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Message: 4750 - Contents - Hide Contents

Date: Mon, 29 Apr 2002 06:34:20

Subject: Re: what's up with the paper?

From: genewardsmith

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

Lately I've been writing music instead of the paper.


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Message: 4751 - Contents - Hide Contents

Date: Tue, 30 Apr 2002 02:25:11

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4179]: >But there's stil quite a difference, so I don't think that it would >be a problem. Note that the change does make the vertical shaft look >more centered in that large 65.3-cent symbol.
OK. I accept the smaller width for the left convex flag.
>By the way, it's been bugging me that we've yet to agree on the >spelling of confusable vs. confusible. I finally looked up the -able >vs. -ible rules. There were two that applied (source: _The Grammar >Bible_, Strumpf & Douglas, Knowledgeopolis, 1999): > >Rule 2: If the base itself is a complete English word, use the >suffix -able. Examples: changeable, flyable > >Which would result in "confuseable". However, see > >Rule 4: If you can add the suffix -ion to the base to make a >legitimate English word, then you should use the suffix -ible. >Examples: corruptible (corruption), perfectible (perfection) > >Which results in "confusible". > >I hope that rule 4 will end the confusion, even if it doesn't >eliminate all of the confusibility.
I've got bad news for you. ;-) It would seem to me that one should only apply such rules when the word itself cannot be found in any dictionary, or when dictionaries disagree, and as such, rule 4 is a good one, since it predicts the dictionary spelling for most such words. Unfortunately I find "confusability" and not "confusibility" in my Shorter Oxford. And the Australian English dictionary that comes with Microsoft Word accepts confusable and confusability, but not confusible or confusibility. Of course it's possible that a US dictionary may have "-ible". Let me know if you find one. I couldn't easily figure out how to switch my Microsoft Word to use a US English dictionary. I have no objection if you wish to continue spelling it "-ible".
>> Which is one thing that causes me to re-propose the convex with the >> slightly squarer corners. Your only comment about them has been >to "leave
>> well enough alone". Can you give a more detailed reason for >rejecting them? >
>Just an aesthetic consideration:
Thought it might be.
>the flags with the squarer corners >tend to look like right-angle flags with rounded corners, as opposed >to flags that are curved along their entire length.
Would that be such a bad thing, if it makes them more distinct from other types? I'll go with your preference for these bitmaps, because I agree they are more pleasing to the eye, but when it comes to designing an outline font I'd be tempted to do something in between the two.
>> I don't want to jump the gun and go to the main list just yet, and >when I
>> do, I'll want a staff showing the odd harmonics of G up to 41, >including
>> all optional spellings (using single shaft symbols with >conventional sharps
>> and flats), as well as the 217-ET notation and a couple of other >ETs. >
>Then I think that we should decide on standard (or preferred) sets of >symbols for as many ET's as we can before doing this.
What would be even better is, after doing a few very different ones the hard way, and therefore thinking about what the issues are, if we could simply give an algorithm for choosing the notation for any ET. I gave two, earlier. They are both undoubtedly too simple. The difference between them was exactly the difference between plan A and Plan B for 217-ET, i.e. whether to favour fewer flags or lower primes. It seems we've decided in favour of lower primes so far. Lets see how that pans out for a few other ETs. One thing we need to decide is how we are going to decide when N-ET's best fifth isn't good enough and instead notate it as every nth step of n*N-ET. I propose that we not accept any notational fifth for which either the apotome or the pythagorean limma vanishes, or is a negative number of steps. This excludes from using their "native" fifth, only the ETs 2 thru 11, 13 thru 16, 18,20,21,23,25,28,30 and 35. I also feel that, if we are using a comma for a prime greater than 9 to notate an ET, then the user would be justified in assuming that the best 4:9s in the tuning are notated as ..., Bb:C, F:G, C:D, G:A, D:E, A:B, E:F#, ... etc. If this is not the case then I suggest that the ET's native fifth is not acceptable for notation, since we have no "9-comma" symbol. This further excludes 32,33,37,40,42,44,45,47,49,52,54,57,59,61,62,64,66,69,71,73,74,76,78,81,83,8 5,86,88,90,93,95,97,98,100,102,103,105,107,110,112,114,115,117,119,122,1124, 126,127,129,131,134,136,138,139,141,143,146,148 and about half of all ETs from then on. Perhaps it simpler if I just list the ones that we _do_ need to try to notate, and beside them list the others that they give us as subsets. 12 (6 4 3 2) 17 19 22 (11) 24 (8) 26 (13) 27 (9) 29 31 34 36 (18) 38 39 41 43 46 (23) 48 (16) 50 (5 10 25) 51 53 55 56 (7 14 28) 58 60 (15 20 30) 63 (21) 65 67 68 70 (35) 72 80 (40) 84 (42) 94 (47) 96 (32) 99 (33) 104 (52) 108 (54) 111 (37) 118 (59) 128 (64) 132 (44 66) 135 (45) 142 (71) 147 (49) 152 (76) 171 (57) 183 (61) 186 (62) 207 (69) 217 This includes every ET up to 72. Here are the first few, showing the accidentals I think are required in addition to # and b in the two-symbol approach. ET 12 17 19 22 24 26 27 29 31 34 36 38 39 steps symbols 1 s|s |x s| s|s |x s| s| |x s| |x s|s s| 2 s|x s|s |w s|s ET 41 43 46 48 50 51 53 55 56 58 60 63 65 steps symbols 1 s| |x s| s| s|x |x s| x| |x s| s| |x s| 2 s|s s|s s|s s|s |x v|x s|s |x s| |s |x s| |x 3 s|x s|s s|s s|s s|s ET 67 68 70 72 80 84 94 96 99 104 108 111 steps symbols 1 x| |x s| s| |x s| w| s| w| v| s| w| 2 |x s| |s |x s| |x s| x| s| |x ss| s| 3 s|s s|s s|s s|s x| s|x x| |x x| x| |x |s 4 s|x s|s s|s s|s s|x w|x v|s w|s 5 s|s s|x s|s I've mostly used a "lowest prime" algorithm but there are definitely problems with this in 38, 50, 51, 55, 56, 67, 68, 99, 108, where either a two flag symbol is fewer steps than one of its component single flags, or symbols represent sizes that are out of order relative to their JI sizes, or a comma has been used that are not 1,3,p-consistent (99-ET), or other problems.
>I would also >like to get the rest of the single symbols taken care of, too. (The >question about the length of the middle shaft of the sesqui-symbols >shouldn't hold us back from designing the flags. There also remains >the question about the design of the X-symbols -- I don't recall that >you replied to my diagonals-of-a-trapezoid answer; we were both >getting a little punchy from overwork, and this is something that we >need to get back to.)
Diagonals of trapezoid is as good as anything, but the main problem with the X's is what you have pointed out yourself; that the crossing of the X seems to be referring to a different note. I also agree that the essential problem of a triple tail is not addressed by making the middle shaft shorter. I suggest you consider other possible tails that have no such extra distracting intersection point and no triples. There are V tails and wavy tails (singly or in pairs with parallel waves or counter waves), and other kinds of curved tails both single and double.
>> You might want to check out >> Shareware.com - * [with cont.] (Wayb.) >cat=247&tag=ex.sa.sr.srch.sa_all&q=truety >> pe+font+editor >
>It looks like there are a few packages that could be used. Do you >have any suggestions or preferences?
Font Lab is the easiest to use and has the most features, but I can't justify paying for it. The demo is limited to saving 20 glyphs and when you actually generate font files, half of those glyphs will be modified to include an "FL" logo. Font Creator (fcreap) is bare bones, but the price is right.
>> Welcome to Sibelius * [with cont.] (Wayb.) >> >> to get the free download which is fully functional except for save. >> -- Dave Keenan >
>This, I presume, would give us a chance to see how a new font would >work with their product.
Yes. You got it right with the 8 pixels between staff lines. The outline fonts are designed for 128 units between staff lines, so it is 32 font units per bitmap pixel. I'm sorry but I'm going to have to take an extended holiday (a week or two) from this stuff until I catch up on a lot of other work I'm supposed to have done. I look forward to great progress when I return. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 4752 - Contents - Hide Contents

Date: Tue, 30 Apr 2002 17:53 +0

Subject: Re: what's up with the paper?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <aaik8g+fl2d@xxxxxxx.xxx>
Is anybody actually planning to write this paper?  There was a lot of talk 
at the start of the year about it, but nobody did anything.  As there 
hasn't been much advance in the methods of calculating the ETs since, it 
probably is time to write that up.  Main things to report are:

Method for calculating a linear temperament from two equal temperaments.

Method for calculating a linear temperament from a set of unison vectors.

Worked examples, including the most important discoveries.


The badness measures still seem to be controversial, so we'll have to 
leave the detailed discussion of them to a later paper.  

emotionaljourney22 wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> > wrote:
>> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> > wrote:
>>> Shall we move on to a full consideration of {2,3,5,7}, >>> preferably with dave keenan and graham breed looking over >>> gene's shoulder? or am i just being a pain in the :-B ? >>
>> I'd prefer to do {2,3,5,7} next. I don't have a good feel for {2,3,7} >
> we have wonder/slendric, which is clearly the best. our paper > need not go into too many more, though many "wonders" lie > within the data gene provided (e.g., slendric seems to have an > equally complex "twin", reminding one of the syntonic-pelogic > pairing).
We don't need to go into any more in the first paper. We do need to cover all limits up to 13. Magic and multiple-29 don't seem to exist in the literature, and it's time they did. Miracle is only mentioned in that one paper that nobody seemed to notice. We also need to give an example that isn't a complete limit, and {2,3,7} will do for that. And an example that isn't octave-based, like the Bohlen-Pierce generalisation. And an example of an inharmonic timbre, like the tubulongs.
>> (or {2,5,7} or {3,5,7}). In fact I'd prefer to do the full 9-limit >> and 11-limit after that, >
> at this rate, maybe we better sacrifice those to help the paper get > done sooner?
No, sacrifice the detailed examination of all 5- and 7-limit temperaments, but don't leave out the very cases where a computer search gives us the advantage.
>> and hopefully by then we'll have figured out how >> to interpolate the cutoffs for the less familiar subsets. >
> sounds like great fodder for a second paper.
No need to do it so soon.
>> So Gene, how about hitting us with a wide-open list of 7-limit > linear
>> temperaments, so we can consider where the cutoffs might
> need to go. > > not a bad idea. also, i'd like to propose that we report the > complexity of the simplest temperament that we left off the end of > each list (in addition to my proposal that we order by complexity).
Report on the ones you want to report on, and leave the rest. For the 7-limit we need miracle and meantone. Maybe 31&6 which is complex but accurate. Does anybody want twintone, 22&31, 27&31 or 15&19? Graham
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Message: 4754 - Contents - Hide Contents

Date: Wed, 1 May 2002 09:35 +01

Subject: Re: what's up with the paper?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <aamsqq+763k@xxxxxxx.xxx>
Me:
>> And an example >> of an inharmonic timbre, like the tubulongs. Paul:
> tricky -- do we assume octave equivalence? we might want to try > assuming it and then not assuming it . . .
Do it both ways so we can explain the difference. Tubulongs may not be the best example of this because the timbre does involve tritones.
> should we go through the same process for 7-limit that we went > through for 5-limit?
Yes, but not for the first paper. Graham
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Message: 4755 - Contents - Hide Contents

Date: Wed, 01 May 2002 03:27:41

Subject: Re: A common notation for JI and ETs

From: David C Keenan

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> Yes. You got it right with the 8 pixels between staff lines. The outline > fonts are designed for 128 units between staff lines, so it is 32 font > units per bitmap pixel.
There's a bit of an arithmetic problem there. I should have written "256 units between staff lines", e.g. from -128 to +128. 32 font units per bitmap pixel is correct. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 4756 - Contents - Hide Contents

Date: Wed, 01 May 2002 17:48:17

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>> Yes. You got it right with the 8 pixels between staff lines. The > outline
>> fonts are designed for 128 units between staff lines, so it is 32 > font
>> units per bitmap pixel. >
> There's a bit of an arithmetic problem there. I should have written "256 > units between staff lines", e.g. from -128 to +128. 32 font units per > bitmap pixel is correct.
Yes, that's more reasonable. A scalable font should work very nicely with that amount of resolution.
> I'm sorry but I'm going to have to take an extended holiday (a week or two) > from this stuff until I catch up on a lot of other work I'm supposed to > have done. I look forward to great progress when I return.
We've been working on this pretty intensively, and it will do you good to get away from it for a little while. In the meantime I have plenty to keep me busy, including catching up on a backlog of tuning list digests. --George
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Message: 4757 - Contents - Hide Contents

Date: Wed, 01 May 2002 20:07:32

Subject: Re: A common notation for JI and ETs

From: jpehrson2

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

Yahoo groups: /tuning-math/message/4193 * [with cont.] 

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>>> Yes. You got it right with the 8 pixels between staff lines. The >> outline
>>> fonts are designed for 128 units between staff lines, so it is 32 >> font
>>> units per bitmap pixel. >>
>> There's a bit of an arithmetic problem there. I should have > written "256
>> units between staff lines", e.g. from -128 to +128. 32 font units > per
>> bitmap pixel is correct. >
> Yes, that's more reasonable. A scalable font should work very nicely > with that amount of resolution. >
>> I'm sorry but I'm going to have to take an extended holiday (a week > or two)
>> from this stuff until I catch up on a lot of other work I'm > supposed to
>> have done. I look forward to great progress when I return. >
> We've been working on this pretty intensively, and it will do you > good to get away from it for a little while. In the meantime I have > plenty to keep me busy, including catching up on a backlog of tuning > list digests. > > --George
***It might be fun to post a "summary" on what has been going on here on the "Main" list... Looks like a lot of interesting stuff has been going on. J. Pehrson
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Message: 4759 - Contents - Hide Contents

Date: Wed, 01 May 2002 23:49:43

Subject: Re: what's up with the paper?

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., graham@m... wrote: >
>>> should we go through the same process for 7-limit that we went >>> through for 5-limit? >>
>> Yes, but not for the first paper.
That process won't take anywhere near as long as it did the first time.
> well, i don't know where you were when it appeared confirmed that the > paper was going to consist of this same process for {2, 3, 5}, {2, 3, > 5, 7}, {2, 3, 7}, {2, 5, 7}, and {3, 5, 7}, and no other cases. you > certainly didn't speak up then.
I don't remember this either, but I think it's a good idea.
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Message: 4760 - Contents - Hide Contents

Date: Thu, 02 May 2002 17:29:17

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > > Yahoo groups: /tuning-math/message/4193 * [with cont.] >
>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>>> I'm sorry but I'm going to have to take an extended holiday (a
week or two)
>>> from this stuff until I catch up on a lot of other work I'm supposed to >>> have done. I look forward to great progress when I return. >>
>> We've been working on this pretty intensively, and it will do you >> good to get away from it for a little while. In the meantime I have >> plenty to keep me busy, including catching up on a backlog of tuning >> list digests. >> >> --George >
> ***It might be fun to post a "summary" on what has been going on here > on the "Main" list... Looks like a lot of interesting stuff has been > going on. > > J. Pehrson
Dave & I need to get a few more things straightened out before we do that. It's been a long and complicated task to agree on as many things as we already have, but it is still premature to put this out on the main tuning list. There are still things that we haven't discussed in sufficient detail, and bringing others into the fray at this point would be counterproductive. Remember, patience comes to those who wait for it. --George
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Message: 4761 - Contents - Hide Contents

Date: Fri, 3 May 2002 20:52:33

Subject: The and groupsps

From: Gene W Smith

I've already looked at <2,3,7> and <2,5,7>, and to this should probably
be added at minimum <2,5/3,7>, <2,3,7/5>, 
<2,5,7/3>, <2,5/3,7/3>, <2,7/3,7/5> and the index two subgroups of the
7-limit given by <2,5,7,9>, <2,5/3,7,9>, <2,5,7/3,9>, <2,5/3,7/3,9>.

A search for ets supporting <2,5/3,7> turns up, for a log-flat badness
measure less than 0.5 and n to 5000, the following:

1   .2630344055
4   .45883938
11   .3949962817
15   .4272823563
57   .1451878461
114   .4106521743
213   .4874803032
270   .3169588282
327   .2215293311
384   .4759848469
3779  .4314215252

The 57-et stands out here. If we go to the index 2 subgroup <2,5/3,7,9>,
we get for badness less than 0.75 up to n = 2000 the following:

1   .3625700797
5   .5389365888
11   .2909521576
16   .7085795380
22   .7331534985
30   .6856035815
41   .7433989479
83   .7334574068
88   .6539733996
171   .3202175071
182   .7238985678
753   .5905099365
924   .4265227361
1095   .6996134010

Here 11 is clearly of special interest. If we call the 7-limit intervals
"blue" which can be expressed in terms of

2^a (5/3)^b 7^c 9^d

and the rest "red", then half of the 7-limit is colored blue (hence,
"index 2"). The MT reduced comma set for the
22-et in the 7-limit is <50/49, 64/63, 245/243>, as it happens, *all* of
these are blue, and so the 11-et has the same
MT reduced set of commas, and the kernel of the map from the 7-limit to
the 22-et is entirely blue.

We have

50/49 = 2 (5/3)^2 7^(-2) 9

64/63 = 2^6 7^(-1) 9^(-1)

245/243 = (5/3) 7^2 9^(-2)

While 50/49 is blue, 7/5 and 10/7 are red, so the meaning of 50/49~1 in
this subgroup has nothing to do with 7/5~10/7. However, we still have the
relationships 9/8~8/7 for 64/63 and (9/7)^2~5/3 for 245/243 as blue. The
connection between the 22-et and the 11-et therefore seems particularly
strong.

The corresponding periodicity blocks are

[1, 21/20, 8/7, 6/5, 9/7, 27/20, 40/27, 14/9, 5/3, 7/4, 40/21]

for the 11-et, and

[1, 28/27, 21/20, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 27/20, 10/7, 40/27,
3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 9/5, 40/21, 27/14]

for the 22-et; the 11-et scale is simply every other note of the 22-et
scale.


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Message: 4762 - Contents - Hide Contents

Date: Fri, 3 May 2002 10:59 +01

Subject: Re: what's up with the paper?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <aapnav+d3g3@xxxxxxx.xxx>
emotionaljourney22 wrote:

> well, i don't know where you were when it appeared confirmed that the > paper was going to consist of this same process for {2, 3, 5}, {2, 3, > 5, 7}, {2, 3, 7}, {2, 5, 7}, and {3, 5, 7}, and no other cases. you > certainly didn't speak up then.
I probably assumed it wouldn't be a paper I'd be working on.
> now that we've shattered this myth of consensus, and now that gene > isn't in the mood to churn out more results for us, there doesn't > look like much hope for a co-authored paper now. am i wrong to be so > pessimistic?
I wouldn't object to publishing three or four papers as one with joint authorship. But why not write up what we know now? If you're still relying on Gene to produce the results, it suggests even you don't understand the method. Graham
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Message: 4764 - Contents - Hide Contents

Date: Sat, 04 May 2002 03:58:00

Subject: Re: The and groupsps

From: genewardsmith

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

> While 50/49 is blue, 7/5 and 10/7 are red, so the meaning of 50/49~1 in > this subgroup has nothing to do with 7/5~10/7.
I forgot to add that (21/20)^2 ~ 9/8 is an entirely "blue" relationship.
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Message: 4766 - Contents - Hide Contents

Date: Sun, 05 May 2002 07:17:21

Subject: Re: badness? [was:] The and groupsps

From: genewardsmith

--- In tuning-math@y..., klaus schmirler <kschmir@z...> wrote:
> hi, > normally i would have read "badness measure", taken it for > what the word says and gone on, but no, not here: the > next-to-nil > badness for 1et baffles me a little. Gene, are you serious? > what > measure is this really? anything to do with music? > my guess: the number of tones itself counts as badness and > is somehow computed with deviations from just; this gives > lower-numbered ets a not quite deserved head start?
It gives it a head start; whether it deserves it or not is a matter of controvery, but you need to go to twice as many notes to the octave to have a chance of beating it, which sounds pretty good, really. I suggest not worrying about the first few entries when considering a log-flat scale, they have some theoretical applications but are not directly usable as temperaments, obviously.
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Message: 4767 - Contents - Hide Contents

Date: Sun, 05 May 2002 17:11:04

Subject: Re: A common notation for JI and ETs

From: David C Keenan

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>>> We've been working on this pretty intensively, and it will do you >>> good to get away from it for a little while. In the meantime I > have
>>> plenty to keep me busy, including catching up on a backlog of > tuning >>> list digests.
And responding to some of my posts in this thread regarding rational apotome complements.
> Remember, patience comes to those who wait for it. Tee hee.
I never did say how much I enjoyed your Justin Tenacious story. I didn't guess what the Leprechauns' solution would be, and of course it's debatable. But fun. On the subject of fiction: I was thinking that mathematics is so unfashionable as a justification for anything musical that we should invent some mythology to introduce our notation. I think the notation should somehow be given by the gods rather than designed (or found mathematically), and of course there's some truth in that if you take numbers (at least the rationals) to be god-given, or as an atheist-mystic like me might prefer, built into the fabric of the universe. I have images of Olympian gods throwing arrows at us in a dream. Or we've discovered a lost parchment from Atlantis or something. Any ideas? Back to mathematics: My obsession is to have no notational schisma greater than 0.5 c and yours is to be able to notate practically everything with a single accidental, and we're doing fairly well at acheiving both. However there seems to be a big hole in the latter when it comes to the relatively common rational subdiminished fifth or augmented fourth 5:7. How will you notate a 5:7 up from C? And its inversion 7:10? I imagine you might want to be able to notate the entire 11-limit diamond (and Partch's extensions of it) (rationally, not in 72-ET or miracle temperament) with single accidentals, so 7:11 and 11:14 might be a problem too. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 4768 - Contents - Hide Contents

Date: Sun, 05 May 2002 20:00:22

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I wrote:
"I imagine you might want to be able to notate the entire 11-limit 

diamond (and Partch's extensions of it) (rationally, not in 72-ET or 

miracle temperament) with single accidentals, so 7:11 and 11:14 might 

be a problem too."

Actually, it looks like 11/5 might be more of a problem than either 11/7 or
7/5.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)


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Message: 4769 - Contents - Hide Contents

Date: Mon, 6 May 2002 23:13:10

Subject: The porcupine-hemithirds complex

From: Gene W Smith

If we take a 7-note porcupine (250/243) MOS, we have a corresponding
5-limit block

1-10/9-6/5-4/3-40/27-8/5-16/9

with step sizes of 10/9 and 27/25, ending with 9/8. In terms of the
22-et, it is 
3333334; whereas 37-et has it as 5555557. If we add another step,
we have instead 33333331 and 55555552 respectively; with the block
becoming 1-10/9-6/5-4/3-40/27-8/5-16/9-48/25.

6/5
        1
8/5
        4/3
                  10/9
        16/9
                   40/27

is the 5-limit lattice, with 40/27~36/25 connecting the top and bottom of
the scale; if we go to the 8-note scale, we get the 48/25-6/5-36/25
triad, filling the connection out with a chord.

This 5-limit (or <2,3,5>-group) scale can be transformed to a
corresponding "no 3s"
<2,5,7>-group scale by means of 2-->2, 3-->20/7, 5-->32/7, which can be
extended to a complete 7-limit transformation by the addition of
7-->48/7. This transformation sends
250/243 to 3136/3125, and 5-limit porcupine to what I've called
hemithirds no-threes.
The map sends 10/9-->28/25, 27/25-->125/112, and the 7-note scale to

1-28/25-5/4-7/5-196/125-7/4-49/25

the addition of 48/25 is then sent to 35/16. In terms of the 31-et, which
does the hemithirds no-threes quite nicely, we now have 5555551 for the
7-note scale and
41555551 for the 8-note scale; the 37-et, which does both of these, has
it as
6666661 and 51666661 respectively.

Finally, we may also send 2-->2, 3-->10/3, 5-->14/3, 7-->32/3; this sends
250/243 to
(1/2)(3087/3125), and the 5-limit tempered by 250/243 to the <2,5/3,7/3>
group tempered by 3125/3087. The 45-et does a good job on this group and
temperament, and in terms of it we have steps of size 25/21 and 147/125
both being sent to 11, and
scales [10,1,10,1,11,11,1] and [10,1,10,1,10,1,11,1]. The amazing 37-et,
which covers
250/243, 3136/3125 and 3125/3087, and hence this entire transformational
complex,
has it as 8181991 and 81818191 respectively.


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Message: 4770 - Contents - Hide Contents

Date: Tue, 07 May 2002 20:49:44

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4204]:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>>>> We've been working on this pretty intensively, and it will do you >>>> good to get away from it for a little while. In the meantime I have >>>> plenty to keep me busy, including catching up on a backlog of tuning >>>> list digests. >
> And responding to some of my posts in this thread regarding > rational apotome complements.
It looks as if this will occur at least partially in this message.
> I never did say how much I enjoyed your Justin Tenacious story. I > didn't guess what the Leprechauns' solution would be, and of course > it's debatable. But fun.
It's been weeks since I put that out there, and you're the first one to say anything about it. I was wondering whether anyone had read it. There are lots of solutions: 13:15:17:20:23 or 12:14:16:18:21 or 26:30:34:39:45 can simulate 5-ET 17:19:21:23:25:28:31 or 18:20:22:24:27:30:33 can simulate 7-ET And here's an application for the 41st harmonic: 22:24:26:28:30:32:35:38:41 or 24:26:28:30:33:36:39:42:45 can simulate 9-ET The lower the ET, the lower the numbers in the scale ratio. As I recall, when I tried these on the Scalatron, I thought that the most successful ones were the 5 and 7-ET ones with the higher primes. As for how well it works? About as well as you could expect, considering that this came from a tricky leprechaun. But I found that I liked the way these sound better than the ET's.
> On the subject of fiction: I was thinking that mathematics is so > unfashionable as a justification for anything musical that we should > invent some mythology to introduce our notation. I think the notation > should somehow be given by the gods rather than designed (or found > mathematically), and of course there's some truth in that if you take > numbers (at least the rationals) to be god-given, or as an > atheist-mystic like me might prefer, built into the fabric of the > universe. > > I have images of Olympian gods throwing arrows at us in a dream. Or > we've discovered a lost parchment from Atlantis or something. Any > ideas?
Yes -- considerably more than you bargained for, and I didn't even have to invent any ideas. From my perspective the pursuit of truth is better than fiction, but even more unfashionable than mathematics. I grew up as an atheist/skeptic and in my late teens came to faith in a personal God by wrestling with the problem of accounting for intricate design in the universe. (I'll try to keep this on topic.) I could not accept the notion that the great variety and complexity of life-forms in existence are nothing more than the product of random processes, i.e., purely by chance. For me this opened up the possibility of belief in a Creator-God that might have revealed himself to us in the course of history. After careful consideration of the writings of the Hebrew prophets and Christian apostles, I concluded that there were more problems in rejecting their testimony than in accepting it, and I became a Christian. And a year or so later I became a microtonalist. Last August I read an article about persons who sustained injury to the part of the brain that processes music, so that even the simplest tunes were now incomprehensible. By this I realized that music is more than just our ability to hear sounds -- it is something that we were designed to be able to enjoy -- nothing less than a gift of God. When Margo Schulter first wrote me last September about my 17-tone well-temperament (only shortly after I had read that article), she included the following words of appreciation:
> It has often been written that music is a gift both priceless and > divine, and your well-temperament is to me a musical offering most > precious, a gift which I hope to honor through use in much joyous > musicmaking.
And my reply (10 Sep 2001) included the following: << Yes, music is a gift, both priceless and divine, and I have never been more keenly aware of that than now. In writing my book, I have found that some of my best ideas just seem to appear suddenly at my mental doorstep, and I sometimes wonder whether I really thought of those things myself or whether God gave me some sort of special I- love-you note to help me along and to remind me that, when I agreed to trust him with everything in my life, that he would guide my steps and direct my path. But then I realize that everything good in this world is ultimately his doing. Whatever gifts we may possess, whether musical or otherwise, as we use them for even the best reasons and highest motivation, it is always good to be reminded that we should not forget to honor the Giver of those gifts. (If this sounds anything like a lecture, then believe me I needed to hear it as much as anyone.) >> Anyway, back to musical notation.
> Back to mathematics: > > My obsession is to have no notational schisma greater than 0.5 c and > yours is to be able to notate practically everything with a single > accidental, and we're doing fairly well at acheiving both. However > there seems to be a big hole in the latter when it comes to the > relatively common rational subdiminished fifth or augmented fourth > 5:7. How will you notate a 5:7 up from C? And its inversion 7:10? > > I imagine you might want to be able to notate the entire 11-limit > diamond (and Partch's extensions of it) (rationally, not in 72-ET or > miracle temperament) with single accidentals, so 7:11 and 11:14 might > be a problem too.
For 7:5 above C it is fortunate that the (17'-17) comma (~6.001 cents) is very close to the difference between the 7 and 5 commas, 5103:5120 (3^6*7:2^10*5, ~5.758 cents). So, according to the complementation rules, 7/5 of C is G lowered by s||x. Likewise, for 11:7 above C the 29 comma (~33.487 cents) is very close to the difference between a Pythagorean G-sharp and 11/7, so that 11/7 of C is G raised by v||w, according to the complementation rules. I don't think that either of these complements is at issue in your concern about rational complementation. --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4205]:
> > Actually, it looks like 11/5 might be more of a problem than either
11/7 or 7/5. I will see about 11/10 when I get a chance. I've been spending a lot of time lately on the rational complementation problem and have made considerable progress. I still need to re-read some of your recent messages so that I can get some additional perspective on my latest work before I post it. (All of this attention to detail is going to have to pay off in the long run.) --George
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Message: 4771 - Contents - Hide Contents

Date: Tue, 07 May 2002 22:51:44

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >It's been weeks since I put that out there, and you're the first one >to say anything about it. I was wondering whether anyone had read it.
When something is perfect, what can anyone say? :-)
>There are lots of solutions: > > 13:15:17:20:23 or 12:14:16:18:21 or 26:30:34:39:45 can simulate 5-ET > 17:19:21:23:25:28:31 or 18:20:22:24:27:30:33 can simulate 7-ET > >And here's an application for the 41st harmonic: > > 22:24:26:28:30:32:35:38:41 or 24:26:28:30:33:36:39:42:45 can >simulate 9-ET
I get the picture. As you said in the story, it's a novel idea to look for harmonic series pieces that approximate ETs instaed of the other way 'round. When I said it's debatable whether the request was satisfied: It's unclear whether Justin wanted all the intervals (dyads) to be justly intoned (I don't find 11:12 to be justly intoned) and if he did, whether he will be satisfied with only 5 notes and that 8:9 and 6:7 are in the same interval class. But I'm splitting hairs.
>The lower the ET, the lower the numbers in the scale ratio. As I >recall, when I tried these on the Scalatron, I thought that the most >successful ones were the 5 and 7-ET ones with the higher primes. As >for how well it works? About as well as you could expect, >considering that this came from a tricky leprechaun. But I found >that I liked the way these sound better than the ET's. OK. >Yes -- considerably more than you bargained for, and I didn't even >have to invent any ideas. From my perspective the pursuit of truth >is better than fiction, but even more unfashionable than >mathematics. I grew up as an atheist/skeptic and in my late teens >came to faith in a personal God by wrestling with the problem of >accounting for intricate design in the universe. (I'll try to keep >this on topic.) I could not accept the notion that the great variety >and complexity of life-forms in existence are nothing more than the >product of random processes, i.e., purely by chance. For me this >opened up the possibility of belief in a Creator-God that might have >revealed himself to us in the course of history. After careful >consideration of the writings of the Hebrew prophets and Christian >apostles, I concluded that there were more problems in rejecting >their testimony than in accepting it, and I became a Christian.
Yikes! That is seriously off-topic. And random processes vs. creator god = personal god, seriously fails to exhaust the possibilities. I'll limit my reply to suggesting two brilliant books. "Darwin's Dangerous Idea" by Daniel Dennett and "A Brief History of Everything" by Ken Wilber.
>And a year or so later I became a microtonalist. > >Last August I read an article about persons who sustained injury to >the part of the brain that processes music, so that even the simplest >tunes were now incomprehensible. By this I realized that music is >more than just our ability to hear sounds -- it is something that we >were designed to be able to enjoy -- nothing less than a gift of God.
We were designed to enjoy it, yes. But to assume that _must_ imply the latter, is either poetic (which is fine), or a serious failure of the imagination. [More seriously off-topic stuff deleted] If you promise to lay off the monotheistic dualism (look where _that's_ got the planet at the moment), I promise to lay off the atheistic mysticism (not that I ever layed any on). What I had in mind for the introduction of the notation was some light -hearted fiction that was obviously fiction, preferably not involving any real extant religion.
>For 7:5 above C it is fortunate that the (17'-17) comma (~6.001 >cents) is very close to the difference between the 7 and 5 commas, >5103:5120 (3^6*7:2^10*5, ~5.758 cents). So, according to the >complementation rules, 7/5 of C is G lowered by s||x. > >Likewise, for 11:7 above C the 29 comma (~33.487 cents) is very close >to the difference between a Pythagorean G-sharp and 11/7, so that >11/7 of C is G raised by v||w, according to the complementation rules. > >I don't think that either of these complements is at issue in your >concern about rational complementation.
That's correct. These complements are uncontroversial.
> >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4205]: >>
>> Actually, it looks like 11/5 might be more of a problem than either
>11/7 or 7/5. > >I will see about 11/10 when I get a chance. > >I've been spending a lot of time lately on the rational >complementation problem and have made considerable progress. I still >need to re-read some of your recent messages so that I can get some >additional perspective on my latest work before I post it. (All of >this attention to detail is going to have to pay off in the long run.) Thanks.
My current thinking is that the rational complements should be based on 665-ET, an ET with an extremely good 1:3 so there is no danger of any size cross-overs with any pairs of symbols. We only need to introduce a |vv symbol (instead of my earlier proposed vw|) as the complement to ss|, the 25 comma symbol. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 4772 - Contents - Hide Contents

Date: Wed, 08 May 2002 15:38:24

Subject: Microtonal Fiction (Was: A common notation for JI and ETs)

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
[Regarding my fictional story about Justin Tenacious]
>> It's been weeks since I put that out there, and you're the first one >> to say anything about it. I was wondering whether anyone had read it. >
> When something is perfect, what can anyone say? :-)
Well, you've made my day! In a conversation, if nobody says anything in response to something that was said, one can usually assume that: 1) They didn't hear and/or understand it; or 2) It was of no interest; or 3) It was either so incredibly ignorant, stupid, or trivial that it would be best to say nothing. So I was a little puzzled during those weeks of silence.
>> ... From my perspective the pursuit of truth >> is better than fiction, but even more unfashionable than >> mathematics. >
> Yikes! That is seriously off-topic. And random processes vs.
creator god =
> personal god, seriously fails to exhaust the possibilities. I'll limit my > reply to suggesting two brilliant books. "Darwin's Dangerous Idea" by > Daniel Dennett and "A Brief History of Everything" by Ken Wilber.
And in return, may I suggest _Darwin on Trial_ by Phillip E. Johnson and _Intelligent Design: The Bridge Between Science & Theology_ by William Dembski.
> [More seriously off-topic stuff deleted] > > If you promise to lay off the monotheistic dualism (look where that's_ got > the planet at the moment), I promise to lay off the atheistic mysticism > (not that I ever layed any on).
I was just indulging in a little introspection illustrating how one's philosophical outlook might be related to the creative process in music (or in this case, music theory). I also submit that it is not religious belief that causes turmoil and misery so much as the inability of those with strongly held beliefs (religious, political, or otherwise controversial) to respect the rights of those who differ, particularly when they are so insecure in their beliefs that they respond irrationally when challenged. Even microtonality has its fanatics, and whatever our passion, we just have to learn to get along. But that's enough of that. --- In tuning-math@y..., "jonszanto" <jonszanto@y...> wrote:
> George and Dave, > > (see, I read this stuff! I'm just not well-versed enough to contribute...) > > --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>> What I had in mind for the introduction of the notation was some >> light-hearted fiction that was obviously fiction, preferably not >> involving any real extant religion. >
> Dave, I remember you doing something like this once before - was it
in connection with the Excel/3D tuning object project? In any event, George, Dave had a great mock-myth behind the stuff (whatever theory/piece/project it was) that was very tongue-in-cheek. Take him up on the offer!
> > Cheers, > Jon
Yes, it's a wonderful idea. Even before reading these latest responses, I showed Dave's suggestion last night to my daughter (who is very knowledgeable about classical mythology), and she offered up the following bit of information: One source of our inspiration could be Apollo, the Greek god of music. Apollo happens to have a certain distinction in classical mythology: most of the Greek gods had a Roman counterpart, but not Apollo -- the Romans had to adopt him from the Greeks. I'll have to ask her if there were any mathematical gods in the Greek pantheon. After all, just as we have been collaborating on this notation, I don't think it is approrpriate that Apollo would be working on this one all by himself. My daughter also loves to write (she has some RPG fan fiction out on the web) and would probably like to get involved in this also. So, Dave, where and when do we get started? (Off-list, I would say. Tuning-math just doesn't seem to be the right place.) --George
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Message: 4773 - Contents - Hide Contents

Date: Wed, 08 May 2002 21:39:16

Subject: Re: A common notation for JI and ETs

From: gdsecor

Dave,

I've put out a file containing my latest proposal for symbols for 
alterations above the half-apotome.

Yahoo groups: /tuning- * [with cont.] 
math/files/secor/notation/Symbols3.bmp

I've paid particular attention to scaling the width of the 2 & 3-
shaft and X symbols.  For these I didn't think it was appropriate to 
make the concave flags as small as you did in your examples 
(comparing the size of the symbols at the left extreme with those at 
the right extreme of the line above), and I find that these and the 
wavy flags are quite readable this way.  I realize that a few of 
these symbols won't be used the way we presently have things figured 
out, but I did all of these just to get a sense of continuity in the 
progression of size moving vertically.

I also tried my hand at ss||, ss|||, and ssX symbols at the far 
right, just to see how those might look.  (I hope that the meanings 
of x and X don't get too confusing.)

--George


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Message: 4774 - Contents - Hide Contents

Date: Wed, 08 May 2002 19:46:58

Subject: Re: A common notation for JI and ETs

From: David C Keenan

There were mistakes in my latest proposal for rational apotome complements.
The 17' and 23' comma complements were wrong. I'll give the whole thing
again with corrections and additions.

Symbol Comp   Comma name  Comments
------------------------------------
 v|    x||w   19
  |v   s||x   (17'-17)
 w|   ww||x   17
 v|v  vw||s
 w|v   x||v   17'
  |w   s||w   23
 v|w   x||    19'
 s|     ||s   5
ww|v   v||x   pythag comma (comp probably not required)
  |x    ||x   7
 v|x  ww||v      (probably not required)
  |s   s||    (11-5)
 x|    v||w   29 or (11'-7)
 v|s  vw||v   31 (comp probably not required)
 w|x  ww||       (probably not required)
 s|w    ||w      (prob not required, 5 comma + 23 comma)
vw|x   none   11/5 (hope comp is not required)
 x|v   w||v   alt 23' (comp is good reason to make this standard 23')
 w|s  vw||    23'
ss|     ||vv  25
 v|wx vv||    37' (comp probably not required)
 s|x    ||v   13
 s|s   x|x    11
sx|     |sx   31'

The above complements correspond to flags being the following numbers of
steps of 665-ET.

v|  2       3  |v
w|  5       9  |w
s|  12     15  |x
x|  19     18  |s

By the way, 217-ET isn't the largest ET we can notate using symbols having
no more than one flag per side. We can do 306-ET as follows. I think it is
the largest.

1   v|
2    |v
3   v|v
4    |w or w|v
5   s|
6   w|w
7    |x
8   v|x
9    |s
10  v|s
11  x|v
12  s|x
13  x|w
14  s|s
15  x|x
16  v||
17   ||v
18  v||v
19   ||w or w||v
20  s||
21  w||w
22   ||x
23  v||x
24   ||s
25  v||s
26  x||v
27  s||x
28  x||w
29  s||s

I haven't shown _all_ the alternatives above. Flag values are

v|  1       2  |v
w|  2       4  |w
s|  5       7  |x
x|  9       9  |s

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


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