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Message: 5126 Date: Wed, 31 Jul 2002 02:07:15 Subject: Four 10-note, 7-limit JI scales From: Gene W Smith If we take (10/9)^2 (15/14)^2 (16/15)^2 (21/20)^3 = 2 as scale steps, and simplify the scale-finding problem by assuming 4/3 and 3/2 both belong to the scale, we obtain four scales, the third and fourth of which are the inverted forms of the first and second. A version of the major/minor transformation, exchanging 10/9 with 16/15, which is equivalent to saying 2-->2, 3-->3, 5-->24/5, 7-->168/25, exchanges the first and second, as well as the third and fourth. The first "decaa", and fourth, "decad", are major versions, having two major tetrads and a minor tetrad, while "decab" and "decac" have two minor and one major tetrad. In any system where 50/49~1 the exchange transform sends tetrads to tetrads and can be considered major/minor. In 22-et in particular, each scale becomes the symmetrical decatonic. All of the scales have 23 intervals, 17 triads and 3 tetrads. ! decad.scl ! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20] inversion of decab 10 ! 15/14 25/21 5/4 4/3 10/7 3/2 5/3 25/14 40/21 2/1 ! decab.scl ! [21/20, 16/15, 15/14, 10/9, 21/20, 15/14, 16/15, 21/20, 10/9, 15/14] (10/9) <==> (16/15) transform of decaa 10 ! 21/20 28/25 6/5 4/3 7/5 3/2 8/5 42/25 28/15 2/1 ! decac.scl ! [15/14, 16/15, 21/20, 10/9, 15/14, 21/20, 16/15, 15/14, 10/9, 21/20] inversion of decaa 10 ! 15/14 8/7 6/5 4/3 10/7 3/2 8/5 12/7 40/21 2/1 ! decad.scl ! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20] inversion of decab 10 ! 15/14 25/21 5/4 4/3 10/7 3/2 5/3 25/14 40/21 2/1
Message: 5127 Date: Wed, 31 Jul 2002 23:34:24 Subject: Tempered versions of Carl's 12-note JI scales From: Gene W Smith I took these five scales and looked for what new 11-limit intervals would appear if we allowed commas of less than 8 cents. This puts us in a range well covered by the 72-et, but none of the resulting scales needed the full power of this, or even of miracle; they all were covered by one or another planar temperament. Except for the case of "lester", which is in the {225/224, 441/440} temperament which doesn't improve from the 72-et values, this meant some improvement in the tuning was possible. Whether it is worthwhile is another question. # lumma in {385/384, 441/440} temperament, 873-et version l873 := [873, 1383, 2026, 2449, 3019]; lum:=[0, 37, 170, 230, 280, 400, 450, 510, 643, 680, 740, 813]; lumd := [37, 133, 60, 50, 120, 50, 60, 133, 37, 60, 73, 60]; 42 ingervals, 58 triads--the least harmony, but the best tuning # {225/224, 385/384} 858-et version l858 := [858, 1359, 1990, 2408, 2967]; # prism prs := [0, 83, 144, 191, 274, 357, 418, 501, 584, 631, 692, 775]; prsd := [83, 61, 47, 83, 83, 61, 83, 83, 47, 61, 83, 83]; 49 intervals, 86 triads--the champ. It's also fairly regular. # stelhex ste := [0, 61, 191, 227, 274, 335, 418, 501, 584, 645, 692, 728]; sted := [61, 130, 36, 47, 61, 83, 83, 83, 61, 47, 36, 130]; 46 intervals, 72 triads # class cla := [0, 61, 108, 227, 274, 335, 418, 501, 548, 645, 692, 775]; clad := [61, 47, 119, 47, 61, 83, 83, 47, 97, 47, 83, 83]; 47 intervals, 80 triads # {225/224, 441/440} in 72-et version # lester les := [0, 5, 12, 16, 23, 30, 35, 42, 46, 53, 58, 65]; lesd := [5, 7, 4, 7, 7, 5, 7, 4, 7, 5, 7, 7]; 46 intervals, 71 triads. Considering this is the least in tune, something of an also-ran.
Message: 5128 Date: Wed, 31 Jul 2002 08:38:24 Subject: Re: Four 10-note, 7-limit JI scales From: Gene W Smith These scales also work well with the {225/224, 441/440} temperament, whose mean square optimal values are essentially those of the 72-et. I give a 72-et version of the first scale below (33 intervals 44 triads); the third and fourth are modes of the first and second, so the second is just a mode of the inversion of the first scale. Qm(3) is not knocked off its perch, but these are a nice suppliment. ! mecaa.scl ! [5, 11, 7, 7, 5, 7, 11, 5, 7, 7] {225/224, 441/440} tempering of decad, 72-et version 10 ! 83.33333333 266.6666667 383.3333333 500.0000000 583.3333333 700.0000000 883.3333333 966.6666667 1083.333333 2/1
Message: 5129 Date: Wed, 31 Jul 2002 09:01:18 Subject: Re: Four 10-note, 11-limit JI scales From: Gene W Smith We can warp in a bit of 11-limit harmony into these scales by means of the {126/125, 441/440} planar temperament. Each scale now has 34 intervals and 50 triads in the 11-limit (compare to Qm(3), with 35 intervals and 52 triads.) ! secad.scl ! [34, 51, 22, 29, 34, 22, 51, 34, 29, 22] {126/125, 176/175} tempering of decad, 328-et version 10 ! 124.3902439 310.9756098 391.4634146 497.5609756 621.9512195 702.4390244 889.0243902 1013.414634 1119.512195 2/1 ! secab.scl ! [22, 29, 34, 51, 22, 34, 29, 22, 51, 34] {126/125, 176/175} tempering of decab, 328-et version 10 ! 80.48780488 186.5853659 310.9756098 497.5609756 578.0487805 702.4390244 808.5365854 889.0243902 1075.609756 2/1 ! secac.scl ! [34, 29, 22, 51, 34, 22, 29, 34, 51, 22] {126/125, 176/175} tempering of decac, 328-et version 10 ! 124.3902439 230.4878049 310.9756098 497.5609756 621.9512195 702.4390244 808.5365854 932.9268293 1119.512195 2/1 ! secad.scl ! [34, 51, 22, 29, 34, 22, 51, 34, 29, 22] {126/125, 176/175} tempering of decad, 328-et version 10 ! 124.3902439 310.9756098 391.4634146 497.5609756 621.9512195 702.4390244 889.0243902 1013.414634 1119.512195 2/1
Message: 5130 Date: Thu, 1 Aug 2002 06:11:51 Subject: Another 12-note scale From: Gene W Smith Here's a 12-note scale which is comparable to the ones I just did by tempering Carl's. I took all the JI scales built from (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 which consisted of two indentical tetrachords separated by a 9/8=15/14 21/20. I got two scales and their inversions, isomorphic by the 21/20 <==> 25/24 transformation. These scales turned out to be adapted to the {225/224, 385/384} temperament, and on tempering I ended up with just one scale (modulo modes) and its inversion. I took this down a fourth to get some dominant harmony, and ended up with this: 1-21/20-9/8-6/5-5/4-21/16-7/5-3/2-8/5-5/3-7/4-28/15 27 (7-limit) intervals, 20 triads Tempering it, I got the following: ! tetra.scl ! [61, 83, 83, 47, 61, 83, 83, 83, 47, 61, 83, 83] {225/224, 385/384} tempering of two-tetrachord 12-note scale ! 858-et version of 1-21/20-9/8-6/5-5/4-21/16-7/5-3/2-8/5-5/3-7/4-28/15 12 ! 85.31468531 201.3986014 317.4825175 383.2167832 468.5314685 584.6153846 700.6993007 816.7832168 882.5174825 967.8321678 1083.916084 2/1 46 (11 limit) intervals 74 triads Something for Carl to think about.
Message: 5131 Date: Sat, 3 Aug 2002 14:25:14 Subject: Optimized 15-note, 7-limit JI scales From: Gene W Smith I did a search on these, using (16/15)^4 (21/20)^3 (25/24)^5 (36/35)^3 as step sizes, and constraining the search by requiring there to be two complete tetrads a iifth apart. I found two optimal solutions: opti15a 42 intervals 37 triads 6 tetrads [1, 21/20, 28/25, 7/6, 6/5, 5/4, 4/3, 7/5, 35/24, 3/2, 8/5, 5/3, 7/4, 28/15, 35/18] opti15b 42 intervals 37 triads 6 tetrads [1, 21/20, 35/32, 7/6, 6/5, 5/4, 4/3, 7/5, 35/24, 3/2, 8/5, 5/3, 7/4, 28/15, 35/18] These are both good candidates for miracle, where they have in the 11-limit 69 intervals and 128 triads. Both tempered and untempered they are graph-isomorphic without being isomorphic.
Message: 5132 Date: Sat, 3 Aug 2002 12:30:31 Subject: Prism plus From: Gene W Smith I finally hit a homer in the search for 12-note, 7-limit JI scales, finding two scales closely related to "prism", but better. I searched scales which contained two tetrachords a fifth apart using (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2 as scale steps. I found ! pris.scl ! [16/15, 21/20, 25/24, 15/14, 16/15, 21/20, 15/14, 16/15, 25/24, 21/20, 16/15, 15/14] optimized (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 scale 12 ! 16/15 28/25 7/6 5/4 4/3 7/5 3/2 8/5 5/3 7/4 28/15 2/1 Then I did another search, which looked for scales containing at least one 7-limit tetradchord, and found another, graph-isomorphic scale (it can be seen as the first scale, taken down a fourth, and transformed so that two of the degrees are changed by 225/224.) "Prism" and similar scales were looked at during this search, but these two have it beat. Here is "prisa": ! prisa.scl ! [21/20, 16/15, 15/14, 25/24, 21/20, 16/15, 15/14, 16/15, 21/20, 25/24, 16/15, 15/14] optimized (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 scale 12 ! 21/20 28/25 6/5 5/4 21/16 7/5 3/2 8/5 42/25 7/4 28/15 2/1 The statistics are prism 30 intervals 24 triads 4 tetrads pris 30 intervals 25 triads 5 tetrads prisa 30 intervals 25 triads 5 tetrads These three scales become the same when tempered by 225/224; in the {225/224, 385/384} temperament, they have 49 11-limit intervals and 86 triads.
Message: 5135 Date: Sat, 10 Aug 2002 12:10:42 Subject: Polynomials and tuning From: Gene W Smith On Fri, 09 Aug 2002 22:06:40 -0000 "wallyesterpaulrus" <perlich@xxx.xxxx.xxx> writes: > --- In tuning-math@y..., "nasnas_100" <nasnas_100@y...> wrote: > > hi i am new student i have a problem > > find the root of f(x)=x^3+30x-30 > > thanks > what does this have to do with tuning? Nothing; it looks like a homework problem. However, for your consideration I present the following. I can represent, and so study, a scale by the polynomial whose roots are the scale elements. To do this right, I want the octave to be represented by a prime number; that is, I want a map to primes h such that h(2)=p, p a prime. In that way I have no zero divisors in the ring mod p, or in other words I am in a field. Suppose I want to study Blackjack, which is a scale in Miracle. I can't do it mod 72, since that is composite, and 31 and 41 are a little small and may give me extraneous relationships. The 103-et would probably be fine, but instead I choose the map [4447, 7039, 10317, 12477], using the prime 4447 to represent 2. This gives me the rms optimal values for Miracle tuning, with a secor of 432/4447. I now take the polynomial with roots 432i where i ranges from -10 to 10, which I can reduce mod 4447 without loss of information: x^21-61*x^19-1724*x^17-1045*x^15-28*x^13+1971*x^11-1724*x^9-1326*x^7+114 *x^5-846*x^3+1260*x I also define a polynomial whose roots correspond to the twelve 7-limit consonant intervals, obtaining x^12-1314*x^10+1560*x^8-1735*x^6-1921*x^4+1244*x^2+1202 If I take the resultant of the first polynomial with x-n substituted for x with the second polynomial and factor mod 4447, I get (n+1601)^5*(n-1550)^3*(n+254)*(n+1169)^5*(n+432)^10*(n+1855)^7*(n+559) ^6*(n+686)*(n+2033)^5*(n-1169)^5*(n-254)*(n+991)^7*(n-1855)^7*(n-610)*(n- 1982)^4*(n-686)*(n-991)^7*(n+305)^6*(n-737)^6*(n-432)^10*(n+1423)^7*(n+15 50) ^3*n^10*(n-1296)^10*(n-864)^10*(n-1423)^7*(n-1601)^5*(n-305)^6*(n+864)^10 *(n +610)*(n-1042)*(n-127)^6*(n+1042)*(n-1118)^2*(n-1728)^9*(n+1296)^10*(n+11 18) ^2*(n+2160)^8*(n-559)^6*(n+127)^6*(n+1728)^9*(n-2160)^8*(n-178)*(n+1982)^ 4*( n+178)*(n+737)^6*(n-2033)^5 This gives me all the places mod 4447 where we have consonant interval relationships to Blackjack--including those in Blackjack. The multiplicities give the number of consonances. The steps of Blackjack, centered at the unison, are [-2160, -1855, -1728, -1423, -1296, -991, -864, -559, -432, -127, 0, 127, 432, 559, 864, 991, 1296, 1423, 1728, 1855, 2160] and the corresponding multiplicities are [8, 7, 9, 7, 10, 7, 10, 6, 10, 6, 10, 6, 10, 6, 10, 7, 10, 7, 9, 7, 8] If we add this up, we get 170, which is twice the number of 7-limit consonances for Blackjack--twice since we count them twice, once for each scale step.
Message: 5137 Date: Sat, 10 Aug 2002 23:45:01 Subject: Re: solution of cubic From: Robert Walker HI there, Just sent a post to the freelists.org - all the posts sent here get echoed there. Anyway in summary it has a link to Quintic equation solution - applet * where I have put up a solution to the quintic that I did basically for fun and because when I looked for on-line pages for solving the quintic I couldn't find any, so it might be somethign of a gap in the range of Web javascripts available right now. Of course you can use it for solving a cubic by setting the first two coefficients to 0. That page also has links to a couple of other pages on the web about the modern solution of the cubic, and its history. I don't know of any relevance of my page to tuning but did use solutions of cubic when exploring fibonacci tonescapes - to find appropriate ratios to use such that if you go up by one ratio and down by another ratio on the long and short beats of a fibonacci rhythm, then in the long term (like hours) you want the pitch to wander not too far from the original 1/1. If you choose the numbers right then even after an hour or so, even with small ratios, you can stay within a fraction of a cent of the original 1/1. If you follow that through you end up with a cubic equation to solve. Robert
Message: 5138 Date: Sat, 10 Aug 2002 23:50:48 Subject: Re:solution of cubic From: Robert Walker HI there, Sorry getting muddled. When you have two beats in the pattern you just need to solve a quadratic. The cubics come in when you look for ratios to use for fibonacci tonescapes with three beat fibonacci rhythms. I expect if one went up to fibonacci patterns built up using four or more beats you would probably need the quartic and quintic - that's just a guess as I haven't worked it out. Robert
Message: 5139 Date: Sat, 10 Aug 2002 17:58:22 Subject: Re: find the root of the function From: Gene W Smith On Sat, 10 Aug 2002 13:59:06 -0700 "M. Edward Borasky" <znmeb@xxxxxxx.xxx> writes: > Well . according to Derive, there are two complex roots and one real > root: > > x = -0.4848069410 - 5.541219478.i ; x = -0.4848069410 + > 5.541219478.i ; x > = 0.9696138820 > > Now that I've given you the answer, your assignment is to look up the > formula (and there is one) There's more than one, and solving it in radicals (as opposed to Chebyshev radicals, for instance) isn't the neatest.
Message: 5141 Date: Tue, 13 Aug 2002 19:19:58 Subject: Re: A common notation for JI and ETs From: David C Keenan At 11:52 13/08/02 -0700, George Secor wrote: >From: George Secor, 8/13/2002 (tuning-math #4577) >Subject: A common notation for JI and ETs > >Note: Dave Keenan has kindly agreed to work with me (off-list) on the >notation project again for a short time to deal with the latest >modifications that I am proposing. (Will there ever be an end to this? > I think there's light at the end of the tunnel.) Otherwise, I expect >that he will continue to take time off from the Tuning List. We will >be posting our correspondence here to maintain a complete record of how >the notation is being developed. That's right. I am not reading any lists. Only CCing my replies to George, to tuning-math. >I have a long reply to his last message, and I will post this in >installments. --GS > >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4532]: >> At 01:03 18/06/02 -0000, you wrote: >> >--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> >--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >> >> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> >> > I would therefore recommend going back to the rational >> >> > complementation system and doing the ET's that way as well. >> >> >> >> Agreed. Provided we _always_ use rational complements, whether >this >> >> results in matching half-apotomes or not. >> > >> >In other words, you would prefer having this: >> > >> >152 (76 ss.): )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~ >/|| ||\ ~||) (||~ /||\ >> > >> >instead of this: >> > >> >152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ >/|| ||\ /||~ /||) /||\ > > > >> >even if it isn't as easy to remember. >> >> OK. I think you've got me there. :-) Remember I said I thought we >shouldn't let complements cause us to choose an inferior set of >single-shaft symbols, because some people won't use the purely saggital >complements. I think we both agree that /|~ is a better choice for >5deg152 than ~|) since it introduces fewer new flags and puts the ET >value closer to the rational value. > >New Rational Complements – Part 1 >--------------------------------- > >Now that I've talked you into this, I'm going to have to try to talk >you out of it (to some extent) because of something that I have come to >realize over these past few weeks. There's nothing like some time off >to create a new perspective: I have come back to this as if I were a JI >composer new to the notation who is asking the question, "How would I >notate a 15-limit tonality diamond?" An excellent question. I think I posed a similar one earlier, but only considering the 11-limit diamond. >And now that I've taken a fresh >look at the notation, I came up with some ideas on how to improve a few >things. > >First of all, here is how I was able to notate all of the 15-odd-limit >consonances taking C as 1/1. (Don't bother to look through all of this >now; I'll be referring to many of these below, so this listing is just >given for reference.) ... That's marvellous, except of course it looks like gobbledygook when up to 5ASCII symbols are being used to represent a single sagittal symbol. How big is the biggest schisma involved? >> I don't think we have defined a rational complement for /|~ because >it isn't needed for rational tunings. > >On the contrary, I found that /|~ is in fact quite useful for rational >tunings (see above table of ratios), but its lack of a rational >complement is a problem. To remedy this, I propose ~||( as its >rational complement. Fair enough, and yes, that would seem the obvious complement. >With C as 1/1, the following ratios would then >use these two symbols (which also appear in the table of ratios above): > >11/10 = D\!~ >20/11 = Bb/|~ or B~!!( >15/11 = F/|~ >13/7 = B\!~ >14/13 = Db/|~ or D~!!( > >In effect, /|~ functions not only as the 5+23 comma (~38.051c), but >also as the 11'-5 comma (~38.906c) and the 13'-7 comma (~38.073c) OK, so a 0.86 c schisma. I can certainly live with that for such obscure ratios. >This would replace (|( <--> ~||( as rational complements. I found that >(|( is not needed for any rational intervals in the 15-odd limit, so >this has no adverse consequences. (However, it leaves the 23' comma >without a rational complement; I will address that problem below.) The >new pair of complements that I am proposing also has a lower offset >(0.49 cents) than the old (-1.03 cents), so, apart from the 23' comma, >I can't think of a single reason not to do this. Me neither. Apart from the 23' comma. We could resurrect ~)||, with two left flags, as the complement of the 23' comma. It isn't like a lot of people really care about ratios of 23 anyway.We already made a good looking bitmap for ~)| with the wavy and the concave making a loop. >The reverse pair of complements, ~|( <--> /||~, would be used for the >following ratios of 17: > > 17/16 = Db~|( or D\!!~ > 17/12 = Gb~|( or G\!!~ > 17/9 = Cb~|( or C\!!~ > 32/17 = B~!( > 24/17 = F#~!( or F/||~ > 18/17 = C#~!( or C/||~ > >All of this is going to affect how we will want to notate not only 152, >but also other ET's, including 217. (More about this later.) If rational complements don't have to be consistent with 217-ET any more, how about making rational complements consistent with 665-ET, as proposed earlier? >> But if we look at complements consistent with 494-ET (as all the >rational complements are) the only complement for /|~ is ~||(. So we >end up with >> >> 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ~||( >/|| ||\ ~||) /||) /||\ >> >> But this is bad because the flag sequence is different in the two >half-apotomes _and_ ~||( = 10deg152 is inconsistent _and_ too many flag >types. So you're right. I don't want to use strict rational complements >for this, particularly with its importance in representing 1/3 commas. >I'd rather have >> >> > 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~ >/|| ||\ /||~ /||) /||\ > >I don't follow the part about ~||( = 10deg152 being inconsistent: The >17' comma ~|( is 2deg, and the apotome (15deg) minus the unidecimal >diesis (7deg) is (|) = 8deg, so (|) + ~|( = ~||( = 10deg. My mistake. Sorry. >So I would replace |~, the 23-comma, with ~|(, the 17' comma, Well of course I think of |~ as 19'-19 when notating ETs. >which gives: > >152 (76 ss.): )| ~|( /| |\ /|~ /|) /|\ (|) (|\ ~||( /|| ||\ > /||~ /||) /||\ (RC w/ 14deg AC) Unfortunately this gives up a desirable property: Monotonicity of flags-per-symbol with scale degree. >This not only uses a symbol ~|( that corresponds to a lower prime >symbol for 2deg, but also uses a rational symbol ~||( that has meaning >for certain ratios of 11 and 13, as also will /|~. The 14deg symbol >/||) is not the rational complement of 1deg )|, but its offset (~1.12 >cents) is small enough that it would have qualified as a rational >complement (RC) if we had no other choice. I'll call this an alternate >complement (AC) -- one that may be used for notating an ET in the >absence of a RC consistent in that ET, but which is not used for >rational notation. Fair enough. >The principle that I am advancing here is that there is another goal or >rule that should take precedence over that of an easy-to-memorize >symbol sequence -- symbols which are used to represent JI consonances >should be used in preference to those that can be expressed only as >sums of comma-flags. These are the symbols that will be used for JI >most frequently, and they will therefore (through repeated use) become >*the most familiar* ones. But many people using ETs couldn't care less about JI, so why should rational approximations take precedence over mnemonics, particularly if they onlyinvolve ratios as uncommon as 5:11 and 7:13? >And these are the symbols that should have >first priority in the assignment of rational complements. Yes. I can accept that. > This is why >I want to eliminate (|( in the rational complement scheme -- it is the >(13'-(11-5))+(17'-17) comma or, if you prefer, the (11'-7)+(17'-17) >comma, neither of which is simple enough to indicate that it would ever >be used to notate a rational interval; and none of the 15-limit >consonances (relative to C=1/1) require it. I'll wait and see where this leads. By the way, I assume we agree that manyof those 15-limit "consonances" are not consonant at all, and are not evenJust, being indistinguishable from the intervals in their vicinity, exceptif they are a subset of a very large otonality or with the most contrived timbre. >This will be continued, following a short digression about 76-ET. > >> I note that 76-ET can also be notated using its native fifth, as you >give (and I agree) below. > >In the process of looking over what we discussed regarding 76 (in >connection with 62 and 69 a bit later in your message #4532), I noticed >that it was given above as a subset of 152. I then noticed how bad the >5-limit is in 76 and wondered why it was being considered on its own. > >I then reviewed our correspondence. In response to a question from >Paul about 76-ET, you told him this (in message #4272): > ><< The native best-fifth of 76-ET is not suitable to be used a >notational fifth because, among other reasons, it is not >1,3,9-consistent (i.e. its best 4:9 is not obtained by stacking two of >its best 2:3s) and I figure folks have a right to expect C:D to be a >best 4:9 when commas for primes greater than 9 are used in the >notation. So 76-ET will be notated as every second note of 152-ET. >> > >It gets even worse than this: not only is 3 over 45 percent of a degree >false, but 5 deviates even more. > >Your next mention of 76-ET was in message #4434, in which you treated >the divisions of the apotome systematically: > >4 steps per apotome ... >69,76: |) ?? (|\ /||\ [13-comma] > >>From that point we had 76 listed both as a subset of 152 and with 69. >So after looking at all this, which will it be? (I would prefer it as >the subset.) If we are proposing a _single_ standard way of notating every ET then 76 should be as a subset of 152-ET. However I think there are several such ETs where some composers may have very good reasons for wanting to notate them based on their native best fifth, (for example because the 76-ET native fifth is the 19-ET fifth), and we should attempt to standardise those too. So Isay give both, but favour the 152-ET subset. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5144 Date: Wed, 14 Aug 2002 19:11:07 Subject: Re: A common notation for JI and ETs From: David C Keenan At 11:27 14/08/02 -0700, George Secor wrote: >Until recently I had a prejudice against //|, because it has two flags >on the same side. But now that I see that other symbols of this sort >haven't popped up all over the place, and since its rational complement >~|| is simple and useful, I would like to include it in the standard >217 notation instead of ~|\ (which is only the 11-5+17 comma, and which >is not needed for any 15-limit consonances). That's fine by me. I totally approve of making more use of //|, but it should only be used in an ET if it is valid as the double 5-comma. >For reference, here is the 217 standard notation as it presently >stands: > >217: |( ~| |~ /| |) |\ ~|) ~|\ /|) /|\ (|) (|\ ~|| ||~ >/|| ||) ||\ ~||) ~||\ /||) /||\ (present) > >Making this change would give us: > >217: |( ~| |~ /| |) |\ ~|) //| /|) /|\ (|) (|\ ~|| ||~ >/|| ||) ||\ ~||) //|| /||) /||\ (all RCs) > >So we would now have true rational complements throughout. > >However, there is a second change that I wish to propose. It >incorporates the change of rational complements from (|( <--> ~||( to >/|~ <--> ~||( that I also proposed above. For 7deg we now have ~|), >which is used for the following ratios, but for nothing in the >15-limit: > >17/10 = Bbb~|) or Bx~ > 17/15 = Ebb~|) or Ex~ > >(For this ascii notation I have used x instead of X to specify a >*downward* alteration of pitch, as we have already done with ! instead >of |. I hope the presence the wavy flag in combination with it is >enough to indicate it is not being used here to indicate a double >sharp. Otherwise, would a capital Y be a suitable alternative?) Little x for downward is fine with me. >The proposed replacement standard symbol /|~ for 7deg217 is used for >11/10, 14/13, and 15/11 (plus their inversions). > >In order to maintain rational complements and a matching symbol >sequence throughout, the symbols for 3, 14, and 18deg217 would also >need to be changed, which would give this for the standard 217 set: > >217: |( ~| ~|( /| |) |\ /|~ //| /|) /|\ (|) (|\ ~|| ~||( >/|| ||) ||\ /||~ //|| /||) /||\ (new RCs) > >The 3deg symbol changes from the 23 comma (or 19'-19 comma, if you >prefer) to the 17' comma. This is a more complicated symbol, but it >symbolizes a lower prime number, making it more likely to be used. >(Besides, it has mnemonic appeal.) Yes I suppose I can give up monotonic flags-per-symbol, but if you don't want to know about JI or don't care about 11/10, 14/13, or 15/11, then that /|~ now seems to come out of nowhere. Why suddenly introduce the right wavy flag. At least ~|) introduces no new flags. >My goal is to minimize the differences between the 217-ET notation and >the rational notation (while maintaining a matched symbol sequence), >with the lowest primes (i.e., the 17 limit) being favored. That's fine so long as it is the 217-ET notation that gets compromised, notthe rational. >This would >make the transition from purely rational symbols to 217-ET standard >symbols as painless as possible in instances where the composer has run >out of rational symbols and has no other choice but to use 217 symbols >to indicate rational intervals. I don't understand why there would be no choice but 217-ET. Is 217-ET really the best ET that we can fully notate? What about 282-ET? It's 29-limit consistent. I've never really understood the deference to 217-ET. ... >So I think it would be best to retain the straight flags in the >standard 217 set, Agreed. > but to have in mind the (| and )||~ symbols as >supplementary rational complements. A composer would have the option >to use (| and )||~ to clarify the harmonic function of the tones which >they represent for either 217-ET or JI mapped to 217. The same could >be said for the rational symbols for ratios of 19 and 23, should one >want to use a higher harmonic limit. (These would be less-used, >less-familiar symbols that would be rarely be needed below the 19 >limit.) > >With these changes in the standard 217 notation, it would be necessary >to memorize only 8 rational complement pairs (half of which use only >straight and convex flags, and half of which are singles, not pairs) to >notate all of the 15-limit consonances and a majority of the ratios of >17 in JI: > >5 and 11-5 commas: /| <--> ||\ and |\ <--> /|| >7 comma: |) <--> ||) >11 diesis: /|\ <--> (|) >13 diesis: /|) <--> (|\ >7-5 comma or 11-13 comma: |( <--> /||) >17 comma and 25 comma: ~| <--> //|| and //| <--> ~|| >17' comma and 11'-5 or 13'-7 comma: ~|( <--> /||~ and /|~ <--> ~||( >19' comma and 11'-7 comma: )|~ <--> (|| and (| <--> )||~ > >(The last pair of RCs are the supplementary symbols that are not part >of the standard 217-ET set.) > >With these symbols you have more than enough symbols to notate a >15-limit tonality diamond (with 49 distinct tones in the octave). Good work. I'd like to see that listed in pitch order. >Notice that I identified |( as something other than the 17'-17 comma. >This is because it is used for the following rational intervals: > >7/5 = Gb!( or G!!!( >10/7 = F#|( or F|||( >13/11 = Eb!( or E!!!( >22/13 = A|( >15/14 = C#|( or C|||( >28/15 = Cb!( or C!!!( > >Thus |( can assume the role of either the 17'-17 comma (288:289, >~6.001c), the 7-5 comma (5103:5120, ~5.758c), or the 11-13 comma >(351:352, ~4.925c). However, there are a limited number of ETs in >which it can function as all three commas (159, 171, 183, 217, 311, >400, 494, and 653) or at least as both the 7-5 and 11-13 commas (130 >and 142). Hmm. It is certainly arguable that we should favour the interpretation of |( as the 7-5 comma when notating ETs. What's the smallest ET that would be affected by this? Is )| still to be interpreted as the 19 comma and what is to be its complement? Is there a lower prime interpretation of |~ now too? It seems to me that what we are discussing here is unlikely to impact on many ETs below 100. Is that the case? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5146 Date: Thu, 15 Aug 2002 19:00:22 Subject: Re: A common notation for JI and ETs From: David C Keenan At 12:31 15/08/02 -0700, George Secor wrote: >> That's fine by me. I totally approve of making more use of //|, but >it should only be used in an ET if it is valid as the double 5-comma. > >Yes, a mandatory test for the use of this symbol in an ET is that the >ET be 1,5,25 consistent. That's a little more strict that what I had in mind, but I guess it's a good idea. I'd be inclined to allow it to represent two 5-commas whether that gives the best 25 or not. >> >217: |( ~| ~|( /| |) |\ /|~ //| /|) /|\ (|) (|\ ~|| >~||( /|| ||) ||\ /||~ //|| /||) /||\ (new RCs) >> > >> >The 3deg symbol changes from the 23 comma (or 19'-19 comma, if you >> >prefer) to the 17' comma. This is a more complicated symbol, but it >> >symbolizes a lower prime number, making it more likely to be used. >> >(Besides, it has mnemonic appeal.) >> >> Yes I suppose I can give up monotonic flags-per-symbol, but if you >don't want to know about JI or don't care about 11/10, 14/13, or 15/11, >then that /|~ now seems to come out of nowhere. Why suddenly introduce >the right wavy flag. At least ~|) introduces no new flags. > >Three reasons: > >1) As I said above, /|~ is used for 3 15-limit ratios (not including >inversions), while ~|) is used for only one ratio of 17. Hence /|~ >will have a wider use. This seems a little circular. If we did not limit ET notations to using only those symbols used for 15-limit JI, but instead tried to minimise the number of different flags each uses (as we have been until recently), then ~|)may well have wider use than /|~, purely due to the number of ETs it is used in. So I don't buy this one. >2) Those who don't care about 11/10 _et al_ will probably be using >tempered versions of these ratios in one way or another if /|~ occurs >in the particular ET they are using. Use of the same symbol in *both* >JI and the ET exploits the *commonality* of the symbols for both >applications. Yes, I agree that is the whole point of our "common notation". However I'm not convinced that there will be many times when somone uses an approximate11:15 or 13:14 _as_ an approximate just interval when the lower note is a natural (or has only # or b). But in the case of a 5:11 I guess it's more likely. So I find this reason to be marginally valid. >3) As I said below, I am now placing a higher priority on minimizing >the number of the most commonly used *symbols* than on minimizing the >number of *flags* used for an ET. This "most commonly used" set of >symbols was summarized in the 8 sets of rational complements that I >listed at the end of my last message. On examing these in more detail I find that I don't understand at all why you chose /|~ as the appropriate symbol for the 11'-5 comma, 44:45 (and the 13'-7 comma). (|( seems the obvious choice to me, since (| is the 11'-7 comma and |( is the 7-5 comma and (| + |( = (|( . (11'-7)+(7-5) = 11'-5. and (11'-7)+(13'-11')=(13'-7). If (|( is the symbol for the 11'-5 comma (or we could more usefully call itthe 11/5 comma) then you don't need to change any rational complements from what we had (the 494-ET-consistent ones) and what's more we don't need tointroduce any more flags into 217-ET when (|( is used for 7 steps. >> I don't understand why there would be no choice but 217-ET. Is 217-ET >really the best ET that we can fully notate? What about 282-ET? It's >29-limit consistent. I've never really understood the deference to >217-ET. > >I never considered 282 before, but I do see some problems with it: > >1) 11 is almost 1.9 cents in error, and 13 is over 2 cents; these >errors approach the maximum possible error for the system. (This is >the same sort of problem that we have with 13 in 72-ET.) You're only looking at the primes themselves. What about the ratios betweenthem. 217-ET has a 2.8 cent error in its 7:11 whereas 282-ET never gets worse than that 2.0 cents in the 1:13. >2) The |) flag is not the same number of degrees for the 7 and 13-5 >commas (which is by itself reason enough to reject 282), nor is (| the >same number of degrees for the 11'-7 and 13'-(11-5) commas. Reason enough to reject 282-ET as what? Reject it as a good way of having afully notatable closed system that approximates 29-limit JI? I seriously disagree. It just means that we should use (| and |) with their non-13 meanings in 282-ET. >3) The following rational complements for the 15-limit symbols are not >consistent in 282: > >)|~ <--> (|| 19' comma > |( <--> /||) as 7-5 comma or 11-13 comma (but 17’-17 is okay) >~| <--> //|| 17 comma > |) <--> ||) 7 comma >//| <--> ~|| 25 comma > (| <--> )||~ 11'-7 comma > >And besides this, there are others that are inconsistent, such as: > > |~ <--> ~||) as both the 19’-19 and 23 comma All this means is that maybe we should consider making our rational complements consistent with 282-ET rather than 217-ET. >What makes 217 so useful is that *everything* is consistent to the 19 >limit, and, except for 23, to the 29 limit. I don't know what you mean by *everything* here. Isn't 282-ET consistent tothe 29-limit with no exceptions? >And I think that the >problems with 23 can be managed, considering how rarely it is likely to >be used. You have to have a way to accommodate the electronic JI >composer who might want to modulate all over the place, and a >consistent ET mapping for JI intervals is the only way to do it with a >finite number of symbols; this is where 217 really delivers the goods! I still fail to see why 217 is better than 282, except that various choiceswe have made along the way, regarding the symbols, have been biased toward217. >> >With these symbols you have more than enough symbols to notate a >> >15-limit tonality diamond (with 49 distinct tones in the octave). >> >> Good work. I'd like to see that listed in pitch order. > >At first I thought you meant listing the symbols like this: > >Symbol set used for 15-limit JI >------------------------------- > )|~ <--> (|| 19' comma (not in standard 217 set) > |( <--> /||) 7-5 comma or 11-13 comma > ~| <--> //|| 17 comma > ~|( <--> /||~ 17' comma > /| <--> ||\ 5 comma > |) <--> ||) 7 comma > |\ <--> /|| 11-5 comma > (| <--> )||~ 11'-7 comma (not in standard 217 set) >//| <--> ~|| 25 comma > /|~ <--> ~||( 11'-5 or 13'-7 comma > /|\ <--> (|) 11 diesis > /|) <--> (|\ 13 diesis No. Although that's interesting too. >But now I think you meant listing the ratios like this: > >Sagittal Notation for 15-limit JI >--------------------------------- ... Yes that was it, but now I realise there are only 6 that are independent and that we haven't already agreed on. Here they are in oder of decreasing importance: 1/1 = C 7/5 = Gb!( or G!!!( 11/5 = D\!~ 11/7 = G#(! or G)||~ 13/5 = F\\! 13/7 = B\!~ 13/11 = Eb!( or E!!!( But I think they should be: 1/1 = C 7/5 = Gb!( or G!!!( 11/5 = D(!( 11/7 = G#(! or G)||~ 13/5 = F\\! 13/7 = B(!( 13/11 = Eb!( or E!!!( >> Hmm. It is certainly arguable that we should favour the >interpretation of |( as the 7-5 comma when notating ETs. What's the >smallest ET that would be affected by this? > >It's hard to say what is the smallest ET in which they differ >consistently. I mean: What's the smallest one we've agreed on that uses |(, where the 7-5comma interpretation of it would be a different number of steps from what we've used it for. >> Is )| still to be interpreted as the 19 comma and what is to be its >complement? > >Yes, and its complement is still (||~. I don't see any lower-prime >interpretations of it without going into rational complements, where we >have only one: 11/7 = G)||~. This is greater than G(|) (2187/1408) by >15309:15488, ~20.125c (vs. the 19' comma, 19456:19683, ~20.082c). But >this is for )|~, so we must subtract |~ from this, but what comma would >|~ be? Since your next question has a positive answer (and since I did >that one first I can peek at the answer), I'll use the 11-limit comma >99:100, which gives 42525:42592 (3^5*5^2*7:2^5*11^3, ~2.725c) as the >11-limit interpretation of )|. > >This is meaningful only if you are using rational complements, i.e., >single-symbol notation. No. Going via complements isn't what I had in mind. Does )| want to be usedas a comma for any of 17/5, 17/7, 17/11, 17/13? >> Is there a lower prime interpretation of |~ now too? > >Hmm, good question! Yes, using /|~ as the 11'-5 comma for 11/10 would >make that symbol 44:45, so |~ would be 99:100, ~17.399 cents. And >using /|~ as the 13'-7 comma for 13/7 would make /|~ 1664:1701, so |~ >would be 104:105, ~16.567 cents. OK. But this is not so, if we adopt (|( as the 7/5-comma symbol. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
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