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Message: 4625 - Contents - Hide Contents Date: Tue, 16 Apr 2002 15:45:49 Subject: an earlier presentation, from September From: Carl Lumma * [with cont.] (Wayb.) * [with cont.] (Wayb.) -C.
Message: 4626 - Contents - Hide Contents Date: Tue, 16 Apr 2002 17:24:44 Subject: more objective From: Carl Lumma Disregard my last message, I think this is the best yet: * [with cont.] (Wayb.) I'd be interested Graham, if you think this is an improvement. -C.
Message: 4627 - Contents - Hide Contents
Date: Tue, 16 Apr 2002 19:23:39
Subject: Re: A common notation for JI and ETs
From: David C Keenan
I'll assume for the moment that you accept the addition of a (17'-17) flag
as the best way of giving us 17', because it's the only such choice that
also gives us 41 (assuming we can only use 1600-ET schismas).
Now if we ignore for the moment the relative sizes of the commas, and
therefore ignore which pairs we might want to have as left and right
varieties of the same type, I get the following left-right assignment of
flags as being the one that minimises flags-on-the-same-side for as far
down the list of prime commas as possible.
By the way, if this list has only one comma for a given prime, the reason
is that the same comma is optimal for both diatonic-based (F to B relative
to G) and chromatic-based (Eb to G# relative to G) notations.
Symbol Left Right
for flags flags
------------------------------
5 = 5
7 = 7
11 = 5 + (11-5)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (11-5)
17 = 17
17' = 17 + (17'-17)
19 = 19
19' = 19 + 23
23 = 23
23' = 17 + (11-5)
29 = 29
31 = 19 + (11-5)
31' = 5 + (17'-17) + 7
or 5 + 23 + 23
37 = 29 + 17
37' = 19 + 23 + 7
or 5 + 17 + 23
or 5+5+19
41 = 17 + (17'-17) + (17'-17)
43 = 19 + 19 + (17'-17)
47 = 19 + 23 + 23
or 5 + 17 + (17'-17)
The story goes like this. The 5 comma must be a left flag so that it works
like the Bosanquet comma slash. Given 5 as a left flag, the 13 symbol then
says the 7 comma must be a right flag and the 11 symbol says (11-5) must
also be a right flag. Given 7 and (11-5) right, both 11' and 13' then say
29 must be left. We accepted this much long ago. Now for the rest.
Given (11-5) right, 23' says 17 must be left and 31 says 19 must be left.
Given 17 left, 17' then says (17'-17) must be right. Given 19 left, 19'
then says 23 must be right.
That's all 8 flags assigned, and it gets us to 31 limit with minimal
same-side flags. There is no other assignment of flags to left and right,
that will do that. Notice that the 37 comma is the only one forced to be
non-minimal by this assignment. It's very convenient that it gives use 4 on
each side.
So my new proposal for the flags is
| Left Right
---------+---------------
Convex | 29 7
Straight | 5 (11-5)
Wavy | 17 23
Concave | 19 (17'-17)
Which just swaps the two concave flags from what I had before.
The only possible alternatives involve choosing which of 17 and 19 is wavy
and which concave, and likewise for 23 and (17'-17), unless you use other
types. The above proposal makes the two larger ones wavy and the two
smaller ones concave.
Note that the above same-side-minimisation process ignores any flag
combinations not listed above, that might be wanted for 217-ET (and other
ETs). This seems right to me, since I take the prime commas as fundamental,
rather than 217-ET. But rest assured that it works out just fine for 217-ET.
Now why did I have 19 as a right flag before?
That was necessary when we wanted to have the 17 and 19 commas being of the
same type. It was also necessary when we were planning to notate 3 steps of
217-ET as 17+19. But now I'm proposing we notate 3 steps as 23 since that
avoids a double flag for such a small increment, and lets us have
number-of-flags increasing monotonically with number-of-steps. i.e. we only
jump from one flag to two at one point in the sequence.
And why did you have 23 as a left flag?
Presumably only because you had 19 as a right flag for the above reasons,
and in that case 19' says 23 should be a left flag. Now that we see that 19
can and should be a left flag, we can see that 23 should be right.
Another consideration for determining which type of flag to use for each
flag-comma is the intuitiveness of the apotome-complement rules in 217-ET,
when the second half-apotome is made to follow the first.
Lets look at the size of all the flags in steps of 217-ET, assuming the
optimum left-right asignment given above, but ignoring my proposed
wavy-concave assignment.
Complementary
Flag Size Size Flag
comma in steps of comma
name 217-ET name
----------------------------
Left
----
29 6 -2 none available with same side and direction
5 4 0 blank
17 2 2 17
19 1 3 none available with same side and direction
Right
-----
7 5 1 (17'-17)
(11-5) 6 0 blank
23 3 3 23
(17'-17) 1 5 7
Assigning 17 and 23 to wavy and 19 and (17'-17) to concave mean that wavy
is always its own complement and, on the right at least, concave and convex
are complements.
On the left, just as we must avoid using the 29-flag below the
half-apotome, we should also avoid using the 19-flag (since it doesn't have
a direct flag-complement in 217-ET). This is easy since we can use the
(17'-17) flag for 1 step. The rule about using the lowest possible prime
would tell us to do this anyway.
What this means is that although I've switched the meaning of concave right
from 19 to (17'-17), my "plan C2" 217-ET notation proposal doesn't change.
Of course you may want to forget about 41 and go back and look at what
happens with those other two choices for a new flag to give 17', i.e.
(17'-19) or 17' itself.
Here's the optimum left-right assignment using a (17'-19) flag.
Symbol Left Right
for flags flags
------------------------------
5 = 5
7 = 7
11 = 5 + (11-5)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (11-5)
17 = 17
17' = 19 + (17'-19)
19 = 19
19' = 19 + 23
or 17 + (17'-19)
23 = 23
23' = 17 + (11-5)
29 = 29
31 = 19 + (11-5)
31' = 5 + 5 + (17'-19)
or 5 + 23 + 23
or (17'-19) + 7 + 23
37 = 29 + 17
or (17'-19) + (11-5)
37' = 19 + 23 + 7
or 5 + 17 + 23
or 17 + 7 + (17'-19)
or 5+5+19
Now lets look at the flag complements for this.
Complementary
Flag Size Size Flag
comma in steps of comma
name 217-ET name
----------------------------
Left
----
29 6 -2 none available with same side and direction
5 4 0 blank
17 2 2 17
19 1 3 none available with same side and direction
Right
-----
7 5 1 none available with same side and direction
(11-5) 6 0 blank
23 3 3 23
(17'-19) 2 4 none available with same side and direction
Looks bad.
Here's the optimum left-right assignment using a 17' flag.
Symbol Left Right
for flags flags
------------------------------
5 = 5
7 = 7
11 = 5 + (11-5)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (11-5)
17 = 17
17' = 17'
19 = 19
19' = 19 + 23
23 = 23
23' = 17 + (11-5)
29 = 29
31 = 19 + (11-5)
31' = 5 + 23 + 23
37 = 29 + 17
37' = 19 + 23 + 7
or 5 + 17 + 23
or 5+5+19
The only reason for putting 17' on the right here is to give 3 left and 3
right flags, since 17' doesn't actually combine with anything else.
Now lets look at the flag complements for this.
Complementary
Flag Size Size Flag
comma in steps of comma
name 217-ET name
----------------------------
Left
----
29 6 -2 none available with same side and direction
5 4 0 blank
17 2 2 17
19 1 3 none available with same side and direction
Right
-----
7 5 1 none available with same side and direction
(11-5) 6 0 blank
23 3 3 17' or 23
17' 3 3 17' or 23
That also looks bad.
It looks to me that including a flag to give us 17' has narrowed our
choices considerably. This is not a bad thing, given that the one choice it
leaves us, works so well (at least in 217-ET).
I think we should describe it as basically 29-limit, and just list the
more-than-one-flag-per-side symbols for 31, 37, 41, 43, 47 (and possibly
higher, I haven't checked) just once, near the end, as a curiosity.
Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4628 - Contents - Hide Contents Date: Tue, 16 Apr 2002 21:31:21 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..., "gdsecor" <gdsecor@y...> wrote:>> Per Ted Mook's criticism (found in your own message #24012 of May > 30,>> 2001 on the main tuning list) about reading new symbols in poor >> lighting at music stand distance, I made the symbols bolder in both >> dimensions, and you will find (for the straight flag symbols only) >> yours (on the second staff) compared with my latest version (on the >> third staff). >> I still believe this is a problem, but I haven't yet found an acceptable > solution. I believe the problem is greatest between the 2, 3 (and in my > version, the X shaft symbols). > > I think there should be a strong family resemblance between the standard > symbols and our new ones, or they will not be found unacceptable on visual > aesthetic grounds.Yes, that is a very valid point.> This is one reason why I find the bold vertical strokes an unacceptable > solution to 2-3 confusability.I added some more to this file: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/symbols1.bmp I think the problem with reading them under poor conditions is a combination of factors -- vertical lines that are rather thin *and* vertical lines that are too close together, with the second factor being more of a problem than the first. I re-did the straight-flag symbols on the fourth staff using single-pixel vertical lines with enough space between them to make them legible at a distance. I also put a couple of them in combination with conventional sharps and flats at the upper right, for aesthetic evaluation. Just above the "conventional accidentals" staff I also added my conventional sharp to the left of yours. Mine (more than yours) looks more like what I found in printed music, and I suspect that Tartini fractional sharps constructed (or written) with too-narrow spacing between the vertical lines (such as we have here) are what led to Ted Mook's observation. (And I do prefer your version.)> Frankly, I think the best solution is to use two symbols side by side > instead of the 3 and X shaft symbols, the one nearest the notehead being a > whole sharp or flat (either sagittal or standard). I think we're packing so > much information into these accidentals that we can't afford to try to also > pack in the number of apotomes. I suggest we provide single symbols from > flat to sharp and stop there. At least you have provided the double- shaft > symbols so you never have to have the two accidentals pointing in opposite > directions.What? Did I understand this correctly? Are you considering using the double-shaft symbols??? (Or are you suggesting that I should do this and forget about the ||| and X symbols?)> The fact that in all the history of musical notation, a single symbol for > double-flat was never standardised, tells me that it isn't very important, > and we could easily get by without a single symbol for double-sharp too.I think that it's because two sharps placed together looks a little weird, but not two flats. If I remember correctly, I think that double-flats are placed in contact with one another, as I put them above the staff (but I will need to check on this.)> With your bold vertical strokes you're taking up so much width anyway, why > not just use two symbols? This would require far less interpretation.I put in a couple of single-symbol equivalents at the upper right, just to show how little space they occupy in comparison to two symbols, even if you put them right up against one another. (And even with the bolder vertical strokes, the single symbols took up less space than the double symbols without any bold vertical strokes.)> Also, the X tail suggests to me that one should start with a double sharp > and add or subtract whatever is represented by the flags, which is of > course not the intended meaning at all.This would represent something within 4/10 of an apotome to a double sharp, but something like this would rarely be used.>> I saw no need to make the vertical strokes as long as >> yours, which enables two new symbols altering notes a fifth apart to >> be placed one above the other (first chord on the third staff). >> You're right. I used the standard flat symbol as my model. You do see flats > on scores directly above one another, a fifth apart. They just let the > stems overlap. But I agree ours should be shorter. > > To do it while maintaining the family resemblance, I've now taken the > standard natural symbol as my model, and shaved two pixels off the tails of > all my symbols. But this still leaves them 2 pixels further from the centre > than yours. > > I also think our flags (except possibly the smallest ones), when > space-centered, should overlap the line above and below by one pixel, just > as the body of the flat symbol does. i.e. The flags should ingeneral be 11> pixels high. Note that the body of the standard natural and sharp symbols > overlap the lines by 2 pixels. I think some overlap is important so as not > to lose the detail of the symbol where the staff lines pass thru it.I did make my new symbols (in the fourth staff) one pixel longer (at the tip of the arrow), which can be seen only when the note is on a line. I didn't think that it was wise to overlap the flag in the other direction, because this would make the nubs at the ends of the flags less visible if a staff line were to pass through them. Where I now have them, the nubs (actually 3x3 pixel squares) are in both cases immediately adjacent to staff lines.>> Let's see what agreement we can come to about how the straight- flag >> symbols should look before doing any more with the rest of them. >> Well I don't think it is possible to consider the straight flags in > isolation from all the others. The full set has to be seen together to be > sure they are sufficiently distinct from one another.Of course. However, at this point we're still establishing general dimensions, etc., and it is easier to change a few symbols than many of them. And you have just stated the reason why I don't want to commit to flag shapes for the higher primes -- I need to see what the final product looks like, which is why I want to do it more than one way.> You are throwing away > a lot of horizontal resolution when you make those vertical strokes wider.Now that I have made them narrower, we have one more pixel of resolution. By the way, if you look at the concave flags in my earlier figures, you will see that one part of the curving flag has postive and another part negative slope (and I am still making them this way); a nub on the end of this sort of flag can be seen very easily. --George
Message: 4631 - Contents - Hide Contents Date: Tue, 16 Apr 2002 03:16:27 Subject: 31 limit unique ets From: genewardsmith There are 541 of these up to 1600; the first ten are: 552 1.637374926 571 1.450447298 612 1.695743633 631 1.240353558 634 1.625692282 653 1.055690210 677 1.673616803 718 1.173415260 730 1.601691329 742 1.667556137 The number following each et is a 31-limit badness score. The following had the lowest log-flat badness among the 541: 653 1.105569021 1395 1.052127118 1600 1.016116129
Message: 4632 - Contents - Hide Contents Date: Tue, 16 Apr 2002 03:30:59 Subject: Re: the harmonic series segment test From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> 3. 0.36 according to Scala 1.8; diatonic scale in 12-et is 0.77 > This is failing. > > 4. R. stability 0.87 by Scala 1.8; diatonic scale in 12 is 0.95. > This passes.How do you get Scala to compute these?
Message: 4633 - Contents - Hide Contents Date: Tue, 16 Apr 2002 00:36:53 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> I noticed that what I was previously using for conventional symbols > was a bit different from what is commonly used, so I also made an > attempt over the weekend to improve on that. I prefer what you have > for the conventional sharp and flat symbols, but I suggest a wider > natural symbol (as shown in the 4th chord, first staff).You're absolutely right. I'll use your natural.> You have > two different double sharp symbols (for the 6th chord); I also came > up with the same as your second one, which looks good on both a line > and a space (7th chord). Ditto. > Per Ted Mook's criticism (found in your own message #24012 of May 30, > 2001 on the main tuning list) about reading new symbols in poor > lighting at music stand distance, I made the symbols bolder in both > dimensions, and you will find (for the straight flag symbols only) > yours (on the second staff) compared with my latest version (on the > third staff).I still believe this is a problem, but I haven't yet found an acceptable solution. I believe the problem is greatest between the 2, 3 (and in my version, the X shaft symbols). I think there should be a strong family resemblance between the standard symbols and our new ones, or they will not be found unacceptable on visual aesthetic grounds. This is one reason why I find the bold vertical strokes an unacceptable solution to 2-3 confusability. Frankly, I think the best solution is to use two symbols side by side instead of the 3 and X shaft symbols, the one nearest the notehead being a whole sharp or flat (either sagittal or standard). I think we're packing so much information into these accidentals that we can't afford to try to also pack in the number of apotomes. I suggest we provide single symbols from flat to sharp and stop there. At least you have provided the double-shaft symbols so you never have to have the two accidentals pointing in opposite directions. The fact that in all the history of musical notation, a single symbol for double-flat was never standardised, tells me that it isn't very important, and we could easily get by without a single symbol for double-sharp too. With your bold vertical strokes you're taking up so much width anyway, why not just use two symbols? This would require far less interpretation. Also, the X tail suggests to me that one should start with a double sharp and add or subtract whatever is represented by the flags, which is of course not the intended meaning at all.> I saw no need to make the vertical strokes as long as > yours, which enables two new symbols altering notes a fifth apart to > be placed one above the other (first chord on the third staff).You're right. I used the standard flat symbol as my model. You do see flats on scores directly above one another, a fifth apart. They just let the stems overlap. But I agree ours should be shorter. To do it while maintaining the family resemblance, I've now taken the standard natural symbol as my model, and shaved two pixels off the tails of all my symbols. But this still leaves them 2 pixels further from the centre than yours. I also think our flags (except possibly the smallest ones), when space-centered, should overlap the line above and below by one pixel, just as the body of the flat symbol does. i.e. The flags should in general be 11 pixels high. Note that the body of the standard natural and sharp symbols overlap the lines by 2 pixels. I think some overlap is important so as not to lose the detail of the symbol where the staff lines pass thru it.> I > also put the symbols above the staves, making it easier to isolate > them for study. Good idea. > Notice that I tried adding some nubs to the right flags to alleviate > the lateral confusibility problem. This could also be done with the > other flags for the larger alteration in each pair.Yes. That's definitely worth a try. But I'd like to see it done for all four flag types, to ensure that it doesn't interfere with making the types distinct from each other. e.g. the straights you've shown with nubs tend to look a little bit concave.> Another altitude consideration is at the upper right: up-arrows used > with flats use up a lot of vertical space when the new symbols have > long vertical lines.Yes. Good point. I'll meet you half-way on that one. My symbols are now 17 pixels high (while yours are 13), which means they overlap each other by one pixel when they are a fifth apart. I'm sure if you look hard enough you'll find some scores that overlap flats like this. They won't overlap naturals or sharps a fifth apart, because that would form a phantom accidental in between, but there's no danger of anything like that with our flags.> Let's see what agreement we can come to about how the straight-flag > symbols should look before doing any more with the rest of them.Well I don't think it is possible to consider the straight flags in isolation from all the others. The full set has to be seen together to be sure they are sufficiently distinct from one another. You are throwing away a lot of horizontal resolution when you make those vertical strokes wider. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4634 - Contents - Hide Contents Date: Tue, 16 Apr 2002 07:44:32 Subject: Re: 31 limit unique ets From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> There are 541 of these up to 1600; the first ten are: > > > 552 1.637374926 > 571 1.450447298 > 612 1.695743633 > 631 1.240353558 > 634 1.625692282 > 653 1.055690210 > 677 1.673616803 > 718 1.173415260 > 730 1.601691329 > 742 1.667556137 > > The number following each et is a 31-limit badness score. > > The following had the lowest log-flat badness among the 541: > > 653 1.105569021 > 1395 1.052127118 > 1600 1.016116129Thanks Gene, that's great. I look forward to similar results re 37, 41, 43, 45 and 47 limit unique, if possible.
Message: 4635 - Contents - Hide Contents
Date: Tue, 16 Apr 2002 03:22:26
Subject: the first six criteria
From: Carl Lumma
Scales that pass all of the first six criteria
----------------------------------------------
07- Diatonic scale in meantone
[0 5 10 15 18 23 28 31]
3. efficiency 0.76
4. strictly proper
5. yes; 5th is 3:2 in 6 of 7 modes
6. yes; 3rd is 6:5 or 5:4 in 7 of 7 modes
10- Paul Erlich's pentachordal decatonic in 22-tet
[0 2 4 7 9 11 13 16 18 20 22]
3. efficiency 0.62
4. strictly proper
5. yes; 7th is 3:2 in 8 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes
07- Dave Keenan's heptatonic MOS in porcupine
[0 3 6 9 12 15 18 22]
3. efficiency 0.76
4. strictly proper
5. yes; 5th is 3:2 in 4 of 7 modes
6. yes; 3rd is 6:5 or 5:4 in 7 of 7 modes
08- Gene Smith's Euclidean-reduced scale in 46-tet
[0 3 12 15 22 27 34 37 46]
3. efficiency 0.55
4. strictly proper
5. yes; 7th is 5:3 in 6 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 7 of 8 modes
10- Carl Lumma's x2 scale in meantone
[0 4 6 10 12 16 18 22 25 28 31]
3. efficiency 0.53
4. strictly proper
5. yes; 9th is 7:4 in 8 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 7 of 10 modes
10- Dave Keenan's decatonic MOS in quadrafourths
[0 3 6 9 11 14 17 20 23 26 29]
3. efficiency 0.73
4. strictly proper
5. yes; 7th is 3:2 in 6 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes
09- Paul Hahn's 4-3-3 trichordal scale in 31-tet
[0 4 7 10 15 18 21 25 28 31]
3. efficiency 0.49
4. strictly proper
5. yes; 4th is 5:4 in 6 of 9 modes
6. yes; 8th is 5:3, 12:7, 7:4 in 9 of 9 modes
Scales that pass most of the first six criteria
-----------------------------------------------
10- Paul Erlich's symmetrical decatonic in 22-tet
[0 2 4 7 9 11 13 15 18 20 22]
3. efficiency 0.51 (ambiguous key)
4. strictly proper
5. yes; 7th is 3:2 in 8 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes
10- Easley Blackwood's decatonic in 15-tet
[0 2 3 5 6 8 9 11 12 14 15]
3. efficiency 0.27 (ambiguous key)
4. strictly proper
5. yes; 7th is 3:2 in 10 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes
08- octatonic scale in 12-tet
[0 1 3 4 6 7 9 10 12]
3. efficiency 0.33 (ambiguous key)
4. strictly proper
5. yes; 3rd is 6:5 in 8 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 8 of 8 modes
06- hexatonic scale in 12-tet
[0 1 4 5 8 9 12]
3. efficiency 0.42 (ambiguous key)
4. strictly proper
5. yes; 3rd is 5:4 in 6 of 6 modes
6. yes; 4th is 4:3 or 3:2 in 6 of 6 modes
09- David Rothenberg's generalized diatonic in 31-tet
[0 5 8 11 14 17 22 25 28 31]
3. efficiency 0.74
4. strictly proper
5. no, but 9th is 15:8 in 7 of 9 modes
6. yes; 8th is 5:3 or 7:4 in 9 of 9 modes
08- Carl Lumma's octatonic subset of kleismic
[0 1 5 6 10 11 15 16 19]
3. efficiency 0.75
4. proper, but R. stability only 0.36
5. yes; 3rd is 6:5 in 6 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 5 of 8 modes
09- Balzano's generalized diatonic in 20-tet
[0 2 5 7 9 11 14 16 18 20]
3. efficiency 0.74
4. strictly proper
5. no.
6. yes; 8th is 5:3 or 7:4 in 9 of 9 modes
9th is 9:5 or 15:8 in 9 of 9 modes
08- Ken Wauchope's minor
[1/1 21/20 7/6 5/4 7/5 3/2 5/3 7/4 2/1]
3. efficiency 0.42
4. strictly proper
5. no, but 6th is 3:2 in 4 of 8 modes
3rd is ~ 6:5 in 6 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 7 of 8 modes
Scales that fail the first six criteria
---------------------------------------
06- Hexatonic scale in MAGIC
[0 11 13 24 26 39 41]
3. efficiency 0.78
4. proper, but R. stability only 0.40
5. yes; 3rd is 5:4 in 4 of 6 modes
6. no.
07- Neutral thirds scale in 31-tet
[0 5 9 13 18 22 27 31]
3. efficiency 0.76
4. strictly proper
5. yes; 5th is 3:2 in 5 of 7 modes
6. no.
08- Ken Wauchope's major
[1/1 21/20 5/4 21/16 7/5 3/2 7/4 15/8 2/1]
3. efficiency 0.42
4. not proper
5. no, but 6th is 3:2 in 5 of 8 modes
6. yes; 7th is 8:5, 7:4, 9:5 in 8 of 8 modes
09- Paul Hahn's 3-3-4 trichordal scale in 31-tet
[0 3 6 10 13 18 21 25 28 31]
3. efficiency 0.45
4. proper, R. stability 0.92
5. no, but 6th is 3:2 in 4 of 9 modes
6. yes; 8th is 5:3, 12:7, 7:4 in 9 of 9 modes
09- Nonatonic MOS in orwell
[0 11 19 30 38 49 57 68 76 84]
3. efficiency 0.74
4. strictly proper
5. yes; 3rd is 7:6 in 8 of 9 modes
6. no.
09- Nonatonic MOS in quadrafourths
[0 3 6 9 12 15 18 21 24 29]
3. efficiency 0.74
4. strictly proper
5. yes; 6th is 3:2 in 5 of 9 modes
6. no.
10- Decatonic MOS in MIRACLE
[0 7 14 21 28 35 42 49 56 63 72]
3. efficiency 0.73
4. strictly proper
5. yes; 9th is 7:4 in 8 of 10 modes
6. no.
-Carl
Message: 4636 - Contents - Hide Contents Date: Wed, 17 Apr 2002 08:49:16 Subject: Re: scala stability logic From: Carl Lumma>> >nd how are you deciding when to say "ambiguous key"? As soon >> as all possible keys are not distinct? > >Yes. >>> For example, the wholetone scale has 1 group of keys, the octatonic >> scale has 2, and the diatonic 7. >>I determine that they have 6, 4 and 1 repeating blocks. So if >that's more than one I add the text "ambiguous key". >>> How are you calculating efficiency in these cases? >>Strictly by R.'s definition. Excellent. >D. Rothenberg, "A Model for Pattern Perception with Musical Applications >Part II: The Information Content of Pitch Structures" Math. Systems >Theory, 1978, p.356 "Note that stability only applies to proper scales."Rats! Looks like Lumma stability is all we have for things like the Pythagorean diatonic, then. -Carl
Message: 4637 - Contents - Hide Contents Date: Wed, 17 Apr 2002 09:49:54 Subject: Re: scala stability logic From: manuel.op.de.coul@xxxxxxxxxxx.xxx Carl wrote:>Any reason you don't display Rothenberg stability for improper >scales?None that I remember, I must have assumed it's not defined for improper scales, but it might well be. I don't have the paper at hand, do you think he didn't make that requirement? Manuel
Message: 4638 - Contents - Hide Contents Date: Wed, 17 Apr 2002 01:11:34 Subject: Re: scala stability logic From: Carl Lumma>None that I remember, I must have assumed it's not defined for >improper scales, but it might well be. I don't have the paper >at hand, do you think he didn't make that requirement?I seem to think he didn't, though I don't have the paper handy either. And how are you deciding when to say "ambiguous key"? As soon as all possible keys are not distinct? For example, the wholetone scale has 1 group of keys, the octatonic scale has 2, and the diatonic 7. How are you calculating efficiency in these cases? -Carl
Message: 4639 - Contents - Hide Contents Date: Wed, 17 Apr 2002 10:25:07 Subject: Re: scala stability logic From: manuel.op.de.coul@xxxxxxxxxxx.xxx>And how are you deciding when to say "ambiguous key"? As soon >as all possible keys are not distinct? Yes. >For example, the >wholetone scale has 1 group of keys, the octatonic scale has 2, >and the diatonic 7.I determine that they have 6, 4 and 1 repeating blocks. So if that's more than one I add the text "ambiguous key".> How are you calculating efficiency in these cases?Strictly by R.'s definition. Manuel
Message: 4640 - Contents - Hide Contents Date: Wed, 17 Apr 2002 09:27 +0 Subject: Re: scala stability logic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <OFF5004A07.0C575491-ONC1256B9E.002AB5D9@xxxxxx.xxxxxxxxx.xx> Manuel wrote:> Carl wrote: >>> Any reason you don't display Rothenberg stability for improper >> scales? >> None that I remember, I must have assumed it's not defined for > improper scales, but it might well be. I don't have the paper > at hand, do you think he didn't make that requirement?D. Rothenberg, "A Model for Pattern Perception with Musical Applications Part II: The Information Content of Pitch Structures" Math. Systems Theory, 1978, p.356 "Note that stability only applies to proper scales." Carl agreed with this a few days ago. Graham
Message: 4641 - Contents - Hide Contents Date: Wed, 17 Apr 2002 11:14 +0 Subject: Re: 41 ET 11-tone diatonic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <B8E24EED.3AAA%mark.gould@xxxxxxx.xx.xx> Mark Gould wrote:> Well, some of the conversation is just about understandable, but I don't > have the Rotheberg definition in front of me, nor do I have Scala.Definitions are all at <Definitions of tuning terms: index, (c) 1998 b... * [with cont.] (Wayb.)>. Propriety and stability are easy enough. Efficiency is hard to calculate, but tends to be high for an MOS anyway. I don't know of anything that can do the calculations on a Macintosh. If I ever understand it myself, I'll try implementing it in Python.> scale patterns > > 19 22122212221 > 30 33233323332 > 41 44344434443 > > The general pattern I get is thus: > > aabaaabaaabSo your scale is really a different tuning of Balzano's 11 from 30? Graham
Message: 4642 - Contents - Hide Contents Date: Wed, 17 Apr 2002 11:13 +0 Subject: Re: more objective From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.2.2.20020416172331.01f61e50@xxxxx.xxx> Carl Lumma wrote:> Disregard my last message, I think this is the best yet:You have a nice 404 page, anyway. It's certainly more objective. Other than that, not so bad. (1b) Immediately fails for any scale without a 3:2. Can we have a range of recognisable fifths? What does "scales with only 1 or 2 unique keys" mean? Is this like whole tone and octatonic scales? I thought something like (2b) should be added, but why mix it with stability? After "Note that stability only applies to proper scales" Rothenberg goes on to say "Also, it does not really measure the degree to which a motif at a given pitch of a scale may be identified with (i.e. recognized as composed of the same interval as) a `modal transposition' of that motif to another pitch in the scale (i.e. a sequence)." So I don't see what it's doing under modal transposition here. He proposed "consistency" as an alternative, although I haven't worked out what he means by it. Whatever, Lumma stability is better than Rothenberg stability for this purpose. Rothenberg stability is certainly important, but should be moved to a different section. I reckon make it (2b) and move the current (2b) to (4b) without the "appears in only one interval class throughout the scale". And maybe without the "strong" as well. What was the point of that? Actually, I think I'll write my own (4b). The ratio f/K where f is the number of consonant intervals within the scale and K is the total number of intervals in the scale, excluding unisons. So for 5-limit diatonic, we have 7 thirds and 6 fourths, plus octave complements. So f/K is 13/21. For 7-limit decimal, 10 2-steps, 3 3-steps, 4 4-steps and 10 5-steps plus complements for everything except 5-steps, giving 44/90. For 11-limit neutral thirds, of course, we have 100%. I think K=k*(k-1). If you are going to use Rothenberg and Lumma stability as alternatives, Lumma stability should be given more weight. It tends to give lower values, doesn't it? I'd also like to see a "tonality" category, where we have something about those characteristic dissonances, and the smallest sufficient (and distinctive) subsets. Efficiency is really the opposite of this. Both are important. Graham
Message: 4643 - Contents - Hide Contents Date: Wed, 17 Apr 2002 11:13 +0 Subject: Re: My Approach Generalized Diatonicity From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.2.2.20020416105902.00a9ad20@xxxxx.xxx> Carl Lumma wrote:> I generally think of 7:6, 7:5 as barely strong enough, and 10:7 too > weak, > to be considered consonant dyads. > > Which scale are you talking about loosing (5) in?The 7 from decimal. The two tritones are the characteristic dissonances in the fourth/fifth category. There could always be a statistical definition where only two of the 6-steps are considered to be 3:2 fifths, but not always the same two. But the same would then have to apply to the 3-steps being 5:4.>> 1 2 1 2 1 2 1 >> 3 3 3 3 3 3 2 >> 4 5 4 5 4 4 4 >> 6 6 6 6 5 6 5 >> 7 8 7 7 7 7 7 >> 9 9 8 9 8 9 8 >> >> In the top row, 2 could be 9:8 or 8:7, and in the second row it could> be >8:7 or 7:6. > > Granted for the sake of argument... >>> So that means 2 and 3 are both consonances, and 2/7 is the >> only interval class where 2/10 is this consonance. > > ??You said one interval class has to contain two consonances that aren't consonant anywhere else. So, the thirds here could be either 5:4 or 7:6. 6:5 isn't allowed because it's too close to 7:6, which is a different interval class, and you lose tonalness. 8:7 isn't allowed because it's ambiguous with the seconds. This is actually fairly reasonable in practice. The 2-step third will tend to be larger than the 2-steps second for melodic reasons. This will increase the statistical stability, but still leave the ambiguity where you need it.>> If major thirds and perfect fifths are both legal consonances, why not >> put them together? >> You've lost me completely.I've decided above that 5:4 and 3:2 are the consonances. Tertian harmony does seem to work best for 7 note diatonicity. However, put them together and you see a 6:5 as well. With a fixed decimal scale this wouldn't happen. With flexible notes, the 6:5 pulls against the decimalness of the scale, because it's a diminished interval. It doesn't affect propriety, but should still be avoided melodically. In harmony, you have to make sure it isn't too obvious. So, like fourths in Common Practice harmony, don't put it in the bass. The 7:9 in a 6:7:9 chord is even worse, because it isn't even a kind of third. But 4:6:7 looks okay, shame it only exists in one place. Although that is useful for cadences.>> The propriety grids are like this: >> >> 3 5 5 5 3 5 5 >> 8 10 10 8 8 10 8 >> 13 15 13 13 13 13 13 >> 18 18 18 18 16 18 18 >> 21 23 23 21 21 23 23 >> 26 28 26 26 26 28 26 >> Isn't this just the diatonic scale?Yes, but not the just diatonic scale.>> 4 7 6 7 4 7 6 >> 11 13 13 11 11 13 10 >> 17 20 17 18 17 17 17 >> 24 24 24 24 21 24 23 >> 28 31 30 28 28 30 30 >> 35 37 34 35 34 37 34 >> Doesn't look very consonant.It's consonant enough. I've found quite a few alternatives. They all have something wrong with them. None have high efficiency. The advantage of using 10 fuzzy pitch classes is that you can have all of them at once, and maintain the sense of a 10 note chromatic. Also that you have all those 11-limit intervals when you want them.>>>> If those consonances can be ambiguous, it does open up new ways of >>>> interpreting the 7 from 10 scale, which is what I'm looking at now. >>>>>> You mean Rothenberg ambiguous, or ambiguous in the sense that they >>> may have more than one harmonic series representation? >> >> Rothenberg ambiguous. >> I'll be anxious to see the results.If we allow 7-limit consonance, thirds and fourths become the intervals with two kinds of consonance. That means 7:8 could be single consonance for seconds, unfortunately it isn't in the majority. But now you've dropped the requirement for a characteristic dissonance, that isn't a problem. However, the way you've worded (2b) means 7:8, 5:7 and 7:10 still drop out because they're ambiguous.>>> Wrong! That's what diatonicity is all about -- tying scale objects >>> to harmonic objects. There are a million ways to have lots of >>> different consonances, but very few to have them make sense in terms >>> of scale intervals. >>>> But isn't that what stability gives us? >> No, stability ties acoustic objects with scales objects. Diatonicity > restricts things further to acoustic objects which are harmonious.Why are you assuming harmonious acoustic objects can't be ambiguous? The tritone plays an important part in tonal harmony because it's rare and ambiguous. Why can't a consonance take on that role, and dissonances tie the harmony to the scale? Graham
Message: 4644 - Contents - Hide Contents Date: Wed, 17 Apr 2002 10:47:56 Subject: Re: the first six criteria From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Scales that pass all of the first six criteria > 08- Gene Smith's Euclidean-reduced scale in 46-tet > [0 3 12 15 22 27 34 37 46] > 3. efficiency 0.55 > 4. strictly proper > 5. yes; 7th is 5:3 in 6 of 8 modes > 6. yes; 4th is 5:4 or 4:3 in 7 of 8 modesThis isn't really Euclidean reduced--I took the h8+v7 map, Euclidean reduced that, mapped it to 46-et, and then permuted the steps to get this scale. Maybe it could be called "Star" since it is related to Starling.
Message: 4647 - Contents - Hide Contents Date: Wed, 17 Apr 2002 21:30:14 Subject: Re: A common notation for JI and ETs From: gdsecor It turns out that I'm replying to my own message, inasmuch as it took over 16 hours to appear on the list after I posted it. --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>> I think there should be a strong family resemblance between the standard >> symbols and our new ones, or they will not be found unacceptable on visual >> aesthetic grounds. >> Yes, that is a very valid point.I did some more work on the symbols in a second file: which I will put out there, once Yahoo gets over its cranky spell and lets me upload it. The more I look at your symbols, the more I like their style, so (assuming that the file is out there) please follow along with me. The fifth staff is a synthesis of features from both of our efforts above that. I made the sesequisharp (|||) and double-sharp (X) group of symbols intermediate in width between what each of us had, while the semisharp (|) and sharp groups (||) are either the same as or very close to your symbols. The biggest problem I had was with the nubs (which I made rather large and ugly) still tending to get lost in the staff lines. I tried one symbol (in the middle of the staff) with a triangular nub, which looks a little neater, I think. To the right of that I copied three of your symbols so I can comment on them. In all three of them the concave or wavy flag is significantly lower than the line or space for its note. I propose using instead the concave style of flag that I described before, for which I prepared a set of symbols on the 7th staff. (Note that the nubs don't get lost, even though they are quite small.) I like your wavy flag, but I would propose waving it a little higher, as I did in the symbols just to the right of yours (back on the 5th staff); I seemed to be getting a better result with a thinner flag, which would also serve to avoid confusion of the wavy with the convex flag. Perhaps the wavy flag would now be most appropriate for the smallest intervals. I put a set of convex flag symbols on the 6th staff, which (like the straight-flag symbols) combines features from both of our previous efforts). At the top right (under altitude considerations) I put my latest version of the symbols in combination with conventional sharps and flats, with single-symbol equivalents included (above the staff). Let me know what you think. I will be going through your latest reply relating commas to flags, now that I have a much better idea what these symbols are going to look like. --George
Message: 4648 - Contents - Hide Contents Date: Wed, 17 Apr 2002 18:01:03 Subject: method for finding balanced sums? From: Carl Lumma All; Say I have a list of positive integers, and I sum them up. Is there a method for assigning signs to them so their sum is as close to zero (either positive or negative) as possible? -Carl
Message: 4649 - Contents - Hide Contents Date: Wed, 17 Apr 2002 19:37:50 Subject: Re: more objective From: Carl Lumma>(1b) Immediately fails for any scale without a 3:2. Can we have a range >of recognisable fifths?You're allowed to approximate intervals, according to harmonic entropy.>What does "scales with only 1 or 2 unique keys" mean? Is this like >whole tone and octatonic scales? Yep. >I thought something like (2b) should be added, but why mix it with >stability?I don't, I mix it with efficiency.>After "Note that stability only applies to proper scales" Rothenberg >goes on to say "Also, it does not really measure the degree to which a >motif at a given pitch of a scale may be identified with (i.e. >recognized as composed of the same interval as) a `modal transposition' >of that motif to another pitch in the scale (i.e. a sequence)."Huh- what does he suggest instead?>Actually, I think I'll write my own (4b). The ratio f/K where f is >the number of consonant intervals within the scale and K is the total >number of intervals in the scale, excluding unisons.That would be a rough measure of how consonant the scale is.>If you are going to use Rothenberg and Lumma stability as alternatives, >Lumma stability should be given more weight. It tends to give lower >values, doesn't it?Yes. I've been thinking about normalizing all these to their values in the diatonic scale.>I'd also like to see a "tonality" category, where we have something >about those characteristic dissonances, and the smallest sufficient >(and distinctive) subsets. Efficiency is really the opposite of this. >Both are important.I punish ambiguous keys, and I allow mode recognition by strong consonance. Other than that, I'm not willing to do anything for tonalness. -Carl
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