Tuning-Math Digests messages 4825 - 4849

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Message: 4825

Date: Wed, 15 May 2002 23:11:15

Subject: Re: graham's linear temperament page

From: monz

> From: "emotionaljourney22" <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, May 15, 2002 2:36 PM
> Subject: [tuning-math] Re: graham's linear temperament page
>
>
> > Definitions of tuning terms: linear temperament, (c) 1998 by Joe Monzo *
> 
> hi joe, sorry if i seemed mean before . . . hope you can forgive me! 


no sweat, paul -- i'm used to your style by now.   :)
consider it water under the bridge.

the main thing is that i want the Dictionary to be as correct
as possible.

> it's just that i've done a lot of work trying to help you correct 
> your et definition and the pages it links to, and you, quite 
> understandably, have had little time to consider my many e-mails, 
> posts, and IM comments to you on these issues. hope you've at least 
> saved my e-mails, and made note of the relevant posts, for later 
> consideration -- that's all i can ask. maybe in 10 years we can get 
> back to this -- i'd be very happy with that.


well, i hope it doesn't take 10 years!  but right now i am
kind of tied up with other projects, and have pretty successfully
weaned myself off the tuning list(s) addiction(s).  i'm just
checking in these days and fighting the urge to reply to everything.


> 
> anyway, on the current topic, it seems you missed an important fix 
> that everyone here agrees on -- please replace "interval of 
> equivalence" with "period" in your definitions.


ok, look at the linear temperament definition now.

NOW -- regarding "equivalence interval" and "periodicity interval",
you'll all see that i provided links to definitions for those two
terms, but still don't have any content in them.  that's because
i'm *still* not clear on how to define them.  can someone(s)
*please* post simply a definition for each of them which i can
put into those Dictionary entries.  if others disagree, then
i'll try to keep updating the webpages the way i've done with
"linear temperament" -- but please give me something to start with.



-monz


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Message: 4834

Date: Wed, 15 May 2002 00:05:04

Subject: Re: graham's linear temperament page

From: dkeenanuqnetau

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

OK. I agree with Gene and Graham. Although it doesn't make any _sense_ 
to call it linear when there is no IoE, we will anyway. If only 
because it _can_ be treated the same mathematically (rank two) whether 
there is an IoE or not. But I wouldn't want a definition to be based 
on that since having no IoE is so rare.

I'd prefer to ignore the question of whether the IoE or the other 
consonances _must_ be rational ratios, and thereby avoid a whole can 
of worms associated with temperaments for inharmonic timbres.


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Message: 4836

Date: Thu, 16 May 2002 02:20:22

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> 
> > It seems to me that the notation for a linear temperament should 
be 
> the
> > same as that for some large ET that represents it well. e.g 
> meantone same
> > as 31-ET, miracle same as 72 ET.
> 
> hmm . . . is there going to be transparancy for cases like 76-equal, 
> which gives you so many good linear-temperament systems within it --  
> can the notation show that? how about 152-equal, in which most of 
the 
> linear-temperament systems use different approximations to the 
primes 
> from what 152-equal as a whole would suggest?

No. The whole basis of the notation is the chain of approximate 
fifths. If two temperaments available within a single ET use different 
sized fifths then how could they possibly be covered by a single 
notation for the ET.

You have already seen, in your adaptive JI example, how 31-ET 
_notation_ cannot continue to exist within 217-ET, despite the fact 
that 31-ET exists within it. The quarter-commas become explicit 
instead of implicit. In exactly the same way, the 1/3 commas must 
become explicit in the notation for 152-ET.

The native best-fifth of 76-ET is not suitable to be used a notational 
fifth because, among other reasons, it is not 1,3,9-consistent (i.e. 
its best 4:9 is not obtained by stacking two of its best 2:3s) and I 
figure folks have a right to expect C:D to be a best 4:9 when commas 
for primes greater than 9 are used in the notation. So 76-ET will be 
notated as every second note of 152-ET.

Here's my proposal for 152-ET.

Steps Symbol
------------
1  )|
2   |~
3  /|
4   |\
5  ~|)
6  (|~
7  /|\
11 B:C, E:F
15 #
26 A:B, C:D, D:E, F:G, G:A

Although it seems a minor problem that the 1/3 comma symbol of 152-ET 
is smaller in rational terms than the 1/4 comma symbol of 217-ET. 
We'll see what George comes up with for 152-ET.


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Message: 4838

Date: Thu, 16 May 2002 02:47:24

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

On second thoughts, here's my revised proposal for 152-ET. There were 
too many different flags in the previous one.

Steps Symbol
------------
1   )|
2    |~
3   /|
4    |\
5   /|~
6   (|~  or  //|  
7   /|\
11 B:C, E:F
15 #
26 A:B, C:D, D:E, F:G, G:A


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Message: 4840

Date: Thu, 16 May 2002 03:36:50

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> Sorry.  I forgot about the 4095:4096 schisma (which needs a 
> distinctive name of some sort).

How about "the 13-schisma" or the "tridecimal schisma".

> I meant to say that we should 
> redefine the xL flag as the 13'-(11-5) comma (which is how I've been 
> calculating everything involving that flag up to this point), and if 
> that's a bit unwieldy, then we could call it the ~29 comma (if we 
can 
> figure out how to pronounce "~" (how about "quasi").

"Quasi" is fine, but (11'-7) is also a quasi-29-comma, so you can't 
call (13'-(11-5)) _the_ ~29 comma.

> I suggest that 37-ET be notated as a subset of 111-ET, with the 
> latter having a symbol sequence as follows:

Yes. That's also what I suggested in a later message (4188).

> 111:  w|, s|, |s, w|s, s|s, x|s, w||, s||, ||s, w||s, s||s.

And that's almost the notation I proposed in the same message (with 
its implied complements), except that I would use x|x (|) as the 
complement of s|s /|\. Surely that is what you would want too, since 
it represents a lower prime and is the rational complement?

> However, a more difficult problem is posed by 74-ET, and the idea of 
> having redefinable symbols may be the only way to handle situations 
> such as this.  Should we do that, then there should probably be 
> standard (i.e., default) ratios for the flags, and the specific 
> conditions under which redefined ratios are to be used should be 
> identified.

I think 74-ET is garbage.

But if someone insisted ... Due to its lack of 1,3,9 consistency and 
the same going for 2*74 = 148, it would need to be notated as every 
third note of 3*74 = 222-ET which is itself garbage and we can't 
notate it anyway. We have no 11 step symbol for it without two flags a 
side. 

I don't think we need to apologise for failing to notate 74-ET. We can 
do every ET up to 72 and many useful ones beyond.

> I think that we'll get more of a feel for this once we start trying 
> to determine symbol sequences for various ET's.

Yes.

> I tried selecting sets of symbols (including complements) for a 
> number of ET's and came to the conclusion that it is not all that 
> obvious what is best.  

I agree. We'll just have to do them all individually. I can't imagine 
there being much disagreement on those using their native fifths until 
we get up to 38-ET. See my message 4188. The complements are implied 
by them having the same sequence of flags in the second half apotome, 
and the complement of /|\ always being itself or (|).

> Among the possible objectives I identified 
are:
> 
> 1)  Consistent symbol arithmetic (a top priority);
> 
> 2)  A matching symbol sequence in the half-apotomes;
> 
> 3)  Choose flags that represent the lower prime numbers;
> 
> 4)  Try not to use too many different types of flags;
> 
> 5)  Use rational complements where possible.

That's an excellent list of (often conflicting) criteria.

> In the same way that a difference of opinion occurs among experts or 
> authorities in the matter of English spelling (as with the 
> word "confusability"), a problem could result when different 
> composers, using the same rules and guidelines, arrive at different 
> sets of symbols for the same ET.  Some composers won't want to use 
> sagittal notation if in involves puzzling with how to notate an ET 
> and uncertainty about the suitability of the outcome, say if, after 
> composing a piece in a certain ET, it turns out that others were 
> already using a different set of symbols.

Yes. A very good point

> I suspect that, in order for us to figure out how the rules should 
be 
> applied, we'll have to do all of the ET's anyway.  So why not just 
do 
> as many as possible and include the symbol sequences along with the 
> specifications of the notation?

OK.

> Notice that in doing 111 (above), I found that giving objectives 2 
> and 4 a higher priority than objective 5 gave me the simplest 
> notation.

If you're talking about (|\ as the complement of /|\ then I must 
disagree. In most ETs that use /|\ its complement would be itself or 
(|) so I think (|) should be exempt from the consideration of too many 
flag types.

> One thing that I thought should be taken into consideration is that, 
> where appropriate, ET's that are subsets of others should make use 
of 
> a subset of symbols of the larger ET.  This would especially be 
> advisable for ET's under 100 that are multiples of 12 -- if you 
learn 
> 48-ET, you have already learned half of 96-ET.

Certainly. It's only the question of how we tell "when appropriate" 
that remains to be agreed. I've proposed two and only two reasons in 
message 4188. You might say what you think of these.

> I previously did symbol sets for about 20 different ET's, but that 
> was before the latest rational complements were determined, so I'll 
> have to review all of those to see what I would now do differently.

Great!


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Message: 4842

Date: Thu, 16 May 2002 12:04 +0

Subject: Re: graham's linear temperament page

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <000f01c1fca0$7d449ca0$af48620c@xxx.xxx.xxx>
monz wrote:

> ok, look at the linear temperament definition now.

Fancy that!

> NOW -- regarding "equivalence interval" and "periodicity interval",
> you'll all see that i provided links to definitions for those two
> terms, but still don't have any content in them.  that's because
> i'm *still* not clear on how to define them.  can someone(s)
> *please* post simply a definition for each of them which i can
> put into those Dictionary entries.  if others disagree, then
> i'll try to keep updating the webpages the way i've done with
> "linear temperament" -- but please give me something to start with.

Dave gave some, and I didn't notice anybody disagreeing.  The only quibble 
I have is that the period definition uses the terminology differently to 
the linear temperament one.  And that it's circular, but that's probably 
unavoidable.

"""
"interval of equivalence" = "equivalence interval" = "formal octave"
is that interval (much larger than a unison) which, when it occurs 
between two pitches, we consider them to be, in some sense, (formally 
if not perceptibly) the same note. For most scales this is the octave 
1:2, and when it is not the octave it is usually some other highly 
consonant interval such as the "tritave" 1:3. But the essential 
feature of the interval of equivalence in relation to definitions of 
scales and types of scales is that when we describe a scale we 
describe only the pitches that fall within a single interval of 
equivalence, and we leave it up to the instrument builder to decide 
the range of the instrument and therefore how many times (including 
fractions) the interval of equivalence should be repeated.

"interval of periodicity" = "periodic interval" = "period"
is that generator of a regular temperament (whether linear, planar, or 
n-dimensional) which generates the interval of equivalence all by 
itself. This means that the period is either equal to the interval of 
equivalence or fits into the interval of equivalence a whole number of 
times.
"""


                  Graham


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Message: 4843

Date: Thu, 16 May 2002 12:04 +0

Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <abul01+9p4m@xxxxxxx.xxx>
Dave:
> > Note that those with an IoE _can_ be treated mathematically as rank 
> > one, provided all arithmetic is modulo the IoE.

Paul:
> i used to think so, but it seems gene was able to convince me 
> otherwise. i think that you can't handle torsion properly unless you 
> express the unison vectors in IoE-specific, rather than IoE-
> equivalent/IoE-invariant terms.

I've never been convinced of this.  Every time it comes up people say they 
aren't interested enough to prove it either way.  So you should state it 
as unproven and not use it to dismiss other ideas.

The biggest outstanding problem is that we don't have an algorithm for 
calculating the optimal generator size for a given mapping and target 
consonances.  But then nobody's seriously looked.  There is a 
specific problem with contorsion, as different generator sizes give 
different, but equally valid results.  But hey, we're not counting 
contorsion anyway.  Adding a range of generator sizes to the definition 
solves this one if there's no other way.

There's also the detail that IoE-equivalent algebra gives a 
period-equivalent result, but you can get round that if you know the (IoE 
equivalent) sizes of the intervals you're approximating.  The real problem 
here is that the IoE-equivalence doesn't make it any simpler if you won't 
accept period-equivalence.

I'm sure the torsion problem will go away when somebody looks at it 
properly.  It doesn't matter anyway if we're only considering linear 
temperaments, because they don't have to be derived from unison vectors.


                       Graham


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Message: 4844

Date: Thu, 16 May 2002 21:42:34

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>Here's what I did [for 152-ET] a couple of weeks back, and after looking
at the 
>rational complements, I would still do it this way:
>
>Steps Symbol
>------------
> 1    |(
> 2    |~
> 3   /|
> 4    |\
> 5   /|~
> 6   /|)
> 7   /|\
> 8   (|)
> 9   (|\
>10    ||~
>11   /||
>12    ||\
>13   /||~
>14   /||)
>15   /||\
>
>Something that you will notice immediately is that I have used the 
>(17'-17) comma as 1 degree.  (In 152 it calculates to zero degrees 
>and would be unusable unless it were redefined as I have chosen to do 
>here.)

But redefined it as what comma? I believe a fundamental tenet of this whole
excercise, one that many people agree on, is that an accidental must never
simply represent a number of steps of the ET, but must represent a rational
comma in a manner consistent with the ETs best approximation of the primes
involved.

>Dave, your solution also redefines a flag, although it is not so 
>obvious: since |~ is 2deg and (|~ is 6deg, then (| must be 4deg.  
>This is so if it is calculated as the 29 comma, but it is 5deg if it 
>is calculated as the 715:729 comma, as I have done.  (But this is not 
>the redefinition to which I refer.)

That was unintentional. Thanks for spotting it. I now agree that using (|~
for 6 steps is completely wrong. It corresponds to 7 steps (but should not
be used). The only possible symbols for 6 thru 9 steps are the ones you
have given. And there's no question about 3 and 4 steps either. (And
therefore also 11, 12, 14, 15).

I've (re-)realised that there is no need to go beyond the 19-prime-limit
for notating any ETs (that we _can_ notate). So, for notating ETs, all the
symbols must be given their 19-limit definitions. The fact that some of
them might also be 23, 29 or 31 commas, when used for rational scales, is
utterly irrelevant (for notating ETs). Then the only remaining ambiguity is
the one involving the 13-schisma.

For notating ETs:
v| is always 512:513
|v is always 288:289
w| is always 2176:2187
|w is always 722:729
s| is always 80:81
|s is always 54:55
|x is always 63:64

but

x| can be either 45056:45927 or 715:729

In some ETs these will be the same number of steps and the choice doesn't
need to be made. But when the choice _is_ made, the meaning of (|( (|~ (|\
and (|) all follow automatically from it.

I now believe that the rational complements beyond the 19-limit (i.e. if
either of the pair is outside the 19-limit in the rational conception) are
very unimportant for notating ETs, and would only be used as a tie-breaker
if all else fails.

Here are the valid options for 1, 2 and 5 steps of 152-ET, from a 19-limit
perspective.

Steps  Symbol  Comma                Comment
-----------------------------------------------------------------------
1       )|   19
1       )|(  19 + (17'-17)

2       ~|   17
2       ~|(  17'
2        |~  19'-19

5       (|   (11'-7)
5 or 4  (|   13'-(11-5)             (5 steps for 1:13, 4 steps for 3:13)
5       )|\  19+(11-5)
5       ~|)  17 + 7
5       /|~  5+(19'-19)
5       (|(  (11'-7)+(17'-17)
5 or 4  (|(  (13'-(11-5))+(17'-17)  (5 steps for 1:13, 4 steps for 3:13)

So we see that 152-ET is not 1,3,13-consistent.

I believe that, if for any prime p, the ET is not 1,3,p-consistent, then
commas involving that prime should not be used for notational purposes
unless there's no other option. I also think that, if possible, all
notational commas should be mutually consistent and consistent with 1,3 and
9. And if the ET can be notated by using more than one such set (unlikely),
then we should use the one with the lowest maximum error in its intervals.

So here's the information about 152-ET that I find most relevant for
deciding the notation. These are its maximal consistent sets of odds in the
19-limit, along with the maximum error of any interval in the set. By
maximally consistent I mean that no other 19-limit odd number can be added
to a set without making it inconsistent.

{1, 13, 17}                  3.7 c
{1, 3, 5, 9, 11, 15, 17}     3.7 c
{1, 3, 5, 7, 9, 11, 15, 19}  2.5 c

I haven't listed any sets that do not include 1, because 
(a) I haven't computed them, and
(b) they would only be relevant if all else fails, which seems very unlikely.

The first set does not contain 3 or 9. The second set does not provide any
way of notating a single step. The third step is just right, and Goldilocks
ate it all up.

The third set works beautifully and happens to have the lowest error. It
says we shouldn't use any 13 or 17 commas, so our choice for 1, 2 and 5
steps is reduced to just these.

Steps  Symbol  Comma
-------------------------
1       )|     19

2        |~    19'-19

5       (|     (11'-7)
5       )|\    19+(11-5)
5       /|~    5+(19'-19)

None of the choices for 5 introduce any new flags, but I consider the
introduction of a new flag prior to the half-apotome to be nearly as bad.
So on that basis I reject (|. Also it seems like it is good to have more
equal numbers of single and double-flag symbols.

They are both 19-limit. None of them are the rational complement of |~. I
choose /|~ because its size in a rational tuning is closer to 5/152 octave
than is )|\.

So here's the full set for 152-ET.

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

We now only disagree on the 1 step symbol.

>You didn't give symbols for 8 and 9deg, but if I assume that 8deg 
>would be (|), so then |) would be 4deg.  In 152 |) calculates to 
>3deg, which is where your redefinition occurs.

As I say, this wasn't an intentional redefinition, it was just dumb.

>So the difference between our two solutions is that the flag that I 
>redefined is associated with a higher prime.

There is no need to redefine any, for 152-ET. And only ever a need to
redefine x|.

>May I assume that you would use matching symbols for the apotome 
>complements?

Yes.

>  That being the case, we both chose a rational 
>complement for 1deg, but a 152-specific complement for 2deg.

Now I choose no rational complement for either of them. It seems that the
_only_ justification for using |v for 1 step is that it is the rational
complement of s|x. To my mind this is not sufficient justification to
violate the definition of |v as the 17 comma 2176:2187. In fact I don't
think anything could be sufficient justification for that.

>I came 
>to the conclusion that a simple (i.e., easy-to-remember) sequence of 
>symbols is more important than using rational complements.

So did I. And I came to the conclusion that 19-limit-comma flag-definitions
are more important than using rational complements.

I don't see how changing 1 step from )| to |( improves ease of remembering.
In fact with )| we have that property that you admired in 217-ET, that the
flags alternate sides as you go up.

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4845

Date: Thu, 16 May 2002 08:29:23

Subject: updated Tuning Dictionary definitions

From: monz

> From: <graham@xxxxxxxxxx.xx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Thursday, May 16, 2002 4:04 AM
> Subject: [tuning-math] Re: graham's linear temperament page
>
>
> In-Reply-To: <000f01c1fca0$7d449ca0$af48620c@xxx.xxx.xxx>
> monz wrote:
>
>
> > NOW -- regarding "equivalence interval" and "periodicity interval",
> > you'll all see that i provided links to definitions for those two
> > terms, but still don't have any content in them.  that's because
> > i'm *still* not clear on how to define them.  can someone(s)
> > *please* post simply a definition for each of them which i can
> > put into those Dictionary entries.  if others disagree, then
> > i'll try to keep updating the webpages the way i've done with
> > "linear temperament" -- but please give me something to start with.
>
> Dave gave some, and I didn't notice anybody disagreeing.  The only quibble
> I have is that the period definition uses the terminology differently to
> the linear temperament one.  And that it's circular, but that's probably
> unavoidable.
>
> <Dave's definitions snipped -- look in the Dictionary!>


thanks much, Graham!  actually, i saw both of those the other day
(in fact, i think Dave posted them in response to my own query),
but as i've been keeping my distance from the tuning lists, i already
lost track of them.  the definitions have been updated:

Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *
Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *


now, doesn't the "planar temperament" definition also need to
be fixed to say "periodicity interval" instead of "equivalence
interval"?

Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *



-monz


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Message: 4846

Date: Thu, 16 May 2002 17:12 +0

Subject: Re: updated Tuning Dictionary definitions

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <000b01c1fcee$75f9b680$af48620c@xxx.xxx.xxx>
monz wrote:

> now, doesn't the "planar temperament" definition also need to
> be fixed to say "periodicity interval" instead of "equivalence
> interval"?
> 
> Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *

What?  Yes, they're like linear temperaments but with one extra generator. 
 And you could also say that the process can be extended to any number of 
generators (however you count them).  Hopefully, anybody doing this will 
mention "planar temperament" somewhere.


                    Graham


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