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Message: 4650 - Contents - Hide Contents
Date: Wed, 17 Apr 2002 21:24 +0
Subject: decimal candidates
From: graham@xxxxxxxxxx.xx.xx
Although some of the fixed pitch 7 from decimal scales have low
efficiency, there still good enough to be in Carl's list by the looks of
it.
The best candidate is probably
0 1 3v 4v 6 7 9v 0
3 5 3 7 3 5 5 /31
4 7 4 9 4 7 6 /41
It's strictly proper. 6 out of 7 thirds are either 5:4 or 6:5. 4 out of
7 fifths are 3:2. For 31-equal, Lumma stability is 35.5% and Rothenberg
efficiency is 52.1%. In 41=, efficiency goes down to 46.6%. Here's a
lattice:
0-----6
/ \ / \
/ \ / \
1-----7-----3v----9v
\ /
\ /
4v
Another one is
0 1 3^ 4^ 6 7 9^ 0
3 7 3 5 3 7 3 /31
4 9 4 7 4 9 4 /41
Improper this time. 6 thirds either 6:5 or 5:4 and 4 fifths 3:2.
Efficiency 50.6% for either tuning, Lumma stability 31.7% in 41=, 29% in
31=. Lattice
3^----9^
/ \ /
/ \ /
4^----0-----6
/ \ /
/ \ /
1-----7
I've also been looking at the 6 from 10 scale 1 2 2 2 1 2. It's a subset
of the 7 from 10 MOS. I don't like it so much, but it works well with the
criteria. The grid is
1 2 2 2 1 2
3 4 4 3 3 3
5 6 5 5 4 5
7 7 7 6 6 7
8 9 8 8 8 9
It'd be nice if the second row could be 5:4 thirds and 4:3 fourths, but
fixed scales don't work out like that. Still, the middle row can be 4:3,
5:7, 7:10 and 3:2 whereas the first row has lots of 9:8 or 8:7 or 7:6s.
There are a couple of fixed scales that work like this.
4 6 6 6 4 5 /31
5 8 8 8 4 7 /41
and
3 6 6 6 3 7 /31
4 8 8 8 4 9 /41
They both depend on two consonances (8:7 and either 9:8 or 7:6) counting
as the same pitch class. Then, they fulfil Carl's original rules.
Rothenberg efficiency is always 49.4%.
It also happens that the scale works as 6 from 26,
3 5 5 5 3 5
8 10 10 8 8 8
13 13 13 13 11 13
18 18 18 16 16 18
21 23 21 21 21 23
5 steps are 8:7, 11 steps are 4:3 and 13 steps are 7:5. It's strictly
proper, Lumma stability of 53.8% and 66.1% efficiency. I thing it should
go on the list, although it'll have trouble with tetrachordality.
Graham
Message: 4651 - Contents - Hide Contents Date: Wed, 17 Apr 2002 09:34:31 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. I'll have to disagree.>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
Message: 4652 - Contents - Hide Contents Date: Wed, 17 Apr 2002 19:35:36 Subject: Re: My Approach Generalized Diatonicity From: Carl Lumma This has been about 10 hours without appearing, so I'm posting again. -C.>>> 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.I did, but I've changed this. Only the 2b interval class must still be un-ambiguous, since this is supposed to aid mode recognition. The "diatonic" interval class can now be ambiguous. I'm also in the process of fixing things so the 3rds here wouldn't qualify as the "diatonic" interval class because there's only one instance of the other interval. The 3rds= 5:4 are instead a bet for 2b, along with 5ths= 3:2. The best bets for a diatonic interval class here would be 4ths= 4:3 or 7:5, and 7ths= 7:4 or 15:8. If you want to make triads, I would think 1-3-7= 4:5:7 or 4:5:15 would be your best bet.>>> 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'll put it on my list of scales to play.>>>> 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?I'm not! I just meant stability is ignorant of harmony. I said "ties scale objects to harmonic objects". You said stability does that. It does, but it doesn't guarantee it, since there are plenty of dissonant, stable scales. I still want the 2b interval to be un-ambiguous, because when you hear the interval, you're supposed to know right away what scale position it's in -- that's what 2b is for.>Why can't a consonance take on that role, and dissonances tie >the harmony to the scale?Because you can't tell dissonances apart from eachother, in the same way that you can tell 5:4, 6:5 apart from eachother. -Carl
Message: 4653 - Contents - Hide Contents Date: Wed, 17 Apr 2002 19:36:11 Subject: Re: scala stability logic From: Carl Lumma Almost 11 hours without appearing... -C.>> 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. 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: 4654 - Contents - Hide Contents Date: Wed, 17 Apr 2002 09:27:37 Subject: Re: My Approach Generalized Diatonicity From: Carl Lumma>>> > 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.I did, but I've changed this. Only the 2b interval class must still be un-ambiguous, since this is supposed to aid mode recognition. The "diatonic" interval class can now be ambiguous. I'm also in the process of fixing things so the 3rds here wouldn't qualify as the "diatonic" interval class because there's only one instance of the other interval. The 3rds= 5:4 are instead a bet for 2b, along with 5ths= 3:2. The best bets for a diatonic interval class here would be 4ths= 4:3 or 7:5, and 7ths= 7:4 or 15:8. If you want to make triads, I would think 1-3-7= 4:5:7 or 4:5:15 would be your best bet.>>> 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'll put it on my list of scales to play.>>>> 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?I'm not! I just meant stability is ignorant of harmony. I said "ties scale objects to harmonic objects". You said stability does that. It does, but it doesn't guarantee it, since there are plenty of dissonant, stable scales. I still want the 2b interval to be un-ambiguous, because when you hear the interval, you're supposed to know right away what scale position it's in -- that's what 2b is for.>Why can't a consonance take on that role, and dissonances tie >the harmony to the scale?Because you can't tell dissonances apart from eachother, in the same way that you can tell 5:4, 6:5 apart from eachother. -Carl
Message: 4655 - Contents - Hide Contents Date: Wed, 17 Apr 2002 21:29:26 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> 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.I like this better, but part of the family resemblance of the existing symbols is that they are all ectomorphs, except for the rarely seen double-sharp being a mesomorph. They are not endomorphs like your ||| and X symbols. Even some of my | symbols are pushing it.> 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.)Maybe, but part of the problem is just that it is hard to tell 2 identical side-by-side things from 3 identical side-by-side things with the same spacing. I've made my ||| and X's wider now, but not as wide as yours, and shortened the middle tail of the 3 by 3 pixels relative to the others, so they are not 3 identical things any more.>> 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??? No. > (Or are you suggesting that I should do > this and forget about the ||| and X symbols?) Yes.>> 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.You're just strengthening my case. I don't find that two of our symbols side by side have the same problem as the conventional sharp symbol (making a third phantom symbol in between). I find them to be more like the conventional flat symbol, for which no single-symbol double has ever been seen as necessary.> 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.)Those I have found have not been touching. I do not propose to have ours touching either.>> 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.)OK. This wasn't a valid point against the single-symbol sesquis and doubles. But other points still stand.>> 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 may not have explained my point well enough. I mean that the musician trained to see an X as a double sharp will tend to see your upward pointing X symbols as a double-sharp _plus_ something, and your downward pointing X symbols as a double sharp minus something. It will be tough for them to unlearn the previous meaning of X. In your notation, the X effectively means _either_ a sesqui sharp or a sesqui flat. I just thing this is too confusing, and the ||| and X symbols are not required anyway.> 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.I don't see what's wrong with the nubs straddling the line in one case, and being in free space in the other case. i.e. put them, not one, but 2 pixels further out than you have them now.> 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.Easier, yes. But inconclusive.> 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. Fine. > 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. True. > 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, Agreed. > 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.I think the nubs will be fine if you make the flags the same height as the body of the standard flat symbol. The line will then pass thru the middle of the nub.> 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.You mean the upward pointing ones? Concave I can understand, but wavy? The horizontally inflected part of the wavy flag is always exactly centred relative to the center of the notehead.> 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 look forward to seing it.> 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).I hope you show some down pointing ones next to a flat, because I think they look strange if their tails are too much shorter than the flat's tail. In fact I'd be in favour of making the tails of down-pointing arrows longer than up-pointing ones.> Let me know what you think. Will do. > 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. Great!-- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4656 - Contents - Hide Contents Date: Wed, 17 Apr 2002 21:45:59 Subject: Re: one from the archives From: Gene W Smith It's been a few days, so I'm joining the resending parade; I hope Carl still remembers what these are about! [1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21] edge connectivity = 3 characteristic polynomial = x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*x^4-8*x^3+304*x^2 +96*x This tells us there are 29 consonant intervals and 22 consonant triads The scale is not epimorphic, but can be extended in various ways, for instance to the h15-epimorphic scale [1, 36/35, 10/9, 8/7, 6/5, 5/4, 4/3, 48/35, 10/7, 3/2, 8/5, 5/3, 12/7, 9/5, 40/21] This also has connectivity 3 and a characteristic polynomial x^15-41*x^13-70*x^12+450*x^11+1316*x^10-899*x^9-6406*x^8-3948*x^7+9034*x^ 6+ 12050*x^5-298*x^4-7243*x^3-3682*x^2-445*x+24 It therefore has 41 consonant intervals and 35 consonant triads. I've attached two files which show the graph for each of these scales. Now I'm wondering what the story is--where does this scale arise from? [This message contained attachments]
Message: 4657 - Contents - Hide Contents Date: Thu, 18 Apr 2002 07:53:45 Subject: Re: one from the archives From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> characteristic polynomial = >> x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968* >> x^4-8*x^3+304*x^2+96*x >> Never did the homework to understand these.The adjacency matrix of a graph is a square matrix labeled by verticies; it has a "1" if they are connected, and a "0" if not (counting the vertex to itself as a "0".) The characteristic polynomial of this is as above, and is a graph invariant. The n-2 term gives the number of edges, and the n-3 term twice the number of triads.>> This tells us there are 29 consonant intervals and 22 consonant >> triads >> That doesn't seem like much to write home about either. What > are some good scales here in your experience, Gene?Not that great.
Message: 4658 - Contents - Hide Contents Date: Thu, 18 Apr 2002 14:26 +0 Subject: Re: 41 Diatonic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4843.194.203.13.66.1019121786.squirrel@xxxxx.xxxxxxx.xx.xx> Me:> So your scale is really a different tuning of Balzano's 11 from 30? Mark: > It could be stated that way, but that's not how it came about. Really, I > just 'discovered' when looking at all the scales I found when manually > enumerating the 2d grids some year ago. The abstract bit about scales > came > later, and not all scales that can be formed from a/b sequences can be > found as segments of a cyclic generator of a Cn.An a/b sequence that's a segment of a cyclic generator is "MOS" or "Well Formed".> Anyway, the point I was trying to make was that, justas the 7-diatonic > has > various 'mean-tone' interpretations in 19 and 31 equal, so this > particular > 11-tone diatonic also has analogous 'mena-tone' interpretations.Yes, that's what I started to work out when you originally posted it to the big list. But you didn't follow up on my comments, so I didn't give the reasoning. In fact, I was using this temperament: 11/30, 443.8 cent generator basis: (1.0, 0.3697965324659695) mapping by period and generator: [(1, 0), (-1, 7), (-1, 9), (-2, 13)] mapping by steps: [(19, 11), (30, 17), (44, 25), (53, 30)] highest interval width: 13 complexity measure: 13 (19 for smallest MOS) highest error: 0.006241 (7.489 cents) unique It can be derived using my cross-platform Python library by>>> h19 = temper.PrimeET(19, temper.primes[:3]) >>> g11 = temper.PrimeET(11, temper.primes[:3]) >>> g11.basis[2]=25 >>> g11.basis[3]=30 >>> g11&h19You'd given the scale in terms of 19-equal then, so this temperament is consistent with 19-equal. You don't get all 7-limit intervals in the 11 note scale, but most of them are there. Only 7:4 and 8:7 are missing, I think. Also 9:8 is missing from the 9-limit. There's also a tuning with a 442.9 cent generator that gets the 5-limit to within 1.5 cents. We're looking at this segment of the scale (or Farey or Stern Brocot) tree 19 11 30 49 41 It's tempting to think the temperament will work with 41-equal, but in fact it doesn't. The temperament which does unify 41 and 19 is what I call "magic" and it's this bit of the scale tree: 19 22 41 60 63 22=11*2 and 60=30*2, so the first temperament only has half the number of notes for some magic scales. That explains my description of it as "half magic" which nobody seemed to notice. The 41 diatonic under discussion now is very half magic, because it has to work with both 41 and 19. That comes form its derivation as alternating intervals which have to be 7:9 and 9:8. The 9:8 is two steps of the 11 note diatonic. There's no way this can work with a general 9-limit temperament. An octave plus the 9:8 has to be two fifths. So two fifths are 11+4=15 diatonic steps. Which means one of the fifths has to be 7 and the other 8 steps. So what's going on? Well, it looks like the 11 note diatonic is every other note from the 22 note magic MOS. Like this 2 1 2 2 2 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2 3 4 4 4 3 4 4 4 3 4 4 You can't describe the whole 9-limit (or even the whole 5-limit) by modulating between 11 note diatonics in this temperament. That's certainly a defect, but not one covered by the current scoring system.> In this connection, it is interesting to note that the 20 equal 11-tone > diatonic, as sugg. Zweifel, and discussed by me, has the pattern > > aaaabaaaaab (9a+2b) > > if a=2 and b=1 chromatic = 20 equal > if a=3 and b=2 chromatic = 31 equal > if a=4 and b=3 chromatic = 42 equal > if a=5 and b=3 chromatic = 51 equal > etc.. > > Strange how these two diatonics sort of 'hang about' each other > > 19 : 8+3 > 20 : 9+2 > 30 : 8+3 > 31 : 9+2 > 41 : 8+3 > 42 : 9+2 > (if it were a sequence, I'd guess at: > 52 : 8+3 > 53 : 9+2 but it sort of isn't, past a=5)It looks like you're fumbling for a scale tree. 9a+2b is 9 2 11 20 13 29 31 44 15 38 49 51 42 55 57 28 17 and the ones you're looking at are between 20 and 11. The similarity between the two diatonics is presumably that the both have 11 notes, and are built on chromatics that differ by 10 notes. To get them on the same tree, you need to go to 2 3 5 7 8 9 12 13 11 11 16 19 17 18 21 19 14 Note that meantone is on there as well. It covers all linear temperaments with an octave period and a generator between a third and a half of an octave. 7 and 9 give pelogic, there will be more of interest. Graham
Message: 4659 - Contents - Hide Contents Date: Thu, 18 Apr 2002 10:25:26 Subject: Re: method for finding balanced sums? From: manuel.op.de.coul@xxxxxxxxxxx.xxx Carl wrote:>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?This is a kind of knapsack problem, which I think is NP-complete. Do you have so many that brute force takes too long a time? With the mail delays, perhaps an answer from Gene is already coming. Manuel
Message: 4660 - Contents - Hide Contents Date: Thu, 18 Apr 2002 14:26 +0 Subject: Re: scala stability logic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.2.2.20020417193556.01ed2d20@xxxxx.xxx> Carl Lumma wrote:> Rats! Looks like Lumma stability is all we have for things like the > Pythagorean diatonic, then.No, the Pythagorean diatonic works fine as a proper subset of a 12 note chromatic. See 13. Distinct Scales and Mistunings on p.365, "That there is only a finite number of equivalence classes of any cardinality is musically significant in that only finitely many significantly differing musical scales may be constructed. Also note that, if a scale is conceived of as an ordering, the use of finer tunings and smaller intervals does not necessarily produce new scales." He does also say the tuning should initially preserve propriety, but the 12 note scale should be familiar enough to most listeners. For Rothenberg stability and efficiency to work properly, you need to define each diatonic on a chromatic. I suggest the chromatic should always be strictly proper, and have high Lumma stability. There should also be a maximum number of notes allowed in a chromatic, somewhere below 53. Graham
Message: 4661 - Contents - Hide Contents Date: Thu, 18 Apr 2002 08:41:46 Subject: Re: method for finding balanced sums? From: genewardsmith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> Carl wrote:>> 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? >> This is a kind of knapsack problem, which I think is NP-complete. > Do you have so many that brute force takes too long a time? > With the mail delays, perhaps an answer from Gene is already > coming.I didn't have anything to say beyond that it looked like a -1/1 knapsack, and so should be NP hard in general. I'd just brute force it.
Message: 4662 - Contents - Hide Contents Date: Thu, 18 Apr 2002 14:26 +0 Subject: Re: My Approach Generalized Diatonicity From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.2.2.20020417084504.01e58740@xxxxx.xxx> Carl Lumma wrote:> The best bets for a diatonic interval class here would be 4ths= > 4:3 or 7:5, and 7ths= 7:4 or 15:8. If you want to make triads, > I would think 1-3-7= 4:5:7 or 4:5:15 would be your best bet.There can't be a 2b interval that way. Thirds are the obvious one, but it's a fudge to expect enough of them to be 5:4. Neutral thirds work in another context, but we don't need the miracle approximations then. When the characteristic dissonances were being enforced, the 8:7 or 7:6 couldn't be it in a 7-limit context. So there are two different diatonic intervals classes in the 7-limit, but no 2b.> I still want the 2b interval to be un-ambiguous, because when you > hear the interval, you're supposed to know right away what scale > position it's in -- that's what 2b is for.That sounds reasonable. I'd like it to be separate points: A consonance exists - in the majority of instances of a diatonic interval class - in a diatonic interval class with no other consonances - in no other diatonic interval classes - and is strong So if a scale fails on one of them, it only loses one point. I think in general each criterion should only enforce one property.>> Why can't a consonance take on that role, and dissonances tie >> the harmony to the scale? >> Because you can't tell dissonances apart from eachother, in the > same way that you can tell 5:4, 6:5 apart from eachother.That looks like the heart of the issue. I find it much easier to tell two dissonances apart than a dissonance from a consonance. With my schismic keyboard setup I found it remarkably difficult to distinguish 5:4 and 6:5 what with them both being well tuned consonances and not far apart. Graham
Message: 4664 - Contents - Hide Contents Date: Thu, 18 Apr 2002 14:26 +0 Subject: Re: My Approach Generalized Diatonicity From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a9klsv+hjop@xxxxxxx.xxx> emotionaljourney22 wrote:> sorry, i thought you were talking about the whole-tone scale. can you > refresh my memory as to what it is you were really talking about?It's the generalisation of what thirds do in a diatonic scale. Carl says a particular diatonic interval class should have two distinct consonances. There are other criteria like the majority of intervals in that class should be consonant, that there have to be more than one of each, and that the same consonances can't occur in other interval classes, although that one's been dropped. I say none of this matters so long as the scale has a variety of consonances that work with the primary consonance. Carl brought up the whole tone scale but I don't think it's a good one because that doesn't have the variety of consonances to start with. Graham
Message: 4665 - Contents - Hide Contents Date: Thu, 18 Apr 2002 17:27 +0 Subject: Re: 41 Diatonic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4843.194.203.13.66.1019121786.squirrel@xxxxx.xxxxxxx.xx.xx> Mark Gould wrote:> In this connection, it is interesting to note that the 20 equal 11-tone > diatonic, as sugg. Zweifel, and discussed by me, has the pattern > > aaaabaaaaab (9a+2b) > > if a=2 and b=1 chromatic = 20 equal > if a=3 and b=2 chromatic = 31 equal > if a=4 and b=3 chromatic = 42 equal > if a=5 and b=3 chromatic = 51 equal > etc..Sorry to give the impression I'm talking to myself, but there are two entries in Carl's list that fit this pattern. 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 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 They should be unified. The "9th" works as 9:5 in 31-equal as well. 29 belongs to the pattern too, as 3 4 3 3 3 4 3 3 3. I don't know if it fits the implied temperament. The efficiencies being identical should have been a clue. Graham
Message: 4666 - Contents - Hide Contents Date: Thu, 18 Apr 2002 16:46 +0 Subject: Re: 41 Diatonic From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4843.194.203.13.66.1019121786.squirrel@xxxxx.xxxxxxx.xx.xx> Mark Gould wrote:> In this connection, it is interesting to note that the 20 equal 11-tone > diatonic, as sugg. Zweifel, and discussed by me, has the pattern > > aaaabaaaaab (9a+2b) > > if a=2 and b=1 chromatic = 20 equal > if a=3 and b=2 chromatic = 31 equal > if a=4 and b=3 chromatic = 42 equal > if a=5 and b=3 chromatic = 51 equal > etc..In which case it's related to this entry from Carl's list: 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 5 3 3 3 3 5 3 3 3 /31 3 2 2 2 2 3 2 2 2 /20 2 1 1 1 1 2 1 1 1 /11 Graham
Message: 4667 - Contents - Hide Contents Date: Thu, 18 Apr 2002 11:50:33 Subject: Re: scala stability logic From: Carl Lumma>> >ats! Looks like Lumma stability is all we have for things like the >> Pythagorean diatonic, then. >>No, the Pythagorean diatonic works fine as a proper subsetThat's funny, since it is improper.>Distinct Scales and Mistunings on p.365, "That there is only a finite >number of equivalence classes of any cardinality is musically >significant in that only finitely many significantly differing musical >scales may be constructed. Right. >Also note that, if a scale is conceived of as an ordering, the use of >finer tunings and smaller intervals does not necessarily produce new >scales." Right. >For Rothenberg stability and efficiency to work properly, you need to >define each diatonic on a chromatic.? What terminology is this? -Carl
Message: 4670 - Contents - Hide Contents Date: Thu, 18 Apr 2002 13:51:38 Subject: Re: My Approach Generalized Diatonicity From: Carl Lumma>> >he best bets for a diatonic interval class here would be 4ths= >> 4:3 or 7:5, and 7ths= 7:4 or 15:8. If you want to make triads, >> I would think 1-3-7= 4:5:7 or 4:5:15 would be your best bet. >>There can't be a 2b interval that way. Thirds are the obvious one, >but it's a fudge to expect enough of them to be 5:4.Don't look at me -- I said the scale fails. I'm just trying to show how it might work.>> I still want the 2b interval to be un-ambiguous, because when you >> hear the interval, you're supposed to know right away what scale >> position it's in -- that's what 2b is for. >>That sounds reasonable. I'd like it to be separate points: > >A consonance exists > - in the majority of instances of a diatonic interval class > - in a diatonic interval class with no other consonances > - in no other diatonic interval classes > - and is strong > >So if a scale fails on one of them, it only loses one point. I >think in general each criterion should only enforce one property.Note "diatonic interval class" should just be "interval class" here. My new scheme keeps the last two, but does away with the second, and turns the first from a binary to more continuous value: | The ratio f/k, where f is the number of modes having a strong | consonance, which appears in only one interval class throughout | the scale, and forms a consonant triad with the equivalence | interval.>>> Why can't a consonance take on that role, and dissonances tie >>> the harmony to the scale? >>>> Because you can't tell dissonances apart from eachother, in the >> same way that you can tell 5:4, 6:5 apart from eachother. >>That looks like the heart of the issue. I find it much easier to tell >two dissonances apart than a dissonance from a consonance.You do? Have you got that backward? Anyway, that isn't the question. The question is: if I randomly play you either 6:5 or 5:4, harmonically, and ask you to identify them, would you perform better than if I had used 11:9 and 9:7?>With my schismic keyboard setup I found it remarkably difficult to >distinguish 5:4 and 6:5 what with them both being well tuned >consonances and not far apart.Really? Maybe I just find consonances easier to recognize because I've trained myself to do it... maybe it isn't innate. -Carl
Message: 4671 - Contents - Hide Contents Date: Thu, 18 Apr 2002 14:27:10 Subject: delays, delays, delays. From: Carl Lumma Can we move this list to columbia? -Carl
Message: 4672 - Contents - Hide Contents Date: Thu, 18 Apr 2002 12:32:55 Subject: Re: one from the archives From: Carl Lumma>>> >haracteristic polynomial = >>> x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968* >>> x^4-8*x^3+304*x^2+96*x >>>> Never did the homework to understand these. >>The adjacency matrix of a graph is a square matrix labeled by >verticies; it has a "1" if they are connected, and a "0" if not >(counting the vertex to itself as a "0".) The characteristic >polynomial of this is as above, and is a graph invariant. The >n-2 term gives the number of edges, and the n-3 term twice the >number of triads. That's crazy.What happened to the n-1 term, above? Does it show tetrads?>>> This tells us there are 29 consonant intervals and 22 consonant >>> triads >>>> That doesn't seem like much to write home about either. What >> are some good scales here in your experience, Gene? >>Not that great.I'd be interested in the stats on the "class" and "stelhex" scales, for comparison. -Carl
Message: 4674 - Contents - Hide Contents Date: Thu, 18 Apr 2002 18:43:38 Subject: Re: method for finding balanced sums? From: Carl Lumma>>> >ay 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? >>>> This is a kind of knapsack problem, which I think is NP-complete. >> Do you have so many that brute force takes too long a time? >> With the mail delays, perhaps an answer from Gene is already >> coming. >>I didn't have anything to say beyond that it looked like a -1/1 >knapsack, and so should be NP hard in general. I'd just brute force >it.I saw knapsack on mathworld last night, but didn't think it applied because of the business about weights. What if I order the elements from largest to smallest and then recursively subtract them in pairs, applying absolute value at the end of each round. So like (define kpsk (lambda (ls) (if (null? (cdr ls)) (car ls) (kpsk (cons (abs (- (cadr ls) (car ls))) (cddr ls)))))) Tests: (5 4 3 2 1) => 1 (10 8 6 4 2) => 2 (16 8 4 2) => 2 (92 91 90 1) => 88 (11 7 5 3 2) => 0 (25 16 9 4 1) => 3 Now, I don't know how hard sorting is. I've got a version of mergesort here that is nlog(n), and IIRC quicksort is often, but not always, better. This doesn't tell you the signs. An extra test at each round might be able to write those out, but all I want is the total, so... This measure doesn't seem to do what I want (see #4 at lumma.org/gd3.txt), since cases like (3 3) will be low (as they should) while (3 3 3) will be high. Back to the drawing board. -Carl
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