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Message: 1600 - Contents - Hide Contents Date: Mon, 17 Sep 2001 19:57:25 Subject: Re: Fwd: Two 21-tone JI scales: detemperings of blackjack From: Paul Erlich --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:> Paul, > > In writing << all possible melodic modes have not the same > importance >> I was at hundred miles to think you could take > that in the sense of historical importance. In the Zarlino > gammier > > 1 5 2 6 > 6 3 0 4 1 > 1 5 2 6 > > there exist 16 modes. One of them is the non-convex > > 1 16/15 6/5 4/3 3/2 5/3 15/8 2 > > . 5 . 6 > . 3 0 4 . > 1 . 2 . > > I wanted to say uniquely that a such mode is less important, > for instance, than the doristi mode > > 1 16/15 6/5 4/3 3/2 8/5 16/9 2 > > . . . . > 6 3 0 4 . > 1 5 2 .Pierre, "mode" in musicians' parlance means rotation -- a different scale degree becomes the first scale degree, and all others rotated by the same amount. All modes of a scale have identical lattice configurations . . . differing only by a translation. My claim is that, in this sense of "mode", one cannot determine "better" or "worse" modes without some reference to the particulars of the musical style they are to be used for. Sonance of the degrees against the 1/1 is irrelevant unless the musical style uses a 1/1 drone, for example.
Message: 1601 - Contents - Hide Contents Date: Mon, 17 Sep 2001 22:56:05 Subject: Re: Ten notes to the octave From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> The question of whether this is a PB turns out to be a matter of > definitions. If instead of partitioning into half-open and non- > intersecting regions to define blocks, we could use closed regions > which intersect on their boundaries. We then can have more than one > candidate for a given scale degree; if we allow ourselves to select > any such candidate we choose then the above can be seen as a block, > but not otherwise. We might call such a thing a periodic semiblock, > or PS.I think I understand this now . . . sorry I didn't at first, Gene. You're talking about the _Fokker_ partitioning into parallelepipeds, right? As you can see in the _Gentle Introduction_, periodicity blocks can have boundaries of different shapes, not just the Fokker specification . . . but I don't mind your suggestion for something like the "PS" terminology . . .
Message: 1602 - Contents - Hide Contents Date: Tue, 18 Sep 2001 04:24:33 Subject: Re: Ten notes to the octave From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: As you can see in the _Gentle Introduction_, periodicity> blocks can have boundaries of different shapes, not just the Fokker > specification . . . but I don't mind your suggestion for something > like the "PS" terminology . . .Interesting--I thought we were committed to paralleopipeds, though any convex region would seem good enough. I still get PBs as "scales along a line", but need to adapt my definition of closeness from its implicit L-infinity norm to other possibilities.
Message: 1604 - Contents - Hide Contents Date: Wed, 19 Sep 2001 18:54:18 Subject: Re: a class of non-octave scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., jon wild <wild@f...> wrote:> I'm not sure what an "equal division" into an irrational number of parts > really means, but if you can swallow it, there it is.What it means is more easily appreciated if you don't look at it in terms of divisions but in terms of generators. Then the maximum generator of this kind is e^(1/4e) = 159.22 cents. One can certainly use this as a scale without reference to a period of repition, but if you want one, you may choose any such period you like; the result will be the same.
Message: 1605 - Contents - Hide Contents
Date: Fri, 21 Sep 2001 05:40:49
Subject: A proposed definition for block
From: genewardsmith@xxxx.xxx
I think I've found a definition for "block" which is both simple and
covers what we would want covered; I'd be interested to hear
criticism either that something is called a block which shouldn't be,
or is not which should be.
Let <b_2,..., b_k> be a (k-1)-tuple of positive rational numbers in
the p-limit, where k=pi(p). Let N be the (k-1)xk matrix whose ith row
corresponds to the factorization of b_i into primes, and suppose the
(k-1)x(k-1) minors of N (that is, the determinants of the (k-1)x(k-1)
square submatricies of N) are relatively prime, so that their gcd is
1, and that the first minor, obtained by excluding the first column,
is n <>0. Let M be the square matrix obtained by adding a top row
[1,0,0,…,0], corresponding to 2, to N, whose determinant n is
therefore not zero, so that M^(-1) is defined.
We define a norm on the k-dimensional real vector space R^k, which we
denote ||v|| where v = [v_1,...,v_k], by taking the product
w = v M^(-1), with coordinates [w_1, …, w_k], and setting
||v|| = max(|w_1|,..., |w_k|). This makes R^k into a normed vector
space and the notes of the p-limit into a lattice. If S is any set of
p-notes (i.e., of positive rational numbers in N_p), then we define
the *diameter* dia(S) of S to be max_{i,j \in S}(||s_i – s_j||) over
all pairs of elements s_i and s_j in S. We define S to be a block for
the set <b_i> if S is maximal subject to the condition that its
diameter be less than one. By maximal I mean no element can be added
to S without the new set having a diameter of at least one.
We could add conditions to <b_i> to make the blocks we obtain more
reasonable—-in particular, the condition that it can be completed to
a valid basis for N_p.
Message: 1606 - Contents - Hide Contents Date: Fri, 21 Sep 2001 17:10:39 Subject: New "old" Scala version From: manuel.op.de.coul@xxxxxxxxxxx.xxx By request, I've made another console (non-graphical) version of Scala, version 1.8 for Windows 9x/NT/2000. It does not contain more functions than the graphical version that will follow the current version 2.02. So if you're happy with the GUI-version you can ignore this post. The file is http://www.huygens-fokker.org/software/scala18win.zip - Type Ok * [with cont.] (Wayb.) Homepage: Scala Home Page * [with cont.] (Wayb.) To install, unpack scala18win.zip in an empty directory and read the readme.txt file. After startup, HELP shows the commands, and @HELP creates a window with the help text. Some of its new features: - Some notation systems were added. - Some chord names were added. - COMPARE can also do approximate scale comparison. - SHOW DATA gives more scale properties. - The following new commands were added: Key/Centre Lattice/Notation/Triangular Set Int_File Show/Notation Scale Show/Interval/Notation Scale Show/Octcps Scale Show Shifts Thanks to Christopher Chapman, Dave Keenan, Sohrab Mohajerin, Margo Schulter and Robert Walker. Manuel
Message: 1607 - Contents - Hide Contents Date: Fri, 21 Sep 2001 18:26:30 Subject: Re: New "old" Scala version From: Carl Lumma> By request, I've made another console (non-graphical) version of > Scala, version 1.8 for Windows 9x/NT/2000. Awesome! -Carl
Message: 1608 - Contents - Hide Contents Date: Fri, 21 Sep 2001 20:06:53 Subject: Re: A proposed definition for block From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I think I've found a definition for "block" which is both simple and > covers what we would want covered; I'd be interested to hear > criticism either that something is called a block which shouldn't be, > or is not which should be.Can you give some examples? I'm not following all the implications. P.S. I'm not sure if you're implying a certain "metric" here, specifically one in octave-equivalent space . . . if you are, I'd warn you that I tend to be adamant about using a "triangular" rather than a "rectangular" lattice, so that 15:8 is a longer distance than 5:3 . . . otherwise, I get very upset :)
Message: 1609 - Contents - Hide Contents Date: Sat, 22 Sep 2001 18:09 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx I found this comment from Paul in my inbox:> What do you mean by miracle generator here? Miracle generator for us > has meant specifically 116.7 cents . . .I don't know about the rest of you, but I take anything between 3/31 and 4/41 octaves as a Miracle generator. That's from 116.1 to 117.1 cents. 16:15 is 111.7 cents and 15:14 is 119.4 cents. Graham
Message: 1610 - Contents - Hide Contents Date: Sat, 22 Sep 2001 19:22:21 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> I don't know about the rest of you, but I take anything between 3/31 > and 4/41 octaves as a Miracle generator. That's from 116.1 to 117.1 > cents. 16:15 is 111.7 cents and 15:14 is 119.4 cents.Splitting the difference gives us sqrt(8/7) = 115.6, a little shy of being a miracle generator; however on the tuning group I pointed out that m = (12/5)^(1/13) = 116.6 cents is what gives us perfect octaves if we iterate a miracle fourth inside of 5/3, and it is something in the vicinity of m I would regard as a miracle generator. The corresponding miracle fourth is f = sqrt(5/3 m) = (2 (5/3)^6)^(1/13) = 500.5 cents, and it is this that I regard as more fundamental, the miracle generator itself being secondary. Incidentally, the difference between sqrt(8/7) and m, which is (2^35 3^4 5^(-2) 7^13)^(1/26), is 1.0007446 cents, which must be the justification for using cents. :)
Message: 1611 - Contents - Hide Contents Date: Sat, 22 Sep 2001 03:13:44 Subject: Re: A proposed definition for block From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote: > Can you give some examples? I'm not following all the implications.I'll give some examples; I had it in mind to continue with the "N" series of scales in any case.> P.S. I'm not sure if you're implying a certain "metric" here, > specifically one in octave-equivalent space . . . if you are, I'd > warn you that I tend to be adamant about using a "triangular" rather > than a "rectangular" lattice, so that 15:8 is a longer distance than > 5:3 . . . otherwise, I get very upset :)There is a metric (on notes, not just on equivalence classes), since a normed vector space entails a metric. A real normed vector space V is a real vector space together with a norm map || ||:V --> R, such that for v in V we have (1) ||v|| >= 0 (2) ||v|| = 0 <==> v = 0 (3) If c is a scalar, then ||c v|| = |c| ||v|| (4) ||u + v|| <= ||u|| + ||v|| Any normed vector space is a metric space in which the norm map is a continuous function. An example would be the taxicap (L1) norm, leading to your favorite taxicap metric. The metric in this case is what it has to be in order for the definition to work, and it depends on the choice of unison vectors. There's no use getting mad at it, whatever it may be in any particular case. :)
Message: 1612 - Contents - Hide Contents Date: Sat, 22 Sep 2001 08:24:38 Subject: Block definition example From: genewardsmith@xxxx.xxx Let's do this for the classic example--the JI diatonic. If we denote by (16/15, 25/24, 81/80) the matrix whose rows are the three intervals expressed in prime factored notation, and by [h7, h5, h3] the matrix whose columns are the 7, 5, and 3 ets, then (16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. We need instead the inverse of (2, 25/24, 81/80), and it is easily checked that (2, 25/24, 81/80)^(-1) = [1/7 h7, h5 - 5/7 h7, h3 - 3/7 h7]; if g_i is the ith column and e_j is the jth row, then g_i(e_j) = d_ij (the Kronecker delta, meaning 1 if i=j and 0 if i<>j.) If therefore [a,b,c] is a note in the [h7,h5,h3] notation, then ||[a,b,c]|| = 1/7 max( |a|, |7b - 5a|, |7c -3a|). In this notation we have 16/15 = [1 0 0], 10/9 = 16/15 * 25/24 = [1 1 0] and 9/8 = 10/9 * 81/80 = [1 1 1]. Using these scale steps we see that the JI diatonic scale is [0 0 0], [1 1 1], [2 2 1], [3 2 1], [4 3 2], [5 4 2], [6 5 3]. If we transform [a,b,c] to [a, 7b - 5a, 7c - 3a] we get [0 0 0], [1 2 4], [2 4 1], [3 -1 -2], [4 1 2], [5 3 -1], [6 5 3]. We note that 5/3, represented by [5 3 -1] is closer to 1 than 15/8, represented by [6 5 3], which may make Paul happy until he notices how much closer 9/8 is to 1 than either. We also notice that the biggest difference is between 15/8 and 4/3 or 1; we have [6 5 3] - [3 -1 -2] = [3 6 5], so that the distance d(15/8, 4/3) = 6/7, and [6 5 3] - [0 0 0] = [6 5 3] and d(15/8, 1) = 6/7; the diameter of the diatonic scale is therefore 6/7, less than 1. To show it is a block under the definition we must show it is maximal; this we may do by looking at the matter modulo 7. The [a, 7b - 5a, 7c - 3a] transformed notation applied to our scale steps gives us 16/15 = [1 -5 -3], 10/9 = 16/15 * 25/24 = [1 -5 -3] + [0 7 0] = [1 2 -3], 9/8 = 10/9 * 81/80 = [1 2 -3] + [0 0 7] = [1 2 4]. The three scale steps are therefore all [1 2 4] mod 7, and so the scale is [n, 2n, 4n] mod 7. Hence we have one representative mod 7 for each coordinate, and adding another note will entail that at least one coordinate will differ by a least 7, so that the diameter will be at least 1/7 7 = 1. We might note also that if we consider instead meantone tempered scales, we simply drop the last coordinate from the definition of the norm, so that any just block leads to a meantone block.
Message: 1613 - Contents - Hide Contents Date: Sun, 23 Sep 2001 16:56 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx Gene wrote:> Splitting the difference gives us sqrt(8/7) = 115.6, a little shy of > being a miracle generator; however on the tuning group I pointed out > that m = (12/5)^(1/13) = 116.6 cents is what gives us perfect octaves > if we iterate a miracle fourth inside of 5/3, and it is something in > the vicinity of m I would regard as a miracle generator. The > corresponding miracle fourth is f = sqrt(5/3 m) = > (2 (5/3)^6)^(1/13) = 500.5 cents, and it is this that I regard as > more fundamental, the miracle generator itself being secondary.That generator is also what gives you a perfect 5:3 when you iterate it within a 2:1. And it's well within the vicinity *I* would regard as a miracle generator. But I don't see what's special about it, or why your definition is secondary to the usual Secors-within-an-octave one. For meantone pair, the transformation would have to be (tone) = (2 -1)(fifth) (oct ) (0 1)(oct ) which has a determinant of 2, because you can't get a fifth by adding and subtracting tones and octaves. The miracle equivalent you give (fourth) = ( -6 1)(Secor) (sixth ) (-13 2)(oct ) is unitary, and so at least works. (Secor) = ( 2 -1)(fourth) (oct ) (13 -6)(sixth ) But why more fundamental? Graham
Message: 1614 - Contents - Hide Contents Date: Sun, 23 Sep 2001 16:50:42 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> That generator is also what gives you a perfect 5:3 when you iterate it > within a 2:1. And it's well within the vicinity *I* would regard as a > miracle generator. But I don't see what's special about it, or why your > definition is secondary to the usual Secors-within-an-octave one.I explained this on Yahoo groups: /tuning/message/28290 * [with cont.] , or at least so I hope. I think the Secors version is secondary for the same reason that iterating a meantone is a less important scale generating process than iterating a meantone tempered fifth--the scales you get from the flat fifth contain all the intervals you normally want, not just some of them. I discussed this analogy in the article Yahoo groups: /tuning/message/28335 * [with cont.] , and the next miracle I mentioned in Yahoo groups: /tuning/message/28465 * [with cont.] . As for the exact value, it is no more magical than the exact 1/4-comma meantone is magical. However, just as that meantone gives you exactly a 5 when interating within 2s, the above value is what you need to get an exact 2 when iterating within 5/3.> (fourth) = ( -6 1)(Secor) > (sixth ) (-13 2)(oct ) > > is unitary, and so at least works.Unimodular is the word you want, unitary means that the inverse matrix is the complex conjugate of the transpose of the matrix.
Message: 1615 - Contents - Hide Contents Date: Sun, 23 Sep 2001 09:07:26 Subject: Tempered blocks and semiblocks From: genewardsmith@xxxx.xxx As I mentioned at the end of my last posting here, we can define a tempered block by omitting some of the vals (other than for 2) for the notation used to define the block. Since JI scales proliferate, and since we probably want to temper anyway, it seems like a good plan to look at the tempered blocks directly, rather than always first constructing a JI block and then tempering. For instance, if we omit h3 from [h7, h2, h3] we are tempering out the interval associated to h3, namely 81/80. The diatonic scale then becomes, in terms of the notation [h7, 7 h2 - 2 h7], [0 0], [1 2], [2 4], [3 -1], [4 1], [5 3], [6 5]. This is a tempered block--in both coordinates, the maximum difference is 6, less than 7. The generator is easily found by sorting according to the second coordinate, giving us [3 -1], [0 0], [4 1], [1 2], [5 3], [2 4], [6 5]; the starting value is [3 -1], corresponding to 4/3, and the generator is [4 1]--the note after [0 0] in this sorted order, corresponding to 3/2. The diatonic scale therefore turns out to be (hold on to your hats!) the scale one gets by starting from 4/3, iterating meantone fifths, and reducing modulo octaves. We also can usefully add a definition of semiblock to our definition of block; we do this by relaxing the condition that the diameter be less than 1 to a condition that it be less than or equal to 1, but less than 1 on the first coordinate; and adding a condition that if n, m are distinct notes in S then the first val gives them distinct values. Let's look again at the ten-note scales where semiblocks came up. Recall that these scales arose from the notation (21/20, 28/27, 64/63, 225/224)^(-1) = [h10, h2, g7, h5] where g7 differs from h7 by the fact that g7(7) = 19. We found from this the JI block 1-15/14-7/6-5/4-4/3-45/32-3/2-5/3-7/4-15/8-(2), in terms of the notation [h10, 10h2 - 2h10, 10g7 - 7h10, 10h5 - 5 h10] this becomes [0 0 0 0] [1 -2 3 5] [2 6 -4 0] [3 4 -1 5] [4 2 2 0] [5 0 -5 5] [6 -2 -2 0] [7 6 1 5] [8 4 -6 0] [9 2 -3 5] Note that since the second coordinate comes from the val 10h2-2h10 it is divisible by 2, and since the last coordinate comes from 10h5-510 it is divisible by 5. Hence we will not be able to completely sort the notes of the scale using these coordinates. However we easily check that this is a block, and by dropping the last two vals we get a tempered block which tempers out 64/63 and 225/224. We can sort by the last coordinate into two groups, giving us [7 6], [8 4], [9 2], [0 0], and [1 -2] in the first group, and [2 6], [3 4], [4 2], [5 0] and [6 -2] in the second group. The generator is [1 -2],corresponding to a semitone of 15/14, 16/15 or 21/20, and we start the first group at [7 6], corresponding to 5/3, and the second at [2 6], corresponding to 7/6. The other JI scale we considered was 1-21/20-9/8-5/4-4/3-7/5-3/2-5/3- 7/4-15/8-(2), corresponding to [0 0 0 0] [1 -2 -7 -5] [2 -4 -4 0] [3 4 -1 5] [4 2 2 0] [5 0 -5 5] [6 -2 -2 0] [7 6 1 5] [8 4 -6 0] [9 2 -3 5] We see that the diameter is 10, since we go from -4 to 6 according to the second val, and from -5 to 5 according to the last. However, it is a semiblock since we can't add a note without the first val giving us a difference of at least 10, which is not allowed. If we temper out 64/63 and 225/224 and sort according to the second coordinate, we get [7 6], [8 4], [9 2], [0 0], [1 -2], and [2 -4] in one group, and [3 4], [4 2], [5 0], [6 -2] in the second group; we have the same generator as before and start the first group from [7 6], corresponding to 5/3 as before, but extend the chain by one more note, to [2 -4] corresponding to 9/8. We make up for this by dropping 7/6 at the start of the second group, and beginning instead from 4/3.
Message: 1616 - Contents - Hide Contents Date: Sun, 23 Sep 2001 20:33 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx Gene wrote:> I explained this on > Yahoo groups: /tuning/message/28290 * [with cont.] , or at least so I > hope.That's "More on Miracles". It explains the arithmetic, but not the reasons for it.> I think the Secors version is secondary for the same reason > that iterating a meantone is a less important scale generating > process than iterating a meantone tempered fifth--the scales you get > from the flat fifth contain all the intervals you normally want, not > just some of them.But in this case *both* methods give all the intervals you want! That's "A meantone analogue of Blackjack". It shows a scale generated by the whole tone. This doesn't make much sense to me.> As for the exact value, it is no more magical than the exact > 1/4-comma meantone is magical. However, just as that meantone gives > you exactly a 5 when interating within 2s, the above value is what > you need to get an exact 2 when iterating within 5/3.But the value for quarter comma meantone is the minimax for the 5-limit. There doesn't seem to be an analogy here.> Unimodular is the word you want, unitary means that the inverse > matrix is the complex conjugate of the transpose of the matrix. Oops, thanks. Graham
Message: 1617 - Contents - Hide Contents Date: Sun, 23 Sep 2001 20:33 +0 Subject: Re: Miracle theory From: graham@xxxxxxxxxx.xx.xx genewardsmith@xxxx.xxx () wrote:> The "miracle" generator splits the difference between 15/14 and > 16/15, and hence is associated to the scales with the jumping jack > 225/224 in the kernel. If we seek other generators with miracle > properties, and obvious place to began is with other jumping jacks. > If we look instead at 81/80, we get the meantone miracle, which is > well-known. If we look higher, we find 2401/2400 = (49/48)(49/50), > 3025/3024 = (55/54)(55/56), and so forth. We therefore might expect a > miracle generator somewhere in the interval between 50/49 and 49/48, > which is to say somewhere between 35 and 35.7 cents.225:224 is also in the kernel for characteristic meantone, schismic and "magic" scales. There's nothing special about associating it with miracles. 225:224, 2401:2400 and 3025:3024, however, completely define miracle temperament. Add 81:80, and you have 31-equal (actually a 62 note periodicity block). So is 31-equal the grand unified jumping jack temperament?> Checking ets with 2401/2400 in the kernel and with good properties, > we find that 18/612 = 35.3 cents and 13/441 = 35.37 cents, which > gives us a pretty good notion of where the next miracle is hiding. On > the high side we have 8/270 = 35.555 cents, and on the low side 5/171 > = 10/342 = 35.08 cents.I don't get this. The temperament consistent with 441- and 612-equal divides the octave into 9 equal parts, with a generator mapping of [-2 -3 -2]. The generator is around 49 cents, 18/441 or 25/612 octaves. It covers the 9-limit with 37 notes, with 0.2 cents accuracy.> The meantone miracle is really better viewed as a flat fifth miracle; > we have 9/8 = (3/2)^2 1/2, so that by extending the circle of fifths > and reducing by octaves we obtain the true miracle of the meantone. > It seems to me it would make sense to do likewise with these other > miracles; we have 16/15 = (4/3)^2 3/5, so that a circle of miracle > fourths in a scale whose interval of repetition is the major sixth > would seem to be the way to take full advantage of the miracle > generator. In the same way, for the 35-cent miracle, we have > 49/48 = (7/4)^2 1/3, so a circle of 7/4's with an interval of > repetition of 3 would make for an interesting collection of scales.I don't see why these are better views, other than for meantone. As I make the last case, the interval of repetition is 3, and a 7:4 is two generators. 5:3 and 7:5 are both a single generator. Graham
Message: 1618 - Contents - Hide Contents Date: Mon, 24 Sep 2001 19:10:11 Subject: Re: semi-periodic scales From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> Posting an open question... > > The normal diatonic scale is said to have octave equivalence. > Which means, you can start anywhere in the scale, move an > octave, and you won't know that anything happened. > > The diatonic scale is semi-periodic at the 3:2. For moves of a > single 3:2, very little changes (say, from LLsLLLs to LLsLLsL). > > How can we quantify this? > > -CarlHi Carl . . . Only an MOS scale generated by approximate 3:2's will have this property (only one note changes with transpositions by a 3:2). Bigger examples include 19-out-of-31 and 29-out-of-41. My omnitetrachordality property is closely related. If omnitetrachordality holds, moves of a 3:2 will only change the notes within a single 9:8 span.
Message: 1619 - Contents - Hide Contents Date: Mon, 24 Sep 2001 19:12:20 Subject: Re: Miracle theory From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> Paul was actually > worried that Jacky's Blackjacky sounded too diatonic, an issue which > would not even arise in the other system.Huh? What do you mean? It's pretty clear that the initial pitch bend messages did not take effect, leading to the beginning of the piece being in 12-tET. What does this have to do with anything?
Message: 1620 - Contents - Hide Contents Date: Mon, 24 Sep 2001 19:17:17 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> No, 7:4 and 3 are two intervals, but they aren't generators of this scale.They are the generators of a system of scales I mentioned, so I need a context for "this scale"--which scale?
Message: 1621 - Contents - Hide Contents Date: Mon, 24 Sep 2001 19:46:20 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> Why is the JI diatonic being taken as the standard?Mostly because of the Blackjacky discussion. What would you suggest as a standard? Even so, your> calculation only seems to cover one octave. Merely to represent two > octaves with the 5:3 equivalence, you need 26 fourths.That is why I suggested beginning the second 5/3 repetion, taking it out to an octave, and then repeating that scale within an octave. I think the result is entirely practical.>> I don't know what you mean by "the" minimax, nor why you could not do >> a similar calculation here.> The 5-limit minimax is the tuning where the largest 5-limit error > (relative to JI) is as small as possible.I take it you are using Paul's definition of 5-limit? The way I use the word, this doesn't mean anything, but you can for instance look at 3^a 5^b where a^2+ab+b^2 < 2, which I recall you doing. I don't see why the result cannot be used for either scale. However, if you add 2 to the mix, you could consider 2^a 3^b 5^c with a^2 + b^2 + c^2 + ab + ac + bc < 2, and get a closely related linear programming exercise; the previous one being what one gets if we fix 2 to its exact value. One could certainly fix 5/3 instead. Similar comments apply to a least squares opitimization.
Message: 1622 - Contents - Hide Contents Date: Mon, 24 Sep 2001 00:10:07 Subject: semi-periodic scales From: Carl Lumma Posting an open question... The normal diatonic scale is said to have octave equivalence. Which means, you can start anywhere in the scale, move an octave, and you won't know that anything happened. The diatonic scale is semi-periodic at the 3:2. For moves of a single 3:2, very little changes (say, from LLsLLLs to LLsLLsL). How can we quantify this? -Carl
Message: 1623 - Contents - Hide Contents Date: Mon, 24 Sep 2001 19:49:47 Subject: Re: semi-periodic scales From: Carl Lumma>Only an MOS scale generated by approximate 3:2's will have this >property (only one note changes with transpositions by a 3:2).Yes but I'm interested in a general measure for any scale. I'm thinking of a function which can take a scale and an interval as an input, and spit out the "extent" of the scale's symmetry at that interval.>My omnitetrachordality property is closely related. If >omnitetrachordality holds, moves of a 3:2 will only change the >notes within a single 9:8 span.Is definition in your paper still the most recent? -Carl
Message: 1624 - Contents - Hide Contents Date: Mon, 24 Sep 2001 00:31:37 Subject: Re: Miracle theory From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:>> I think the Secors version is secondary for the same reason >> that iterating a meantone is a less important scale generating >> process than iterating a meantone tempered fifth--the scales you get >> from the flat fifth contain all the intervals you normally want, not >> just some of them.> But in this case *both* methods give all the intervals you want!The miracle generator certainly does better than the meantone, which may be why it is a miracle, but I am not convinced. The JI diatonic scale requires 25 miracle steps from 5/3 at -13 to 9/8 at 12, whereas the compass is 14 for the fourth-in-5/3 version. Paul was actually worried that Jacky's Blackjacky sounded too diatonic, an issue which would not even arise in the other system.>> As for the exact value, it is no more magical than the exact >> 1/4-comma meantone is magical. However, just as that meantone gives >> you exactly a 5 when interating within 2s, the above value is what >> you need to get an exact 2 when iterating within 5/3.> But the value for quarter comma meantone is the minimax for the 5- limit. > There doesn't seem to be an analogy here.I don't know what you mean by "the" minimax, nor why you could not do a similar calculation here.
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