).  Neither of our options has rational
>complementation throughout.


OK. I'll go with yours. 159a.


>>> 176a:  |(  |~  /|  |)  |\  ~|)  /|)  /|\  (|)  (|\  ||~  /||  ||)

>||\  ~||)  /||)  /||\    (RC & MS)

>>> 176b:  |(  ~|  /|  |)  |\  ~|)  /|)  /|\  (|)  (|\  ~||  /||  ||)

>||\  ~||)  /||)  /||\    (MS & MM)
>> 

>> Of those two, I prefer 176a, but I like these single-shafters better
>> 176c:  |(  |~  /|  |)  |\  //|  /|)  /|\  (|)  (|\
>> 176d:  |(  ~|  /|  |)  |\  //|  /|)  /|\  (|)  (|\
>

>This is another of the half-dozen larger divisions in which it is
>possible to have both matching symbols and complete rational
>complementation (version 176a), but it is at the price of using a
>couple of relatively unimportant symbols.  Evidently you didn't care
>too much for them.
>
>Your versions differ only in using //| for 6deg.  This time for //|
>it's only 1 out of 3: as the 5+5 comma, but not as the 25 or 5:13
>commas.  For the more nondescript symbol ~|) it's 1 out of 2: as the
>7+17 comma, but not as the 5:17 comma; but this is of little
>significance -- it's just a symbol to match ~||), the rational
>complement of |~.
>
>For (|(, a symbol that neither of us chose, it's 3 out of 3: as 5:11,
>7:13, and 11:17 commas, but its unidecimal-diesis complement ~|( does
>not have the same number of degrees for the |( flag, so ~|( can't be
>used.  With this many degrees in the apotome I thought it advisable to
>use matching symbols, so if I were to pick the best single-shaft
>symbols and duplicate the flags in the double-shaft symbols, I would
>have this:
>
>176e:  |(  ~|  /|  |)  |\  (|(  /|)  /|\  (|)  (|\  ~||  /||  ||)  ||\
>(||(  /||)  /||\    (MS)
>
>On the other hand, using the same single-shaft symbols along with their
>rational complements would give this:
>
>176f:  |(  ~|  /|  |)  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||)  ||\
>  //||  /||)  /||\    (RC)
>
>I'm beginning to wonder whether it would be more meaningful to have
>rational complements (instead of matching flags) for the double-shaft
>symbols whenever there is a good set of single-shaft symbols.  (I'll
>have to try experimenting with the second half-apotome of some of these
>larger divisions to see how often that will work without the symbol
>arithmetic going to pieces.)
>
>Anyway, what do you think of the single-shaft symbols in those last
>two?


I like em.


>>> 181a:  |(  ~|  |~  /|  /|(  ~|)  /|~  /|)  (|~  (|\  ||(  ~||  ||~

>||\  /||(  ~||)  /||~  /||\    (MM)

>>> 181b:  |(  ~|  ~|(  /|  /|(  (|  /|~  /|)  (|~  (|\  ||(  ~||  ~||(

>||\  /||(  (||  /||~  /||\    (MM)
>> 

>> These are both wrong if |( is the 5:7 comma, since the 5:7 comma
>vanishes

>> in this tuning.
>

>You're right; what was I thinking of, anyway?
>
>> I prefer

>> 181c:  )|  ~|  |~  /|  )|)  (|  /|~  /|)  (|~  (|\  )||  ~||  ||~

>||\  )||)  (||  /||~  /||\    (MM)
>
>It appears that you're just trying to minimize the number of flags.
>However, |~ is not the 23 comma here, but that's what the symbol is
>supposed to indicate.  I would rather use something else for 3deg.  The
>best choice appears to be ~|(, which adds the |( flag back into the
>notation.  With that, there doesn't seem to be any point in replacing
>/|( with )|).  So now I get this:
>
>181d:  )|  ~|  ~|(  /|  /|(  (|  /|~  /|)  (|~  (|\  )||  ~||  ~||(
>||\  /||(  (||  /||~  /||\    (MM)

OK.


>>> 183a:  |(  ~|(  /|  |)  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||)

>||\  (||(  /||)  /||\
>> 
>> I prefer

>> 183b:  |(  ~|(  /|  |)  |\  //|  /|)  /|\  (|)  (|\  ~||(  /||  ||)

>||\  //||  /||)  /||\
>
>For 6deg my decision is a matter of which symbol is valid in the
>greater number of roles.  For //| it's 2 out of 3: as 5+5 and 25
>commas, but not as 5:13.  For (|( it's 3 out of 3: as 5:11, 7:13, and
>11:17 commas.  That, plus the fact that ~|( <--> (||( and (|( <--> ~||(
>are rational complements, makes this one of the few larger divisions
>that can have both matching symbols and complete rational
>complementation.

OK. 183a


>>> 193:  )|  ~|  ~|(  /|  |\  ~|)  ~|\  /|)  /|\  (|)  (|\  ~||  ~||(

>/||  ||\  ~||)  ~||\  /||)  /||\
>> 
>> I prefer

>> 193b:  )|  ~|  ~)|  /|  |\  (|  ~|\  /|)  /|\  (|)  (|\  ~||  ~)||

>/||  ||\  (||  ~||\  /||)  /||\

>> 193c:  )|  ~|  ~|(  /|  |\  (|  ~|\  /|)  /|\  (|)  (|\  ~||  ~||(

>/||  ||\  (||  ~||\  /||)  /||\
>
>Yes, (| will work here.  I prefer 193c.

Agreed.


The following is from a different message of yours but it seemed best to 
address it here.


>By the way, in looking at some of the divisions you mentioned I
>happened to notice 100-ET:
>
>100-ET (apotome=6, limma=10) requires 5 symbols
>100:  )|)  /|)  )|\  (|\  )||\  /||\

Agreed.

>We're also doing 100 as a subset of 200, but I didn't give a notation
>for 200, so here it is:
>
>200:  |(  ~|  |~  /|  |\  ~|)  ~|\  /|)  /|\  (|)  (|\  ~||  ||~  /||
>||\  ~||)  ~||\  /||)  /||\    (MS)


I prefer these symbols for 5 and 6 degrees because they represent much 
simpler commas.
200b:  |(  ~|  |~  /|  |)  (|(  ~|\  /|)  /|\  (|)  (|\  ~||  ||~  /||
||)  (||(  ~||\  /||)  /||\   (MS)


>> Here are some others for your consideration:
>> 1   2   3   4    5  6    7    8   9    10  11   12   13   14
>15

>> 282: )|  ~|  ~)|  |~  /|  |)  )|)  (|  (|(  //|  /|)  (|~  /|\  (|)
>> |(      ~|(                       /|~  ~|\  |~)
>> 
>> 11deg282 is the difficult one. /|) is only correct as the 5-comma +
>> 7-comma, not the 13-comma, and |~) is a two-flags-on-the-same-side
>symbol

>> I'm proposing to stand for the 13:19-comma (and possibly the
>5:13-comma).

>> But if you'd rather, I'll just accept that 282-ET and 294-ET are not
>notatable.
>

>Yes, I think that there are too many problems.
>> 

>> However, 306-ET _is_ notatable without using any
>two-flags-on-the-same-side

>> symbols. Alternatives for some degrees are given on the line below.
>> 
>> 306: )|  |(  )|(  ~|(  /|  ~|~  |)  (|  |\  //|  ~|\  /|)  (|~  /|\
>(|)

>> ~|  ~)|   |~               )|) ~|) (|(  |~)
>

>(|( is a better choice than //| for the comma roles it fulfills.


I guess so. Since //| only works as 5+5 comma and (|( works in all its 
possible roles.


>   (|~
>and ~|~ look like they may be a little shaky in the flag arithmetic for
>|~.  (A wavy flag becomes a shaky flag?)


I hadn't noticed that, thanks. But in cases like this (where the only 
alternative is incomplete notation, I don't think we should let flag 
arithmetic stop us.


>> 318 is notatable if you accept (/| (the 31' comma) for 15 steps.
>

>Neither 306 nor 318 are 7-limit consistent, so I don't see much point
>in doing these, other than they may have presented an interesting
>challenge.


Good point. Forget 318-ET, but 306-ET is of interest for being strictly 
Pythagorean. The fifth is so close to 2:3 that even god can barely tell the 
difference. ;-)

If we can accept fuzzy arithmetic with the right wavy flag, and the 
addition of the 13:19 comma symbol |~) then the 31-limit-consistent 388-ET 
can be notated (but surprisingly, not 311-ET).

       1  2    3    4   5   6    7   8   9   10  11   12   13   14
388: )|  |(  ~|  ~)|  ~|(  |~  /|  ~|~  |)  |\  (|  ~|)  ~|\  //|

15   16   17   18   19   20   21   22
|~)  /|)  /|\  (/|  |\)  (|)  (|\  ||(  ... (MS)

The symbols (/| and |\) are of course the 31-comma symbols we agreed on 
long ago.

Here's another one I think should be on the list, 494-ET, if only because 
of the fineness of the division, and because it shows all our rational 
complements*. It is 17-limit consistent. Somewhat surprisingly, it is fully 
notatable with the addition of the 13:19 comma symbol |~). It has the same 
problem as 306 and 388, with right-wavy being fuzzy, taking on values 6, 7 
and 8 here.

       1  2    3    4    5   6   7    8    9  10   11  12   13  14  15
494: )|  |(  )|(  ~|  ~)|  ~|(  |~  )|~  /|  ~|~  |)  )|)  |\  (|  ~|)

  16   17   18   19   20   21   22   23   24   25   26   27  28
(|(  ~|\  //|  |~)  /|)  (|~  /|\  (/|  |\)  (|)  )||  (|\  )||( ...
(RC* & MS)

* It agrees with all our rational complements so far, except that we'd need 
to accept
  ~|~  <--->  )|)     [where the |~ flag corresponds to 6 steps of 494]
instead of
  ~|~  <--->  /|(
which might become an alternative complement.

and we'd need to add

  )|(  <--->  |~)     [where the |~ flag corresponds to 8 steps of 494]

In all other symbols above, the |~ flag corresponds to 7 steps of 494.

My interpretations are
~|~    5:19 comma
)|)    7:19 comma
)|(    19 comma + 5:7 comma
|~)    13:19 comma

Obviously these symbols should be the last to be chosen for any purpose.

So we see that the addition of that one new symbol |~) for the 13:19 comma 
and the acceptance of a fuzzy right wavy flag, lets the maximum notatable 
ET leap from 217 to 494, more than double!

So who cares about notating 282, 388 and 494? I dunno, but here's a funny 
thing: The difference between them is 106. 176 is the next one down.

Here's another big one we can notate this way. Only 11-limit consistent, 
but its relative accuracy at that limit is extremely good. 342 = 2*3*3*19.

342: 
)|  |(  )|(  ~|(  )|~  /|  ~|~  |)  |\  ~|)  (|(  //|  |~)  /|)  /|\  (/| 
(|)  (|\

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

Message: 5228 - Contents - Hide Contents

Date: Fri, 20 Sep 2002 02:24:06

Subject: Re: A common notation for JI and ETs

From: monz

hi Gene,



> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Thursday, September 19, 2002 9:28 PM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote 
> 
>> An irrational-number frequency as a tuning standard?
>> I thought that we could do better than that.
> 

> If simplicity is what you want, I suggest the Verdi
> middle C of 256 Hz, beloved of physics teachers.
> Why do you need a tuning standard, BTW?




i'm interested in this "Verdi middle C" of which you speak.
can you please give more details? (who's Verdi?)


i too proposed a middle-C of 256 Hz = n^0 (= 1/1) as one of
two alternates for a reference frequency, in the original
paper i wrote about my notational system:

Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)


the other alternative was C n^0 = 1 Hz, which still gives
a middle-C of 256 Hz, but in this case middle-C = n^8.  this
reference has been adopted by a few other microtonalists
(a couple of whom wrote to me to say so).


i can't really give an answer as to why a reference is needed ...
just seemed the right thing to do to me.



-monz
"all roads lead to n^0"

Message: 5229 - Contents - Hide Contents

Date: Fri, 20 Sep 2002 15:10:02

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> hi Gene,
> 

>> From: "Gene Ward Smith" <genewardsmith@j...>
>> To: <tuning-math@y...>
>> Sent: Thursday, September 19, 2002 9:28 PM
>> Subject: [tuning-math] Re: A common notation for JI and ETs
>> 
>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote 
>> 
>>> An irrational-number frequency as a tuning standard?
>>> I thought that we could do better than that.
>> 

>> If simplicity is what you want, I suggest the Verdi
>> middle C of 256 Hz, beloved of physics teachers.
>> Why do you need a tuning standard, BTW?
> 

> i'm interested in this "Verdi middle C" of which you speak.
> can you please give more details? (who's Verdi?)
> 
> i too proposed a middle-C of 256 Hz = n^0 (= 1/1) as one of
> two alternates for a reference frequency, in the original
> paper i wrote about my notational system:
> 
> Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)
> 
> the other alternative was C n^0 = 1 Hz, which still gives
> a middle-C of 256 Hz, but in this case middle-C = n^8.  this
> reference has been adopted by a few other microtonalists
> (a couple of whom wrote to me to say so).
> 
> i can't really give an answer as to why a reference is needed ...
> just seemed the right thing to do to me.


You guys didn't get my statement in its complete context.  The 
problem that we discussed involves notating some ETs as subsets of 
others, in which case their native fifths might not be notated as 
such.  It would therefore be necessary to specify which natural note 
would be kept, since all of the other tones related to it by native 
fifths would be modified by symbols from the superset ET.  Hence the 
notation for the subset ET would probably not contain any natural 
note other than the one chosen.

Dave proposed that "D" be the standard natural note for any and all 
of these.  I then observed that most pitch standards are geared 
to "A" or "C" and that there might be some difficulty arriving at an 
appropriate pitch standard for D.

So your comments, while well-intentioned, do not address the problem.

In response to your comments, C=256 would be fine if we were still in 
a previous century when the prevailing musical pitch was close to 
that, but the forces of evil have driven it progressively higher.

In the 1970s there was an unofficial consensus of C=264 among most of 
the microtonalists that I was in contact with.  This is a 3:5 
relationship with A=440, and all of the frequencies of a "just" C 
major scale starting on 264 are integers, which would make it easy to 
present in a music theory class.

--George

Message: 5230 - Contents - Hide Contents

Date: Fri, 20 Sep 2002 10:02:34

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 07:48 AM 19/09/2002 -0700, George Secor wrote:

>From:  George Secor,  9/19/2002 (#4663)
>Subject: A common notation for JI and ETs
>
>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>> At 10:24 AM 13/09/2002 -0700, George Secor wrote:
>> 

>>> ET Notation Proposals
>>> ---------------------
>

>I missed one of these:
>> 

>>> 135a:  ~|  |~  /|  (|  /|~  /|\  (|)  ~||  ||~  ||\  (||  /||~  /||\
>    (MM)

>>> 135b:  ~|  ~|(  /|  (|  /|~  /|\  (|)  ~||  ~||(  ||\  (||  /||~
>/||\    (MM)
>> 

>> I prefer 135a.
>

>Now that I've had a chance to look this over again, I will agree with
>you for the single-shaft symbols in 135a, but I want to change the
>others to this:
>
>135c:  ~|  |~  /|  (|  /|~  /|\  (|)  )||(  )||~  ||\  ~||)  //||  /||\
>    (RC/AC)
>
>Since there is no opportunity to match the symbol sequence, we should
>be trying to maximize the rational complementation.  I realized that
>the rational complement of (| is )||~, not ~||(, and that the RC of |~
>can also be used: ~||).  The symbol arithmetic is correct for both of
>these.
>
>For the rational complement of ~|, I am using //||; the symbol
>arithmetic is not correct for the two straight flags, but there is no
>/|| symbol here to conflict with it, so I think I might be able to
>justify forcing the symbol into use here, as I attempted with 87 and 94
>a couple of messages ago:
>
>87f, 94f: ~|  /|  (|(  /|\  (|)  ~||(  ||\  //||  /||\    (RC)
>
>For 5deg135 there is no choice but to use /|~.


I see that's /|~ as 5:23 comma, and not 5+(19'-19). Why couldn't ~|\ (as 
23' comma) be used?


>   This doesn't have a
>rational complement, and the two alternate complements nearest in size,
>||~ and )||~ are the wrong number of degrees,


I get ||~ and ~||( as the two nearest in size for /|~, and ~||( is the 
right size if ~|( is interpreted as 17 comma + 5:7 comma (but not as 17'). 
~||( also agrees with 494-ET (as a complement for /|~).

And the RC for ~|\ is ~)|| which is the right number of steps (17:19 comma).


>  and you will notice that
>)||~ is already being used.  That leaves, in order of nearest size,
>)||(, ~||, and ||(.  I hesitate to use ~|| because it gives the
>impression that it would represent /||\ minus //|, which is not valid
>as an apotome less either a 5+5 or 5:13 comma.  I decided to use )||(,
>not only because it has the closest size, but also because its
>obscurity is comparable to that of /|~, i.e., both symbols are
>practically meaningless from a harmonic standpoint, so it is fitting
>that they complement one another.


That might be ok, but note that I recently proposed that |~) and )||( be 
rational complements.

How about:
135d:  ~|  |~  /|  (|  /|~  /|\  (|)  ~||(  )||~  ||\  ~||)  //||  /||\
    (RC/AC)


>This sort of complementation is a bit different from what I have
>previously done.  I am starting to concentrate more on the goal that
>the double-shaft symbols should function in more of these divisions as
>true rational complements, so that they can be remembered by their
>association with a harmonic function rather than their position in a
>symbol sequence.  I am beginning to see that, as we did for ||) in
>72-ET, an occasional single-degree discrepancy in symbol arithmetic can
>be tolerated, as long as it is not so noticeable as to be disruptive.
>
>Do you think I'm on the right track here?


I think this goal is OK. It's just the execution of it in this case 
(135-ET) that I'm having trouble with. Although I'm really not sure whether 
to give more weight to harmonic meaning or pitch-order meaning of the 
symbols. The former may be of more interest to composers and the latter to 
performers. I think performers should be favoured. The composer only has to 
translate it once.


>I just read your message #4662, so I want to quote and respond to the
>following portion before sending this:
>
>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>> At 06:19 PM 17/09/2002 -0700, George Secor wrote:

>>> ...  So this would give us:
>>> 
>>> 87f, 94f: ~|  /|  (|(  /|\  (|)  ~||(  ||\  //||  /||\    (RC)
>>> 
>>> The only problem I have with this is whether we can get away with
>>> forcing //|| as 8deg.  If not, then I would use ~||\ as an
>alternate

>>> complement (valid in both 87 and 94):
>>> 
>>> 87g, 94g: ~|  /|  (|(  /|\  (|)  ~||(  ||\  ~||\  /||\    (8degAC)
>>> 
>>> But if I consider 94 apart from 87, I would prefer my first
>version,

>>> because all of the flag usages, comma roles, and rational
>complements

>>> are free of any problems:
>> 

>> I accept 87g.
>

>The above discussion of 135 assumed that the relaxing of symbol
>arithmetic to justify the use of //|| as a rational complement of ~| in
>87 would be acceptable.  I really would prefer 87f, because I think
>that //|| as /||\ minus ~| is a much more meaningful symbol than ~||\.
>
>So does your response mean that you didn't accept 87f?  And after this
>further discussion is that still the case?


I can accept 87f if you have a strong preference for it, although in 
general I find it hard to divorce the double left-straight flags from an 
association with a double 5 comma, despite the double shaft.


>(If this has any bearing on the matter, I read this:
>

>> So we see that the addition of that one new symbol |~) for the 13:19
>comma

>> and the acceptance of a fuzzy right wavy flag, lets the maximum
>notatable

>> ET leap from 217 to 494, more than double!
>

>and am favorable to allowing fuzzy-right-wavy-flag logic if that will
>give us 494.)

Good.


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

Message: 5232 - Contents - Hide Contents

Date: Fri, 20 Sep 2002 12:00:35

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 02:16 PM 19/09/2002 -0700, George Secor wrote:

>From:  George Secor,  9/19/2002 (#4664)
>Subject: A common notation for JI and ETs
>
>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote (#4662):

>> At 06:19 PM 17/09/2002 -0700, George Secor wrote:

>>> From:  George Secor (9/17/02, #4626)
>>> Subject:  A common notation for JI and ETs
>>> 
>>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>>>> At 10:24 AM 13/09/2002 -0700, George Secor wrote:

>>>>> ET Notation Agreed Upon
>>>>> -----------------------
>>>> ...

>>>> I also realise we need to say _which_ subset to use. I think we
>should

>>>> always specify the subset that contains D natural, for reasons I
>expect are

>>>> obvious to you.
>>> 

>>> Well, D is the center of symmetry for the 7 naturals.  But pitch
>>> standards are usually set for A or C, so how did you intend to
>handle

>>> that if the ET doesn't have either A or C in the notation?  Do you
>have

>>> a particular pitch standard for D in mind that the rest of the world
>>> might be willing to accept?
>> 

>> Yes, the D of 12-equal when its A is 440 Hz.
>

>An irrational-number frequency as a tuning standard?  I thought that we
>could do better than that.


Actually, I don't think we should be proposing any tuning standard at all. 
That is a completely separate issue to the notation and I wouldn't like to 
see the notation sink because it's tied to some frequency standard that 
someone doesn't like. One can use this notation with any frequency standard 
or with none.


>>> (Come to think about it, D isn't a bad
>>> choice for a pitch standard once you consider the 5 notes
>corresponding

>>> to the open strings of the violin family.)
>> 

>> Yes except cello is CGDA (not a problem). Guitars are GDAEB (not a
>problem).
>

>I was thinking of the violin family overall: CGDAE, for which D is in
>the middle.


I see. Yes.


>>> )|  <-->  (||~    19' comma
>> 

>> You mean 19 comma.
>

>Right.  For something like this I frequently copy and paste a previous
>line, then edit it, which works fine as long as I edit everything that
>needs to be edited.  This time I unfortunately find that I was
>unknowingly past my prime.

Hee hee.

>

>>> (|~ <-->  )||     various complex dieses
>> 

>> The 11:19 comma, 171;176, is probably the simplest of them. But
>really this

>> symbol is just the half-apotome of last resort. I agree it's
>pointless to

>> give its comma value. We can always come up with one if challenged.
>> 

>>> |~ <-->  ~||)    23 comma
>> 
>> Also 19'-19.
>

>I would prefer not to use this flag alone as the 19'-19 comma, because
>it's not going to find any practical use that way and will compromise
>the meaning of this symbol as the 23 comma.  This is the same reason I
>don't want to see |) by itself as the 13-5 comma (if it's not also
>valid as the 7 comma) or |( as the 17'-17 comma (if it's not also valid
>as the 5:7 comma).


OK. That makes good sense.


>>> ~|) <-->   ||~    7+17 comma
>

>I should start calling this the 5:17 comma.


Oh right! I hadn't caught up with that one either.


>>> ~|\ <--> ~)||     23' comma
>> 

>> Also the 11':19' comma, 297;304.
>

>Although this symbol can be used to notate 19/11 (as A~|\ for C=1/1),
>it won't be used very often, since it treats 11:19 as a sixth (which
>corresponds to 16:19 as a raised second) rather than the more usual
>seventh, as Bb(!~, which corresponds to 16:19 as a lowered third.  And
>at least half of the good higher-numbered ETs don't even allow the
>symbols to be used for this, including 217, 224, 311, and 494.  So I'm
>not giving that role much importance.


Fair enough. It's product complexity is way higher than 1:23 anyway.


>>> ~)|  <-->  ~||\    17+19 comma
>> 

>> Can also be described as 17:19 comma for what that's worth.
>

>Yes, although that way it is actually the 19'-17 comma (152:153,
>~11.352 cents), which is not necessarily the same number of degrees as
>the symbols would indicate (e.g., different in 217 and 311, but same in
>270 and 494).


Aha! I didn't know about 152:153. I was only looking at 1114112:1121931 
which is of course ridiculous _as_ an actual 17:19 comma, given the 
existence of 152:153. For consistency's sake, think we should take it to be 
152:153. I don't think this causes any problems in any of those in which 
we've agreed to use ~)|.


>>> /|( <-->  ~||~    5+(17'-17) comma
>>> ~|~ <-->  /||(    17+23 comma
>> 

>> Primarily the 5:19 comma.
>

>For ~|~, that is.

Yes.


>>> So if there is a choice between (|( and //|, for example, it will
>come

>>> down to how many of their assigned roles they are able to play.
>> 

>> I disagree. I think that //| is so obvious a symbol for a double 5
>comma,

>> and double 5 commas will be in far greater demand than any ratio of
>11,

>> that I think it should have priority. I'm even prepared to use it
>when it

>> isn't the 25-comma, i.e. when the ET isn't 1,5,25 consistent.
>

>I can hardly even imagine doing microtonality without going to at least
>the 11 or 13 limit, because it's there that you get the unusual
>intervals that make it clearly evident that this isn't a 12-tone octave
>you're using.  But I guess I'm of the school of thought that says I
>want to do something different, whereas I view the need for a double 5
>comma to be more in line with the need to have 5-limit harmony in
>better intonation.
>
>But I'm not arguing about which interpretation of a symbol is more
>important, because I think they're both important, since one composer
>may favor one and another composer the other.  And likewise I think
>that this holds for any symbol having multiple roles, which is why I
>would like to choose symbols that fulfill all or most of their roles
>over ones that don't, so that the recommended symbol sets are the ones
>that are most valid for use in a truly *general* sense.

Fair enough.

>In choosing so-called "standard" symbols for ETs, I don't think that we
>should be sending the message that these are the ones to use, and no
>others.  All of the symbols have meanings in any ET, and perhaps we
>should list recommended alternate symbols below the standard ones for
>optional use, where appropriate and/or helpful for indicating
>particular harmonic functions.  This practice would in fact be very
>useful for notating compositions that could, under certain conditions,
>be "ported" from one tuning to another with minimal need to make
>adjustments to the symbols.


I think we should not give alternate symbols for steps in a given ET. That 
goes against the whole idea of standardisation. I think instead we should 
list all the commas that are sufficiently accurately represented in the ET 
and then give the symbol that serves for it in that ET. For example there 
are many ETs in which the 7 comma is the same number of steps as the 5 
comma. So we must let the user know that in this ET, the symbol /| is also 
symbol for the 7 comma.


>In summary, what I am trying to arrive at with these recommended symbol
>sets are the "safest" choices for the composer who doesn't care to be
>bothered with mathematical ratios, but just wants a decent way to get
>the intervals in an ET down on paper.

Indeed.


But I thought standardisation was what this effort was all about. The 
current problem is not a lack of notations, but too many of them, all 
designed for a specific purpose.

The ratio-oriented composer working in an ET, might start off using the 
rational symbols fort all the commas, but I feel she should finally 
translate them to the standard notation for the ET, so that she does not 
use two different symbols to represent the same number of steps and confuse 
the poor performer.


>> 80-ET is of interest for being the smallest 19-limit-consistent
>division,
>

>A dubious honor, considering that three odd harmonics (7, 9, 15)
>deviate by over 40 percent of a degree.
>

>> however its 7s are relatively bad, so I could accept
>> 80c:  )|  /|  (|(  /|\  (|)  )||  ||\  (||(  /||\   (MM)
>> 80d:  )|  /|  (|(  /|\  (|)  (||~  ||\  ~||(  /||\  (RC)
>

>Okay, 80d will work, except that it should be:
>
>80d:  )|  /|  (|(  /|\  (|)  ~||(  ||\  (||~  /||\    (RC)

Agreed


>> 125-ET has very good 7s. Wouldn't it be a travesty not to provide the
>7

>> comma symbol, and instead to use a symbol that only means 11 comma
>minus 5

>> comma? Likewise, if //| fulfills all its possible roles, how can we
>use (|(

>> in its place?
>
>Okay.


You just slipped that one in to see if I was really checking, didn't you? :-)


>> Even when the mutishaft version of the notation is used, surely the
>> single-shaft symbols will be used more often than any others and so
>should

>> not be made less useful or memorable thereby. Isn't this just an
>> application of your own principle that you mustn't make the simple
>things

>> more complex in the process of making the complex things simpler?
>

>I seem to remember saying something like that.
>

>> Do what you like with the complements but I still prefer the single
>shaft

>> symbols in 125b.
>

>Suppose I give you 2 out of 3 by choosing the single-shaft symbols that
>are usable for all of their possible roles, along with the 7-comma, and
>use all rational complements:
>
>125c:  ~|(  /|  |)  //|  /|\  (|)  ~||  ||)  ||\  (||(  /||\    (RC)

Agreed.


>>>>> 152a:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\

>(||(  /||)  /||\    (MS; 14deg AC)

>>>>> 152b:  )|  |~  /|  |\  ~|)  /|)  /|\  (|)  (|\  ||~  /||  ||\

>~||)  /||)  /||\    (MS; 14deg AC)

>>>>> 152c:  )|  ~|  /|  |\  ~|)  /|)  /|\  (|)  (|\  ~||  /||  ||\

>~||)  /||)  /||\    (MS; 10,13,14deg AC)
>>>> 

>>>> I prefer 152b.
>>> 

>>> I'm rather surprised by your choice -- one that uses both wavy flags
>>> and that prefers the 23 comma over either of the 17 commas.  It
>looks

>>> very much like the set you chose in your message (#4272) of 15 May
>>> (which you quickly revised).  So you need to explain this one to me.
>> 

>> I think I screwed up.
>> 

>>> I discussed the above options in a previous message (#4596 of 28
>Aug),

>>> which I will repeat here (with comma designations updated):
>>> 
>>> <<  In version a, (|( as 6deg152 is valid as the 5:11 and 11:17
>commas,

>>> but not the 7:13 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

>>> 5:17) commas.  Version c uses the simplest matching symbols, and I
>am

>>> inclined to go with that.
>> 

>> Now I'm liking this one.
>> 152d:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  ||\  /||~

>/||)  /||\    (MS)

>> or maybe this
>> 152e:  )|  |~  /|  |\  (|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||~(

>/||)  /||\    (MS)
>
>It looks as if an extraneous character got in there.  For matching
>symbols it would have to be:
>
>152d:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  ||\  /||~
>/||)  /||\    (MS)
>152e:  )|  |~  /|  |\  (|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||(
>/||)  /||\    (MS)

Yes. Sorry.

>I'm still trying to figure out why you're using a 23 comma symbol for
>2deg when either 17 comma will work, and /|~ is an even more obscure
>choice than ~|).  Maybe you should tell me why you're liking those now.
>  (Could it have something to do with using only one kind of wavy flag?)
>
>If I were doing it to get the most useful single-shaft symbols for each
>degree that fulfilled all of their comma roles and used their rational
>complements, then it would be one of these:
>
>152f:  )|  ~|(  /|  |\  ~|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||(
>(||~  /||\    (RC)
>152g:  )|  ~|  /|  |\  ~|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  //||
>(||~  /||\    (RC)
>
>But if I replace ~|( with (|( inasmuch as it is good for 2 out of 3
>roles (in addition to having good symbol arithmetic for its rational
>complement), then I get these:
>
>152h:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  (||(
>(||~  /||\    (RC)
>152i:  )|  ~|  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  //||
>(||~  /||\    (RC)
>
>In these last two, the single-shaft symbols differ from your version e
>only in the 2deg position.  Version h almost has matching symbols,
>which would probably make it easier to remember than version i.  If I
>modify version h to give matching symbols, I get:
>
>152a:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  (||(
>/||)  /||\    (MS; 14deg AC)
>
>which was the one I started with some 3 weeks ago.
>
>But please let me know why you prefer the 23 comma.


I think it was just to have monotonic flags-per-symbol. I've run out of 
time now. I may accept ~|( for 2deg152.  But one thing I'd like you to look 
at is whether we have a good progression of symbols when 152 and 217 are 
overlaid, since these relate to 1/3-comma and 1/4-comma meantone  and are 
both important for adaptive JI.


>>> (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.)  >>
>> 

>> I think I agree with that principle. But choose the best
>single-shafters

>> first and only let complements alter that choice if it does very
>little damage.
>

>But now that I'm finding more leeway with the rational complement
>symbol arithmetic, "most memorable" is starting to translate into "most
>harmonically meaningful."


I'd be careful about that. Some people have no interest in harmony, or at 
least JI harmony. We shouldn't forget about trying to maintain a 
progression in the size of the symbols, and monotonic flags-per-symbol.


>>> 200:  |(  ~|  |~  /|  |\  ~|)  ~|\  /|)  /|\  (|)  (|\  ~||  ||~

>/||  ||\  ~||)  ~||\  /||)  /||\    (MS)
>> 

>> I prefer these symbols for 5 and 6 degrees because they represent
>much
>> simpler commas.
>> 200b:  |(  ~|  |~  /|  |)  (|(  ~|\  /|)  /|\  (|)  (|\  ~||  ||~

>/||  ||)  (||(  ~||\  /||)  /||\   (MS)
>
>I was about to say that your choices for these are excellent, and then
>I noticed that |) is 5deg by itself, but was already used as 4deg in
>/|).  Also, |( is 1deg by itself, but in (|( it would have to vanish,
>because (| is already 6deg, since (|) is 10deg.
>
>Anyway, nice try!


If we used (|~ for 8deg200 then we'd have consistent arithmetic.
|( 1
~| 2
|~ 3
/| 4
|\ 6
(| 5
|) 5

So that's
200c: |(  ~|  |~  /|  |)  (|(  ~|\  (|~  /|\  (|)  ||(  ~||  ||~
/||  ||)  (||(  ~||\  /||)  /||\   (MS)

Just a thought.


>I will have to leave the rest of these (the ones above 200) until I
>have more time to look at them.  Things start getting very complicated
>with these, and I don't want to draw any hasty conclusions.

Fair enough.

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

Message: 5233 - Contents - Hide Contents

Date: Fri, 20 Sep 2002 16:02:40

Subject: a reference pitch (was: A common notation for JI and ETs)

From: monz

hi George,


good to be back in contact with you.

i proposed my "C" standards in the interests
of mathematical simplicity, and still stand by
them because of that reason.  it doesn't get
much simpler than C n^0 = 1 Hz.


-monz
"all roads lead to n^0"



----- Original Message -----
From: "gdsecor" <gdsecor@xxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Friday, September 20, 2002 8:10 AM
Subject: [tuning-math] Re: A common notation for JI and ETs



> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>> hi Gene,
>> 

>>> From: "Gene Ward Smith" <genewardsmith@j...>
>>> To: <tuning-math@y...>
>>> Sent: Thursday, September 19, 2002 9:28 PM
>>> Subject: [tuning-math] Re: A common notation for JI and ETs
>>> 
>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>>>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote
>>> 
>>>> An irrational-number frequency as a tuning standard?
>>>> I thought that we could do better than that.
>>> 

>>> If simplicity is what you want, I suggest the Verdi
>>> middle C of 256 Hz, beloved of physics teachers.
>>> Why do you need a tuning standard, BTW?
>> 

>> i'm interested in this "Verdi middle C" of which you speak.
>> can you please give more details? (who's Verdi?)
>> 
>> i too proposed a middle-C of 256 Hz = n^0 (= 1/1) as one of
>> two alternates for a reference frequency, in the original
>> paper i wrote about my notational system:
>> 
>> Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)
>> 
>> the other alternative was C n^0 = 1 Hz, which still gives
>> a middle-C of 256 Hz, but in this case middle-C = n^8.  this
>> reference has been adopted by a few other microtonalists
>> (a couple of whom wrote to me to say so).
>> 
>> i can't really give an answer as to why a reference is needed ...
>> just seemed the right thing to do to me.
>

> You guys didn't get my statement in its complete context.  The
> problem that we discussed involves notating some ETs as subsets of
> others, in which case their native fifths might not be notated as
> such.  It would therefore be necessary to specify which natural note
> would be kept, since all of the other tones related to it by native
> fifths would be modified by symbols from the superset ET.  Hence the
> notation for the subset ET would probably not contain any natural
> note other than the one chosen.
>
> Dave proposed that "D" be the standard natural note for any and all
> of these.  I then observed that most pitch standards are geared
> to "A" or "C" and that there might be some difficulty arriving at an
> appropriate pitch standard for D.
>
> So your comments, while well-intentioned, do not address the problem.
>
> In response to your comments, C=256 would be fine if we were still in
> a previous century when the prevailing musical pitch was close to
> that, but the forces of evil have driven it progressively higher.
>
> In the 1970s there was an unofficial consensus of C=264 among most of
> the microtonalists that I was in contact with.  This is a 3:5
> relationship with A=440, and all of the frequencies of a "just" C
> major scale starting on 264 are integers, which would make it easy to
> present in a music theory class.
>
> --George
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@xxxxxxxxxxx.xxx
>
>
>
> Your use of Yahoo! Groups is subject to Yahoo! Terms of Service * [with cont.]  (Wayb.)
>
>

Message: 5234 - Contents - Hide Contents

Date: Fri, 20 Sep 2002 04:28:46

Subject: Re: A common notation for JI and ETs

From: Gene Ward Smith

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote 

> An irrational-number frequency as a tuning standard?  I thought that 
> we could do better than that.


If simplicity is what you want, I suggest the Verdi middle C of
256 Hz, beloved of physics teachers. Why do you need a tuning standard, BTW?

Message: 5236 - Contents - Hide Contents

Date: Sat, 21 Sep 2002 11:28:40

Subject: Re: Combinatorics and Tuning Systems?

From: Gene Ward Smith

--- In tuning-math@y..., paul.hjelmstad@u... wrote:

> 
> I see how to calculate the Steiner system S(5,6,12) to obtain 132 hexads
> from r=66. And I see how that is connected with M12 Matthieu group. But
> what I don't see is how M12 has order 2^6*3^3*5*11, the number of elements
> in the group. How does this relate to the Steiner system? Thanks


|M12|
= 132*6! = 132 |S6|. M12 is 5-transitive, so all n-tuples aside from
6-tuples are permutable. For 6-tuples, M12 sends a hexad of the
Steiner system to another hexad of the Steiner system; since it is a
simple group this gives a permutation representation of M12 on the 132
hexands.

Message: 5237 - Contents - Hide Contents

Date: Sat, 21 Sep 2002 11:30:50

Subject: Fwd: Re: Combinatorics and Tuning Systems?

From: Gene Ward Smith

--- In tuning-math@y..., <Josh@o...> wrote:


> I've also yet to see much pc set theory for
> 19tet, which I suspect would interest a few people
> other than me.


There was a discussion of difference sets in relation to the 19-et
here recently.

Message: 5238 - Contents - Hide Contents

Date: Sat, 21 Sep 2002 11:32:21

Subject: Re: Combinatorics and Tuning Systems?

From: Gene Ward Smith

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:


> |M12|
= 132*6! = 132 |S6|. M12 is 5-transitive, so all n-tuples aside from
6-tuples are permutable. For 6-tuples, M12 sends a hexad of the
Steiner system to another hexad of the Steiner system; since it is a
simple group this gives a ^ permutation representation of M12 on the
132 hexands.            faithful

Message: 5239 - Contents - Hide Contents

Date: Sat, 21 Sep 2002 09:20:36

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I previously posted:

George Secor:

>>>>> 152a:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\

>(||(  /||)  /||\    (MS; 14deg AC)

>>>>> 152b:  )|  |~  /|  |\  ~|)  /|)  /|\  (|)  (|\  ||~  /||  ||\

>~||)  /||)  /||\    (MS; 14deg AC)

>>>>> 152c:  )|  ~|  /|  |\  ~|)  /|)  /|\  (|)  (|\  ~||  /||  ||\

>~||)  /||)  /||\    (MS; 10,13,14deg AC)
>>>> 

>>>> I prefer 152b.
>>> 

>>> I'm rather surprised by your choice -- one that uses both wavy flags
>>> and that prefers the 23 comma over either of the 17 commas.  It
>looks

>>> very much like the set you chose in your message (#4272) of 15 May
>>> (which you quickly revised).  So you need to explain this one to me.
>> 

>> I think I screwed up.
>> 

>>> I discussed the above options in a previous message (#4596 of 28
>Aug),

>>> which I will repeat here (with comma designations updated):
>>> 
>>> <<  In version a, (|( as 6deg152 is valid as the 5:11 and 11:17
>commas,

>>> but not the 7:13 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

>>> 5:17) commas.  Version c uses the simplest matching symbols, and I
>am

>>> inclined to go with that.
>> 

>> Now I'm liking this one.
>> 152d:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  ||\  /||~

>/||)  /||\    (MS)

>> or maybe this
>> 152e:  )|  |~  /|  |\  (|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||~(

>/||)  /||\    (MS)
>
>It looks as if an extraneous character got in there.  For matching
>symbols it would have to be:
>
>152d:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  ||\  /||~
>/||)  /||\    (MS)
>152e:  )|  |~  /|  |\  (|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||(
>/||)  /||\    (MS)

Me:
"Yes. Sorry."

George Secor:
>I'm still trying to figure out why you're using a 23 comma symbol for
>2deg when either 17 comma will work, and /|~ is an even more obscure
>choice than ~|).  Maybe you should tell me why you're liking those now.
>  (Could it have something to do with using only one kind of wavy flag?)
>
>If I were doing it to get the most useful single-shaft symbols for each
>degree that fulfilled all of their comma roles and used their rational
>complements, then it would be one of these:
>
>152f:  )|  ~|(  /|  |\  ~|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||(
>(||~  /||\    (RC)
>152g:  )|  ~|  /|  |\  ~|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  //||
>(||~  /||\    (RC)
>
>But if I replace ~|( with (|( inasmuch as it is good for 2 out of 3
>roles (in addition to having good symbol arithmetic for its rational
>complement), then I get these:
>
>152h:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  (||(
>(||~  /||\    (RC)
>152i:  )|  ~|  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  //||
>(||~  /||\    (RC)
>
>In these last two, the single-shaft symbols differ from your version e
>only in the 2deg position.  Version h almost has matching symbols,
>which would probably make it easier to remember than version i.  If I
>modify version h to give matching symbols, I get:
>
>152a:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  (||(
>/||)  /||\    (MS; 14deg AC)
>
>which was the one I started with some 3 weeks ago.
>
>But please let me know why you prefer the 23 comma.

Me:

"I think it was just to have monotonic flags-per-symbol. I've run out of 
time now. I may accept ~|( for 2deg152.  But one thing I'd like you to look 
at is whether we have a good progression of symbols when 152 and 217 are 
overlaid, since these relate to 1/3-comma and 1/4-comma meantone  and are 
both important for adaptive JI."


Now that I've looked at this myself, I definitely agree that 2deg152 should 
be ~|(. There's more of an argument for 3deg217 being |~. I can accept 
either /|~ or (|( for 5deg152. I also realised that we should not be using 
13-comma symbols in 152. It has inconsistent 13s. The symbol //| is quite 
valid (in all its roles) for 6deg152.

152j:  )|  ~|(  /|  |\  (|(  //|  /|\  (|)  )||  ~||(  /||  ||\  (||(
//||  /||\  (MS)
152k:  )|  ~|(  /|  |\  /|~  //|  /|\  (|)  )||  ~||(  /||  ||\  /||~
//||  /||\  (MS)

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

Message: 5243 - Contents - Hide Contents

Date: Mon, 23 Sep 2002 18:32:24

Subject: Re: a reference pitch (was: A common notation for JI and ETs)

From: gdsecor

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> hi George,
> 
> 
> good to be back in contact with you.

Likewise.

> i proposed my "C" standards in the interests
> of mathematical simplicity, and still stand by
> them because of that reason.  it doesn't get
> much simpler than C n^0 = 1 Hz.


And how does that translate into a pitch standard for ETs (including 
12-ET)?

--George

Message: 5244 - Contents - Hide Contents

Date: Mon, 23 Sep 2002 18:34:03

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote (#4665):

> At 07:48 AM 19/09/2002 -0700, George Secor wrote:

>> From:  George Secor,  9/19/2002 (#4663)
>> Subject: A common notation for JI and ETs
>> 
>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>>> At 10:24 AM 13/09/2002 -0700, George Secor wrote:
>>> 

>>>> ET Notation Proposals
>>>> ---------------------
>> 

>> I missed one of these:
>>> 

>>>> 135a:  ~|  |~  /|  (|  /|~  /|\  (|)  ~||  ||~  ||\  

(||  /||~  /||\    (MM)

>>>> 135b:  ~|  ~|(  /|  (|  /|~  /|\  (|)  ~||  ~||(  ||\  

(||  /||~  /||\    (MM)

>>> 
>>> I prefer 135a.
>> 

>> Now that I've had a chance to look this over again, I will agree 
with
>> you for the single-shaft symbols in 135a, but I want to change the
>> others to this:
>> 
>> 135c:  ~|  |~  /|  (|  /|~  /|\  (|)  )||(  )||~  ||\  

~||)  //||  /||\    (RC/AC)

>> 
>> Since there is no opportunity to match the symbol sequence, we 
should
>> be trying to maximize the rational complementation.  I realized 
that
>> the rational complement of (| is )||~, not ~||(, and that the RC 
of |~
>> can also be used: ~||).  The symbol arithmetic is correct for both 
of
>> these.
>> 
>> For the rational complement of ~|, I am using //||; the symbol
>> arithmetic is not correct for the two straight flags, but there is 
no
>> /|| symbol here to conflict with it, so I think I might be able to
>> justify forcing the symbol into use here, as I attempted with 87 
and 94
>> a couple of messages ago:
>> 
>> 87f, 94f: ~|  /|  (|(  /|\  (|)  ~||(  ||\  //||  /||\    (RC)
>> 
>> For 5deg135 there is no choice but to use /|~.
> 

> I see that's /|~ as 5:23 comma, and not 5+(19'-19). 


Yes, since 2deg is already |~.  The /|~ is not very meaningful, and 
if I could find something better, I would probably use it.


> Why couldn't ~|\ (as 23' comma) be used?


Because ~| is already used for 1deg and |\ for 3deg, and they don't 
add up to 5deg.

I'm putting this here again so it's easier to refer to:

135c:  ~|  |~  /|  (|  /|~  /|\  (|)  )||(  )||~  ||\  
~||)  //||  /||\    (RC/AC)


>> This [i.e., /|~] doesn't have a
>> rational complement, and the two alternate complements nearest in 
size,
>> ||~ and )||~ are the wrong number of degrees,
> 

> I get ||~ and ~||( as the two nearest in size for /|~, and ~||( is 
the 
> right size if ~|( is interpreted as 17 comma + 5:7 comma (but not 
as 17'). 
> ~||( also agrees with 494-ET (as a complement for /|~).
> And the RC for ~|\ is ~)|| which is the right number of steps 
(17:19 comma).
> 

>> and you will notice that
>> )||~ is already being used.  That leaves, in order of nearest size,
>> )||(, ~||, and ||(.  I hesitate to use ~|| because it gives the
>> impression that it would represent /||\ minus //|, which is not 
valid
>> as an apotome less either a 5+5 or 5:13 comma.  I decided to use )
||(,
>> not only because it has the closest size, but also because its
>> obscurity is comparable to that of /|~, i.e., both symbols are
>> practically meaningless from a harmonic standpoint, so it is 
fitting
>> that they complement one another.
> 

> That might be ok, but note that I recently proposed that |~) and )||
( be 
> rational complements.
> 
> How about:
> 135d:  ~|  |~  /|  (|  /|~  /|\  (|)  ~||(  )||~  ||\  

~||)  //||  /||\    (RC/AC)

I'd have no problem using ~||( for 8deg if /|~ and (|(, its true 
rational complement, were the same number of degrees in 135 (as they 
are in 494), but in order for the rational complement symbols to be 
meaningful, I don't think that ~||( should be used if it's not valid 
as the apotome minus the 5:7 comma, just as I wouldn't want to use (|
( if it weren't valid as the 5:7 comma.

In the second half-apotome I have felt that we could take a little 
liberty with the symbol arithmetic now and then to get a rational 
complement in the proper place, but I would rather not compromise the 
meaning of an important rational-complement symbol by putting in a 
position that invalidates that meaning, if it could be avoided 
without creating any other significant problems.


>> This sort of complementation is a bit different from what I have
>> previously done.  I am starting to concentrate more on the goal 
that
>> the double-shaft symbols should function in more of these 
divisions as
>> true rational complements, so that they can be remembered by their
>> association with a harmonic function rather than their position in 
a
>> symbol sequence.  I am beginning to see that, as we did for ||) in
>> 72-ET, an occasional single-degree discrepancy in symbol 
arithmetic can
>> be tolerated, as long as it is not so noticeable as to be 
disruptive.
>> 
>> Do you think I'm on the right track here?
> 

> I think this goal is OK. It's just the execution of it in this case 
> (135-ET) that I'm having trouble with. Although I'm really not sure 
whether 
> to give more weight to harmonic meaning or pitch-order meaning of 
the 
> symbols. The former may be of more interest to composers and the 
latter to 
> performers. I think performers should be favoured. The composer 

only has to 

> translate it once.


I thought that in a division where /| and |\ are not both used (such 
as 135), and where we therefore can't have a matching symbol 
sequence, that rational complementation would be the organizing 
principle for the selection of the second half-apotome.

Hopefully the printed score and parts will eventually be computer-
generated so that the composer could use any symbols when composing, 
and there could be options to print the music using whatever sets of 
symbols might be desired separately for each and every part, 
including single and double-symbol options.  (I can dream, can't I?)

--George

Message: 5245 - Contents - Hide Contents

Date: Mon, 23 Sep 2002 19:16:00

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote (#4666):

> At 02:16 PM 19/09/2002 -0700, George Secor wrote:

>> From:  George Secor,  9/19/2002 (#4664)
>> Subject: A common notation for JI and ETs
>> 
>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote 
(#4662):

>> [GS:]
>> In choosing so-called "standard" symbols for ETs, I don't think 
that we
>> should be sending the message that these are the ones to use, and 
no
>> others.  All of the symbols have meanings in any ET, and perhaps we
>> should list recommended alternate symbols below the standard ones 
for
>> optional use, where appropriate and/or helpful for indicating
>> particular harmonic functions.  This practice would in fact be very
>> useful for notating compositions that could, under certain 
conditions,
>> be "ported" from one tuning to another with minimal need to make
>> adjustments to the symbols.
> 

> I think we should not give alternate symbols for steps in a given 
ET. That 
> goes against the whole idea of standardisation. I think instead we 
should 
> list all the commas that are sufficiently accurately represented in 
the ET 
> and then give the symbol that serves for it in that ET. For example 
there 
> are many ETs in which the 7 comma is the same number of steps as 
the 5 
> comma. So we must let the user know that in this ET, the symbol /| 
is also 
> symbol for the 7 comma.


The paragraph that I had in a previous message about computer-
generated parts would be valid here also.  A composer would be able 
to write a piece more than one (or with no particular) tuning in 
mind, using the harmonically correct symbols, and then the software 
could translate it into different ETs (with the appropriate standard 
symbols), and the composer might only have to do a little cleanup on 
each one to get the desired result.  In this situation a harmonically 
literate player might prefer to have the original symbols instead of 
an ET notation.

The point is that the notation doesn't prevent you from doing any of 
this.  The limitations are more likely to occur with microtonally 
inexperienced or acoustically ignorant players.


>> In summary, what I am trying to arrive at with these recommended 
symbol
>> sets are the "safest" choices for the composer who doesn't care to 
be
>> bothered with mathematical ratios, but just wants a decent way to 
get
>> the intervals in an ET down on paper.
> 
> Indeed.
> 

> But I thought standardisation was what this effort was all about. 
The 
> current problem is not a lack of notations, but too many of them, 
all 
> designed for a specific purpose.


But at least we're eliminating a lot of the chaos by providing a 
common superset of symbols for everything, symbols that don't mean 
one thing in one tuning and something else in another tuning.  I just 
don't want standardization to go so far that users might think that 
we were discouraging them from exploiting the full versatility of the 
notation.


> The ratio-oriented composer working in an ET, might start off using 
the 
> rational symbols fort all the commas, but I feel she should finally 
> translate them to the standard notation for the ET, so that she 
does not 
> use two different symbols to represent the same number of steps and 
confuse 
> the poor performer.


Okay, it looks like we're on the same wavelength.  There are some 
performers who can handle it and some who can't, or won't, or don't 
have time.  Ideally we would like everybody to be able to know how 
many degrees each of the important commas is in each tuning being 
used, but that's not always possible, so we provide standard symbol 
subsets that everybody would learn.


> ...

>>> 125-ET has very good 7s. Wouldn't it be a travesty not to 

provide the 7

>>> comma symbol, and instead to use a symbol that only means 11 

comma minus 5

>>> comma? Likewise, if //| fulfills all its possible roles, how 

can we use (|(

>>> in its place?
>> 
>> Okay.
> 

> You just slipped that one in to see if I was really checking, 

didn't you? :-)

Really, I didn't!  You caught me in an instance where I made the 
selection of the symbol (a little while back) on the basis of which 
rational complements were more arithmetically correct.  But lately 
I've been paying more attention to the best selection of single-shaft 
symbols (most harmonically meaningful in the greatest percentage of 
roles) and allowing a little slack with the complement symbol 
arithmetic.  So it's nice that you were checking on me.


>>> Even when the mutishaft version of the notation is used, surely 
the
>>> single-shaft symbols will be used more often than any others 

and so should

>>> not be made less useful or memorable thereby. Isn't this just an
>>> application of your own principle that you mustn't make the 
simple things
>>> more complex in the process of making the complex things 
simpler?
>> 

>> I seem to remember saying something like that.
>> 

>>> Do what you like with the complements but I still prefer the 
single shaft
>>> symbols in 125b.
>> 

>> Suppose I give you 2 out of 3 by choosing the single-shaft 
symbols that
>> are usable for all of their possible roles, along with the 7-
comma, and
>> use all rational complements:
>> 
>> 125c:  ~|(  /|  |)  //|  /|\  (|)  ~||  ||)  ||\  (||(  /||\    
(RC)
> 
> Agreed.

Hooray!  


>>>>>> 152a:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||

(  /||  ||\  (||(  /||)  /||\    (MS; 14deg AC)

>>>>>> 152b:  )|  |~  /|  |\  ~|)  /|)  /|\  (|)  (|\  ||~  /||  

||\  ~||)  /||)  /||\    (MS; 14deg AC)

>>>>>> 152c:  )|  ~|  /|  |\  ~|)  /|)  /|\  (|)  (|\  ~||  /||  

||\  ~||)  /||)  /||\    (MS; 10,13,14deg AC)

>>>>> 
>>>>> I prefer 152b.
>>>> 

>>>> I'm rather surprised by your choice -- one that uses both wavy 
flags
>>>> and that prefers the 23 comma over either of the 17 commas.  
It looks
>>>> very much like the set you chose in your message (#4272) of 15 
May
>>>> (which you quickly revised).  So you need to explain this one 
to me.
>>> 

>>> I think I screwed up.
>>> 

>>>> I discussed the above options in a previous message (#4596 of 
28 Aug),
>>>> which I will repeat here (with comma designations updated):
>>>> 
>>>> <<  In version a, (|( as 6deg152 is valid as the 5:11 and 
11:17 commas,
>>>> but not the 7:13 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
>>>> 5:17) commas.  Version c uses the simplest matching symbols, 

and I am

>>>> inclined to go with that.
>>> 

>>> Now I'm liking this one.
>>> 152d:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  

||\  /||~  /||)  /||\    (MS)

>>> or maybe this
>>> 152e:  )|  |~  /|  |\  (|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  

(||~(  /||)  /||\    (MS)

>> 
>> It looks as if an extraneous character got in there.  For matching
>> symbols it would have to be:
>> 
>> 152d:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  

||\  /||~  /||)  /||\    (MS)

>> 152e:  )|  |~  /|  |\  (|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||

(  /||)  /||\    (MS)

> 
> Yes. Sorry.
> 

>> I'm still trying to figure out why you're using a 23 comma symbol 
for
>> 2deg when either 17 comma will work, and /|~ is an even more 
obscure
>> choice than ~|).  Maybe you should tell me why you're liking those 
now.
>> (Could it have something to do with using only one kind of wavy 
flag?)
>> 
>> If I were doing it to get the most useful single-shaft symbols for 
each
>> degree that fulfilled all of their comma roles and used their 
rational
>> complements, then it would be one of these:
>> 
>> 152f:  )|  ~|(  /|  |\  ~|(  /|)  /|\  (|)  (|\  ||~  /||  ||\  (||

(  (||~  /||\    (RC)

>> 152g:  )|  ~|  /|  |\  ~|(  /|)  /|\  (|)  (|\  ||~  /||  

||\  //||  (||~  /||\    (RC)

>> 
>> But if I replace ~|( with (|( inasmuch as it is good for 2 out of 3
>> roles (in addition to having good symbol arithmetic for its 
rational
>> complement), then I get these:
>> 
>> 152h:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  

(||(  (||~  /||\    (RC)

>> 152i:  )|  ~|  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  

||\  //||  (||~  /||\    (RC)

>> 
>> In these last two, the single-shaft symbols differ from your 
version e
>> only in the 2deg position.  Version h almost has matching symbols,
>> which would probably make it easier to remember than version i.  
If I
>> modify version h to give matching symbols, I get:
>> 
>> 152a:  )|  ~|(  /|  |\  (|(  /|)  /|\  (|)  (|\  ~||(  /||  ||\  

(||(  /||)  /||\    (MS; 14deg AC)

>> 
>> which was the one I started with some 3 weeks ago.
>> 
>> But please let me know why you prefer the 23 comma.
> 

> I think it was just to have monotonic flags-per-symbol. I've run 
out of 
> time now. I may accept ~|( for 2deg152.  But one thing I'd like you 
to look 
> at is whether we have a good progression of symbols when 152 and 
217 are 
> overlaid, since these relate to 1/3-comma and 1/4-comma meantone  
and are 
> both important for adaptive JI.


That's a very good point.  Since we've been looking at symbols for 
494 lately, and since 494 is a multiple of 19, I wondered if that 
would have any bearing on this.  It turns out it might.  In 494 the 5 
comma is 9 degrees, which is divisible in thirds as )|(  ~|(  /|.  
These symbols are also 1, 2, and 3 degrees, respectively in 152, so 
it's possible to use them.  The rational symbol sizes are around 9.4, 
14.7, and 21.5 cents, and 1deg152 is ~7.9 cents.  But the symbols for 
1/3 and 2/3 comma are not as easy to distinguish as )| and |~ or )| 
and ~|(, so I don't this that gets us anywhere.

The 2/3-comma symbol in 494 ~|( is the same as the 3/4-comma symbol 
in 217, while the 1/3-comma symbol is larger than the 1/2-comma 
symbol, so the size correlation between divisions isn't very good.  
And )| for 1/3 comma in 152 is smaller than |( for 1/4 comma in 217, 
while |~ for 2/3 comma in 152 is larger than ~|( for 3/4 comma in 
217, which is no better.


>>>> (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.)  >>
>>> 

>>> I think I agree with that principle. But choose the best single-
shafters
>>> first and only let complements alter that choice if it does 

very little damage.

>> 
>> But now that I'm finding more leeway with the rational complement
>> symbol arithmetic, "most memorable" is starting to translate 
into "most
>> harmonically meaningful."
> 

> I'd be careful about that. Some people have no interest in harmony, 
or at 
> least JI harmony. We shouldn't forget about trying to maintain a 
> progression in the size of the symbols, and monotonic flags-per-
symbol.


You followed up that thought in a subsequent message (#4673), which I 
will include here:


> Now that I've looked at this myself, I definitely agree that 
2deg152 should 
> be ~|(. There's more of an argument for 3deg217 being |~. I can 
accept 
> either /|~ or (|( for 5deg152. I also realised that we should not 
be using 
> 13-comma symbols in 152. It has inconsistent 13s. The symbol //| is 
quite 
> valid (in all its roles) for 6deg152.
> 
> 152j:  )|  ~|(  /|  |\  (|(  //|  /|\  (|)  )||  ~||(  /||  ||\  (||

(  //||  /||\  (MS)

> 152k:  )|  ~|(  /|  |\  /|~  //|  /|\  (|)  )||  ~||(  /||  

||\  /||~  //||  /||\  (MS)

You have a couple of good points there.  Using ~|( agrees with its 
use in 494 as 2/3 of the 5 comma.

If those who attach harmonic meaning to the symbols recognize that 
ratios of 13 are compromised in 152, then they would readily accept 
the fact that (|( is not valid as the 7:13 comma. So I have no 
objections to using the single-shaft symbols of 152j.  I have some 
reservations about the meaning of //|| being misleading, since it's 
not valid as the complement of the 17 comma, but at this point I can 
provide neither a good alternative proposal nor a good rationale for 
using something else, so I can't disagree with what you have.

I will have to see if there are any other divisions for which //| 
might be more appropriate than /|) and if there is any advantage in 
changing those from what we already have.


>>>> 200:  |(  ~|  |~  /|  |\  ~|)  ~|\  /|)  /|\  (|)  (|\  ~||  

||~  /||  ||\  ~||)  ~||\  /||)  /||\    (MS)

>>> 
>>> I prefer these symbols for 5 and 6 degrees because they 
represent much
>>> simpler commas.
>>> 200b:  |(  ~|  |~  /|  |)  (|(  ~|\  /|)  /|\  (|)  (|\  ~||  

||~  /||  ||)  (||(  ~||\  /||)  /||\   (MS)

>> 
>> I was about to say that your choices for these are excellent, and 
then
>> I noticed that |) is 5deg by itself, but was already used as 4deg 
in
>> /|).  Also, |( is 1deg by itself, but in (|( it would have to 
vanish,
>> because (| is already 6deg, since (|) is 10deg.
>> 
>> Anyway, nice try!
> 

> If we used (|~ for 8deg200 then we'd have consistent arithmetic.
> |( 1
> ~| 2
> |~ 3
> /| 4
> |\ 6
> (| 5
> |) 5
> 
> So that's
> 200c: |(  ~|  |~  /|  |)  (|(  ~|\  (|~  /|\  (|)  ||(  ~||  

||~  /||  ||)  (||(  ~||\  /||)  /||\   (MS)

> 
> Just a thought.


I think that the 13-comma symbols should be kept because 13 is very 
accurate, while the 7-comma symbol should be eliminated, because 7 is 
almost 1/2 degree off.

--George

Message: 5246 - Contents - Hide Contents

Date: Mon, 23 Sep 2002 21:33:43

Subject: Re: A common notation for JI and ETs

From: monz

----- Original Message -----
From: "David C Keenan" <d.keenan@xx.xxx.xx>
To: "George Secor" <gdsecor@xxxxx.xxx>
Cc: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Monday, September 23, 2002 7:03 PM
Subject: [tuning-math] Re: A common notation for JI and ETs



> Hi George,
>
> got your latest, thanks. I don't think there's anything that needs my
> reply. I just wanted to say that there's probably no point in going public
> until we've got an actual font that folks can use with Sibelius. I
> understand we have 29 flag-combinations * 2 directions * 4 shaft-types + 1
> natural + 1 conventional sharp + 1 conventional flat = 235 symbols
>
> It might be a good idea to map the 29 single-shaft down symbols to the
> characters a-z[]\ and the 29 single-shaft up symbols to the characters
> A-Z{}|, in order of rational size. Double-shafts could be obtained with
the
> Alt key, and triple and X shafts with the Ctrl key and Ctrl and Alt keys.
> -- Dave Keenan
> Brisbane, Australia
> Dave Keenan's Home Page * [with cont.]  (Wayb.)




i think that's a fantastic idea!


-monz

Message: 5247 - Contents - Hide Contents

Date: Tue, 24 Sep 2002 12:41:18

Subject: Re: a reference pitch (was: A common notation for JI and ETs)

From: monz

hi George,



> From: "gdsecor" <gdsecor@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, September 23, 2002 11:32 AM
> Subject: [tuning-math] Re: a reference pitch
>          (was: A common notation for JI and ETs)
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>> hi George,
>> 
>> 

>> good to be back in contact with you.
> 
> Likewise.
> 

>> i proposed my "C" standards in the interests
>> of mathematical simplicity, and still stand by
>> them because of that reason.  it doesn't get
>> much simpler than C n^0 = 1 Hz.
> 

> And how does that translate into a pitch standard for ETs
> (including 12-ET)?
> 
> --George




umm ... well ... it only means that "middle-C" is 256 Hz.
this would make the 12edo "A" = ~430.5 Hz.

it was just my thinking that since this is not too far
off from most of the pitch-standards already in use today,
it makes more sense as a basis from a logical point of view.
we commonly use "C" as the reference anyway instead of "A",
so why not simply equate it with 1 Hz?



-monz
"all roads lead to n^0"

Message: 5248 - Contents - Hide Contents

Date: Tue, 24 Sep 2002 12:03:08

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Hi George,

got your latest, thanks. I don't think there's anything that needs my 
reply. I just wanted to say that there's probably no point in going public 
until we've got an actual font that folks can use with Sibelius. I 
understand we have 29 flag-combinations * 2 directions * 4 shaft-types + 1 
natural + 1 conventional sharp + 1 conventional flat = 235 symbols

It might be a good idea to map the 29 single-shaft down symbols to the 
characters a-z[]\ and the 29 single-shaft up symbols to the characters 
A-Z{}|, in order of rational size. Double-shafts could be obtained with the 
Alt key, and triple and X shafts with the Ctrl key and Ctrl and Alt keys.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 5249 - Contents - Hide Contents

Date: Wed, 25 Sep 2002 14:18:58

Subject: Re: a reference pitch (was: A common notation for JI and ETs)

From: gdsecor

--- In tuning-math@y..., "monz" <monz@a...> wrote:

>> From: "gdsecor" <gdsecor@y...>
>> --- In tuning-math@y..., "monz" <monz@a...> wrote:

>>> i proposed my "C" standards in the interests
>>> of mathematical simplicity, and still stand by
>>> them because of that reason.  it doesn't get
>>> much simpler than C n^0 = 1 Hz.
>> 

>> And how does that translate into a pitch standard for ETs
>> (including 12-ET)?
>> 
>> --George
> 

> umm ... well ... it only means that "middle-C" is 256 Hz.
> this would make the 12edo "A" = ~430.5 Hz.
> 
> it was just my thinking that since this is not too far
> off from most of the pitch-standards already in use today,
> it makes more sense as a basis from a logical point of view.
> we commonly use "C" as the reference anyway instead of "A",
> so why not simply equate it with 1 Hz?
> 
> -monz
> "all roads lead to n^0"


The two main obstacles are 1) getting wind instruments to play in 
tune and 2) getting others to accept this.  I have a feeling that the 
second one is the more formidable obstacle.

--George

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